Dimensionless Physical Quantities in Science Kunes... · · 2018-03-15Dimensionless Physical...

Dimensionless Physical Quantities in Scienceand Engineering

Dimensionless PhysicalQuantities in Scienceand Engineering

Josef KunesDepartment of PhysicsUniversity of West BohemiaPlzenCzech Republic

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I dedicate this book to the memory of my teachersProf. Vladimır Marcelli

andProf. Josef Hosek.

They took me and my workplace,half a century ago,

on the path of modelling and experiment.

PrefaceAll things are numbers.

Pythagoras of Samos (570�495 BC)

The century of informatics, characterized by the remarkable growth of information,

is closely connected with the increasing importance of the model experiment. The

specific important has computer modelling, in practically all scientific spheres.

Modelling is a powerful tool not only in terms of the scale assignment of a model

and its original, as it was for a long time in the case of physical models, but also in

terms of the compression of the information flow on a model and by the processing

of the modelling results. In this sense, the importance of dimensionless quantities �and especially of similarity criteria � has, in fact, increased. The dominant opinion,

still held by many even today, that these quantities are connected primarily with

physical modelling, is mistaken. On the contrary, their importance has grown

significantly in today’s dominant mathematical and computer modelling. The

dimensionless formulation of a mathematical or computer model does not lose its

physical importance; on the contrary, it is magnified. Simplification and generaliza-

tion of modelling results to the dimensionless forms are also other significant attri-

butes. One example is the generation of experimental mathematical models used

whenever it is not possible to use an exact mathematical model. It follows that

dimensionless quantities and similarity theory have an even wider importance in the

contemporary development of modelling than they did in the past.

The use of dimensionless quantities in science matches only the growth of

globalization in the world. Its use in science is, above all, in interlinking dimen-

sional physical and other quantities in dimensionless groups and sinks so growth of

information flow in modelling.

Dimensionless Physical Quantities in Science and Engineering presents in nine

chapters approximately 1200 dimensionless quantities from several types of fields

in which modelling plays an important role. It is probably the most extensive col-

lection of these quantities involving both classic and newly developing fields. In

addition to traditional fields like fluid mechanics and heat transfer, in which using

dimensionless quantities has long been common, this book includes many others,

in particular newly developing fields such as solid phase mechanics, electromagne-

tism, physical macro- and nanotechnology, technology and mechanical engineering,

geophysics and ecology. Each dimensionless quantity is presented with both its

physical characteristics and its significance in the relevant field.

This book is not only a simple summary of these quantities, but also features

a clarification of their physical principles, areas of use and other specific

properties. The book also facilitates the retrieval of dimensionless quantities for

practical use. Furthermore, it provides citations to important sources, also facilitat-

ing the use of dimensionless quantities in their appropriate fields.

The wide range of different spheres in which dimensionless quantities have

extraordinary importance presented considerable difficulty. First of all, for their

specific dissimilarities in such physically different spheres. Moreover, there are

many problems with terminology, and the unsystematic designation of single non-

traditional dimensionless quantities by various authors. The book presents the

established or most often used designations, and recommends preferred

designations.

The book is an attempt to systematically present dimensionless physical quanti-

ties from different scientific and engineering fields. It concerns especially the fields

and processes related to physical substance. Therefore, this book does not include

some fields, such as economics, in which modelling has considerable importance,

but in which the use of dimensionless quantities is out of physical line. The use of

dimensionless quantities in medicine, biology, physiology and other spheres is

increasing along with the fast development of modelling in those fields. However,

this book is mostly concerned with typical dimensionless quantities or their modifi-

cations. A similar situation exists in other fields.

In this book, the presented bibliography is divided into four parts, including a

part discussing books and proceedings (A) and section on articles in journals (B).

Many information sources are presented on Internet sites (C), and there is also a

list (D) of some simple calculators for determination of dimensionless quantities,

and information related to simple software.

A mutual connection of relevant dimensionless quantities by means of cross-

references is an important advantage of this book.

Josef Kunes

x Preface

Foreword

The most fascinating feature of the book Dimensionless Physical Quantities in

Science and Engineering is the presentation of about 1200 dimensionless quantities

relevant for modelling in several research areas, both the traditional and newly

developing ones, which hopefully will contribute to the further development of

model experiments. For this reason, we are indeed fortunate to have Professor

Kunes to guide the thoughts and activities of many scientists and students.

The reason for presenting this relatively great number of dimensionless quanti-

ties in this book is their extraordinary significance for modelling. They are relevant

not only to scale models but also to almost all types of models, including computer

models. The prevalent opinion that these quantities are primarily connected to

physical modelling is erroneous. On the contrary, we cannot ignore the fact that

their importance continues to grow, especially in today’s predominant mathemati-

cal and computer modelling. With the dimensionless expressions used in a mathe-

matical or computer model, the physical meaning is not lost, but is, in fact,

intensified. Moreover, the process investigated becomes more transparent, and the

number of variable quantities and the intensity of the flow of information are sig-

nificantly reduced. This is also true of the number of output quantities according to

which the modelling results are generalized. The dimensionless quantities have fun-

damental significance for experimental mathematical models (phenomenological

mathematical models) generally, when more exact asymptotic mathematical models

cannot be used.

The English edition of this book will thus certainly make a significant contribu-

tion to possible future achievements in the relevant fields of research.

Professor J. Vlcek

Head of the Department of Physics

University of West Bohemia

Plzen, Czech Republic

1 Introduction

The main goal of physics is to describe a maximum of phenomena with a minimum

of variables.CERN Courier [A40]

The dimensionless quantity expresses either a simple ratio of two dimensionally

equal quantities (simple) or that of dimensionally equal products of quantities in the

numerator and in the denominator (composed). The dimensionless quantities can

be divided into several groups. The most important group consists of the physical

similarity criteria obtained by some of the similarity theory methods. They are also

called generalized variable quantities. The dimensionless physical constants belong

to another group. In addition, the approximate ratio quantities can also be included

among the dimensionless quantities. They usually come from experimental results

and from the experimenter’s intuition, i.e. without using any of the similarity theory

methods. Usually, the extent of the validity of these quantities is limited only to

a certain area. Other dimensionless ratio quantities can be created as well, which do

not have any full-value importance from the modelling point of view.

Each of the similarity criteria can be expressed in the form of a mutual relation

between, for example, two forces, momentums or energies acting in a process.

Therefore, by observing the size of the criterion, an idea can be obtained from the

character of the investigated process. This fact is well known, for example, with

the Reynolds number Re, which expresses the dynamic-to-viscous force ratio and

characterizes viscous fluid flow. According to the value of the Re number, the flow

can be distributed into three fundamental characteristic types: laminar, transient

and turbulent. This is similar to the use of the Weber number for single-phase and

two-phase fluids, where this number expresses the ratio of the surface tension force

to the inertial one. More details are in [A23] and [A24], where examples using the

Weber number are given for condensation, boiling and motion of gas bubbles in a

fluid or interaction of a drop with a warm wall.

Sometimes, however, it seems as though the transition from dimensional physical

quantities to dimensionless ones would obscure the view of the investigated process. In

fact, the contrary is true because the reduced number of variable quantities � expressed

in the dimensionless way � enables one to understand the mutual physical contexts in

the investigated process more deeply. A good example is the Fourier number, which

is very often used to express dimensionless time. In fact, it is used in all unsteady

processes occurring in various fields. For example, in thermomechanics � in the

case of heat conduction � it expresses the coupling of time with the characteristic

Dimensionless Physical Quantities in Science and Engineering. DOI: 10.1016/B978-0-12-416013-2.00001-4

© 2012 Elsevier Inc. All rights reserved.

http://dx.doi.org/10.1016/B978-0-12-416013-2.00001-4

geometrical dimension and the thermal diffusivity. It is described by a single

dimensionless variable which is expressive of the influence of all three dimensional

quantities on the temperature field.

Because of a misunderstanding of the physical significance of the similarity

criteria and the functions thereof as generalized variables, dimensional quantities

are often applied even if an experiment or other research should result in an expres-

sion of the information obtained in a condensed and generalized way, so that it can

be utilized for all physically similar systems. With dimensional quantities, on the

contrary, the immediate perception of the process is advantageous in measurement

and identification of an investigated system. The dimensionless similarity criteria

impose a limitation on repeating solutions for tasks of similar character.

Physical similarity criteria are divided into composed criteria and simple (parame-

trical) criteria. Among other things, the importance of similarity criteria is that they

keep a deep physical significance despite the fact that they are outwardly dimension-

less. That is to say, they express the ratios of diverse physical quantities such as forces,

energies and momentums, which especially enable one to understand the acting mech-

anism of individual quantities in analyzing complex physical processes. The physical

similarity criteria can be obtained either by the dimensional analysis method or by the

similarity analysis method, either of a phenomenological physical model or by exact

mathematical model analysis. These methods have been described in the author’s

book Similarity and Modeling in Science and Engineering (CISP, Cambridge, 2012)

in more detail.

This book is primarily focused on physical similarity criteria in various fields

which are characteristics of the development of contemporary science and engi-

neering. Eight chapters summarize about 30 independent fields or spheres in which

modelling plays an important role. The most widespread application of similarity

criteria has already occurred in the era of pre-computer modelling in fluid mechan-

ics and in heat transfer. The present number of criteria in these fields corresponds

to this, and so do the systematic descriptions and surveys presented in the literature,

such as [B11] and [B12]. Of course, the origin of new fields and the increasing

importance of existing ones, together with the entrance of computer modelling, has

resulted in the emergence of many other similarity criteria and modifications of

original ones. However, the literature lacks both adequate descriptions and analyses

of criteria in individual fields and magnetism are partial exceptions. A survey of

the similarity criteria of several fields is given in [A32] and [A33].

The dimensionless quantities from tens of fields are summarized in eight chapters.

Among them, new original and modified dimensionless quantities are presented,

which have been introduced and used in the workplaces of the author.

This especially concerns chapters 5, 6, 7 and 8. Included in each of the chapters

are brief profiles of many important scientists and engineers who worked in the fields

surveyed and have similarity criteria named after them. This should contribute to

the recognition that the dimensionless physical quantities have a human intellectual

dimension in addition to their physical significance.

2 Dimensionless Physical Quantities in Science and Engineering

2 Physics and Physical Chemistry

The basic laws of physics and chemistry are like each other.Dmitri Ivanovich Mendeleev (1834�1907)

2.1 Physics, Mathematics and Geometry

In physics, dimensionless physical quantities and constants have been widely used,

in thermodynamics, optics, radiation and other spheres of physics, especially in appli-

cations in various natural scientific and technical branches, and have become

an important tool in their development. In mathematics, dimensionless quantities

have their theoretical base in the theory of groups and also in linear algebra and

matrix calculus. At the same time, the fundamental theorem to determine the simi-

larity criteria is the dimensional homogeneity of equations of mathematical physics

as defined by Fourier. The similarity criteria are practically important in numerical

mathematics and computer modelling, e.g. not only in generalized dimensionless

expressions of numerical solution stability of mathematical physics equations

but also in other spheres of mathematics. Among the best known physical dimen-

sionless quantities are the following numbers: Abbe, Fresnel and Snellen numbers in

optics; Bejan, Boyle, Carnot, Gay-Lussac, Pitzer and Van der Walls numbers in ther-

modynamics; and the Planck number for radiation.

In mathematics, for example, diverse dimensionless numbers express the

stability conditions of the numerical solution, such as the Courant, Damkohler,

Neumann and Peclet numbers and other mathematical and geometrical dimen-

sionless numbers.

2.1.1 Abbe Number V

V 5nD 21

nF 2 nC

nD, nF, nC (�) � refractive indices of the material at the wavelength of the

Fraunhofer D-, F- and C spectral lines (589.2, 486.1 and 656.3 nm, respectively).

It is used to classify the glasses in the dispersion measurement in the visible radia-

tion band. Low-fracture glasses have high values of V, e.g. for lead crystal glass it is

V, 50, whereas for crown glass it is V. 50. For heavy flint glasses, the common

extent of V is about 20. Very light crown glasses have values of V up to 60.

Info: [C2].



http://dx.doi.org/10.1016/B978-0-12-416013-2.00002-6

Ernst Abbe (23.1.1840�14.1.1905), German physicist and

astronomer.

He was engaged in physics, mathematics and meteorol-

ogy but in optics and astronomy above all. He deduced the

mathematical theory of a light microscope. He designed and

fabricated high-quality lenses for scientific purposes. He

manufactured special instruments. He was the co-founder

of the Carl Zeiss works in Jena.

2.1.2 Energy Accommodation Coefficient rE, σ, α

rE 5Ein 2Ere

Ein 2EW

Ein, Ere (J) � incident and reflected energy flux; EW (J) � reflected energy flux

obtained if the molecules are in thermal equilibrium.

It characterizes the mutual energetic effects of gas molecules with a solid body

surface with heat passing in diluted gases. It expresses the energy of that part

of the total number of gas molecules which come in contact with the surface

and the energy which, after rebound or reemission, is reduced because of being

accommodated to the surface temperature. Besides, it is the measure of the

thermal energy transfer perfection. With the complete transfer, rE5 1 is valid;

and with a complete (mirrored) reflection of the energy, rE5 0. Its size depends

on the physical properties of the surroundings and usually does not differ very

much from the number one.

Info: [A33].

2.1.3 Avogadro Number NA

NA 5 6:02203 1023 mol21

The Avogadro number expresses the number of particles (e.g. atoms, molecules

or ions) in a chemically homogeneous body with the substance quantity of one

mole (mol). The mole is the substance quantity of the set which contains exactly

as many elementary individual units (e.g. atoms, molecules or other particles) as

the atoms in 0.012 kg of the nuclide of the carbon isotope 126C: It is widely applied

in physics, chemistry and other branches.

Info: [C5].


Lorenzo Romano Amedeo Carlo Avogadro (9.8.1776�9.7.1856), Italian mathematician, physicist and chemist.

He was engaged in statistics, physics and chemistry, and

in meteorology as well. In the year 1811, he published the

hypothesis known later as the Avogadro law, which expresses

the fact that equal gas volumes contain equal numbers of

molecules under equal temperature and pressure. However,

Avogadro could not prove the hypothesis precisely by experi-

ment and did not live to see its acceptance. To honour his

work in the molar mass theory, the number of molecules in

one mole of particles was named after him.

2.1.4 Bejan Thermodynamical Number Be

Be5S1

S1 1 S2

S1 (J K21) � entropy generation contribution by heat transfer; S2 (J K21) � entropy

generation contribution by fluid friction.

It expresses the ratio of heat transfer unreturnability to the total unreturnability

caused by heat transfer and fluid friction.

Info: [C6]

Adrian Bejan (born 24.9.1948), American engineer of

Romanian origin.

His research activity involves a wide area of thermal

engineering and thermodynamics. He was engaged in

the entropy minimization problem; the energy conversion

analysis (exergy); natural convection; convection in porous

materials; heat and mass transfer; and problems of turbu-

lence, melting, solidification, condensation, contamination,

solar energy conversion, cryogenic engineering, applied

superconductivity and tribology. He created the constructive

form and structure theory in nature.

2.1.5 Boltzmann Distribution, Boltzmann Factor NB, Pn

NB 5Ni

N5 e2

γBhkT

Ni (m23) � number of states having energy Ei; N (m23) � total number of particles;

γ (�) � magnetogyroscopic ratio; B (T) � magnetic induction; h (J s) � Planck

constant; k (J K21) � Boltzmann constant; T (K) � sample temperature.

5Physics and Physical Chemistry

In physics, it represents the prediction of the function of particle distribution

of which each one has the energy Ei. Alternatively, it expresses the volume magne-

tizing vector as well.

Info: [C8].

Ludwig Boltzmann (p. 205).

2.1.6 Boyle Number Bo

Bo5TB

Tcrit

TB (K) � Boyle temperature; Tcrit (K) � critical temperature.

This number expresses the ratio of the thermodynamic temperature of Boyle’s

point, corresponding to the zero isobar, to the critical temperature. BoA 2:3; 3h i:Info: [C11].

Robert Boyle (25.1.1627�30.12.1691), Irish chemist and

mathematician.

He is called the father of chemistry. He applied experi-

mental and quantitative methods. He was the first to deliver

the modern definition of chemical elements, and he used

it to measure the acidity of colour indicators. He discovered

the indirect proportionality between gas pressure and volume

under constant temperature. This is Boyle’s law.

2.1.7 Bulk Concentration Nbc

Nbc 5cb

R

cb (kg m23) � concentration of bulk particles; R (kg m23) � liquid density.

It characterizes the relative concentration of solid particles in a solution. Filtration.

2.1.8 Carnot Number Ca

Ca5T2 2 T1

T2

T1, T2 (K) � absolute temperatures.


It characterizes the theoretical efficiency of the Carnot circulation occurring

between two thermal states, limited by the thermodynamic temperatures T1 and T2.

Thermodynamics.

Info: [B20],[C16].

Nicolas Leonard Sadi Carnot (1.6.1796�24.8.1832), French

physicist.

He was engaged in thermodynamics above all. In the year

1824, he elaborated the thermal machines theory in his work

Reflections sur la Puissance Motrice de la Feu (Considerations

on the Driving Power of Fire). He designed the Carnot reversal

thermal circulation and found that thermal machine efficiency

depends only on the inlet and outlet temperatures.

2.1.9 Coefficient of Variation C

C5σμ

σ (�) � standard deviation2; μ (�) � mean value.

In probability and statistical theories, it expresses the dispersion size of the

probability distribution. It is often used to evaluate the normal distribution with

a positive mean value and with a standard deviation which is less than the mean

deviation expressively. In compliance with the distribution character of the standard

deviation, the variation coefficient size can be greater or less than one. Mathematics,

statistics.

Info: [C21].

2.1.10 Compressibility Factor Z

Z5pv

r Tð1Þ; Zcrit 5Wa21 ð2Þ

p (Pa) � pressure; v (m3 kg21) � specific volume; r (J kg21 K21) � specific gas

constant; T (K) � temperature; Wa (�) � Van der Waals number (1.) (p. 32).

This factor characterizes the mutual molecular coupling in a substance for a

certain thermodynamic state. At the thermodynamic critical point, the relation (2)

is valid. In ideal gases with Z5 1, the deviations from this value express the size of

the intermolecular coupling.

Info: [C83].


2.1.11 Courant Numerical Number Cou, CFL, ν

Cou5wΔτΔx

,C; where C# 1 ð1Þ;

Cou5wxΔτΔx

1wyΔτΔy

,C ð2Þ;

Cou5Δτ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiwx

Δx

� �2

1wy

Δy

� �2s

# 1 ð3Þ;

CFL5ΔτΔx4

,C ð4Þ

w (m s21 � velocity; Δτ (s) � time step; Δx, Δy (m) � mesh increment in the

x and y axis; C (�) � constant (time stepping parameter) depends on the particular

equation to be solved and not on Δτ and Δx; wx, wy (m s21) � flow (fluid) velocity

component in the x and y axis.

It expresses the stability criterion of a numeric process solution of one-dimen-

sional fluid flow (1), two-dimensional (2) fluid flow and two-dimensional flow

through a porous material (3). For certain fourth-order partial differential equations,

the form (4) can be used. The CFL condition can be a very limiting constraint on

the time step Δτ. Numerical mathematics.

Info: [B114].

2.1.12 Courant Wave Number Cou

Cou5w2Δτ2

Δ2#

1

2; if Δ5Δx5Δy

w (m s21) � velocity of wave propagation;Δτ (s) � time step;Δ,Δx,Δy (m) � mesh

increment.

It expresses the numerical solution stability condition for wave propagation.

Numerical mathematics.

Info: [A11]

2.1.13 Courant�Friedrichs�Lewy Numerical Number CFL, NCFL, Cou

CFL5Δτk

w

ΔxwBL ð1Þ;

CFL5Δτk

wx

Δx1

wy

Δy1

wz

Δz

� �wBL ð2Þ


Δτ (s) � time step; k (m s21) � filtration coefficient; w, wx, wy, wz (m s21) � flow

velocity and its components; Δx, Δy, Δz (m) � space step; wBL (m s21) � saturation

velocity.

This number represents the numerical criterion to determine optimal time

step in a non-stationary, two-phase, one-dimensional (1) filtering flow or in a

three-dimensional (2) one. The stability condition is CFL# 1.

Info: [B114].

2.1.14 Crank�Nicolson Parameter μC

μC 5 0:5 ð1Þ; μC 5 0 ð2Þ; μC 5 1 ð3Þunj (�) � dimensionless-dependent variable; ΔFo (�) � Fourier difference number

(p.13); j (�) � geometric nodal point; n (�) � time level in the numeric solution.

It occurs in the Crank�Nicolson differential diagram for a one-dimensional

solution of the parabolic Fourier’s equation:

un11j 2 unj 5μCΔFo ðun11

j11 2 2un11j 1 un11

j21 Þ1ðunj112 2unj 1 unj21Þh i

:

In case relation (1) is valid, the Crank�Nicolson solution is stable unconditionally.

With relation (2) holding, the explicit solution (FTCS) is obtained. In the case of (3),

the implicit solution is gained.

Info: [C24].

John Crank (6.2.1916�3.10.2006), English mathematician.

Together with the mathematician Phyllis Nicolson, he

worked in the sphere of the numerical solution of second-

order partial differential equations � especially those of

unsteady heat conduction � and the solution thereof by the

finite differences method. They were engaged in the problem

of numerical solution instability and of the related choice

of optimal geometric and time steps. The Crank�Nicolson

method is stable numerically, but a simple system of linear

equations must be solved at every time level.

Phyllis Nicolson (21.9.1917�6.10.1968), English mathematician.

Together with the mathematician John Crank, she worked

on the solution of second-order partial differential equations

numerically. As a mathematician, she was engaged in the

magnetron theory and its interpretation.


2.1.15 Damkohler Numerical Number Danum

Danum 5kΔx

w

k (s21)� disintegration unit rate;Δx (m)� mesh increment; w (m s21)� flow velocity.

This number expresses the stability criterion in a digital process of the one-

dimensional modelling of a chemical conversion in fluid flow. It characterizes

the hydrodynamic influence in chemical reactions.

Info: [C98].

Gerhard Friedrich Damkohler (1908�1944), German physical chemist.

2.1.16 Dimensionless Heat Capacity NC

NC 5C

nR5

C

Nk

C (J K21) � heat capacity of a body; n (mol) � amount of matter in the body;

R (J mol21 K21) � molar gas constant; N (�) � number of molecules in the body;

k (J K21) � Boltzmann constant.

It expresses the body heat capacity in a dimensionless shape. Thermomechanics.

Info: [C123].

2.1.17 Eddington Number Ed

Ed5 1363 2256 � 1:5753 1079

It expresses the exact number of protons in the Universe, where 136 represents

the inverse value of the fine structure constant α (p. 13) as it could be stated by

measurement in the given time.

Info: [C43].

Arthur Stanley Eddington (28.12.1882�22.11.1944),

American physicist and astronomer.

Probably, he is the most prominent astrophysicist of the

twentieth century. He examined the stellar system interior. The

basic contribution consisted in the verification of gravitational

curvature influence on the bending of rays around the Sun.

He explained the pulsation of stars theoretically. He calculated

the Sun’s nucleus temperature in millions of Kelvin. He was

one of the first who defined Einstein’s relativity theory with

more precision.


2.1.18 Eddington�Dirac Number EdD

EdD � 1040

This hypothetical number follows from the fact that: (i) the ratio of the mutual

force of the electron and the proton equals 2.273 1039, (ii) the ratio of the elemen-

tary length (radius of the electron) to the radius of the Universe is 33 1040 and

(iii) the ratio of the elementary time (electron radius to light velocity) to the

Universe’s age is 63 1040. It represents only the mysterious dimensionless number

in the extent of 1040 approximately, which was presented by Dirac with respect to

digital relations between the microscopic and macroscopic scales and the effects

of various forces. Considering Dirac’s hypothesis of extensive numbers, obviously

this numerical relation between very different phenomena was given a deeper

cosmologic significance. At present, this idea is not set greater physical store

mostly.

Info: [C129].

Arthur Stanley Eddington (see above).

2.1.19 Energy Efficiency ηt

ηt 5W

E; ηtAh0; 1i

W (J) � mechanical work or energy released by the process; E (J) � quantity of work

or energy used as input to run process.

It is an important technical indicator for economic utilization of processes

and facilities. According to thermodynamic law, the efficiency ηt5 1 cannot be

reached.

Info: [C46].

2.1.20 Entropy Generation Number NS

NS 5L2T0EG

λðTw 2 T0Þ2;

where EG 5λT20

ðrxTÞ2 1 ðryTÞ2� �

1ηT0

1 ðryuÞ2

L (m) � characteristic length (wall thickness); T0 (K) � input fluid temperature;

EG (W m23 K21) � entropy change (volume density of heat flux) by tempera-

ture change 1 K; λ (W m21 K21) � wall thermal conductivity; Tw (K) � wall


temperature; η (Pa s) � dynamic viscosity of the fluid; rxT, ryT (K m21) � tempera-

ture gradient in the direction of x and y axis; ryu (s21) � velocity gradient in the

direction of y axis.

It characterizes the fluid entropy change in laminar streaming of viscous

incompressible fluid through an inclined canal with isothermic walls. It was

determined from the analysis of the second law of thermodynamics. Fluid

mechanics.

Info: [B76].

2.1.21 Feigenbaum Delta δ

δ5 limn-N

Tn 2 Tn21

Tn11 2 Tn

Tn (s) � value of the nth bifurcations period.

It characterizes the convergence velocity of the cascade redoubling period. It

is determined experimentally and has the value δ. 1. It is used in the dissipation

theory of non-linear systems in the phase transfer measurement � for example,

in electronic circuits, lasers, chemical reactions and in fluid mechanics � if

approaching the turbulent state.

For example, Feigenbaum showed that all non-linear dynamic systems, showing

periodic doubling, tend towards chaos and usually have a value of δ5 4.669.

Chaos theory.

2.1.22 Fermi’s Paradox K

K5 eTτ � 10433 106

T (year) � age of the Universe (T5 1010); τ (year) � specific time of exponential

development of our civilization (τ5 102 years).

This gigantic dimensionless number, exceeding the framework of theoretical

physics � see Eddington number Ed (p. 10)� characterizes the growth of technologi-

cal civilization during the Universe’s existence. This number is so large that the total

number of elementary particles in the Universe is very small as compared to it.

According to Fermi, ‘If there were any civilisations in the Universe, their spaceship

would have been in our Solar system long ago’. The absence probability of ‘space

miracles’ is 102433 106 in our Universe, or virtually zero. Unfortunately, nobody

has discovered them. Fermi’s paradox consists in the idea that our miracleless world

is fantastic and does exist.

Info: [C54].


2.1.23 Fine Structure Constant, Sommerfeld Fine-Structure Constant α

α5e2

4 π ε0h c

e (C) � elementary charge (1.602193 10219 C); ε0 (F m21) � permittivity of vacuum;

h (J s) � Planck constant h5 h (2π)21 (1.05458873 10234 J s); c (m s21) � speed

of light.

It is the basic dimensionless physical constant characterizing the electromagnetic

interaction intensity. It was introduced into physics by A. Sommerfeld in the year

1916. It is used in analyzing Feynman’s quantum electrodynamic diagrams. Its

exact value, determined on a physical basis is α215 137 but α215 137.0399976

if determined by experimental procedure.

Info: [A29].

Arthur Stanley Eddington (p. 10).

2.1.24 Fourier Difference Number ΔFo, Fomesh

ΔFo5Fomesh 5aΔτðΔxÞ2 ð1Þ; ΔFo, 0:5 ð2Þ;

ΔFo#1

4ð3Þ; ΔFo5

1

6ð4Þ

a (m2 s21) � thermal diffusivity; Δτ (s) � time step; Δx (m) � finite increase of

distance coordinate x.

In equation (1), it expresses the relation between the amount of time and the

geometric steps in the numerical solution of the parabolic Fourier equation. For

the explicit 1-D task, condition (2) is valid for solution stability. For the 2-D task,

condition (3) is valid; for the 3-D task, condition (4) is valid. Numerical mathematics.

The final difference method.

Info: [A10].

Jean Baptiste Joseph Fourier (p. 175).

2.1.25 Fresnel Number F

F5L2

λs

L (m) � characteristic size (radius) of the aperture; λ (m) � wavelength; s (m) �distance of the screen from the aperture.


It characterizes Frauenhofer diffraction equation (F{1) and that of Fresnel

(F$ 1). Optics.

Info: [C59].

Augustin Jean Fresnel (10.5.1788�14.7.1827), French

physicist and mathematician.

Due to his works in optics, he became one of the foun-

ders of light wave theory. He showed the reason for optical

diffraction, consisting in transversal light undulation. He

created the mathematical theory of refraction and polariza-

tion in anisotropic materials. From this theory, conical

refraction was predicted and discovered soon afterwards.

Joseph von Fraunhofer (6.3.1787�7.6.1826), German

physicist.

In the year 1814, he investigated the solar spectrum and

discovered dark spectral lines called the Fraunhofer lines.

He is well known for his work on light diffraction in systems

with small Fresnel numbers. This is called Fraunhofer’s

diffraction to honour him.

2.1.26 f-Stop Number Nf

Nf 5f

D

f (m) � focal length of the lens or mirror; D (m) � aperture diameter;

In optics, the Nf 5f2; f3; . . . is used currently. In film exposition, it is P~ 1

N2f

:Optics, film, photography.

2.1.27 Gay-Lussac Number Gc

Gc51

βΔT

β (K21) � coefficient of bulk expansion; ΔT (K) � temperature difference.

In the dimensionless form, it characterizes the relative thermal volume expansi-

bility of substances.

Info: [A29].


Joseph Louis Gay-Lussac (1778�1850), French chemist

and physicist.

For ideal gases, he expressed the law of gas pressure

dependence on temperature, so-called the Gay-Lussac law.

Later, this led to the introduction of the thermodynamic

temperature scale and to the formulation of the ideal gas

state equation. When only a 26-year-old chemist, he executed

many courageous high-altitude atmospheric measurements

by means of a balloon and with instruments he designed.

He studied terrestrial magnetism as well.

2.1.28 Geometric Coordinates X, Y, Z

X5x

L; Y 5

y

L; Z5

z

L

x, y, z (m) � dimensional coordinates; L (m) � characteristic length; S (m2) � surface

area; V (m3) � volume.

They express the ratio of the coordinate of a point in space to the characteristic

length of the system. In the dimensionless form, it characterizes the position of the

place M (X, Y, Z) in space. For a plate, the semi-thickness (symmetrical case) or

the thickness (unsymmetrical case) is usually chosen as the characteristic length.

In the case of a cylinder or a ball, the radius is chosen.

Sometimes for a general shape, it is convenient to choose the volume�surface

ratio of a body (module) L5V � S21. In the case of a plate, an unlimited cylinder

and a ball, the ratio 3:1.5:1 is obtained, which is also the ratio of mutually corre-

sponding process times according to the analytic heat transfer theory. However, the

relative length, determined in this manner, does not agree with the geometric coef-

ficient influence. Therefore, the generalized relative length L5KtV � S21 is used

sometimes where Kt is the relative shape coefficient. For a ball, an unlimited plate

and a cylinder, Kt5 1 holds, with Kt. 1 for other bodies.

Info: [A23].

2.1.29 Gravitational Coupling Constant αg

αg 5Gm2

e

hc5

me

mP

� �2

� 1; 7523 10245

G (N m2 kg22) � Newtonian constant of gravitation; me (kg) � electron mass; h� (J s) �Planck constant; c (m s21) � speed of light in vacuum; mP (kg)� Planck mass.

It represents a basic physical constant and characterizes the gravitational force

between typical elementary particles. It is related to gravitation as the fine structure

constant α (p. 13) is to electromagnetism and quantum electrodynamics.

Info: [C69],[C68].


2.1.30 Guldberg Number Gu

Gu5Tn

Tcrit

Tn (K) � thermodynamic temperature of saturation by pressure p5 105 Pa; Tcrit (K) �critical thermodynamic temperature.

This number expresses the relation between the thermodynamic saturation

temperature at an atmospheric isobar and the critical thermodynamic temperature.

GuAh0.37; 0.8i.Info: [A23].

Cato Maximilian Guldberg (p. 42).

2.1.31 Hadamard Number Hd

Hd53ηb 1 3ηf3ηb 1 2ηf

ηb, ηf (Pa s) � dynamic viscosity of fluid in bubble (b) and viscosity of ambient

fluid (f).

In the dimensionless form, it expresses the resulting dynamic viscosity of a

mixture of a fluid and bubbles contained therein.

Info: [A29].

Jacques Salomon Hadamard (8.12.1865�17.10.1963),

French mathematician.

He introduced the correct task concept in the partial differ-

ential equation theory. Hadamard’s matrices and the Hadamard

transformation are named for him, representing an example

of the generalized class of Fourier transformations. He edited

many publications, e.g. publications on geodesy and matrix

theory. After the WWII, his long and eventful life led to peace

activities and support for mathematicians all over the world.

2.1.32 Hamilton Numeric Number H

H5XNi.j

1

Rij

e2kRij 1XNi

R2i ;

where

Ri 5ri

rref; rref 5

ffiffiffiffiffiffiffiffiffiffiffi2q2

mεω2

3

r; k5

rref

lse


N (�) � number of the particles; k (�) � inverse dimensionless screening length;

Rij (�) � interparticle relative position of ith and jth particle; Ri (�) � relative

position of the ith particle; ri (m) � position of the ith particle; rref (m) � reference

position; q (C) � particle charge; m (kg) � mass of the particle; ε (F m2 1) �dielectric constant of the medium; ω (s21) � angular frequency of the particle;

lse (m) � screening length.

It characterizes the interaction of particles due to the Coulomb potential, depend-

ing on physical parameters and plasma background, and can have various forms.

Description by the Hamiltonian (H) starts from the isotropic potential between

particles and from the purely repelling and approximated potential. With k5 0,

the interaction between particles is purely the Coulomb potential. Mathematical

physics.

2.1.33 Kolmogorov Microscale λKol

λKol 5

ffiffiffiffiffiν3

ε4

rð1Þ; τ5

νε

ð2Þ

ν (m2 s21) � kinematic viscosity; ε (m2 s23) � speed of energy dissipation relative

to the mass unit; τ (s) � time.

It expresses the nominal length with which the viscous dissipation occurs in

three-dimensional turbulent flow. Together with the timescale (2), it characterizes

the turbulence beginning in a certain place in the flow. Physics. Fluid mechanics.

Info: [C75].

Andrey Nikolaevich Kolmogorov (25.4.1903�20.10.1987), Russian mathema-

tician (p. 104).

2.1.34 Lautrec Number for Fluid NL

NL 5L

δf; where δf 5

ffiffiffiffiffiffiffi2af

ω

r

L (m) � separated half-thickness of the plate; δf (m) � sound penetration in fluid

material; af (m2 s21) � thermal diffusivity of fluid; ω (s21) � angular frequency of

the acoustic oscillations.

It characterizes the thermoacoustic process originating in the thermal and acous-

tic energy transformation in a fluid. It expresses the influences in the acoustics, in

which the heat transfer and fluid entropy changes play an important role.

Thermoacoustics.


2.1.35 Lautrec Number for Solid NL

NL 5L

δf; where δs 5

ffiffiffiffiffiffiffi2af

ω

r

L (m) � plate half-thickness; δf (m) � sound penetration in solid material; as(m2 s21) � thermal diffusivity of material; ω (s21) � angular frequency of the

acoustic oscillations.

This number expresses the thermoacoustic process of acoustic wave penetration

into a material. Alternatively, it describes the thermal and acoustic energy transfor-

mation in a material. Thermoacoustics.

2.1.36 Lobachevsky Number Lo

Lo51

OA

ððOAÞ

dbmed

dblocdli ð1Þ; Lo5

1

SA

ððSAÞ

dbmed

dblocdSi ð2Þ

OA (m), SA (m2) � circumference and area of cross section; bmed, bloc (m) � mean and

local distance between isolines; dli (m) � elementary length; dSi (m2) � elementary

area.

In heat and mass transfer and in aero-hydrodynamics, it characterizes the isolines

shape. It represents the ratio, centred on the line of force, of the elementary distance

between the isolines to the local distance between two points of the same lines of

force. With the isolines and isoareas not changing their shape, the Lo number keeps

an equal value. In heat and mass transfer, it expresses the influence of a capillary

porous body on the heat and mass transfer.

Info: [A23],[A24].

Nikolay Ivanovich Lobachevsky (1.12.1792�24.2.1856),

Russian mathematician.

He founded non-Euclidean geometry, a geometry in which

Euclid’s fifth postulate is not true. He published a whole

range of books, starting with the Foundation of Geometry

(1835�1838) to Pangeometry (1855). Geometry with a con-

stant negative curvature is called Lobachevskian geometry.

2.1.37 Logarithmical Decrement Λ, ϑ

Λ5 lnsðtÞ

sðt1 TÞs(t) (m) � displacement of oscillation; t (s) � time; T (s) � time period of oscillation.


It is the logarithm of the ratio of the vibrating movement deviation s(t) to a

deviation in the time t1 T.

Info: [C28].

2.1.38 Lorentz Number Lo, L

Lo � ðΔUÞ2γΔT λ

ΔU (V) � voltage difference; γ (S m21) � specific electrical conductivity; ΔT (K) �temperature difference; λ (W m21 K21) � specific thermal conductivity.

It expresses the dependence between thermal and electrical conductivities

for metals. It starts with the Wiedemann�Franz law, according to which

both conductivities are proportional to the absolute temperature and depend

on the movement of free electrons. Physics. Thermomechanics. Electrical

engineering.

Info: [C79].

Hendrik Antoon Lorentz (p. 323).

2.1.39 Ludolph’s Number π

π5U

D

U (m) � circle circumference; D (m) � circle diameter.

It expresses the circumference-to-diameter ratio of a circle. Probably, it is

the oldest parametric criterion, known about the year 2000 BC, and approximated

by the Egyptians afterwards. Mathematics. Geometry.

Info: [C82].

Ludolph van Ceulen (28.1.1540�31.12.1610), Dutch math-

ematician and engineer.

In the year 1600, he was appointed as the first professor

of mathematics at Leyden University. He calculated the num-

ber π to 35 decimal places by applying the same methods as

Archimedes about 2000 years earlier. With this irrational

transcendental number, he confirmed that the classic circle

quadrature task is unsolvable. He wrote several works, of

which On the Circle (Van den Circkel) is the most popular.

The number is called Ludolph’s number after him.


2.1.40 Mass-to-Charge Ratio MZ

MZ5m

z

m (�) � atom (molecule) mass in atomic mass units; z (�) � atom (molecule)

electric charge.

This is an important quantity used in mass spectrometry, for example, in

depicting the ion signal dependence on this dimensionless parameter. In the case of

multi-degree mass spectrometry, this method can be applied to chemical structure

determination.

Info: [C86].

Joseph John Thomson (18.12.1856�30.8.1940), English

physicist. Nobel Prize in Physics, 1906.

He discovered the electron in cathode radiation and

presented the atom as having its own internal structure.

He clarified the properties of ions. He determined the mass-

to-charge ratio of the electron. He observed the first data

confirming the existence of isotopes. He received the Nobel

Prize for his research in electricity conduction in gas.

2.1.41 Mechanical Efficiency ηmech

ηmech 5P2

P1

5u F

P1

P2 (W) � work output; P1 (W) � mechanical advantage (work input); u (m s21) �speed; F (N) � force.

It expresses the effective input power ratio.

Info: [C88].

2.1.42 Mendeleev Number Me

Me5patm

pcrit

patm (Pa) � atmospheric pressure; pcrit (Pa) � critical pressure.

It expresses the atmospheric to critical pressure ratio on a static surface.

MeAh0.4303 1022; 0.4433 1022i.Info: [A23].


Dmitriy Ivanovich Mendeleev (7.2.1834�2.2.1907), Russian

chemist.

He assembled and published (1869) 63 elements,

unknown before, in the periodic table based on the atomic

number. Thus, he became the discoverer of the periodic law

of the elements, which was confirmed later by subsequent

discovery of other elements and by studying X-ray spectra

and quantum mechanics. He wrote about 400 published

works, concerning physics and technology, besides chemistry.

He wrote the textbook Foundations of Chemistry.

2.1.43 Moment of Inertia Ratio γ

γ5I

m r2

I (kg m2) � moment of inertia of the body; m (kg) � mass of the body; r (m) �radius.

The ratio γ represents the normalized dimensionless inertia moment.

Info: [C93].

2.1.44 Monin�Obukhov Length L

L52Rcpw3

gαK

R (kg m23) � air density; cp (J kg21 K21) � specific heat capacity; w (m s21) � air

velocity; g (m s22) � gravitational acceleration; α (K21) � coefficient of linear

thermal expansion; K (m kg K22) � Karman constant.

The length L corresponds to the measurement at which the Richardson

number Ri (p. 83) is Ri5 1 for the flow near a heated wall with free and forced

convections.

Info: [C95],[C94].

2.1.45 Neumann Numerical Number Neu

Neu5D1

ðΔxÞ2 11

ðΔyÞ2� �

Δτ#χ

D (m2 s21) � diffusion coefficient; Δx, Δy (m) � grid spacing in both coordinate

axes; Δτ (s) � time step; χ (�) � time stepping parameter.


This number expresses the stability criterion for a numerical solution, for

example, of two-dimensional diffusive flow (Neu# 1). Together with the Courant

numerical number Cou (p. 8), it characterizes the oscillating convection flow

through porous material. Numerical mathematics. Two-phase flow.

Info: [B44].

John von Neumann (28.12.1903�8.2.1957), American

mathematician of Hungarian origin.

He was one of the greatest mathematicians of the twentieth

century. He substantially influenced the development of the

structural and functional principles of computers. His scientific

sphere of interest was extraordinarily wide. He participated in

elaborating theoretical foundations for atomic energy utiliza-

tion. He founded the theory of sets, quantum theory and theory

of games, too, which represent important areas of mathematics

and economics.

2.1.46 Normalized Frequency, V number V

V 52πrλ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin21 2 n22

q

r (m) � core radius of optical fibre; λ (m) � wavelength in vacuum; n1 (�) � maxi-

mum refractive index of the core; n2 (�) � refractive index of the homogeneous

cladding.

It is important in telecommunication, to quantify the optical fibres especially,

to determine the actual to relative or nominal frequencies ratio. It is applied in

optoelectronics and telecommunications. For single-mode operations it is V, 2 and

for multi-mode ones V. 5. Optoelectronics, physics and telecommunications.

Info: [C97].

2.1.47 Optical Thickness τ, h

τ5 k L ð1Þ; Ix;λ 5 I0;λ exp 2x

τ

� �ð2Þ

k (m21) � monochromatic absorption coefficient; L (m) � characteristic length,

thickness; Ix,λ, I0,λ (�) � luminous intensity at the depth x and on the material

surface at the wavelength λ; x (m) � depth below the surface; λ (m) � wavelength.

It expresses the material depth at which the radiation intensity (of various wave-

lengths, e.g. of visible light) of a given frequency is reduced by a coefficient 1e.

In the optical thickness depth, about 23of the radiation is absorbed. Physics. Optics.

Atmospheric radiation.

Info: [C99].


2.1.48 Peclet Difference Number Penum

Penum 5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�wx

Δx

�2

1wy

Δy

� �2s

Dxx

ðΔxÞ2 1Dyy

ðΔyÞ2 2Dxy

Δx Δy

� �21

wx, wy (m s21) � flow velocity components in pores; Δx, Δy (m) � mesh increments

in x and y axes; Dxx, Dyy (m2 s21) � components of the dispersion coefficient;

Dxy (m2 s21) � transverse dispersion coefficient.

This number expresses the stability criterion of a numerical solution, for example,

of an underground water flow through dispersive surroundings. Physically, it is about

dynamic and dispersive forces acting. Numerical mathematics. Two-phase flow.

Info: [C98].

Jean Claude Eugene Peclet (p. 180).

2.1.49 Peclet Grid Number Pemesh

Pemesh 5w Δx

D, 2 ð1Þ;

Pemesh 5maxwx;maxΔx

D;wz;maxΔz

D

� �ð2Þ

w (m s21) � velocity; wx,max, wz,max (m s21) � maximum value of the velocity

components; Δx, Δz (m) � mesh increments in the x and z axes; D (m2 s21) � dif-

fusion coefficient.

Equation (1) is the stability criterion of a numerical process solution for one-

dimensional flow or transfer. Equation (2) is the limiting condition to choose a dif-

ference step for solution a two-dimensional diffusive process with a moving mem-

ber. Numerical fluid mechanics. Mathematics.

Info: [A47],[B77],[B44].


2.1.50 Pitzer Number Pi

Pi � P0:7 5p

pref5 0:7 at Tred 5

T

Tref5 0:7

p, pref (Pa) � pressure, reference pressure; T, Tref (K) � temperature, reference

temperature; P0.7 (�) � reduced pressure; Tred (�) � reduced temperature.

It serves to determine the reduced steam pressure at reduced temperature.

Thermodynamics.


Kenneth Sanborn Pitzer (6.1.1914�26.12.1997), American

theoretical chemist.

Pitzer was the founder of modern theoretical chemistry.

He used quantum and statistical mechanics to explain the

thermodynamic and conformational properties of molecules,

and he pioneered quantum scattering theory for describing

chemical reactions at the most fundamental level. He also

made contributions to relativistic effects in chemical bonding

and the theory of fluids and electrolyte solutions.

2.1.51 Planck Number, Radiation Parameter Pl

Pl5λβ

4n2σ0T3ref

λ (W m21 K21) � intrinsic ambient thermal conductivity; β (m21) � parameters

vector of the absorption coefficient; n (�) � refractive index; σ0 (W m22 K24) �Stefan�Boltzmann constant; Tref (K) � reference temperature.

This number expresses radiation effects on a convective boundary layer with a

uniformly heated surface. An increasing Planck number leads to the reduction of

the thickness of the layer. On the contrary, an increasing temperature increases the

thickness of the layer. For gases, the Pl number depends on the Prandtl number Pr

(p. 197) and the temperature and is in the range of 100�150. For water steam with

the temperature 100�500�C and Pr5 1, it is PlAh30; 200i. It is analogous to the

radiation number (2.) N (p. 211).

Info: [B3].

Max Karl Ernst Ludwig Planck (23.4.1858�4.10.1947),

German physicist. Nobel Prize in Physics, 1918.

He was engaged in research related to the radiation of the

perfect black body and discovered the Planck radiation law,

which determines the dependence of the volume radiation

density of this body on the radiation frequency and body tem-

perature. This law is based on the hypothesis of discontinuous

radiation of electromagnetic energy in doses. For his discovery

of the electric quanta, he won the Nobel Prize. He was one of

the first who accepted and evolved Einstein’s relativity theory.

2.1.52 Porosity Φ, ϕ, p, n, e

Φ5Vf

Vt

Vf (m3) � volume of void space filled up by fluid; Vt (m

3) � total or bulk volume

of material including the solid and void components.


It characterizes the ability of a porous material to receive a fluid. The porosity

ΦAh0; 1i or ΦAh0; 100%i. Fluid mechanics. Geophysics. Sedimentation. Drying.

Info: [C105].

2.1.53 Reduced Boiling Temperature Nrbt

Nrbt 5Pi

4RePr

D

L

D (m) � circular pipe diameter; L (m) � characteristic length; Pi (�) � Pitzer number

(p. 23); Re (�) � Reynolds number (p. 81); Pr (�) � Prandtl number (p. 197).

It expresses the reduced normal boiling temperature. It is used for steam transfer

through a pipeline. Thermodynamics.15

2.1.54 Reduced Pressure pred

pred 5p

pcrit

p (Pa) � liquid pressure; pcrit (Pa) � critical pressure.

It serves to predict the properties (p, v, T) of gases from critical values.

Thermodynamics.

Info: [C110].

2.1.55 Reduced Volume vred

vred 5v

vcrit

v (m3 kg21) � gas or vapour specific volume; vcrit (m3 kg21) � gas or vapour spe-

cific volume at critical state.

It serves to compare and predict the properties (p, v, T) of gases from critical

values. Thermodynamics.

2.1.56 Relative Atomic (Molecular) Mass Ar, Mr

Ar 5mA

uð1Þ; Mr 5

mM

uð2Þ

mA (kg) � atomic mass; mM (kg) � molecular mass; u (kg) � atomic mass unit

(1.660573 10227 kg); mM (kg mol21) � molar mass.


The relative atomic mass of an atom is a dimensionless number indicating

how many times the atomic mass is greater than the atomic mass unit. Similarly,

the relative molecular mass of a molecule is a dimensionless number indicating

how many times the molecular mass is greater than the atomic mass unit. The

relative molecular mass can be calculated as a sum of all relative atomic masses of

all atoms of a molecule. The atomic mass unit is defined as 1/12 of the mass of the

carbon isotope nuclide 126C;

u5m 12

6C 12

:

The molar mass gives the 1-mole quantity substance mass. The pure substance

molar mass can be calculated as a quotient of its mass and its mass quantity or as a

product of the atom (molecular) mass and the number of atoms (molecules) in one

mole of the substance,

Mm 5mMNA5MruNA 5 0:000999995Mr

which is valid for the molar mass in the kg mol21. For the molar mass in g mol21

units, the relation Mm5Mr is valid. Therefore, the relative molecular (atomic)

mass corresponds numerically to the molar mass expressed in the g mol21 units.

For example, for the relative molecular mass of carbon dioxide CO2, one obtains

(12123 15.9994)5 43.9988. Hence, the CO2 molecule has the molecular mass of

43.9988u and the molar mass of 43.9988 g mol21.

The relative molecular (atomic) masses and the molar masses can be used espe-

cially in stochiometric calculations.

Info: [C91].

2.1.57 Relative Humidity ϕ

ϕ5φφv

φ (kg kg21) � specific absolute humidity; φv (kg kg21) � saturation-state-specific

absolute humidity.

It expresses the relation of the gas humidity to the saturation-state humidity, spe-

cifically under the same temperature and pressure. If saturated, the gas has the dew

point temperature.

Info: [A24].

2.1.58 Riedel Number Rie

Rie5T

p

dp

dT

� �crit

T (K) � temperature; p (Pa) � pressure.


It expresses the temperature�pressure relation of fluids under critical

conditions.

2.1.59 Shape Equivalence of System A

A5S

Sekv

S (m2) � heat transfer surface of warmed-up or cooled-down body; Sekv (m2) �surface area of equivalent body by the same volume; O, Oekv (m) � cross section

perimeters of the body; Vekv (m3) � sphere equivalent volume.

It characterizes the relation between the geometric shape of a body and its

thermal field. It is used to replace geometrically complicated bodies with an unlim-

ited finite thickness plate, a cylinder or a ball. For an infinitely long cylinder,

A5 S S21ekv5O O21

ekv, and for a ball

A5Sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

36πV2ekv

3p

For example, for a boundary condition, it is αekv 5α S S21ekv.

Info: [A23].

2.1.60 Size Factor x

x52πaλ

5 ka

a (m) � particle size; λ (m) � wavelength; k (m21) � wave number.

It expresses the relation between the size of a moving particle, the wavelength

and the wave number.

Info: [C121].

2.1.61 Snellen Number Sn

Sn5a

bð1Þ; Sn5

a

b5 2913 1026

a

hð2Þ

a (m) � distance at which an object can be recognized; b (m) � distance at which the

same object can be seen at a subtended angle of 1/60� to the eye; h (m) � object size.

It expresses the perfection of sight. For perfect sight it is Sn5 1 and for imperfect

sight it is Sn, 1. Optics. Ophthalmology.

Info: [A43].


Herman Snellen (19.2.1834�18.1.1908), Dutch ocularist

and physicist.

Above all, he was engaged in the physiology of the

eye. He created a very complex body of work in this special

sphere. Together with F.C. Donders he did great numbers

of experiments and participated in the development of oph-

thalmological instruments. He focused on corrective eye

glasses calculations. The visual Snellen chart, consisting of

variously sized letters, is still used to determine the quality

of sight all over the world.

2.1.62 Sound Emissivity NI

NI 5Ia

Ii

Ia, Ii (W m22) � intensity of absorbed (a) and incident (i) sound.

It expresses the ratio of the absorbed sound intensity to that of the incident

sound.

Info: [A24].

2.1.63 Sound Pressure Level L

L5 20 logp

p05 10 log

I

I0

p (Pa) � acoustic pressure; p0 (Pa) � reference pressure (p05 23 1025 Pa);

I (W m22) � acoustic intensity; I0 (W m22) � threshold acoustic intensity.

It expresses the logarithm of the acoustic-to-reference pressure ratio. Acoustics,

noise measurement.

2.1.64 Specific Heat Ratio k, γ

k5cp

cv

cp (J kg21 K21) � specific heat capacity at constant pressure; cv (J kg21 K21) �specific heat capacity at constant volume.

It expresses the ratio of the heat capacity at constant pressure to the heat capac-

ity at constant volume. Alternatively, it expresses the ratio of the enthalpy to the

internal energy. It characterizes thermodynamic relations in compressible fluid

flow. It is called also the adiabatic criterion, Poisson’s constant or the adiabatic

exponent. For air, it is γ5 1.4. Physics. Mechanics. Aerodynamics.


Info: [C108].

Simeon Denis Poisson (p. 143).

2.1.65 Strehl Ratio S

S5 expð22ð2πσÞ2Þ

σ (�) � root-mean-square deviation of the wavefront measured in wavelengths.

The Strehl ratio is expressed by the relation of the focus intensities peak in

deviated and ideal points of diffraction. Optics, light interference and diffraction.

Info: [C125].

2.1.66 Surface Elasticity Number NSE

NSE 52CS

DS

L

η@σ@CS

CS (K s21) � surface concentration of surfactant in undisturbed state; DS (m2 s21) �surface diffusivity; L (m) � film thickness; η (Pa s) � dynamic viscosity;

σ (N m21) � surface tension.

This number expresses the surface elasticity of solutions in diffusive mass

transfer.

Info: [A29].

2.1.67 Surface Tension Number NST

NST 5η2

hσR

η (Pa s) � dynamic viscosity; h (m) � ratio of surface area to perimeter; σ (N m21) �surface tension; R (kg m23) � density.

It characterizes the surface tension in mass transfer.

Info: [A31].

2.1.68 Surface Viscosity Number NSV

NSV 5ηSη h

ηS (N s m21) � surface viscosity; η (Pa s) � dynamic viscosity; h (m) � film depth

(thickness).


This number is used if there is a convective transfer zone in a fluid layer with

a surface-active substance.

Info: [A29].

2.1.69 Temkin Number Es

Es5

ffiffiffiS

pffiffiffiffiV3

p 5LS

LVð1Þ; Es5

OffiffiffiA

p ð2Þ

S (m2), V (m3) � surface and volume of thermal system; LS, LV (m) � characteristic

dimensions; O (m), A (m2) � perimeter and cross-sectional area.

It characterizes the body shape influence on its thermal field. However, in

comparing diverse bodies, the difference of physical parameters is not considered.

For limited bodies, expression (1) is valid and so is expression (2) for in-one-direction

non-limited bodies which have two finite shapes.

The lengths LS and LV depend on the prolongation and shape of the body. To

consider also the body size in the Es number, the auxiliary parameter Es0 5 L L21Vis used.

In general, the Es number is not a similarity criterion because the set of bodies

corresponds to its one value. However, from the equality of the criteria, no conclu-

sions can be made about their physical similarity even if their shapes are subject to

the isomorphism condition. The mutually unambiguous relation between the body

shape and the Es criterion can be determined if the body shape and size are the

functions of two physical dimensions only. For example, Es5 2.35 is valid for a

cylinder of a radius r and height 2r. For a ball of radius r, Es5 2.21 and Es5 2, 51

for a cone of the height of rffiffiffi3

pand a base of a radius r.

With the same values of the Biot number Bi (p. 173) and Fourier number Fo

(p. 175), the increasing of the number Es leads to faster cooling of the body, to

decreasing its surface temperature and to increasing the temperature non-uniformity.

The influence of the Es number on the temperature field of bodies, other than convex

ones, appears most expressively. It is also called the shape criterion or the geometric

cooling down criterion.

Info: [A23].

A.G. Temkin.

2.1.70 Timescale Number Nτ, Go

Nτ 5wτL

ð1Þ; Go5τDτR

ð2Þ

w (m s21) � speed; τ (s) � time scale; L (m) � length scale; τD (s) � dynamic time;

τR (s) � radiation time.


In expression (1), it is the dimensionless expression of time of a moving body

or fluid. The controlling timescale number for the atmosphere or the interior

of non-rotating planets (2) is called the Golitsyn number Go (p. 397). Dynamic pro-

cesses. Astronomy.

Info: [C134].

2.1.71 Tortuosity τ

τ5l

L

l (m) � mean length of the free path; L (m) � characteristic length.

It characterizes permeable substances and expresses the ratio of a true

mean length of free path, for example, of the flow, to the characteristic length

(thickness) of the substance. Besides the application in fluid mechanics in flow

through crystallic or porous materials, it is used in hydrogeology, in physical

technologies, in mathematics in functional minimizing by cubic splines and in

solving curvature problems. For example, in physical technologies, it expresses the

ratio of the molecule path, which must pass through the deposited layer tortuously,

to the thickness of the layer. Mathematics. Physics. Fluid mechanics. Geophysics.

Physical technology.

Info: [C135].

2.1.72 Tzou Number, Stability and Convergency Criterion Tz

Tz 5Δxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2aΔtð2τT 1ΔtÞ2τq 1Δt

s ð1Þ;

Tz 5Δx

vΔt

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi11

Δt

2τq

s; v5

ffiffiffiffiffia

τq

sð2Þ

Δx (m) � mesh increment; a (m2 s21) � thermal diffusivity; Δt (s) � time step;

τT (s) � thermalization time; τq (s) � relaxation time; v (m s21) � thermal wave

speed.

For the finite differences method it determines the stability and convergence zone

of a solution if Tz$ 1. In the case of unbalanced heat propagation with the defined

relaxation time τq and the thermalization time τT, the criterion is in form (1). In the

case of wave-diffusive heat propagation (τT5 0), the criterion can be expressed in

form (2). In the case of balanced diffusive heat propagation (τT5 τq), the criterion

is reduced on the Crank�Nicolson parameter (p. 9).

Info: [A46],[B116].


2.1.73 Van der Waals Number (1.) Wa

Wa5rRcritTcrit

pcrit5

rTcrit

pcritvcrit

r (J kg21 K21) � specific gas constant; Rcrit (kg m23) � substance density at critical

state; Tcrit (K) � critical thermodynamic temperature; pcrit (Pa) � critical pressure;

vcrit (m3 kg21) � critical specific volume.

It is the fundamental criterion of thermodynamics. It expresses the state law of

fluids for a critical point and characterizes one substance or a whole group thereof in

accordance with their thermodynamical similarity. For various substances, its value

is various in general. However, in van der Waal’s equation its value is Wa5 8/3, but

it is WaAh3.3; 4.5i alternatively. It is called the critical coefficient as well.

Info: [A23].

Johannes Diderik van der Waals (23.11.1837�9.3.1923),

Dutch physicist. Nobel Prize in Physics, 1910.

He was engaged in research in gaseous and liquid states

of a mass. He started from the idea that the ideal gas state

equation can be deduced from the kinetic gas theory, pro-

vided the volume of molecules and the force between them

is neglected. With it, he succeeded in creating a new theory.

Subsequently, in the year 1881, he formulated a new state

equation named the van der Waals equation, the validity of

which he extended to all substances.

2.1.74 Van der Waals Number (2.) Wa

Wa5A

kT

A (J) � Hamaker constant; k (J K21) � Boltzmann constant; T (K) � absolute

temperature.

It characterizes the ratio of the van der Waals interaction energy to the heat

energy of particles. Thermodynamics of two-phase fluids. Filtration. Deposition.

Info: [A42].

Johannes Diderik van der Waals (see above).

2.1.75 Void Fraction ε

ε5Va

Vm

Va (m3) � volume of void space; Vm (m3) � total or bulk volume of material

including the solid and void components.


It expresses the ratio of the void space volume in the material to the total material

volume. Physics. Fluid mechanics. Flow through a porous material. Drying.

2.1.76 Wave Number K, k

K5L

λð1Þ; K5

2πLλ

5ωLa

ð2Þ; K5Krefn ð3Þ

L (m) � characteristic length; λ (m) � wavelength; ω (s21) � angular frequency;

a (m s21) � sound velocity; Kref (�) � wave number in the reference medium;

n (�) � medium’s index of refraction.19

In a dimensionless shape it expresses the wave length. The expressions (1) and

(2) are different forms only. For example, the expression (3) is used with electro-

magnetic waves. In acoustics, it is called the substance wave number (medium

wave number).

Info: [A2],[B7],[C139].

2.1.77 Zhukovsky Shape Number Ju

Ju5A

16Es2Es20 ð1Þ; Julam 5 32LoΨ ð2Þ;

Jurel 51

64Julam ð3Þ

Es, Es0 (�) � Temkin number and parameter (p. 30); A (�) � shape equivalence

of system (p. 27); Lo (�) � Lobachevsky number (p. 18).

It expresses the influence of clear area on the hydraulic resistance for fluid flow in

non-circular canals and pipelines. By using the Lobachevsky number Lo (p. 18) and

the non-uniformity criterion of the rate field Ψ5wmaxw21, the Zhukhovski laminar

number (2) is obtained. The expression (3) is the relative shape criterion. The corre-

spondence between the expressions (1) and (3) is given by various modifications of

Hagen�Poiseuille’s law for the laminar flow in a circular, cross-sectional tube.

Info: [A23],[A33].

Nikolay Jegorovich Zhukovsky (17.1.1847�17.3.1921),

Russian physicist and mechanicist.

He founded the Russian school of hydromechanics and

aerodynamics and is called the father of Russian aeronautics.

He was engaged in improving the wing profile (Zhukhovski’s

profile) and published two works (1906) in which he introduced

a mathematical expression for wing lift (Kutta�Zhukhovski’s

theorem). He devoted himself also to high-velocity aerodynam-

ics and to hydraulic shocks. In mathematics, he is known thanks

to his conformal mapping in a complex plane, which is known

as Zhukhovski’s transform.


2.2 Physical Chemistry

In physical chemistry, the dimensionless quantities mostly express the physical

and chemical conversions in homogeneous or heterogeneous processes or systems.

The focus is the non-isothermic or isothermic processes such as burning and

combustion, dynamics of flame propagation, baking, and heat and mass transfers

with chemical or phase reactions. Further, there are mixing, drying and corrosion

processes and physical and chemical processes in chemical reactors, mass diffusion

processes in catalyzers, granulation processes, dissociation of electrolytes in solu-

tions, separation of solid particles in solutions and others.

The Arrhenius, Damkohler, Frank-Kamenetskii, Guldberg�Waage, Knudsen

diffusion and Prater numbers are among the best known dimensionless numbers.

2.2.1 Arrhenius Number (1.) Energetic Ah1

Ah1 5E

RTð1Þ; Ah1 5U21 5

E

RTNð2Þ

E (J mol21) � molar activation energy; R (J mol21 K21) � molar gas constant;

T (K) � absolute temperature; TN (K) � ambient temperature.

This number expresses the activation-to-potential energy ratio of a fluid. It

characterizes the energetic conditions in a chemical�thermal reaction. For example,

it relates to the ability of a fuel to self-ignite. It is used, for example, to describe the

heat transfer in combustion chambers. Usually, the inverse value of equation (2) is

called the combustion parameter U (p. 35).

Info: [A29],[A33],[B17].

Svante August Arrhenius (19.2.1859�2.10.1927), Swedish

physical chemist. Nobel Prize in Chemistry, 1903.

He was engaged in chemical reaction description and

discovered the Arrhenius velocity law, which describes the

speed of chemical reactions. In addition, he elaborated on

the theory of electrolytes and made clear the electrical

conductivity of ion solutions. He received the Nobel Prize

for his electrolytic dissociation theory.

2.2.2 Arrhenius Number (2.) Velocity Ah2

Ah2 � Bd � Pem 5wL

Da

w (m s21) � speed; L (m) � characteristic length; Da (m2 s21) � axial diffusivity;

Bd (�) � Bodenstein number (p. 244); Pem (�) � Peclet mass number (p. 258).


It characterizes the mass diffusion process in chemical reactors. It is the measure

of the convective mass transfer related to the molecular transfer in flowing sur-

roundings. It expresses the degree to which an investigated system approaches an

idealized model of the reacting flow structure. With Ah2-N, it relates to a reactor

or system with total separation. With Ah2-0, it relates to a completely mixing sys-

tem. In intermediate models, N.Ah2. 0 is valid. It concerns diffusive, combined

or cellular models. Often, it is called the Bodenstein number Bd (p. 244). In

essence, it is the modified Peclet mass number Pem (p. 258).

Info: [A33].

Svante August Arrhenius (see above).

2.2.3 Arrhenius Number (3.) Time Ah3

Ah3 � Fo21ch 5uL2

RDa

5L2

Daτch

u (kg m23 s21) � chemical reaction rate; L (m) � characteristic length; R (kg m23) �density; Da (m2 s21) � axial diffusivity; τch (s) � chemical reaction time;

Foch (�) � Fourier chemical number (p. 39).

In the dimensionless shape, its reciprocal value characterizes the chemical reaction

time. Alternatively, it is also the Fourier chemical number Foch (p. 39). Sometimes it

is called the diffusion contact number Ko (p. 38), namely in a mononuclear reaction

in a stationary process with a simple conversion.

Info: [A33].

Svante August Arrhenius (see above).

2.2.4 Burnout Parameter γ

γRT2

N

E

cV RQ Ci

R (J mol21 K21) � molar gas constant; TN (K) � ambient temperature; E (J mol21) �molar activation energy; cV (J kg

21 K21)� specific heat capacity; R (kg m23)� density;

Q (J)� heat reaction effect; Ci (m23)� reagent numerical molecule concentration.

It characterizes the burnout in fuel combustion.

Info: [A23].

2.2.5 Combustion Parameter U

U � Ah211 5RTN

E

R (J mol21 K21) � molar gas constant; TN (K) � ambient temperature; E (J mol21) �molar activation energy; Ah1 (�) � Arrhenius number (1.) energetic (p. 34).


This parameter characterizes the temperature originating in a thermal system as

a chemical reaction result. See the Arrhenius number (1.) energetic Ah1 (p. 34).

Info: [A23].

2.2.6 Chemical Reaction Rate Γ

Γ 5 δγη expΘ

11UΘ

� �

δ (�) � Frank-Kamenetskii number (p. 35); γ (�) � burnout parameter (p. 40); η (�)

� critical reagent concentration; Θ (�) � reagent temperature (p. 47); U (�) �combustion parameter (p. 35).

It characterizes the reaction kinetics in a physical and chemical conversion

process.

Info: [A23].

2.2.7 Damkohler Number Da

Da5τDτR

τD (s) � characteristic diffusion timescale; τR (s) � characteristic reaction timescale.

It expresses the relation of diffusion and relaxation times in chemical processes.

It is used for turbulent combustion. With Da{1, the turbulence is much faster than

the chemical reaction.

Info: [A23],[A33].


2.2.8 Damkohler Number (1.) Hydrodynamic Da1

Da1 5uL

wRið1Þ; Da1 5

τcτr

ð2Þ

u (kg m23 s21) � chemical reaction rate; L (m) � characteristic length; w (m s21) �flow velocity; Ri (kg m

23) � mass concentration; τc, τr (s) � chemical conversion

and reaction time.

This number expresses the ratio of chemical conversion rate to that of the fluid

flow. Alternatively, it is the ratio of the number of moles entering the reaction to

that of moles induced by the fluid flow. It characterizes the hydrodynamic influence

in chemical reactions. Alternatively, together with other Damkohler numbers, it

characterizes the reaction time in general with transport phenomena.

Info: [A2],[A26],[A29],[B11].

Gerhard Friedrich Damkohler (see above).


2.2.9 Damkohler Number (2.) Diffusion Da2

Da2 5u L2

D Ri

u (kg m23 s21) � chemical reaction rate; L (m) � characteristic length; D (m2 s21) �diffusivity; Ri (kg m

23) � mass concentration.

It characterizes the diffusion relations in chemical reactions. It expresses the

ratio of the chemical conversion rate in a fluid flow to the molecular mass diffusion

rate; alternatively, it is the ratio of the number of reacting moles in a fluid flow to

the number of moles brought by the molecular diffusion.

Info: [A2],[A23],[A26],[A29],[B11].


2.2.10 Damkohler Number (3.) (1. Heat) Da3

Da3 5huL

cpRwT

h (J kg21) � specific enthalpy; u (kg m23 s21) � chemical reaction rate; L (m) �characteristic length; cp (J kg

21 K21) � specific heat capacity; R (kg m23) � density;

w (m s21) � flow velocity; T (K) � temperature.

This number expresses the ratio of the heat liberated in a chemical reaction in

the fluid flow to that transferred by convection. It characterizes the heat transfer by

convection in chemical reactions in fluids.

Info: [A2],[A23],[A26],[A29],[B11].


2.2.11 Damkohler Number (4.) (2. Heat) Da4

Da4 � Th2 5huL2

λTð1Þ;

Da4 � PoV � Os � Th2 � huL2

λTrefð2Þ

h (J kg21) � specific enthalpy; u (kg m23 s21) � chemical reaction rate; L (m) �characteristic length; λ (W m21 K21) � thermal conductivity; T (K) � temperature;

Tref (K) � reference temperature; Th (�) � Thiele modulus (1.) (p. 267); PoV (�) �Pomerantsev heat number (p. 181); Os (�) � Ostrogradsky number (p. 179).

It expresses the ratio of the heat liberated by a chemical reaction to that trans-

ferred by conduction. It characterizes the heat transfer process in chemical reactions.

Substantially, the matter is the Pomerantsev heat number PoV (p. 181), also


called the Ostrogradsky number Os (p. 179) in Russian literature. Its second root

corresponds to the Thiele modulus (1.) Th (p. 267).

Info: [A2],[A23],[A26],[A29],[B11].


2.2.12 Damkohler Number (5.) Hydrodynamic Da5

Da5 � Re5w L

vð1Þ;

Da2

Da1Da55 Sc ð2Þ; Da4

Da3Da55Pr ð3Þ

w (m s21) � flow velocity; L (m) � characteristic length; ν (m2 s21) � kinematic

viscosity; Re (�) � Reynolds number (p. 81); Sc (�) � Schmidt number (p. 263);

Pr (�) � Prandtl number (p. 197); Da1 (�) � Damkohler number (1.) hydrody-

namic (p. 36); Da2 (�) � Damkohler number (2.) diffusion (p. 37); Da3 (�) �Damkohler number (3.) (1. heat) (p. 37); Da4 (�) � Damkohler number (4.) (2.

heat) (p. 37).

For flowing fluid, it expresses the inertia-to-friction ratio. It characterizes the

hydrodynamic influence of the inertia and friction forces of the streaming fluid in

a chemical reaction. The relation of the Schmidt number (p. 263) and the Prandtl

number (p. 197) to the Damkohler numbers (p. 36, 38) is obvious from the expres-

sions (2) and (3).

Info: [A4],[A23],[A26],[A29],[B11].


2.2.13 Diffusion Contact Number Ko

See Arrhenius number (3.) time Ah3 (p. 35).

2.2.14 Djakonov Number (1.) Contact Dj, Ko

Dj5 k1ϕ1ðckÞτ5ττr

k1 ϕ1 (ck) (s21) � unit speed at a given concentration; k1 (�) � virtual speed constant

of direct reduction process; ck (�) � concentration; τ (s) � contact time; τr (s) �decay time.

Together with the thermodynamic equilibrium of chemical reduction Nreac

(p. 48), it expresses the general similarity condition for the physical and chemical

conversion process. It is the measure of the contact-to-decomposition time ratio

of exiting products of the chemical conversion. The contact time means the time

during which the corresponding mixture composition is in the reaction zone.

Info: [A13].


German Konstantinovich Djakonov (11.6.1907�1953),

Russian physical chemical engineer.

His work represents a new dimension in complex process

modelling. He was engaged especially in the theory and

methods of modelling physical and chemical processes and

conversions. He solved the problem of thermodynamic equi-

librium. From the analysis of all gained similarity conditions

and criteria, he determined that they cannot be fulfilled at

the same time. Therefore, he was engaged in approximate

modelling and created an original method of solution.

2.2.15 Djakonov Number (2.) Thermodynamic Dj, Po

Dj5k2ϕ2ðckÞk1ϕ1ðckÞ

k2 ϕ2 (ck) (s21) � unit rate of indirect change; k1 ϕ1(ck) (s

21) � unit rate of direct

change; k1, k2 (�) � virtual rate constant of direct and indirect reaction; ck (�) �critical concentration.

It expresses the thermodynamic equilibrium in the physical and chemical con-

version process. It represents the return-to-direct conversion rates ratio. Together

with the Djakonov number (1.) contact Dj (p. 38), it represents the condition for

the general physical and chemical conversion.

Info: [A13].

German Konstantinovich Djakonov (see above).

2.2.16 Djakonov Number (3.) Thermodynamic Di

Di5q

n

q (m3 s21 m23) � specific volume flux; n (s21) � rotational frequency.

It expresses the ratio of the substance, mixed in a unity reaction volume during

a time unit, to the amount of the substance conveyed into a reacting volume. It

characterizes the physical and chemical conversion process in the mixing of fluids.

Alternatively, it can also be expressed as the ratio of the substance conversion rate

to the rate of its feeding into the chemical fluid mixing facility.

Info: [A13].

German Konstantinovich Djakonov (see above).

2.2.17 Fourier Chemical Number Foch

See Arrhenius number (3.) time Ah3 (p. 35).



2.2.18 Fraction Granulation Number Pfrac

Pfrac 5V

VV

V (m3) � particle volume; VV (m3) � vessel volume.

It represents the parameter expressing the ratio of the volume of particles to that

of the vessel in the granulation process. Physical chemistry. Granulation. Mixing.

Info: [C26].

2.2.19 Frank-Kamenetskii Number δ

δ5Q k CiL

2

λ TN

E

RTNexp 2

E

R TN

� �5

qv L2

λTNE

RTNexp 2

E

RTN

� �

5PoV

Uexp 2

1

U

� �5PoV Ah1 expð2Ah1Þ

Qk (J s21)� volume reaction heat flow rate; Ci (m23)� initial numerical concentration

of reagent molecules; L (m) � characteristic length; λ (W m21 K21) � thermal con-

ductivity of reacting mixture; TN (K)� ambient temperature; E (J mol21)� activation

energy per unit molar mass; R (J mol21 K21) � molar gas constant; qV (W m23) �volume heat flux at the beginning of chemical reaction; PoV (�) � Pomerantsev

heat number (p. 181); U (�) � combustion parameter (p. 35); Ah1 (�) � Arrhenius

number (1.) energetic (p. 34).

In the dimensionless shape, this number characterizes the reaction heat and the

heat transfer in reacting systems, for example, in burning.

Info: [A29],[A35].

David Albert Frank-Kamenetskii (born 1910), Russian engineer.

2.2.20 Frank-Kamenetskii Temperature Φ

Φ5EðT 2 T0Þ

RT2N

E (J mol21) � molar activation energy of reaction; T (K) � temperature after

reaction; T0 (K) � initial temperature of the reacting chemicals; R (J mol21 K21) �molar gas constant; TN (K) � reference temperature.

It characterizes the temperature or temperature stains (hot spots, higher tempera-

ture zones) originating in an exothermic reactive filling of a reservoir in which a

thermal explosion can occur if the heat generated in the reservoir is greater than

that let out of it. Physical chemistry. Ecology.

Info: [B106].

David Albert Frank-Kamenetskii (see above).


2.2.21 Froude Granulation Number Frgran

Frgran 5n2d

g

n (s21) � engine revolutions; d (m) � engine diameter; g (m s22) � acceleration

of gravity.

It expresses the centrifugal-to-gravitational energy ratio in mixing and granulating

processes. The criterion was deduced from empirical experience. Physical chemistry.

Granulation.

Info: [C67].

William Froude (p. 63).

2.2.22 Granulation Fluid Number Ngran

Ngran 5mτVR

m (kg s21) � quantity of granulating fluid added at time unit; τ (s) � time; V (m3) �volume of particles; R (kg m23) � density of particles.

This number expresses the specific quantity of granulating fluid in granulation.

Physical chemistry. Granulation.

Info: [C67].

2.2.23 Guldberg�Waage Number Gw

Gw5 yn11 yn22 . . . 2ðxm11 xm22 . . .Þ 1k

y1, y2, ... (�) � mole fractions of reaction products; x1, x2,... (�) � mole fractions

of reactants; n1, n2, ... (�) � stoichiometric coefficients of individual reaction

products; m1, m2, ... (�) � stoichiometric coefficients of individual reactants;

k (�) � dimensionless partial equilibrium constant.

It characterizes the similarity of chemical reactions in progress, for example,

in blast furnaces. It is expressed by the relation of volumes of reacting gases and

products of the reaction.

Info: [A23],[A29].

Cato Maximilian Guldberg (11.8.1836�14.1.1902), Norwegian

mathematician.

Together with his cousin Peter Waage, in the year 1864, he

formulated the law of mass behaviour in chemical reactions.

This law expresses in detail the influence of concentration,

mass and temperature on the speed of chemical reactions.


Peter Waage (29.6.1833�13.1.1900), Norwegian chemist.

In the year 1864, together with his cousin Cato

Maximilian Guldberg, he formulated the law of mass

behaviour in chemical reactions. He was also engaged in

mineralogy and crystallography.

2.2.24 Hatta Number Ha, Ht

Ha5γ

tanh γ; where γ5

ðknRn21i DÞ12β

ð1Þ;

Ha5k1D

β2ð2Þ

kn (m3 (n21) kg2 (n21) s21) � reaction rate constant for nth order chemical reaction;

Ri (kg m23) � mass concentration; D (m2 s21) � diffusivity; β (m s21) � mass

transfer coefficient.

Generally, in form (1), it characterizes the gas absorption with a chemical

reaction being under way. In form (2), it is about the first order of a chemical reac-

tion, for example, in cleaning (washing) the gas in fluid. Flow in geologic layers,

silts and deposits.

Info: [A39],[A50].

Shiroji Hatta (born 1895), Japanese chemist.

2.2.25 Helmholtz Pulsation Number, Helmholtz Resonator Group K

K5L3

V

� �12

M21

L (m) � characteristic length, diameter; V (m3) � volume; M (�) � Mach number

(p. 73).

It characterizes the processes in pulsating combustion.

Info: [A43].

Hermann Ludwig Ferdinand von Helmholtz

(31.8.1821�8.9.1894), German physiologist, physicist and

philosopher.

In the year 1847, he explained the energy conservation law

mathematically and was first to extend Joule’s results into a

general principle. In the year 1850, he measured the impulse

propagation velocity in a nerve tissue. He was engaged in the

physiology of eyesight and hearing. Furthermore, he devoted

himself to electromagnetism, acoustics and thermodynamics.


2.2.26 Henry Number He

He5R1R2

R1, R2 (kg m23) � mass concentration.

In a chemical reaction, it expresses the ratio of the mass concentration and the

phase interface.

Info: [A33].

William Henry (12.12.1775�2.9.1836), English chemist.

He was focused on research on the chemistry of gas, espe-

cially on gas absorption in water under diverse temperatures

and pressures. He formulated Henry’s law. At only 23 years

of age, he published his best works describing the experi-

ments he executed. In other works, gas analysis, humidity

in burning and other chemical problems are treated. The

book Elements of Experimental Chemistry (1799) was his

most important work.

2.2.27 Hess Number Hs

Hs5knL

2Rn21ini

am

kn (m3 (n21) kg2 (n21) s21) � reaction rate constant for nth order of chemical

reaction; L (m) � characteristic length; Rini (kg m23) � initial concentration of

solid material; am (m2 s21) � mass diffusivity of reaction products.

It characterizes the heat and mass transfer in processes with chemical and phase

reactions.

Info: [A35].

Germain Henri Hess (1802�1850), Russian chemist of Swiss origin.

2.2.28 Karlowitz Number (1.) Ka

Ka5dw

dy

L

w; where L5

λcpRgw

w (m s21) � gas velocity; y (m) � coordinate; L (m) � characteristic length;

λ (W m21 K21) � thermal conductivity; cp (J kg21 K21) � specific heat capacity;

Rg (kg m23) � gas density.

In the burning process, it characterizes the flame extending or increasing its

surface in flowing gas, the rate of which is strongly higher than the burning rate.

Due to the surface extending, the front of the flame front is curved. The criterion


expresses the critical degree of the flame divergence with which the equilibrium

between heat generation and thermal loss, in the reactive zone, can be destroyed

and the flame extinguished. With the burning wave extended on the path L being

longer than the critical path, the flame is extinguished.

Info: [A23],[A33].

2.2.29 Karlowitz Number (2.) Ka

Ka5τcharτKol

5w2Kol

w2lam

; where wKol 5ffiffiffiffiffivε4

p

τchar (s) � characteristic flame timescale; τKol (s) � smallest Kolmogorov turbulence

time; wKol (m s21) � Kolmogorov velocity; wlam (m s21) � laminar flame velocity;

ν (m2 s21) � kinematic viscosity; ε (m2 s23) � turbulence dissipation velocity.

It characterizes a thin reactive zone in the combustion area. It is used if the least

turbulent whirling in the flow is less than the flame width.

Info: [A103].

2.2.30 Karman Ratio N

N5λ

R cp

κ M um

R w25

at

ap

a2

w25 Le21p M22;

where

ap 5p

Mum; at 5

λcpR

; a5

ffiffiffiffiffikp

R

s

λ (W m21 K21) � gas thermal conductivity; Rg (kg m23) � gas density;

cp (J kg21 K21) � specific heat capacity; M (kg mol21) � molar density;

um (mol m23 s21) � molar rate of chemical reaction; w (m s21) � tributary velocity

of fuel mixture; at, ap (m2 s21) � temperature and pressure diffusivity; a (m s21) �

sound velocity; p (Pa) � pressure; k (�) � specific heat ratio (p. 28); Lep 5 ap a21t

(�) � Lewis pressure number; M (�) � Mach number (p. 73).

This ratio characterizes the steady-state burning of a gas mixture flow. It indicates

the relation between the flame propagation rate and the quantities that characterize

burning.

Info: [A23].

Theodore von Karman (p. 67).


2.2.31 Knudsen Number (2.) Diffusion Kn

K n � Sm21 5μ DAB

ζ DKA

μ (�) � porosity; DAB (m2 s21) � coefficient of binary volume diffusion of AB

system; ζ (�) � diffusion tortuosity; DKA (m2 s21) � Knudsen diffusion coefficient;

Sm (�) � Smoluchowski number (p. 86).

It expresses the ratio of the volume diffusion to that of Knudsen diffusion in

a granulated layer. It characterizes the diffusion in continuous granulated layers

of a catalyzer. It is especially applied in chemical technology. Its inverse value is

called the Smoluchowski number Sm (p. 86).

Info: [A29],[A35].

Martin Hans Christian Knudsen (p. 420).

2.2.32 Mass Fraction M

M5mAPmi

mA (kg) � mass of substance A; mi (kg) � mass of individual substances.

It characterizes the mass fraction of partial components, for example, in mixing.

Physical chemistry.

Info: [C85].

2.2.33 Mole Fraction xi

xi 5ni

n5

Ni

N

ni (�) � number of moles of ith substance; n (�) � total number of moles in a

mixture; Ni (�) � number of molecules of ith substance; N (�) � total number of

molecules in a mixture; NA (�) � Avogadro number (p. 4).

The mole fraction is one of the ways to express the concentration of various

chemical components in a mixture. It is about approximation to an ideal mixture,

but corrections (active coefficients) must be introduced in practice.

n5Xj

nj; Ni 5 niNA;Xi

xi 5 1


Physical chemistry. Thermodynamics.

Info: [C90].

2.2.34 Prater Number Pra, β

Pra52ΔH DeCA0

T λe

ΔH (kJ kmol21) � heat of reaction; De (m2 s21) � effective diffusivity; CA0

(kmol m23) � concentration of species A; T (K) � temperature; λ (W m21 K21) �effective thermal conductivity.

This number characterizes the non-isothermic chemical reaction with diffusion

and thermal transfer resistance in pores or the isothermic internal reaction with

the diffusion resistance in pores. It expresses the joining of physical and chemical

processes such as diffusive and convective heat and mass transfers. Above all, it is

important in designing, starting and controlling chemical reactors.

Info: [A49].

2.2.35 Reaction Enthalpy Number N

N5ΔhAΔcA

cpΔT

ΔhA (J kg21) � specific reaction enthalpy change of A product; ΔcA (�) � change

of mass fraction of A product; cp (J kg21 K21) � specific heat capacity; ΔT (K) �

temperature difference.

It expresses the ratio of the reaction energy change to the thermal energy

change. It characterizes the inter-phase transfer with chemical reactions.

Info: [A35].

2.2.36 Reaction Enthalpy Number (2.) Nre

Nre 5ðΔHÞAΔMA

CΔT

(ΔH)A (J kg21) � reaction enthalpy; ΔMA (kg) � mass of fraction A; C (J K21) �heat capacity; ΔT (K) � temperature difference.

It expresses the ratio of reactive energy change to that of thermal energy.

A phase transfer with a chemical reaction.

Info: [A29].


2.2.37 Reagent Concentration η, χ

η5C

Cini

ð1Þ; χ5R

E

Q

RC

cVð2Þ

C, Cini (m23) � numerical initial molecule concentration of reagent; R (J mol21 K21) �

molar gas constant; E (J mol21) � molar activation energy; Q (J) � thermal effect

of reaction; Rg (kg m23)� density; cV (J kg

21 K21) � specific heat capacity.

In burning, it characterizes the reagent concentration or the chemical enthalpy.

Info: [A48].

2.2.38 Reagent Temperature Θ

Θ5E

RT2N

ðT2 TNÞ

E (J mol21) � molar activation energy; R (J mol21 K21) � molar gas constant;

TN (K) � ambient temperature; T (K) � reagent temperature.

It characterizes the reagent temperature in fuel burning.

Info: [A23].

2.2.39 Rejection Coefficient R

R5cr 2 cp

cr; where RAh0; 1i

cr (kg m23) � separating and suspension with a solid concentration; cp (kg m23) �at the same time measure the solid concentration of the permeate.

It characterizes the solid substances separation process in suspensions.

Essentially, it expresses the diaphragm (of filters) efficiency. Physical chemistry.

Separation and filtration of materials.

Info: [C81],[C112].

2.2.40 Semenov Number N

N5βwch

β (m s21)� mass transfer coefficient; wch (m s21)� chemical reaction progress rate.

It characterizes the reaction kinetics in a physical and chemical conversion

process.

Info: [A23].


Nikolay Nikolajevich Semenov (3.4.1896�25.9.1986),

Russian physicist and chemist. Nobel Prize in Chemistry, 1956.

He was engaged in the mechanisms of chemical transforma-

tion involving the complete analysis of chain theory application

for diverse chemical reactions and combustion processes. He

proposed the degenerate branching theory, which explained the

phenomenon related to inducing periods in oxidation processes.

He wrote great numbers of works in the sphere of molecular

physics and investigated shock wave propagation and electron

phenomena.

2.2.41 Thermal Capacity of Combustion Chamber N

N5σ T4k

k u

σ (W m22 K24) � specific thermal absorption; T (K) � flue gas temperature;

k (m21) � coefficient of environment reduction in combustor; k (J kg21) � specific

combustion heat; u (kg m23 s21) � chemical reaction rate.

It expresses the ratio of the heat absorbed in a chamber filled with the mediums to

all heat released by fuel combustion. It characterizes the complicated heat transfer in

combustion chambers with physical and chemical processes of fuel burning.

Info: [A23].

2.2.42 Thermodynamic Equilibrium of Chemical Reduction Nreac

Nreac 5k2ϕ2ðRÞk1ϕ1ðRÞ

ð1Þ; Nreac 5k2R

Pα2i21

2i

k1RP

α1i21 1i

ð2Þ

k1, k2 (�) � constants of chemical reaction; ϕ1(R), ϕ2(R) (�) � mass concentration

function of chemical reaction products; R, R1i, R2i (kg m23) � densities of individual

components; α1i, α2i (�) � stoichiometric number of component change.

In the form (1), it expresses the ratio of outlet products of physical and chemical

reduction to inlet ones. It characterizes the internal process thereof. It represents

a local criterion of the thermodynamic equilibrium or the deviation degree from

the state of the thermodynamic equilibrium alternatively. For simple homogeneous

processes, (2) is valid. For quasi-static processes, Nreac5 1 holds and so does

Nreac, 1 for dynamic ones. For continuously changing states of the internal equi-

librium, it is Nreac-1. For the process rate approaching the change rate, Nreac-0

holds, whereas Nreac5 0 is valid for limiting irreversible processes.

Info: [A33].


2.2.43 Thermochemical Bond Ntch

Ntch 5lRmcpRT

l (J kg21) � latent heat; Rm (kg m23) � mass concentration; cp (J kg21 K21) � specific

heat capacity; R (kg m23) � density; T (K) � temperature.

It expresses the ratio of thermochemical heat reduction to that caused by a molar

movement of reacting surroundings. It characterizes the bond between the thermal

and concentrating fields. Heat transfer in physical and chemical processes.

Info: [A33].

2.2.44 Thiele Modulus (2.) Φ, Th

Φ5 L

ffiffiffiffiffik1

D

r5

ffiffiffiffiffiffiffiffiDa2

p

L (m) � effective particle diameter of porous catalyst (equals 6/S)26; k1 (s21) �reaction rate constant; D (m2 s21) � diffusivity; Da2 (�) � Damkohler number (2.)

diffusion (p. 37).

It expresses the isothermic reaction with the diffusion resistance in pores.

Diffusion in porous catalyzers.

Info: [A33],[A49].

Ernest William Thiele (p. 267).

2.2.45 Thring�Newby Number Tn

Tn 5Qmt 1QmP

QmP

r

L

Qmt (kg s21) � fluid mass flux in nozzle; QmP (kg s21) � ambient mass flux; r (m) �

equivalent nozzle radius; L (m) � characteristic length, wall half-thickness.

It characterizes the fuel combustion process in combustion spaces.

Info: [A23],[A35].

Meredith Wooldridge Thring (17.12.1915�15.9.2006), English engineer.

Maurice Purcell Newby (born 1917), English physicist.

2.2.46 Time of Chemical Reaction Kτ

Kτ 5τchτL

τch (s) � chemical reaction time necessary to reach the equilibrium state in flowing

fluid; τL (s) � time necessary to fluid flow through the characteristic length L.


It characterizes the time to reach the equilibrium state in chemical reactions

in flowing fluids. For a flow with homogeneous chemical reactions, with Kτ5 0,

the chemical equilibrium state occurs in the flow. It is the case in which the

chemical reaction rates are so high that the equilibrium state occurs in every

place of the surroundings with given mass densities. With Kτ-N, the chemical

reactions are so small that in every place of the surroundings, the mixture

composition is determined by mass flows of various components. It is the case

of frozen flow.

In flow with heterogeneous reactions with Kτ5 0, the chemically reacting

mixture, adhering to the surface, is in a chemical equilibrium state with the

temperature and pressure on the surface. This situation is called the ideally cata-

lytic surface. The non-catalytic surface case occurs for Kτ-N, in which the

catalytic reaction rate is so small that the mixture surface composition is deter-

mined by diffusion and convection processes not coupled with the reactions on

the wall.

Info: [A23].

2.2.47 Van ’t Hoff Factor i, Ni

i5 11 aðq21Þ

a (�) � degree of ionization; q (�) � number of ions formed for each dissociated

molecule.

It expresses the number of moles in a water solution which are created by a

mole of a solid admixture. The matter is the electrolyte dissociation into ions in the

solution. For example, NaCl-Na1 1Cl2 (i5 2), MgBr2-Mg21 1 2Br2 (i5 3).

For glucose it is i5 1 because one mole of it dissolved in water leads to the result

of one diluted mole.

Info: [C137].

Jacobus Henricus van ’t Hoff, Jr. (30.8.1852�1.3.1911),

Dutch physical chemist. Nobel Prize in Chemistry, 1911.

He was engaged in research on chemical kinetics, chemi-

cal equilibrium, osmotic pressure and crystallography. He

contributed to elaborating the branch of physical chemistry.

He was engaged also in organic chemistry. In the year 1874,

he described the optical activity phenomenon. He studied

three-dimensional chemical structures and other problems

of stereochemistry. In addition, he worked out Svante

Arrhenius’ theory of electrolytic dissociation.


2.2.48 Volume Fraction ϕ

ϕ5VAPVi

VA (kg) � volume of the A component; Vi (kg) � volumes of individual compo-

nents in a mixture.

It characterizes the volume fraction of partial components, for example, in mix-

ing. Physical chemistry.

Info: [C138].

2.2.49 Wagner Number Wa

Wa5γd

@U

@JA

γ (S m21) � electric conductivity of solid phase; U (V) � upper potential for reduc-

tion of O2; JA (A m22) � surface density of electric current; d (m) � thickness of

catalytic layer.

In catalytic layers it characterizes the ohmic drop, specifically by comparing

the charge transfer with the ohmic resistances. For Wa, 1, the ohmic resistance

predominates. It is used to design diaphragm�electrolyte systems, to solve cathodic

catalytic layers and gas diffusers and to fuel elements if need be. It can also be

applied to determine the current density distribution in electrochemical cells.

Info: [B42].

2.2.50 Weisz Modulus Wz, Ψ0

Wz5 ηkΦ2 5 ηkDa2; where ηk 5

uef

uðTS;CSÞ

uef (s21) � effective chemical reaction rate; u (s21) � chemical reaction rate; TS (K) �

outside surface temperature of catalyzer; ηk (�) � degree of required porosity; Φ (�) �Thiele modulus (2.) (p. 49); Da2 (�) � Damkohler number (2.) diffusion (p. 37);

CS (�)� gas constant for catalyzer surface.

It is applied to characterize the isothermal reaction with the diffusive resistance

of pores. Catalyzers. Diffusion in porous catalyzers.

Info: [A49].


3 Fluid Mechanics

The quantitative character of objects, one of the most general laws of the

existence.Leonhard Euler (1707�1783)

3.1 One-Phase Fluid Mechanics

In this field, the range of similarity criteria used is great. It involves the flow of

potential, compressible and incompressible fluids; viscous and non-viscous flows;

steady and unsteady flows; and laminar, turbulent and other flows. Some criteria

concern the related phenomena arising in flow, such as the boundary layer, friction,

acting forces, stroke waves, the sound barrier, surface tension and swirls. Among

the fundamental and most widespread similarity criteria are the Froude, Knudsen,

Mach, Prandtl, Richardson and Strouhal numbers.

3.1.1 Acceleration Number Ac

Ac5E3

R g2 η25Re2 Fr2 Ho23

E (Pa) � modulus of elasticity; R (kg m23) � density; g (m s22) � gravitational

acceleration; η (Pa s) � dynamic viscosity; Re (�) � Reynolds number (p. 81);

Fr (�) � Froude number (1.) (p. 62); Ho (�) � Hooke number (p. 138).

This number characterizes the acceleration flow. It depends on physical proper-

ties and gravitational acceleration only.

Info: [B20].

3.1.2 Archimedes Hydrodynamic Number Ar

Ar5g L3

v2Rs 2 R

R5Ga

Rs 2 RR

ð1Þ; Ar5g L3

v2εB

12 εBð2Þ

g (m s22) � gravitational acceleration; L� d (m) � particle diameter; ν (m2 s21) �fluid kinematic viscosity; Rs, R (kg m23) � particle and fluid density; εB (�) �volume fraction; Ga (�) � Galileo number (p. 123).



http://dx.doi.org/10.1016/B978-0-12-416013-2.00003-8

It expresses the ratio of product of a floating body weight and the inertia force

to the square of the viscosity force. It characterizes the free flow of fluid caused by

the density difference of partial components. For example, it is about the floating of

solid particles, drops or bubbles (2) � or possibly the whole composition thereof �caused by uplift forces in the fluid volume. It is an unusual modification of the

Galileo number Ga (p. 123). It occurs in the heat transfer between the particles,

for example, in fluidization and in material bundles. The characteristic feature is

the heat transfer by convection and conduction in a wall-particle system.

Info: [A14],[A29],[B20].

Archimedes of Syracuse (287 BC�212 BC), Greek mathematician.

From our present point of view, he was a physicist and

engineer. He formulated the physical principle of upward

force and the weight of drawn fluid. This theorem became

known as the Archimedes law. He utilized Euclidean geome-

try to calculate the surfaces and volumes of bodies. He deter-

mined the approximate value of π.

3.1.3 Bairstow Number NBa

See Mach number M (p. 73).

Info: [A29].

Leonard Bairstow (1880�1963), English aerospace engineer.

3.1.4 Bejan Convective Number Be

Be5ΔpL2

ηa

Δp (Pa) � pressure drop; L (m) � characteristic length; η (Pa s) � dynamic viscos-

ity; a (m2 s21) � thermal diffusivity.

In fluid mechanics, it represents the dimensionless expression of the pressure

drop along an L-long canal. In convection, its role resembles the Rayleigh number

(2.) (heat instability) Ra2 (p. 187) in natural convection.

Info: [C6].

Adrian Bejan (p. 9).

3.1.5 Beranek Number Be

Be5w3sR

2l

ηlgðRs 2 RlÞ


ws (m s21) � fall velocity of solid particles; Rl, Rs (kg m23) � density of liquid and

solid particles; ηl (Pa s) � dynamic viscosity of liquid; g (m s22) � gravitational

acceleration.

It expresses the ratio of the inertia force square to the product of the viscosity

and the gravitational force. It characterizes the bypassing of free falling particles in

immobile unlimited fluid.

Info: [A29].

Josef Beranek, Czech chemist.

3.1.6 Blasius Shape of Boundary Layer δ=x

δx5

4; 9ffiffiffiffiffiffiffiRex

p ; where Rex 5wc

v

δ (m) � boundary layer thickness; x (m) � distance from entering edge; w (m s21) �flow velocity; ν (m2 s21) � kinematic viscosity; Rex (�) � local Reynolds number

relative to the x position.

It characterizes the viscous fluid flow in a laminar boundary layer above a float-

ing plate. The expression is used to monitor the boundary layer development and to

simplify the solution.

Info: [A21].

3.1.7 Boussinesq Number Bs

Bs5w2

2gL

w (m s21) � flow velocity; g (m s22) � gravitational acceleration; L (m) � charac-

teristic length, hydraulic radius.

It expresses the inertia-to-weight forces ratio. Weight influence on surface

waves in open canals.

Info: [A29], [B17], [B20].

Valentin Joseph Boussinesq (13.3.1842�19.2.1929), French

physicist and mathematician.

He was engaged in many areas of mathematical physics,

but especially in hydrodynamics, the undulation and flowing

of fluids, resistance of bodies in flowing fluid, the cooling

down effect of fluids and the problems of turbulence and

fluid elasticity.

55Fluid Mechanics

3.1.8 Boussinesq Approximation Parameter Bs

Bs5g0

g5

R1 2 R2Rref

g0 (m s22) � approximated gravitational acceleration; g (m s22) � gravitational

acceleration; R1, R2 (kg m23) � density of warm and cold liquid; Rref (kg m23) �reference density.

Compared to the often used parametric expression R1/R2, the flow tasks become

more precise by making use of this parameter. Fluid mechanics.

Info: [C109].

Valentin Joseph Boussinesq (see above).

3.1.9 Buoyancy Number NB

NB 5L2FβΔT

ηVw

L (m) � characteristic length; F (N) � buoyancy force; β (K21) � volume thermal

expansion coefficient; ΔT (K) � temperature difference; η (Pa s) � dynamic vis-

cosity; V (m3) � volume; w (m s21) � velocity.

It is the buoyancy-to-viscous forces ratio. It describes the influence of viscous

fluid natural convection on buoyancy.

3.1.10 Buoyancy Parameter NB

NB 5ΔT

T

gL

w25Gr1Re

22 5ΔT

TFr21

ΔT (K) � temperature difference; T (K) � environment temperature; g (m s22) �gravitational acceleration; L (m) � characteristic length; w (m s21) � flow velocity;

Gr1 (�) � Grashof heat number (p. 185); Re (�) � Reynolds number (p. 81);

Fr (�) � Froude number (1.) (p. 62).

It characterizes the buoyancy in free and forced convections. It is analogous to

the Archimedes thermodynamic number Ar2 (p. 184).

Info: [A35].

3.1.11 Capillary Number (2.) Ca

Ca5η2ERσ2

5ηwσ

� �25We21Ho

21Re22


η (Pa s) � dynamic viscosity; E (Pa) � bulk modulus of fluid; R (kg m23) � liquid

density; σ (N m21) � surface tension; w (m s21) � flow velocity; We1 (�) �Weber number (1.) (p. 91); Ho (�) � Hooke number (p. 138); Re (�) � Reynolds

number (p. 81).

It represents the influence of the surface tension acting on the flowing fluid and

its capillarity. It depends on physical properties of the fluid.

Info: [A29],[A35].

3.1.12 Centrifuge Number Ncf

Ncf 5Rr2hω2

σ

R (kg m23) � density; r (m) � radius; h (m) � fluid depth; ω (s21) � angular fre-

quency; σ (Pa) � surface tension.

It equals the centrifugal-to-capillary forces ratio. The flow in curved canals. The

centrifugal forces of stroke and surface waves. Hydromechanics.

Info: [B20].

3.1.13 Coefficient of Velocity ϕ

ϕ5w

wt

; whereas ϕ, 1

w (m s21) � real velocity; wt (m s21) � theoretic velocity.

This coefficient expresses the ratio of the actual fluid flow velocity, with friction

and without contraction, to the theoretical velocity without friction. It depends on

the bypassed wall roughness. Usually ϕAh0, 95; 1i. Together with the contraction

coefficient α (p. 57), it determines the outflow coefficient μ (p. 77). Aero-

hydrodynamics.

Info: [A39].

3.1.14 Contraction Coefficient α, ε

α5A

A0

A, A0 (m2) � cross section of nozzle spray and outlet area.

It is the ratio of the out-flowing beam diameter to the orifice area.

Hydromechanics.

Info: [A39].

57Fluid Mechanics

3.1.15 Correlation Coefficient R, .

R5wx1wx2ffiffiffiffiffiffiffiffiffiwx1

2p ffiffiffiffiffiffiffiffiffi

wx22

pwx1, wx2, wx1 ;wx2 (m s21) � pulsation velocities measured at the same time in posi-

tions 1 and 2 and their mean values.

It expresses the flow turbulence measure. Aerodynamics.

Info: [C22].

3.1.16 Crispation Number Ncr

Ncr 5ηaσL

η (Pa s) � dynamic viscosity; a (m2 s21) � thermal diffusivity; σ (Pa m) � undis-

turbed surface tension of the layer; L (m) � characteristic length, layer thickness.

This number characterizes the influence of the penetration of viscous fluid con-

vective flow in the layer thickness while considering the thermal diffusion and the

surface stress. One-phase fluid mechanics.

Info: [C25].

3.1.17 Crocco Number Cr

Cr5w

wmax

5 112

ðk21ÞM2

� �212

w (m s21) � local flow velocity; wmax (m s21) � velocity at adiabatic expansion;

κ (�) � specific heat ratio (p. 28); M (�) � Mach number (p. 73).

This number expresses the ratio of local flow rate to the highest possible fluid

rate in outlet into a vacuum (adiabatic expansion). It characterizes the compressible

fluid flow. It is also called the modified Mach number M.

Info: [A29], [B20].

3.1.18 Darcy Number (1.) Dc

Dc52ghd

w2L

g (m s22) � gravitational acceleration; h (m) � pressure loss; d (�) � pipe diame-

ter; w (m s21) � flow velocity; L (m) � characteristic pipe length of constant cross

section.


It expresses the pressure loss of fluid flowing through a pipeline. The Darcy

friction number fD (p. 119) or the Fanning friction number fF (p. 163) are modi-

fications of this. Fluid mechanics. Hydraulics. Tribology.

Info: [A26], [B20].

Henry Philibert Gaspard Darcy (p. 98).

3.1.19 Darcy Number (2.) Dc

Dc52ghrh

w2ð1Þ; Dc5

8ghrh

w2ð2Þ

g (m s22) � gravitational acceleration; h (m) � mean hydraulic depth; rh (�) �hydraulic gradient; w (m s21) � flow velocity.

It characterizes the fluid flows in open canals and over falls. It is analogous to

the Darcy friction number fD (p. 119). Hydraulics and hydromechanics.

Info: [A21],[B20].


3.1.20 Dean Number Dn

Dn5wd

v

ffiffiffir

R

r5Re

ffiffiffir

R

r

w (m s21) � flow velocity; r5 d/2 (m) � canal radius; ν (m2 s21) � kinematics

viscosity; R (m) � radius of canal curvature; Re (�) � Reynolds number (p. 81).

It expresses the mutual relation between centrifugal, friction and inertia forces

and characterizes the flow in curved canals.

Info: [A14],[B17],[B20].

William Reginald Dean (1896�1973), British mathematician and physicist.

3.1.21 Degree of Turbulence E

E5w

wT

w (m s21) � mean square velocity of pulsation; wT (m s21) � mean velocity of tur-

bulence flow.

This degree expresses the ratio between the mean-quadratic pulsation velocity

and the mean turbulent flow velocity. Aero-hydrodynamics.

59Fluid Mechanics

3.1.22 Density Number N.

NR 5ΔRR0

ΔR (kg m23) � density difference of matter; R0 (kg m23) � initial density.

It expresses the ratio of the density difference, before and after loading, to the

initial density of solid, liquid or gaseous surroundings. For example, it is used in

fluid mechanics with the Boussinesq approximation when the original gravitational

acceleration g is transformed to g5 g(ΔR/R) in controlled buoyancy fluid flow.

Mechanics. Fluid mechanics. Buoyancy flow.

Info: [C10].

3.1.23 Derjaguin Number De

De5 L

ffiffiffiffiffiffiRg2σ

r

L (m) � characteristic length, film thickness; R (kg m23) � density; g (m s22) �gravitational acceleration; σ (N m21) � surface tension.

It expresses the ratio of the film thickness to the capillary length. It characterizes

the process of coatings and films generation of fluids on a body surface. Fluid

mechanics. Physical technology.

Info: [A35].

3.1.24 Drag Coefficient CD

CD 5FD

1

2Rw2A

ð1Þ; CD 5ðRS 2 RFÞLSg

RFw2ð2Þ

FD (N) � drag force on body; R, RS, RF (kg m23) � density, density of solid parti-

cles and fluid; w (m s21) � flow velocity; A (m2) � cross-sectional area of body

perpendicular to flow velocity; LS (m) � characteristic small dimension of parti-

cles; g (m s22) � gravitational acceleration; _m (kg s21) � mass flux.

In expression (1), this coefficient expresses the ratio of the drag force FD acting

perpendicularly to the dynamic force at a stagnation point on a transversal area A. It

characterizes the aerodynamic or hydrodynamic resistance in fluid flow. The expres-

sion (2) is valid in case of flow with small particles and steady distribution thereof.

From the relation (1), an equation expressing the drag force follows, which is

necessary to overpower the aerodynamic resistance and to set an object into

movement.


FD 5CD

1

2Rw2A ð3Þ

The expression (3) for the drag force is valid provided the flow is turbulent,

where CD is approximately constant. For laminar flow, it is

FD � _mw ð4Þand the force is, therefore, proportional to the velocity. With a turbulent flow, the

power is

P5FDw51

2Rw3A ð5Þ

The drag coefficient CD is important especially in the aircraft and car industries.

One tries to reach the least value of CD. For example, for automobiles it is CDAh0.35;0.45i and for sports cars CDAh0.25; 0.30i. For standard cars, it is CD�0.5, for a rac-

ing bicycle CD�0.9, and for a motorcycle CD�1.8. The coefficient CD is the physical

similarity criterion which enables � in the case of its equality between the model and

the object (CDM5CD), the provision of the behaviour similarity condition, for exam-

ple, with various geometric dimensions of the model and the object.

Info: [A21],[A35].

3.1.25 Euler Number (1.) Eu

Eu5Δp

Rw25

F

Rw2L2ð1Þ; Eu5

1

κM2ð2Þ

Δp (Pa) � pressure drop due to friction; R (kg m23) � density; w (m s21) � flow

velocity; F (N) � frictional force; L (m) � characteristic length; κ (�) � specific

heat ratio (p. 28); M (�) � Mach number (p. 73).

This number expresses the pressure gradient due to friction related to dynamic

pressure; alternatively, the ratio of the pressure surface force to the inertia force. It

characterizes the hydrodynamic pressure, pressure loss, hydraulic resistance and

fluid friction, and their influence on the proceeding process. With a compressible

environment flow, the expression (2) is used. For the pressure flow, it is Eu5Ne.

Info: [A29],[A35],[A43],[B20].

Leonhard Euler (15.4.1707�18.9.1783), Swiss mathemati-

cian and physicist.

He shifted the boundaries in modern analytical geometry

and trigonometry. He was characterized by his extraordinary

memory and, despite his initially partial and later total blind-

ness, he was most prolific mathematician of all time with his

more than 800 works. Great numbers of his works concern

optics, mechanics, electricity and magnetism.

61Fluid Mechanics

3.1.26 Euler Curved Number Eucurv

Eucurv 58Δpr3RDn2η2R

Δp (Pa) � pressure drop in the curved pipe; r (m) � pipe diameter; R (kg m23) �liquid density; η (Pa s) � dynamic viscosity; R (m) � curvature radius; Dn (�) �Dean number (p. 59).

It expresses the ratio of the pressure gradient due to friction in a curved tube to

the dynamic pressure. It characterizes the pressure loss, hydraulic resistance and

friction in curved canals. Fluid mechanics. Flow in curved canals.

Leonhard Euler (see above).

3.1.27 Force Coefficient CF

CF 5τw

1

2Rw2

5F

1

2Rw2A

5 2Ne

τw (Pa) � stress component of circumfluenced body; R (kg m23) � fluid density; w

(m s21) � flow velocity; F (N) � force; A (m2) � drag area; Ne (�) � Newton

number (p. 75).

It is important mainly in aerodynamics and expresses the resistance-to-inertia

forces ratio. As a vector, the force F has a drag component FD and that of uplift FL.

Info: [A20].

3.1.28 Fourier Hydrodynamic Number (Fourier Flow Number) Foh

Foh � Zh5vτL2

5ητRL2

ν (m2 s21) � kinematic viscosity; τ (s)� time; L (m) � characteristic time; η (Pa s)�dynamic viscosity; R (kg m23) � density; Zh (�) � Zhukovsky number (p. 93).

In the dimensionless shape, it characterizes the non-stationary flow time of a

viscous incompressible fluid. It is called the Zhukovsky number Zh (p. 93).

Info: [A29].


3.1.29 Froude Number (1.) Fr

Fr5Ri21 5w2

gL5Ga21Re2 ð1Þ; Fr5

w2RgLΔR

ð2Þ


w (m s21) � flow velocity; g (m s22) � gravitational acceleration; L (m) � charac-

teristic length; R (kg m23) � liquid density; ΔR (kg m23) � density difference

between liquid and other phase; Ri (�) � Richardson number (p. 83); Ga (�) �Galileo number (p. 123); Re (�) � Reynolds number (p. 81).

This number expresses the ratio of the inertia force to the gravitational or uplift

force; alternatively, the ratio of the kinetic energy to the potential energy. For

example, it characterizes the surface phenomena in free-level fluid flow. It is

applied in hydraulics, wherever the weight influence must be considered. It can be

neglected in gases. With the bubble floating process in a fluid, the bubble radius

(L� r) is inserted for the characteristic length. The inverse value of the Froude

number (1.) is called the Richardson number Ri (p. 83) or the Reech number Ree

(p. 81) sometimes. The Boussinesq number (1.) Bs (p. 55) is its analogue, as well.

Info: [A7],[B20].

William Froude (28.11.1810�4.5.1879), English engineer

and naval architect.

He was engaged in hydrodynamics and hydraulics and

especially in surface phenomena in fluid flowing on free sur-

faces and in open canals, influenced by gravity. He also stud-

ied processes of the rising of bubbles in a fluid and other

phenomena connected to fluid flow and floating bodies.

3.1.30 Froude Number (2.) External Fr

Fr5wffiffiffiffiffiffigL

p 5ffiffiffiffiffiffiffiFr�

pð1Þ; Fr5

wffiffiffiffiffigh

p 5w

cð2Þ

w (m s21) � flow velocity; c (m s21) � velocity of the water level wave propaga-

tion at the position, where is the sea depth h; Fr� (�) � Froude number (1.) Fr

(p. 63); for other quantities see Froude number (1.) Fr (p. 62).

It expresses the relation of the inertia force to the gravitation force; alternatively,

the ratio of the flow rate in an open canal to the propagation rate of slight gravita-

tion waves. It characterizes the free-level fluid flow in open canals. It is also used

to determine the character of the flow. With Fr, 1, the flow is subcritical, with

Fr5 1, it is critical and with Fr. 1, it is supercritical or accelerated.

Info: [A43],[B20].

William Froude (see above).

3.1.31 Froude Internal Number (1.) Frint

Frint 5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRw2

gL2ðdR=dzÞ

s

63Fluid Mechanics

R (kg m23) � unperturbed density; w (m s21) � flow velocity; g (m s21) � gravita-

tional acceleration; L (m) � characteristic length; dR/dz (kg m24) � vertical density

gradient.

This number expresses the influence of the kinetic and potential energy and of

the vertical density gradient in sedimentation.

Info: [C73].


3.1.32 Froude Internal Number (2.) Frint

Frint 5w

cg5

w

fBh

w (m s21) � flow velocity; cg (m s21) � velocity of gravitation wave propagation;

fB (s21) � buoyancy frequency; h (m) � depth.

It expresses the ratio of the current fluid acceleration to the wave-induced

acceleration.

Info: [C73].


3.1.33 Froude Rotation Number Frrot

Frrot 5Ln2

g

L (m) � characteristic length, L� d � diameter of pipe, bubble, drop, shaft or

wheel; n (s21) � rotation speed; g (m s22) � gravitational acceleration.

It characterizes the forced fluid flow due to mixing, vibrating or other

movement.

Info: [A29].


3.1.34 Gortler (Goertler) Number, Parameter Goe

Goe5Reδ

ffiffiffiδR

rð1Þ; whereas Reδ 5

wδv

ð2Þ

δ (m) � thickness of momentum boundary layer; R (m) � radius of wall edge; w

(m s21) � flow velocity; ν (m2 s21) � kinematic viscosity; Reδ (�) � Reynolds

boundary layer number (p. 82).

In the form (1), it characterizes the instability of laminar boundary layers on

curved concave walls. It expresses the origin of Karman swirls in the bypassing of


solid bodies. It is a modification of the Reynolds boundary layer number Reδ(p. 82) given by the expression (2). Therefore, it is also called the flow stabilization

criterion in a boundary layer.

Info: [A29].

Henry Gortler (born 1909), German mathematician.

3.1.35 Goucher Number Go

Go5r2Rg2σ

r (m) � radius of body, wire, etc.; R (kg m23) � density; g (m s22) � gravitation

acceleration; σ (N m21) � surface tension.

It is the gravitation-to-surface-tension forces ratio. The surface phenomena in

fluids. The tension. Making of films and coatings. See the Deryagin (p. 60), Bond

(p. 95) and Eotvos (p. 99) numbers.

Info: [A29],[B20].

Frederick Shand Goucher (30.9.1888�21.8.1973), Canadian physicist.

3.1.36 Hagen Number Hg

Hg52 Rdp

dx

L3

v2

rp52 dp/dx (Pa m21) � pressure gradient, for natural convection holds

rp5 gRβΔT; L (m) � characteristic length; ν (m2 s21) � kinematic viscosity;

g (m s22) � gravitational acceleration; R (kg m23) � liquid density; β (K21) � coef-

ficient of volume expansion at constant pressure; ΔT (K) � temperature difference.

It is used for forced flow. The Hg number approaches the Grashof heat number

Gr (p. 185) for a natural flow (Hg-Gr).

Info: [C70].

3.1.37 Hagen�Poiseuille Number Ha

Ha52dp

ds

L2

ηw

dp/ds (Pa m21) � pressure gradient; L (m) � characteristic length; η (Pa s) �dynamic viscosity; w (m s21) � flow velocity.

This number characterizes laminar viscous fluid flow. It is also called the

Poiseuille number.

Info: [A35].

Jean Louis Marie Poiseuille (p. 78).

65Fluid Mechanics

3.1.38 Hodgson Number Hs

Hs5VfΔp

QVpS

V (m3) � system volume; f (s21) � flow gas frequency; Δp (Pa) � pressure fall

due to friction loss, etc.; pS (Pa) � mean static pressure; QV (m3 s21) � mean vol-

ume flux.

It expresses the non-uniformity of the delivered amount in pulsating flow.

It states the ratio of a set time constant to the pulsation period time. Pulsating flow.

Info: [A29],[A33],[B20].

John Lawrence Hodgson (1881�1936), English engineer.

3.1.39 Hydraulic Resistance Group NRH

NRH 5Δp

RlgL5Eu Fr

Δp (Pa) � pressure drop across liquid on distillation tray; Rl (kg m23) � fluid den-

sity; g (m s22) � gravitational acceleration; L (m) � depth of liquid layer; Eu (�) �Euler number (1.) (p. 61); Fr (�) � Froude number (1.) (p. 62).

This number expresses the pressure drop, for example, in distillation columns.

Info: [A29].

3.1.40 Jeffrey Number Je

Je5RgL2

ηw5Re Fr21 5 Stk211

R (kg m23) � density; g (m s22) � gravitational acceleration; L (m) � characteristic

length; η (Pa s) � dynamic viscosity; w (m s21) � velocity; Re (�) � Reynolds

number (p. 81); Fr (�) � Froude number (1.) (p. 62); Stk1 (�) � Stokes number

(p. 130).

It expresses the slow fluid flow.

Info: [A26].

3.1.41 Karman Number (1.) Ka

Ka5 Rd3 2dp

dx

� �η22 5

U Rpd3

η2L5 2f

1=2F Re2


R (kg m23) � density; d (m) � pipe diameter; dp/dx (Pa m21) � pressure gradient;

η (Pa s) � dynamic viscosity; L (m) � characteristic pipe length; fF (�) � Fanning

friction number (p. 163); Re (�) � Reynolds number (p. 81).

It expresses the flow in a pipeline with friction.

Info: [A35],[B17].

Theodore von Karman (11.5.1881�6.5.1963), American

engineer of Hungarian origin.

His scientific work is characterized by the use of applied

mathematics in engineering. His work is very extensive and

includes fluid mechanics, turbulence theory, supersonic

flights, aircraft structures and wind erosion of soil. He was

engaged in the theory of missile movement in the atmosphere

and was a co-founder of magneto-hydrodynamics.


Ka5h

v

ffiffiffiffiffiτsR

r

h (m) � mean height of surface roughness; ν (m2 s21) � kinematic viscosity;

τs (Pa) � shear stress near the wall surface; R (kg m23) � density.

It expresses the ratio of the shear stress and the viscous force in a fluid. It char-

acterizes the friction in floating fluids.

Info: [A29].

Theodore von Karman (see above).

3.1.43 Karman Turbulent Number Katur, Tu

Katur � Tu5

ffiffiffiffiffiffiffiffi2e

3w2r

s5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiw2x 1w2

y 1w2z

3w2r

s

e (m2 s22) � specific kinetic energy of turbulent flow; wr (m s21) � characteristic

flow velocity, for example, input flow; wx, wy, wz (m s21) � pulsation velocity com-

ponents in any points of turbulent flow; Tu (�) � turbulence number (p. 90).

It expresses the ratio of a mean-quadratic component of a local flow rate pulsa-

tion vector to the mean value of the relative flow rate. It characterizes the flow tur-

bulence and its influence on the heat transfer within forced convection. It is also

called the turbulence number Tu (p. 90).

Info: [A33].


67Fluid Mechanics

3.1.44 Karman�Tsien Correction Factor Cp

Cp 5Cp0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

12M2N

p1

M2N

11ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12M2

N

p !

Cp0

2

Cp0 (�) � pressure coefficient for incompressible fluid, see pressure coefficient Cp

(p. 80); MN (�) � Mach number for high subsonic flow, see Mach number M

(p. 73).

This rule expresses the compressibility correction for the pressure distribution on

a bypassed body surface in high subsonic flow. Fluid mechanics. Aerodynamics.

Info: [A21].


3.1.45 Keulegan Number Ke

Ke5wffiffiffiffiffi

gv3p

5ffiffiffiffiffiffiffiffiffiffiffiffiFr Re3

p

w (m s21) � flow velocity; g (m s22) � gravitational acceleration; ν (m2 s21) �kinematic viscosity; Fr (�) � Froude number (1.) (p. 62); Re (�) � Reynolds

number (p. 81).

This number expresses the dimensionless fluid flow rate, hydrodynamic stability

and structure and dynamics of the near-surface layer of the ocean. Hydraulics.

Oceanography.

Info: [A41].

3.1.46 Keulegan�Carpenter Number (1.) Kec, KC

Kec52πAL

A (m) � amplitude of sinusoidal flow; L (m) � characteristic length, height of the

circumfluenced body.

It characterizes the ratio of drag force to inertia for a bluff object in oscillatory

flow. It is analogous to the Strouhal number Sh (p. 87). A low value of KecAh0.4; 2ishows the inertial effect to be more important than the viscosity effect.

Hydrodynamics and aerodynamics. Oceanography. Flow oscillations in turbines and

pipelines.

Info: [A8],[A21].


3.1.47 Kirchhoff Number Kh

Kh5RpL2

η25Eu Re2

R (kg m23) � density; p (Pa) � pressure; L (m) � characteristic length; η (Pa s) �dynamic viscosity; Eu (�) � Euler number (1.) (p. 61); Re (�) � Reynolds

number (p. 81).

It expresses the pressure-to-molecular-friction forces ratio. It characterizes the

fluid flow process in flow-through canals.

Info: [A23],[A33].

Gustav Robert Kirchhoff (12.3.1824�17.10.1887), German

physicist.

He was engaged in mechanics and electricity. In the year

1859, he constructed a microscope by means of which he dis-

covered two new elements. In addition to applied spectral

analysis, he was engaged in electric circuits and in thermal

radiation. He formulated two Kirchhoff laws enabling the

calculation of the currents and voltages in electric circuits.

His work on black body radiation contributed to the develop-

ment of quantum theory.

3.1.48 Kirpitchev Hydrodynamic Number Kih

Kih 5

ffiffiffiffiffiffiffiffiffiRFR

η23

s5

ffiffiffiffiffiffiffiffiffiffiffiffiffiRe2CF

3p

R (kg m23) � density; FR (kg m s22) � hydrodynamic resistance; η (Pa s) � dynamic

viscosity; Re (�) � Reynolds number (p. 81); CF (�) � force coefficient (p. 62).

This number expresses the flow around obstacles and the resistance acting on a

body submerged in a viscous fluid flow.

Info: [A26],[A29].

Michail Viktorovich Kirpichev (1879�1955), Russian engineer and physicist

(p. 177).

3.1.49 Knudsen Number (1.) Kn

Kn5 Sm21 5lm

L5 1:28

ffiffiffiκ

pη

aRLð1Þ;

Kn5M Re21 ð2Þ; Kn5M Re212 ð3Þ;

69Fluid Mechanics

Kn5 1:28ffiffiffiκ

p M

Reð4Þ

lm (m) � mean free path; L (m) � characteristic length; η (Pa s) � dynamic viscosity;

a (m2 s21) � thermal diffusivity; R (kg m23) � density; Sm (�) � Smoluchowski

number (p. 86); κ (�) � specific heat ratio (p. 28); M (�) � Mach number (p. 73);

Re (�) � Reynolds number (p. 81).

It is also called the flow continuity criterion. It expresses the ratio of the average

free path of molecules in gas to the characteristic length dimension of a body.

It characterizes the aerodynamic gas dilution degree or, alternatively, its deflection

from the continuum state. It determines the conditions under which the phenomena

connected to gas dilution occur. With Kn. 10 the molecules flow freely, with

0.1,Kn, 10 transient flow occurs, and with 0.01,Kn, 0.1 slipping on a wall

sets in. With Kn, 0.01, a continual flow is achieved. The flow of artificial satel-

lites in an orbit without collisions above the earth is an example for Knc1. With

Re# 1 the expression (2) is applied, as is expression (3) for Rec1. The inverse

value of the Knudsen number is called the Smoluchowski number Sm (p. 86). In

kinetic theory, this number is used in the form (4).

Info: [A21],[A43],[B11],[B20].


3.1.50 Lagrange Number (3.) Lg3

Lg3 5ΔpL

wη5Eu Re

Δp (Pa) � pressure difference; L (m) � characteristic length; w (m s21) �mean

velocity; η (Pa s) � dynamic viscosity; Eu (�) � Euler number (1.) (p. 61);


This number expresses the ratio of the pressure force, under the action of the

hydraulic resistance, to the viscosity force. With fluid flowing in direct smooth and

even rough flow canals, it characterizes the relation between the pressure and

velocity fields. It is used in magneto-hydrodynamics especially.

Info: [A1],[A4],[A43],[B110].

Joseph-Louis Lagrange (25.1.1736�10.4.1813), French

mathematician and physicist of Italian origin.

He was the greatest mathematician of the nineteenth cen-

tury. He was engaged in the theory of numbers, algebra, the

theory of analytic functions, differential equations, variation

calculus, celestial mechanics and mathematical cartography.

The Lagrange equations represent the foundation of theoretical

mechanics. His Mecanique Analytique (Analytic Mechanics,

1788) was his principal work.


3.1.51 Laplace Number Lp

Lp5ΔpL

σ5Eu We1 ð1Þ; Lp5

σRLη2

5Ca Re ð2Þ;

Lp5

ffiffiffiffiffiffiffiffiffiffiσ

RgL2

r5Fr We

212

1 ð3Þ

Δp (Pa) � pressure difference; L (m) � characteristic length; σ (N m21) � surface

tension; R (kg m23) � density; η (Pa s) � dynamic viscosity; g (m s22) � gravita-

tional acceleration; Eu (�) � Euler number (1.) (p. 61); Ca (�) � capillary num-

ber (2.) (p. 96); Re (�) � Reynolds number (p. 81); Fr (�) � Froude number (2.)

external (p. 63); We1 (�) � Weber number (1.) (p. 91).

In expression (1), it expresses the ratio of the pressure force to the surface

stress. In expression (2), it expresses the ratio of the surface stress to the momen-

tum transfer in a fluid or to the gravitation force in a flowing mixture. In case (3),

it describes the gas�fluid mixture flow in a vertical pipeline or in distillation

plants.

Info: [A29],[A35],[C76].

Pierre-Simon, Marquis de Laplace (23.3.1749�5.3.1827),

French physicist, mathematician, astronomer and statesman.

He was engaged in mathematical analysis, probability

theory and celestial mechanics, in which he summarized

the results of his predecessors in the five-volume work

Mecanique Celeste (Celestial Mechanics). He formulated a

hypothesis on the origin of the solar system from a rotating

nebula. He was engaged in the integral transform known

today as the Laplace transform. He contributed to the calori-

metric theory and the determination of the thermal capaci-

ties of many materials. He discovered the gravitational

potential.

3.1.52 Laval Number La

La5w

acritð1Þ; La5w

2κκ1 1

rT

� �2 12

ð2Þ

w (m s21) � flow velocity; acrit (m s21) � critical sound velocity; r (J kg21 K21) �specific gas constant; T (K) � temperature; κ (�) � specific heat ratio (p. 28).

It expresses the ratio of the flow rate to the critical sound velocity. It charac-

terizes incompressible fluid flow. It is also called the critical Mach number Mcrit

and sometimes is denoted as M�. It is analogous to the Crocco number Cr (p. 58).

Info: [A26],[A29],[A35].

71Fluid Mechanics

Carl Gustaf Patrik de Laval (9.5.1845�2.2.1913), Swedish

engineer of French origin.

In 1887, he built small steam turbines to show how these

plants could be constructed on a large scale. Then, in the

year 1890, he developed the Laval nozzle enabling the

increase of the steam velocity entering a turbine. At present,

the principle of this nozzle is still utilized, for example, in

missile construction. He contributed significantly to the

development of centrifugal separators and other devices.

3.1.53 Lift Coefficient CL

CL 5FL

1

2Rw2A

FL (N) � lift force representing vertical aerodynamic drag; R (kg m23) � fluid den-

sity; w (m s21) � flow velocity; A (m2) � cross-sectional area of body.

Its main importance is in aerodynamics. It expresses the ratio of the vertical

component of the resistance force to the inertia force. It depends strongly on the

inclination angle. For the lift force perpendicular to the drag force, the following

equation for object lifting is valid:

FL 5CL

1

2Rw2A

Info: [A7],[A20],[A21].

3.1.54 Lift-to-Drag Ratio CLD

CLD 5FL

FD

5CL

CD

FL (N) � buoyancy force; FD (N) � drag force; CL (�) � lift coefficient (p. 72);

CD (�) � drag coefficient (p. 60).

It expresses the efficiency measure of an aerodynamic profile. The greater this

number is, the better the wing is.

Info: [A8],[A21].

3.1.55 Loss Factor ζ, ζc, K

ζ5ez

ekð1Þ; ζc 5 2

Xni5 1

ezi

w2ð2Þ; ζ5

Δp12Rw2

ð3Þ

ez (J kg21) � specific loss energy; ek (J kg

21) � specific kinetic energy; w (m s21) �flow velocity; Δp (Pa) � head loss; R (kg m23) � fluid density.


This factor expresses the ratio of the specific loss energy to the specific kinetic

energy (1). In expression (2), it is about the relation of the total energy loss to the mean

kinetic energy. In equation (3), it is the loss coefficient for pipelines. Fluid mechanics.

Info: [A21].

3.1.56 Lyashenko Number Ly

Ly5Re3Ar21

Re (�) � Reynolds number (p. 81); Ar (�) � Archimedes hydrodynamic number

(p. 53).

This number expresses the influence of the inertia force, gravitation force and

viscosity. Fluidization.

Info: [A4],[A35].

3.1.57 Mach Number M, Ma

M5w

að1Þ; M5

wffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðκ21ÞcpTp ð2Þ;

where a5ffiffiffiffiffiffiffiffiffiE=R

pfor any liquid, a5

ffiffiffiffiffiffiffiffiffikRT

pfor ideal gas; w (m s21) � local flow

velocity; a (m s21) � local sound velocity; κ (�) � specific heat ratio (p. 28); cp(J kg21 K21) � specific heat capacity; T (K) � gas temperature; E (Pa) � modulus

of elasticity; R (kg m23) � density.

In form (1), it expresses the ratio of the local flow rate to the sound velocity:

elastic deformation propagation or the mean velocity of the thermal molecule

movement in gases. It characterizes the fluid flow in a given place or, alternatively,

the enthalpy conversion into the kinetic energy in a flowing ideal gas. For M, 1 it

is about subsonic velocities, for M 5U 1 about sonic ones, for M. 1 about super-

sonic ones, for 0.8,M, 1.3 about transonic ones and for Mc1 about hypersonic

flow. With a hypersonic flow (M. 5) in the flow core, the fluid starts to behave as

a plasma due to high temperature. In the subsonic zone, it is a measure of the ideal

gas compressibility (2). The compressibility influence appears over M .~

0:3:In France, it is also called the Sarrau number Sa (p. 85) or the Bairstow number

NBa (p. 54) in England, though this is not yet used in general practice. Sometimes,

it is called the Maievski number Ma (p. 74).

Info: [A2],[B20].

Ernst Mach (18.2.1838�19.2.1916), Austrian physicist of

Moravian origin.

He influenced considerably the development of science in the

twentieth century. The idea that recognition of the laws of nature

is mediated by human senses forms the basis of his philosophy.

The Mach principle expresses inertia effects as a property of the

whole universe. Using this concept, Einstein began to formulate

the general relativity theory. He was engaged, too, in experimen-

tal physics and the construction of devices.

73Fluid Mechanics

3.1.58 Mach Angle μ

μ51

sinðM21Þ ; for M-1 it holds μ-90�

M (�) � Mach number (p. 73).

It expresses the Mach cone angle in supersonic fluid flow. High velocity

aerodynamics.

Info: [A21].

Ernst Mach (see above).

3.1.59 Mach Characteristic Number Machar, M0

Machar 5w

a0

w (m s21) � flow velocity; a0 (m s21) � sound velocity at Mchar5 1.

The Mchar is not the usual sonic Mach number M (p. 73), but the number related

to the sound velocity reached with the sonic Mach number. This Mach number form

is used to simplify the basic equations only; see the Prandtl relation M2 (p. 79).

Info: [A21].


3.1.60 Mach Number Across a Shock M

M5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðγ21ÞM�2 1 2

2γM�2 2 ðγ2 1Þ

s

γ (�) � specific heat ratio (p. 28); M� (�) � Mach number of inclination wave,

see Mach number M (p. 73).

For a normal shock wave, the Mach number across a shock expresses the

changes across the shock, which are a function of the drain Mach number only. For

the Mach number of inclination wave M� 5 1, the Mach number across a shock is

M5 2. For M�-N, it is M5ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðγ21Þ=2γ

p: Fluid mechanics. Aerodynamics of

high velocities.

Info: [A21].


3.1.61 Maievski Number Ma

See Mach number M, Ma (p. 73).

Nikolaj Vladimirovic Majevskij (1823�1892), Russian physicist and

mechanist.


3.1.62 Mass Ratio Nm

Nm 5m

πRL3

m (kg) � mass; R (kg m23) � fluid density; L (m) � characteristic length.

It characterizes the ratio of a submerged body weight to that of the surrounding

fluid.

Info: [A29],[B20].

3.1.63 Moment Coefficient CM

CM 5M

12Rw2Ac

M (N m) � total moment acting on wing; R (kg m23) � fluid density; w (m s21) �flow velocity; A (m2) � drag area c (m) � mean chord of aerofoil section.

This moment is important in the aerodynamic stability of aircraft and other

objects. For the moment necessary for tilting, it is

M5CM

1

2Rw2Ac

Info: [A20].

3.1.64 Momentum Number NM

NM 5MNδRηΔw

MN (m2 s22) � specific energy of momentum flux; δ (m) � layer thickness; R (kg m23)

� density; η (Pa s)� dynamic viscosity;Δw (m s21)� velocity difference.

It expresses the ratio of the specific energy of a momentum flux to the momen-

tum change of viscous fluid.

Info: [B20].

3.1.65 Newton Number Ne

Ne5F

Rw2L25 0; 5CF

F (N) � hydrodynamic force; R (kg m23) � fluid density; w (m s21) � flow veloc-

ity; L (m) � characteristic length; CF (�) � force coefficient (p. 62).

75Fluid Mechanics

This number expresses the loading-to-inertia forces ratio. It characterizes the

friction in flowing fluids. In the case of viscous fluid flow, it expresses the turbuli-

zation process in flowing fluid. In a pressure flow, it is Ne5Eu; in a gravitation

flow, it is Ne5Fr2 1, and when considering the internal friction the Ne5Re2 1 is

valid. It also expresses dynamic force processes in vibration, mixing, material

deformation and other processes. The Fanning friction number fF (p. 163) and

the force coefficient CF (p. 62) are also analogues to the Newton number.

Hydrodynamics.

Info: [A4],[A29],[A35],[B17].

Isaac Newton (4.1.1643�31.3.1727), English physicist,

mathematician, astronomer and philosopher.

He laid the foundations of differential and integral calcu-

lus. His works on optics and gravitation made him one of

the greatest scientists of the world. In his book, Principia

(Principles), he formulated three movement laws: of inertia,

of action and reaction, and of acceleration proportional to

force. He was ingenious not only in mathematics but also in

experimentation, which enabled him to formulate new and

simpler laws of mechanics and hydromechanics.

3.1.66 Number of Velocity Heights Nvh

Nvh 5F

RL2

� �w2

2

� �21

F (N) � force; R (kg m23) � density; L (m) � characteristic length; w (m s21) �fluid velocity.

It expresses the ratio of the utilized pressure height to the velocity height.

Friction in pipelines and other water systems.

Info: [A35].

3.1.67 Nusselt Film Thickness Parameter Nuft

Nuft 5 L

ffiffiffiffiffig

v23

r� Ga

13

L (m) � characteristic length (layer thickness); g (m s2) � gravitational accelera-

tion; ν (m2 s21) � kinematic viscosity; Ga (�) � Galilei number (p. 123).

This parameter represents a special case of the Galilei number Ga (p. 123) for

thin film creation.

Info: [B20].

Ernst Kraft Wilhelm Nusselt (p. 196).


3.1.68 Oberbeck Number Ob

Ob5L3w2RQ

L (m) � characteristic length; w (m s21) � flow velocity; R (kg m23) � fluid den-

sity; Q (J) � heat.

It expresses the ratio of flowing fluid kinetic energy to heat transfer.

Info: [A6].

3.1.69 Outflow Coefficient μ

μ5αϕ; whereas μ, 1

α (�) � contraction coefficient (p. 57); ϕ (�) � coefficient of velocity (p. 57).

Using this, the velocities and contractions are considered in fluid outflow or pas-

sage. Aero-hydrodynamics.

3.1.70 Pipe Friction Coefficient Λ, Cf, cf

Λ5ez

ek

D

L

ez (J kg21) � friction loss energy; ek (J kg

21) � specific kinetic energy; D (m) �pipe diameter; L (m) � pipe length.

It expresses the friction loss in fluid flow in pipelines. Hydraulics.

Hydrodynamics.

3.1.71 Pipeline Parameter Np, .

Np 5wpwini

2ghs

wp (m s21) � pressure wave velocity; wini (m s21) � initial fluid velocity; g (m s22) �gravitational acceleration; hs (m) � static head.

It expresses the ratio of the highest pressure in a water stroke to the static pres-

sure. It characterizes the pressure originating in a pipeline during the water stroke.

Hydraulics.

Info: [B11],[B20].

77Fluid Mechanics

3.1.72 Pohlhausen Number Ph

Ph5δ2

vRwN

dp

dL

δ (m) � boundary layer thickness; ν (m2 s21) � kinematic viscosity; R (kg m23) �density; wN (m s21) � rise velocity; p (Pa) � pressure; L (m)� characteristic length.

In tasks on a boundary layer, it expresses the influence of the outer flow proper-

ties on the velocity distribution in a certain boundary layer cross section and on the

velocity profile change in the flow direction. It is the measure of the pressure-

to-viscous forces ratio in flowing fluid.

Info: [A19].

E. Pohlhausen.

3.1.73 Poiseuille Number (1.) Ps

Ps5D2Δp

ηwL

D (m) � pipe diameter; Δp (Pa m2 1) � loss of pressure along the pipe; η (Pa s) �dynamic viscosity; w (m s21) � flow velocity; L (m) � characteristic length.

It expresses the pressure-to-friction forces ratio in fluid laminar flow in a pipeline.

Info: [B20].

Jean Louis Marie Poiseuille (22.4.1797�26.12.1869),

French doctor and physiologist.

He was educated in physics and mathematics. He was

engaged in blood flowing in narrow tubes. In 1840 and 1846,

he formulated and published the Poiseuille law, which he

had deduced by experiment in 1838. This describes the

stationary flowing of incompressible viscous fluid through

cylindrical tubes. He applied the results to blood flowing

through capillaries, arteries, pulmonary alveolas and the like.

3.1.74 Prandtl Dimensionless Distance X

X5xffiffiffiffiffiffiffiffiRτw

pη

x (m) � distance from wall surface.; R (kg m23) � fluid density; τw (Pa) � surface

tension on the wall; η (Pa s) � dynamic viscosity.

It is used to study turbulent flow.

Info: [A29].

Ludwig Prandtl (p. 197).


3.1.75 Prandtl�Glauert rule Cp, CL, CD

Cp 5Cp0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12M2

N

p ð1Þ; CL 5CL0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12M2

N

p ð2Þ;

Cp 52ϑffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

M2N 2 1

p ð3Þ; CL 54αffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

M2N 2 1

p ð4Þ;

CD 54α2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2

N 2 1p ð5Þ;

where MN 5wN

að6Þ

Cp0 (�) � uncorrected pressure coefficient for incompressible fluid, see pressure

coefficient Cp (p. 80); CL0 (�) � uncorrected buoyancy coefficient for incompress-

ible fluid, see lift coefficient CL (p. 72); α (�) � incidence angle of thin aerofoil

section; ϑ (�) � fluid incidence angle; wN (m s21) � velocity of free-stream con-

ditions; a (m s21) � local sound velocity; MN (�) � Mach number of free-stream

conditions, see Mach number M (p. 73).

The dimensionless quantities Cp, CL and CD are the pressure, lift and drag coef-

ficients, respectively, corrected for a compressible fluid. See the pressure coeffi-

cient Cp (p. 80), the lift coefficient CL (p. 72) and the drag coefficient CD (p. 60).

The expressions (1) and (2) express the corrected relations for the fluid compress-

ibility in high subsonic flow which is characterized by the extent (0.3,MN, 0.8),

where MN is the Mach number of free-stream conditions, defined by the relation

(6). The expressions (3)�(5) are valid for supersonic flow.

Info: [A21].

Ludwig Prandtl (see above).

3.1.76 Prandtl Relation M2

M1;critM2;crit 5 1 ð1Þ;

M1;crit 5w1

acritð2Þ; M2;crit 5

w2

acritð3Þ

w1, w2 (m s21) � flow velocity before and after shock; acrit (m s21) � critical sound

velocity; M1,crit (�) � Mach number before shock considering the critical value

instead of local sound velocity, see Mach number M (p. 73); M2,crit (�) � Mach

number after shock considering the critical value instead of local sound velocity,

see Mach number M (p. 73).

It expresses the case of a fluid hit by a permanent shock wave. It simplifies both

Mach numbers (2) and (3) determination significantly because the sound velocity is

79Fluid Mechanics

equal on both sides of the shock, compared to the local velocity, and the velocities

can be measured very easily. In comparison to it, the pressure, temperature, density,

enthalpy and entropy increase. The expressions (1)�(3) are valid for a perfect gas,

and for supersonic and subsonic conditions. Fluid mechanics. High velocity aerody-

namics. Shocks.

Info: [A21].


3.1.77 Prandtl Velocity Ratio Pr, u1

Pr5w

ffiffiffiRτ

r

w (m s21) � flow velocity; R (kg m23) � density; τ (Pa) � shear stress.

This is the ratio of the inertia force to the shear force arising on a wall by the

action of a flowing fluid. It characterizes the degree of fluid flow turbulence.

Hydromechanics.

Info: [A29],[B20].


3.1.78 Pressure Coefficient Cp

Cp 5Δp

12RNw2

N

5p2 pN12RNw2

N

Δp (Pa) � pressure difference in fluid; p (Pa) � static pressure; pN (Pa), RN(kg m23), wN (m s21) � pressure, density and velocity of free flow.

It expresses the ratio of the overpressure in a fluid to the dynamic pressure in a

free flow. It is a special case of the Newton number Ne (p. 75).

Info: [A7],[A21],[A29].

3.1.79 Pressure Number (1.) Np

Np 5pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

gηðRl 2 RgÞp

p (Pa) � total pressure in system; g (m s22) � gravitational acceleration; η (Pa s) �dynamic viscosity; Rl, Rg (kg m23) � liquid and gas density.

It expresses the ratio of the total system pressure to the pressure gradient on a

fluid surface.

Info: [A29].


3.1.80 Pressure Recovery Coefficient Cp,rec

Cp;rec 5pout 2 pin

1

2R2in

pout (Pa) � output pressure; pin (Pa) � inlet flow pressure; R (kg m23) � liquid

density; win (m s21) � inlet flow velocity.

It expresses the ratio of the difference of static inlet and outlet pressures to the

inlet dynamic pressure. It characterizes the pressure rise in flow which is typical

for diffusers. Fluid mechanics. Aeromechanics.

Info: [A21].

3.1.81 Reech Number Ree

Ree5gL

w25Fr21

g (m s22) � gravitational acceleration; L (m) � characteristic length; w (m s21) �velocity; Fr (�) � Froude number (1.) (p. 62).

This number expresses the weight-to-inertia forces ratio. It can be neglected for

gases. It represents the movement of solid bodies in fluid, waves and surface phe-

nomena. Gravitation influence on the movement of surface ships.

Info: [A29],[A35].

Ferdinand Reech (1805�1880), French naval instructor.

3.1.82 Relative Amplitude of Fluid Displacement NA

NA 52wmax

ωL5

D

2L

Remax

Va

wmax (m s21) � maximal velocity (amplitude) at fluid flow oscillation; ω (s21) �angular frequency; L (m) � characteristic pipe length; D (m) � pipe diameter; Remax

(�) � Reynolds maximum number (p. 83); Va (�) � Valensi number (p. 90).

It characterizes the pulsation flow in a pipeline. With NA. 1, all initial fluid

passes outwards from a pipeline during one cycle. With NA5 1, all initial fluid is

moving in a pipeline. With NA, 1, a part of the initial fluid volume contained in a

pipeline does not leave the pipeline during one cycle.

Info: [A1],[A19].

3.1.83 Reynolds Number Re

Re5wLRη

5wL

vð1Þ; Rerot 5

ωL2

v5

RnL2

ηð2Þ

81Fluid Mechanics

w (m s21) � flow velocity; L (m) � characteristic length; R (kg m23) � density; η(Pa s) � dynamic viscosity; ν (m2 s21) � kinematic viscosity; ω (Hz) � angular

frequency; n (s21) � rotational frequency.

This number expresses the ratio of the fluid inertia force to that of molecular

friction (viscosity). It characterizes the hydrodynamic conditions for viscous fluid

flow. It determines the character of the flow (laminar, turbulent and transient flows).

For a laminar flow Re, 2000 is valid, for a transient flow 2000,Re, 4000, and

for a turbulent flow it is Re. 4000. With low values of the Re number, the viscous

friction muffles the originating dynamic influence of the flow relatively quickly and

intensively, due to which the streamlines and elementary fluid volumes cannot be

deformed substantially and the flow remains laminar. With large Re numbers, the

dynamic flow effect cannot be equalized by viscous friction and the flow stability is

lost, which is manifested by swirls and turbulence in the fluid. The expression (2) is

valid for flow in rotating canals and is often called the Reynolds rotary number.

Info: [A21],[A23],[A29],[A35],[B20].

Osborne Reynolds (23.8.1842�21.2.1912), English engineer

and physicist.

He opened the era of convective heat transfer (1838). He

showed that the flow of fluid through a tube depends on the

relation of the inertial force to the viscosity (Re number);

so he expressed the basic criterion for forced viscous fluid

convection and determined its critical value. In addition,

Reynolds’s studies relate to vapour condensation and fluid

boiling, heat transfer between solid materials and fluid,

lubrication theory (1886) and resistance laws in streaming

(1883).

3.1.84 Reynolds Boundary Layer Number Reδ

Reδ 5wδv

w (m s21) � velocity; δ (m) � layer thickness; ν (m2 s21) � kinematic viscosity.

It characterizes the viscous fluid flow in the boundary layer near a bypassed

body surface.

Osborne Reynolds (see above).

3.1.85 Reynolds Curved Number Re

Re5wfL

2v; where wf 5

ffiffiffiffiτRl

r


wf (m s21) � friction velocity; L (m) � characteristic length, curvature radius of layer;

ν (m2 s21) � kinematics viscosity; τ (Pa) � shear stress; Rl (kg m23) � fluid density.

It expresses the friction influence on the heat transfer in a curved fluid layer.


3.1.86 Reynolds Entry Number ReE

ReE 5L

DRe

L (m) � input length; D (m) � diameter; Re (�) � Reynolds number (p. 81).

It is applied to solve the flow in inlet flow-through parts.

Info: [B34].


3.1.87 Reynolds Maximum Number Remax

Remax 5wmaxL

v5

4Va

Sh

wmax (m s21) � maximal velocity amplitude at oscillation flow; L (m) � character-

istic length, pipe diameter; ν (m2 s21) � kinematic viscosity; Va (�) � Valensi

number (p. 90); Sh (�) � Strouhal number (p. 87).

It characterizes a pulsating flow. Fluid mechanics.

Info: [A1].


3.1.88 Reynolds Turbulence Number Returb

Returb 5e2

vε

e (J kg21) � turbulent specific kinetic energy; ν (m2 s21) � kinematic viscosity;

ε (s21) � dissipative velocity of turbulence.

For a turbulent flow, it expresses the kinetic-to-friction energy ratio. Fluid

mechanics.

Info: [A1].


3.1.89 Richardson Number Ri

Ri5gh

w25

1

Frð1Þ; Ri5

gredh

w2ð2Þ

83Fluid Mechanics

g (m s22) � gravitational acceleration; h (m) � characteristic length, vertical;

w (m s21) � characteristic flow velocity; gred (m s22) � reduced gravitational

acceleration; Fr (�) � Froude number (1.) (p. 62).

This number expresses the potential-to-kinetic energies ratio. More often, its

inverse value is used: the Froude number (1.) Fr (p. 62). With small flow density

changes (for example in the ocean or in the atmosphere), the Richardson number is

used in expression (2).

Info: [A21].

Lewis Fry Richardson (p. 403).

3.1.90 Roshko Number Ro

Ro5 Sh Re5fL2

v

f (s21) � whirl frequency; L (m) � characteristic length, for example, hydraulic

radius; ν (m2 s21) � kinematic viscosity; Sh (�) � Strouhal number (p. 87);


It describes the mechanism of a swirl oscillating flow. Compared to the Strouhal

number Sh (p. 87), it describes the character and viscosity of the fluid flow. Its

validity has been verified by bypassing a cylinder in the range ReAh40; 10000i.Fluid mechanics. Turbulence.

Info: [A37],[C118].

Anatol Roshko (born 1911), American physical engineer of

Canadian origin.

He is an authority in aircraft and space engineering, espe-

cially in the spheres of turbulence, swirl flows, fluid mechan-

ics and gas dynamics. In turbulence research, he initiated

new trends primarily with his work on coherent structures. As

a co-author, he wrote the book Elements of Gasdynamics

(1956), which is well known all over the world. He worked

for 40 years as a professor at the California Institute of

Technology and gave remarkable lectures on fluid mechanics.

3.1.91 Russell Number Ru

Ru5w

fh; where f 5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

g

RdRdz

s

w (m s21) � wind velocity; f (s21) � natural vertical frequency of the fluid part;

h (m) � height of obstruction; g (m s22) � gravitational acceleration; R (kg m23) �density; z (m) � vertical coordinate.


This number expresses the inertia-to-uplift forces ratio. It concerns buoyancy,

gravitation effects, two-phase flow, and shock and surface waves in hydromechan-

ics and waves in stratified flow.

Info: [B20].

3.1.92 Sarrau Number Sa

See Mach number M (p. 73).

Jacques Rose Ferdinand Emile Sarrau, French engineer.

3.1.93 Shear Stress Number NS

NS 5RwIF

R (kg m23) � fluid density; w (m s21) � flow velocity; IF (kg m22 s21) � force

impulse.

It characterizes the shear stress in fluid flow.

Info: [A49].

3.1.94 Schiller Number (1.) Sch

Sch5wL

ffiffiffiffiffiffiffiffiR2

2ηF3

s5

ffiffiffiffiffiffiRe

CD

3

r

w (m s21) � flow velocity; L (m) � characteristic length; R (kg m23) � gas or

liquid density; η (Pa s) � dynamic viscosity; F (N) � resistance force; Re (�) �Reynolds number (p. 81); CD (�) � drag coefficient (p. 60).

This number characterizes the flow around obstacles and expresses the ratio of

velocity to the force acting on a submerged body. It also characterizes the inertia

force of certain sized particles in relation to the resistance coefficient. It also

expresses the ratio of forces acting on a diving body.

Info: [A25],[A29],[A35].

L. Schiller.

3.1.95 Slosh Time NS

NS 5 τffiffiffiffiffiffiffiσgr3

3

r

τ (s) � time; σ (N m21) � surface tension; g (m s22) � gravitational acceleration;

r (m) � pipe radius.

85Fluid Mechanics

It expresses the dimensionless time of fluid strokes in pipelines and in surface

oscillation. Hydraulics.

Info: [B 20].

3.1.96 Smoluchowski Number Sm

It expresses the low pressure gas flow, see Knudsen number (1.) Kn (p. 69).

Info: [A29],[A35].

Marian Smoluchowski (28.5.1872�5.9.1917), Polish physicist.

He was engaged in the phenomena connected with the

thermal movement of molecules and atoms. Independently of

Einstein, he formulated the expression explaining the Brown

movement. He worked on the statistical expression of the

second law of thermodynamics. The whole range of his work

was an important contribution to science. It was not only

about physics, but also about physical chemistry, geophysics

and geology. He was the first to calculate the zeta potential

of electrokinetic movement.

3.1.97 Speed Number Pw, W

Pw 5w

wmax

w, wmax (m s21) � local and maximal velocity.

It expresses the local-to-maximum velocities ratio. Aero-hydrodynamics.

Info: [A24].

3.1.98 Stokes Number (6.) Stk

Stk5 1:042mFgR 12RRF

� �R3

η2

mF (kg) � float mass; g (m s22) � gravitational acceleration; R (kg m23) � fluid

density; RF (kg m23) � float density; R (�) � pipe and float radii ratio; η (Pa s) �dynamic viscosity.

It is used to calibrate the rotameters used in measuring fluid passage. Measuring

techniques.

Info: [A35].

George Gabriel Stokes (p. 131).

3.1.99 Stratification Parameter S

S5ω

2ωC


ω (s21) � angular frequency; ωC (s21) � Coriolis frequency.

It characterizes the fluid flow with rotation and action of the Coriolis force. For

S{1, stratified flow occurs (laminar flow, etc.). For Sc1, the rotation influence

prevails. Fluid mechanics. Geophysics.

Info: [C124].

3.1.100 Strouhal Number Sh

Sh � Th5wτL

ð1Þ; Sh5w

nLð2Þ;

Sh5fL

wð3Þ; Sh5

w

fLð4Þ

w (m s21) � velocity; τ (s) � time; L (m) � characteristic length; n (s21) � rota-

tional frequency; f (Hz) � frequency; Th (�) � Thomson number (p. 90).

This number describes the oscillating flow mechanism and expresses the vibration-

to-flow velocities ratio; alternatively, it relates the ratio of local force, caused by a

non-stationary process, to the inertia force. It represents the universal dynamic simi-

larity criterion of non-stationary processes in systems which are similar in terms of

geometry and kinematics. It determines the non-stationary to convective momentum

transfer in a system. It is the measure for movement non-stationarity. With Sh$ 1, the

local accelerations are commensurate with the convective ones and the movement is

non-stationary. On the contrary, in stationary movement, Sh 5_ 0 is valid.

It is known as the concept of reduced frequency. Sometimes, it is called

the Thomson number Th (p. 90), or the acceleration (accelerating) ratio in expres-

sion (2).

Info: [A29],[A43],[A95],[B20].

Vincenc Strouhal (10.4.1850�23.1.1922), Czech experi-

mental physicist.

He was engaged in hydrodynamic phenomena, acoustics

and electric and magnetic properties of steel. In the sphere of

gas and fluid dynamics, his experimental work led to the

determination of the dimensionless parameter which is called

the Strouhal number. He wrote several publications on exper-

imental physics: Mechanika (Mechanics, 1901), Akustika

(Acoustics, 1902), Thermika (Thermics, 1908) and Optika

(Optics, 1919).

3.1.101 Strouhal Turbulence Number Shturb

Shturb 5jðwrÞwj

@w

@τ

� wτcorLturb

~τcorτturb

87Fluid Mechanics

w (m s21) � flow velocity; τ (s) � running time; τcor (s) � correlation time; τturb(s) � fluctuation time; Lturb (m) � characteristic length of turbulence.

In the dimensionless form, it expresses the ratio of the correlation time to the

fluctuation time of turbulent flow. Fluid mechanics. Astrophysics.

Info: [B58].

Vincenc Strouhal (see above).

3.1.102 Swirl Number Sw

Sw5w

u

w (m s21) � tangential velocity component; u (m s21) � axial velocity component.

It expresses the swirl intensity in flow. It is given by the tangential-to-axial

momentum ratio. The tangential velocity is the mean tangential velocity component

in the outlet plane of a passage canal. The axial velocity is the longitudinal move-

ment velocity in the canal plane.

Info: [C127].

3.1.103 Szebehely Number Sz, Σ

Sz5j@w=@τjjðwUrÞwj

w (m s21) � flow velocity; τ (s) � time.

The Szebehely number characterizes the unsteadiness measure of a fluid flow. It

expresses the ratio of the fluid flow local acceleration to the convective accelera-

tion. Conforming to this criterion, the flow can be divided into the following

categories:

� steady flow with acceleration (Sz5 0),

� unsteady flow without acceleration (Sz5 1),

� steady flow without acceleration (Sz not determined),

� unsteady flow with acceleration (0, Sz#N, Sz 6¼ 1). Sz5N corresponds to unsteady

flow with zero convective acceleration.

The Szebehely number follows from two definitions for the velocity field:

� a steady flow is that for which @w=@τ5 0 is valid and

� a flow without acceleration is that for which Dw=Dt5 ð@w=@tÞ1ðwUrÞw5 0:

Therefore, an arbitrary velocity field can be assigned to one of the four catego-

ries together with the corresponding Szebehely number value.

Info: [B5].


Victor G. Szebehely (1921�13.9.1997), American space

engineer of Hungarian origin.

He was among the key persons in the US Apollo space

program, which resulted in the landing of a man on the

moon. He was engaged in applied mathematics, especially in

the dynamics of seaborne and space ships. In 1956, he used

the dimensionless number to express time-dependent nonsta-

tionary flowing, which is named the Szebehely number.

Similarly, the Szebehely equation, which determines the

gravitational potential of the earth, planets, satellites and

galaxies, is named after him.

3.1.104 Taylor Number Ta

Ta52wL2

v

� �2ð1Þ; Ta5

f 2L4

v25Re2rot ð2Þ;

Ta5ωr

121ðr2 2 r1Þ

32

vð3Þ; Ta5Re

12d

23v21 ð4Þ;

where Tacrit 5 41:3

ω (Hz) � angular frequency; L (m) � characteristic length (thickness of flow gap

or liquid); ν (m2 s21) � kinematic viscosity; f (s21) � frequency; r2, r1 (m) � outer

and inner gap radii; d (m) � gap width between the cylinders (d5 r22 r1); Rerot(�) � Reynolds rotation number (p. 383).

This number expresses the centrifugal and viscous forces; alternatively, the sec-

ond power of the Coriolis force and the viscous force. It characterizes the rotation

influence on free convection. It is used in expressions (1) or (2). In expression (3),

it expresses a flow with viscous fluid swirl instability originating in a ring canal

between two concentric circular cylinders of which the inner one rotates and the

outer one is stable. In this case, with Ta, 41.3 the flow is laminar, for

41.3# Ta, 400 the flow is laminar with steady origination of Taylor swirls, and

for Ta. 400 the flow is turbulent.

Info: [A21],[A35],[B11],[B20].

Geoffrey Ingram Taylor (7.3.1886�27.6.1975), English

mathematician, physicist, aerodynamicist and meteorologist.

Having started with the theoretical study of shock waves, he

continued with the work of J.J. Thompson. Taylor was engaged

in his proposals and verified quantum theory experimentally.

His other spheres of interest included dynamic meteorology �including his work on atmospheric turbulence � and

oceanography.

89Fluid Mechanics

3.1.105 Thien Parameter NT

NT 5Maϑ

ϑ (�) � flow angle; Ma (�) � local Mach number (p. 73).

It characterizes the hypersonic flow for cases of sin ϑ-0 (NT{1). In these cases,

it enables linearization of complicated non-linear equations of the supersonic flow.

Info: [A21].

3.1.106 Thoma Number (1.) Th

See Cavitation number Ncav (p. 366).

Dieter Thoma (1881�1942), German hydraulic engineer.

3.1.107 Thomson Number Th

Th � Sh5wτL

w (m s21) � flow velocity; τ (s) � time; L (m) � characteristic length; Sh (�) �Strouhal number (p. 87).

It describes oscillation flow. It is identical with the Strouhal number Sh (p. 87).

Info: [A29],[A35].

James Thomson (1882�1992), Irish engineer.

3.1.108 Transiency Group Nt

Nt 51

@p

@L

@@p

@L

� �@ðFoÞ ð1Þ; Nt 5

1

Re

@ðReÞ@ðFoÞ ð2Þ

@p=@L (Pa m21) � pressure gradient in the flow direction; Fo (�) � Fourier

number (p. 175); Re (�) � Reynolds number (p. 81).

It expresses transient changes of flowing fluid behaviour.

Info: [A35].

3.1.109 Turbulence Number Tu

See Karman turbulent number Ka, Tu (p. 67).

3.1.110 Valensi Number Va

Va5r2ωv


r (m) � inner pipe radius; ω (s21) � angular oscillation frequency; ν (m2 s21) �kinematic viscosity.

This number expresses the ratio of the inner tube radius to the viscous diffuse

length in fluid flowing through a tube with a periodic boundary condition. Fluid

dynamics.

Info: [A1],[A29],[A35].

Jacques Valensi (born 1903), French engineer.

3.1.111 Vedernikov Number Ve

Ve5ζξw

wA 2w

ζ (�) � exponent of hydraulic radius; ξ (�) � form factor of cross section;

w (m s21) � mean flow velocity; wA (m s21) � absolute velocity of wave

disturbance.

It describes the flow instability in an open canal. Hydraulics.

Info: [A33],[A35].

3.1.112 Wave Period NT

NT 5Twαh

T (s) � period of water wave; w (m s21) � mean velocity of steady flow; α (�) �canal slope; h (m) � steady water depth in canal.

It is applied to determine the opening time of irrigation canal gates. Hydraulics.

Info: [A47].

3.1.113 Weber Number (1.) and (2.) We

We1 5Rw2L

σð1Þ; We2 5 ðWe1Þ

12 5w

RLσ

� �12

ð2Þ

R (kg m23) � fluid density; w (m s21) � movement velocity; L (m) � characteristic

length; σ (N m21) � surface tension.

These numbers express the ratio of the inertia force to the surface stress force.

The Weber number is connected with surface stress waves, as the Froude number

(2.) external Fr (p. 63) is with gravitational waves. The We number characterizes

the originating process of bubbles or drops during fluid boiling or steam condensa-

tion. Sometimes, it is used in expression of the second root (2) and is called the

Weber number (2.).

Info: [A7],[A21],[A43],[B20].

91Fluid Mechanics

Ernst Heinrich Weber (24.6.1795�26.1.1878), German

psychophysicist, anatomist, physiologist.

He laid the foundation for psychophysics, which involves

psychological reactions to physical stimuli. Together with his

younger brother, Wilhelm Eduard Weber, he was engaged in

research on fluid flow. The well-known Weber criterion for

viscous liquid flow is named after them. Together, they

wrote the book Wave Theory and Fluidity.

Wilhelm Eduard Weber (24.10.1804�23.6.1891), German

physicist.

Already at only 20 years of age, he was engaged in fluid

flow research, with his elder brother Ernst Heinrich Weber.

He also devoted himself to the study of magnetism and mag-

netic tension measurement. He formulated a logical system

of units for electricity and tried to unify electricity and mag-

netism into a unique basic law of forces. He described elec-

tric current as a flow of electrons. The magnetic flux unit

was named after him.

3.1.114 Weber Rotation Number Werot

Werot 5n2L3Rσ

n (s21) � rotation frequency; L (m) � characteristic length, diameter; R (kg m23) �fluid density; σ (N m21) � surface tension.

It is the inertia-to-capillary forces ratio. It characterizes the process of forced

fluid mixing. Hydromechanics. Blade machines.

Info: [A35].

Ernst Heinrich Weber (see above).

Wilhelm Eduard Weber (see above).

3.1.115 Womersley Number Wo

Wo5 L

ffiffiffiffiωv

r5

ffiffiffiffiffiffiffiffiffiffiffiffiRe Sh

pð1Þ; Wo5 L

ffiffiffiffiffiffiffiffiffi2πfnv

rð2Þ

L (m) � characteristic length, half-thickness; ω (s21) � angular frequency (ω5 2πf);ν (m2 s21) � kinematic viscosity; f (s21) � pulsation frequency; n (�) � degree of

harmonic frequency; Re (�) � Reynolds number (p. 81); Sh (�) � Strouhal number

(p. 87).


It characterizes the non-stationary oscillating flow which manifests itself as the

response to pressure gradient oscillations. With composed polyharmonic pulsating

flow, the expression (2) is valid. The Wo number shows whether the flow is quasi-

stationary or not. It is also important in biomechanics in the flow of biologically

significant fluids such as air, water and blood. With Wo, 1, a quasi-stationary

flow can be presumed, with an oscillating pressure gradient and a parabolic veloc-

ity profile with the greatest amplitude. With Wo. 1 the flow is phase shifted in

time as compared with pressure gradient oscillations, and the fluid oscillation

amplitude can either grow or drop. Fluid mechanics. Aeroelasticity. Biomechanics.

Info: [B69].

3.1.116 Zhukovsky Number Zh

See Fourier hydrodynamic number Foh (p. 62).

Nikolay Yegorovich Zhukovsky (p. 33).

3.2 Multiphase Fluid Mechanics

Usually, this concerns the similarity criteria for two-phase flow in which mutual

chemical interference does not occur. It is about various combinations of two

phases, such as, for example, water�steam, water�air and solid particles (dust) in

the air or in fluid. Flow with more than two phases occurs, for example, with explo-

sions. The criteria concern such things as the solid particle movement in fluid, the

dynamics of increasing transfer and collapse of bubbles in fluids, flow accompanied

with boiling or condensation and granulation flow. The Bagnold, Camp, Darcy,

gravitational, capillary, Kolmogorov, Leibenzonov, Leverette, Morton, Ohnesorge

and Stokes numbers are among the wide range of similarity criteria applied.

3.2.1 Accelerating Frequency Nf

Nf 5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΔRω2L3

σ

r

ΔR (kg m23) � density difference; ω (s21) � angular velocity; L (m) � characteris-

tic length; σ (N m21) � surface tension.

This frequency characterizes the fluid flow acceleration in microgravitation

conditions. Essentially, it is an analogue to the Weber rotation number Werot(p. 92), which compares inertia and capillary forces. Two-phase flow.

3.2.2 Bagnold Number (1.) Bg

Bg53CDRgw

2

4dgRs

93Fluid Mechanics

CD (�) � drag coefficient (p. 60); Rg, Rs (kg m23) � density of fluid and solid

particles; w (m s21) � flow velocity; d (m) � particle diameter; g (m s22) � gravi-

tational acceleration.

It is the ratio of the front resistance of bodies (particles) to their weight. It char-

acterizes the carrying along of the solid particles by a fluid. Hydromechanics.

Ecology. Soil erosion.

Info: [A26],[B20].

Ralph Alger Bagnold (1896�1990), English geophysicist,

soldier and traveller.

His observations of sand movement while travelling in the

African desert led to his interest in the physical substance

and movement of sand dunes. His efforts resulted in the

monograph The Physics of Blown Sand and Desert Dunes

(1941). Later he also devoted himself to solid material water

stream transport. He was noted not only for his study of sand

dunes on earth, but also those on Mars.

3.2.3 Bagnold Number (2.) Bg

Bg5VS

12VS

RSL2 _τ

ηð1Þ; Bg5

V13S

V13max 2V

13S

0B@

1CA

12

RsL2 _τ

ηð2Þ

VS (m3) � volume of granular particles; RS (kg m23) � particle density; L (m) �characteristic length, grain diameter; _τ (s21) � unit speed of shear deformation;

η (Pa s) � fluid dynamic viscosity in pores; Vmax (m3) � maximal possible volume

grain concentration.

It expresses the relation between the number of collisions and the viscous tension

in steady and uniform shear flow. With Bg, 40, the macroviscous mode occurs, in

which the tensions are proportional to the shear tension velocity. With Bg. 200,

the collision mode prevails and the tension value is proportional to the second

power of the shear tension velocity. The expression (2) is the improvement of (1).

Info: [A7],[B13].

Ralph Alger Bagnold (see above).

3.2.4 Blake Number Bl

Bl5wRL

ηð12 eÞ

w (m s21) � flow velocity; R (kg m23) � density; L5V S2 1 (m) � characteristic

length of particle; V (m3) � volume of particle; S (m2) � surface of particle;

η (Pa s) � dynamic viscosity; e (�) � porosity (p. 24).


This number expresses the inertia-to-viscous forces ratio in momentum and mass

transfers through loose material layers. It characterizes the flow in loose material

transport. Essentially, it is a modification of the Reynolds number Re (p. 81).

Info: [A35],[B20].

J.R. Blake.

3.2.5 Bond Number Bo

Bo5Eo5ΔRgL2

σ5We1Fr

21

ΔR5 Rl2 Rg (kg m23) � difference between liquid and gas or vapour density; g

(m s22) � gravitational acceleration; L5 dB (m) � diameter of water drop or bub-

ble; σ (N m21) � surface tension; Eo (�) � Eotvos number (p. 99); We1 (�) �Weber number (1.) (p. 91); Fr (�) � Froude number (1.) (p. 62).

It expresses the ratio of the gravity force to the surface stress force. It charac-

terizes the movement of free bubbles or little drops in a stationary fluid, further the

vapour strokes arising due to energy release in fast mixing of heated and cold evap-

orating fluids. It applies to the processes in nuclear reactors and in combustion

engines, for example. Generally, it characterizes the fluid spraying process in a

two-phase environment flow. It is often called the Eotvos number Eo (p. 99).

See also Goucher number Go (p. 65). Fluids atomization. Capillary flow. Stroke

and surface waves. Bodies immersed in fluid. Spraying of fluids.

Info: [A35],[B20].

Wilfrid Noel Bond (1897�1937), English physicist.

3.2.6 Boussinesq Number (2.) Bob

Bob 5ηsηrb

ηs (Pa m s) � surface shear viscosity; η (Pa s) � liquid dynamic viscosity; rb (m) �bubble diameter.

It expresses the characteristic radius of bubbles in flowing foamed fluid. Two-

phase flow.

Info: [B109].

Valentin Joseph Boussinesq (p. 55).

3.2.7 Brownell�Katz Number BK

BK5Ca Bo5ηwRgL2

σ2

95Fluid Mechanics

η (Pa s) � dynamic viscosity; w (m s21) � flow velocity; R (kg m23) � density;

g (m s22) � gravitational acceleration; L (m) � characteristic length, diameter;

σ (N m21) � surface tension; Ca (�) � capillary number (1.) (p. 97); Bo (�) �Bond number (p. 95).

This number characterizes the flow through the porous material. It expresses the

combination of the capillary number (1.) Ca (p. 97) and the Bond number Bo (p. 95);

alternatively, it is the ratio of viscous and gravitational forces product to the second

power of the surface stress force. Two-phase flow. Filtration. Draining. Drying.

Info: [C38].

3.2.8 Camp Number Ca

Ca5

ffiffiffiffiffiffiffiffiPV

ηQ2

s

P (W) � energy dissipation in fluid volume V ; V (m3) � fluid volume; η (Pa s) �dynamic viscosity; Q (m3 s21) � volume flow.

It expresses the ratio of the elapsed time to the average shear off value in a

fluid. It is the criterion for free clustering of suspended particles.

Info: [A29].

Thomas Ringgold Camp (born 1895), American engineer.

3.2.9 Capillary Buoyancy Number CaB

CaB 5gη4

Rσ35We31Fr

21Re24

g (m s22) � gravitational acceleration; η (Pa s) � dynamic viscosity; R (kg m23) �density; σ (N m21) � surface tension; We1 (�) � Weber number (1.) (p. 91);

Fr (�) � Froude number (1.) (p. 62); Re (�) � Reynolds number (p. 81).

It characterizes the influences of the surface tension, viscosity and acceleration

in two-phase flow when fluid balls move in another fluid. It depends on physical

properties only. See the Morton number (p. 107). In the case of solid particles in

flowing fluid, the modified number (property number) K is used, which provides

better correlation with the experiment. Capillary flow.

Info: [A29],[B20].


Ca5σffiffiffik

p

ηwL


σ (N m21) � surface tension; k (m2) � permeability; η (Pa s) � dynamic viscosity;

w (m s21) � velocity; L (m) � characteristic length dimension.

It is the capillary-to-filtration forces ratio. Capillary flow and porous materials.

Info: [B20].


Ca5ηwσ

5We1Re21 ð1Þ; Ca5

ηwpcr

2cos ϑ ð2Þ;

where pc 52σ cos ϑ

r

η (Pa s) � dynamic viscosity; w (m s21) � characteristic velocity; σ (N m21) �surface or half-surface tension between two fluid phases; pc (Pa) � capillary pres-

sure; r (m) � diameter of pore throat; ϑ (�) � wetting angle, for hard

wettable material is valid cos ϑ5 1; We1 (�) � Weber number (1.) (p. 91);


This number expresses the ratio of the viscosity force to the surface stress which

acts across the interface between a fluid surface and a gas or between two mixable

fluids. It characterizes the fluid spraying process in the flow of two-phase surround-

ings, in capillary tubes and porous material layers. With the pore structure and wet-

tability considered, expression (2) is valid. Capillary flow and porous materials.

Info: [A29],[B20].


Ca5L2Rgσ

L� d (m) � diameter (e.g. of pipe, particle, bubble, impeller or shaft); R (kg m23) �density; g (m s22) � gravitational acceleration; σ (N m21) � undisturbed surface

tension.

This number equals the ratio of the gravitational force to the surface tension

force. Two-phase medium flow.

Info: [B6].

3.2.13 Capillary Multiphase Number Ca

Ca5wDηpCR

; where pC 5 2σ cos ϑR21

wD (m s21) � characteristic velocity (Darcy velocity); η (Pa s) � dynamic viscos-

ity; pC (Pa) � capillary pressure; R (m) � mean pore radius; σ (N m21) � surface

tension; ϑ (�) � contact angle.

97Fluid Mechanics

It characterizes the capillary flow in porous material. For a fully wetted material,

it is cos ϑ5 1. With constant velocity, the capillary number is almost constant.

With the velocity changing in time, the number Ca becomes the dynamic parame-

ter. An example is petroleum driven out with gas.

Info: [A29].

3.2.14 Darcy Granulation Number (1.) Dc

Dc5η

VSRS _τk

η (Pa s) � fluid dynamic viscosity; VS (m3) � volume of grain particles;

RS (kg m23) � density of solid particles; _τ (s21) � unit speed of shear strain;

k (m21) � hydraulic permeability.

It expresses the ratio of tension caused by interaction to that due to particle inertia.

Info: [A29],[B13].

Henry Philibert Gaspard Darcy (10.6.1803�3.1.1858),

French engineer.

He executed research on flow and friction loss in a pipe-

line. This research resulted in the Darcy�Weisbach equation.

He improved the pitot tube structure and was the first to

express the boundary layer in flowing fluid. In the last years

of his life, he carried out many experiments leading to the

Darcy law for flow in sand. Since that time, this law has been

generalized and extended to unsaturated and multiphase flow.

3.2.15 Darcy Granulation Number (2.) Dc

Dc5wL

D0

w (m s21) � flow velocity; L (m) � characteristic length; D0 (m2 s21) � permeabil-

ity of granulated material.

It expresses the inertia-to-permeability forces ratio. It characterizes the flow in

porous material. It is an analogy of the Reynolds number Re (p. 81) and the Peclet

heat number Pe (p. 180).

Info: [A29].

Henry Philibert Gaspard Darcy (see above).

3.2.16 Driftage Elements Number ND

ND 5wvw

gL


wv (m s21) � free fall velocity of particle; w (m s21) � flow velocity; g (m s22) �gravitational acceleration; L (m) � characteristic length.

This number describes the drifting and depositing of fine particles in a fluid

flow. Creation of sediments in canals and pipelines. Pneumatic transport.15

3.2.17 Eotvos Number Eo

See the Bond number Bo (p. 95).

Info: [A29],[B7].

Lorand Baron von Eotvos (27.7.1848�8.4.1919),

Hungarian physicist.

Initially, he was engaged in capillary phenomena. Later

he devoted himself to gravitation. He used torsion balance to

prove that the ratio of the gravitational and inertia mass is

constant with the accuracy of 53 1029, which is fundamen-

tally important for general relativity theory because it con-

firms that both masses are equivalent.

3.2.18 Expansion Bubbling Number Ex

Ex5gd

w2

Rl 2 RvRl

5Fr21PR

g (m s22) � gravitational acceleration; d (m) � diameter of gas or vapour bubble;

w (m s21) � bubble velocity; Rl, Rv (kg m23) � liquid and gas density; Fr (�) �Froude number (1.) (p. 62); PR (�) � relative change of density, equivalent of

Boussinesq approximation number Bs (p. 56).

It expresses the lift-to-inertia forces ratio. It characterizes the process of gas or

vapour bubble propagation in a fluid.

Info: [A29].

3.2.19 Filtration Energy Gradient J

J52@H

@x5

w2

gLf ðReÞ

H (m) � energetic height; x (m) � axis x; w (m s21) � filtration velocity; g (m s22) �gravitational acceleration; L (m) � characteristic length of grains; Re (�) � Reynolds

filtration number, see Reynolds number (p. 81).

It expresses the energy gradient in the x-axis direction. Two-phase flow.

Hydraulics. Filtration.

Info: [A48].

99Fluid Mechanics

3.2.20 Fluidization Number Nfluid

Nfluid 5ζVsðRs 2 RfÞ

ηVf

ffiffiffig

L

r

ζ (m2) � permeability of porous material; Vs, Vf (m3) � volume of solid particles

and fluid; Rs, Rf (kg m23) � density of solid particles and fluid; η (Pa s) � dynamic

viscosity; g (m s22) � gravitational acceleration; L (m) � characteristic length, par-

ticle diameter.

It expresses the fluidization measure in relation to the velocity. For Nfluid{1, it

is used for the majority of flows. Two-phase fluid mechanics.

Info: [B54].

3.2.21 Frequency Number Nf

Nf 5ωLw

5 2πSh

ω (s21) � angular frequency; L (m) � characteristic length, thickness; w (m s21) �velocity of fluid infiltration; Sh (�) � Strouhal number (p. 87).

It characterizes the flow in condensed or suspension layers.

Info: [A29].

3.2.22 Friction Number Nf

Nf 5Vs

12Vs

nðRs 2 RfÞgL tan ϕ_τη

Vs (m3) � volume of solid particles; n (�) � grain number on the grain surface; Rs,

Rf (kg m23) � density of solid part and fluid; g (m s22) � gravitational accelera-

tion; L (m) � characteristic length (diameter); ϕ (�) � angle of inner friction; _τ(s21) � unit velocity of shear deformation; η (Pa s) � dynamic viscosity.

It expresses the ratio of friction tensions, originating with steady contact of

grains, to viscous shear tensions. It resembles the Bingham number Bm (p. 118),

but the characteristic tensions originate for solid and liquid parts differently.

Dominating viscous forces arise with Nf. 100.

Info: [B13].

3.2.23 Froude Multiphase Number Frmp

Frmp 5wgffiffiffiffiffiffigL

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiRg

Rl 2 Rg

s


wg (m s21) � velocity of moist gas flow; g (m s22) � gravitational acceleration;

L (m) � characteristic length; Rl, Rg (kg m23) � density of fluid and gas.

It expresses the inertia-to-gravitation forces ratio in two-phase fluid flow.

Multiphase fluid mechanics. Wet gas flow.

Info: [D3].

3.2.24 Gibbs Number Gb

Gb5E

kT

E (J) � local energy of perturbation; k (J K21) � Boltzmann constant; T (K) �absolute temperature.

This number expresses the kinetic theory of molecules. For example, it charac-

terizes cavitation processes and bubble dynamics in fluids. It is about the final

period in the homogeneous nucleon theory, in which the deposition energy E acts

and the bubbles originate. Two-phase flow. Boiling. Cavitation.

Info: [A7].

Josiah Willard Gibbs (11.2.1839�28.4.1903), American

mathematician, theoretical physicist and chemist.

He became an authority in theoretical thermodynamics. He

was a Professor in the Department of Mathematical Physics at

Yale University. He deduced the Gibbs phase rule. Free

enthalpy is denoted as the Gibbs thermodynamic potential.

Gibbs distribution, which is a basic law of statistical physics,

is connected with his name as well. His most important book

is Elementary Principles in Statistical Mechanics (1902).

3.2.25 Granulometric Number Gm

Gm5dmax

dekv

� �0:6

; where dekv 5Xni5 1

ðgid21i Þ

" #21

dmax (m)� maximal particle diameter; dekv (m)� equivalent particle diameter of layer

that polydisperses; gi (�), di (m) � weighted part and diameter of i-kind particles.

It expresses the fluid flow in dispersed layers.

Info: [A33].

3.2.26 Gravitational Number (1.) Ng

Ng 5kgΔRηw

5kgðRp 2 RfÞ

ηw

101Fluid Mechanics

k (m2) � permeability of environment; g (m s22) � gravitational acceleration;

ΔR (kg m23) � density difference between particles and fluid; Rp, Rf (kg m23) �density of particles and fluid; η (Pa s) � dynamic viscosity; w (m s21) � flow

velocity.

It expresses the gravitational-to-filtration forces ratio. Two-phase flow. Filtration.

Info: [B20],[B115].


Ng 54πr4pðRp 2 RfÞg

3kT

rp (m) � particle radius; Rp, Rf (kg m23) � density of particle and fluid; g (m s22) �

gravitational acceleration; k (J K21) � Boltzmann constant; T (K) � fluid absolute

temperature.

It expresses the ratio of the gravitational potential of a particle to its thermal

energy. Physically chemical filtration in saturated porous materials.

Info: [B115].


Ng 5ηw

gΔRL25

Ca

Bo

η (Pa s) � fluid dynamic viscosity; w (m s21) � flow velocity; g (m s22) � gravita-

tional acceleration; ΔR (kg m23) � density difference between two non-miscible

liquids; L (m) � characteristic length, thickness, diameter; Ca (�) � capillary

number (1.) (p. 97); Bo (�) � Bond number (p. 95).

It expresses gravitational force as the viscous-to-capillary forces ratio. It acts in

the penetration of two non-miscible fluids through a porous material. Two-phase

porous flow. Filtration.

3.2.29 Gravity Number NG

NG 5kgΔRηw

ð1Þ; NG 52

9

r2gðRp 2 RfÞηw

ð2Þ

k (m2) � permeability; g (m s22) � gravitational acceleration; ΔR (kg m23) � den-

sity difference of phases; η (Pa s) � dynamic viscosity; w (m s21) � flow velocity;

r (m) � particle radius; g (m s22) � gravitational acceleration; Rp, Rf (kg m23) �

density of particle and fluid.


This number denotes the gravity-to-filtration forces ratio or, alternatively, the

ratio of the Stokes rate of particle sedimentation to the fluid rate. It is expressed in

expression (2) as well. Flow through porous material. Two-phase flow. Filtration.

Info: [B20].

3.2.30 Haberman�Morton Number Hab, Hm

Hab5gη4lRlσ3

12mp

RlVp

� �5

We31Fr2Re4

g (m s22) � gravitational acceleration; ηl (Pa s) � fluid dynamic viscosity;

Rl (kg m23) � fluid density; σ (N m21) � surface tension; mp (kg) � particle mass;

Vp (m3) � particle volume; We1 (�) � Weber number (1.) (p. 91); Fr (�) �Froude number (1.) (p. 62); Re (�) � Reynolds number (p. 81).

This number expresses the deformation of moving particles, as drops, bubbles or

fine deforming (granular) materials. In the case of bubbles mp{Rl, mp=ðRlVpÞ-0

holds. Then the bubble dimension influence on the number Hab can be neglected.

For filtered water Hmi5 0:253 10210; for mineral oil Hmi5 1:453 1022 and for

syrup Hmi5 0:923 106:Info: [A2],[A7],[A8].

3.2.31 Inertia Parameter Ninert

Ninert 5d2RPw18ηL

d (m) � diameter (such as of pipe, particle, bubble or drop); RP (kg m23) � particle

density; w (m s21) � fluid velocity; η (Pa s) � dynamic viscosity; L (m) � charac-

teristic length (such as of channel width, distance from wall or height of liquid

layer).

It expresses the inertia to friction forces in particle flow in a fluid.

Info: [C72].

3.2.32 Keulegan�Carpenter Number (2.) KC

KC5wAτp2L

wA (m s21) � speed of body oscillating; τp (s) � time period; L (m) � characteristic

length, diameter.

It characterizes non-steady oscillation phenomena joined with the particle move-

ment in fluid. For KC, 5, the inertial phenomena prevail. Two-phase flow.

Info: [A4],[A7].

103Fluid Mechanics

3.2.33 Knudsen Number (3.) Diffusion Kn

Kn53

4

hDAB

ζKOAwA

h (m) � height of roughness; DAB (m2 s21) � binary bulk diffusion coefficient for

system AB; ζ (�) � diffusion tortuosity; KOA (m) � Knudsen coefficient of perme-

ability; wA (m s21) � equilibrium mean molecular speed of species A.

In contrast to the Knudsen number (2.) diffusion Kn (p. 45), it involves the influ-

ence of surface roughness and diffusive tortuousness. It expresses the gas diffusion

in bundles.

Info: [A29].


3.2.34 Kolmogorov Number Kol

Kol5Rp 2 R

Rgwg

w2dyn

@w

@z

� �21

; where wdyn 5

ffiffiffiτR

r

Rp, R (kg m23) � density of particles and liquid; g (m s22) � gravitational accelera-

tion; wg (m s21) � free fall velocity of a particles in liquid; wdyn (m s21) � dynamic

velocity;@w

@z(s21) � vertical gradient of velocity flow; τ (m21 kg s22) � shear fric-

tion tension.

It expresses the relative loss of the turbulent energy expended to maintain the

equilibrium of particles in a fluid flow. It is a criterion of the dynamic balancing

activity, for example, the balanced particles’ influence on the flowing fluid dynam-

ics. It is analogous to the Kolmogorov parameter Kol (p. 399).

Info: [B6],[C75].

Andrey Nikolaevich Kolmogorov (25.4.1903�20.10.1987),

Russian mathematician.

He contributed substantially to the development of proba-

bility theory, topology and other mathematical branches.

However, he also worked in the sphere of classic mechanics,

especially on turbulence theory. He formulated the algorith-

mic complex theory. Beginning in the 1940s, he was engaged

in dynamic systems theory in relation to interplanetary flight

and other problems in space science. Besides mathematics,

he had an extraordinary interest in poetry, especially the

Russian poet Pushkin.


3.2.35 Kozeny Number Kz, k

Kz5Δp

ηLμ3S

ð12ϕ2Þw

Δp (Pa) � piezometric pressure drop in the thickness L of porous material; η (Pa s) �dynamic viscosity; L (m) � characteristic length, thickness; ϕ (�) � porosity (p. 24);

S (m2) � area; w (m s21) � mean velocity component in direction L.

It expresses the ratio of the pressure and volume deformation forces to the fluid

momentum. It characterizes the mass transfer through a porous material.

Info: [A29].

Josef Alexander Kozeny (born 1889), Austrian engineer of Czech origin.

3.2.36 Kubo Number of Percolation Kub

Kub5w

Lω

w (m s21) � characteristic velocity; L (m) � characteristic length, necking; ω (s21) �characteristic perturbation frequency.

It expresses the accidentality level in the percolation process of a moderately

compressible fluid under turbulent transfer conditions. Two-phase fluid mechanics.

Turbulent diffusion.

Info: [B4],[B53].

3.2.37 Leibenson Number (2.) Lb

Lb5RR3

HΔp

η2L

R (kg m23) � fluid density; RH (m) � hydraulic radius; Δp (Pa) � pressure drop;

η (Pa s) � dynamic viscosity; L (m) � characteristic length.

It characterizes the fluid filtration in a porous environment. It expresses the

porous environment layer resistance, which depends on its structure and the physi-

cal properties of the filtering environment. It is a measure of the ratio of pressure

forces under the action of hydraulic resistance to the viscosity forces.

Info: [A33].

Leonid Samuilovich Leibenzon (1879�1951), Russian physicist and

mechanist.

105Fluid Mechanics

3.2.38 Leverett Number Lt

Lt5

ffiffiffik

p

spc

σ

k (m2) � permeability; p (�) � porosity (p. 24); pc (Pa) � capillary pressure;

σ (Pa) � surface tension.

It expresses the ratio of the characteristic dimension of an inter-area curvature to

the characteristic dimension of pores. Two-phase flow in porous materials.

Info: [A29],[B20].

Miles Corrington Leverett (born 1910), American chemical engineer.

3.2.39 Lockhart�Martinelli Parameter χ

χ5ml

mg

ffiffiffiffiffiRgRl

r

ml, mg (kg s21) � liquid and gas mass flux; Rl, Rg � liquid and gas density.

This parameter is used for internal two-phase flow. It characterizes the influence

of the fluid fraction portion on the flow. It occurs mainly in pressure spraying and

heat transfer in boiling and condensation.

Info: [A12],[C9].

3.2.40 Lyashenko Number Lj

Lj5Re3Ar21

Re (�) � Reynolds number (p. 81); Ar (�) � Archimedes hydrodynamic number

(p. 53).

This number characterizes the hydraulic effect of the fluid flowing through a

granular material (fluidization).

Info: [A29].

3.2.41 Martinelli Parameter X

X2 5dp

dz

fr

� �liq

dp

dz

fr

� �21

g

dp

dz(N m23) � pressure gradient; subscripts: fr � friction; liq � liquid; g � gas or

vapour.


It expresses the ratio of the pressure caused by the friction of drops and the gas

flow. It is used for two-phase models to express the pressure of diversely shaped

drops.

Info: [A7].

3.2.42 Mass Number Nm

Nm 5VS

Vf

RSRf

VS, Vf (m3) � volume of solid and liquid components; RS, Rf (kg m23) � density of

solid and liquid components.

It expresses the volume to density fractions ratio of solid and liquid particles or,

alternatively, the inertia thereof in mixing. It quantifies the effect of the volume tol-

erance. For Nm. 1, the momentum transfer of the granules in the fluid prevails.

Two-phase fluid mechanics.

Info: [A4],[B53],[B54].

3.2.43 Mobility Number of Grains Nmov

Nmov 5Rw2

fVL

R (kg m23) � fluid density; w (m s21) � characteristic flow velocity; fV (N m23) �specific grain force relative to volume unit; L (m) � characteristic length of grains.

In a two-phase flow, it characterizes the ratio of the fluid flow dynamic force to

the dynamic force of flowing grains. Two-phase flow. Hydrodynamics. Filtration.

Info: [A48].

3.2.44 Morton Number Mo

Mo5gη4

Rσ35

We31Fr2Re4

ð1Þ; Mo5gη4ðR1 2 RgÞ

R2l σ3ð2Þ;

Mo5gη4ðRf 2 RsÞ

R2fσ3ð3Þ

g (m s2) � gravitational acceleration; η (Pa s) � dynamic viscosity; Rl, Rg, Rf, Rs(kg m23) � density of liquid, gas, fluid and solid particles; σ (N m21) � surface

tension; W e1 (�) � Weber number (1.) (p. 91); Fr (�) � Froude number (1.)

(p. 62); Re (�) � Reynolds number (p. 81).

107Fluid Mechanics

This number expresses the ratio of the gravitational acceleration to the molecular

acceleration of a fluid. It depends on fluid properties only. In expression (1), it is

called the Morton fluid number, which expresses the relations arising in the process

of immersing solid bodies into viscous fluid. It corresponds to the capillary buoy-

ancy number CaB (p. 96). Expression (2) describes the free bubble movement in

a steady state fluid. Expression (3) describes the solid particle movement in the

fluid. It is also called the property groups NP (p. 110) or the Haberman�Morton

number Hab (p. 103). For water at 20�C, its value is approximately 33 10211.

Info: [A26],[B20].

3.2.45 Non-dimensional Thermal Diffusivity Na

Na 5agf

c2g

ag (m2 s21) � gas thermal diffusivity; f (s21) � bubble frequency; cg (m s21) � gas

sound velocity; κ (�) � specific heat ratio (p. 28).

It expresses the internal thermal conductivity, including the thermal dumping

influence of bubbles, in two-phase surroundings. The effective polytropic exponent

k depends on internal thermal conductivity, and 1, k,κ holds. For high frequen-

cies, the mean free path in gas can be comparable with the bubble dimension and

the k can be outside the range mentioned as well.

3.2.46 Ohnesorge Number Oh, Z

Oh � Z5ηffiffiffiffiffiffiffiffiffiRLσ

p 5We121Re

21 ð1Þ; Oh5 Su12 ð2Þ

η (Pa s) � dynamic viscosity; R (kg m23) � density; L (m) � characteristic length;

σ (Pa) � surface tension; We1 (�) �Weber number (1.) (p. 91); Re (�) �Reynolds number (p. 81); Su (�) � Suratman number (p. 115).

It is the ratio of the viscous force to the surface stress force. It characterizes the

processes of fluid spraying (atomization) in a two-phase flow. Sometimes, the

Suratman number Su (p. 115) is used instead of it. Stroke and surface waves.

Spraying. Splashing.

Info: [A26],[B20],[B56].

Wolfgang von Ohnesorge (8.9.1901�26.5.1976), German hydrodynamics

specialist.

3.2.47 Pavlovsky Number Pa

Pa5d2Δp

wηδ


d (m) � particle diameter; Δp (Pa) � pressure drop; w (m s21) � characteristic

flow velocity; η (Pa s) � dynamic viscosity; δ (m) � layer thickness.

It characterizes the fluid flow through a granular material layer. It expresses the

pressure force, with consideration of the hydraulic resistances, to the viscosity force.

Info: [A33].

3.2.48 Poiseuille number (2.) Ps

Ps5wv

ðRs 2 RfÞgD2

w (m s21) � flow velocity; ν (m2 s21) � kinematic viscosity; Rs, Rf (kg m23) � den-

sity of solid particles and liquid; g (m s22) � gravitational acceleration; D (m) �diameter of solid particle.

It expresses the fraction and gravitation forces in a fluid flow with mass parti-

cles. Two-phase flow.

Info: [A29].

Jean Louis Marie Poiseuille (p. 78).

3.2.49 Porous Flow Number NP

NP 5wηLffiffiffik

pσ cos ϑ

w (m s21) � flow velocity; η (Pa s) � dynamic viscosity; L (m) � characteristic

length; k (m2) � permeability; σ (N m21) � surface tension; ϑ (�) � angle of cap-

illary depression.

It is the viscous-to-capillary pressures ratio. It is the inverse value of the capil-

lary number (1.) Ca (p. 56). Filtration with a porous material.

Info: [B20].

3.2.50 Porous Pressure Number NP

NP 5D

h2

ffiffiffiL

g

s

D (m2 s21) � porous pressure diffusivity; h (m) � height; L (m) � characteristic

length; g (m s22) � gravitational acceleration.

This number expresses the ratio of the gravity flow time to the time of the pore

pressure diffusion which is perpendicular to the flow direction. The small value

Npor{1 shows that the porous pressure and the diffusion act at different times.


109Fluid Mechanics

3.2.51 Property Groups NP

NP 5gη4ðRf 2 RsÞ

R2fσ3

g (m s22) � gravitational acceleration; η (Pa s) � dynamic viscosity; Rf, Rs (kg m23) �

density of fluid, solid particles or bubbles; σ (N m21) � surface tension.

It is a widely used and specifying modification of the capillary buoyancy

number CaB (p. 96).

Info: [A29].

3.2.52 Rabin Number Rab

Rab5We1ffiffiffiffiffiffiRe

p ð1Þ; Rab5

ffiffiffiffiffiffiffiffiffiWe31Lp

4

sð2Þ

d (m) � drop diameter; R (kg m23) � drop density; σ (N m21) � surface tension;

η (Pa s) � dynamic viscosity; We1 (�) � Weber number (1.) (p. 91); Re (�) �Reynolds number (p. 81); Lp (�) � Laplace number (p. 71).

In expression (1), it serves to determine various drop decomposition. It can be rewrit-

ten into expression (2), which is essentially analogous to the Weber number (1.) We1(p. 91), with certain corrections for the drop dimension expressed by the Laplace

number Lp (p. 71). Two-phase flow. Condensation. Getting frostbitten. Spraying.

Info: [A15].

3.2.53 Radial Frequency Parameter (1.) Pfr

Pfr 5ωrD

w2

ωr (s21) � angular frequency; D (m2 s21) � mass diffusion or dispersion;

w (m s21) � fluid velocity.

It expresses the cyclic transfer of molecular mass diffusion with dynamic fluid

acting on a rotating system. It serves to monitor the resonating frequency and is

related to control. With its value increasing, the radial characteristics strengthen

repeatedly. Fluidized coating.

Info: [A35].


Pfr 5ωrL

2

a


ωr (s21) � angular frequency; a (m2 s21) � thermal diffusivity; w (m s21) � fluid

velocity.

It expresses the heat diffusion cyclic transfer with dynamic fluid acting on rotat-

ing systems. It is also called the wave thermal parameter. It serves to monitor the

resonating frequency. An increasing value leads to repeated strengthening.

Fluidized coating.

Info: [A35].


Pfr 5ω2rDL

w35 ðP�

frÞ2Pe21m

ωr (s21) � angular frequency; D (m2 s21) � mass diffusivity or dispersion; L (m) �

characteristic length; w (m s21) � fluid velocity; P�fr (�) � radial frequency param-

eter (1.) (p. 110); Pem (�) � Peclet mass number (p. 258).

It characterizes the cyclic transfer of molecular mass diffusion with dynamic

fluid acting on a rotating system. Fluidized coating.

Info: [A35].


Pfr 5 L

ffiffiffiffiffiffiωr

2D

r

L (m) � characteristic length; D (m2 s21) � diffusivity or dispersion; ωr (s21) �

angular frequency.

It characterizes the cyclic transfer of molecular mass diffusion. Fluidized

coating.

Info: [A35].

3.2.57 Rayleigh Pore Flow Number Rapor

Rapor 5ϕgβL3ΔT

av

ϕ (�) � porosity (p. 24); g (m s22) � gravitational acceleration; β (K21) � volume

thermal expansion coefficient; L (m) � characteristic length, thickness; ΔT (K) �temperature difference; a (m2 s21) � thermal diffusivity; ν (m2 s21) � kinematic

viscosity.

111Fluid Mechanics

This number expresses the viscous fluid infiltration through a porous layer

caused by the thermal gradient, gravitation force and diffusion heat propagation. It

is analogous to the Rayleigh modified number Ramod (p. 382).

Info: [A35].

Lord Rayleigh (p. 187).

3.2.58 Relative Filtration Velocity Pw

Pw 5c

w

c (m s21) � velocity propagation of surface wave; w (m s21) � filtration velocity.

It expresses the influence of the water wave propagation velocity on the filtra-

tion velocity. Two-phase flow. Filtration. Geophysics.

Info: [A48].

3.2.59 Reynolds Quasi Number Req

Req 5Rh

ffiffiffiffiffiffigL

p

Vfη

R (kg m23) � density; h (m) � layer thickness; L (m) � characteristic length;

g (m s22) � gravitational acceleration; Vf (m3) � fluid volume; η (Pa s) � dynamic

viscosity.

In infiltration, it represents the dynamic coefficient analogous to the Reynolds

number Re (p. 81) in Newtonian flow. Usually, it reaches the value Req. 106.


Info: [B12],[B13],[B505].

Osborne Reynolds (p. 82).

3.2.60 Savage Number Sav

Sav5_γRsL

ðRs 2 RfÞghð1Þ; Sav5

_γRsLnðRs 2 RfÞgh tan ϕ

ð2Þ

_γ (s21) � unit speed of shear deformation; Rs, Rf (kg m23) � density of particles

and fluid; L (m) � characteristic length, grain diameter; g (m s22) � gravitational

acceleration; h (m) � layer thickness; n (�) � number of surface grains; ϕ (�) �angle of inner friction.

This number expresses the ratio of tensions caused by grain collision to gravita-

tion tensions arising due to the contact friction of grains. With Sav. 0.1 in the

usual layer depth, the tensions originating due to collision have great influence.

Info: [B13].


3.2.61 Schiller Number (2.) Sch

Sch5w

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3R

4gvðRs 2 RÞ3

s5

ffiffiffiffiffiffiffiffiReδ

W

r

where W 5wδ

w; Reδ 5

wδv

w (m s21) � fluid flow velocity; R, Rs (kg m23) � density of fluid and solid parti-

cles; g (m s22) � gravitational acceleration; ν (m2 s21) � kinematic viscosity; wδ

(m s21) � boundary layer velocity; Reδ (�) � Reynolds boundary layer number,

see Reynolds number Re (p. 81); W (�) � relative velocity.

It expresses the equilibrium rate of ball-shaped particles floating in flowing

fluid. Fluidization.

Info: [A25],[A35].

L. Schiller.

3.2.62 Spray Number NA

NA 5gðR2 RfÞL2

σ

g (m s22) � gravitational acceleration; R, Rf (kg m23) � density of bubble and

ambient fluid; L (m) � characteristic length; σ (N m21) � surface tension.

It expresses the ratio of the weight force to the surface stress force. It corre-

sponds to the Bond number Bo (p. 95). Atomization of fluids.

Info: [B7].

3.2.63 Stokes Foam Drainage Number St

St5ηws

Rgr2b5 fðε;ReÞ

η (Pa s)� dynamic fluid viscosity; ws (m s21) � liquid surface velocity; R (kg m23) �density; g (m s22) � gravitational acceleration; rb (m) � bubble diameter; ε (�) �volume liquid fraction in foam; Re (�) � Reynolds number (p. 81).

It expresses the fluid foaming in steady state flowing through sewers or other

canals, and the fact that the flow is influenced by the relative volume fluid fraction

and the Reynolds number Re (p. 81). Two-phase fluid mechanics.

Info: [B109].

George Gabriel Stokes (p. 131).

113Fluid Mechanics

3.2.64 Stokes Particle Number St

St5τrxτk

5R

18

d

η

� �2

where τrx 5Rd2

18v; τk 5

ffiffiffiv

ε

r; η5 v

34ε

14

τrx (s) � relaxation time of particle; τk (s) � Kolmogorov time; R (�) � density

ratio of particle and fluid; d (m) � particle diameter; η (m) � Kolmogorov length;

ν (m2 s21) � kinematic viscosity; ε (m2 s21) � dissipation rate of turbulent energy.

This number characterizes the particles moving by inertia in a turbulent flow

and clustering in a low swirl zone of the fluid due to the inertia equilibrium among

denser particles and the lighter surrounding fluid. The clustering peaks occur with

St�1. Particle clustering in turbulent aerosols. Two-phase flow.

George Gabriel Stokes (see above).

3.2.65 Stokes Time Number Stk

Stk5τpτf

5Rd2wCc

18ηdc

τp (s)� time response; τf (s)� characteristic flow time; R (kg m23)� density; d (m)�particle diameter; w (m s21) � flow velocity; Cc (�) � Cunningham�Stokes correction

factor; η (Pa s) � dynamic viscosity; dc (m)� cylinder diameter.

It expresses the ability of aerosol particles to collide. It represents the ratio of

the response time to the characteristic slow time. Sometime, it is called the inertia

parameter.

George Gabriel Stokes (see above).

3.2.66 Strain Rate of Grain N, B

N5 _ε

ffiffiffiffiffiffiffiffiffiffiRsD3

k

r

_ε (s21) � speed of relative elongation; Rs (kg m23) � particle density; D (m) � par-

ticle diameter; k (kg s22) � linear stiffness of particle.

It characterizes the deformation rate of grains in the two-phase slide flow of

granulated fluid where the deformation can be understood as plastic or viscous.

Usually, real granulated flow is between these two extremes. Two-phase flow.

Info: [B4].


3.2.67 Suratman Number Su

Su5Oh22 5RlLση2l

5Re2We211

Rl (kg m23)� fluid density; ηl (Pa s)� liquid dynamic viscosity; L (m)� characteristic

length; σ (N m21) � liquid surface tension; Oh (�) � Ohnesorge number (p. 108);

Re (�)� Reynolds number (p. 81);We1 (�) � Weber number (1.) (p. 91).

It expresses the ratio of the product of the inertia force and the surface stress to

the square of the viscous force. Alternatively, it expresses the relation between the

surface stress and the friction of plunged particles. It characterizes the fluid spray-

ing process. Sometimes, the Ohnesorge number Oh (p. 108) is used instead of it.

Hydromechanics. Dynamics of particles.

Info: [A29],[B20].

3.2.68 Thermal Diffusivity of Gas Ag

Ag 5agωc2g

ag (m2 s21) � gas thermal diffusivity; ω (s21) � bubble frequency; cg (m s21) �gas sound velocity.

This diffusivity expresses the influence of the frequency of the bubbles of

diverse gases in a fluid on the change of thermal diffusivity and the polytropic gas

exponent k. At low frequencies, the time is sufficient for thermal isothermal diffu-

sion (k5 1). With higher frequencies, the time for the thermal diffusion is insuffi-

cient and the behaviour causes an isentropic tendency. With a constant high

frequency, the mean free path in the gas is comparable to the bubble size and

the polytropic exponent can reach a value outside the range kAh1; κi, where κ is

the specific heat ratio (p. 28). Thermodynamics. Multiphase fluids mechanics.

Cavitation.

Info: [A2].

3.2.69 Two-Phase Flow Number NP2

NP2 5ηdwσL

η (Pa s) � dynamic viscosity; d (m) � bubble or particles diameter; w (m s21) �flow velocity; σ (N m21) � surface tension; L (m) � characteristic length.

It is the ratio of the viscous (friction) force to that of surface stress. Capillary

flow. Dispersion materials.

Info: [B20].

115Fluid Mechanics

3.2.70 Two-Phase Porous Flow Number NPP

NPP 5wηffiffiffiffiffiffiffiffiffi

k1k2p

gΔR

w (m s21) � flow velocity; η (Pa s) � dynamic viscosity; k1, k2 (m2) � longitudinal

and transversal permeability; g (m s22) � gravitational acceleration; ΔR (kg m23) �density difference of both phases.

It is the viscous-to-gravitational pressures ratio. Capillary flow in porous

materials.

Info: [B20].

3.2.71 Valensi Two-Phase Number Va

Va5ωL2

v

ω (s21) � angular oscillation frequency of object in fluid with zero dynamic viscos-

ity; L (m) � characteristic length; ν (m2 s21) � kinematic viscosity.

Physically, it characterizes the ratio of the viscous diffusion time L2/(4ν) to the

oscillating frequency ω2 1. It expresses the oscillation of objects, drops and bubbles

in a fluid especially.

Info: [A35].

Jacques Valensi (born 1903), French engineer.

3.2.72 Viscous Inverse Number Nvis

Nvis 5

ffiffiffiffiffiffiffiffigD3

t

pv

5

ffiffiffiffiffiffiffiffiEo3

Mo

4

r

g (m s22) � gravitational acceleration; Dt (m) � characteristic pipe diameter;

ν (m2 s21) � kinematic viscosity; Eo (�) � Eotvos number (p. 99); Mo (�) �Morton number (p. 101).

This number characterizes the influence of the viscosity and other properties of

a two-phase fluid with the generation of individual Taylor bubbles and formation

of wake in a vertical pipeline or canal. With high gas velocities (Nvis. 1500), the

liquid part can be pushed to the pipeline wall and the gas flows through the core

continuously. With 500,Nvis, 1500, the end of bubbles is nearly in plane and

the wake part is closed immovably. It tends to lose symmetry with respect to the

tube axis and provides periodic waves, the frequency of which grows with increas-

ing Nvis. With Nvis, 500, the wake part is divided into two closed ring swirls

which act mutually on each other.

Info: [B112].


3.2.73 Weber Spray Number Wespray

Wespray 5Rq2VLσ

R (kg m23) � fluid density; qV (m s21) � volume flux density; L (m) � characteristic

length, diameter; σ (N m21) � surface tension.

It characterizes the shower sprinkler spraying of a fluid. Among other things, it

applies to cooling in electrical engineering, diverse industrial, apartment or office

rooms, gardening and agriculture.

Info: [B7].

Ernst Heinrich Weber (p. 92).

Wilhelm Eduard Weber.

3.3 Rheology

In rheology, the similarity criteria express the deformation and flow of non-

Newtonian fluids and plastic materials with various resistances thereof. It concerns

the mechanical properties of these substances, such as relations between stress and

deformation and deformation rate. The applications of rheology involve a wide

range of fields, from engineering to geophysics to physiology. These include, for

example, diverse polymer and biopolymer systems, terrestrial lava movement and

blood flow. The Bingham number, characterizing the flow of Bingham plastics, is a

fundamental similarity criterion. As for other criteria, these include the Deborah,

elastic, Elvis, Galilei, Hedstrom and Weissenberg numbers.

3.3.1 API Gravity Degree API

API5RrefR

2131:5

Rref (kg m23) � reference density (Rref5 1.4153 105 kg m23); R (kg m23) � liquid

density at temperature 15.55�C.The API gravity degree characterizes the density of liquid petroleum products.

Info: [A43].

3.3.2 Bingham Compression Number Bn

Bn5σminH

2

Fð1Þ; Bn5

σminHn

Kð2Þ

σmin (Pa) � minimal yield strength; H (m) � initial height of material sample;

F (N) � constant loading force; n (�) � constant of time independent deformation

speed; K (N) � dynamic loading force.

117Fluid Mechanics

It expresses the state of a semi-solid Bingham material if compressed. In

expression (1), it is about a constant load; in expression (2), it is about the condi-

tion with a constant deformation rate. Rheology. Bingham fluid.

Info: [B2],[C7].

Eugene Cook Bingham (1878�1945), American physicist.

3.3.3 Bingham Growth Exponent M

M5m

ffiffiffiffiffiffiffiffiffiffiF

H2K

n

rð1Þ; M5

mu

Hð2Þ

m (s) � exponent of stress raising; n (�) � power index of non-linear fluid behav-

iour; F (N) � loading force; H (m) � initial sample height; K (m21 kg s21) � con-

sistent index corresponding to dynamic viscosity; u (m s21) � compression speed.

This exponent characterizes the rheological behaviour dynamics of semi-solid

pulps. The expression (1) holds for a constant load; the expression (2) is valid for a

constant deformation rate. The index n has the value of n, 1 for a thinner fluid

layer and n. 1 for a thicker one. For the Bingham model, n5 1 is valid.

Info: [B2],[C7].

Eugene Cook Bingham (see above).

3.3.4 Bingham Number Bm

Bm � P5σkL

ηw

σk (Pa) � yield strength; L (m) � channel width; η (Pa s) � dynamic viscosity of

plastic; w (m s21) � plastic flow rate; P (�) � plasticity number (p. 129).

This number expresses the ratio of the yield point stress to the viscous tension.

It characterizes the creep of Bingham plastics and the rheological phenomena in

viscous material flow, and material creep and forming. It is connected with the

Hedstrom number (1.) Hd1 (p. 124). Sometimes, it is called the plasticity number

P (p. 129). Its inverse value is the plastic deformation K (p. 128).

Info: [A13],[B20],[C7].

Eugene Cook Bingham (see above).

3.3.5 Brinkman Rheological Number Br

Br5w2ηλΔT

ð1Þ; Br � Gn5ηw2

λΔTprocð2Þ


w (m s21) � flux velocity; η (Pa s) � dynamic viscosity; λ (W m21 K21) � thermal

conductivity; ΔT (K) � reference temperature difference; ΔTproc (K) � tempera-

ture difference between fluid and boundary; Gn (�) � heat build-up number

(p. 124); Pr (�) � Prandtl number (p. 197).

It characterizes the mutual relation between the dissipation and heat conduction

in non-isothermal rheological systems, such as molten polymers. With the relative

temperature gradient represented by the difference in a thermal process

(ΔT5ΔTproc), it is called the Heat build-up number Gn (p. 124) or the Eckert

rheological number Ecrh (p. 121), Ecrh5GnPr2 1. Large values of the numbers

Gn or Ecrh mean that, in the flowing polymer, the heat development is determined

by the viscous dissipation, but very little by the conduction.

Info: [A23].

Henri Coenraad Brinkman, German physicist.

3.3.6 Darcy Friction Number fD

fD 52dp

dx

DH

R w2ð1Þ; fD 5 4 fF 5

8 Δp

R w2

DH

Lð2Þ;

fD 58τwR w2

Nð3Þ; where DH 5

4 A

U

dp

dx(Pa m21) � pressure gradient; DH (m) � hydraulic diameter; R (kg m23) �

liquid density; w (m s21) � flow velocity; Δp (Pa) � pressure difference; L (m) �characteristic length; A (m2) � cross-section area; U (m) � wetted perimeter of

cross-section area; τw (Pa) � shear stress on the wall; wN (m s21) � free flow

velocity; fF (�) � Fanning friction number (p. 163).

In expression (1) or (2), it describes the pressure loss caused by the friction in

the flow in a constant cross-section pipeline. In expression (3), it is analogous to

the Darcy�Weisbach friction coefficient. See Fanning friction number fF (p. 163).

Info: [A2],[A35].


3.3.7 Deborah Generalized Number De

De5 τffiffiffiffiffiffiffiffiffiffiffiffiffiIe 2 Iv

pτ (s) � natural time of viscoelastic material; Ie (s

22) � invariant of rate of strain

tensor; Iv (s22) � invariant of vorticity tensor.

In the dimensionless form, it characterizes the viscoelastic process time. See

also the Weissenberg generalized number Ws (p. 132).

Info: [A29],[A35].

119Fluid Mechanics

3.3.8 Deborah Mixing Number Demix

Demix 5τmix

τpol

τmix (s) � mixing time; τpol (s) � polymerization time.

It expresses the ratio of the characteristic mixing time to that of a polymerizing

process. It is analogous to the Deborah number (1.) De (p. 120). Rheology. Mixing

of melted polymers.

Info: [A29].

3.3.9 Deborah Number (1.) De

De5τrxΔτ

ð1Þ; De5τmΔτ

ð2Þ

De{1 ðliquid-likeÞ ð3Þ;

De-0 ðviscoelasticÞ ð4Þ;

Dec1 ðsolid-likeÞ ð5Þ;

τrx (s) � relaxation time; Δτ (s) � natural time process interval (experimental or

observation time); τm (s) � molecular time.

This number expresses the ratio of the relaxation time to the process observation

(experimental) time interval. It characterizes the elastic behaviour � depending on

the relaxation time length � of the polymeric fluid. A long relaxation time is con-

nected with further stress preservation, whereas a short relaxation time means purely

viscous behaviour of the fluid. A Newtonian fluid is the limit case (De5 0). With a

great value of the number De, the material behaves as elastically solid; with extremely

small De numbers, it behaves as a purely viscous fluid. In expression (2), for example,

it holds for the flow of nematic liquid crystals. Rheology of viscoelastic fluids.

Info: [A29].

3.3.10 Deborah Pressurization Number Dep

Dep 5τrxvh

ð1Þ; Dep;crit 5 τrxωcrit ð2Þ

τrx (s) � relaxation time; v (m s21) � moving plate velocity; h (m) � gap width

between two plates; ωcrit (s21) � critical angular velocity.

In expression (1), it characterizes the dynamics of viscoelastic fluid pressuriza-

tion in a gap between two parallel plates. The critical value is given by expression

(2). Rheology.

Info: [A35],[B98].


3.3.11 Eckert Rheological Number Ecrh

See Heat build-up number Gn (p. 124).

Ernst Rudolf Georg Eckert (p. 192).

3.3.12 Elasticity Number (1.) El

El5τrxvL2

5De Re21 � Forx

τrx (s) � relaxation time; ν (m2 s21) � kinematic viscosity; L (m) � characteristic

length; De (�) � Deborah number (1.) (p. 120); Re (�) � Reynolds number

(p. 81); Forx (�) � Fourier viscoelastic relaxation number (p. 121).

This number expresses the elastic-to-inertia forces ratio. Alternatively, it is the

ratio of the relaxation (polymerization) time to the diffusion time. It characterizes

the non-isotropic flow of polymer fluids. Elc1. Rheology.

Info: [A26],[B20].

3.3.13 Elasticity Number (2.) Nelast, El

Nelast 5cpRβK

5Gc Ho Du21

cp (J kg21 K21) � specific heat capacity; R (kg m23) � density; β (K21) � volume

thermal expansion coefficient; K (Pa) � bulk modulus of elasticity; Gc (�) � Gay-

Lussac number (p. 14); Ho (�) � Hooke number (p. 138); Du (�) � Dulong num-

ber (p. 191).

It expresses the elasticity effect in the viscoelastic fluid flow process. It depends

only on the physical properties of the fluid.

Info: [A26],[B20].

3.3.14 Elasticity Number (3.) Nelast

Nelast 5cp

βa2

cp (J kg21 K21) � specific heat capacity; β (K21) � volume thermal expansion

coefficient; a (m s21) � sound velocity.

It expresses the elastic-to-inertia forces ratio or, alternatively, the elasticity

effect in fluid flow processes.

Info: [A152],[B20].

3.3.15 Elasto-capillary Number Nec, Γ0

Nec 5τrxσηL

5De Ca21

121Fluid Mechanics

τrx (s) � relaxation time; σ (N m21) � surface tension; η (Pa s) � dynamic viscos-

ity of viscoelastic fluid; De (�) � Deborah number (1.) (p. 120); Ca (�) � capil-

lary number (1.) (p. 97).

It characterizes the elastic and capillary properties of adhesive and other non-

Newtonian fluids in the production of microfibres. Rheology. Microbiology.

Info: [C136].

3.3.16 Ellis Number El

El5ηiniwτd

ηini (Pa s) � initial limiting value of viscosity as shear nears to zero; w (m s21) �flow velocity; τ (Pa) � shear stress by condition η5 ηini/2; d (m) � pipe diameter.

It characterizes the non-Newtonian fluid flowing through a pipeline. Rheology.

Info: [A35],[B20].

Samuel Benjamin Ellis (born 1904), American chemist.

3.3.17 Ericksen Number Er

Er58wLτMUR2

58

U

L

R

� �2

De

w (m s21) � velocity of movement of crystal wall; L�Δs (m) � dislocation; τM(s) � molecular time; U (�) � dimensionless molecular concentration parameter;

R (m) � space position; De (�) � Deborah number (1.) (p. 120).

This number expresses the ratio of the viscous moment of rheological fluid flow

to the moment of elastic flow and, alternatively, it quantifies the hydrodynamic and

elastic couplings. For example, it describes the Poiseuille flow of nematic liquid

crystals with diverse degrees of space orientation. Here, it expresses the intensity

of the narrow extent of nematic potential, this intensity being related to the broad

extent of the elastic potential.

Info: [C51].

3.3.18 Fourier Rheology Number Forh

Forh 5στηL

σ (N m21) � surface tension; τ (s) � natural process time; η (Pa s) � liquid

dynamic viscosity; L (m) � characteristic length.

It expresses the dimensionless rheological process time expressed by means of

the elastic and capillary properties of fluid. Rheology. Microrheology.

Info: [C136].



3.3.19 Fourier Viscoelastic Relaxation Number Forx

Forx � Zhrx 5vτrxL2

ν (m2 s21) � kinematic viscosity; τrx (s) � relaxation time; L (m) � characteristic

length; Zhrx (�) � Zhukovsky viscoelastic number (p. 133).

It expresses the elastic-to-inertia forces ratio and, alternatively, the relaxation-to-

diffusion time ratio. In the dimensionless form, it characterizes the stroke flow of vis-

coelastic fluids. In addition, it characterizes the stroke wave propagation time and

viscoelastic vibration of mechanical stress. Sometimes, it is called the Zhukovsky vis-

coelastic number Zhrx (p. 133) or the viscoelastic number (1.) Nη (p. 131).

Info: [A23].

Jean Baptiste Joseph Fourier (see above).

3.3.20 Galilei Number Ga

Ga1 5gL3

v25Re2Fr21 ð1Þ;

Ga2 5Ga1

βΔTð2Þ; Ga3 5

RΔR

Ar ð3Þ

g (m s22) � gravitational acceleration; L (m) � characteristic length; ν (m2 s21) �kinematic viscosity; β (K21) � volume thermal expansion coefficient;ΔT (K) � tem-

perature difference; R, ΔR (kg m23) � liquid density and their difference in various

places of non-isothermic flow; Re (�) � Reynolds number (p. 81); Fr (�) � Froude

number (1.) (p. 62); Ar (�) � Archimedes hydrodynamic number (p. 53).

This number characterizes the molecular friction and gravity forces in flowing

fluid. In expression (2), it expresses the mutual action of the molecular friction

force and the buoyancy force in a non-isothermal process. In expression (3), it is

about the mutual action of the molecular friction force and the Archimedes force

determined by different densities of the fluid in a system. Rheology.

Info: [B17],[B20].

Galileo Galilei (15.2.1564�8.1.1642), Italian mathematician,

astronomer and physicist, originally doctor.

In addition to extensive astronomical observations with a

telescope he had improved, he was also engaged in the

dynamics of bodies. For example, he discovered inertia, the

law of superposition of velocities, the free fall law and

the pendulum swing time dependence on its length. With his

telescope, he was the first to discover the craters on the

Moon, solar spots, the four moons of Jupiter and the Milky

Way as an aggregation of remote stars.

123Fluid Mechanics

3.3.21 Graetz Rheological Number Gzr

Gzr 5RcpwL2nλL

R (kg m23) � density; cp (J kg21 K21) � specific heat capacity; w (m s21) � refer-

ence velocity; L, Ln (m) � length of heat transfer in front and vertical flow;

λ (W m21 K21) � thermal conductivity.

It expresses the ratio of heat transferred by convection to that transferred by con-

duction. It characterizes the mutual ratio of the heat transfer by convection to the

transfer by conduction in rheological systems such as molten polymers.

Info: [A29],[B17].

Leo Graetz (p. 193).

3.3.22 Heat Build-up Number Gn

See Brinkman rheological number Br (p. 118).

3.3.23 Hedstrom Number (1.) Hd1

Hd1 5σkL

2

Rv25Re Bm

σk (Pa) � yield limit; L (m) � characteristic length; R (kg m23) � fluid density;

ν (m2 s21) � kinematic viscosity of plastic; Re (�) � Reynolds number (p. 81);

Bm (�) � Bingham number (p. 118).

It characterizes non-Newtonian fluid flow (the creep of Bingham plastics). See

the Bingham number Bm (p. 118) and the plastic deformation K (p. 128).

Info: [A26],[A29],[B17].

Bengt Olaf-Arvid Hedstrom (born 1926), Swedish chemist.

3.3.24 Hedstrom Number (2.) Hd2

Hd2 5σEL

ηw

σE (Pa) � elastic limit stress; L (m) � characteristic length; η (Pa s) � dynamic

viscosity in the plastic state; w (m s21) � mean velocity.

It relates to Bingham plastics flow and corresponds to the Bingham number Bm

(p. 118).

Info: [A26].

Bengt Olaf-Arvid Hedstrom (see above).


3.3.25 Ilyushin Number Il

Il54

3

Lτηw

5Re4

3

τRw2

� �

L (m) � characteristic length, channel diameter; τ (Pa) � maximum dynamic slip

stress; η (Pa s) � dynamic viscosity; w (m s21) � mean flow velocity; R (kg m23) �density; Re (�) � Reynolds number (p. 81).

It characterizes the viscoplastic fluid flow under the action of a pressure gradient

in a ring canal consisting of two immovable cylinders. With Il-0, a Newtonian

fluid flows.

Info: [A29].

Alexey Antonovich Ilyushin (born 1911), Russian engineer.

3.3.26 Mackley�Sherman Number MS

MS5βw

5 tgϑ

β (m s21) � mass transfer coefficient of particle; w (m s21) � velocity of incident

particle; ϑ (�) � particle impact angle.

It expresses the ratio of the particle mass transfer velocity to its velocity of

impact on a diaphragm surface which equals the tangent of the particle impact

angle. The Stanton mass number Stm (p. 265) is another analogue of the number

MS for a cluster of particles.

Malcolm Robert Mackley (born 1947), English chemical

engineer.

He is engaged in the flow of rheological fluids and its

industrial applications in the polymerization process in the

food industry. Especially, his research concerns the oscillation

mixing of liquids and the cool processing thereof in production

lines. It includes the drawing-through of molten polyethylene

and other industrial and nutritional materials. In his work, he

uses both direct experiment and numerical modelling.

3.3.27 Magnetic and Thermal Energy Ratio λ

λ5Emg

Et

5Emg

kT5

μ0μm2

16πd3kT; where m5

4

3πd3H

Emg (J) � magnetic energy; Et (J) � thermal energy; k (J K21) � Boltzmann con-

stant; T (K) � absolute temperature; μ0 (H m21) � vacuum magnetic permeability;

125Fluid Mechanics

μ (H m21) � dissolvent magnetic permeability; d (m) � particle diameter; m

(A m2) � magnetic moment; H (A m21) � magnetization.

It describes the microstructure development in magnetorheological suspensions,

which represent artificial suspensions of magnetized particles immersed in non-

magnetic fluid. Together with the Mason number Mas (p. 126), it represents the

basic criterion in magnetorheology. Rheology.

Info: [B74].

3.3.28 Mach Viscoelastic Number Ma

Ma5wffiffiffiffiffiffiffiffiffiη

Rτrx

r

w (m s21) � flow velocity; η (Pa s) � dynamic viscosity; R (kg m23) � fluid den-

sity; τrx (s) � time of stress relaxation.

It expresses the viscoelastic fluid flow rate at the inlet of a pipeline or canal. It

is important in the extruding or injecting of plastics, particularly with geometrically

simple input parts when complex singularities arise in the fluid flow. Fluid mechan-

ics. Rheology.

Info: [B34].

Ernst Mach (p. 73).

3.3.29 Mason Number Mas

Mas5122ηωμμ0H

2

η (Pa s) � solvent dynamic viscosity; ω (s21) � rotating frequency; μ (H m21) �solvent magnetic permeability; μ0 (H m21) � vacuum magnetic permeability;

H (A m21) � particle magnetization.

This number expresses the viscous-to-magnetic forces ratio. It characterizes the

ability of particles suspended in a non-magnetic fluid to be magnetized (the diame-

ter 1�10 μm approximately) with respect to a rotating magnetic field under the

influence of modulation according to the Mason number. Magnetorheological pro-

cesses of sedimentation.

Info: [B74].

3.3.30 Nahme Number Na

Na5w2η

λΔTrh; where ΔTrh 52

@ ln Ah1

@T

� �21

Tini


w (m s21) � characteristic velocity; η (Pa s) � dynamic viscosity; λ (W m21 K21) �thermal conductivity; ΔTrh (K) � temperature difference in the rheological process;

T (K) � temperature; Ah1 (�) � Arrhenius number (1.) energetic (p. 34).

It characterizes the mutual relation between the heat transfer by dissipation and

conduction in non-isothermal rheological systems such as molten polymers. It is a

special case of the Heat build-up number Gn (p. 124) pro ΔT5ΔTrh.

Info: [A23].

3.3.31 Nahme Nonisothermal Number Na

Na5ηβw2

λT; where w5 L _γ; β5

T

ηdηdT

η (Pa s) � dynamic viscosity in the limit of zero shear rate; β (�) � thermal sensi-

tivity of the fluid viscosity; w (m s21) � characteristic velocity; λ (W m21 K21) �thermal conductivity; T (K) � absolute temperature; L (m) � characteristic length

(L5 h); _γ (s21) � shear rate.

It characterizes the non-isothermal instability in the torsion flow of polymeric

fluids and expresses the influence of viscous fluid heating. NaAh0; 1i.Info: [B98].

3.3.32 Nahme�Griffith Number Na

Na5ΔTdis

ΔTvis5Br Ps

ΔTdis (K) � temperature difference developed by viscous dissipation of fluid;

ΔTvis (K) � temperature difference developed by temperature-dependent viscosity;

Br (�) � Brinkman number (p. 174); Ps (�) � Pearson number (p. 127).

This number expresses the viscous heating influence in the action of both vis-

cous dissipation and fluid viscosity depending on temperature. Nahme�Griffith

number is usually in the range 0,Na, 200, for 0.1,Na, 0.5 viscosity is unaf-

fected by temperature. Rheology.

Info: [A9].

3.3.33 Pearson Number Ps

Ps5ΔTop

ΔTvis5αðTS 2 TNÞ

ΔTop (K) � temperature difference imposed by operating conditions; ΔTvis (K) �temperature difference required to change the viscosity; TS (K) � surface wall

127Fluid Mechanics

temperature; TN (K) � temperature of thermal undisturbed fluid area; α (K21) �thermal expansion coefficient; η0, η (Pa s) � basic and real dynamic viscosities.

This number expresses the ratio of the operating temperature range to the tem-

perature gradient necessary to change the viscosity. It characterizes the viscosity

temperature dependence in cases when the temperature gradient must be considered

in boundary conditions. For the viscosity exponential dependence on the tempera-

ture, it is

η5 η0 expð2αΔTopÞ

Rheology.

Info: [A23].

3.3.34 Peclet Rheological Number Perh

Perh 5L2 _γa

L (m) � characteristic length, gap width; _γ (s21) � sound velocity; a (m2 s21) �thermal diffusivity.

It characterizes the diffusion heat transfer in non-isothermal polymer fluid flow.

PerhBh0; 1i.Info: [B25].


3.3.35 Plastic Deformation K

See Bingham number Bm (p. 118).

3.3.36 Plastic Viscosity Ratio Pη, k

Pη 5ηp

ηs 1 ηp

ηs, ηp (Pa s) � dynamic viscosities of dissolvent and plastic.

It expresses the total viscosity as the ratio of the Newtonian fluid (solvent) vis-

cosity to the polymeric viscosity. It is used, for example, in drawing or injecting

plastics. Fluid mechanics. Rheology.

Info: [B34].


3.3.37 Plasticity Number Npl

Npl 5τELηw

τE (Pa) � yield limit; L (m) � characteristic length; η (Pa s) � dynamic viscosity;

w (m s21) � velocity.

It is used in rheology and the creep of plastics. It is an equivalent of the

Bingham number Bm (p. 118) and analogous to the Hedstrom number (2.) Bm

(p. 124).

Info: [A35].

3.3.38 Prandtl Generalized Number Prgener

Prgener 5wL

aðw22 nLnkR21Þ2 1

11 n

w (m s21) � flow velocity; L (m) � characteristic length; a (m2 s21) � thermal dif-

fusivity; n (�) � exponent of non-Newtonian fluid; κ (kg m21 s21) � consistency

coefficient of pseudoplastic and dilating fluids; R (kg m23) � density.21 It expresses the relation between the velocity and temperature processes in

flowing non-Newtonian fluids.

Info: [A33].


3.3.39 Processability Parameter Np, P

Np 5βησ

5 Sh CaðRe ScÞ21 5 StmCa

β (m s21) � mass transfer coefficient; η (Pa s) � dynamic viscosity; σ (N m21) �surface tension; Sh (�) � Sherwood number (p. 264); Ca (�) � capillary number

(1.) (p. 97); Re (�) � Reynolds number (p. 81); Sc (�) � Schmidt number

(p. 263); Stm (�) � Stanton mass number (p. 265).

In microfibre rheometry, it is applied to predict the fibre shape and to support

the process using adhesive and other non-Newtonian fluids. This parameter serves

to determine the critical time within which the fibre is broken. Above all, this time

depends on the rheological properties of the fluid and the mass transfer as

expressed by this parameter. Rheology. Microrheology.

Info: [C136].

129Fluid Mechanics

3.3.40 Reynolds Generalized Number for Non-Newtonian Fluids Renon

Renon 58Rw2

τcharð1Þ; Renon 5

Rw2

τcharð2Þ

R (kg m23) � fluid density; w (m s21) � fluid velocity; τchar (Pa) � characteristic

shear stress depending on Boltzmann constant and absolute temperature.

It is used in form (1) for the flow of non-Newtonian fluids in tubes of circular

cross section. In form (2) to liquid crystals flow, it expresses the inertia-to-

characteristic tensions ratio which originates in liquid crystal flow.

Info: [A29].


3.3.41 Reynolds Rheological Number (1.) Rerh

Rerh 5L2 _γv0

L (m) � characteristic length, gap width; _γ (s21) � shear velocity; ν0 (m2 s21) �kinematic viscosity at zero shear velocity.

It characterizes the convective transfer in the non-isothermic flow of polymer

fluids. Rerh{1. Rheology.

Info: [B98].


3.3.42 Stokes Number Stk

Stk1 5vw

gL25Ca Bo21 ð1Þ; Stk2 5

vτL2

5 Sh21 Re21 ð2Þ;

Stk3 5ωLv

ð3Þ; Stk4 5LΔp

ηwð4Þ

ν (m2 s21) � kinematic viscosity; w (m s21) � flow velocity; g (m s22) � gravita-

tional acceleration; L (m) � particle characteristic dimension; τ (s) � particle

vibration time in fluid flow; ω (s21) � angular frequency of particle; Δp (Pa) �pressure difference; η (Pa s) � dynamic viscosity; Ca (�) � capillary number (1.)

(p. 97); Bo (�) � Bond number (p. 95); Sh (�) � Strouhal number (p. 87);


In expression (1), it expresses the viscosity-to-gravity forces ratio. In cases (2)�(4),

it expresses the inertia-to-friction forces ratio. Hydromechanics. Dynamics of particles.

Rheology.

Info: [A7],[A29],[A35],[B20].


George Gabriel Stokes (13.8.1819�1.2.1903), Irish mathe-

matician and physicist.

He was engaged in the mechanics of continua, especially

in hydrodynamics, waves in elastic bodies, acoustics and dif-

fraction. The fundamental continuum mechanics equation is

called the Navier�Stokes equation after him. In addition to

continuum mechanics, he measured gravitational field

changes on the surface of the earth. He was engaged in vec-

tor analysis, and in chemistry and botany as well.

3.3.43 Thermoelastic Rheological Number Nelast

Nelast 5 τrxL

ffiffiffiffiffiffiffiffiη0βλT

r

τrx (s) � relaxation time of viscoelastic stresses; L (m) � characteristic length; η0(Pa s) � dynamic viscosity at zero shear rate; β (�) � temperature sensibility of fluid

viscosity; λ (W m21 K21) � thermal conductivity; T (K)� absolute temperature.

It characterizes the non-isentropic flow of polymer fluids. It expresses the ratio

of the heat propagation time to the polymerization time. NelastAh0; 1i.Info: [B98].

3.3.44 Truncation Number, Shearing Failure Parameter NT

NT 5ηωp

η (Pa s) � dynamic viscosity; ω (s21) � angular speed of shear deformation; p (Pa) �pressure.

It is the ratio of the shear stress to the normal pressure. It relates to viscous non-

Newtonian fluid flow. Viscoelasticity. See the Hersey number He (p. 165).

Info: [A35].

3.3.45 Viscoelastic Number (1.) Nη, E

Nη 5τrxηRL2

ð1Þ; Nη 5M2Re22 ð2Þ

τrx (s) � stress relaxation time; η (Pa s) � dynamic viscosity; R (kg m23) � den-

sity; L (m) � characteristic length; M (�) � Mach number (p. 73); Re (�) �Reynolds number (p. 81).

In expression (1) or (2), it expresses the elastic properties of viscoelastic sur-

roundings and their flow. In the case of viscometry, L denotes the capillary tube

131Fluid Mechanics

diameter. It is also called the Fourier viscoelastic relaxation number Forx (p. 123)

or the Zhukovsky viscoelastic number Zhrx (p. 133).

Info: [A23].

3.3.46 Viscoelastic Number (2.) Nη

Nη 5G

ηω

G (Pa) � shear modulus; η (Pa s) � dynamic viscosity; ω (s21) � angular speed.

It is the elastic-to-viscous forces ratio. It describes the flow (outflow) of viscous

fluids. Viscometry.

Info: [A24],[B20],[B59].

3.3.47 Viscoelastic Parameter Pη

Pη 5ηSηP

5ηPwLτchar

ηS (Pa s) � dynamic viscosity of dissolvent matter; ηP (Pa s) � characteristic

dynamic viscosity of the polymer; w (m s21) � motion velocity; L (m) � wall dis-

placement; τchar (Pa) � characteristic stress.

It expresses the ratio of the diluting material viscosity to the characteristic vis-

cosity of the polymer or, alternatively, the ratio of the shear rate to the characteris-

tic tension. It appears in the flow of polymer liquid crystals.22

3.3.48 Weissenberg Generalized Number Wi

Wi5 τffiffiffi_γ

pτ (s) � time scale for viscoelastic stress relaxation; _γ (s22) � shear rate.

It expresses the characteristic material time (relaxation time) and the shear

velocity. It characterizes the velocity and time relations in rheological processes in

viscoelastic shear flow. See also the Deborah generalized number De (p. 119).

Info: [A23],[A35].

Karl Weissenberg (born 1893), German physicist.

3.3.49 Weissenberg Number (1.) Ws

Ws5Rw2

τ5Re El ð1Þ; Ws5

wτrxL

5Re Forx ð2Þ;


Ws5ðt1 2 t2Þw

Lð3Þ; where τ1 t1 _τ5 η0ðε1 t2 _εÞ;

Ws5Ma2Re21 ð4Þ

R (kg m23) � fluid density; w (m s21) � flow velocity; τ (Pa) � shear stress; _τ(Pa s21) � shear stress rate; τrx (s) � relaxation time; L (m) � characteristic length,

diameter; t1, t2 (s) � time constants; η0 (Pa s) � dynamic viscosity; ε (�) � rela-

tive elongation; _ε (s21) � relative elongation rate; Re (�) � Reynolds number

(p. 81); El (�) � Ellis number (p. 122); Forx (�) � Fourier viscoelastic relaxation

number (p. 123); Ma (�) � Mach viscoelastic number (p. 126); Re (�) �Reynolds number (p. 81).

It characterizes the influence of the viscoelastic fluid properties on proceeding

processes. In expression (1), it expresses the ratio of the dynamic viscoelastic force

to the viscous force. In expression (2), it expresses the time change of the visco-

elastic fluid flow close to a bypassed wall. The expression (3) is analogous to (2),

and expresses more exactly the ratio of viscoelastic force � considering the relaxa-

tion modulus � to the elastic force only. In expression (4), it expresses that the

shear wave propagation rate is analogous to the sound propagation velocity in com-

pressible fluid flow. The Ws is a special case of the Strouhal number Sh. Fluid

mechanics. Rheology.

Info: [A29],[A33],[B20],[B34].

Karl Weissenberg (see above).

3.3.50 Zhukovsky Viscoelastic Number Zhrx

See the Fourier viscoelastic relaxation number Forx (p. 123), or the viscoelastic

number (1.) Nη (p. 131).

Nikolay Yegorovich Zhukovsky (p. 33).

133Fluid Mechanics

4 Solid Mechanics

I hold a single experiment higher than thousands of opinions originated by

imagination only.Mikhail Vasilyevich Lomonosov (1711�1765)

4.1 Linear and Non-Linear Solid Mechanics

In solid mechanics, the dimensionless quantities involve the elastic, viscoelastic

and plastic fields. For the elastic, the linear tension-dependent deformation is valid

and for the viscoelastic�plastic sphere, the non-linear behaviour is valid. The rela-

tive prolongation, Hooke and Johnson numbers are among the most widespread

similarity criteria.

4.1.1 Angular Displacement Number ND

ND 5EJkαqL3

E (Pa) � modulus of elasticity; Jk (m4) � shear modulus; α (rad m21) � angular

displacement; q (N m21) � length loading; L (m) � characteristic length.

It expresses the angular displacement of an elastically laid beam.

Info: [A24].

4.1.2 Beam Loading N

N5q

ELð1Þ;

N5qA

Eð2Þ

q (N m21) � length loading; qA (N m22) � surface loading; E (Pa) � modulus of

elasticity; L (m) � characteristic length.

It expresses longitudinal (1) and planar (2) beam loading. Elasticity, statics.

Info: [A24].

Dimensionless Physical Quantities in Science and Engineering. DOI: 10.1016/B978-0-12-416013-2.00004-X


http://dx.doi.org/10.1016/B978-0-12-416013-2.00004-X

4.1.3 Crater Scaling Relationship Ncra

Ncra 5hmax

r5αFr2β 5α

gr

w2im

� �β

hmax (m) � maximum crater depth; r (m) � radius of the projectile; α, β (�) �empirically or numerically determined constants; g (m s22) � gravitational acceler-

ation; wim (m s21) � impact velocity; Fr (�) � Froude number (1.).

It expresses the dynamics of projectile penetration into a purely hydrodynamic

target. The data of the constants (α, β), determined by experiment and numerically,

are presented in the literature.

Info: [B122].

4.1.4 Criterion of Stringed Musical Instruments Nsmi

Nsmi 5RL2f 2

σ

R (kg m23) � string density; L (m) � characteristic length of string; f (s21) � string

frequency; σ (Pa) � mechanical stress of string.

It expresses the dynamic similarity criterion for string instruments provided the

string tension and density are equal for all instruments (violin, viola, violoncello

and double bass). When building these instruments, it has the same value, for

example, Nsmi5 0.2 usually. The criterion has very deep musical and physical

meaning. Its value is deduced from the instrument prototype and involves all previ-

ous experiences. Diverse relative length of the string instruments is 0.9 for the vio-

lin, 1 for the viola, 2 for the violoncello and 2.8 for the double bass. With this,

corresponding frequencies are obtained for certain musical tones.

Info: [A24].

4.1.5 Decibel Scales SPL, Lv, La, LF, LP, LI , LE , Lw

SPL5 20 logp

p0

� �; where p0 5 20 μPa ð1Þ;

Lv 5 20 logv

v0

� �; where v0 5 1 nm s21 ð2Þ;

La 5 20 loga

a0

� �; where a0 5 1 μm s22 ð3Þ;

LF 5 20 logF

F0

� �; where F0 5 1 μN ð4Þ;


LP 5 20 logP

P0

� �; where P0 5 1 pW ð5Þ;

LI 5 20 logI

I0

� �; where I0 5 1 pW m22 ð6Þ;

LE 5 20 logE

E0

� �; where E0 5 1 pJ ð7Þ;

Lw 5 20 logw

w0

� �; where w0 5 1 pJ m23 ð8Þ;

P0 (Pa) � reference pressure value; v0 (m s21) � reference speed value; a0 (m s22) �reference acceleration value; F0 (N) � reference force value; P0 (W) � reference

power value; I0 (W m22) � reference intensity value; E0 (J) � reference energy

value; w0 (J m23) � reference energy density value.

In the area of dynamic quantities measurement, the decibel scales express the loga-

rithmic dimensionless useful-to-threshold signals ratio denoted hereafter by a zero

index. Usually the quantities are sound, noise, velocity, acceleration, force, power,

intensity, energy and density thereof. Then, the sound (SPL), velocity (Lv), accelera-

tion (La), force (LF), power (LW), intensity (LI), energy (LE) and energy density (Lw)

levels are the corresponding logarithmic scales. The logarithmic scale introduction is

based on the fact that the human ear perceives logarithmically due to which a loga-

rithmic unit can be applied advantageously (the bel or the decibel as the tenth part of

it). Most frequently, the decibel scale is used in acoustics and in measuring instru-

mental techniques, electronics and other ranges to express the signal-to-noise ratio.

Info: [C34].

4.1.6 Explosion Number Ex

Ex5 rpE

R

� �215

τ225

rp (m) � blast wave radius; E (J) � explosive energy; R (kg m23) � density; τ (s) �time.

It characterizes the pressure stroke waves in detonations. Special techniques.

Info: [A24], [B20].

4.1.7 Force Numbers NF

NF 5Fτ2

mLð1Þ;

NF 5F

qLð2Þ;

137Solid Mechanics

NF 5FL2

GJð3Þ;

F (N) � force; τ (s) � time; m (kg) � density; L (m) � characteristic length;

q (N m21) � length loading; G (Pa) � shear modulus; J (m4) � square moment of

cross section.

In equation (1), they express the dynamic force in damped mass movement. In

equation (2), they express the static force with longitudinal beam load and � in

equation (3), they express the static force with static torsion stress.

Info: [A24].

4.1.8 Fourier Thermoelastic Number Fote

Fote 5wteτL

; where wte 5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiλ1 2μ

R

s

wte (m s21) � velocity propagation of the thermoelastic deformation; τ (s) � time;

L (m) � characteristic length; λ, μ (Pa) � Lame functions; R (kg m23) � density.

It expresses the ratio of the thermoelastic wave penetration depth to the charac-

teristic system length. In the dimensionless form, it characterizes the thermoelastic

process propagation time in a system.

Info: [A23].


4.1.9 Fourier Wave Number Fow,mech

Fow;mech 5c2τ2

L2; where c5

ffiffiffiffiE

R

s

c (m s21) � velocity propagation of the mechanical wave; τ (s) � time; L (m) �characteristic length; E (Pa) � modulus of elasticity; R (kg m23) � density of

material.

This number expresses the dimensionless time of the mechanical wave propaga-

tion in the material under strokes, for example. Solid phase mechanics. Stroke

phenomena.

Info: [A24].


4.1.10 Hooke Number Ho

Ho � Cau5RσσF

ð1Þ;


Ho5Rw2

Eð2Þ

R (kg m23) � density; σ (Pa) � stress; σF (Pa) � stress from the loading force;

w (m s21) � velocity; E (Pa) � modulus of elasticity; Cau (�) � Cauchy number

(aeroelasticity parameter) (p. 155).

It is the ratio of the yield point or the breaking strength σ to the tension due to a

loading force (1), which can be the dynamic pressure of the fluid (2). Elasticity

hydrodynamics.

Info: [A23],[A29],[B17].

Robert Hooke (18.7.1635�3.3.1703), English scientist.

The Hooke’s law is his important discovery, according to

which the body deformation is proportional to the acting

force. This is valid for small deformations to the extent of

the elasticity limit. In the year 1678, he tried to prove that

the Earth’s orbit around the Sun is an ellipse. In the same

year, he discovered that the gravity force drops with the sec-

ond root of the distance. He informed Newton about this in

writing, but subsequently an authorship conflict arose

between them which led to extreme hostility.

4.1.11 Impact Factor Nimp, I

Nimp 5Mw2

ini

σcd3

M (kg) � mass of a projectile; wini (m s21) � impact velocity of a projectile;

σc (Pa) � unconfined compressive strength of a concrete target; d (m) � diameter

of a projectile.

This factor expresses the ratio of the projectile energy to the inverse deforming

pressure energy in a target. Impact mechanics. Ballistics.

Info: [B66].

4.1.12 Johnson’s Damage Number Dn

Dn5Rwini

σk

R (kg m23) � material density; wini (m s21) � initial impulse velocity; σk (Pa) �yield stress of the material.

This number is the measure of the inertia loading force related to the material

resistance against loading. It represents the basic parameter in impact mechanics to

solve the material plastic response especially with dynamic loading. It is the

139Solid Mechanics

deformation state measure in the zone where plastic deformation arises abruptly.

The Cauchy number (aeroelasticity parameter) Cau (p. 155) is a similar criterion

for the dynamic elastic response. Non-linear mechanics.

Info: [B127],[B47].

4.1.13 Johnson’s Damage Number Jo

Jo5w2ini Rtσc

5Nimp

NA

wini (m s21) � impact velocity of a projectile; Rt (kg m23) � density of a concrete

target; σc (Pa) � unconfined compressive strength of a concrete target; Nimp (�) �impact factor (p. 139); NA (�) � slenderness factor (p. 145).

It expresses the degree of the relative damage in the projectile impact on a tar-

get. Usually, it is applied to classify the projectile impact. Stroke mechanics.

Ballistics.

Info: [B66].

4.1.14 Loss Coefficient η

η5Δu

2πuð1Þ;

where u5

ðσmax

0

dεσ2max

2Eð2Þ;

Δu5

Iσ dε ð3Þ

Δu (J m23) � change of specific volume energy; u (J m23) � specific volume

energy; σmax (Pa) � maximal stress; σ (Pa) � stress by loading; E (Pa) � modulus

of elasticity; ε (�) � relative elongation (p. 144).

It expresses the material dynamic memory function and, alternatively, the degree

to which the material dissipates the energy of vibration. With the elastic load up to

the highest stress σmax, expression (2) is valid for the elastic energy related to the

volume unit. Expression (3) concerns the loading and following unloading in which

the energy dispersion corresponds to the area of the hysteresis stress deformation

loop. Usually, the loss coefficient is time and cycling frequency dependent.

Info: [C65].

4.1.15 Mach Thermoelastic Number Mte

Mte 5wt

wte

; where wt 5a

L; wte 5


R

s


wt, wte (m s21) � velocity propagation of thermal and thermoelastic wave in the

system; a (m2.s21) � thermal diffusivity; L (m) � characteristic length; λ, μ (Pa) �Lame functions; R (kg m23) � density.

This number expresses the ratio of the heat propagation rate in elastic surround-

ings to the propagation rate of mechanical waves (sound rate). It characterizes the

mutual relation between the heat propagation rate and the thermal stress in a ther-

mal system.

Info: [A23].

Ernst Mach (p. 73).

4.1.16 Mass of the Projectile Nm

Nm 5M

Rd3

M (kg) � mass of a projectile; R (kg m23) � material density of a target; d (m) �projectile diameter.

In the dimensionless shape, it expresses the ratio of the projectile mass to the

target density. Solid mechanics. Ballistics.

Info: [B22].

4.1.17 Noise Number Noi

Noi 5NiT12refS

212

Ni (m) � mutually uncorrelated stochastic processes with zero mean value;

Tref (s) � dimensionless reference time; S (m2 s) � power density spectrum of the

stochastic process.

It characterizes a random process. It appears in a dimensionless mathematical

model to solve the stochastic equations set by means of the continuous hybrid

Monte Carlo method.

Info: [A24],[B36].

4.1.18 Nose Factor Pn

Pn 51

3Ψ2

1

24Ψ2; where Ψ5

R

dð1Þ; ðogive noseÞ

Pn 51

11 4Ψ2; where Ψ5

h

dð2Þ; ðconical noseÞ

Pn 5 121

8Ψ2; where Ψ5

r

dð3Þ; ðblunt=spherical noseÞ

141Solid Mechanics

Ψ (�) � calibrated radius head (CRH); R (m) � calibrated radius or diameter of

the ogive; d (m) � projectile diameter; h (m) � projectile nose length; r (m) �projectile nose radius.

This factor expresses the influence of the projectile nose shape on penetration.

The expression (1) is valid for the calibrated head radius or a broken Gothic arch,

the expression (2) holds for a conical peak, whereas the expression (3) holds for a

spherical top. The projectile nose factor becomes values in the range 0,Pn# 1.

For a flat top of a projectile, Pn5 1 is valid. For a semispherical projectile point,

Pn5 0.5 is valid. Stroke mechanics. Ballistics.

Info: [B66].

4.1.19 Oscillation Amplitude NA

NA 5EJy

qL4

E (Pa) � modulus of elasticity; J (m4) � square moment of cross section; y (m) �displacement of oscillation; q (N m21) � length loading; L (m) � characteristic

length.

This number expresses the amplitude of an oscillating beam. Elasticity.

Dynamics.

Info: [A24].

4.1.20 Oscillation Frequency Nf

Nf 5RL4f 2

EJð1Þ;

Nf 5ω2L

gð2Þ;

Nf 5ω2I

GL3ð3Þ;

R (kg m23) � density; L (m) � characteristic length; f (s21) � frequency; E (Pa) �modulus of elasticity; J (m4) � square moment of cross section; ω (s21) � angular

frequency; g (m s22) � gravitational acceleration; I (kg m2) � moment of inertia;

G (Pa) � shear modulus.

In form (1), it expresses the oscillating beam frequency. In form (2), it expresses

the mathematical pendulum oscillation frequency and in form (3), it means the

eigenfrequency of shaft torsion oscillations. Dynamics. Elasticity. Physics.

Info: [A24].


4.1.21 Peclet Thermoelastic Number Pete

See Vernotte thermoelastic number Vete (p. 283).


4.1.22 Penetration Depth Hp

Hp 5h

d5 f ðNimp; NA; PnÞ

h (m) � penetration depth; d (m) � diameter of a projectile; Nimp (�) � impact

factor (p. 139); Na (�) � slenderness factor (p. 145); Pn (�) � nose factor.

It expresses the projectile penetration depth of a non-deformable projectile strik-

ing a target. It is the function of the impact factor Nimp (p. 139), of the slenderness

factor NA (p. 149) and of the nose factor Pn (p. 141). Stroke mechanics. Ballistics.

Info: [B66].

4.1.23 Poisson’s Ratio ν

ν52ε1ε2

5E

2G21

ε1 (�) � relative deformation in the cross section (contraction) by uniaxial stress;

ε2 (�) � relative deformation in axial direction (elongation) by uniaxial stress;

E (Pa) � modulus of elasticity; G (Pa) � shear modulus.

It characterizes the relative deformation in single axis stress. For perfect non-

compressible materials, it would be ν5 0.5 in theory; for most technical materials

it is νA 0; 0:5h i; for cork it is ν-0; for most steels it is ν � 0.3; for rubber it is

almost 0.5. For the Poisson constant, it is μ5 ν21. Mechanics of solid bodies.

Elasticity.

Info: [A23],[A29].

Simeon Denis Poisson (21.6.1781�25.4.1840), French math-

ematician and physicist.

He was one of the founders of mathematical physics and

one of Laplace’s students. Especially, he was engaged in

mathematical analysis, variation calculus, probability theory

and hydromechanics. By extending and generalizing the

Laplace equation with source members, he formed the

Poisson equation describing steady-state fields with internal

sources and sinks.

143Solid Mechanics

4.1.24 Ratio of the Plug Mass to the Projectile Mass NΔm

NΔm 5πd2RH4M

d (m) � projectile diameter; R (kg m23) � material density of a plate; H (m) pene-

tration depth of a projectile; M (kg) � projectile mass.

It expresses the ratio of the material mass loss of a thin plate, after a projectile

impact, to the projectile mass. Solid mechanics. Ballistics.

Info: [B22].

4.1.25 Relative Elongation ε

ε5σE5

ΔL

L5

F

EA

σ (Pa) � stress; E (Pa) � modulus of elasticity; ΔL (m) � absolute elongation;

L (m) � characteristic length; F (N) � force; A (m2) � cross-sectional area.

It expresses the relative elongation of a body.

Info: [A24].

4.1.26 Relative Stiffness Ns

Ns 5ksy

F

ks (kg s22) � stiffness of elastic element, spring constant; y (m) � compression of

elastic component; F (N) � force.

In the dynamics of mechanical sets, it expresses the ratio of elastic element com-

pression to the acting force.

4.1.27 Relative Volume Change PV

PV 5ΔV

V0

ΔV (m3) � volume change; V0 (m3) � initial volume.

It expresses the ratio of the volume change to the original volume before defor-

mation. Elasticity, mechanics.

Info: [A24].


4.1.28 Sachs Number Sa

Sa5 r

ffiffiffiffiffiPa

E

3

r

r (m) � blast wave radius (distance from explosion place to reference one); pa (Pa) �atmospheric pressure; E (J) � explosive energy.

It expresses the superficial explosion propagation. See the analogous explosion

number Ex (p. 137).

Info: [A24].

4.1.29 Slenderness Factor NA

NA 5M

Rchard3

M (kg) � mass of a projectile; Rchar(kg m23) � characteristic density of environ-

ment between projectile and target; d (m) � projectile diameter.

It expresses the ratio of the pressure cross section of a projectile to the charac-

teristic density zone for an unlimited target. It represents the projectile slenderness

for a solid characteristic density of the environment between a projectile and a tar-

get. Stroke mechanics. Ballistics.

Info: [B66].

4.1.30 Statical Moment Number NMS

NMS 5M

EL3ð1Þ;

NMS 5M

GL3ð2Þ;

NMS 5M

qL2ð3Þ

M (N m) � moment of force; E (Pa) � modulus of elasticity; G (Pa) � shear mod-

ulus; L (m) � characteristic length; q (N m21) � length loading.

In form (1), it expresses the moment for the static tensile or pressure stress; in

form (2), it expresses the torsion stress. The bending moment of a beam, laid on a

common elastic base and loaded longitudinally, is given by form (3).

Info: [A24].

4.1.31 Structural Property Number NS

NS 5σLE

145Solid Mechanics

σ (N m23) � volume loading; L (m) � characteristic length; E (Pa) � modulus of

elasticity.

This number expresses the mass-to-stiffness ratio of a structure. In a complex

mechanical system, it expresses the resulting value of its partial subsystems.

4.1.32 Thermoelastic Coupling Number (1.) N1

N1 5ð3λ1 2μÞα

cR5

Eαð12 2νÞcR 5

3KαcR

λ, μ (Pa) � Lame functions; α (K21) � linear thermal expansion coefficient;

cR (J m23 K21) � specific heat capacity; E (Pa) � modulus of elasticity; K (Pa) �volume modulus of elasticity; ν (�) � Poisson’s ratio (p. 143).

This number expresses the ratio of the specific thermal stress to the specific vol-

ume thermal capacity. It characterizes the relation between the thermoelastic stress

and the thermal capacity in a compound thermal and thermal stress process. In a

non-compound process, α5 0.

Info: [A23].

4.1.33 Thermoelastic Coupling Number (2.) (Gay-Lussac ThermoelasticNumber) N2, Gcte

N2 � Gcte 5ð3λ1 2μÞðλ1 2μÞ αTref 5βTref

λ, μ (Pa) � Lame functions; α (K21) � linear thermal expansion coefficient;

Tref (K) � reference temperature; β (K21) � volume thermal expansion coefficient.

In an elastic environment, it expresses the ratio of the thermal deformation prop-

agation velocity to the electromagnetic waves propagation velocity. It characterizes

the velocity relation between thermal and mechanical deformation energy propaga-

tion. With α5 0, the thermal and deformation processes do not depend on each

other. It enables the solution of temperature rise connected to the body deformation

in tasks on thermal energy dissipation.

Info: [A23].

Joseph Louis Gay-Lussac (p. 15).

4.1.34 Thermoelastic Coupling Number Resultant Nυ

Nυ 5N1N2 5ð3λ1 2μÞ2α1α2Tref

ðλ1 2μÞcR 5Eα1βTrefð12 2νÞcR 5

3Kα1βTrefcR


λ, μ (Pa) � Lame functions; α1, α2 (K21) � linear thermal expansion coefficient

in system 1 and system 2; Tref (K) � reference temperature; cR (J m23 K21) � spe-

cific volume heat; E (Pa) � modulus of elasticity; β (K21) � volume thermal

expansion coefficient; K (Pa) � volume modulus of elasticity; ν (�) � Poisson’s

ratio (p. 143); N1 (�) � thermoelastic coupling number (1.) (p. 146); N2 (�) �thermoelastic coupling number (2.) (p. 146).

It characterizes a compound thermal and thermal stress process in linear sys-

tems. The independence of both processes is expressed by the value of Kυ, because

α15 0 or the reference temperature Tref5 0.

Info: [A23].

4.1.35 Thermoelastic Deformation Nε

Nε 5ε

αTrefð1Þ;

Nε 5ε

ð11 νÞαTrefð2Þ;

Nε 5ð12 νÞε

ð11 νÞαTrefð3Þ

α (K21) � linear thermal expansion coefficient; Tref (K) � reference temperature;

ν (�) � Poisson’s ratio (p. 143); ε (�) � relative elongation (p. 144);

This quantity expresses the relative thermoplastic deformation with single axis (1),

planar (2) and space (3) thermal stresses.

Info: [A23].

4.1.36 Thermoelastic Displacement Nu

Nu 5u

αTrefLð1Þ;

Nu 5u

αLTrefð11 νÞ ð2Þ;

Nu 5uð12 νÞ

ð11 νÞαTrefLð3Þ

u (m) � displacement; α (K21) � linear thermal expansion coefficient; Tref (K) � ref-

erence temperature; L (m) � characteristic length; ν (�) � Poisson’s ratio (p. 143).

It characterizes the dimensionless thermoelastic displacement with single axis (1),

planar (2) and space (3) thermal stresses.

Info: [A23].

147Solid Mechanics

4.1.37 Thermoelastic Displacement Potential NΦu

NΦu 5Φ

βTrefL25

ð12 νÞΦð11 νÞαTrefL2

5ðλ1 2μÞΦ

ð3λ1 2μÞαTrefL2

Φ (m2) � thermoelastic potential of the displacement; β (K21) � volume thermal

expansion coefficient; Tref (K) � reference temperature; L (m) � characteristic

length; α (K21) � linear thermal expansion coefficient; λ, μ (Pa) � Lame func-

tions; ν (�) � Poisson’s ratio (p. 143).

It characterizes the dimensionless thermoelastic displacement potential.

Info: [A23].

4.1.38 Thermoelastic Potential NΦ

NΦu 5ΦuL

ð1Þ;

NΦε 5ΦεL2

ð2Þ;

NΦσ 5GΦσL2

ð3Þ

Φ (m2) � thermoelastic potential of the displacement; u (m) � displacement; L (m)�characteristic length; G (Pa) � shear modulus; σ (Pa) � stress; ε (�) � relative elon-

gation (p. 144).

It characterizes the dimensionless thermoelastic potential related to displacement (1),

deformation (2) or stress (3).

Info: [A23].

4.1.39 Thermoelastic Stress Function NF

NF 5ð12 νÞFEαTrefL2

F (N) � thermoelastic stress function; E (Pa) � modulus of elasticity; α (K21) lin-

ear thermal expansion coefficient; Tref (K) � reference temperature; L (m) charac-

teristic length; ν (�) � Poisson’s ratio (p. 143).

In the dimensionless form, it expresses the thermoelastic stress function.

Info: [A23].


4.1.40 Thermoelastic Stress Number Nσ

Nσ 5σ

αETrefð1Þ;

Nσ 5σð12 νÞαETref

ð2Þ;

Nσ 5σk

αEΔTlsð3Þ

σ (Pa) � stress; α (K21) � linear thermal expansion coefficient; E (Pa) � modulus

of elasticity; Tref (K) � reference temperature; σk (Pa) � yield limit; ΔTls (K) �temperature difference between liquidus and solidus; ν (�) � Poisson’s ratio

(p. 143).

In the dimensionless form, this number characterizes the thermoelastic stress for

planar (1) and space (2) linear stresses. In expression (3), it is about the criterion of

internal residual thermoelastic stress.

Info: [A23].

4.1.41 Viscous Damping Coefficient Nd, .

Nd 5kdffiffiffiffiffiffiffiffiksm

p

kd (kg s21) � coefficient of viscous damping; ks (kg s22) � stiffness of elastic com-

ponent, spring constant; m (kg) � mass.

In the dynamics of mechanical sets, it expresses the ratio of the viscous damping

force to the elastic element stiffness.

4.2 Fracture Mechanics and Micromechanics

In fracture mechanics, the physical similarity criteria are focused to prevent struc-

ture defects from originating, especially those which could cause cracks and mate-

rial fractures. They characterize the material resistance. This relates to the

elastoplastic and/or fully plastic material loading conditions. Well-known dimen-

sionless numbers include the Fick number, resistance against crack generation,

fracture strength and the crack propagation number.

149Solid Mechanics

4.2.1 Crack Filling Number N

N5pmτνR

5EuFoPrRe2

pm (Pa) � metallostatic pressure; τ (s) � local solidification time; ν (m2 s21) �kinematic viscosity; R (kg m23) � density; Eu (�) � Euler number (1.) (p. 61); Fo

(�) � Fourier number (p. 175); Pr (�) � Prandtl number (p. 197); Re (�)

Reynolds number (p. 81).

It characterizes the possibility to fill the crack with residual interdendrite molten

material or alternatively to liquefy the existing crack. It is the fracture mechanics

criterion.

Info: [A23].

4.2.2 Crack Propagation Number, Ice Number NCR

NCR 5Rw2

E

� �2ffiffiffiffiffiffiEL

R

r; where R5

πσ2a

E

R (kg m23) � density; w (m s21) � crack propagation velocity; E (Pa) � modulus

of elasticity; L (m) � characteristic length; R (kg s21) � mechanical resistance;

σ (Pa) � stress; a (m) � half-length of a crack.

It is the extension of the Cauchy number (aeroelasticity parameter) (p. 155) in

the fracture mechanics. It enables one to follow the evolution of crack propagation

and the influence of resistance on crack freezing.

Anthony G. Atkins, American engineer.

Robert M. Caddell, American engineer.

4.2.3 Energy Release Number Nσ,res

Nσ;res 5NF

qEf

σ2resh

; where Ef 5E

ð12 ν2Þ

NF (�) � acting force on the notch forehead; q (N m21) � length loading; Ef (Pa) �plane elasticity modulus of layer and substrate; σres (Pa) � residual stress; h (m)

distance in the layer thickness; E (Pa) � modulus of elasticity; v (�) � Poisson’s

ratio (p. 143).

It expresses the inner-to-outer stress ratio created due to loosening in the loca-

tion of a longitudinal notch, crack or cutting in a surface layer.

Info: [B50].


4.2.4 Fick Number Fi

Fi � Fo21D 5

L2

Dτ

L (m) � characteristic length; D (m2 s21) � diffusivity; τ (s) � time; FoD (�) �Fourier mass number (p. 249).

It characterizes the diffusion intensity of the elements in the solid phase in rela-

tion to the distance of primary or secondary dendritic axes and to the local solidifi-

cation time. With increasing Fi, the mixing off of the elements during

solidification rises, the solidification interval increases and the trend to cracks

between dendrites increases (e.g. in steel especially). Of the metallurgic factors, the

steel temperature magnifies the value of the Fi. By refining the dendrite structure,

for example, by modifying the steel at the end of the reduction time in a furnace or

in a pan in tapping, the unfavourable influence of the number Fi can be limited

substantially. It is a criterion of fracture mechanics.

Info: [A23].

Adolf Eugen Fick (3.9.1829�21.8.1901), German

physiologist.

After an initial study of mathematics, he devoted himself

primarily to physiology. He formulated the diffusion laws

and an equation for neutral particles (1855). With this he laid

an important base for subsequent research of diffusion pro-

cesses, which led later to contact lens development, among

others. In addition, he was engaged in studying the activity

of the sense organs and in examining electric signals under

the excitation of nerves.

4.2.5 Fracture Strength Nf

Nf 5σfrac

σth

5 60

ffiffiffiffiffiffiσEh

r

σfrac (Pa) � fraction stress of material; σth (Pa) � theoretical stress of material

(σth5E/30); σ (N m21) � surface stress; E (Pa) � modulus of elasticity; h (m) �thickness of material.

This number characterizes fraction stress in nanomaterials. Fracture mechanics

and nanomaterials.

Info: [B38].

4.2.6 Interval of Solidification N

N5ΔTls

Tc

ΔTls (K) � temperature interval of solidification; Tc (K) � casting temperature.

151Solid Mechanics

It expresses the relation of the thermal solidification interval to the pouring tem-

perature. It characterizes the material trend to cracks between the dendrites during

solidification. With optimal pouring temperature, the tendency increases with the

solidification interval. Then, the crack tendency increases with the increasing crite-

rion value. It is a fracture mechanics criterion.

Info: [A23].

4.2.7 Metallostatic Pressure N

N5pm

gRL5EuFr

pm (Pa) � metallostatic pressure; g (m s22) � gravitational acceleration;

R (kg m23) � density; L (m) � characteristic length; Eu (�) � Euler number (1.)

(p. 61); Fr (�) � Froude number (1.) (p. 62).

It characterizes the ratio of the metallostatic pressure in a critical place of a cast-

ing to its characteristic wall thickness. The criterion application is joined with the

building of risers on castings. The greater its value, the less the probability of a

crack originating. For certain characteristic casting thicknesses, the tendency to cre-

ate interdendrite cracks can be suppressed by increasing the metallostatic height

(e.g. riser heightening, riser filling or overpressure riser application). It is a fracture

mechanics criterion.

Info: [A23].

4.2.8 Microcrack Initial Number Ncrini

Ncrini 5FiniEf

σ2A

Fini (N) � acting dynamic force; Ef (Pa) � elasticity modulus of thin layer; σ (Pa) �tension stress in thin layer; A (m2) � opening surface in the crack size of surface

layer.

This number characterizes the crack evolution in a thin surface layer or on an

elastic substrate or a viscous sublayer. It expresses the relation between an outer

initializing force acting on the layer and an internal stress force in the layer. For

example, it relates to electronic parts such as solar cells and flat panel display tech-

nology. Microelectronics in macroelectronic systems. Fracture mechanics.

Info: [B110].

4.2.9 Microcrack Steady Number Ncrst

Ncrst 5FstEf

σ2A


Fst (N) � steady dynamic force; Ef (Pa) � elasticity modulus of thin layer; σ (Pa) �tension stress in thin layer; A (m2) � opening surface in the crack size of sur-

face layer.

It characterizes a steady crack in a thin surface layer or on an elastic substrate

or a viscous sublayer. It expresses the relation between an outer steady-state force

acting on a layer and an internal stress force in a layer. Microelectronics in macro-

electronic systems. Fracture mechanics.

Info: [B110].

4.2.10 Relative Fracture Strain εfrac

εfrac 5 2

ffiffiffiffiffiffiEs

Eh

r

Es (J m22) � density of surface energy; E (Pa) � modulus of elasticity; h (m) �

depth of the crack opening.

It characterizes the fracture deformation in nanomaterials. Fracture mechanics.

Nanotechnology.

Info: [B38].

4.2.11 Relative Strain Index Pε

Pε 5εyieldεfrac

εyield (�) � yield strain; εfrac (�) � fracture strain.

This index expresses the relation of a relative yield point deformation to its

material destruction value. Fracture mechanics.

4.2.12 Resistivity Against Crack Generation (1.) N

N5pmL

νRgτ5EuFrReSh21

pm (Pa) � metallostatic pressure; L (m) � critical defect size; ν (m2 s21) � kinematic

viscosity; R (kg m23) � density; g (m s22) � gravitational acceleration; τ (s) �solidification time; Eu (�) � Euler number (1.) (p. 61); Fr (�) � Froude number

(1.) (p. 62); Re (�) � Reynolds number (p. 81); Sh (�) � Strouhal number (p. 87).

It characterizes the resistivity against crack generation in a solidifying casting. It

is a fracture thermomechanics criterion.

Info: [A23].

153Solid Mechanics

4.2.13 Resistivity Against Crack Generation (2.) N

N5σ

νRgτ5 LeFrðFoWeScÞ21

σ (Pa) � surface stress; ν (m2 s21) � kinematic viscosity; R (kg m23) � density;

g (m s22) � gravitational acceleration; τ (s) � solidification time; Le (�) � Lewis

number (p. 254); Fr (�) � Froude number (1.) (p. 62); Fo (�) � Fourier number

(p. 175); We (�) � Weber number (p. 389); Sc (�) � Schmidt number (p. 263).

It characterizes the resistivity against crack generation as a function of the

molten stuff surface energy, viscosity and solidification time. It is a fracture ther-

momechanics criterion.

Info: [A23].

4.2.14 Strain at Fracture εfrac

εfrac 53fr sin α

L2

f (m) � deflection of the lateral twig at breakage; r (m) � radius of the lateral

twig, distance from the neutral plane; α (�) � angle just before breakage; L (m) �distance from the twig base to the point where the force acts.

It expresses the deformation at the point where the fracture arises. Fracture

mechanics.

4.2.15 Stress Index Pσ

Pσ 5σyield

σfrac

σyield (Pa) � yield stress; σfrac (Pa) � fraction stress.

It expresses the ratio of the yield point stress to the material destruction stress.

Fracture mechanics.

4.3 Aeroelasticity

In aeroelasticity, the dimensionless quantities express the influence of the interac-

tion between inertia, elasticity and aerodynamic forces. In static aeroelasticity, the

dimensionless quantity relates to the problems of divergence and regressive control

of a bypassed surface. In dynamic aeroelasticity, the dimensionless quantity relates

to the problems of fluttering and the dynamic response of mechanical parts which

are bypassed. The aeroelasticity parameter, Cauchy’s, Connor’s, Frueh’s, Regier’s


and other numbers are among the well-known dimensionless quantities in

aeroelasticity.

4.3.1 Aeroelasticity Parameter Ae

See the Cauchy number (aeroelasticity parameter) Cau (p. 155).

Info: [A29].

4.3.2 Aeroelasticity Stiffness NAE

NAE 52E

Rw2

E (Pa) � modulus of elasticity; R (kg m23) � density; w (m s21) � flow velocity.

It expresses the ratio of the structure stiffness to the aerodynamic force. It is the

inverse value of the aeroelasticity parameter AE (p. 155). Aeroelasticity.

4.3.3 Cauchy Number (Aeroelasticity Parameter) Cau

Cau � M2 � Ho5R w2

Kð1Þ;

Cau5R w2

Eð2Þ

R (kg m23) � density; w (m s21) � velocity; K (Pa) � volume modulus of elastic-

ity; E (Pa) � modulus of elasticity; M (�) � Mach number (p. 73); Ho (�) �Hooke number (p. 138).

This parameter expresses the ratio of the inertia force to the material compress-

ibility force. It characterizes the compressible fluid flow (1) and the dynamic mate-

rial strain (2) by inertia forces. It is often called the aeroelasticity parameter as

well. Aerodynamics, aeroelasticity, dynamics.

Info: [A23],[A43],[B127],[C37].

Augustin Louis Cauchy (21.8.1789�23.5.1857), French

mathematician.

He was known for his precision and consistency in mathe-

matics. He introduced many concepts such as the determi-

nant, limit, continuity and convergence. He founded complex

analysis and deduced the Cauchy�Riemann conditions with

Riemann. He was very prolific, publishing nearly 800 works.

155Solid Mechanics

4.3.4 Connors Number Con

Con5w

fL5K

ffiffiffiffiffiffiffiffimδRL2

s

w (m s21) � mean velocity of two-phase flow; f (s21) � oscillation frequency;

L (m) � characteristic length, pipe diameter; K (�) � instability factor;

m (kg m21) � length density of pipe material; δ (�) � damping decrement;

R (kg m23) � fluid density.

It relates to the fluid elastic vibration characteristics in the two-phase flow of

fluids. For example, it relates to tubes exposed to a dynamic two-phase flow of air

and water in heat exchangers. The number Con serves to determine the critical

flow rate and is based on the mean flow rate, mean fluid density and damping in a

two-phase flow. Aeroelasticity, two-phase flow, heat exchangers.

Info: [G28].

H. J. Connors, American scientific researcher.

Together with P.M. Moretti, Connors is among the significant personalities in

the aeroelasticity field, in dynamic induced oscillation and fluttering.

Peter M. Moretti, American scientist.

His scientific work is very wide-ranging and involves the

dynamics of mechanical systems, especially those of induced

vibrations excited by fluid streaming. Primarily, this relates

to the fluid stream dynamic acting on the nest of tubes in

heat exchangers, further about the instability solution in

transversal streaming through these nests and the eigenfre-

quency determination and damping of nests. He was also

engaged in research related to thermal stratification in lakes,

in supersonic streaming and heat transfer problems.

4.3.5 Flutter Number F

F5weq

wR

5M

Rg

weq (m s21) � equivalent air velocity near the sea level; wR (m s21) � Regier sur-

face velocity index (flutter parameter); M (�) � Mach number (p. 73); Rg (�) �Regier number (p. 157).

High velocities aerodynamics. Aeroelasticity. Flutter.

Info: [B94].

4.3.6 Frequency Parameter Pf

Pf 5ωLw

5 2πSh


ω (s21) � angular velocity; L (m) � characteristic length (e.g. channel width or dis-

tance from a wall); w (m s21) � flow velocity; Sh (�) � Strouhal number (p. 87).

It characterizes the unsteady flow in bundles, fluidization and other dynamic

processes acting on mechanical systems, as an example.

Info: [A29],[A35].

4.3.7 Frueh Number Fh

Fh5Lω1

a

ffiffiffiffiffiffiNm

cL

r

L (m) � characteristic length, (e.g. half-length of wing chord); ω1 (s21) � first har-

monic frequency of torsional vibration; a (m s21) � sound velocity; cL (�) � uplift

curve inclination; m (kg m21) � mass to the wing length; R (kg m23) � air density;

Nm (�) � mass ratio (p. 75).

Aeroelasticity, transonic flutter of wings.

Info: [A24].

Frank J. Frueh, American aerodynamic engineer.

4.3.8 Mass Ratio Nm

Nm 5m

πRL3

m (kg) � mass of body; R (kg m23) � fluid density; L (m) � characteristic length

of a body.

Mechanical systems dynamics. Aircraft fluttering and stability. Aeroelasticity.

Info: [A24].

4.3.9 Regier Number Rg

Rg5Lωa

ffiffiffiffiffiffiffiffiffiffiRl

πRL2

s

L (m)� characteristic length; ω (s21)� angular velocity; a (m s21) � sound velocity;

Rl (kg m23) � fluid density; R (kg m23) � material density.

It expresses aeroelastic relations, transonic flutter of wings and blades of flow

machines. It is a modification of the Strouhal number Sh (p. 87). Aeroelasticity.

Info: [A24].

157Solid Mechanics

4.3.10 Regier Surface Number Rgs

Rgs 5wR

a

wR (m s21) � Regier surface velocity index (flutter parameter); a (m s21) � sound

velocity.

It is a simpler expression of the Regier number Rg (p. 157) and represents the

elastic-to-aerodynamic forces ratio at sea level. High velocity aerodynamics.

Aeroelasticity. Flutter.

Info: [A24],[B94].

4.4 Tribology

The tribologic dimensionless quantities involve especially the subjects of friction,

lubrication and wearing. Usually, they express the interaction of surfaces in their

relative movement. For example, they relate to the friction and lubrication of bear-

ings and the friction and wear of tools in machining. The dimensionless quantities

especially apply to new micro- and nanotechnologies, in which the tribological pro-

cesses represent a basic problem. Among the well-known tribologic dimensionless

quantities are Gumbel’s, Hersey’s and Sommerfeld’s numbers, as well as wear

intensity and friction factor.

4.4.1 Adhesion Parameter Nadh,Θ

Nadh 5Eh

32

r12aΔσ

; where Δσ5 σ1 1σ2 2σ1;2

E (Pa) � modulus of elasticity; h (m) � standard deflection of peak heights; ra (m) �curve radius of surface roughness; Δσ (N m21) � Dupre adhesion, adhesion surface

stress; σ1, σ2, σ1,2 (N m21) � surface stress of two spheres and their boundary line.

It expresses the ratio of the elastic energy to the adhesion work, provided a con-

tact has occurred. With Nadhc1; there is a partial contact in which the elastic mate-

rial has contact only on the highest roughness peaks. A full contact occurs for

Nadh{1: The parameter is based on the Gauss transversal roughness distribution on

the elastic material surface. Micro and nanotechnology. Tribology.

Info: [B126].

4.4.2 Coefficient of Bearing Friction f

f1 5 2π2 ηnp

r

cð1Þ


f2 5η n

pð2Þ

η (Pa s) � dynamic viscosity; n (s21) � rotational speed; p (Pa) � pressure induced

by weight or loading force on bearing surface (2rl ); r (m) � radius of shaft journal;

l (m) � length of shaft journal; c (m) � angular clearance.

Expression (1) serves to estimate the bearing friction or the energy loss in a

bearing approximately. For stable lubrication (boundary, hydrodynamic or mixed),

relation (2) is valid. For fully hydrodynamic lubrication in a plain bearing, it is

ηnp

$ 1:73 1026

The course of the friction coefficient dependence on the Sommerfeld number Sm

(p. 385), expressed with the Stribeck curve, can be replaced by three zones �boundary, mixed and hydrodynamic lubrication � in each of which the effective

coefficient value can be determined.

Sm{1 fef 5 fs ðboundary lubricationÞ

Sm � 1 fef 5 fsð12 SmÞ1 SmhL

ðmixed lubricationÞ

Smc1 fef 5 SmhL

ðhydrodynamic lubricationÞ

Tribology, plain bearings.

4.4.3 Coefficient of Kinetic Friction fk,μk

fk 5F

Fn

ð1Þ;

fk # fs ð2Þ

F (N) � friction force; Fn (N) � normal force pressing friction surfaces together;

fs (�) � coefficient of static friction (p. 160).

It expresses the friction in movement. In contrast to the coefficient of static fric-

tion (p. 160), its size depends on the surface roughness and conditions thereof. In

general, expression (2) holds for the rate range of cm s21 to m s21, and it is sup-

posed to approximate a constant. For comparison, for an iron casting, for example,

the coefficient is fs5 1.1, fk5 0.15; for steel/bronze, it is fs5 0.51, fk5 0.44; for

steel/steel, it is fs5 0.74, fk5 0.57. Mechanics. Tribology.

159Solid Mechanics

4.4.4 Coefficient of Static Friction fs,μs

fs 5Ff

Fn

Ff (N) � friction force; Fn (N) � normal force pressing standing friction surfaces

together.

This coefficient expresses the friction of solid bodies at a standstill. It depends

on the surface roughness and is greater than the coefficient of kinetic friction fk(p. 157), generally ( fk , fs ). The rolling coefficient fr belongs here, which occurs

in rolling operations, such as wheel contact with the rolling surface of vehicles.

It expresses the friction of solid bodies at a standstill. Its size does not depend

on the contact surface area. The static friction force is a reaction the size of which

depends on the size of some outer forces tending to shift the surfaces mutually.

The static friction force rises to the maximum value (1), with which the object

starts to move. The static friction coefficient is equal to or greater than the kinetic

friction coefficient (2). For example, for a few materials and dry friction the values

are as follows: for wood/wood fs A h0.25; 0.5i, for steel/steel fs A h0.5; 0.8i, foraluminium/steel fs A h0.6; 0.8i, for glass/glass fs A h0.9; 1.0i. The oxidation sur-

face cover strongly influences this coefficient, for example, for clean surfaces of

steel/steel fs5 0.78; for an oxidized surface fs5 0.27; and for a sulfidated surface

fs5 0.39. A survey of the static and sliding friction coefficient values is given in

[C103] for various combinations of materials. Mechanics. Tribology.

Info: [C57].

4.4.5 Critical Frictional Temperature Θf

Θf 5η2wsut

L2σ2f

ð1Þ;

Θf 5η2wsut

L2σ2m

ð2Þ

η (Pa s) � dynamic viscosity at the oil temperature in input to the sliding part;

ws (m s21) � sliding velocity; ut (m s21) � peripheral velocity; L (m) � character-

istic length (disc radius); σf (Pa) � contact stress considering the surface rough-

ness; σm (Pa) � maximum contact stress according to Hertz definition at arbitrary

loading and without seizing.

It characterizes the heating and seizing of discs in rolling movement and lubrica-

tion. The expressions (1) and (2) are linearly dependent, approximately.

Info: [B16].


4.4.6 Dimensionless Film Pressure NP; p

NP 5p

pHZ5

πbLp2F

; where pHZ5

ffiffiffiffiffiffiffiffiffiffiffiFE

2πLR

r5

2F

πbL

p (Pa) � oil-film pressure; pHZ (Pa) � maximal dry (Hertzian) friction in contact;

b (m) � half width of dry contact; L (m) � characteristic length (bearing width);

F (N) � force loading of bearing; E (Pa) � reduced modulus of elasticity; R (m) �reduced curve radius.

It expresses the dimensionless oil-film pressure.

Info: [B117].

4.4.7 Dimensionless Film Thickness h

h5πLEh4F

; where E5 212 ν21E1

112 ν22E2

� �

L (m) � characteristic length (bearing width); E (Pa) � reduced modulus of elastic-

ity; h (m) � film thickness; F (N) � force loading of bearing; E1, E2 (Pa) � elas-

ticity modulus of bearing parts; ν1, ν2 (�) � Poisson’s ratio (p. 143) of bearing

parts.

The dimensionless oil-film thickness. Tribology, plain bearings.

Info: [B117].

4.4.8 Drag Friction Factor fD

fD 5Ff 1Fd

12ApRw2

N

Ff (N) � friction force; Fd (N) � drag force; Ap (m2) � cross-section area;

R (kg m23) � fluid density; wN (m s21) � free flow velocity.

This factor characterizes friction under the action of dynamic drag and friction

forces. Fluid mechanics. Tribology.13

4.4.9 Duty Cycle Group NC

NC 524η0TE

π2

� �32 ELR

F

� �32

η0 (Pa s) � dynamic viscosity at normal conditions; T (s) � period of load cycle;

E (Pa) � reduced modulus of elasticity; L (m) � characteristic length (bearing

width); R (m) � reduced curve radius; F (N) � force bearing load.

161Solid Mechanics

It expresses the influence of the bearing duty cycle on the thickness and pressure

of an oil film in a bearing. The elastohydrodynamic lubrication is considered.

Tribology, lubrication, plain bearings.

Info: [B117].

4.4.10 Dynamic Force NFd

NFd 5Rw2E

σ2P

R (kg m23) � density; w (m s21) � sliding velocity; E (Pa) � modulus of elasticity;

σP (Pa) � tensile strength.

It expresses the ratio of the dynamic loading force to the material strength force.

Together with the static force NFs (p. 170), it represents two basic parameters influ-

encing the coefficient size of the friction between a rotating wheel and a support

(e.g. road or rail).14

4.4.11 Energetic Wear Intensity NE

NE 5ΔV

FLλ

ΔV (m3) � volume of wearout layer; F (N) � friction force; L (m) � length of

sliding; λ (J m23) � volume density of deformation energy.

It expresses the ratio of the energy consumed for wearing to the friction energy.

Tribology, material wearing.

4.4.12 Euler Number (2.) Eu2

Eu2 52dp

dl

d

Rw25 2fF

dp

dl(Pa m21) friction pressure gradient; d (m) � pipe diameter; R (kg m23) � fluid

density; w (m s21) � flow velocity; fF (�) � Fanning friction number (p. 163).

This number expresses the fluid friction if flowing through a pipeline.

Info: [A35].

Leonhard Euler (p. 61).

4.4.13 Fanning Number Fa

Fa52 τR w2

τ (Pa) � shear stress; R (kg m23) � fluid density; w (m s21) � flow velocity.


It expresses the ratio of the shear stress to the fluid dynamic pressure. It charac-

terizes the friction loss for flow around bodies or in a pipeline. See also the

Fanning friction number fF (p. 163).

Info: [B20].

John Thomas Fanning (1837�1911), American engineer.

4.4.14 Fanning Friction Number fF, cF

fF 52Δp

Rw2

DH

L5

fD

4

Δp (Pa) � pressure difference; R (kg m23) � fluid density; w (m s21) � flow veloc-

ity; DH (m) � hydraulic diameter; L (m) � characteristic pipe length; fD (�) �Darcy friction number (p. 119).

This number characterizes the friction in turbulent isothermic flow in a bare

pipeline. In the range of the Reynolds number (p. 81) Re A h53 103; 2003 103i,the value fF is given by the relation fF5 0.046 Re20.2.

Info: [A2],[B17].

John Thomas Fanning (see above).

4.4.15 Flexible Number NF, F

F51

Ered

R

α4ηus

� �13

Ered (Pa) � reduced modulus of elasticity; R (m) � effective radius curve;

α (m2 N21) � piezoviscosity coefficient; η (Pa s) � input dynamic viscosity;

us (m s21) � sum of tangential surface velocities.

It influences the minimum thickness of a bearing film. Elastohydrodynamic

lubrication of a bearing.

4.4.16 Friction Factor f

f 52dp

dxDH

1

2Rw2

� �21

2dp

dx(Pa m21) � pressure drop per unit length; DH5 4 SO21 (m) � hydraulic diam-

eter; S (m2) � surface area; O (m) � perimeter; R (kg m23) � fluid density;

w (m s21) � fluid density.

163Solid Mechanics

It represents the general definition of the friction factor. The Fanning friction

number fF (p. 163) and the Darcy friction number fD (p. 119) are practical expres-

sions thereof.

Info: [C29].

4.4.17 Gumbel Number (1.) Gu

Gu5Fh2

2ηuR2

F (N m21) � force on length of bearing; h (m) � angular clearance; η (Pa s) �dynamic viscosity; u (m s21) � peripheral velocity; R (m) � shaft radius.

It expresses the ratio of the bearing loading force to the dynamic friction force.

Tribology, lubrication, bearings.

Info: [A29].

Ludwig Karl Friedrich Gumbel (1874�1923), German naval engineer.

4.4.18 Gumbel Number (2.) Gu

Gu5ηωD2

F

η (Pa s) � dynamic viscosity; ω (s21) � angular velocity; D (m) � diameter of

bearing journal; F (N) � loading force.

It expresses the ratio of the dynamic friction force to the loading force of a plain

bearing. Tribology, lubrication, bearings.

Info: [A29].

Ludwig Karl Friedrich Gumbel (see above).

4.4.19 Harrison Number Hr

Hr56ηuLpaδ2

η (Pa s) � dynamic viscosity; u (m s21) � peripheral velocity; L (m) � length of

bearing pad in direction of motion; pa (Pa) � ambient of inflow pressure; δ (m) �bearing film thickness by output.

It expresses the ratio of the dynamic friction force to the static pressure force in

loading a bearing. Tribology. Lubrication. Bearings.

Info: [A29].

William John Harrison (1884�1969), English mathematician.


4.4.20 Hersey Number He

He5FL

ηað1Þ;

He5F

ηuLð2Þ;

Fl (N m21) � length force loading (bearing); η (Pa s) � dynamic viscosity;

a (m s21) � sound space; F (N) � load force; u (m s21) � peripheral velocity;

L (m) � bearing length.

It expresses the loading-to-viscosity forces ratio and characterizes the force rela-

tion in tribology (e.g. for bearing loading and lubrication).

Info: [A29],[B20].

Mayo Dyer Hersey (born 1886), American engineer.

4.4.21 Hydrodynamic Load Group NH

NH 5 3π2 η0uER

ELR

F

� �2

η0 (Pa s) � dynamic viscosity at normal conditions; u (m s21) � peripheral velocity

E (Pa) � reduced modulus of elasticity; R (m) � reduced radius curve; L (m) �characteristic length (bearing width); F (N) � bearing load.

It expresses the hydrodynamic bearing load influence on the thickness and pres-

sure of the oil film in a bearing. Elastohydrodynamic lubrication. Tribology, lubri-

cation, plain bearings.

Info: [B117].

4.4.22 Mass Wear Intensity Nm

Nm 5Δm

LAR

Δm (kg) � mass of wearout layer; L (m) � length of sliding; A (m2) � nominal

area; R (kg m23) � material density.

It represents the mass wear of friction surfaces. Tribology, material wearing.

4.4.23 Material Abradability NA

NA 5ΔV

fΔmL5

kAQA

f

165Solid Mechanics

ΔV (m3) � volume wearout of material by abrasion; f (�) � coefficient of sliding

friction; Δm (kg) � volume wearout of material by sliding; L (m) � length of

sliding; kA (m2 kg�1) � abrasivity factor; QA (kg m22) � surface loading limit.

It expresses the ratio of the volume decrease due to abrasion, to the mass loss

due to sliding friction. Tribology, material wear.

4.4.24 Material Wear Resistance NR

NR 5N 21A 5

f

kAQA

f (�) � coefficient of sliding friction; kA (m2.kg21) � abrasivity factor;

QA (kg m22) � surface loading limit.; NA (�) � material abradability (p. 165).

It expresses the material wear resistance. It is the inverse value of the material

abradability NA (p. 165). Tribology, material wear.

Info: [B92].

4.4.25 Mechanical Severity of Contact MS

MS5ð11 10 fsÞσmax

ffiffiffid

p

K

σmax (Pa) � maximum occurring Hertzian stress18; d (m) � crack length;

K (m21/2 kg s22) � fracture toughness of the material; fs (�) � coefficient of static

friction (p. 160).

It expresses the contact mechanical resistivity under force loading. Tribology.

Info: [B75].

4.4.26 Minimal Film Thickness NH,H

HH 5 hmin

1

α2η2u2sR

hmin (m) � minimal film thickness; α (m2 N21) � piezoviscosity coefficient;

η (Pa s) � input dynamic viscosity; us (m s21) � the sum of tangential surfaces

velocities; R (m) � effective curve radius.

It expresses the minimum thickness of a lubricating film. In general, it increases

with increasing flexibility, though this is not valid in some applications.

Elastohydrodynamic lubrication, bearings.

Info: [B31].


4.4.27 Moody Friction Factor fM

fM 5 fD 5 4fF

FD (�) � Darcy friction number (p. 119); fF (�) � Fanning friction number

(p. 169).

It corresponds to the Darcy friction number and fourfold of the Fanning friction

number (p. 169).

4.4.28 Piezoviscous Number NL , L

NL 5αηusE3

red

R

� �14

α (m2 N21) � piezoviscosity coefficient; η (Pa s) � dynamic viscosity; us (m s21) �the sum of tangential surface velocities; Ered (Pa) � reduced modulus of elasticity;

R (m) � effective curve radius.

This number expresses the influence of the lubricating film physical properties,

such as viscosity, velocity, shape geometry and elasticity in the sliding lubrication

process. Electrodynamic lubrication, plain bearings.

Info: [B77].

4.4.29 Reduced Frequency Nf,red

Nf ;red 5ωLc

ω (s21) � angular frequency; L (m) � characteristic length; c (m s21) � undisturbed

sound velocity.

It is the measure for the ratio of the gap width to the wavelength of sound propa-

gation. Tribology, bearings.

4.4.30 Relative Pipe Roughness εrel

εrel 5εD

ε (m) � mean height, roughness height of inner pipe surface; D (m) � inner pipe

diameter.

It is analogous to the roughness factor fr (p. 168). It serves to determine the fric-

tion factor. Usually, this is determined from the Moody diagram, which represents

167Solid Mechanics

its dependence on the Reynolds number Re (p. 81) and the relative pipeline rough-

ness. For example, for Re5 33 104 and εrel5 1022, the pipeline fracture coeffi-

cient is f5 0.02.

4.4.31 Relative Roughness εrel

εrel 5Ag

Arough

; where εrelAh0; 1i

Ag (m2) � geometric sample surface; Arough (m2) � surface determined by

roughness of the sample.

It expresses the relative planar roughness of a specimen. Tribology.

4.4.32 Reynolds Friction Number Rer,δ1

Reτ 5wτδν

; where wτ 5w0

ffiffiffifficf

2

r5

ffiffiffiffiffiffiτwR

r

wτ (m s21) � friction velocity; δ (m) � boundary layer thickness; ν (m2 s21) �kinematic viscosity; w0 (m s21) � reference velocity of the undisturbed flow in

pipes and channels; cf (�) � friction coefficient; τw (Pa) � shear stress on circum-

fluenced wall.

It characterizes the viscous fluid flow in a boundary layer close to walls, in a

pipeline and canals. Tribology.


4.4.33 Roughness Factor, Roughness Ratio fr

fr 5εL

ð1Þ;

fr 5Ar

Ag

ð2Þ

ε (m) � height of roughness; L � d (m) � characteristic length, pipe diameter;

Ar (m2) � real surface (interface) area ; Ag (m

2) � geometric surface (interface) area.

It expresses the ratio of the real interface height to the geometric one (1) or

the ratio of the real surface area (of the interface) to the geometric surface (2).

In the case of a circular cross-section pipeline, the diameter is the characteristic

dimension; the equivalent diameter is that for the non-circular one. It characterizes

the roughness influence of a bypassed surface on the friction of fluids.

Info: [A29].


4.4.34 Shear Wave Number Nws

Nws 5 L

ffiffiffiffiων

r

L (m) � characteristic length (thickness); ω (s21) � angular frequency; ν (m2 s21) �kinematic viscosity.

It expresses the inertia-to-friction forces ratio. For low values of the number, the

viscosity influence predominates. On the contrary, for high values of the number,

the inertia influence dominates. From the physical point of view, it represents the

ratio between the gap width and the boundary layer thickness. Tribology, bearings.

4.4.35 Skin Friction Coefficient, Wall Shear Stress Coefficient Cf

Cf 5τw

12Rw2

N

ð1Þ; where τw 5 ηdw

dyy50:��

Cf 50:664ffiffiffiffiffiffiffiRex

p ð2Þ

Cf 5 0:0576ffiffiffiffiffiffiffiRex

5p ð3Þ

τw (Pa) � shear stress on the wall (y5 0); R (kg m21) � fluid density; wN (m s21) �free fluid flow; η (Pa s) � dynamic viscosity; y (m) � coordinate perpendicular to

the wall surface; Rex (�) � Reynolds number.

It expresses the dynamic friction resistance originating in viscous fluid flow

around a fixed wall. The expression (2) is valid for laminar flow along a flat plate

(Blasius’ boundary layer) and the expression (3) holds for turbulent flow.

Tribology. Aerohydrodynamics.

Info: [A21].

4.4.36 Squeeze Number Sq

Sq512ηωpa

L

δ0

� �2

η (Pa s) � dynamic viscosity; ω (s21) � oscillation frequency; pa (Pa) � ambient

pressure; L (m) � characteristic length, radius; δ0 (m) � initial gap thickness.

It characterizes the damping of a thin compressible film squeezed between a

fixed and movable oscillating plate. It expresses the friction and damping condi-

tions in bearings and lubrication. Tribology.

Info: [A26],[B20].

169Solid Mechanics

4.4.37 Stanton Pannell Friction Factor fSP

fSP 51

2Fa

Fa (�) � Fanning number (p. 162).

It corresponds to half of the Fanning number (p. 162).

Info: [C49].

Thomas Edward Stanton (p. 201).

4.4.38 Static Force NFs

NFs 5FE

σ2P

F (N) � static force; E (Pa) � modulus of elasticity; σP (Pa) � tensile strength.

It expresses the ratio of the static loading force to the material strength force.

Together with the dynamic force NFd (p. 162), it represents two basic parameters

acting on the coefficient size of the friction between a rotating wheel and a support

(road, rail, etc.).

4.4.39 Thermal Severity of Contacts TS

TS5fsFu

ΔTsbλef

F (N) � contact load; u (m s21) � velocity;ΔTs (K) � temperature difference; b (m) �half width of the elliptic contact in the direction of sliding; λef (W m21 K21) � effective

thermal conductivity; fs (�)� coefficient of static friction (p. 160).

It characterizes the thermal resistivity of a shock loaded contact. Tribology.

Info: [B75].

4.4.40 Torque Frictional Coefficient CM

CM 52Mf

r3ω2AR

Mf (N m) � friction moment; r (m) � characteristic radius; ω (s21) � angular

velocity; A (m2) � surface; R (kg m23) � fluid density.

It expresses the ratio of a friction moment in rotation to the moment due to

dynamic force. Hydromechanics.

Info: [A8].


4.4.41 Transfer Number (4.) Friction Nf

Nf 5τwLwNη

τw (Pa) � shear stress on circumfluenced wall; L (m) � characteristic length

(e.g. radius of pipe, wall and curvature); wN (m s21) � free flow fluid velocity;

η (Pa s) � dynamic viscosity.

This number expresses the friction-to-viscous forces ratio or dimensionless sur-

face friction alternatively. Laminar flow in a boundary layer.

4.4.42 Wear Intensity PW

Pw5h

Lð1Þ;

Pw5ΔV

LAð2Þ

h (m) � thickness of wearout layer; L (m) � length of sliding; ΔV (m3) � volume

of wear layer; A (m2) � nominal area.

In expression (1), it expresses linear wear intensity. In expression (2), it

expresses volume intensity. Tribology, material wearing.

171Solid Mechanics

5 Thermomechanics

The theory becomes the more attractive the simpler are its premises and the

expressive is the heterogeneity of phenomena it involves and the wider is the area

of applicability thereof.Albert Einstein (1879�1955)

5.1 Heat Conduction

In a thermal system, the heat conduction manifests itself as a thermal field. The

thermal field can be steady or unsteady, linear or non-linear, with inner sources

or without them, with movable or stationary borders, with basic, composed or

combined boundary conditions, or with a phase conversion. Conjugate processes

of heat and mass transfers, heat and thermally induced stress, as well as non-

equilibrium heat propagation processes belong to this area. Dimensionless quanti-

ties also correspond to this classification. Among them are the following numbers:

the Biot, Brinkmann, Fourier, Kirpichev, Peclet, Pomerantsev and Stefan two-phase

conduction number.

5.1.1 Biot Number Bi

Bi5αLλ

ð1Þ;

Bi5αð2πfλcRÞ21=2 ð2Þ

α (W m22 K21) � heat transfer coefficient; L (m) � characteristic length; λ(W m21 K21) � thermal conductivity; f (Hz) � frequency; c (J kg21 K21) � spe-

cific heat capacity; R (kg m23) � density; V (m3), S (m2) � body volume and sur-

face; TS, TP (K) � temperature of the body and environment.

This number expresses the ratio of the heat flow transferred by convection on a

body surface to the heat flow transferred by conduction in a body. It characterizes

the third-type boundary condition. The equation Bi5 L (λα21)21 expresses the ratio

of a characteristic body length to an equivalent environmental thickness adhering

to the surface. With Bi$ 100 in the heat transfer on the surface, the thermal resis-

tance is slight in comparison to the heat transfer by conduction. The temperature



http://dx.doi.org/10.1016/B978-0-12-416013-2.00005-1

field is characterized by considerable non-uniformity. On the contrary, with

Bi# 0.1, difference (TP2 TS) is great on the body surface and slight inside the

body and the field is uniform. With the general shape of body, the ratio L5VS21 is

inserted for the characteristic length. With periodically changing heat transfer, unit

amplitude and frequency, the modified Biot number is used in expression (2).

Info: [A23],[A29],[A31],[A35],[B20].

Jean-Baptiste Biot (21.4.1774�3.2.1862), French physicist.

He was engaged in a wide range of physics � especially

in optics � specifically in polarized light, changes in polari-

zation and optical gas refractions. He also studied heat, con-

duction analysis and the external convection influence on

conduction. He was noted for requiring precision in mathe-

matical formulations and experiments. With Gay-Lussac, he

collaborated to design the first balloon for scientific purposes

(1804) and discovered the Biot�Savart law (1870) with

Felix Savart.

5.1.2 Brinkmann Number Br

Br5ηw2

λΔT5

ηw2

λðTS 2 TNÞ

η (Pa s) � dynamic viscosity; w (m s21) � flow velocity; λ (W m21 K21) � thermal

conductivity; ΔT (K) � temperature difference; TS (K) � surface temperature;

TN (K) � temperature in the thermal undisturbed fluid area.

It expresses the ratio of the heat arising due to viscous friction of a fluid to the

heat transferred by molecular conduction. It characterizes the heat conduction in

viscous fluid flow. For high fluid viscosity values and low thermal conductivity

values (e.g. molten polymers), the value is Brc1.

Info: [A23],[A29],[A31],[B20].

Henri Coenraad Brinkmann, German physicist.

5.1.3 Clausius Number Cl

Cl5w3LRλΔT

w (m s21) � velocity of fluid or solid body relative to undisturbed fluid; L (m) �characteristic length; R (kg m23) � density; λ (W m21 K21) � thermal conductivity;

ΔT (K) � temperature difference.


It is the ratio of the kinetic energy propagation rate to the heat propagation rate

by conduction. It characterizes the heat conduction in forced fluid flow or in a

moving environment generally. The characteristic length can be the canal width,

distance from the wall or fluid layer thickness.

Info: [A23],[A29],[A31],[A35],[B20].

Rudolf Julius Emanuel Clausius (2.1.1822�24.10.1888),

German mathematician and physicist.

He worked in the field of thermodynamics and formulated

the first and second thermodynamic theorems. The first is

about the equivalence of heat and work. By introducing the

entropy concept, he discovered that entropy can never drop

during physical processes, but can only stay constant in

reversal processes, as expressed by the second thermody-

namic theorem. Together with Maxwell, he formulated the

kinetic theory of gases.

5.1.4 Fourier Number Fo

Fo5aτL2

ð1Þ;

Fo5λLτcRL3

ð2Þ

a (m2 s21) � thermal diffusivity; τ (s) � time; L (m) � characteristic length;

λ (W m21 K21) � thermal conductivity; c (J kg21 K21) � specific heat capacity;

R (kg m23) � density.

It expresses the ratio of the time of a proceeding thermal process to that of the

molecular diffusion of the heat. In this time, expressed in a dimensionless way, the

relations between the thermal field change rate, physical parameters and thermal

system dimensions appear. In form (2), it expresses the ratio of the heat transferred

by conduction in a system to the heat accumulated in the system.

Info: [A23],[A31],[A19],[A35],[A43],[B20].

Jean Baptiste Joseph Fourier (21.3.1768�16.5.1830),

French mathematician and physicist.

With his work Theorie Analytique de la Chaleur

(Analytic Theory of Heat) of 1822, he created the founda-

tions for the mathematical heat conduction theory. First, he

formulated the important property of mathematical physics

equations: their dimensional homogeneity. His basic contri-

bution was very important for the theory of trigonometric

arrays and that of real variable functions.

175Thermomechanics

5.1.5 Heat Capacities Relation K

K5cR

ðcRÞref

cR, (cR)ref (J m23 K21) � specific volume heat capacity and its reference value.

It is the parametric criterion expressing the dimensionless material thermal

capacity of a body. It can be used in non-linear heat conduction. Usually cR 5c(T)R(T) is valid.

Info: [A23].

5.1.6 Heat Diffusivities Relation K

K5a

aref

a, aref (m2 s21) � thermal diffusivity and its reference value.

It is the parametric criterion expressing the dimensionless material thermal dif-

fusivity of a body. It can be used in non-linear heat conduction. Usually a5 a(T) is

valid.

Info: [A23].

5.1.7 Irregularity of Temperature Field Ψ

Ψ5Tp 2 Ts

Tp 2 Tmed

Tp (K) � temperature of external environment; Ts (K) � wall surface temperature;

Tmed (K) � mean integral volume body temperature.

It expresses the ratio of the environment and surface temperature difference of a

body to the difference between the environment temperature and the mean integral

volume temperature. It characterizes the temperature field non-uniformity in a

body.

Info: [A23].

5.1.8 Kirpichev Heat Number Ki

Ki � PoA 5qAL

λΔTð1Þ;

Ki5BiTS 2 TP

ΔTð2Þ


qA (W m22) � surface heat flux density; L (m) � characteristic length;

λ (W m21 K21) � thermal conductivity; ΔT (K) � reference temperature differ-

ence or temperature; TS, TP (K) � body surface and environment temperatures;

PoA (�) � Pomerantsev heat number (p. 181); Bi (�) � Biot number (p. 173).

This number expresses the ratio of the thermal heat flow surface density to the

density of the thermal heat flow conducted in a body. It characterizes the second-

kind boundary condition or, alternatively, a planar heat source.

Info: [A4],[A23],[A26],[A29],[A31],[A20],[B20].

Mikhail Viktorovich Kirpichev (1879�1955), Russian

engineer and physicist.

Using thermal similarity theory, he laid the foundations of

systematic modelling research for heat transfer, especially in

liquid metals, and elaborated the regular mode theory in ther-

mal systems with internal heat sources. He was engaged in

research of the heat transfer in boiling and condensation.

Essentially, he interpreted similarity theory as theory of an

experiment. His book Teorija podobija i rnodelirovanije

(The Theory of Similarity and Modeling, 1956) has substan-

tial significance.

5.1.9 Kondratiev Number Kd

Kd5mL2

að1Þ;

Kd5BiΨ5αλLΨ ð2Þ

m (s21) � unit speed of cooling; L5VS2 1 (m) � characteristic length dimension;

a (m2 s21) � thermal diffusivity; α (W m22 K21) � heat transfer coefficient;

λ (W m21 K21) � thermal conductivity; Bi (�) � Biot number (p. 173); Ψ (�) �irregularity of temperature field (p. 176).

It characterizes the regular state of the heat transfer between a fluid and a body

in which the temperature space distribution in the body preserves the thermal simi-

larity in time. In this state, the temperature is delayed regularly at an arbitrary point

of the body against the external environment temperature. It expresses the intensity

of mutual action of the body surface with the surrounding environment and, further,

the non-uniformity of the body temperature field in the regular state. With uniform

temperature distribution (Bi-0, for Bi, 0.1, essentially), Ψ5 1 and Kd5Bi are

valid. On the contrary, with the greatest thermal field non-uniformity, Ψ5 0 holds

(Bi-N, for Bi. 100, essentially), and then Kd is constant. The number Kd

involves the zone from zero to a value given by the body shape. It is often called

the thermal inertia criterion as well.

Info: [A23],[A29],[A35].

Georgy Mikhailovich Kondratiev, Russian engineer.

177Thermomechanics

5.1.10 Local Temperature Θ

Θ5T 2 Tmin

Tmax 2 Tmin

T, Tmin, Tmax (K) � local, minimal and maximal temperatures.

It expresses the ratio of the difference between the local and minimal tempera-

ture to that between the maximal and minimal temperatures. It is the parametric cri-

terion expressing the dimensionless temperature in a system. Usually, it represents

the dependent variable in the solution of thermal tasks. ΘAh0; 1i.Info: [A23].

5.1.11 Maxwell Number Ml

Ml5ληcV

5 kPr21

λ (W m21 K21) � thermal conductivity; η (Pa s) � dynamic viscosity;

cV (J kg21 K21) � specific heat capacity; κ (�) � specific heat ratio (p. 28);

Pr (�) � Prandtl number (p. 197).

It expresses the relation between gas thermal conductivity and gas viscosity.

It is a modification of the Prandtl number Pr (p. 197).

Info: [A23].

James Clerk Maxwell (p. 197).

5.1.12 Mikheyev Number Mi

Mi5ατcRL

5BiFo ð1Þ;

Mi5α2τλcR

5Bi2Fo ð2Þ

α (W m22 K21) � heat transfer coefficient; τ (s) � time; c (J kg21 K21) � specific

heat capacity; R (kg m23) � density; L (m) � characteristic length; λ (W m21 K21) �thermal conductivity; Bi (�) � Biot number (p. 173); Fo (�) � Fourier number

(p. 175).

This number characterizes the dependence of the mean integral quantities in

space and time on a body shape. It expresses the ratio of the heat passed by the

body to the volume heat. For finite volume bodies L5ffiffiffiS

pis valid, and for longitu-

dinally unlimited bodies it is L � 0: In equation (2), it expresses the ratio of the

heat transferred by convection to the body thermal activity ε5ffiffiffiffiffiffiffiffiλcR

p: It is applied

in cases in which L5λα2 1 is a characteristic length dimension.


Info: [A23].

Mikhail Alexandrovich Mikheyev (1902�1970), Russian engineer.

5.1.13 Number of Phase Changes Nph, K

Nph 51

cpΔT5 Sf 21 ð1Þ;

Nph 5rRwq

5NuPe21 ð2Þ;

Nph 52rRνqL

5NuPe21 ð3Þ

l (J kg21) � specific latent heat of phase change; cp (J kg21 K21) � specific heat

capacity; ΔT (K) � temperature difference; r (J kg21) � reaction heat of chemical

change; R (kg m23) � fluid density; w (m s21) � flow velocity; q (W m22) � sur-

face heat flux; ν (m2 s21) � kinematic viscosity; Sf (�) � Stefan two-phase con-

duction number (p. 183); Nu (�) � Nusselt number (p. 196); Pe (�) � Peclet heat

number (p. 180).

It expresses phase conversions. It is the inverse value of the Stefan two-phase

conduction number Sf (p. 183). Expression (1) is the ratio of the thermal flow nec-

essary to the phase conversion in a material, to the overheating or subcooling heat

of one of the phases. Expressions (2) and (3) represent the number Nph with phase

transformations in flowing fluid. Expression (2) represents the ratio of the phase

conversion rate to the flow rate. Expression (3) is the ratio of an inertia force in

flowing fluid, originating due to the phase conversion, to the inner friction force.

Therefore, it represents a specific form of the Reynolds number Re (p. 81). It is

analogous to the Jakob evaporation number Jak (p. 220).

5.1.14 Ostrogradsky Number Os

See Pomerantsev heat number PoV (p. 181) and Damkohler number (4.) (2. heat)

Da4 (p. 37).

Mikhail Vasilyevich Ostrogradsky (24.9.1801�1.1.1862),

Russian mathematician and physicist.

He was the first mathematician to publish the proof of

the divergence theorem. In the year 1822 he went to Paris,

where he worked on heat theory and on hydrodynamic pro-

blems. He applied the divergence theorem as a tool to con-

vert the volume integral to the planar one. He wrote many

articles on partial differential equations, algebra, elasticity,

hydromechanics, heat and electricity. He is considered the

founder of the Russian school of theoretical mechanics.

179Thermomechanics

5.1.15 Peclet Heat Number Pe

Pe5wL

a5RePr

w (m s21) � motion velocity; L (m) � characteristic length; a (m2 s21) � thermal

diffusivity; Re (�) � Reynolds number (p. 81); Pr (�) � Prandtl number (p. 197).

This number is the criterion for the mutual action of the convective and molecu-

lar heat transfers in flowing fluid. It expresses the ratio of the convective thermal

flow transferred by the fluid to that transferred by conduction. It characterizes the

thermal process with consideration of the movement of the environment and physi-

cal properties. It can be understood as the ratio of the enthalpy changes of the fluid

flowing in the axial direction to the heat penetrating into the flow by conduction in

the normal direction. With the Pe number increasing, the heat conduction portion

decreases and the convective heat portion grows. Thermal motion processes.

Info: [A4],[A23],[A35],[A43],[B20].

Jean Claude Eugene Peclet (10.2.1793�6.12.1857),

French physicist.

He was engaged in research on heat transfer by flow and

conduction especially, and in radiation and mass transfer as

well. His publications are well known for his clear and dis-

tinct opinions and well-executed experiments. His principal

work is the Traite de la Chaleur Consideree dans ses

Applications (Essay on Heat and its Applications in Crafts

and Manufacture, 1829).

5.1.16 Peclet Non-Linear Number of Solidification Pe

Pe5ωLλðTÞ

dh

dΘ1

TS 2 TC; where Θ5

T 2 TC

TS 2 TC

w (m s21) � interface motion velocity; L (m) � characteristic length; λ(W m21 K21) � thermal conductivity; h (J m23) � specific volume enthalpy of

solid; TS, TC, T (K) � solidifying interface, cooler and local temperatures.

It characterizes the motion of the isothermic solid�liquid interface, for example,

in continuous casting and in solidification of castings.

Info: [A23].

Jean Claude Eugene Peclet (see above).

5.1.17 Phase Change of Enthalpy N

N5h2 hs

lslð1Þ;


N5cðTs 2 TslÞ

lslð2Þ;

N5cðTi 2 TslÞ

lslð3Þ

h (J kg21) � specific enthalpy; hs (J kg21) � specific enthalpy of solid phase; lsl

(J kg21) � specific latent heat of melting; c (J kg21 K21) � specific heat capacity;

Ts (K) � solid phase temperature; Tsl (K) � temperature of solidification or melt-

ing; Ti (K) � initial (casting) temperature of overheated melting.

In the dimensionless form, it characterizes phase conversion enthalpy. It is anal-

ogous to the Stefan two-phase conduction number Sf (p. 183).

Info: [A23].

5.1.18 Pomerantsev Heat Number, Heat Generation Term Po, G

PoV � Os5qVL

2

λΔTrð1Þ;

PoA 5Ki5qAL

λΔTrð2Þ;

PoL 5qL

λΔTrð3Þ;

PoB 5qB

λLΔTrð4Þ

qV (W m23), qA (W m22), qL (W m21), qB (W) � volume, area, linear and point

density of heat flux (source); L (m) � characteristic length; λ (W m21 K21) �thermal conductivity; ΔTr (K) � characteristic temperature difference; Os (�) �Ostrogradsky number (p. 179).

It characterizes volume (1), area (2), linear (3) or point (4) heat sources in a sys-

tem. Sometimes in expression (1), it is called the Ostrogradsky number Os (p. 179),

whereas it is called the Kirpichev number Ki (p. 176) in expression (2).

Info: [A4],[A23],[A26],[A35],[A42],[B20].

Alexey Alexandrovich Pomerantsev, Russian engineer.

5.1.19 Predvoditelev Number Pd

Pd5dΘP

dFo

� �max

ð1Þ;

181Thermomechanics

Pd5bL2

aTP;ið2Þ;

Pd5b2L

2

að3Þ;

Pd5 2πfL2a21 ð4Þ

ΘP (�) � dimensionless environment temperature; b (K s21) � constant; L (m) �characteristic length; a (m2 s21) � thermal diffusivity; TP,i (K) � initial tempera-

ture; b2 (s21) � constant; f (Hz) � oscillation frequency of external temperature;

Fo (�) � Fourier number (p. 175).

This number characterizes the temperature time change of the outer environment

or of internal heat sources. For a linear change, TP5 TP,i1 bτ holds for expression

(2), and for T(τ)5 T(N)2 (T(N)2 Ti) exp (2b2τ), expression (3) is valid. For

the thermal flow density changing exponentially from an internal source, q(τ)5 qiexp (2b2τ) holds, when the constant b2 expresses the maximum change rate of the

thermal source. Expression (4) holds for the environmental temperature T(τ)�Ti5 Tm cos (2πfτ), which changes in a harmonic way.

Info: [A4],[A23],[A26],[B20].

Alexander Savviich Predvoditelev (1891�1973), Russian physicist.

5.1.20 Stefan�Fourier Number N

N5cðTls 2 TsÞ

lls

aτL2

5 Sf Fo ð1Þ;

N5cðTs 2 TslÞ

lsl

aτL2

5 Sf Fo ð2Þ

c (J kg21 K21) � specific heat capacity at solidification or melting temperature;

Tls, Tsl (K) � temperature of solidification or melting; Ts (K) � surface temperature

of solidification body (ingot); lls, lsl (J kg21) � specific latent heat of solidification

or melting; a (m2 s21) � thermal diffusivity; τ (s) � time; L (m) � characteristic

length; Sf (�) � Stefan two-phase conduction number (p. 183); Fo (�) � Fourier

number (p. 175).

In the dimensionless form, it characterizes the time of the thermal two-phase

process of solidification (1) or melting (2).

Info: [A123].

Josef Stefan (see below).



5.1.21 Stefan Two-Phase Conduction Number Sf

Sf 5cðTls 2 TsÞ

llsð1Þ;

Sf 5cðTs 2 TslÞ

lslð2Þ

c (J kg21 K21) � specific heat capacity at solidification or melting temperature; Tls,

Tsl (K) � temperature of solidification or melting; Ts (K) � surface temperature of

solidification body (ingot); lls, lsl (J kg21) � specific latent heat of solidification or

melting.

It expresses the ratio of the specific enthalpy of an ingot to the specific latent

heat of solidification. It characterizes the sensitivity to the thermal process latent

energy of solidification (1) or melting (2) in a solidus�liquidus system under the

condition of linear heat conduction, neglecting the difference of densities of both

phases and considering the phase conversion at an equal temperature.

Info: [A23].

Josef Stefan (p. 214).

5.1.22 Thermal Conductivity Relation Λ

Λ5λλref

λ, λref (W m21 K21) � thermal conductivity and its reference value.

It is the parametric criterion expressing the dimensionless thermal conductivity

of a material. It can be used in non-linear heat conduction.

Info: [A23].

5.1.23 Two-Phase Heat Conduction K

K5λlðT1 2 TsÞλsðTs 2 T2Þ

λl, λs (W m21 K21) � thermal conductivity of liquid and solid phases; T1, T2, Ts(K) � input liquid casting, output and solid phase temperatures.

It expresses the thermal flows ratio in liquid and solid phases. It characterizes

the heat conduction process in a two-phase environment during solidification or

melting.

Info: [A23].

183Thermomechanics

5.2 Free Convection

In free (natural) convection, the dimensionless quantities express the spontaneous

heat flow in fluids or gases due to the thermal difference caused by the difference

of fluid densities. The force fields setting the fluid in motion originate directly in

the fluid. Free convection occurs in the atmosphere, in oceans and near the surface

of heated or cooled bodies, for example. For free convection, the Rayleigh,

Grashof, Marangoni, Archimedes, Grigull and Schwarzschild numbers are

characteristic.

5.2.1 Archimedes Thermodynamic Number Ar

Ar5ΔT

T0

gL

w25ΔΘFr21 ð1Þ;

Ar2 5gL

w2βΔT 5

L3gβΔT

ν2ν2

w2L25GrRe22 ð2Þ

ΔT (K) � temperature difference; T0 (K) � steady-state temperature; g (m s22) �gravitational acceleration; L (m) � characteristic length; w (m s21) � flow velocity;

β (K21) � thermal volume expansion coefficient; ν (m2 s21) � kinematic viscosity;

Θ (�) � relative temperature; Fr (�) � Froude number (1.) (p. 62); Gr (�) �Grashof heat number (p. 185); Re (�) � Reynolds number.

This number characterizes the free and forced convection caused by the temper-

ature gradient and the fluid flow rate. In equation (2), it is often called the buoy-

ancy parameter NB (p. 56).

Info: [A4],[A12],[A29],[B20].

Archimedes of Syracuse (p. 54).

5.2.2 Bejan Pressure Number Be

Be5ΔpL2

ηa

Δp (Pa) � pressure difference in channel; L (m) � characteristic length of circum-

fluenced channel; η (Pa s) � dynamic viscosity; a (m2 s21) � thermal diffusivity.

It expresses the pressure gradient in a flow-through canal with consideration of

the heat diffusion. It has an analogous role as the Rayleigh number (2.) (heat insta-

bility) Ra (p. 187) in natural convection. Natural convection.

Info: [C23].

Adrian Bejan (p. 5).


5.2.3 Crispation Number Cr

Cr5ηaσL

η (Pa s) � dynamic viscosity; a (m2 s21) � thermal diffusivity; σ (N m21) �surface

tension; L (m) � characteristic length, layer thickness.

This number expresses the ratio of a viscous force and diffusive heat action to a

surface tension force. It characterizes the influence of convective flows in heat and

momentum transfers.

Info: [A26],[A29].

5.2.4 Grashof Heat Number Gr

Gr5L3gΔRRν2

5L3gβΔT

ν25 βΔTGa ð1Þ;

Gr5L3gΔT

ν2Tð2Þ;

Grrot 5Gr�ω2L

gð3Þ

L (m) � characteristic length dimension; g (m s22) � gravitational acceleration;

ΔR (kg m23) � change of fluid density due to temperature change ΔT; R (kg m23) �density; ν (m2 s21) � kinematic viscosity; β (K21) � volume thermal expansion coef-

ficient; T (K) � temperature; ω (Hz) � angular frequency; Ga (�) � Galilei number

(p. 123); Gr� (�) � Grashof heat number defined in expression (1).

It expresses the buoyancy-to-viscous forces ratio and its action on a fluid. It

characterizes the free non-isothermal convection of the fluid due to the density dif-

ference caused by the temperature gradient in the fluid. It is often used in expres-

sion (2) for an ideal gas. In form (3), it is used for convective transfer in rotating

canals of which the axis is parallel with the rotation axis. With this modified Gr

number, the centrifugal force influence on the heat transfer is considered. Usually,

the Coriolis force influence is negligible.

Info: [A23],[A26],[A43],[B20].

Franz Grashof (11.7.1826�26.10.1893), German engineer.

Within the framework of applied mechanics and general

machinery, he was engaged in research on material stiffness,

hydraulics and heat theory. As for heat transfer, he was

engaged in the problem of free convection caused by the

density difference due to the temperature gradient in a fluid.

185Thermomechanics

5.2.5 Grashof Modified Number Grmod

Grmod 5gR2βL4

λη2

g (m s22) � gravitational acceleration; R (kg m23) � liquid density; β (K21) � vol-

ume thermal expansion coefficient; L (m) � characteristic length; λ (W m21 K21) �thermal conductivity; η (Pa s) � dynamic viscosity.

It is a modification of the Grashof heat number Gr (p. 185). It expresses the natural

convective transfer from a heated or cooled wall into the surrounding viscous fluid.

Info: [B60].

Franz Grashof (see above).

5.2.6 Grigull Number Gg

Gg5βgqAτ2

ηcp

β (K21) � volume thermal expansion coefficient; g (m s22) � gravitational acceler-

ation; qA (W m22) � heat flux density; τ (s) � time; η (Pa s) � dynamic viscosity;

cp (J kg21 K21) � specific heat capacity.

It characterizes the conditions for the origin of the development of free convec-

tion in a horizontal layer of a fluid or gas which is heated from below.

Info: [A23],[A33].

Ulrich Grigull (12.3.1912�20.10.2003), German engineer.

He was engaged in research on thermodynamics and heat

transfer, and in research on turbulent film condensation. He

published the book Grundgesetze der Warmeubertragung

(Basic Laws of Heat Transfer). His research included the

investigation of water and water steam properties, thermal

physical properties � such as viscosity under high pressure

and temperature � and determination of critical water steam

properties.

5.2.7 Marangoni Number Mr

Mr5dσdT

ΔTL

ηa5

ΔσΔT

ΔT

ΔL

L2

ηa

σ (N m21) � surface tension; ΔT (K) � temperature difference between upper surface

layer and lower one; L (m) � characteristic length (layer thickness); η (Pa s) �dynamic viscosity; a (m2 s21) � thermal diffusivity.

It is the criterion for mutual influence of capillary forces and molecular friction

forces. It characterizes the thermal capillary phenomena on a fluid-free surface and


the surface tension influence on the origin of the free convection in a thin fluid

layer. The Mrcrit5 80 is the critical value leading to instability.

Info: [A4],[A29],[A35],[B20],[C7].

Carlo Guiseppe Matteo Marangoni (1840�1925), Italian physicist.

5.2.8 Rayleigh Number (2.) (Heat Instability) Ra2

Ra2 5gL3ΔTβ

νa5

gL3ΔRηa

5GrPr

g (m s22) � gravitational acceleration; L (m) � characteristic length; ΔT (K) �temperature difference; β (K21) � volume thermal expansion coefficient;

ν (m2 s21) � kinematic viscosity; a (m2 s21) � thermal diffusivity; ΔR (kg m23) �density difference; η (Pa s) � dynamic viscosity; Gr (�) � Grashof heat number


It characterizes the free convection heat transfer along a heat-exchanging sur-

face. It expresses the buoyancy-to-diffusion ratio or, alternatively, the free convec-

tion thermal instability in fluids. In a closed space with Ra, 103, the heat is

transferred between warmer and cooler walls by conduction only.

Info: [A4],[A23],[A29],[A43],[B20].

Lord Rayleigh, John William Strutt (12.11.1842�30.6.1919), British physicist. Nobel Prize in Physics, 1904.

He was engaged in research on optics, acoustics, electro-

magnetism and molecular light dispersion. He discovered

one of the radiation laws for perfect black body radiation in

the long-wave zone. He intensively studied Maxwell’s elec-

tromagnetism theory, both theoretically and experimentally.

He researched the density of rare gases and discovered

argon.

5.2.9 Rayleigh Number (3.) Ra3

Ra3 5qAL

5gβνλax

5GrNuPrX21

qA (W m22) � heat flux density; L (m) � characteristic length, pipe diameter;

g (m s22) � gravitational acceleration; β (K21) � volume thermal expansion coef-

ficient; ν (m2 s21) � kinematic viscosity; λ (W m21 K21) � thermal conductivity;

a (m2 s21) � thermal diffusivity; x (m) � distance from inlet to vertical tube;

Gr (�) � Grashof heat number (p. 185); Nu (�) � Nusselt number (p. 196);

Pr (�) � Prandtl number (p. 197); X (�) � geometrical coordinates (p. 15).

187Thermomechanics

It characterizes the combined free and forced convections in vertical tubes.

Info: [A4],[A23].

Lord Rayleigh (see above).


Ra4 5qVL

5gβvλa

5GrPrPoV

qV (W m23) � volume density of heat flux; L (m) � characteristic length;

g (m s22) � gravitational acceleration; β (K21) � volume thermal expansion coef-

ficient; v (m2 s21) � kinematic viscosity; λ (W m21 K21) � thermal conductivity;

a (m2 s21) � thermal diffusivity; Gr (�) � Grashof heat number (p. 185); Pr (�) �Prandtl number (p. 197); PoV (�) � Pomerantsev heat number (p. 181).

It characterizes the free convection in a horizontal fluid layer with a constant

internal volume heat source.

Info: [A4],[A23],[A29],[A35].


5.2.11 Schwarzschild Number Sch

Sch5gT

cpv

@v

@T

� �p

dT

dl

� �21

g (m s22) � gravitational acceleration; T (K) � temperature; cp (J kg21 K21) � specific

heat capacity; v (m3 kg21) � specific volume; l (m) � length.

This number characterizes the stability conditions for a fluid mechanical balance

with its longitudinally changing temperature or the condition under which no ther-

mal convection occurs in the fluid. In a fluid, over the horizontal surface of a wall,

of which the temperature changes perpendicularly to the surface up to the free fluid

level, the following condition is valid to cause convection

dT

dl

��. gT

cpv

dv

dT

� �p

Info : [A23],[A33].


Karl Schwarzschild (9.10.1873�11.5.1916), German

astronomer, physicist and mathematician.

He was focused on electrodynamics, optics and the radi-

ation of stars related to stellar atmosphere research. He

applied photography to measure the instability of stars and

investigated internal star structures and star gas equilibrium.

He devoted himself also to relativity theory, and the

Schwarzschild radius, which represents the critical gravita-

tional solid-body radius at which a body becomes a black

hole according to general relativity theory, is named after

him.

5.2.12 Thompson Number Th

See Marangoni number Mr (p. 186).

Info: [A29].

5.3 Forced Convection

The dimensionless quantities expressing forced convection heat transfer are among

the most widely used. Forced convection is characterized by a flow with mutual

action of inertia and viscous forces. The predominance of the viscous flow charac-

terizes a laminar flow. On the contrary, turbulent flow arises under the action of

external influences and its development depends on the viscous-to-inertia forces

ratio exclusively. Among the numerous dimensionless quantities, the Reynolds,

Prandtl, Colburn, Dulong, Eckert, Nusselt, Fliegner, Graetz, Stanton and other

numbers are especially important.

5.3.1 Bansen Heat Number Ba

Ba5T1 5 T2

T 2 Ts

T1, T2 (K) � input and output temperatures in channel; T, Ts (K) � fluid flow and

channel wall temperatures.

This number expresses the ratio of the temperature gradient between a channel

inlet and outlet to the temperature gradient between the fluid and the channel wall.

It characterizes the convection heat transfer intensity for fluid flowing in channels.

Info: [A14],[A23],[A29].

R.V. Bansen.

189Thermomechanics

5.3.2 Boundary Layer Number N

N5τsLwNη

τs (N m21) � shear surface stress; L (m) � characteristic length; wN (m s21) �flow velocity far from boundary layer; η (Pa s21) � dynamic viscosity.

It characterizes the surface friction in flow in a laminar boundary layer. It is

analogous to the Bingham number Bm (p. 118).

Info: [A23].

5.3.3 Colburn Number JQ, Jh

JQ � Jh 5α

Rcpwcpηλ

� �23

5 StPr23 5NuRe21Pr2

13

α (W m22 K21) � heat transfer coefficient; R (kg m23) � density; cp (J kg21 K21) �

specific heat capacity; w (m s21) � fluid flow velocity; η (Pa s) � dynamic viscosity;

λ (W m21 K21) � thermal conductivity; St (�) � Stanton number (p. 201);

Pr (�) � Prandtl number (p. 197); Nu (�) � Nusselt number (p. 196); Re (�) �Reynolds number (p. 81); Jh (�) � J-heat transfer factor (p. 195).

It characterizes the heat transfer in forced and free flows of a viscous fluid. The

Colburn number is identical with the J-heat transfer factor (p. 195).

Info: [A23],[A29].

Allan Philip Colburn (8.6.1904�1955), American chemi-

cal engineer.

He was engaged in water steam condensation from a sat-

urated air flow, which was his main interest during his

entire life. It was due to him that the common foundations

of momentum, heat and mass transfer, following from ther-

modynamic principles, were formulated in the US for the

first time.

5.3.4 Crispation Group Ncr

Ncr 5ηaσL

η (Pa s) � dynamic viscosity; a (m2 s21) � thermal diffusivity; σ (kg s22) � undis-

turbed surface tension; L (m) � characteristic length (thin layer thickness).

It expresses the possibility of fluid crispation (tearing) originating in convective

flow and heat transfer.

Info: [A23],[A35].


5.3.5 Dulong Number Du

Du � r21h 5w2

cpΔT

w (m s21) � flow velocity; cp (J kg21 K21) � specific heat capacity; ΔT (K) �temperature difference.

This number expresses the ratio of the flowing fluid kinetic energy to the

enthalpy change in adiabatic compression. It characterizes the convection heat

transfer between a wall and fast flowing compressible gas with viscous energy dis-

sipation and adiabatic compression. It represents the volume density of the heat

flow that arises and the heating of the bypassed surface. It corresponds to the

Eckert number Ec (p. 191). Its inverse value corresponds to the restitution enthalpy

coefficient rh (p. 198).

Info: [A23],[A29],[B20].

Pierre Louis Dulong (12.2.1785�19.7.1838), French phys-

icist and chemist.

Above all, he is well known for the Dulong�Petit law

(1819), which joins the specific thermal capacity of metals

with their molar mass. It is a classic relation that expresses

the specific thermal crystal capacity dependence on the grid

oscillating frequency. Despite its simplicity, this relation pro-

vides a very good result for solid materials with relatively

simple crystal structure in the high-temperature range.

5.3.6 Eckert Number Ec

Ec5w2N

cpΔT5 2Rf ð1Þ;

Ec5 ðκ21ÞM2N

TN

ΔT5 2

ΔTad

ΔT5 2

Tad;N 2 TN

TN 2 Tð2Þ

wN (m s21) � fluid flow velocity far from body; cp (J kg21 K21) � specific heat

capacity of fluid; ΔT (K) � temperature difference; T, Tad, TN (K) � static, adia-

batic and far from body temperatures; k (�) � specific heat ratio (p. 28); Rf (�) �temperature recovery factor (p. 202).

It expresses the ratio of kinetic energy to a thermal energy change. Ec1 is a

special case of the Dulong number Du (p. 191) for w5wN. Ec2 is valid for

a compressible fluid.

Info: [A23],[A26],[A29],[A35].

191Thermomechanics

Ernst Rudolf Georg Eckert (13.9.1904�8.7.2004),

American engineer of Czech-German origin.

He researched a wide range of heat transfer problems,

from cryogenic temperatures to plasma. His research

involves basic studies on humidity migration in porous mate-

rials and on related problems of gas turbine cooling. His

principal works are Introduction to the Transfer of Heat and

Mass (1959) and Analysis of Heat and Mass Transfer (1972).

5.3.7 Fliegner Number Fl

Fl5Rwpst

ffiffiffiffiffiffiffiffiffifficpTst

pð1Þ;

Fl5Rwp

ffiffiffiffiffiffiffifficpT

pð2Þ;

Fl5Rw

ðps 1 Rw2ÞffiffiffiffiffiffiffifficpT

pð3Þ;

Fl5Qm

ffiffiffiffiffiffiffifficpT

pAðps 1 Rw2Þ 5

kMffiffiffiffiffiffiffiffiffiffik21

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi11

ðk21ÞM2

2

rð4Þ

R (kg m23) � density; w (m s21) � flow velocity; pst (Pa) � stagnation absolute

pressure of gas flow; cp (J kg21 K21) � specific heat capacity; Tst (K) � stagnation

temperature; p (Pa) � static absolute pressure; T (K) � absolute temperature;

ps (Pa) � static pressure; Qm (kg s21) � mass flux; A (m2) � clear area; k (�) �specific heat ratio (p. 28); M (�) � Mach number (p. 73).

It characterizes the coupling between the fluid flow and the heat transfer. It

expresses the ratio of the heat related to the velocity unit to the fluid momentum

flow. In expression (1), it is the stagnation number; in expression (2), it is the static

number; and in expression (3), it is the impulse number. Generally, the Fl5 f(M)

expressed in form (4) holds for the compressible fluid flow.

Info: [A23],[A29],[A35],[B20].

Albert Friedrich Fliegner (1842�1928), Swiss engineer.

5.3.8 Frossling Heat Number Fs

Fs5Nu22

Re12Pr

13

ð1Þ;


Fs5Nu

Re12

ð2Þ

Nu (�) � Nusselt number (p. 196); Re (�) � Reynolds number (p. 81); Pr (�) �Prandtl number (p. 197).

It expresses the influence of the fluid molecular properties and character of the

flow in a boundary layer on the average value of the specific thermal transference.

Expression (1) holds for turbulent flow around a cylinder or a ball. Expression (2)

is valid for an isothermic plate with laminar flow in a boundary layer.

Info: [A23],[A29],[A35].

Karl Gustav Frossling (born 1913), Swedish engineer.

5.3.9 Graetz Number Gz

Gz5Qmcp

λL5

cpRwAλL

ð1Þ;

Gz5wD2

aL5RePr

D

Lð2Þ

Qm (kg s21) � mass flux; cp (J kg21 K21) � specific heat capacity; λ (W m21 K21) �

thermal conductivity; L (m) � characteristic length; ρ (kg m23) � density; w (m s21)

� velocity; A (m2) � area; D (m) � pipe diameter; a (m2 s21) � thermal diffusivity;

Re (�) � Reynolds number (p. 81); Pr (�) � Prandtl number (p. 197).

This number expresses the ratio of the fluid thermal capacity to its thermal con-

ductivity. It characterizes the heat transfer in the laminar flow of fluids. The rela-

tion (2) is valid for a circular pipeline. A large Gz number represents a

predominating convection influence. A small Gz number indicates to a predominat-

ing heat conduction influence. In the last case, the temperature does not change

along the path and the thermal field is fully developed. The Gz number is used

mainly in the American science texts, in which it often replaces the Peclet heat

number Pe (p. 180).

Info: [A23],[A29],[A35],[B20].

Leo Graetz (26.9.1856�12.11.1941), German physicist.

His scientific work was focused on heat transfer by con-

duction and radiation, and on the problems of friction and

elasticity. For a long time, he was devoted to problems of

electromagnetic waves and cathode radiation. His publica-

tions Electricity and Magnetism and Electricity and its

Applications are especially significant.

193Thermomechanics

5.3.10 Heat Non-Stationarity Number (1.) NT

NT 5@TS@τ

L2

ðTS 2 TmedÞaTS (K) � wall temperature; τ (s) � time; L (m) � characteristic length; Tmed (K) �mean fluid temperature; a (m2 s21) � thermal diffusivity.

This number characterizes the influence of the time-dependent temperature of a

wall on the non-stationary heat transfer of single-phase fluids in passageways.

Info: [A23].

5.3.11 Heat Non-Stationarity Number (2.) NZ

NZ 5@TS@z

L

TS 2 Tmed

Ts (K) � wall temperature; z (m) � axis along the pipeline; L (m) � characteristic

length; Tmed (K) � mean fluid temperature.

It characterizes the influence of the temperature gradient change in a wall on the

heat transfer of single-phase fluids in passageways.

Info: [A23].

5.3.12 Heat Source Effect Number F

F5DqVL

2RcpνΔT

D (m) � tube diameter; qV (W m23) � volume density of heat flux; L (m) � char-

acteristic length; R (kg m23) � density; cp (J kg21 K21) � specific heat capacity;

ν (m2 s21) � kinematic viscosity; ΔT (K) � temperature difference.

It characterizes the heat source influence on the heat transfer by convection in

wholly developed laminar fluid flow from below upwards in inclined tubes with

uniformly distributed inner sources and uniform thermal flow on the walls. With

increasing parameter F, the Nusselt number Nu (p. 196) drops. With F 5 const, the

heat transfer intensity depends on the Rayleigh number (2.) (heat instability) Ra2.

With large Ra2 (p. 187) numbers and arbitrary F, the tube orientation influences

the heat transfer intensity.

Info: [A23].

5.3.13 Heat Transfer Efficiency N

N5αLRw2

λΔp5BiEu21


α (W m22 K21) � heat transfer coefficient; L (m) � characteristic length;

R (kg m23) � density; w (m s21) � velocity; λ (W m21 K21) � thermal conductiv-

ity; Δp (Pa) � pressure difference; Bi (�) � Biot number (p. 173); Eu (�) �Euler number (1.) (p. 61).

It characterizes the external heat transfer intensity related to system hydraulic

resistance.

Info: [A23],[A33].

5.3.14 Hydrodynamic Non-Stationarity Number NG

NG 5@Qm

@τL2

νQm

Qm (kg s21) � mass flux; τ (s) � time; L (m) � characteristic length; ν (m2 s21) �kinematic viscosity.

It expresses time change in the passage of a mass which influences the non-

stationary heat transfer of single-phase fluid.

Info: [A23].

5.3.15 J-Heat Transfer Factor Jh

Jh � JQ 5α

cpqm

cpηλ

� �23

α (W m22 K21) � heat transfer coefficient; cp (J kg21 K21) � specific heat capac-

ity; qm (kg m22 s21) � mass flux density; η (Pa s) � dynamic viscosity;

λ (W m21 K21) � thermal conductivity; JQ (�) � Colburn number (p. 190).

Generally, it characterizes the heat transfer in free and forced flows. It is identi-

cal with the Colburn number JQ (p. 190).

Info: [A35],[B20].

5.3.16 Leibenzon Number (1.) Lb

Lb5NuPr1 2

Pr

Nu (�) � Nusselt number (p. 196); Pr (�) � Prandtl number (p. 197).

It is an extended modification of the Nusselt number Nu (p. 196), with which

convective and diffuse heat transfers are considered.

Info: [A23],[A33].

Leonid Samuilovich Leibenzon (1879�1951), Russian physicist and

mechanist.

195Thermomechanics

5.3.17 Margoulis Number Mg

Mg5 St21 5cpRwα

5RePr

Nu

cp (J kg21 K21) � specific heat capacity; R (kg m23) � fluid density; w (m s21) �fluid flow velocity; α (W m22 K21) � heat transfer coefficient; St (�) � Stanton

number (p. 201); Re (�) � Reynolds number (p. 81); Pr (�) � Prandtl number (p.

197); Nu (�) � Nusselt number (p. 196).

It expresses forced convection. It is also called the thermal resistance number.

Info: [A23],[A35].

Wladimir Margoulis (born 1886).

5.3.18 Nusselt Number Nu

Nu5αLλ

5ReStPr5PeSt

α (W m22 K21) � heat transfer coefficient; L (m) � characteristic length;

λ (W m21 K21) � thermal conductivity; Re (�) � Reynolds number (p. 81); St (�)

� Stanton number (p. 201); Pr (�) � Prandtl number (p. 197); Pe (�) � Peclet

heat number (p. 180).

It expresses the ratio of the total heat transfer in a system to the heat transfer by

conduction. In characterizes the heat transfer by convection between a fluid and the

environment close to it or, alternatively, the connection between the heat transfer

intensity and the temperature field in a flow boundary layer. It expresses the

dimensionless thermal transference. The physical significance is based on the idea

of a fluid boundary layer in which the heat is transferred by conduction. If it is not

so, the criterion loses its significance. In the expression α (λ/L)21, it expresses the

ratio of the heat transfer intensity to heat conduction intensity in a boundary layer.

In the expression L (λ/α)21, it represents the ratio of the characteristic length to the

boundary layer thickness. In the expression L/λ (1/α)21, it expresses the ratio of

thermal resistances by conduction to those by convection in a boundary layer.

Info: [A4],[A29],[A35],[A43],[B20].

Ernst Kraft Wilhelm Nusselt (25.11.1882�1.9.1957),

German engineer.

In 1915, he published an important work on basic heat

transfer laws. In this work, he first suggested the dimension-

less thermal similarity criteria. He was engaged in research

on the analogy between the transfer of heat and mass, and

convective heat transition. He formulated the basic theory of

heat regenerators. He was also engaged in work related to

thermal processes in membrane steam condensation and in

powder fuel combustion.


5.3.19 Prandtl Number Pr

Pr5ηcpλ

5νa5PeRe21

η (Pa s) � dynamic viscosity; cp (J kg21 K21) � specific heat capacity;

λ (W m21 K21) � thermal conductivity; ν (m2 s21) � kinematic viscosity; a (m2 s21) �thermal diffusivity; δ, δt (m) � thicknesses of thermodynamic and thermal boundary

layer; Pe (�) � Peclet heat number (p. 180); Re (�) � Reynolds number (p. 81).

This number expresses the ratio of the momentum diffusivity (viscosity) to the

thermal diffusivity. It characterizes the physical properties of a fluid with convec-

tive and diffusive heat transfers. It describes, for example, the phenomena con-

nected with the energy transfer in a boundary layer. It expresses the degree of

similarity between velocity and diffusive thermal fields or, alternatively, between

hydrodynamic and thermal boundary layers. With Pr5 1 and grad p5 0, the ther-

mal and hydrodynamic fields are similar. For example, if diverse molten materials

have equal Prandtl numbers, they have similar velocity and temperature fields in

crystallization. The following are valid:

Pr{1 δ{δt;Pr61 δ5 δt;Prc1 δcδt:

With small Pr numbers (Pr, 1), the molecular heat transfer by conduction

predominates over that by convection. With Pr. 1, it is the opposite case.

For example, for air and many other gases, the Pr value is about 0.7, for

coolants PrAh2; 10i, for machine oils PrAh102; 43 104i, and for mercury it is

approximately 153 1023.

Info: [A23],[A29],[A43],[B20].

Ludwig Prandtl (4.2.1875�15.8.1953), German physicist.

His principal works relate to the hydrodynamic and ther-

mal boundary layers, turbulent boundary layer analysis and,

especially, heat transfer. He published (1910) his elabora-

tion of Reynolds’ original concept (1874) of the analogy

between turbulent momentum and heat transfer. He applied

modelling methods extensively, above all the hydraulic

analogy to visualize flow.

5.3.20 Prandtl Resultant (Effective, Total) Number Pref

Pref 5εM 1 νεT 1 a

197Thermomechanics

εM (m2 s21) � vortex momentum diffusivity; εT (m2 s21) � vortex thermal diffu-

sivity; ν (m2 s21) � kinematic viscosity; a (m2 s21) � thermal diffusivity.

This number expresses the ratio of the resulting momentum diffusivity to the

resulting thermal diffusivity in mixed laminar and turbulent flow. It characterizes

the influence of the physical properties of a fluid on convective heat transfer and

momentum transfer under turbulent flow conditions.

Info: [A23],[A29],[A33].


5.3.21 Prandtl Turbulent Number Prtur

Prtur � Scv 5εMεT

5L

LT

εM (m2 s21) � vortex momentum diffusivity; εT (m2 s21) � vortex thermal diffusiv-

ity; L, LT (m) � characteristic lengths for momentum and heat transfer by mixing;

Scv (�) � Schmidt turbulent number (p. 264).

It expresses the ratio of turbulent momentum diffusivity to turbulent thermal dif-

fusivity. It characterizes the influence of turbulent flow and turbulence in heat

transfer.

Info: [A23],[A29],[A33].



Ra1 5w

ffiffiffiffiffiffiRLσ

r

w (m s21) � flow velocity; R (kg m23) � liquid density; L (m) � characteristic

length; σ (N m21) � surface tension.

It characterizes the condition of fluid jet decomposition. It is equivalent to the

Weber number (2.) We2 (p. 91) and it is often called the Rayleigh parameter.

Info: [A23].


5.3.23 Restitution Enthalpy Coefficient rh

rh 5ha 2 hl

hst 2 hlð1Þ;


rh 5cpðTaS 2 TlÞ

w2ð2Þ

ha (J) � specific adiabatic enthalpy of high-speed fluid flow on the wall; hl (J) � spe-

cific enthalpy of fluid flow; hst (J) � specific enthalpy of retarded flow;

cp (J kg21 K21) � specific heat capacity; TaS (K) � adiabatic temperature of circum-

fluenced surface; Tl (K) � static temperature of fluid flow; w (m s21) � flow velocity.

This coefficient expresses the ratio of the fluid flow adiabatic enthalpy on a wall

to the sum of the total flow enthalpy and the flow kinetic energy or, alternatively,

the ratio of the real conversion temperature to the theoretical conversion tempera-

ture. It characterizes the change of the specific fluid flow enthalpy when a high

velocity flow bypasses a wall; see the Eckert number Ec (p. 191) and the Dulong

number Du (p. 191). It expresses the ratio of the real flow heating due to conver-

sion to the ideal gas heating. It is used for the convective heat transfer in compress-

ible fluid flow.

Info: [A23].

5.3.24 Restitution Temperature Coefficient r

r5Taw 2 Tl

Tst 2 Tl

Taw (K) � adiabatic wall temperature; Tst (K) � stagnation temperature of fluid

flow; Tl (K) � static fluid temperature.

It expresses the ratio of the heating of a real wall due to flowing fluid kinetic

energy to the highest theoretical heating of the wall. It characterizes the heating

degree of a wall surface due to flowing fluid kinetic energy in relation to the high-

est theoretical heating.

Info: [A23].

5.3.25 Shukhov Number Su

Su5πkdLQcpR

k (W m22 K21) � heat passage coefficient; d (m) � pipe diameter; L (m) � charac-

teristic length; Q (m3 s21) � volume flow; cp (J kg21 K21) � specific heat capacity;

R (kg m23) � density.

It characterizes the influence of the radial and axial temperature gradients, in

a heated or cooled fluid flowing through a pipeline, on the hydraulic resistance

(pressure loss) in a considered pipeline section.

Info: [A23],[A33].

Vladimir Grigoryevich Shukhov (1853�1939), Russian engineer.

199Thermomechanics

5.3.26 Spalding Function Sp

Sp52@Θ@u1

� �u150

where Θ5T 2 TN

Tw 2 TN; u15w

ffiffiffiRτ

r

T (K) � local temperature; TN (K) � free stream temperature; Tw (K) � wall tem-

perature; w (m s21) � velocity; τ (Pa) � shear stress; R (kg m23) � density;

u1 (�) � Prandtl velocity ratio (p. 80).

This function characterizes the dimensionless temperature gradient in a wall

with forced convection.

Info: [A23].

Dudley Brian Spalding (see above).

5.3.27 Spalding Number (1.) Sp

Sp 5αν

wλffiffiffiffiffiffiffiffiffiffi0:5fF

p 5ανλ

τR

0@1A

21

2

5 StPrNe21

2

5NuRe21Ne21

25NuPe21PrNe

21

2

α (W m22 K21) � heat transfer coefficient; ν (m2 s21) � kinematic viscosity;

w (m s21) � velocity; λ (W m21 K21) � thermal conductivity; fF (�) � Fanning

friction number (p. 163); St (�) � Stanton number (p. 201); Pr (�) � Prandtl num-

ber (p. 197); Ne (�) � Newton number (p. 75); Nu (�) � Nusselt number (p. 196);

Re (�) � Reynolds number (p. 81); Pe (�) � Peclet heat number (p. 180).

It characterizes the convective heat transfer intensity in a fully developed boundary

layer.

Info: [A23].

Brian Spalding (born 1923), English engineer.

Initially, he devoted himself to the liquid fluid combus-

tion problem and to thermodynamics. He published a whole

range of original scientific works concerning this sphere.

He has also worked on the transfer of heat and mass,

boundary layer theory and the fluid dynamics simulations,

making use of the rapid evolution of calculation techniques

in recent years.


5.3.28 Stanton Number St

St � Mg21 5α

Rcpw5

Nu

RePr5NuFo ð1Þ;

St5αS

qmVVcpð2Þ;

St5qA

RwcpðTr 2 TsÞð3Þ

α (W m22 K21) � heat transfer coefficient; R (kg m23) � density; cp (J kg21 K21) �

specific heat capacity; w (m s21) � fluid velocity; S (m2) � surface area;

qmV (kg m23) � volume density of mass flux; V (m3) � volume; qA (W m22) �surface heat flux density; Tr (K) � reference temperature; TS (K) � wall tempera-

ture; Nu (�) � Nusselt number (p. 196); Re (�) � Reynolds number (p. 81);

Pr (�) � Prandtl number (p. 197); Fo (�) � Fourier number (p. 175).

It expresses the ratio of the heat transferred in a system by convection to the

thermal capacity of heat-bearing surroundings. It characterizes the forced convec-

tion heat transfer. In form (2), it is used to express the convective heat transfer in

burning chambers and in steam boiler spaces. In form (3), it expresses the heat

flow through a laminar boundary layer when solid bodies are passed by with fluid.

In form (1), it is also called the Margoulis number Mg (p. 196).

Info: [A4],[A21],[A43],[B11],[B20].

Thomas Edward Stanton (12.12.1865�1931), English

engineer.

His principal interest was the flowing of fluids and the

friction problem, with its related heat transfer. From 1902

to 1907, he was engaged in wide research on the influence

of wine on structures such as bridges and roofs. After the

first flight of the Wright brothers in an aircraft in 1908, he

devoted himself enthusiastically to airplane and airship

design, and above all to the heat transfer in air-cooled

engines.

5.3.29 Temperature Factor ΘS

ΘS 5TS

Trð1Þ;

ΘS 5TS

TaSð2Þ

201Thermomechanics

TS (K) � surface temperatures of body wall; Tr (K) � characteristic temperature of

fluid flow; TaS (K) � adiabatic temperature of body surface.

It expresses the ratio of the absolute surface temperature of a body wall to the

characteristic absolute temperature of a flowing fluid. It characterizes the influence

of the flowing fluid temperature on a body. Instead of the relative temperature Tr,

sometimes the adiabatic temperature of a solid, ideally insulated and non-radiating

wall is considered, which is passed by a fluid with heat development caused by

energy dissipation. With outer bypassing of a body, usually the outer flow tempera-

ture is chosen as the characteristic temperature Tr. The mean flow temperature is

chosen as the characteristic temperature in flow in tubes and canals.

Info: [A23].

5.3.30 Temperature Recovery Factor Rf

Rf 52cpΔT

w25 2Ec21

cp (J kg21 K21) � specific heat, capacity of fluid flow; ΔT (K) � difference

between static temperature of flowing gas and adiabatic temperature of circum-

fluenced wall; w (m s21) � fluid velocity; Ec (�) � Eckert number (p. 191).

For an ideal gas, this factor expresses the ratio of the real temperature of a

bypassed wall to the theoretical temperature.

Info: [A15].

5.3.31 Thermal Coupling Number K, χ

K5 0:5λl

λs

Re12; KA 0;Nh Þ

λl, λs (W m21 K21) � specific thermal conductivity of fluid and solid; Re (�)

Reynolds number (p. 81).

This number characterizes the influence of thermally physical properties on the

convective heat transfer in compound tasks. It occurs in compressible fluid flow

when considering the mechanical energy dissipation and heat propagation in a solid

body. Usually, its value is approximately K5 1. The case of K-N (λs-0) occurs

when a perfect thermal insulator is inserted into the fluid flow.

Info: [A23].

5.3.32 Heat Transfer Number Nq

Nq 5q

Rw3L2


q (W) � heat flux; R (kg m23) � density; w (m s21) � flow velocity; L (m) � char-

acteristic length.

It expresses the ratio of the heat transferred in a system to the kinetic energy of

the flowing environment. Heat transfer in flowing fluid.

Info: [A23],[A29],[A33].

5.3.33 Unsteady Heat Transfer Number N

N5Nu

Nu0

Nu (�) � Nusselt unsteady number, see Nusselt number (p. 196); Nu0 (�) �Nusselt quasi-stationary number, see Nusselt number (p. 196).

It expresses the ratio of unsteady heat transfer to quasi-stationary heat transfer.

It characterizes the dimensionless transfer in unsteady flow.

Info: [A23].

5.4 Radiation

Here radiation denotes thermal radiation, of which the main aspect is electromag-

netic radiation. Besides conduction and convection, it describes the third kind of

heat transfer, which is of use with high surface temperatures especially. Among the

basic dimensionless quantities are the Boltzmann and Stefan numbers. The former

expresses the relation of the heat transferred by convection and radiation, whereas

the latter expresses that transferred by radiation and conduction. The Bansen,

Bouguer, Nusselt radiation, Peclet radiation and Schuster numbers are also among

the dimensionless values, and so are the radiation pressure and radiation viscosity

numbers.

5.4.1 Bansen Radiation Number Ba

Ba5αRS

Qmcp

αR (W m22 K21) � radiation heat transfer coefficient; S (m2) � surface area of

channel wall; Qm (kg s21) � mass flux; cp (J kg21 K21) � specific heat capacity.

It expresses the ratio of the heat transferred by radiation to the fluid thermal

capacity.

Info: [A4],[A35],[B20].

R.V. Bansen.

203Thermomechanics

5.4.2 Biot Radiation Number BiR

See the Stefan number Sf (p. 213).

Jean-Baptiste Biot (p. 174).

5.4.3 Boltzmann Number Bo

Bo � Tg5cpRwεσ0T3

ð1Þ;

Bo5 4NRePrτ21 5 4NPeτ21 ð2Þ;

Bo53

16τBo1 ð3Þ;

Bo5Bo1

2τð4Þ;

Bo5Rwðh1 2 h2Þ

εσ0T4ð5Þ;

Bo5εσ0SRT

3

Qmcpð6Þ

cp (J kg2 1 K2 1) � specific heat capacity; R (kg m2 3) � flue gases density;

w (m s21) � flue gases velocity; σ0 (W m22 K24) � Stefan�Boltzmann constant;

T (K) � flue gas thermodynamic temperature; τ (�) � optical thickness, ratio of

characteristic length to mean free path; h1, h2 (J kg21) � specific enthalpy of flue

gases; SR (m2) � radiation surface; Qm (kg s21) � mass flux; ε (�) � emissivity

(p. 206); Tg (�) � Thring radiation number (p. 214); N (�) � radiation number

(2.) (p. 211); Re (�) � Reynolds number (p. 81); Pr (�) � Prandtl number

(p. 197); Pe (�) � Peclet heat number (p. 180).

This number expresses the ratio of the heat transferred by forced convection to

that transferred by radiation. It characterizes the relation between the convective

energy transfer (in the flow direction) and the heat radiation transfer. It expresses

also the ratio of the absorbing surroundings’ enthalpy to the thermal flow radiated

by a surface which limits the surroundings. In heat transfer by radiation, it has sim-

ilar significance to the Peclet heat number Pe (p. 180) in heat conduction.

Characteristic optical thickness is important for determining the relative ratio of the

heat radiation transfer to the convective transfer for optically thick (τc1) or

optically thin (τ{1) layers. This ratio is expressed by the modified Bo number (3)

for an optically thick layer or by (4) for a thin one. Sometimes, the modified

enthalpy expression (5) is used. For radiation in combustion spaces, expression (6)

is applied.

Info: [A23],[B20].


Ludwig Boltzmann (20.2.1844�5.10.1906), Austrian

physicist.

He founded gas kinetic theory and statistical physics. He

was engaged in entropy and probability. He deduced one of

the radiation laws now known as the Boltzmann law. His

works on electromagnetism and thermodynamics are espe-

cially important.

5.4.4 Bouguer Number (1.) Bu1

Bu1 53RilR2Rmd

Ri (kg m23) � mass concentration of particles in reference gas volume; lR (m) �

mean length of radiation path; Rm (kg m23) � density of mass particles; d (m) �diameter of mass particles.

It relates to fluid flow with mass particles. It characterizes the heat transfer by

radiation into mass particles in flowing gas.

Info: [A23],[A29],[B20].

Pierre Bouguer (16.2.1698�15.8.1758), French mathema-

tician, astronomer and physicist.

He was engaged in research on earth density, photome-

try, astronomical photometry and general light propagation

problems. He is often considered the founder of photome-

try. The Bouguer law expresses the relation between radia-

tion energy absorption and the absorbing surroundings.

5.4.5 Bouguer Number (2.) Bu2

Bu2 5κL

κ (m21) � attenuation coefficient of environment including absorption and disper-

sion coefficients; L (m) � reference length.

It expresses the ratio of the characteristic relative length of a ray path to the free

ray path caused by absorption and radiation scatter. It characterizes the degree of

attenuation (absorbance) of the surroundings in heat radiation transfer in a gas. It is

analogous to the Knudsen number (1.) Kn (p. 69) in gas dynamics.

Info: [A23],[A29].

Pierre Bouguer (see above).

205Thermomechanics

5.4.6 Emissivity ε

ε5εgbεbb

5 12 r; where εAh0; 1i

εgb (�) � grey body absorptivity; εbb (�) � ideal black body absorptivity; r (�) �surface reflectance; Q (W m22) � heat flux transferred by radiation;

σ0 (W m22 K24) � Stefan�Boltzmann constant; T1, T2 (K) � temperature of radi-

ating and absorbing surfaces.

It expresses the relative radiativity (absorbance) of the surface of grey bodies. It

is a non-linear function of the surface temperature and depends on the radiation

wave length. For the amount of heat transferred by radiation, the

Stefan�Boltzmann law is valid.

Q5 εσ0ðT41 2 T4

2 Þ

Thermomechanics. Radiation. Contactless measurement of surface temperatures.

Info: [C4].

5.4.7 Fourier Radiation Number FoR

FoR 5 aκ2τ

a (m2 s21) � thermal diffusivity; κ (m21) � attenuation coefficient; τ (s) � time.

It expresses the ratio of the radiation heat transferred in a system of molecular

diffusion to the radiation heat absorbed in the system. In the dimensionless form, it

characterizes the time of non-stationary heat transfer by radiation. It determines the

relation between the velocity due to temperature field radiation and the physical

parameters of the system.

Info: [A23].


5.4.8 Hottel Number Hot

Hot5α

εσ0T3

α (W m22 K21) � convective heat transfer coefficient; σ0 (W m22 K24) �Stefan�Boltzmann constant; T (K) � absolute temperature; ε (�) � emissivity

(p. 206).

It expresses the ratio of heat flows by convection to those by radiation. Heat

transfer by radiation. Thermomechanics. High-temperature devices. Industrial

furnaces.

Info: [A2].

H.C. Hottel.


5.4.9 Kirpichev Radiation Number KiR

KiR � N21 5σT3

λκ5 SfBu212

σ (W m22 K24) � absorptance (absorption capacity, radiating capacity); T (K) �temperature; λ (W m21 K21) � thermal conductivity; κ (m21) � attenuation coeffi-

cient; N (�) � radiation number (2.) (p. 211); Sf (�) � Stefan number (p. 213);

Bu2 (�) � Bouguer number (2.) (p. 205).

It expresses the ratio of the heat transferred by radiation to that transferred by

molecular conduction in a turbulent flow. It characterizes the heat transfer by radia-

tion in an environment with attenuation effect. Its inverse value is called the radia-

tion number (2.) N (p. 211).

Info: [A23],[A33].

Mikhail Viktorovich Kirpichev (p. 177).

5.4.10 Local Radiation Heat Transfer ξ

ξ5εσ0T

3κxRcpwr

5κxBo

5KReEc21 5KηEi2Re

Bo2Ec

σ0 (W m22 K24) � Stefan�Boltzmann constant; T (K) � temperature; κ (m21) �attenuation coefficient; x (m) � coordinate; R (kg m23) � density; cp (J kg

21 K21) �specific heat capacity; wr (m s21) � reference velocity; ε (�) � emissivity (p. 206);

Bo (�) � Boltzmann number (p. 204); K (�) � radiation high-temperature number

(p. 210); Re (�) � local Reynolds number, see Reynolds number (p. 81); Ec (�) �Eckert number (p. 191); Kη (�) � radiation viscosity (p. 212); Ei (�) � Einstein

number (p. 317).

This transfer expresses the ratio of the heat transferred by radiation in absorbing

surroundings to that transferred by convection. It characterizes the influence of the

radiation and convective heat transfer on the local heat transfer in high-temperature

processes, where the hydrodynamic and radiation gas viscosities and gas flow rate

are considered. The thermal radiation dissipation by gas molecules is presumed to

be negligible.

Info: [A23],[A33].

5.4.11 Nusselt Radiation Number NuR

NuR 5qAκL

ðT 2 TSÞσT3

qA (W m22) � heat flux density; κ (m21) � attenuation coefficient; L (m) � char-

acteristic length; T, TS (K) � radiating environment and wall surface temperature;

σ (W m22 K24) � absorptance (absorption capacity, radiating capacity).

207Thermomechanics

It characterizes the radiation heat transfer intensity of the radiating turbulent

flow into a canal wall. It expresses the size of the thermal impact on the interface

between the radiating gas and a wall.

Info: [A23],[A33].


5.4.12 Optical Thickness of Heat Boundary Layer τ

τ5 kδ ð1Þ;

τ5 5κffiffiffiffiffiνxwr

r5 5

ffiffiffiffiffiffiffiffiffiffiffiPrNξ

pð2Þ

κ (m21) � attenuation coefficient; δ (m) � thickness of thermal boundary layer;

ν (m2 s21) � kinematic viscosity; x (m) � coordinate; wr (m s21) � reference

velocity; Pr (�) � Prandtl number (p. 197); N (�) � radiation number (2.)

(p. 211); ξ (�) � local radiation heat transfer (p. 207).

It expresses the ratio of the thickness of the thinned thermal boundary layer to

that of the unthinned thermal boundary layer. In the dimensionless form, it charac-

terizes the optical thickness of the thermal boundary layer. With the expressionffiffiffiffiffiffiNξ

pc1; it is about an optically thick boundary layer, but with

ffiffiffiffiffiffiNξ

p {1; it isabout an optically thin one. The size of the τ depends strongly on pressure. It drops

with increasing pressure and vice versa.

Info: [A23].

5.4.13 Peclet Radiation Number PeR

PeR 5wRcpεσ0T3

κL

w (m s21) � velocity; R (kg m23) � density; cp (J kg21 K21) � specific heat capacity;

σ0 (W m22 K�4) � Stefan�Boltzmann constant; T (K) � temperature; κ (m21) �attenuation coefficient; L (m) � characteristic length; ε (�) � emissivity (p. 206).

It characterizes the relation between the convective and radiation heat transfers

in the turbulent flow of a radiating media.

Info: [A23],[A33].


5.4.14 Pomerantsev Radiation Number PoR

PoR 5qRV

κσT4ð1Þ;


PoR 5qVL

RwcpTð2Þ

qRV (W m23) � volume density of radiation heat flux; κ (m21) � attenuation coef-

ficient; σ (W m22 K24) � absorptance (absorption capacity, radiating capacity);

T (K) � temperature; qV (W m23) � volume density of heat flux; L (m) � charac-

teristic length; R (kg m23) � density; w (m s21) � velocity; cp (J kg21 K21) � spe-

cific heat capacity.

This number expresses the ratio of the internal volume heat source to the heat

transferred by radiation. It characterizes the internal volume heat source in radia-

tion heat transfer.

Info: [A23].


5.4.15 Radiation Heat Flow (1.) QR1

QR1 5qRA

σT45

qRA

pR

qRA (W m22) � radiation heat flux density; σ (W m22 K24) � absorptance (absorp-

tion capacity, radiating capacity); T (K) � temperature of outer environment;

pR (W m22) � radiation pressure.

It expresses the ratio of the heat flow transferred by radiation to that absorbed

by a surface. It characterizes the resulting radiation heat flow in the heat transfer in

a system.

Info: [A23].

5.4.16 Radiation Heat Flow (2.) QR2

QR2 5DRσT3

λw5

DRaRT3

λ

DR (m2 s21) � radiation diffusion coefficient; σ (W m22 K24) � fluid absorptance

(absorption capacity, radiating capacity); T (K) � temperature; λ (W m21 K21) �thermal conductivity; w (m s21) � flow velocity; aR (J m23 K24) � radiation

constant.

It expresses the ratio of the heat transferred by flowing fluid radiation to that

transferred by conduction. It characterizes the heat transfer in various high-temper-

ature heat-exchanging devices.

Info: [A23].

209Thermomechanics

5.4.17 Radiation Heating Up (1.) N1

N1 5σT3τcRL

σ (W m22 K24) � absorptance (absorption capacity, radiating capacity); T (K) �temperature; τ (s) � time; cR (J m23 K21) � specific volume heat; L5 h (m) �depth of warming through.

It expresses the ratio of the heat, carried to a body by radiation, to the thermal

content of the body. It characterizes the radiation heat transfer on a body surface.

Info: [A23].

5.4.18 Radiation Heating Up (2.) N2

N2 5ðσT3Þ2τλcR

σ (W m22 K24) � absorptance (absorption capacity, radiating capacity); T (K) �flue gases temperature; τ (s) � time; λ (W m21 K21) � thermal conductivity;

cR (J m23 K21) � specific volume heat.

It characterizes the velocity of the radiation heating of bodies, where not the

geometrical body size but other thermal parameters act strongly. These parameters

are expressed by the body thermal conductivity and the boundary condition of sur-

face heat transfer.

Info: [A23].

5.4.19 Radiation High-Temperature Number K

K5σT3ηκR2c2p

5ηc2κσT4

σT3

Rcpc

0@

1A

2

5ηηR

σT3

Rcpc

0@

1A

2

5ηηR

w

c

0@1A

2

σT3

Rcpw

0@

1A

2

5KηEi

Bo

0@

1A

2

σ (W m22 K24) � absorptance (absorption capacity, radiating capacity); T (K) �temperature; η, ηR (Pa s) � dynamic (hydrodynamic) viscosity, radiation dynamic

viscosity; κ (m21) � attenuation coefficient; R (kg m23) � density; cp (J kg21 K21) �

specific heat capacity; c (m s21) � light velocity; w (m s21) � flow velocity;

Kη (�) � radiation viscosity (p. 212); Ei (�) � Einstein number (p. 317); Bo (�) �Boltzmann number (p. 204).


This criterion expresses the ratio of the energy transferred by hydrodynamic and

radiation viscosities to the total energy transferred by radiation and convection. It

characterizes the influence of the hydrodynamic and radiation viscosities and the

flow rate on the resulting energy transfer by radiation and convection in high-

temperature processes.

Info: [A23].

5.4.20 Radiation Number (1.) K

K5λE

εσ0σT35WeHo21Sf 21

λ (W m21 K21) � thermal conductivity; E (Pa) � modulus of elasticity;

σ0 (W m22 K24) � Stefan�Boltzmann constant; σ (N m21) � surface tension;

T (K) � temperature; ε (�) � emissivity (p. 206); We (�) � Weber number

(p. 91); Ho (�) � Hooke number (p. 138); Sf (�) � Stefan number (p. 213).

It characterizes the radiation heat transfer (Sf ) under conditions of inertia force

and surface tension (We) and elasticity (Ho) acting on the flowing fluid.

Info: [A23].

5.4.21 Radiation Number (2.) N

N5λκσT3

λ (W m21 K21) � thermal conductivity; k (m21) � attenuation coefficient;

σ (W m22 K24) � absorptance (absorption capacity, radiating capacity); T (K) �temperature.

It expresses the ratio of heat transferred by conduction to that transferred by

radiation. It characterizes the radiation heat transfer in attenuating the surroundings.

It is analogous to the Planck number (p. 24).

Info: [A23],[A29].

5.4.22 Radiation Parameter Φ

Φ5εσ0T

3Lh

λð1Þ;

Φ5f εσ0T

3Lh

λð2Þ

σ0 (W m22 K24) � Stefan�Boltzmann constant; T (K) � gas temperature;

Lh5AO21(m) � hydraulic diameter; A (m2) � channel cross-section surface area;

211Thermomechanics

O (m) � channel perimeter with fluid contact; λ (W m21 K21) � thermal conduc-

tivity; ε (�) � emissivity of inner channel wall (p. 206); f (�) � function of mean

surface wall temperature.

This parameter expresses the ratio of the heat transferred by radiation in a pas-

sageway to that transferred by conduction in a channel wall. It characterizes the

relation between the radiation and conduction heat transfers in passageways.

Alternatively, it expresses the influence of the radiation on the convective transfer.

Info: [A23],[A29].

5.4.23 Radiation Pressure Rp

Rp 5σT4

cpð1Þ;

Rp 5ηRckp

ð2Þ

σ (W m22 K24) � absorptance (absorption capacity, radiating capacity); T (K) �temperature; c (m s21) � light velocity; p (Pa) � gas pressure; ηR (Pa s) �radiation dynamic viscosity; k (m21) � attenuation coefficient.

It expresses the ratio of the radiation pressure to the static pressure of a gas. It

characterizes the relation between the radiation heat transfer energy and the gas

pressure energy. The value of the number Rp is very small and so is the signifi-

cance of the radiation pressure if the temperature T is not greater than approxi-

mately 105 K or if the pressure p is not too low. In most technical applications, the

Rp number can be neglected. However, it is very important, for example, in high-

temperature plasma physics.

Info: [A23],[B20].

5.4.24 Radiation Viscosity Kη

Kη 5ηηR

5ηc2κσT4

η, ηR (Pa s) � dynamic (hydrodynamic) viscosity, radiation dynamic viscosity;

c (m s21) � light velocity; κ (m21) � attenuation coefficient; σ (W m22 K24) �absorptance (absorption capacity, radiating capacity); T (K) � temperature.

It characterizes the relation between hydrodynamic and radiation dynamic

viscosities.

Info: [A23].


5.4.25 Schuster Number Sch

Sch5κp

κc

κp (m21) � attenuation coefficient by absorption of radiation; κc (m

21) � attenua-

tion coefficient by absorption and dispersion on particles in gas flow.

This number expresses the ratio of the radiation attenuation by dissipation to the

total radiation attenuation by dissipation and absorption. It characterizes the rela-

tion between absorbing and dissipating properties of particles floating in a radiating

gas flow. It states the dispersion portion of the total attenuation of radiation due to

surroundings.

Info: [A23],[A33].

Sir Franz Arthur Friedrich Schuster (1851�1934),

British physicist of German origin.

He is well known for his work in spectroscopy, in which

he is a renowned authority. In addition, he was engaged in

research on electricity behaviour in gases, terrestrial magne-

tism, radiometry, calorimetry and the mathematical period-

icity theory. He wrote the book An Introduction to the

Theory of Optics (1904).

5.4.26 Stark Number Sk

See the Stefan number Sf (p. 213).

5.4.27 Stefan Number Sf

Sf � Sk5εσ0T

3L

λ

σ0 (W m22 K24) � Stefan�Boltzmann constant; T (K) � temperature; L (m) �characteristic length; λ (W m21 K21) � thermal conductivity; Sk (�) � Stark num-

ber (p. 213); ε (�) � emissivity of inner channel wall (p. 206).

It expresses the ratio of the heat transferred by radiation to that transferred by

conduction. It characterizes the coupling between the thermal field in a solid body

and the heat transfer by radiation on the body surface. It is analogous to the Biot

number Bi (p. 173) and therefore is called the Biot radiation number BiR (p. 204)

sometimes. It is often called the Stark number Sk (p. 213).

Info: [A4],[A23],[B20].

213Thermomechanics

Josef Stefan (24.3.1835�7.1.1893), Slovenian�Austrian

mathematician and physicist.

He was engaged in research on the kinetic theory of

gases, thermodynamics and heat transfer by radiation. In

the year 1891, he published his famous work on iceberg

melting in polar seas, in which he laid the foundation for

the non-linear heat conduction theory with phase change:

the eponymous Stefan problem. Contrary to Kirchhoff,

Stefan showed that radiated energy exists in all wavelengths

and is governed by the 4th power of the absolute

temperature.

5.4.28 Thring Radiation Number Tg

See the Boltzmann number Bo (p. 204).

Meredith Wooldridge Thring (17.12.1915�15.9.2006), English engineer.

5.5 Boiling

Boiling represents a phase conversion with fast fluid evaporation within the entire

fluid mass. It passes through three stages involving the initial creation of vapour

kernels, a transient zone and a film boiling. The complicated phase conversion

process is described by a lot of dimensionless quantities. Among the basic ones are

the following: the Bond, Archimedes, Jakob, Kutateladze, Lockhart�Martinelli,

Stanton, Sterman and other numbers, including classic criteria modified for boiling.

5.5.1 Archimedes Steam Number Ar

Ar5R12l 2σ

32

η2l g12

RvRl

� �21

Rl, Rv (kg m23) � liquid and vapour density; σ (N m21) � surface tension; ηl (Pa s) �

liquid dynamic viscosity; g (m s22) � gravitational acceleration.

It expresses the ratio of the buoyancy force (the Archimedes force) of the vapour

phase in a non-isotropic fluid to the molecular friction force in the fluid. It charac-

terizes the mutual action of capillary, viscous and gravity forces in a vapour�fluid

mixture. It influences the origin and development of gravity capillary waves in a

viscous fluid. It is a special modification of the Galilei steam number (p. 217).

Heat transfer in boiling.



5.5.2 Beginning of Boiling Instability N

N5ðTS 2 TvÞllvRlL

Tvσ

TS (K) � wall temperature before rising of vapour bubbles (initial maximal super-

heating of liquid); Tv (K) � vapour temperature; llv (J kg21) � specific latent heat

of vaporization; Rl (kg m23) � density of liquid; L (m) � characteristic length;


It characterizes the origin of the instability of a fluid as it begins boiling.

Info: [A23].

5.5.3 Boiling Number Boi

Boi5qA

llvRlwð1Þ;

Boi5L2qA

Qm llv5

qA

qm llvð2Þ

qA (W m22) � area heat flux density; llv (J kg21) � specific latent heat of vaporiza-

tion; Rl (kg m23) � liquid density; w (m s21) � flow velocity; L (m) � characteristic

length; Qm (kg s21) � mass flux; qm (kg s21 m22) � area mass flow density.

It expresses the ratio of the specific vapour heat flow perpendicular to a

bypassed wall to the total heat flow parallel with the wall. It characterizes the pro-

cess of creating vapour kernels in fluid boiling and the process of heat transfer

intensification. For flow through a pipeline, with the internal diameter d, L5 d

holds.

Info: [A23],[B56].

5.5.4 Boiling Stability K

K5qcrit

llvR12v½σgðRl 2 RvÞ�

14

qcrit (W m22) � surface density of critical heat flux; llv (J kg21) � specific latent

heat of vaporization; Rv, Rl (kg m23) � density of vapour and liquid; σ (N m21) �surface tension; g (m s22) � gravitational acceleration.

It expresses the ratio of the evaporation rate on a heated surface to the mean

velocity of vapour bubble growth. It characterizes the degree of the dynamic action

of the evaporation process on the vapour�fluid system stability in boiling.

Sometimes, it is called the criterion of the heat transfer hydrodynamic crisis in

boiling.

Info: [A23].

215Thermomechanics

5.5.5 Convection Number Ncon

Ncon 512 x

x

� �0:9 RgRl

� �0:5

x (�) � mass ratio of vapour fraction; Rg, Rl (kg m23) � density of gas (vapour)

and liquid.

Essentially, it represents the modified Martinelli parameter X (p. 106) used for

vapour bubble flow in boiling fluid.

Info: [A23],[B56].

5.5.6 Frequency Evolution of Steam Bubbles N

N5RlRv

clσdllvpβ

RlRl 2 Rv

Rl, Rv (kg m23) � liquid and vapour; cl (J kg21 K21) � liquid and vapour density;

σ (N m21) � surface tension; d (m) � characteristic dimension of vapour bubble;

llv (J kg21) � specific latent heat of vaporization; p (Pa) � pressure; β (K21) �thermal expansivity of vapour.

It expresses the ratio of the surface stress force to the thermogravitational force.

It characterizes the evolution frequency of vapour bubbles in boiling fluid.

Info: [A23].

5.5.7 Froude Boiling Number Frb

Frb 52

R3g

ðTS 2 TvÞclRlllvRv

� �4ðπalÞ2 ð1Þ;

Frb2 5w2

gR12

RvRl

� �21

ð2Þ

R (m) � radius of vapour bubble or drop; g (m s22) � gravitational acceleration;

TS, Tv (K) � temperature of heated wall and vapour; cl (J kg21 K21) � specific

heat capacity of liquid; Rl, Rv (kg m23) � density of liquid and vapour; llv (J kg

21) �specific latent heat of vaporization; al (m2 s21) � thermal diffusivity of liquid;

w (m s21) � motion velocity.

This number expresses the inertia-to-buoyancy forces ratio. It characterizes the

influence of force relations on the origin and evolution of the vapour bubbles in

boiling. In form (1), it expresses the condition for bubbles originating; in form (2),

it expresses the condition for vapour bubble motion. The vapour bubble radius can


be determined in a linear scale R5V1/3, where V denotes the bubble volume, or an

effective radius R5 (3V/4πn)1/3 can be used.

Info: [A23].

William Froude. (p. 63)

5.5.8 Froude Heat Number Frt

Frt 5gL3

a2

g (m s22) � gravitational acceleration; L (m) � characteristic length; a (m2 s21) �thermal diffusivity.

It expresses the gravitation-to-thermodiffusion forces ratio. It characterizes the

mean intensity of the heat transfer under forced convection and fluid boiling in a

tube.

Info: [A23].


5.5.9 Galilei Steam Number Ga

Ga5g

ν2l

σgðRl 2 RvÞ

� �32

g (m s22) � gravitational acceleration; νl (m2 s21) � kinematic viscosity of liquid;

σ (N m21) � surface tension; Rl, Rv (kg m23) � density of liquid and vapour.

It characterizes the heat transfer in closed two-phase thermosiphon systems

(a fluid boiling in an evaporator, vapour motion in a transfer part, condensation,

condensate return to the evaporator), all under the influence of gravity and inertia

forces.

Info: [A23].

Galileo Galilei (p. 123).

5.5.10 Heat Activity K

K5ðRcλÞlðRcλÞw

R (kg m23) � density; c (J kg21 K21) � specific heat capacity; λ (W m22 K21) �thermal conductivity; subscripts: l � liquid; w � channel wall.

This activity expresses the ratio of the fluid thermal activity to the canal wall

thermal activity. It characterizes the influence of material thermal properties on the

non-stationary heat transfer in fluid boiling. Together with the off liquid heating Ψ,

217Thermomechanics

(p. 222) it expresses the temperature of the film boiling crisis. In bubble boiling, it

is of use similarly as the characteristic roughness dimension.

Info: [A23].

5.5.11 Impulse Heating NP

NP 5qA

Rlllv

RvRl

Rb� �2

1k

qA (W m22) � heat flux density; Rl, Rv (kg m23) � density of liquid and vapour;

R (kg m23) � density of generated vapour nuclei; llv (J kg21) � specific latent heat

of vaporization; b (�) � mean value of coefficient depending on liquid superheat-

ing; k (�) � boiling constant.

This quantity expresses the conditions to bring a fluid to boiling with impulse

heating when the vapour origin is caused spontaneously by originating random-

character kernels. For k. 1, the temperature of intensive kernel nuclei origin is

reached. For kc1, the vapour phase nuclei are created in a fluctuating way even

for small masses of the fluid, and the created vapour conversion centres do not

play any role. With k5 3 in an experiment, the growth of bubbles is delayed by

inertia forces (Rayleigh’s case). With k5 0.5, the growth rate is limited by the heat

supply. Heat transfer in boiling.

Info: [A33].

5.5.12 Jakob Evaporation Speed Number Ja

Ja � To5qA

llvRvdf

qA (W m22) � heat flux density; llv (J kg21) � specific latent heat of vaporization;

Rv (kg m23) � density of saturated vapour; d (m) � diameter of vapour bubble;

f (Hz) � generation frequency of vapour bubbles; To (�) � Tolubinsky number

(p. 226).

It expresses the ratio of the vapour phase generation (bubbles) rate, on a heated

surface, to the mean growth rate of vapour bubbles. It characterizes the heat trans-

fer intensity in boiling. It is the equivalent of the modified Peclet boiling speed

number Pewb (p. 223). It expresses also the ratio of the maximum vapour bubble

diameter to the overheated fluid film thickness. Sometimes, it is called the

Tolubinsky number To (p. 226).

Info: [A23].

Max Jakob (see below).


5.5.13 Jakob Number Ja1

Ja1 5c1ΔT

llv

RlRv

cl (J kg21 K21) � specific heat capacity of liquid; ΔT5 Tl�Tn (K) � liquid super

heating; Tl, Tn (K) � temperature of liquid and saturated vapour; llv (J kg21) � spe-

cific latent heat of vaporization; Rl, Rv (kg m23) � liquid and vapour density.

This number expresses the ratio of the heat supplied to overheat the unit volume

of a fluid to the specific volume latent heat of evaporation. It characterizes the heat

transfer intensity in fluid boiling. It depends on the fluid pressure and overheating.

It drops with the pressure increasing because the vapour density increases, and vice

versa. With the pressure greater than the atmospheric pressure (Ja1# 20), the

vapour bubbles grow in size due to the heat supplied from a heated surface over

the adhering fluid layer. With low pressures (Ja1$ 20), the heat that led to a

vapour bubble is transferred from the overheated fluid on the inter-phase surface.

To determine the vapour bubble diameter, the formula is in the first case

R5ffiffiffiffiffiffiffiffiffiffiffiffi2βJa1

p ffiffiffiffiffiaτ

p

and in the second case

R5 2γJa1ffiffiffiffiffiaτ

p

and generally

R5 γJa1 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiγ2Ja21 1 2βJa1

q� � ffiffiffiffiffiaτ

p

The last expression is valid for Ja15 0.1�1000, which corresponds to pressures

of 0.01�10 MPa. The factors β and γ depend on geometric parameters. For exam-

ple, for the wettability angle ϑ5 40�90�, it is γ5 0.1�0.49, β66.

Info: [A23],[B20],[B56].

Max Jakob (20.7.1879�1955), American physicist of

German origin.

After the year 1910, he was engaged in research on ther-

modynamics and heat transfer and carried out a great deal

of work in this area. His work concerned, especially, steam

and air properties under high pressure, devices to measure

thermal conductivity, explanations of the boiling mecha-

nism and condensation, and flow in pipelines and channels.

In this sphere, he published approximately 500 works.

219Thermomechanics

5.5.14 Kutateladze Evaporation Number Ku

Ku5llv

clΔTð1Þ;

Ku5llvRlwqA

5Ku�PeNu21 ð2Þ;

Ku5llvRlνlqAL

5Ku�PrNu21 ð3Þ;

Ku4 5 Ja211

RlRv

ð4Þ

llv (J kg21) � specific latent heat of vaporization; cl (J kg21 K21) � specific heat

capacity of liquid; ΔT5 Tv2 Ts (K) � difference of vapour and wall temperatures;

Rl, Rv (kg m23) � density of liquid and vapour; w (m s21) � liquid velocity;

qA (W m22) � heat flux density; vl (m2 s21) � kinematic viscosity of liquid; L (m) �

characteristic length; Ku� (�) � Kutateladze evaporation number defined by the rela-

tion (1); Pe (�) � Peclet heat number (p. 180); Nu (�) � Nusselt number (p. 196);

Pr (�) � Prandtl number (p. 197); Ja1 (�) � Jakob number (p. 219).

It expresses the ratio of the specific evaporation heat to the heat content increase

in the overheating of one of the phases. In form (2), it expresses the ratio of the

phase conversion rate to the phase flow rate. In form (3), it expresses the ratio of

flow inertia forces originating due to the phase conversion process to viscous

forces. It is a special expression of the Reynolds number Re (p. 81). In form (1), it

is also called the Trawton number Tr (p. 232), and in form (4) it is the Jakob

vapour number.

Info: [A23].

Samson Semenovich Kutateladze (p. 237).

5.5.15 Local Boiling Stability K

K5Rvw

2vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

σgðRl 2 RvÞp ð1Þ;

K5Rlw

2lffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

σgðRl 2 RvÞp ð2Þ

Rv, Rl (kg m23) � density of vapour and liquid; wv, wl (m s21) � velocity of vapour

and liquid; σ (N m21) � surface tension; g (m s22) � gravitational acceleration.

It characterizes the local vapour�fluid set stability under dynamic and capillary�gravitational mutual action. Equation (1) expresses the approximate transition of the

non-viscous fluid bubble boiling to the film boiling, and equation (2) expresses the


effect of non-viscous fluid being driven out from the microporous surface in the

barbotage.

Info: [A23].

5.5.16 Lockhart�Martinelli Number (1.) LM1, Lp

LM1 5RvσLη2v

Rv (kg m23) � vapour density; σ (N m21) � surface tension; L (m) � characteristic

length; ηv (Pa s) � dynamic viscosity of vapour.

It expresses the ratio of the fluid surface stress force to the vapour viscosity

force. It characterizes the influence of hydrodynamic forces on the heat transfer in

film boiling, for example, of cryogen fluids in a pipeline.

Info: [A23].

5.5.17 Lockhart�Martinelli Number (2.) LM2, Xtt

LM2 512 x

x

� �0:9 RvRl

� �0:5 ηlηv

� �0:1

x (�) � relative mass flux of vapour fraction; Rv, Rl (kg m23) � density of vapour

and liquid; ηv, ηl (Pa s) � dynamic viscosity of vapour and liquid.

It characterizes the convective fluid boiling in vertical tubes and constant cross-

section canals. Together with the boiling number Boi (p. 215), it determines the

heat transfer. The formula

ααl

5C1 Boi1C2

1

Xtt

� �C3

( )C4

is valid, where C15 6.73 103; C25 3.53 1024; C35 0.67; C45 1; αl is the heat

transfer coefficient on the fluid side of a channel; α5αk1αv, where αk is the con-

vection part of heat transfer and αv is the part corresponding to the formation of

vapour nuclei.

Info: [A23].

5.5.18 Number of Steam Nucleus N

N5dp

dT

� �Tn

d2ðRl 2 RvÞqAλlσRl

221Thermomechanics

p (Pa) � pressure; T, Tn (K) � temperature and saturation temperature; d (m) �characteristic dimension of vapour bubble; Rl, Rv (kg m23) � density of liquid and

vapour; qA (W m22) � area heat flux density; λl (W m21 K21) � thermal conduc-

tivity of liquid; σ (N m21) � surface tension.

It characterizes the number of vapour nuclei originating in the vapour creation

in fluid.

Info: [A23].

5.5.19 Nusselt Boiling Number Nub

Nub 5αdλl

5αλl

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiσ

gðRl 2 RvÞr

α (W m22 K21) � heat transfer coefficient; d (m) � characteristic dimension of

vapour bubble; λl (W m21 K21) � thermal conductivity of liquid; σ (N m21) �surface tension; g (m s22) � gravitational acceleration; Rl, Rv (kg m23) � density

of liquid and vapour.

This number characterizes the convective heat transfer in the heating and boiling

of fluids.

Info: [A23].


5.5.20 Off Liquid Heating Ψ

Ψ5clðTn 2 TlÞ

llv

cl (J kg21 K21) � specific heat capacity of liquid; Tn, Tl (K) � temperature of satu-

ration and liquid; llv (J kg21) � specific latent heat of vaporization.

It expresses the ratio of the specific heat required to finish the fluid heating to

the boiling point to the specific latent heat of evaporation. It characterizes the

degree of insufficient fluid heating in boiling. With increasing Ψ, the effect of heat

flow intensification is reduced.

Info: [A23].

5.5.21 Peclet Boiling Number Peb

Peb 5qAcl

llvλl


gðRl 2 RvÞr

qA (W m22) � area heat flux density; cl (J kg21 K21) � specific heat capacity of

liquid; λl (W m21 K21) � thermal conductivity of liquid; llv (J kg21) � specific


latent heat of vaporization; σ (N m21) � surface tension; g (m s22) � gravitational

acceleration; Rl, Rv (kg m23) � density of liquid and vapour.

It characterizes the heat transfer by convection and conduction in a fluid bound-

ary layer in boiling.

Info: [A23].


5.5.22 Peclet Boiling Speed Number Pewb

Pewb �fLRvcld

λl

5wRvclλl


gðRl 2 RvÞr

ð1Þ;

Pewb � Sh21 5df

wt

ð2Þ

f (Hz) � frequency of rising bubbles; L (m) � characteristic length; Rv, Rl (kg m23) �

density of vapour and liquid; cl (J kg21 K21) � specific heat capacity of liquid;

λl (W m21 K21) � thermal conductivity of liquid; d (m) � characteristic dimension of

vapour bubble; w5 qA (llvRv)21 (m s21) � boiling velocity; qA (W m22) � heat flux

density; llv (J kg21) � specific latent heat of vaporization; σ (N m21) � surface ten-

sion; g (m s22) � gravitational acceleration; wt (m s21) � heat diffusion velocity;

Sh (�) � Strouhal number defined by the relation (3) (p. 87).

It expresses the ratio of the velocity of vapour bubbles originating in boiling to

the heat diffusion rate in the fluid. It characterizes the rate at which vapour bubbles

originate in boiling. In equation (2), it represents the inverse value of the Strouhal

number Sh defined by equation (3) (p. 87).

Info: [A23].


5.5.23 Rayleigh Steam Number Ra

Ra5GrPr5gL3βlΔT

νlalð1Þ;

Ra5gL3critνvav

Rl 2 RvRv

ð2Þ

where Lcrit 5 2πσ

gðRl 2 RvÞ

� �12

g (m s22) � gravitational acceleration; L (m) � characteristic length; βl (K21) �

volume thermal expansion coefficient of liquid; ΔT (K) � temperature difference;

νl, νv (m2 s21) � kinematic viscosity of liquid and vapour; al, av (m

2 s21) � thermal

223Thermomechanics

diffusivity of liquid and vapour; Rl, Rv (kg m23) � density of liquid and vapour; σ(N m21) � surface tension; Lcrit (m) � critical wavelength; Gr (�) � Grashof heat

number (p. 185); Pr (�) � Prandtl number (p. 197).

It characterizes the heat transfer by free convection in boiling and thermal insta-

bility in fluid. In the basic form (1), it is valid for fluid heating, and in the modified

form (2), it holds for the boiling and resulting instability of a fluid. In the latter

case, the critical wavelength is taken as the characteristic length.

Info: [A23],[A29].

Lord Rayleigh, John William Strutt (p. 187).

5.5.24 Reynolds Boiling Number Reb

Reb 5qA

llvRvνl


gðRl 2 RvÞr

5NubPr21Ku21 Rl

Rvð1Þ;

Reb 5qAd

llvRvνlð2Þ

qA (W m22) � heat flux density; llv (J kg21) � specific latent heat of vaporization;

Rv, Rl (kg m23) � density of vapour and liquid; νl (m2 s21) � kinematic

viscosity of liquid; σ (N m21) � surface tension; g (m s22) � gravitational acceler-

ation; d (m) � characteristic tube dimension; Nub (�) � Nusselt boiling number

(p. 222); Pr (�) � Prandtl number (p. 197); Ku (�) � Kutateladze evaporation

number (p. 220).

It expresses the ratio of the inertia force arising in flowing fluid with phase con-

version to the internal friction force. It characterizes the heating and boiling of a

fluid. It is the hydrodynamic boiling criterion. In equation (2), it expresses the ratio

of the pipe surface heat loading to the specific latent evaporating heat. It is the

measure for pipe wall thermal loading in fluid boiling.

Info: [A33].


5.5.25 Reynolds Boiling Speed Number Rewb

Rewb 5d

νlRl

π6d3Rv fn

�

d (m) � characteristic dimension of vapour bubble; νl (m2 s21) � kinematic viscos-

ity of liquid; Rl, Rv (kg m23) � density of liquid and vapour; f (Hz) � generation

frequency of vapour bubbles; n (�) � number of vapour nuclei passing through

surface unit.

It characterizes the fluid boiling rate.


Info: [A23].


5.5.26 Reynolds Bubbling Evaporation Number Rebb

Rebb 5Rlwdηl

5wd

νl

Rl (kg m23) � liquid density; νl (m

2 s21) � kinematic viscosity of liquid; ηl (Pa s) �dynamic viscosity of liquid; w (m s21) � propagation velocity of vapour bubble;

d (m) � characteristic dimension of vapour bubble.

It expresses the ratio of the vapour bubble inertia force to the fluid friction force.

It characterizes the hydrodynamic relations in fluid bubble boiling.

Info: [A23].


5.5.27 Stanton Boiling Number Stb

Stb 5αllvqAcl

ð1Þ;

Stb 5qA

RvclwlðTS 2 TnÞð2Þ

α (W m22 K21) � heat transfer coefficient; llv (J kg21) � specific latent heat of

vaporization; qA (W m22) � surface density of heat flux; cl (J kg21 K21) � specific

heat capacity of liquid; Rv (kg m23) � vapour density; wl (m s21) � liquid motion

velocity; TS, Tn (K) � wall and saturation temperatures.

This number characterizes the convective heat transfer in fluid boiling. In equa-

tion (2), it is valid for the shooting process4 in film boiling.

Info: [A23],[A29].


5.5.28 Steam Pressure Np

Np 5p

σ


gðRl 2 RvÞr

p (Pa) � pressure; σ (N m21) � surface tension; g (m s22) � gravitational accelera-

tion; Rl, Rv (kg m23) � density of liquid and vapour.

It characterizes the pressure in heat transfer, for example, in closed two-phase

thermosiphon systems in which the boiling and condensation processes proceed

under the action of gravitation and inertia forces.

Info: [A23],[A29].

225Thermomechanics

5.5.29 Steam Superheating Number Ψ

Ψ5cpvðTv 2 TnÞ

llv

cpv (J kg21 K21) � specific heat capacity at constant pressure; Tv (K) � vapour

temperature; Tn (K) � saturation temperature; llv (J kg21) � specific latent heat of

vaporization.

It expresses the superheated vapour specific heat to the specific latent heat of

evaporation. With Ψ-0, the heat flow from a wall causes vapour generation and

superheating.

Info: [A23].

5.5.30 Sterman Number Sn

Sn5llv

cpvTn

llv (J kg21) � specific latent heat of vaporization; cpv (J kg21 K21) � specific heat

capacity of saturated vapour; Tn (K) � temperature of saturated vapour.

It expresses the ratio of the specific evaporation heat to the specific heat of satu-

rated steam. It characterizes the developed bubble boiling.

Info: [A23].

5.5.31 Tolubinsky Number To

See the Jakob evaporation speed number Ja (p. 218).

5.5.32 Wallis Number Ws

Ws51

η½d3gðRl 2 RvÞRl�

12

η (Pa s) � dynamic viscosity of liquid; d (m) � tube diameter; g (m s22) � gravita-

tional acceleration; Rl, Rv (kg m23) � density of liquid and vapour or gas.

It characterizes the fluid viscosity influence on stroke flow in boiling.

Info: [A23].

5.5.33 Weber Boiling Number Web

Web 5Rvðw2

v 2w2l ÞL

σ


Rv (kg m23) � vapour density; wv, wl (m s21) � flow velocity of vapour and liquid;

L (m) � characteristic length dimension (wall thickness, boundary layer thickness,

drop or bubble diameter); σ (N m21) � surface tension.

It expresses the inertia-to-capillary forces ratio. It characterizes the hydrody-

namic forces influence on the process of bubbles originating in fluid boiling.

Info: [A23].



5.6 Evaporation

Evaporation is the phase conversion in which a fluid is converted to vapour on its

surface. The evaporation occurs under an arbitrary fluid temperature. The higher

the temperature is, the faster the evaporation. Further, the evaporation is influenced

by the fluid’s properties, by gas pressure and motion over the fluid and by the

evaporating area size. In addition to several evaporation numbers, some of the fun-

damental dimensionless quantities are the Borishanskyi, Bulygin, Gukhman, Jakob

evaporation, Lomonosov, Richman, Spalding and Trawton numbers.

5.6.1 Borishansky Number Bs

Bs5pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

σðRl 2 RvÞgp

p (Pa) � pressure; σ (N m21) � surface tension; Rl, Rv (kg m23) � density of liquid

and vapour; g (m s22) � gravitational acceleration.

This number expresses the ratio of the absolute pressure in the system to the

pressure gradient on the interface of phases. It characterizes the pressure conditions

in fluid evaporation.

Info: [A23].

Veniamin Mironovich Borishansky (born 1917), Russian engineer.

5.6.2 Bulygin Number Bu

Bu5llvcmpp

clðT 2 TiÞ

llv (J kg21) � specific latent heat of evaporation; cmp (Pa

21) � pressure mass capac-

ity of vapour; p (Pa) � pressure; cl (J kg21 K21) � specific heat capacity of liquid;

T, Ti (K) � material and initial temperatures.

It expresses the ratio of the evaporation heat to the fluid heat (the heat necessary

to bring the fluid to boiling). It characterizes the evaporation heat transfer and the

227Thermomechanics

body accumulating ability in high-intensity heat transfer. The influence of the heat

which is lost during evaporation in molecular transfer is considered.

Info: [A23],[A29].

N.P. Bulygin.

5.6.3 Drag Evaporation Coefficient CD,ev

CD;ev 5FD

12Rcw2

FD (N) � drag force acting on particle; Rc (kg m23) � density of cold liquid;

w (m s21) � velocity difference between particle and cold liquid.

It expresses the resistance of the vapour layer arising on the surface of a hot par-

ticle during its penetration into cold fluid. It is analogous to the Dalton number Dal

(p. 394). Evaporation.

Info: [B15].

5.6.4 Euler Modified Number Eumod

Eumod 5HLRlgw2GRg

HL (m) � pressure height of liquid; Rl (kg m23) � liquid density; Rg (kg m23) �vapour density; g (m s22) � gravitational acceleration; WG (m s21) � vaporization

velocity from free surface.

It expresses the vapour flow across a mass transfer.

Info: [A29].


5.6.5 Evaporation Flow Number Ev

Ev5wevDp

η

wev (kg m22 s21) � vaporization velocity; Dp (m) � particle diameter; η (Pa s) �dynamic viscosity.

It expresses the dynamic-to-viscous forces ratio. It characterizes the vapour flow

inside a vapour film, with the vapour being induced during cold fluid evaporation

at the interface between the cold fluid and the vapour. Evaporation.

Info: [A23].


5.6.6 Evaporation Number (1.) Nev

Nev 5w2

llv

w (m s21) � flow velocity; llv (J kg21) � specific latent heat of vaporization.

It characterizes the evaporation process. It expresses the ratio of the flow kinetic

energy to the fluid latent heat.

Info: [A23].

5.6.7 Evaporation Number (2.) Nev

Nev 5cl

βllv5N�

evGcDu21

cl (J kg21 K21) � specific heat capacity of liquid; β (K21) � volume thermal

expansion coefficient; llv (J kg21) � specific latent heat of evaporation; N�

ev (�) �evaporation number (1.) (p. 229); Gc (�) � Gay-Lussac number (p. 14); Du (�) �Dulong number (p. 191).

It characterizes the evaporation process. It expresses the ratio of the heat neces-

sary to heat the fluid to the evaporation temperature to the specific latent heat of

evaporation.

Info: [A23].

5.6.8 Evaporation Number (3.) Elasticity Nev

Nev 5K

llvRl5

a2

llv5N�

evCau

K (Pa) � bulk modulus; llv (J kg21) � specific latent heat of vaporization;

Rl (kg m23) � liquid density; a (m s21) � sound velocity; N�ev (�) � evaporation

number (1.) (p. 229); Cau (�) � Cauchy number (aeroelasticity parameter)

(p. 155).

It characterizes the fluid evaporation process. It expresses the ratio of the energy

of forces acting with fluid evaporation to the specific latent evaporation heat. It is

called the elasticity evaporation number as well.

Info: [A23].

5.6.9 Gravitational Evaporation Number Ngv

Ngv 5D3

pR2cg

η2c

Dp (m) � particle diameter; Rc (kg m23) � density of cooling agent; g (m s22) �gravitational acceleration; ηc (Pa s) � dynamic viscosity of cooling agent.

229Thermomechanics

It describes the particles in a cooling system under the influence of gravitation.

It expresses the gravity-to-viscosity forces ratio. Together with the evaporation

flow number Ev (p. 228) and the Reynolds number Re (p. 81), it is among the fun-

damental criteria influencing the force acting on a cooled particle under laminar

flow conditions during film evaporation.6

5.6.10 Gukhman Evaporation Number Gu

Gu5tg 2 ts

Tg

tg, ts (K) � hot gas temperature, moist material surface (liquid level) temperature;

Tg (K) � thermodynamic gas temperature.

With evaporation from a free fluid surface, it expresses the ratio of the outer

mass transfer intensity (moisture), expressed by the volume fluid evaporation inten-

sity (moisture) in the air boundary layer, to the outer heat transfer intensity. It is

the thermodynamic criterion of evaporation in isobaric, adiabatic conditions. It is

used to describe the evaporation from a fluid-free surface or moist material drying.

With non-adiabatic evaporation, an additional criterion Θ5 T/Tl must be intro-

duced to express the thermal conditions’ influence if the fluid surface temperature

Tl differs from the adiabatic air saturation temperature close to the moist body sur-

face. In moist material drying, it is called the Gukhman drying number Gu (p. 251).

Info: [A4],[A23].

Alexander Adolfovich Gukhman, Russian chemical engineer.

5.6.11 Jakob Evaporation Number Jak

Jak5llv

cpΔT

llv (J kg21) � specific latent heat of vaporization; cp (J kg21 K21) � specific heat

capacity; ΔT (K) � temperature difference.

It expresses the ratio of the specific latent heat of evaporation to the fluid heat

capacity. It is analogous to the number of phase changes N (p. 179).

Info: [B20].

Max Jakob (p. 219).

5.6.12 Leidenfrost Number Lei

Lei5Bi

Pe2

Bi (�) � Biot number (p. 173); Pe (�) � Peclet heat number (p. 180).


It characterizes the heat transfer by evaporation with hot wall cooling by a thin

fluid film flowing over its surface. The hardening of bodies in fluids and other

technological processes are examples. The Leidenfrost phenomenon.

Info: [B30].

Johann Gottlob Leidenfrost (27.11.1715�2.12.1794),

German physicist and chemist.

Above all, his basic contribution consisted in the discov-

ery of the Leidenfrost phenomenon. With this, in the vicin-

ity of a hot body being much hotter than the boiling

temperature, which is the so-called Leidenfrost point, the

fluid forms a thin lagging steam layer. This steam layer

forms a thermal barrier which prevents the fluid from get-

ting into fast boiling. Therefore, it takes much longer for

the fluid to evaporate. With the temperature under the

Leidenfrost point, this thermal barrier does not exist, due to

which the fluid evaporates swiftly.

5.6.13 Lomonosov Number Lo

Lo5gΔRLRw2

5ArRe22

g (m s22) � gravitational acceleration; ΔR (kg m23) � vapour and air density dif-

ference; L (m) � characteristic length; R (kg m23) � air density; w (m s21) � air-

flow velocity; Ar (�) � Archimedes hydrodynamic number (p. 53); Re (�) �Reynolds number (p. 81).

It expresses the ratio of the buoyancy force (Archimedes force) to the inertia

force. It acts in processes of non-isothermal water evaporation from surfaces of

vessels and tanks into flowing air.

Info: [A23],[A33].

Mikhail Vasilyevich Lomonosov (19.11.1711�15.4.1765),

Russian scientist, encyclopedist, writer and poet.

He was a founder of both Russian science and culture.

He studied physics, chemistry, metallurgy, history and arts,

and he was interested in other fields as well. By creating

the mechanical heat theory, expressing and proving the law

of mass conservation, he laid the first foundations for gen-

eral heat theory. He showed that heat is a form of move-

ment of particles. He wrote a whole range of works,

including the first history of Russia.

231Thermomechanics

5.6.14 Richman Number Ri

Ri5qmL

Rν5

wL

ν

qm (kg s21 m22) � mass flow density; L (m) � characteristic length; R (kg m23)

liquid density; ν (m2 s21) � kinematic viscosity; w (m s21) � flow velocity.

It expresses the ratio of inertia forces to viscous forces which act in condensate

evaporation. It characterizes the influence of hydrodynamic conditions on the evap-

oration process. Sometimes, it is called the transversal flow criterion for parallel

running heat and mass transfers during evaporation or vapour condensation in a

vapour�gas mixture. It can be called the Reynolds mass number Rem (p. 262).

Info: [A33].

Georg Vilhelm Richman (1711�1753), Russian physicist.

5.6.15 Spalding Number (2.) Sp, B’

Sp5 cpΔT llv 2ΦQm

� �21

cp (J kg21 K21) � specific heat capacity of surroundings; ΔT (K) � surroundings

and liquid level temperature difference; llv (J kg21) � specific latent heat of vapori-

zation; Φ (W) � radiation heat flux; Qm (kg s21) � mass flux.

This number expresses the ratio of the received heat to the latent heat of an

evaporating material. It characterizes the heat transfer in the evaporation of fluid

drops.

Info: [A23].

Brian Spalding (born 1923), English engineer (p. 200).

5.6.16 Trawton Number Tr

Tr � Ku5llv

ΔTcp

llv (J kg21) � specific latent heat of vaporization; ΔT (K) � temperature differ-

ence; cp (J kg21 K21) � specific heat capacity; Ku (�) � Kutateladze evaporation

number defined by equation (1) (p. 220).

It expresses the ratio of the specific latent evaporation heat to that delivered to

heat a fluid volume unit. It characterizes the fluid evaporation process with

the temperature corresponding to normal conditions. It is a special case of the

Kutateladze evaporation number Ku defined by relation (1) (p. 220).

Info: [A23].

Frederic Thomas Trawton (1863�1922), Irish physicist.


5.7 Condensation

For condensation, several dimensionless quantities are used, to represent the gas

or vapour transformation of the fluid. For example, there is a set of quantitative

condensation criteria expressing energetic, force and other influences which

are connected with condensation. Other dimensionless quantities are the following:

the Archimedes, Galilei, Kapica, Grigul, Kirbride, Kutateladze, Laplace,

McAdams, Nusselt, Prandtl, Reynolds, Sherwood, Stanton and Weber numbers. In

addition, some dimensionless quantities are used for such things as various binary

mixtures and thermocapillary processes.

5.7.1 Archimedes Condensation Number Ark

Ark 5gL3

ν2l12

RvRl

� �5Gak 12

RvRl

� �

g (m s22) � gravitational acceleration; L (m) � characteristic length; νl (m2 s21) �

kinematic viscosity of liquid; Rv, Rl (kg m21) � density of vapour and liquid; Gak(�) � Galilei condensation number (p. 235).

This number expresses the ratio of the product of the static buoyancy, arising

from the difference of densities, and the inertia force to the square of the friction

force. It characterizes the influence of the free two-phase vapour�condensate envi-

ronment on the heat transfer during condensation.

Info: [A23].


5.7.2 Condensation Number (2.) K

K5L3R2glvlληΔT

L (m) � characteristic length; R (kg m23) � condensation liquid density; g (m s22) �gravitational acceleration; lvl (J kg21) � specific latent heat of condensation;

λ (W m21 K21) � thermal conductivity of condensation liquid; η (Pa s) � dynamic

viscosity of condensation liquid; ΔT (K) � temperature difference during

condensation.

It characterizes the condensation process on a vertical wall.

Info: [A23].

5.7.3 Condensation Number of Binary Steam Mixtures Kb

Kb 5 2σ@σ@cm

ðlvlη2l Þ21

233Thermomechanics

σ (N m21) � surface tension; cm (�) � mass fraction; lvl (J kg21) � specific latent

heat of condensation; ηl (Pa s) � dynamic viscosity of liquid.

It characterizes the influence of a vapour binary mixture on drop condensation.

The surface stress is presumed to be determined by the condensation liquid first.

Info: [A23].

5.7.4 Condensation Number of Flowing Steam Kw

Kw 52w2

v

ðgνlÞ23

RvRl

wv (m s21) � vapour velocity; g (m s22) � gravitational acceleration; νl (m2 s21) �

kinematic viscosity of liquid; Rl, Rv (kg m23) � density of liquid and vapour.

It characterizes the influence of clean vapour velocity on the heat transfer during

drop condensation on a vertical surface. With it, the mutual vapour friction force

on the condensate film surface, the gravity force and the molecular film friction are

considered.

Info: [A23].

5.7.5 Condensation Number of Thermocapillary Movement Kk

Kk 5βσΔTcritσRcrit

Rlν2l5βσΔTcritLak

where βσ 51

σ@σ@T

; Rcrit 52σTn

RllvlΔTcrit

βσ (K21) � temperature coefficient of surface tension; ΔTcrit (K) � critical temper-

ature difference; σ (N m21) � liquid surface tension; Rcrit (m) � critical radius

of condensation nucleus; Rl (kg m23) � liquid density; νl (m2 s21) � kinematic

viscosity of liquid; T (K) � temperature; Tn (K) � saturation temperature; lvl(J kg21) � specific latent heat of condensation; Lak (�) � Laplace condensation

number (p. 238).

It expresses the thermocapillary-to-viscosity forces ratio. It characterizes the

influence of thermocapillary motion in immovable, clean vapour condensation.

Info: [A23].

5.7.6 Condensation Number of Thermocapillary Separation Kδ

Kδ 5βpΔTE

Rlν2l Rcrit

; where Rcrit 52σTn

RllvlΔTcrit


βp (K21) � thermal coefficient of displaced pressure; ΔT (K) � temperature differ-

ence between saturation temperature and surface wall temperature; E (J) � pressure

energy of separation; Rl (kg m23) � liquid density; νl (m2 s21) � kinematic viscos-

ity of liquid; σ (N m21) � surface tension; Tn (K) � saturation temperature;

lvl (J kg21) � specific latent heat of condensation; ΔTcrit (K) � critical temperature

difference.

It expresses the ratio of mutually acting forces between condensate molecules

and those of molecularly active surface material on a condensation surface. It char-

acterizes the influence of the wall surface material on the drop condensation of

immobile clean vapour.

Info: [A23].

5.7.7 Degree of Vapour Condensation ε

ε5Qm1 2Qm2

Qm1

Qm1, Qm2 (kg s21) � vapour mass fluxes on input and output of vapour condenser.

It expresses the ratio of the vapour mass flow on a condenser inlet to that on its

outlet. It characterizes the condensation efficiency of the condenser.

Info: [A23].

5.7.8 Galilei Condensation Number Gak

Gak 5gL3

ν2lð1Þ;

Gak 5gx3

ν2lð2Þ

g (m s22) � gravitational acceleration; L (m) � characteristic length (wall height);

νl (m2 s21) � kinematic viscosity of liquid; x (m) � distance in direction of con-

densation liquid layer generation.

It expresses the ratio of the molecular friction force during condensation to the

gravitation force. It characterizes the influence of friction and gravitation on the

heat transfer in condensation. Expression (1) holds for developed condensation, and

expression (2) is valid for a developing condensation layer.

Info: [A23].

Galileo Galilei (p. 123).

235Thermomechanics

5.7.9 Grigull Condensation Number Ggk, Z

Ggk 5g

ν2l

� �13 λlΔTL

lvlηl5Ga

13k

λlΔT

lvlηl

g (m s22) � gravitational acceleration; νl (m2 s21) � kinematic viscosity of conden-

sate liquid; λl (W m21 K21) � thermal conductivity of condensate liquid

ΔT5 Tn�TS (K) � temperature difference during condensation; Tn (K) � saturation

temperature; TS (K) � condensation wall surface temperature; L (m) � characteristic

length; lvl (J kg21) � specific latent heat of condensation; ηl (Pa s) � dynamic

viscosity of condensation liquid; Gak (�) � Galilei condensation number (p. 235).

It characterizes the condensate film flow. It represents the relative effective

length of the condensate flow-down surface. The condensate film flow is laminar

in horizontal tubes with Ggk, 3900 and in vertical ones with Ggk, 2300. A turbu-

lent flow occurs with Ggk. 3900 in horizontal tubes and with Ggk. 2300 in the

case of vertical tubes.

Info: [A23].

Ulrich Grigull (p. 186).

5.7.10 Kapica Number Ka

Ka5σ3

gR3l ν4l

σ (N m21) � surface tension; g (m s22) � gravitational acceleration; Rl (kg m23) �liquid density; νl (m

2 s21) � kinematic viscosity of liquid.

This number expresses the ratio of the surface stress force to the product of the

volume force and the viscosity force. It characterizes fluid film thermal conductiv-

ity increase due to wave periodic flow on a vertical wall.

Info: [A23].

Peter Leonidovich Kapica (8.7.1894�8.4.1984), Russian


He was engaged in research on magnetism under low

temperatures and helium superfluidity problems. He

designed several unique devices during his stay in England,

where he collaborated with Ernst Rutherford, and in the for-

mer Soviet Union as well. His research on microwave gen-

erators led him to the study of controlled nuclear reaction

starting in the 1960s.


5.7.11 Kirbride Condensation Number Kb

Kb5αλl

ν2lg

� �13

5NuGa213

k

α (W m22 K21) � heat transfer coefficient; λl (W m21 K21) � thermal conductivity

of condensation liquid; νl (m2 s21) � kinematic viscosity of condensation liquid;

g (m s22) � gravitational acceleration; Nu (�) � Nusselt number (p. 196); Gak (�)

� Galilei condensation number (p. 235).

It characterizes the heat transfer in membrane clean vapour condensation and

turbulent condensate flow. It is often called the condensation number (1.).

Info: [A23].

5.7.12 Kutateladze Condensation Number Ku

Kup 5lvl

cpvðTv 2 TlÞð1Þ;

Kuk 5lvl

cplðTl 2 TSÞð2Þ

lvl (J kg21) � specific latent heat of condensation; cpv, cpl (J kg

21 K21) � specific

heat capacity of vapour and condensation liquid; Tv (K) � vapour temperature;

Tl (K) � mean temperature of condensation liquid; TS (K) � external surface wall

temperature.

This number expresses the ratio of the condensation specific latent heat to the

specific heat cooling of one phase. In equation (1), it characterizes the vapour con-

densation process, and in equation (2), it characterizes the condensate subcooling.

Therefore, it is called the vapour (1) or condensate (2) Kutateladze number.

Info: [A23].

Samson Semenovich Kutateladze (18.7.1914�20.3.1986),

Russian engineer and physicist.

In the year 1936, he formulated the basic similarity condi-

tions for heat-exchange processes with the change of state

phase of material. His experimental work with the heat trans-

fer in liquid metals in nuclear engineering is extensive.

He formulated the basic ideas of the hydrodynamic theory of

the boiling crisis. His books Osnovy teorii teploobmena

(Fundamentals of Heat Exchange Theory, 1979) and Analiz

podobija v teplofizike (Similarity Analysis in Thermophysics,

1982) are significant in the evolution of the field.

237Thermomechanics

5.7.13 Laplace Condensation Number Lak

Lak 5ΔpL

σð1Þ;

Lak 5RlσLη2l

ð2Þ

Δp (Pa) � pressure difference across condensation liquid layer; L (m) � thickness

of condensation layer; σ (N m21) � surface tension; Rl (kg m23) � density of

condensation liquid; ηl (Pa s) � dynamic viscosity of condensation liquid.

It expresses the ratio of the pressure force to the surface stress force. It charac-

terizes the force relations in the membrane or drop condensation of vapour.

Info: [A23].

Pierre-Simon Laplace (p. 71).

5.7.14 McAdams Number Mc

Mc5α4LηlΔT

λ3l R

2l glvl

α (W m22 K21) � heat transfer coefficient; L (m) � characteristic length; ηl (Pa s) �dynamic viscosity of condensation liquid layer; λl (W m21 K21) � thermal conduc-

tivity of condensation liquid layer; Rl (kg m23) � density of condensation liquid;

ΔT (K) � mean temperature difference between vapour and wall g (m s22) �gravitational acceleration; lvl (J kg

21) � specific latent heat of condensation.

It characterizes the heat transfer in fluid membrane condensation. It is constant

for a specific surface orientation.

Info: [A23].

William Henry McAdams (15.3.1892�2.5.1975), American

chemical engineer.

He was engaged in the heat transfer, distillation and flow-

ing of viscous fluids. His publication Heat Transmission is

widely known. In addition, he edited many of technical

papers. His work on the Manhattan project and his original

work on nuclear submarines and jet aircraft are well known.

5.7.15 Middle Mass Temperature Θ

Θ5T 2 T1

Tn 2 T1


T (K) � mean temperature in condenser; T1 (K) � temperature in input in con-

denser; Tn (K) � temperature of saturated vapour.

It characterizes the mean mass temperature of the dispersed water flow in a con-

denser (water and water vapour).

Info: [A23].

5.7.16 Nusselt Condensation Number (1.) Nukl

Nukl 5L3R2l glvlλlηlΔT

5PrKuGak ð1Þ;

Nukl 5qAL

λlΔTð2Þ

L (m) � characteristic length; Rl (kg m23) � condensation liquid density;

λl (W m21 K21) � thermal conductivity of condensation liquid; ηl (Pa s) �dynamic viscosity of condensation liquid; g (m s22) � gravitation acceleration;

lvl (J kg21) � specific latent heat of condensation; ΔT (K) � temperature difference

during condensation; qA (W m22) � surface heat flux density; Pr (�) � Prandtl

number (p. 197); Ku (�) � Kutateladze evaporation number (p. 220); Gak (�) �Galilei condensation number (p. 235).

It characterizes the influence of molecular friction and gravity acting on the heat

transfer in membrane condensation.

Info: [A23].


5.7.17 Nusselt Condensation Number (2.) Nuk2

Nuk2 5λl

αR2l gη2l

� �13

5R2l gη2l

� �13

5Ga13k

λl (W m21 K21) � thermal conductivity of condensation liquid; Rl (kg m23) � den-

sity of condensation liquid; ηl (Pa s) � dynamic viscosity of condensation liquid;

g (m s22) � gravitational acceleration; α (W m22 K21) � heat transfer coefficient;

δ (m) � thickness of condensation liquid layer; Gak (�) � Galilei condensation

number (p. 235).

It characterizes the heat transfer in membrane condensation on a vertical wall

under the mutual influence of friction force, gravity and molecular friction in

flowing-down condensate film.

Info: [A23].

Ernst Kraft Wilhelm Nusselt (see above).

239Thermomechanics

5.7.18 Phase Transformation Number N

N5lvl

clðTn 2 T1Þlvl (J kg

21) � specific latent heat of condensation; cl (J kg21 K21) � specific heat

capacity of condensation liquid; Tn (K) � saturation temperature of vapour; T1 (K) �steam condenser input temperature.

It expresses the ratio of the specific latent condensation heat to that withdrawn

from the saturated vapour by a unit quantity of the cooling water. It is among the

set of quantitative condensation criteria F, G, J, K, L, M, R (p. 240).

Info: [A23].

5.7.19 Prandtl Condensation Number Prk

Prk 5νlal

5PeRe21

νl (m2 s21) � kinematic viscosity of condensation liquid; al (m

2 s21) � thermal dif-

fusivity of condensation liquid; Pe (�) � Peclet heat number (p. 180); Re (�) �Reynolds number (p. 81).

It characterizes the relation between the temperature and velocity fields in con-

densation or, alternatively, between the thickness of the temperature and hydrody-

namic boundary layers in gradientless bypassing of solid bodies. In liquid metals, it

is often Prk{1, which can be of use both in drop condensation (mercury) and

membrane condensation (sodium, potassium).

Info: [A23].


5.7.20 Proportion of Vapour Content x

x5Qmv

Qmv 1Qml

Qmv, Qml (kg s21) � vapour and condensation liquid mass fluxes.

It expresses the ratio of the vapour mass flow to the mass flow of a

vapour�condensate mixture. It characterizes the relative vapour content in the mix-

ture, for example, at the inlet and outlet of a condenser tube.

Info: [A23].

5.7.21 Quantitative Condensation Criteria F, G, J, K, L, M, R

F5gdηllvl

w2NλlΔT

5Fr21PrN ð1Þ;


G5ΔTλl

ηllvl

ffiffiffiffiffiffiffiffiffiRlηlRvηv

r5Pr21N 21M5RM ð2Þ;

J5glvl

qA

ffiffiffiffiffiffiffiffiffiffiRlηldw3N

s5Re

12ðFrLÞ21 ð3Þ;

K5qA

lvl

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid

wNRvηv

s5MLRe2

12 ð4Þ;

L5qAd

ηllvlð5Þ;

M5

ffiffiffiffiffiffiffiffiffiRlηlRvηv

rð6Þ;

R5Pr21N 21 ð7Þ

g (m s22) � gravitation acceleration; d (m) � outer tube diameter; ηl, ηv (Pa s) �dynamic viscosity of condensation liquid and vapour; lvl (J kg

21) � specific latent

heat of condensation; wN (m s21) � velocity of vapour free flow; λl, λv(W m21 K21) � thermal conductivity of condensation liquid and vapour; ΔT (K) �temperature difference; Rl, Rv (kg m23) � density of condensation liquid and

vapour; qA (W m22) � surface density of heat flux; Fr (�) � Froude number (1.)

(p. 62); Pr (�) � Prandtl number (p. 197); N (�) � phase change (p. 180); Re (�)

� Reynolds number (p. 81).

These criteria characterize the energetic, force and other quantitative relations

under certain conditions, which are, for example, constant density of the through-

wall heat flow (criteria J, K, L), constant wall temperature (criteria F, G) and physi-

cal parameters of phases (M). The phase change N (p. 180) is also among the quan-

titative criteria.

Info: [A23].

5.7.22 Reynolds Number of Film Condensation Rebk

Rebk 5αΔTL

lvlηl11ϕ

clΔT

lvl

� �21ð1Þ;

Rebk 5Qm

2πrηlð2Þ

α (W m22 K21) � heat transfer coefficient; ΔT (K) � temperature difference dur-

ing condensation; L (m) � characteristic length; lvl (J kg21) � specific latent

heat of condensation; ηl (Pa s) � dynamic viscosity of condensation liquid;

241Thermomechanics

cl (J kg21 K21) � specific heat capacity of condensation liquid; ϕ (�) � relative

subcooling of condensation liquid; Qm (kg s21) � mass flux in condensation layer;

r (m) � inner tube radius.

It expresses the ratio of the condensate film inertia force to the molecular fric-

tion in a condensate film. It characterizes the hydrodynamic force relations in a

thin condensate film. In equation (2), it is used with partial condensation in tubes.

Info: [A23].


5.7.23 Reynolds Number of Two-Phase Condensation Rek

Rek 5wNL

νv5

λlðTn 2 TSÞlvlηl

ð1Þ;

Rep 5wvx

νvð2Þ

wN (m s21) � vapour free flow velocity; L (m) � characteristic length; νv (m2 s21) �

kinematic viscosity of vapour; λl (W m21 K21) � thermal conductivity of condensa-

tion liquid; ηl (Pa s) � dynamic viscosity of condensation liquid; Tn (K) � temperature

of saturated vapour; TS (K) � temperature of wall surface; lvl (J kg21) � specific latent

heat of condensation; wv (m s21) � local vapour velocity; x (m) � distance in direction

of condensation liquid layer generation.

It expresses the ratio of the flowing fluid inertia force to the molecular flowing

fluid friction force. It characterizes the hydrodynamic force relations in vapour con-

densation. In expression (1), it represents a resultant criterion, and in expression

(2), it represents a local criterion.

Info: [A23].


5.7.24 Sherwood Condensation Number Shk

Shk 5L3gRllvl4νlλlΔT

� �14

L (m) � characteristic length; g (m s22) � gravitational acceleration; Rl (kg m23) �density of condensation liquid; νl (m

2 s21) � kinematic viscosity of condensation

liquid; λl (W m21 K21) � thermal conductivity of condensation liquid; lvl (J kg21) �

specific latent heat of condensation; ΔT (K) � temperature difference between

vapour and wall.

It characterizes the influence of molecular friction and gravity on the heat trans-

fer in vapour membrane condensation on a vertical wall.

Info: [A23].

Thomas Kilgore Sherwood (p. 265).


5.7.25 Stanton Condensation Number Stk

Stk 5α

Rlclw5NuPe21 ð1Þ;

Stk 5αlvlqAcl

ð2Þ

α (W m22 K21) � heat transfer coefficient; Rl (kg m23) � density of condensation

layer; cl (J kg21 K21) � specific heat capacity of condensation layer; w (m s21) �

velocity growth of condensation film; lvl (J kg21) � specific latent heat of condensa-

tion; qA (W m22) � surface heat flux density; Nu (�) � Nusselt number (p. 196); Pe

(�) � Peclet heat number (p. 180).

It expresses the ratio of the condensate temperature change along a wall to the

temperature drop between the wall and the condensate. It characterizes the convec-

tive heat transfer process in drop and membrane condensations. Its significance is

similar to that of the Nusselt number Nu (p. 196).

Info: [A23].


5.7.26 Weber Condensation Number Wek

Wek 5ðRl 2 RvÞgL2

σð1Þ;

Wek 5RlgL

2

σð2Þ

Rl, Rv (kg m23) � density of liquid and vapour; g (m s22) � gravitational accelera-

tion; L (m) � characteristic length; σ (N m21) � surface tension.

It expresses the ratio of the gravity force to the surface stress force. It charac-

terizes the force relations in water condensation which appear, for example, in the

formation of the arising condensate drop. Sometimes, it is used in form (2), in the

description of the surface distillation process.

Info: [A23].



5.8 Heat and Mass Transfer

Compound heat and mass transfer is characterized by a mathematical model in the

expression of second-order partial equations set with composed boundary condi-

tions. A great number of dimensional quantities are used to describe heat and mass

243Thermomechanics

transfer. Among the most important are the Bulygin, Fedorov, Fourier, Kirpichev,

Kossovitch, Lykov and Posnov numbers, and also many mostly modified, dimen-

sionless quantities for heat or mass transfer, such as the Sherwood, Schmidt and

Stanton numbers.

5.8.1 Absorption Number Ab

Ab5 βffiffiffiffiffiffiffix

Dw

rð1Þ;

Ab5 βffiffiffiffiffiffiffiffiffixL

Dw1

rð2Þ

β (m s21) � absorption coefficient of liquid transfer; x (m) � surface length with

liquid layer; D (m2 s21) � gas diffusivity in liquid layer; w (m s21) � mean liquid

velocity in the layer; L (m) � layer thickness; w1 (m2 s21) � volume velocity.

In the dimensionless form, it expresses the mass transfer coefficient, for exam-

ple, in gas absorption on wet walls of chambers. In expression (2), the fluid film

thickness is considered.

Info: [A23].

5.8.2 Biot Mass Number Bim

Bim 5αmL

λm

5βLam

αm (kg m22 s21) � specific mass transfer; L (m) � characteristic length;

λm (kg m21 K21) � specific mass transfer; β (m s21) � mass transfer coefficient;

am (m2 s21) � mass diffusivity.

This number expresses the ratio of the mass transfer on a fluid�body interface

to the mass transfer inside the L-thick wall of the body. It characterizes the relation

between the mass transfer on the outer surface of the body by convection and that

by conduction inside the body. The third-type boundary condition is used in solving

heat transfer tasks, for example, in the drying or wetting of various materials.

Info: [A23].


5.8.3 Bodenstein number Bd

Bd � Pem 5w L

D

w (m s21); L (m) � characteristic axial length; D (m2 s21) � effective axial diffu-

sivity; Pem (�) � Peclet mass number (p. 258).


It expresses the ratio of the convective heat transfer rate to the diffusion rate. It

characterizes the diffusion process in chemical reactors. In substance, it is about

expressing the degree of the approximation of a system being examined to ideal-

ized models of reacting flows. With Bd-N, it is about total displacement of a

system and with Bd-0, it is about total mixing. For real diffusion and combined

models, N,Bd. 0 is valid. It is a special case of the Peclet mass number Pem(p. 258) for mass transfer. It describes the diffusion in granulated material layers in

forced convection.

Info: [A23],[A33],[B20].

Max Ernst August Bodenstein (15.7.1871�3.9.1942),

German physical chemist.

In the year 1916, he explained the velocity of certain

chemical reactions with the chain reaction concept. He was

convinced that in those cases the process starts by random

collision of two molecules of different chemicals. This col-

lision generates heat, causing another couple of molecules

to react and generate further heat, and so on. In later years,

this chain reaction concept was applied to designing the

atomic reactor and the atomic bomb.

5.8.4 Bulygin Number Bu

Bu2llvδpΔp

clΔT5RmKo

llv (J kg21) � specific latent heat of evaporation; δp (Pa21) � Soret pressure coeffi-

cient; p (Pa) � pressure difference; cl (J kg21 K21) � specific heat capacity of liquid;

ΔT (K) � temperature difference; Rm (�) � Ramzin number (p. 261);

Ko (�) � Kossovitch number (p. 253).

It expresses the ratio of the heat consumed to evaporate a portion of the mois-

ture in a molecular vapour�gas mixture transfer to that consumed to heat the wet

material to the boiling point. It characterizes the phase conversion in high-tempera-

ture mass transfer especially. It is about a filtering vapour transfer in wet material

and the wet material’s ability to accumulate heat, connected with this transfer.

With it, the heat which is lost to change the moisture to vapour participating in the

molecular transfer is considered.8

Info: [A23],[A33].

N.P. Bulygin.

5.8.5 Colburn Mass Number Jm, Com

Jm � Sc � Prm 5νD

5Sh

ReffiffiffiffiffiSc3

p

245Thermomechanics

ν (m2 s21) kinematic viscosity; D (m2 s21) diffusion coefficient; Sc (�) � Schmidt

number (p. 263); Prm (�) � Prandtl mass number (p. 260); Sh (�) � Sherwood

number (p. 264); Re (�) � Reynolds number (p. 81).

This number characterizes the mass transfer in forced viscous fluid flow. It is

also called the Schmidt number Sc (p. 263) or the Prandtl mass number Prm(p. 260). Heat and mass transfers.

Info: [A20].

Allan Philip Colburn (p. 190).

5.8.6 Cooling of Porous Bodies Np

Np 5qmcpL

λð12μÞ

qm (kg s21 m22) � mass flux density; cp (J kg21 K21) � specific heat capacity;

L (m) � characteristic length; λ (W m21 K21) � thermal conductivity of porous

wall; μ (�) � porosity (p. 24).

It expresses the ratio of heat transferred by convection inside a porous wall to

that transferred by conduction with an equal temperature gradient.

Info: [A23].

5.8.7 Coupling Energy Number E

E5l

llv

l (J kg21) � specific bond heat; llv (J kg21) � specific latent heat of evaporation.

It expresses the ratio of the heat consumed to overcome the moisture-material

coupling to the specific evaporation heat. It characterizes the energy consumed to

release the moisture in hygroscopic material drying.

Info: [A23],[A33].

5.8.8 Drew Number Dr

Dr5ðMA 2MBÞxA 1MB

xA 2 xASðMB 2MAÞlnMv

MS

MA, MB (kg mol21) � molar mass (gram molecule) of components A, B; xA (�)

molar fraction of A in diffusion flow; xAS (�) � molar fraction of component A in

the wall; Mv, MS (kg mol21) � molar mass (gram molecule) of the mixture in

vapour and wall.


It characterizes the mass transfer intensity in a boundary layer with said inten-

sity causing the change in the velocity profile in binary systems. For these systems,

it also represents the drag coefficient CD (p. 60).

Info: [A33].

5.8.9 Dufour Number Du

Du5r

cpLe

ðN1N21Þ0

ðN1N 21ÞðN2N 21Þ 5r

cpLe

ðp1p21Þ0ðp1p21Þðp2p21Þ

ð1Þ;

Du5qD

cpTð2Þ

r (J kg21 K21) � specific gas constant; cp (J kg21 K21) � specific heat capacity;

N1, N2 (�) � molecule number of components 1 and 2 in volume of binary gas

mixture, N5N11N2; p1, p2 (Pa) � partial pressures, p5 p11 p2; qD (J kg21) �heat transferred by diffusion in gas mixture; T (K) � gas mixture temperature;

Le (�) � Lewis number (p. 254).

It characterizes the ratio of the diffuse heat and mass transfers, in a binary mix-

ture of gases under isotropic conditions, to the enthalpy of the unit mixture mass,

provided the linear diffusion rate equals that of conduction. Thermodiffusion.

Info: [A23],[A33].

Louis Dufour (1832�1892), Swiss physicist.

5.8.10 Fedorov Number (1.) Fe

Fe5εδtllvcp

5 εKoPn

δt (K21) � Soret thermal coefficient; llv (J kg

21) � specific latent heat of evapora-

tion; cp (J kg21 K21) � specific heat capacity of damp material; ε (�) � liquid

phase conversion in vapour (p. 255); Ko (�) � Kossovitch number (p. 253);

Pn (�) � Posnov diffusion number (p. 259).

This number expresses the ratio of the heat necessary to evaporate the moisture

in a material internally to the average specific thermal capacity of the wet material.

It characterizes the heat and mass transfer process, especially with respect to the

drying of wet material, and defines the transition between quiet and boiling fluid

layers. The criterion does not depend on the choice of heat and mass transfer poten-

tials, but only on ε, δt coefficients and the thermodynamic characteristics. It is

applied as a generalized variable in the analytical solution of material drying

problems.9

Info: [A23],[A33].

Igor Mikhailovich Fedorov, Russian physicist.

247Thermomechanics


Fe5 δtβ� 5NtrPn

δt (K21) � Soret thermal coefficient; β� (K) Dufour coefficient; Ntr (�) � transfer

number (p. 268); Pn (�) � Posnov diffusion number (p. 259).

It characterizes the transfer processes. It is analogous to the Posnov diffusion

number Pn (p. 259) for mass transfer.

Info: [A23].

Igor Mikhailovich Fedorov (see above).


Fe5 dekv

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4gR2g3η2

RsRg

21

!3

vuut 5

ffiffiffiffiffiffiffiffiffi4

3Ar

3

r

dekv (m) � equivalent particle diameter; g (m s22) � gravitational acceleration;

Rg, Rs (kg m23) � density of gas and particles; η (Pa s) � dynamic viscosity;

Ar (�) � Archimedes hydrodynamic number (p. 53).

It characterizes the transition from the immobile layer to the uplifting one in

fluidizing processes. The criterion is analogous to the Archimedes hydrodynamic

number Ar (p. 53).

Info: [A23],[A33].

Igor Mikhailovich Fedorov (see above).

5.8.13 Fourier Filtration Number Fop

Fop 5apτL2

5FoLuf ; where ap 5kh

μ

ap (m2 s21) � pressure filtration velocity; τ (s) � time; L (m) � characteristic

length; k (m s21) � filtration coefficient; h (m) � mean static height; μ (�) �porosity (p. 24); Fo (�) � Fourier number (p. 175); Luf (�) � Lykov filtration

number (p. 256).

It characterizes the time of non-stationary filtration of moisture in the porous

material.

Info: [A23].



5.8.14 Fourier Mass Number FoD, Fom

FoD 5DτL2

ð1Þ;

Fom 5amτL2

5FoLu ð2Þ

D (m2 s21) � diffusivity; τ (s) � time; L (m) � characteristic length; am (m2 s21) �mass diffusivity; Fo (�) � Fourier number (p. 175); Lu (�) � Lykov number

(p. 256).

This number expresses the ratio of the intensity of the volume mass diffusion

change to that of the time change of moisture content in a material. In a dimension-

less expression, it characterizes the non-stationary mass (moisture) transfer. It

determines the relation between the change rate of the mass transfer potential field

and the physical properties and size of the system in molecular material transfer in

a non-stationary state.

Info: [A23],[A33].


5.8.15 Frossling Mass Number Fsm

Fsm 5Sh22

Re12Sc

13

Sh (�) � Sherwood number (p. 264); Re (�) � Reynolds number (p. 81);

Sc (�) � Schmidt number (p. 263).

It characterizes the convective mass transfer for a ball located in flowing fluid.

Info: [A23].

Karl Gustav Frossling (born 1913), Swedish engineer.

5.8.16 Geometric Number Ge

Ge5h

H

h (m) � bundle perimeter; H (m) � bundle height.

It expresses a dimensionless bundle height. It is used to determine the mass

transfer in bundles, for example, in drying wood.

249Thermomechanics

5.8.17 Graetz Mass Number Gzm

Gzm 5RemPrm 5PemL

dH

Rem (�) � Reynolds mass number (p. 262); Prm (�) � Prandtl mass number (p.

260); Pem (�) � Peclet mass number (p. 258); L (m) � characteristic length; dH(m) � hydraulic diameter.

It expresses the ratio of the convective mass transfer to the diffusion heating or

cooling in fluid flow inside closed pipelines and other systems.

Info: [A23].

Leo Graetz (p. 193).

5.8.18 Granulation Number Au

Au5lls

llvΔu

lls (J kg21) � specific heat of crystallization; llv (J kg21) � specific latent heat of

evaporation; Δu (�) � moisture difference.

It expresses the ratio of the heat arising in crystallization and drying to the total

heat necessary to evaporate moisture from a dried solution. With it, the influence

of the phase conversion (crystallization) of the solution (molten material) layer on

the moisture evaporation intensity on the granule surface is considered. It deter-

mines the heat flow direction among the granules (grains) and bypassing gas. With

the granulation in a fluidization layer, it determines the boundary between the gran-

ulation zones in drying (Au, 1) and cooling of molten material (Au. 1).

Info: [A23],[A33].

5.8.19 Grashof Mass Number Grm

Grm 5gL3βc

ν2lðRs 2 RÞ5 gL3

ν2l

MNTS

MTP21

� �ð1Þ;

Grm 5gL3

ν2l

ΔRR

ð2Þ

g (m s22) � gravitational acceleration; L (m) � characteristic length; βc (m3 kg21) �

volume expansion coefficient inducing solution; ν1 (m2 s21) � kinematic viscosity of

fluid; Rs (kg m23) � mass concentration of dissolved matter at surface; R (kg m23) �

mass concentration of dissolved matter; M, MN (kg mol21) � molar mass of

steam�gas mixture at liquid surface and in sufficient distance from surface; TS, TP(K) � liquid surface temperature and external environment temperature.


It expresses the ratio of the product of inertia and buoyancy forces to the square

of a viscous force. It characterizes the mass transfer by natural convection in a

non-isotropic vapour�gas mixture, with the transfer being caused by the tempera-

ture difference and vapour concentration in the air of the vapour�gas mixture.

Info: [A23].

Franz Grashof (p. 185).

5.8.20 Gukhman Drying Number Gu

See the Gukhman evaporation number Gu (p. 230).

Alexander Adolfovich Gukhman, Russian chemical engineer.

5.8.21 Heating Up of Moist Particles Nt, Nm

Nt 5

ffiffiffiffiffiffiffiffiffiffibw

aL

rð1Þ;

Nm 5

ffiffiffiffiffiffiffikL2

a

rð2Þ

w (m s21) � motion velocity of material in dryer; a (m2 s21) � thermal diffusivity;

L5Vd S21 (m) � ratio of absolute dry body volume to its surface; Vd (m

3) � vol-

ume of absolute dry body; S (m2) � surface of absolute dry body; k (s21) � dry

coefficient; b (�) � logarithmical decrement (p. 18).

It characterizes the heating intensity of dispersive spherical particles in gas flow

drying with convective and radiation heat supply of constant value and temperature

changing T5 To exp (28wτ). It also characterizes material motion velocity in a

drying stove w(Nt) and evaporation rate Nm.

Info: [A33].

5.8.22 Internal Heat and Mass Transfer N

N5Bi

εKo

Bi (�) � Biot number (p. 173); ε (�) � liquid phase conversion in vapour

(p. 255); Ko (�) � Kossovitch number (p. 253).

It expresses the ratio of the heat supplied to a body material to the heat con-

sumed in internal mixture evaporation in the material.

Info: [A23],[A33].

251Thermomechanics

5.8.23 J-Mass Factor Jm

Jm 5βRqm

νD

� 235

βw

νD

� 235NumRe

21Pr213

m 5βwSc

23

β (m s21) � mass transfer coefficient; R (kg m23) � density; qm (J kg21 m22) �mass flux density; ν (m2 s21) � kinematic viscosity; D (m2 s21) � diffusivity;

w (m s21) � flow velocity; Num (�) � Nusselt mass number (p. 258); Re (�) �Reynolds number (p. 81); Prm (�) � Prandtl mass number (p. 260); Sc (�)

Schmidt number (p. 263).

It characterizes the convective transfer of mass, heat and momentum.

Info: [A23].

5.8.24 Jermakov Number Er

Er5kqS

Rl

k (m s21) � energy absorbing coefficient by freeze-drying; qS (W m22) � surface

heat flux density; R (kg m23) � density of liquid or vapour; l (J kg21) � specific

latent heat.

This number characterizes the sublimation drying process in electromagnetic

heating with a constant drying rate. It is the measure of the ratio of the heat amount

delivered to the drying material in a certain time to that consumed to evaporate

unit volume of drying material in the same amount of time.

Info: [A33].

E.A. Jermakova.

5.8.25 Kirpichev Mass Number Kim

Kim 5qmL

λmtΔϑ5

qmL

amtRΔuΔϑð1Þ;

Kim 5BimϑS 2ϑP

Δϑð2Þ

qm (kg s21 m22) � mass flux density; L (m) � characteristic length; λmt(kg m21 s21 K21) � thermal mass conductivity; Δϑ (K) � potential difference of

mass transfer; amt (m2 s21 K21) � thermal mass diffusivity; R (kg m23) � density;

Δu (�) � moisture content difference; ϑS, ϑP (K) � mass transfer potential of wall

and environment; Bim (�) � Biot mass number (p. 244).

It characterizes the relation between outer and inner mass flow intensities. In

mass transfer tasks, it represents the second-type boundary condition.


Info: [A23].


5.8.26 Kossovitch Number Ko

Ko5llvcm

cpΔT5

llvΔu

cpΔT

llv (J kg21) � specific latent heat of evaporation; cm (�) � mass fraction;

cp (J kg21 K21) � specific heat capacity of damp material; ΔT (K) � temperature

difference; Δu (�) � dampness difference.

It is the phase conversion criterion. It expresses the ratio of the evaporation heat

for moisture evaporation to the heat for wet body heating. It characterizes the con-

vective heat transfer in evaporation. It represents the ratio of the specific heat con-

sumed to evaporate the whole amount of moisture to that consumed by wet

material when heating to maximum temperature.

Info: [A23].


Lg1 5D1 εD

D

D (m2 s21) � molecular diffusivity; εD (m2 s21) � vortex mass diffusivity.

It expresses the ratio of the sum of molecular and swirl mass transfer intensities

to the molecular heat transfer intensity. It characterizes the mass transfer in a turbu-

lent flow.

Info: [A23].

Joseph-Louis Lagrange (p. 70).

5.8.28 Lebedev Number Lb

Lb5μbðTP 2 TÞRscmpp

ð1Þ;

Lb5TR

TPð2Þ;

Lb5umed

ucritð3Þ

μ (�) � porosity (p. 24); b5 @ðmvV21v Þ=@Tðkg m23 K21Þ � temperature gradient

of vapour concentration; mv (kg) � vapour mass; Vv (m3) � vapour volume;

253Thermomechanics

T, TP (K) � initial temperature of material and environment; Rs (kg m2 3) �density of solid phase; cmp (kg kg21 Pa2 1) � pressure mass capacity of vapour

(vapour mass per unit mass of dry gas and per unit of pressure difference); p (Pa) �pressure; TR (K) � temperature of radiating source; umed (�) � mean moisture

content in a body; ucrit (�) � critical moisture content.

This number expresses the ratio of the vapour flow by molecular expansion to

the total macroscopic vapour flow from porous material. It characterizes thermal

processes in drying intensification. In expression (2), it represents the volume evap-

oration of dispersed fluid in a boundary layer. It determines the heat transfer

increase due to boundary layer thickness reduction with the temperature increase of

a radiating source. In expression (3), it represents the ratio of a material’s mean

moisture with drying rate reduction to a material’s mean critical moisture. With it,

the heat transfer coefficient reduction due to material moisture reduction is consid-

ered. Sometimes expression (2) is called the Gukhman modified criterion.

Info: [A23].

Peter Nikolaevich Lebedev (8.3.1866�14.3.1912),

Russian physicist.

He founded the Russian school of physicists. He was the

first to discover and investigate mm-range electromagnetic

waves (1895) and addressed himself to the nature of the ter-

restrial magnetic field. He was engaged in the action of

light on solid substances (1900) and gases (1908). He con-

firmed the electromagnetic theory of light from the quanti-

tative point of view. In his last years, he devoted himself to

the problem of ultrasound.

5.8.29 Lewis Number Le

Le5λ

RcpD5

a

D5 ScPr21 5PrmPr

21 ð1Þ;

Le5D

að2Þ

λ (W m21 K21) � thermal conductivity; R (kg m23) � density; cp (J kg21 K21) �specific heat capacity; D (m2 s21) � molecular diffusivity; a (m2 s21) � thermal

diffusivity; Sc (�) � Schmidt number (p. 263); Prm (�) � Prandtl mass number


It expresses the thermal-to-molecular diffusivities ratio. It characterizes the

mutual relation of heat and mass transfers in various materials. It is the ratio of

chemical potential field change to a thermal field in the flow of gas or fluid mix-

tures. For gases, it is LeAh0.8; 1.2i, for fluids LeAh70; 100i. In expression (2), it is

usually called the Lewis�Semenov number, or sometimes only the Lewis number.

Info: [A23].


Warren Kendall Lewis (21.8.1882�9.3.1975), American

chemical engineer.

He is called the father of modern chemical engineering.

He was engaged in researching distillation and evaporation,

petroleum flow modelling, predicting the lifetime of petro-

leum deposits and developing methods for exploiting them.

His work concerning powder fluidization and the control of

its movement in combustion chambers, and in heat and cat-

alytic cracking, was very significant.

Nikolay Nikolayevich Semyonov (p. 48).

5.8.30 Lewis Turbulence Number Letur

Letur 5R cp εD

λT

5lD

lT5

εDεT

R (kg m23) � density; cp (J kg21 K21) � specific heat capacity; εD, εT (m2 s21) �vortex mass and thermal diffusivity; λT (W m21 K21) � vortex thermal diffusivity;

lD, lT (m) � length of mixing for mass and heat.

This number expresses the ratio of the molecular mass transfer intensity of a

given component in a mixture to the turbulent heat transfer intensity. It charac-

terizes the kinematic properties in heat and mass transfers in gas mixture flow. It

expresses the relation between thermal and concentration fields.

Info: [A23].

Warren Kendall Lewis (see above).

5.8.31 Liquid Phase Conversion in Vapour ε

ε5duf

duð1Þ;

ε5qmv

qmv 1 qmlð2Þ

0# ε# 1

uf (�) � body moisture content with phase change; u (�) � body moisture content;

qmv, qml (kg s21 m22) � mass flux density of vapour and liquid.

It expresses the ratio of the relative moisture content change by evaporation to

the total moisture content change in a given place of a body under conditions of

non-stationary moisture transfer in drying. It characterizes the vapour amount, dif-

fusing in capillary porous material, with respect to fluid and vapour flow without

255Thermomechanics

considering the convective transfer. For the moisture content change in an arbitrary

place, the following limits are valid:

ε5 0 � the moisture is changed by fluid transfer only,

ε5 1 � the moisture is changed only by evaporation.

In hygroscopic materials drying, it characterizes, in equation (2), the ratio of the

absolute value of the resulting vapour mass flow density to the sum of densities of

the vapour and fluid mass flows in non-stationary moisture transfer.

Info : [A23],[A33].

5.8.32 Lukomsky Number Lk

Lk � Lu21 5a

am

a (m2 s21) � thermal diffusivity; am (m2 s21) � mass diffusivity; Lu (�) � Lykov

number (p. 256).

It characterizes combined heat and mass transfer. It is analogous to the Lykov

number Lu (p. 256).

Info: [A29].

5.8.33 Lykov Filtration Number Luf

Luf 5ap

a

ap, a (m2 s21) � pressure filtration and thermal diffusivity.

It expresses the pressure to temperature fields propagation intensities ratio. It

characterizes the pressure to thermal diffusion ratio in heat and mass transfers.

Usually, it is Lufc1, but it is 100�1000 most frequently.

Info: [A23].

Alexey Vasilievich Lykov (see below).

5.8.34 Lykov Number Lu

Lu5am

a

am, a (m2 s21) � mass and thermal diffusivity.

This number expresses the ratio of the mass to temperature field propagation

intensities. It characterizes the relation between the temperature field propagation

and the mass (moisture) field. It expresses the mutual heat-to-mass transfer ratio in

a capillary porous material and in dispersive materials. It is the degree of the poten-

tial molecular field relaxation of the materials. The difference between the Lykov


and Lewis number Le (p. 254) is analogous to that between the Biot Bi (p. 174) and

Nusselt number Nu (p. 196), for example, as the intensities of corresponding phe-

nomena ratio.

Info: [A23].

Alexey Vasilievich Lykov (20.9.1910�28.6.1974), Russian

physicist.

In the year 1935, he discovered the thermodiffusion of

the humidity in capillary porous material. It was called the

Lykov effect. He was engaged in solving complicated pro-

blems of heat and mass transfer in capillary porous material

and dispersal environments under acting of phase and

chemical conversions. Of his many monographs, the

Teorija suski (Theory of Drying, 1951) and Teorija teplo-

provodnosti (Theory of Thermal Conductivity, 1969) are his

most important works.

5.8.35 Margoulis Mass Number Mgm

See the Stanton mass number Stm (p. 256).

Wladimir Margoulis (born 1886).

5.8.36 Merkel Number Me

Me5qmS

Qm

qm (kg s21 m22) � mass flux density; S (m2) � total surface of water level that is

in contact with air; Qm (kg s21) � mass flux.

It expresses the ratio of water mass, transferred in cooling, to dry gas mass. It

characterizes the mass transfer in basin coolers and cooling towers.

Info: [A23],[B11].

Friedrich Merkel (born 1892), German engineer.

5.8.37 Mikhailov Number Mi

Mi5 δpdp

du5BuRb

δp (Pa21) � Soret pressure coefficient; p (Pa) � pressure; u (�) � moisture con-

tent; Bu (�) � Bulygin number (p. 227); Rb (�) � Rebinder number (p. 262).

It characterizes the filtering mass transfer kinetics in intense heat and mass

transfer in drying. It expresses the degree to which pores are filled by a fluid at a

257Thermomechanics

given temperature or, alternatively, the total pressure change of the vapour�gas

mixture or a fluid inside a material.

5.8.38 Miniovich Number Mn

Mn5Sr

Vp

S (m2) � particle surface, V (m3) � particle volume, r (m) � pore radius; p(�) �porosity (p. 24).

In heat and mass transfer processes in a very small area of a layer of material, it

expresses the layer geometry influence on temperature and moisture fields in the

layer. Drying of porous materials.

Info: [A33].

J.M. Miniovich.

5.8.39 Nusselt Mass Number Num

See the Sherwood number Sh (p. 264).


5.8.40 Peclet Mass and Vortex Number Pemu

Pemu 5wL

εD

w (m s21) � velocity; L (m) � characteristic length; εD (m2 s21) � vortex mass

diffusivity.

It expresses the ratio of the convective mass propagation to the propagation rate

caused by molecular vortex diffusion. It characterizes the convective-vortex diffu-

sive mass transfer in flowing fluid.

Info: [A23].

Jean Claude Eugene Peclet (p. 180)

5.8.41 Peclet Mass Number Pem

Pem 5wL

D5RePrm 5ReSc ð1Þ; Pem 5

J

kBDð2Þ

w (m s21) � velocity; L (m) � characteristic length; D (m2 s21) � diffusivity;

J (kg s21) � convective mass flux through the membrane; kBD (kg s21) � total


coefficient of back diffusion; Re (�) � Reynolds number (p. 81); Prm (�) �Prandtl mass number (p. 260); Sc (�) � Schmidt number (p. 263).

In expression (1), it expresses the ratio of the volume mass transfer rate,

with forced convection, to the diffusive mass transfer rate. It characterizes the

convective�diffusive mass transfer in flowing fluid. It is also called the Reynolds

mass number Rem (p. 262).

For diaphragm systems, the dimensionless mass transfer parameter (2) is used,

which expresses the ratio of convective flow through a diaphragm to a phenomenon

which causes return diffusion expressed by the total coefficient kBD.

Info: [A23].


5.8.42 Pomerantsev Mass Number Pom

Pom 5qmVL

2

amRΔu

qmV (kg s21 m23) � volume density of mass flux; L (m) � characteristic length;

am (m2 s21) � mass diffusivity; R (kg m23) � material density; Δu (�) � differ-

ence of moisture content.

This number expresses the ratio of unit mass development by an internal source

to the maximum possible unit mass amount transferred by conduction. It charac-

terizes an internal mass source in a dimensionless way.

Info: [A23].


5.8.43 Posnov Diffusion Number Pn

Pn5δtΔT

Δu5

amtRΔT

amtRΔu5

amtRΔT

λmtΔϑð1Þ;

Pn5cmt

@ϑ@T

�uΔT

ðcmtÞmedΔϑ5

@ ϑΔϑ

�@ T

ΔT

�" #

u

ð2Þ

δt (K21) � Soret thermal coefficient; ΔT (K) � temperature difference; Δu (�) �

difference of moisture content; amt (m2 s21 K21)� thermal mass diffusivity; R (kg m23)�

density of material; am (m2 s21) � mass diffusivity; λmt (kg m21 s21 K21) � thermal

mass conductivity; Δϑ (K) � difference of mass transfer potential; cmt (K21) � thermal

mass capacity.

It expresses the ratio of the thermodiffusive moisture transfer intensity to the dif-

fusive moisture transfer intensity. It characterizes the internal moisture transfer by

conduction under non-isentropic conditions. The relative non-uniformity of a mois-

ture field in a material is caused by heat and mass transfer in a steady state. With

259Thermomechanics

small Δu or linear dependence of u(ϑ), the Pn number becomes a dynamic transfer

parameter � a relative temperature product of the mass transfer potential expressed

by expression (2). In this form, it expresses the ratio of the potential change of

mass (moisture content) to the relative temperature change with a given mass steady-

state content in a body. It is analogous to the Fedorov number (2.) Fe (p. 248).

Info: [A23].

5.8.44 Posnov Filtration Number Pnp

Pnp 5δpΔp

Δu5

λmpΔp

RamΔu

δp (Pa21) � Soret pressure coefficient; Δp (Pa) � pressure difference; Δu (�) �difference of moisture content; λmp (kg m21 s21 Pa21) � pressure mass conductiv-

ity; R (kg m23) � density; am (m2 s21) � mass diffusivity.

It expresses the ratio of the filtration to diffusive moisture transfer intensities. It

characterizes the internal moisture transfer in material during filtration. The relative

moisture field non-uniformity in the material is caused by a pressure gradient

(filtration potential difference).

Info: [A23].

5.8.45 Prandtl Mass Number Prm

See the Schmidt number Sc (p. 263).

Ludwig Prandtl (p. 197)

5.8.46 Predvoditelev Mass Number Pdm

Pdm 52dðqmq21

mi ÞdFo

� �max

ð1Þ;

Pdm 5kL2

amð2Þ

qm, qmi (kg s21 m23, or kg s21 m22) � volume and surface mass flow intensity and

its initial value; k (s21) � drying coefficient; L (m) � characteristic length;

am (m2 s21) � mass diffusivity; Fo (�) � Fourier number (p. 175).

It expresses the ratio of the material concentration change rate in the surround-

ings to that in a body. It characterizes the mass flow density time change, for exam-

ple, that of an internal source (qm� qmV) or of the second-type boundary condition

(qm� qmA). In tasks involving wet material heating in constant temperature sur-

roundings, when the surface moisture evaporation occurs in the second drying


period according to the law qm5 qmi exp (2 kτ), expression (2) is valid, and the

first period of drying Pdm5 0 is valid.

Info: [A23].

Alexander Savviich Predvoditelev (1891�1973), Russian physicist.

5.8.47 Psychrometric Relation K

K5αcmqmc

α (W m22 K21) � heat transfer coefficient; cm (�) � mass ratio; qm (kg s21 m22) �mass flux density; c (J kg21 K21) � specific heat capacity.

It expresses the ratio of the heat transferred by convection to that transferred by

the mass transfer. It characterizes wet air as a heat transfer means and a drying

environment simultaneously. Wet and dry thermometer well.

Info: [A23].

5.8.48 Ramzin Number Rm

Rm5δpΔp

Δu5

δpΔp

δtΔϑ5BuKo21

δp (Pa21) � Soret pressure coefficient; Δp (Pa) � pressure difference; Δu (�) �difference of moisture content; δt (K

21) � Soret thermal coefficient; Δϑ (K) � dif-

ference of moisture transfer potential; Bu (�) � Bulygin number (p. 227); Ko (�) �Kossovitch number (p. 253).

In molecular transfer, it characterizes the vapour content described as a mass

content of the material which takes part in the molecular transfer. See the Posnov

filtration number Pnp (p. 260).

Info: [A23].

5.8.49 Rayleigh Mass Number Ram

Ram 5XgL3Δc

νD5

gL3

νDΔRR

5GrSh

X (�) � dimensionless diffusion number; g (m s22) � gravitational acceleration;

L (m) � characteristic length; Δc (�) � difference of initial and final mass ratio of

chemical reaction products; ν (m2 s21) � kinematic viscosity; D (m2 s21) � diffu-

sivity; ΔR (kg m23) � density difference; R (kg m23) � density; Gr (�) � Grashof

heat number (p. 185); Sh (�) � Sherwood number (p. 264).

It characterizes the natural convection influence on the mass transfer in fluids,

for example, the concentration and temperature propagation influence on the

261Thermomechanics

thermodiffusion separation of liquid mixtures in a fluid thermodiffusion separating

column.

Info: [A23].


5.8.50 Rebinder Number Rb

Rb5cb

llv5

dTmed

dumed

� �c

llv5BKo21

c (J kg21 K21) � specific heat capacity; b5 dTmed/dumed (K) � change of mean

integral temperature to change of mean integral moisture content ratio; llv (J kg21) �

specific latent heat of vaporization; B5ΔubΔT21 (�) � dimensionless drying

coefficient; Ko (�) � Kossovitch number (p. 253).

This number expresses the ratio of the heat consumed in material heating to that

consumed in moisture evaporating during an infinitely short time interval. It relates

to the kinematic characteristics of integral properties of heat and mass transfer. It

expresses the characteristics of local changes of temperature and moisture content.

Info: [A23].

Peter Aleksandrovich Rebinder (21.9.1898�12.7.1972),

Russian physical chemist.

He was engaged in the problems of the origin, stability,

surface phenomena and structure formation in disperse sys-

tems. In addition, he explained the idea of the molecular

mechanism of surface active substances and their applica-

tions in diverse technological processes. In the year 1928,

he clarified the influence of absorption in reducing the

rigidity of solid bodies.

5.8.51 Reynolds Mass Number Rem

See the Peclet mass number Pem (p. 258), Richman number Ri (p. 232).


5.8.52 Romankov Number Ro

Ro5Tg 2 Ts

Tgð1Þ;

Ro5Tg

Tsð2Þ


Tg (K) � hot gas temperature used for drying; Ts (K) � temperature of dried-up

material.

This number expresses the ratio of the temperature gradient between gas and

material in drying to the heated gas temperature at the drying stove outlet. In dry-

ing, it characterizes the potential wet gas possibilities to heat the dried material.

Sometimes, it is used in the form of equation (2).

Info: [A23].

Peter Grigorievich Romankov (17.1.1904�1.10.1990),

Russian physical chemist.

He was engaged in the analysis of kinetic laws and the

generalization of the methods of similarity theory and

dimensional analysis. He applied the results on heat and

mass transfer processes and hydromechanical ones in

chemical technology. His principal work is the monograph

Gidravliceskije processy v chimiceskoj technologii

(Hydraulic Processes in Chemical Engineering). He was

also engaged in fluid mixing, filtration, centrifugal separa-

tion, cleaning of gas, drying, absorption, extraction and

distillation.

5.8.53 Schmidt Effective Number Scef

Scef 5εM 1 νεD 1D

εM, εD (m2 s21) � vortex momentum diffusivity and vortex mass diffusivity; ν(m2 s21) � kinematic viscosity; D (m2 s21) � diffusivity.

It expresses the ratio of the total momentum diffusivity to the total mass diffu-

sivity in the mass transfer in combined laminar and turbulent flow. It characterizes

the mass transfer in combined turbulent and laminar flow.

Info: [A23].

Ernst Schmidt (see below).

5.8.54 Schmidt Number Sc

Sc � JQ � Prm 5νD

ν (m2 s21) � kinematic viscosity; D (m2 s21) � diffusivity; JQ (�) � Colburn

number (p. 190); Prm (�) � Prandtl mass number (p. 260).

This number expresses the ratio of the kinematic viscosity, or momentum trans-

fer by internal friction, to the molecular diffusivity. It characterizes the relation

between the material and momentum transfers in mass transfer. It provides the

263Thermomechanics

similarity of velocity and concentration fields in mass transfer. For example, mol-

ten materials with an equal Schmidt number have similar velocity and concentration

fields. Higher Sc number values characterize slower mass exchange and higher

values of dividing coefficients. This leads to higher mixing and a tendency to crack

in a solidified casting. The criterion was first introduced by Schmidt in 1929. It

is also called the Colburn number Jq (p. 190) or the Prandtl mass number Prm(p. 260).

Info: [A23],[A43].

Ernst Schmidt (11.2.1892�22.1.1975), German engineer.

He was a pioneer in engineering thermodynamics, espe-

cially in heat and mass transfer. Initially, he was engaged in

measuring the radiation properties of solid materials, which

was the reason to design an effective thermal shield of alu-

minium foil. In addition, he was engaged in solving non-

stationary thermal fields by a graphic difference method, in

determining the local heat transfer coefficient and in anal-

ogy between heat and mass transfer.

5.8.55 Schmidt Turbulent Number Scv

Scv 5εMεD

εM, εD (m2 s21) � vortex momentum and mass diffusivity.

It expresses the ratio of the vortex momentum diffusivity to the vortex mass dif-

fusivity. It characterizes the mass transfer in a turbulent flow.

Info: [A23].

Ernst Schmidt (see above).

5.8.56 Sherwood Number Sh

Sh � Num � Tam 5βLD

β (m s21) � mass transfer coefficient; L (m) � characteristic length; D (m2 s21) �molecular diffusivity; Num (�) � Nusselt mass number (p. 258); Tam (�) � Taylor

mass number (p. 266).

It expresses the ratio of the heat transfer to the molecular diffusion. It charac-

terizes the mass transfer intensity at the interface of phases. It is also called the

Nusselt mass number Num (p. 258) or the Taylor mass number Tam (p. 266).

Info: [A23].


Thomas Kilgore Sherwood (25.7.1903�14.1.1976),

American chemical engineer.

He was engaged primarily in research on mass transfer

and its interactions with flow, chemical reactions and indus-

trial applications in which these processes play a principal

role. After having published Absorption and Extraction

(1937), which was the first, sui generis in this field, he

became a world famous personality.

5.8.57 Soret Thermodiffusion Number So

So5 LeðN2N

21ÞiniðN1N 21ÞðN2N 21Þ

N1, N2 (�) � molecule number of components 1 and 2 in binary gas mixture,

N5N11N2; Le (�) � Lewis number (p. 254).

It characterizes the thermodiffusion effect in mass transfer. In binary gas mix-

tures with equal initial content of components, it equals half of the thermodiffusion

constant. It expresses the coupling between heat and mass transfers. It represents

the dimensionless thermodiffusion coefficient.

Info: [A23].

Charles Soret (1854�1904), Swiss physicist.

5.8.58 Spalding Transfer Number Spp

Spp 5 cpΔT llv 2ΦR

Qm

� �21

cp (J kg21 K21) � specific heat, capacity; ΔT (K) � temperature difference;

llv (J kg21) � specific latent heat of vaporization; ΦR (W) � radiation heat flux;

Qm (kg s21) � mass flux.

It expresses the ratio of the thermal energy change to the latent evaporation heat

of material. It characterizes heat and mass transfers under conditions of thermal

radiation and phase conversion.

Info: [A23].

Brian Spalding (p. 200).

5.8.59 Stanton Mass Number Stm

Stm � Mg5βw

5 ShRe21Sc21 ð1Þ;

265Thermomechanics

Stm 5qm

Rw5

1

αmpwð2Þ

β (m s21) � mass transfer coefficient; w (m s21) � flow velocity; qm (kg s21 m22) �mass flux density; R (kg m23) � fluid density; αmp (kg m22 s21 Pa21) � pressure

specific mass transfer; Mg (�) � Margoulis number (p. 196); Sh (�) � Sherwood

number (p. 264); Re (�) � Reynolds number (p. 81); Sc (�) � Schmidt number

(p. 263).

It expresses the ratio of the mass transfer perpendicular to a solid phase surface

to the mass transfer by flowing in parallel with the solid phase surface. It charac-

terizes the mass transfer at the interface between the solid and fluid phases. It is

called the Margoulis number Mg (p. 196) as well.

Info: [A23].


5.8.60 Stefan Parameter Sf

Sf 5p

ppS 2 ppPlnp2 ppP

p2 ppS

p (Pa) � local absolute static pressure; ppS (Pa) � partial vapour pressure above

the surface of evaporation or condensation; ppP (Pa) � partial vapour pressure in

environment.

It expresses the influence of the transversal material flow on the size of the

mass transfer coefficient in evaporation or vapour condensation from a vapour�gas

mixture.

Info: [A23].


5.8.61 Taylor Mass Number Tam

See the Sherwood number Sh (p. 264).

Geoffrey Ingram Taylor (p. 89).

5.8.62 Temkin Heat and Mass Transfer Number Di

Di5qmllvL

λΔT5KoLuKim

qm (kg s21 m22) � density of mass flux; llv (J kg21) � specific latent heat of evapo-

ration; L (m) � characteristic length; λ (W m21 K21) � thermal conductivity of

material; ΔT (K) � initial and final temperature difference; Ko (�) � Kossovitch


number (p. 253); Lu (�) � Lykov number (p. 256); Kim (�) � Kirpichev mass

number (p. 252).

This number expresses the ratio of the external heat transition intensity, caused

by the external mass transition, to the internal heat transfer intensity by conduction

in wet material. It characterizes the mean thermal gradient value in wet material.

Info: [A23].

A.G. Temkin.

5.8.63 Temperature Criterion N

N5Ta 2 Ts

Ta 2 Tad

Ta (K) � temperature of moist air; Ts (K) � material temperature after drying-up;

Tad � air temperature in a state of adiabatic saturation (temperature of a wet

thermometer).

It characterizes the potential possibilities of wet air as a heating material and

drying agent as well.

Info: [A23].

5.8.64 Thiele Modulus (1.) Th, mT

Th � Da4 5euL2

λT

e (m2 s22) � specific energy; u (kg m23 s21) � chemical reaction rate; L (m) �characteristic length; λ (W m21 K21) � thermal conductivity; T (K) � temperature;

Da4 (�) � Damkohler number (4.) (p. 37).

It expresses the ratio of the surface reaction rate to the thermal diffusion rate. It

characterizes the non-isothermal heat and mass transfer in physical, chemical and

biotechnical processes. Catalysis. Diffusion in porous catalysers.

Info: [A29].

Ernest William Thiele (8.12.1895�29.11.1993), American

chemical engineer.

Initially, he was engaged in work related to carbon reac-

tions and evolved an idea leading to the McCabe�Thiele

method of fractional column design. During the war, he

participated in the atomic energy development program and

was engaged in heavy water extraction. After the war

(1948), he worked on a project to exploit the nuclear drive

in aeronautics. Then he was engaged in refining processes

and the distillation of hydrocarbon mixtures. In connection

with this, he solved the heat and mass transfer problems.

267Thermomechanics

5.8.65 Transfer Number Ntr

Ntr 5β�Δn

ΔTð1Þ;

Ntr 5β�Δu

ΔTð2Þ

β�(K) � Dufour coefficient; Δn (�) � difference of mass concentration; ΔT (K) �

temperature difference; Δu (�) � difference of relative moisture content.

In equation (1), it expresses the general transfer phenomena which represent the

mass concentration change due to a temperature change, and in equation (2) it

expresses a moisture change. Thermomechanics. Heat and mass transfer.

Info: [A23].

5.8.66 Transfer Number (1.) Thermodynamic Ntr , ε

Ntr 5βc

r

β (K21) � volume thermal expansion coefficient; c (J kg21 K21) � specific heat

capacity; r (J kg21) � specific heat of evaporation.

It characterizes the thermodynamic heat and mass transfer processes. It is analogous

to the Posnov diffusion number Pn (p. 259) and the Fedorov number (2.) Fe (p. 248).

Info: [A23].

5.8.67 Transfer Number (2.) Concentration Ntr

Ntr 5 δtΔcm

ΔT

δt (K21) � Soret thermal coefficient; Δcm (�) � difference of relative substance

concentration; ΔT (K) � temperature difference.

It characterizes the mass concentration process in heat and mass transfer.

Info: [A23].

5.8.68 Transfer Number (3.) Diffusion Ntr , ε

Ntr 5Dx

Qδ5Ab2Sh22


D (m2 s21) � molecular diffusivity; x (m) � wetted length; Q (m3 s21) � volume

flow; δ (�) � relative film thickness; Ab (�) � absorption number (p. 244);

Sh (�) � Sherwood number (p. 264).

It characterizes dimensionless diffusivity. It expresses the gas absorption on wet

chamber walls. Mass transfer.

Info: [A23].

5.9 Non-Equilibrium Thermomechanics

The dimensionless quantities for non-equilibrium thermomechanics involve the

processes of wave heat propagation and thermal stress in material and parallel

heat and mass propagation. The wave propagation is closely related to thermal

strokes and the action of intensive heat sources on the material. A survey of the

wave propagation theory is given in [A46]. Parallel heat and mass propagation is

applied, for example, in the transfer in porous and capillary materials and is related

to new technologies and materials especially. Most of the dimensionless quantities

are more complicated modifications of basic criteria and represent the influence of

the heat and mass propagation velocities, relaxation action, thermalization and

internal heat transfer. The following numbers are important: the Fourier, Biot,

Nusselt, Peclet, Vernotte, Sparrow and others. In this section, the more detailed

explanation of some dimensionless quantities relates, above all, to their specific

significance in non-equilibrium thermomechanics and in the actual complexity of

wave and wave-diffusion processes.

5.9.1 Biot Volumetric Number of Internal Heat Transfer BiV

BiV 5αVL

2

λð1Þ;

BiV 5τ2α2

V

c1c25Fo2ðBi�V Þ2Pc ð2Þ

αV (W m23 K21) � volume coefficient of internal heat transfer; L (m) � character-

istic length; λ (W m21 K21) � thermal conductivity; τ (s) � time; c1, c2(J m23 K21) specific volume heat capacity of material components; Fo (�) �Fourier number (p. 175); Bi�V (�) � Biot volumetric number of internal heat trans-

fer defined by the relation (1); Pc (�) � relative parallel heat capacity (p. 279).

The Biot volumetric number of internal heat transfer expresses the ratio of the

internal inter-component conductivity (heat transfer between the material compo-

nents) and the standard volume heat conductivity.

269Thermomechanics

With great internal heat transfer, the heat propagates as in homogeneous mate-

rial and only one diffusive heat conduction equation can be used for the

description.

With the internal heat transfer comparable to the material thermal conductivity,

the heat propagates in parallel within the framework of material components and

heat exchange occurs among them. To describe it mathematically, one equation of

parallel heat propagation must be used or, alternatively, two diffusive equations

with mutual heat transfer coupling.

With less intensive internal heat transfer, the heat propagates in the material in

parallel within the framework of the components, between which thermal interac-

tion scarcely occurs. Then, the mathematical description of the heat propagation

consists of two independent equations for the heat conduction in material compo-

nents. If the thermal conductivities of the material components do not differ too

much, only one single equation can be used for heat conduction, as in a homoge-

neous material.

In contrast to the standard expression (1), with the characteristic dimension

being replaced by the product of the thermal process time and the equivalent paral-

lel heat propagation rate, the Biot volumetric number of internal heat transfer can

be defined as in expression (2).

Info: [A17],[B45].


5.9.2 Cattaneo Number Cat

Cat5τλw2

ν2Rcp

τ (s) � cas; λ (W m21 K21) � thermal conductivity of fluid flow; w (m s21) �flow velocity; ν (m2 s21) � kinematic viscosity; R (kg m23) � fluid density;

cp (J kg21 K21) � specific heat capacity.

It expresses the existence and propagation of a discontinuity in the velocity gra-

dient and temperature of a flowing fluid along a wall, heated with an impulse

method or randomly by an intensive heat source. Non-equilibrium

thermomechanics.

Info: [B93].

5.9.3 Coupling Factor Ncoup

Ncoup 5GL2

λref

G (W m23 K211) � electron�phonon coupling factor; L (m) � characteristic

length; λref (W m21 K21) � thermal conductivity.


It represents the ratio of the square of the convective electron heat transfer to

the heat transferred by conduction. It expresses an energetic coupling between elec-

tron gas and a grid. Wave heat propagation. Non-equilibrium thermomechanics.

Physical technology.

Info: [A70],[B63].

5.9.4 Dominance Number K

K5a

c2@2T

@t2@T

@t

� �21

a (m2 s) � thermal diffusivity; c (m s21) � velocity propagation of thermal wave;

T (K) � temperature; t (s) � time; τ (s) � relaxation time ðτ5 a=c2Þ:The wave character of heat transfer predominates for Kc1; i.e. (1) if the critical

frequency c2/a is small (it is the inverse value of the relaxation time τ), (2) if theprocess time t is very short or (3) if the heating or cooling rate (caused by the heat

flow) is high. The first condition involves the solid material properties only,

whereas the second and third depend also on the combined influence of the geo-

metric arrangement and system thermal load. This criterion is more general than

the usual wave propagation one.

Info: [A46].

Robert Tzou (born 1955), American physical engineer.

He is engaged in unbalanced macro- and microscopic

heat propagation, in shock heat processes and their model-

ling. Furthermore, he was engaged in the diffusion anomaly

in amorphous surroundings, control of fast shock thermal

processes, thermomechanical properties of composites, ther-

momechanical interactions in fractures and high-velocity

penetration through surroundings. His monographs Macro-

to Microscale Heat Transfer (1997) and Ultrafast Heating

and Thermomechanical Coupling Induced by Femtosecond

Lasers (2007) are especially significant.

5.9.5 Fourier Mass Relaxation Number Form

Form 5amτrxL2

am (m2 s21) � mass diffusivity; τrx (s) � relaxation time of material; L (m) � char-

acteristic length.

It characterizes the relaxation time of a non-stationary mass transfer.

Info: [A23].


271Thermomechanics

5.9.6 Fourier Porous Number (1.) Fop

Fop 5λekvτtcL2

5cf

cSp21

where τt �Rhcf

α; c5 εcf 1ð12 εÞcs; ε5

Vf

V

λekv (W m21 K21) � equivalent thermal conductivity; τt (s) � time delay of tem-

perature; c, cf, cs (J m23 K21) � specific volume heat capacity: final, fluid and

solid material; L (m) � thickness of porous layer; Rh (m) � hydraulic diameter;

α (W m22 K21) � internal heat transfer; ε (�) � volume ratio; Vf, V (m3) � fluid

volume, total volume; Sp (�) � Sparrow number (p. 281).

This number characterizes the dimensionless time in heat transfer by fluid flow-

ing through porous material. It expresses the influence of the non-equilibrium heat

propagation.

Info: [B76].


5.9.7 Fourier Porous Number (2.) Fop

Fop 5c2fw

2τcλekv

5λekvτcL2

cfwL

λekv

5Fo�pPe2p

cf (J m23 K21) � fluid specific volume heat capacity; w (m s21) � constant mean

velocity; τ (s) � time; c (J m23 K21) � final specific volume heat capacity; λekv(W m21 K21) � equivalent thermal conductivity; L (m) � porous layer thickness;

τq, τt (s) � time delay of heat flux and temperature; Fo�p (�) � Fourier porous

number (1.) (p. 272); Pep (�) � Peclet porous number (p. 279).

It characterizes the dimensionless time in forced flow and heat transfer in a

porous material. The equilibrium state, constant velocity and times τq5 τt are

considered.

Info: [B79].


5.9.8 Fourier Relaxation Heat Number Forx

Forx 5aτrxL2

5 ShPe21

a (m2 s21) � thermal diffusivity; τrx (s) � relaxation time; L (m) � characteristic

length; Sh (�) � Strouhal number (p. 87); Pe (�) � Peclet heat number (p. 180).


It characterizes the inertia of temperature equalizing in a thermal system. It

represents the part of the Fourier number Fo (p. 175) corresponding to the thermal

field relaxation time in a system. The relaxation time is that during which all para-

meters of the considered thermal system are equalized.

The influence of the Fourier relaxation heat number can be expressed, analo-

gously to the molecular heat and mass transfers, in three bands: wave conduction

for Forx. 102; wave-diffusive conduction for 1024,Forx, 102; and diffusive con-

duction for Forx, 1024. Non-equilibrium thermomechanics. Physical technology.

Info: [A23],[B45].


5.9.9 Fourier Relaxation Parallel Number Foq

Foq 5aτqL2

5Bi21V P1c ð1Þ;

Foq 51

L2λ1 1λ2

c1 1 c2

� �1

αV

1

c11

1

c2

� �21ð2Þ

a (m2 s21) � thermal diffusivity; τq (s) � relaxation time of parallel propagation;

L (m) � characteristic length; λ1, λ2 (W m21 K21) � thermal conductivity of mate-

rial components; c1, c2 (J m23 K21) � specific volume heat capacity of material

components; αV (W m23 K21) � volume coefficient of internal heat transfer;

BiV (�) � Biot volumetric number of internal heat transfer (p. 269); Pc (�) �relative parallel heat capacity (p. 279).

It determines the timescale for material transition from the non-equilibrium

state, which is caused by different temperatures of its individual components in one

place. The velocity, with which the material reaches equilibrium by the action of

internal heat transfer, depends on the relaxation time value of parallel heat propaga-

tion, where the equilibrium is expressed by equal temperatures of all transition

components. The relaxation time value is defined as the product of the material

thermal capacity and the internal transfer flow of unit energy. The relaxation time

represents the effective time during which the whole material thermal content could

be transported among its components. The Fourier relaxation parallel number

expresses the ratio of the series and parallel material thermal capacities and the

inner heat transfer (1). In the case of two-component material, it can be expressed

as in expression (2).

Info: [A17].


5.9.10 Fourier Relaxation Wave Number Foq

Foq 5aτqL2

5l

L

� �2

273Thermomechanics

a (m2 s21) � thermal diffusivity; τq (s) � relaxation time of wave heat propaga-

tion; L (m) � characteristic length; l (m) � mean free path of heat carrier.

The relaxation time of heat wave propagation describes the effect of inelastic

collisions with which the thermal wave energy converts to the system internal

energy and is damped in passing through the material. The Fourier relaxation

wave number equals the second power of the ratio of the mean free path and the

characteristic system dimension.

By making use of the analogy with low pressure gas flow, the influence of Foqon the propagation character can be determined. With Foq. 102, almost no colli-

sions occur in the considered material volume, there is no diffusion, and the energy

carriers transfer the heat freely directly inside the material. With 1024,Foq, 102,

a transient state exists. With the Foq value decreasing, the heat transfer wave char-

acter weakens steadily because of the effect of collisions, damping the wave propa-

gation, increases, and, therefore, the propagation character changes to diffusive

propagation. For Foq, 1024, the collisions break the propagation of heat and dis-

tribute it in all directions, resulting in the known character of diffusive propagation.

Info: [A17],[B45].


5.9.11 Fourier Thermalization parallel Number (1.) FoPT

FoPT 5aPτTL2

5Bi21V P21λ ð1Þ;

FoPT 51

L2λ1λ2

λ1c2 1λ2c1

� �1

αV

λ1c2 1λ2c1

λ1 1λ2

� �ð2Þ

aP (m2 s21) � parallel thermal diffusivity; τT (s) � thermalization time of parallel

heat propagation; L (m) � characteristic length; λ1, λ2 (W m21 K21) � thermal

conductivity of material components; c1, c2 (J m23 K21) � specific volume heat

capacity of material components; αV (W m23 K21) � volume coefficient of internal

heat transfer; BiV (�) � Biot volumetric number of internal heat transfer (p. 269);

Pλ (�) � relative parallel thermal conductivity (p. 280).

It expresses the ratio of series and parallel thermal material conductivities and

internal heat transfer (1). In the case of two-component material, it can be

expressed as in equation (2).

Info: [A17],[B45].


5.9.12 Fourier Thermalization Parallel Number (2.) FoT

FoT1 5aτTL2

5Bi21V P21λ P21

1 5FoPTP21a ð1Þ;


FoT 51

L2λ1 1λ2

c2 1 c1

� �1

αV

λ1c2 1λ2c1

λ1 1λ2

� �ð2Þ

a (m2 s21) � thermal diffusivity; τT (s) � thermalization time of parallel heat prop-

agation; L (m) � characteristic length; λ1, λ2 (W m21 K21) � thermal conductivity

of material components; c1, c2 (J m23 K21) � specific volume heat capacity of

material components; αV (W m23 K21) � volume coefficient of internal heat transfer;

BiV (�) � Biot volumetric number of internal heat transfer (p. 269); Pλ (�) � rela-

tive parallel thermal conductivity (p. 280); Pa (�) � relative parallel thermal diffu-

sivity (p. 280); FoPT (�) � Fourier thermalization parallel number (1.) (p. 274).

It expresses the relation between internal heat transfer, series and parallel ther-

mal material conductivities and diffusivity (1). It is a modification of the Fourier

thermalization parallel number (1.) (p. 274) by the mutual ratio of the series and

parallel material thermal diffusivities. In the case of two-component material, it

can be expressed as in equation (2).

Info: [A17],[B45].


5.9.13 Fourier Thermalization Wave Number FoT

FoT 5vτqL

vτTL

5aτTL2

v (m s21) � heat carrier velocity propagation; τq (s) � relaxation time of wave

heat propagation; τT (s) � thermalization time of wave heat propagation; L (m) �characteristic length; a (m2 s21) � thermal diffusivity.

The thermalization time of the wave heat propagation describes the effect of

elastic collisions with which the thermal wave momentum is not lost, but the veloc-

ity distribution of heat carriers is formed. The Fourier thermalization wave number

expresses the product of the ratio of the path covered by a heat carrier during the

relaxation time to the characteristic length of the observed material zone and

the ratio of the path covered by a heat carrier during the thermalization time to the

characteristic length of the observed material zone.

Info: [A17],[B45].


5.9.14 Fourier Wave Number Fo

Fo5vτL

vτqL

v (m s21) � heat carrier velocity propagation; τ (s) � time; τq (s) � relaxation

time of wave heat propagation; L (m) � characteristic length of a system.

275Thermomechanics

It characterizes the dimensionless time of the wave heat propagation process. It

is the product of the ratio of the path covered by a heat carrier during the time τ to

the characteristic system length and the ratio of the path covered by a heat carrier

during the relaxation time to the characteristic system length.

Info: [A17],[B45].


5.9.15 Heat Transfer in Micropores Kτ

Kτ 5τekvτt

5ð12 εÞcs

c

where τt �Rhcf

α; τekv 5

ð12 εÞcsτtc

; ε5Vf

V

τekv (s) � equivalent time; τt (s) � time delay, thermalization time of non-equilib-

rium heat propagation; ε (�) � volume ratio; Vf, V (m3) � fluid volume and total

volume; c, cs, cf (J m23 K21) � specific volume heat capacity: total, solid and fluid;

Rh (m) � hydraulic radius of nano or microchannels; α (W m22 K21) � heat trans-

fer coefficient.

It limits the character of non-equilibrium heat propagation in fluid flow through

micropores.

5.9.16 Knudsen Phonon Number Knph

Knph 5

ffiffiffiffiffiffiffiffiffiffi3aτd

p

h5

ffiffiffiffiffi3τ

p; where τd 5

h2

a

a (m2 s21) � thermal diffusivity; τd (s) � diffusion time; h (m) � thickness; τ (s) �time constant of Cattaneo�Vernotte.

It characterizes heat wave propagation and determines when the classic

Cattaneo�Vernotte mathematical model can be used. This model cannot be used in

cases in which it exceeds the limits of KnphAh0.35; 0.5i.Info: [B35].


5.9.17 Mikheyev Volumetric Number of Internal Heat Transfer MiV

See the non-equilibrium parallel heat transfer number Knep (p. 277).

Mikhail Alexandrovich Mikheyev (1902�1970), Russian engineer.


5.9.18 Number of Non-Equilibrium Heat Transfer Kτ

Kτ 5τTτq

5FoT

Foq

τT (s) � thermalization time of non-equilibrium (wave or parallel) heat propaga-

tion; τq (s) � relaxation time of non-equilibrium (wave or parallel) heat propaga-

tion; FoT (�) � Fourier thermalization wave number (p. 275) or Fourier

thermalization parallel number (2.) (p. 274); Foq (�) � Fourier relaxation wave

number (p. 273) or Fourier relaxation parallel number (p. 273).

This number defines the physically allowed zone of wave and parallel non-equi-

librium heat propagation. The character of non-equilibrium heat propagation is

determined by the thermalization to relaxation time ratio of material. The value

Kτ5 1 represents equilibrium diffusion propagation. With Kτ, 1, it expresses the

non-equilibrium heat wave propagation which changes to diffusion with increasing

time. For Kτ. 1, it expresses the non-equilibrium parallel heat propagation.

Info: [A17],[B45].

5.9.19 Number of Non-Equilibrium Parallel Heat Transfer Knep

Knep 5Fo

Foq5

ταV

cS5FoBiVPc

τ (s) � time; αV (W m23 K21) � volume coefficient of internal heat transfer;

cS (J m23 K21) � series specific volume heat capacity; Fo (�) � Fourier number

(p. 175); Foq (�) � Fourier relaxation parallel number (p. 273); BiV (�) � Biot

volumetric number of internal heat transfer (p. 269); Pc (�) � relative parallel heat

capacity (p. 279).

This criterion is defined as the ratio of the Fourier number (p. 175) to the

Fourier relaxation parallel number (p. 273). It determines the time zone of

the action of parallel non-equilibrium heat propagation. It expresses the ratio of the

heat transferred inside a material between its components to the total thermal con-

tent of a body. The Biot volumetric number of internal heat transfer BiV (p. 269)

and the relative parallel heat capacity Pc (p. 279) decide whether in the time Fo,

since the origin of the non-equilibrium state, the material is already in an equilib-

rium state or still in a non-equilibrium state. The criterion is also called the

Mikheyev volumetric number of internal heat transfer MiV (p. 276).

With Knep, 1022, the internal energy exchange between material components

with unequal temperatures is not able to manifest itself in the short time, since the

origin of the non-equilibrium state with the action of a heat source. In this phase,

the thermal process can be described by a set of independent diffusive heat conduc-

tion equations for individual material components. If 1022,Knep, 101, the energy

transfer appears between the components inside the material. With increasing time,

the temperatures of individual components come near to the equilibrium values.

277Thermomechanics

The heat propagation process must be expressed by the parallel heat propagation

equation or by the set of diffusive equations with mutual coupling. For Knep. 101,

sufficient time has elapsed already since the origin of the non-equilibrium state that

the material may get, due to internal redistribution of energy, into the equilibrium

state with equal temperature of all of its components. Then, the heat propagation

can be expressed as a rule by one diffusion heat propagation equation for the mate-

rial as a whole.

Info: [A17],[B45].

5.9.20 Number of Non-Equilibrium State of Matter Knsm

Knsm 5Fo

Forx

Fo (�) � Fourier number Fo (p. 175); Forx (�) � Fourier relaxation heat number

(p. 272).

It expresses the time zone of non-equilibrium wave heat conduction. It is the cri-

terion which enables the division of the wave-diffusive heat conduction process

into three time zones: the non-equilibrium heat conduction with Knsm, 1021; the

transient zone with 1021,Knsm, 101; and the equilibrium diffusive heat conduc-

tion with Knsm. 101. Non-equilibrium thermomechanics. Physical technology.

Info: [A17],[B45].

5.9.21 Number of Non-Equilibrium Wave Heat Transfer Knev

Knev 5Fo

Foq

Fo (�) � Fourier wave number (p. 275); Foq (�) � Fourier relaxation wave

number (p. 273).

This criterion is defined as the ratio of the Fourier wave number (p. 275) to the

Fourier relaxation wave number (p. 273). It determines the time of acting of the

wave non-equilibrium heat propagation.

With Knev, 1022, the heat carriers forming the thermal wave propagate through

a material without collisions. In the short time since the origin of non-equilibrium

state due to a heat source, no collision process creating material thermal resistance

occurred yet. In this phase, the heat propagation process can be described by a

wave propagation equation. If 1022,Knev, 101, the wave propagation damping

appears with increasing time and the material gets into the equilibrium state gradu-

ally. The heat propagation process is expressed by a wave-diffusion equation. With

Knev. 101, sufficient time has elapsed since the origin of the non-equilibrium state

that the wave propagation character may be damped. Then, the heat propagation

can be expressed by a heat conduction diffusion equation.

Info: [A17], [B45].


5.9.22 Nusselt Porous Number Nup

Nup 5αRp

λf

α (W m22 K21) � heat transfer coefficient; Rp (m) � hydraulic radius expressed as

pore volumes ratio to their surface (ΔVp/ΔAp); λf (W m21 K21) � thermal conduc-

tivity of fluid.

This number characterizes the convection heat transfer in porous material. It is

applied especially if the heat conduction predominates, due to which its size is con-

stant. It occurs in cases where the Rayleigh pore flow number Rapor (p. 111) is

small and the natural convection in pores predominates in the heat conduction.

Equilibrium and non-equilibrium heat transfers. Two-phase flow.

Info: [B55].


5.9.23 Peclet Porous Number Pep

Pep 5cfwL

λekv

cf (J m23 K21) � specific volume heat capacity of fluid; w (m s21) � mean flow

velocity; L (m) � characteristic length, porous layer thickness; λekv (W m21 K21) �equivalent thermal conductivity; Sp (�) � Sparrow number (p. 281); Pe (�) �Peclet heat number (p. 180).

It characterizes the velocity of fluid flowing through porous material. The vari-

able velocity frequency has a strong influence on the heat transfer. The ratio

SpPe21 shows approximately whether there is a local equilibrium state in the

porous material.

Info: [B76].


5.9.24 Relative Parallel Heat Capacity Pc

Pc 5cp

cs5X

ciX 1

ci

cp (J m23 K21) � parallel heat capacity ðcp 5

PciÞ; csðJ m23 K21Þ � series specific

volume heat capacity cs 5P

1ci

� 21� �; ciðJ m23 K21Þ � specific volume heat

capacity of ith material component.

It expresses the parallel-to-series thermal capacities ratio of a material, with said

capacities being determined by the heat capacities of individual material components.

Info: [A17],[B45].

279Thermomechanics

5.9.25 Relative Parallel Heat Transfer Rate Pv

Pv 51

wt

ffiffiffiffiffia

τq

rð1Þ;

Pv 51

wt

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiαV

λ1 1λ2

c1c2

rð2Þ

wt (m s21) � heat propagation velocity; a (m2 s21) � thermal diffusivity; τq (s) �relaxation time of parallel heat propagation; αV (W m23 K21) � volume coefficient

of internal heat transfer; λ1, λ2 (W m21 K21) � thermal conductivity of material

components; c1, c2 (J m23 K21) � specific volume heat capacity of material

components.

It expresses the ratio of the apparent parallel heat transfer rate to the heat propa-

gation rate in a material (1). In the case of a two-component material, it can be

expressed as in equation (2).

Info: [A17],[B45].

5.9.26 Relative Parallel Thermal Conductivity Pλ

Pλ 5λp

λs

5X

λi

X 1

λi

λp (W m21 K21) � parallel thermal conductivity ðλp 5P

λiÞ;λs (W m21 K21) � series thermal conductivity ðλs 5 ðP 1

λiÞ21Þ;

λi (W m21 K21) � thermal conductivity of ith material component.

It expresses the ratio of the parallel thermal material conductivity to the series

conductivity, with said conductivities being determined by the thermal conductivi-

ties of individual material components.

Info: [A17],[B45].

5.9.27 Relative Parallel Thermal Diffusivity Pa

Pa 5ap

a

a (m2 s21) � thermal diffusivity; ap (m2 s21) � parallel thermal diffusivity.

It expresses the character of non-equilibrium heat propagation by means of the

parallel-to-standard heat diffusivities ratio.

The relative parallel diffusivity has two limits. In the case Pa-0, i.e. ap-0, the

heat propagates in material mostly by mediation of one kind of heat carrier. With

Pa-1, i.e. ap-a, the heat propagates nearly exclusively by a parallel mechanism.


In materials with two kinds of heat carriers, the thermal capacity of one of the

carriers is often many times greater than that of the second carrier and, on the con-

trary, the thermal conductivity of the first carrier is substantially less than that of

the second one.

Info: [A17],[B45].

5.9.28 Relaxation Time Nτ,rx

Nτ;rx 5τrxw2

ν5CatPr

τrx (s) � relaxation time; w (m s21) � fluid flow velocity; ν (m2 s21) � kinematic

viscosity; Cat (�) � Cattaneo number (p. 270); Pr (�) � Prandtl number (p. 197).

In non-equilibrium thermodynamics, this number expresses the relaxation time �in dimensionless form � in flowing fluid along a plate wall, heated by impulses or

strokes. It is used, for example, in observing the discontinuities of velocity gradients

and temperatures.

Info: [B77].

5.9.29 Sound Speed Number K

K5vτqL

� 2v (m s21) � adiabatic sound velocity in liquid; τq (s) � relaxation time; L (m) �characteristic length.

It expresses the ratio of the product of the relaxation time characterizing the

hyperbolic fluid behaviour and the in-fluid propagation rate in the fluid to the char-

acteristic dimension of the system.

Info: [B43].

5.9.30 Sparrow Numbers Sp

Sp5αL2

λekvRh

5Nuλf

λekv

� �L

Rh

� �2

α (W m22 K21) � equivalent heat transfer coefficient; L (m) � characteristic

length, porous layer thickness; λekv (W m21 K21) � effective thermal conductivity;

λf (W m21 K21) � fluid thermal conductivity; Rh 5ΔVp=ΔApðmÞ � hydraulic

radius; ΔVp (m3) � pore volume; ΔAp (m2) � pore surface; Nu (�) � Nusselt

number (p. 196); Pe (�) � Peclet heat number (p. 180).

281Thermomechanics

It characterizes the non-equilibrium heat transfer in porous material. Essentially,

it expresses the extension of the Nusselt number Nu (p. 196) for equilibrium heat

propagation to non-equilibrium heat propagation, which is of use in the flow in

porous materials. Usually λf,λekv holds. The number Sp enables the estimation of

the time delay due to the process non-equilibrium. The fraction SpPe21 enables

estimating local equilibrium in fluid flow through porous material.

Info: [B76].

5.9.31 Thermal Wave Speed Number K

K5Lv

að1Þ;

K5Ll

aτq5

1ffiffiffiffiffiffiffiffiFoq

p ð2Þ

L (m) � characteristic length; v (m s21) � velocity of the heat wave propagation;

a (m2 s21) � thermal diffusivity; l (m) � mean free path; τq (s) � relaxation time;

Foq (�) � Fourier relaxation wave number (p. 273).

It expresses the ratio of the thermal wave propagation rate to the characteristic

system dimension and the thermal diffusivity of the material or surroundings. With

the propagation rate expressed by the mean free path and the wave propagation

relaxation time v5 1=τq; the criterion is simplified to equation (2). Then, it

expresses the inverse root of the Fourier relaxation wave number Foq (p. 273).

Info: [B24].

5.9.32 Vernotte Heat Number Ve

Ve5Pe21 5a

wtLð1Þ;

Ve5a

wtrLð2Þ;

where wtr 5

ffiffiffiffia

τr

r

a (m2 s21) � thermal diffusivity; wt (m s21) � velocity of heat propagation;

wtr (m s21) � relaxation velocity of heat propagation; L (m) � characteristic

length; τr (s) � relaxation time; Pe (�) � Peclet heat number (p. 180).

It characterizes the influence of the finite heat propagation rate in a body. Wave

heat transfer in stroke phenomena. Non-equilibrium thermomechanics. The action

of intensive heat sources on material. New technologies.

Info: [A23].

Pierre Vernotte (born 18.5.1898), French physicist.


5.9.33 Vernotte Thermoelastic Number Vete

Vete 5Pe21te 5

a

wteL; where wte 5


R

s

a (m2 s21) � thermal diffusivity; L (m) � characteristic length; wte (m s21) �velocity of thermoelastic deformation propagation; λ, μ (Pa) � Lame functions;

R (kg m23) � density; Pete (�) � Peclet thermoelastic number (p. 143).

It characterizes the wave propagation rate of the thermoelastic stress or deforma-

tion in heat systems with stroke phenomena. Its inverse value is called the Peclet

thermoelastic number Pete (p. 143). Thermomechanics of stroke thermal processes.

New technologies.

Info: [A23].

Pierre Vernotte (see above).

5.9.34 Wave Penetration Depth (1.) H

H5FoffiffiffiffiffiffiffiffiffiForx

p ð1Þ;

ΘðFoÞ5Θð0Þexp 2Fo

2Forx

� �ð2Þ

Fo (�) � Fourier number Fo (p. 175); Forx (�) � Fourier relaxation heat number

(p. 272).

It characterizes a material zone in which the heat wave propagation influence

predominates. In the time Fo, it expresses the heat wave penetration depth

into material with wave-diffusive heat propagation and impulse action of surface

thermal flow. The front of the arising thermal wave is damped gradually (2).

Non-equilibrium thermomechanics. Physical technology.

Info: [A17],[B63].

5.9.35 Wave Penetration Depth (2.) Effective Hef

Hef 5 2ffiffiffiffiffiffiffiffiffiForx

p

Forx (�) � Fourier relaxation heat number (p. 272).

It expresses the heat wave penetration depth into a material with which the

wave temperature drops e times in passing through the material. It also confirms

that the penetration depth is proportional to the mean free path of the heat carriers.

Non-equilibrium thermomechanics. Physical technology.

Info: [A17],[B63].

283Thermomechanics

6 Electromagnetism

Finally, some observations of the momentum transfer from the Sun to the Planets

have been carried out and have fundamental significance for the theory. The signif-

icance of magneto-hydrodynamic waves is remarkable from this aspect.Hannes Olof Gosta Alfven (1908�1955)

Electromagnetism represents the fundamental interaction between electricity and

magnetism. The electric dimensionless quantities involve electric fields and cir-

cuits, electronics, illumination, electrical energy and a wide range of applications

in various fields. The magnetic dimensionless quantities express, above all, basic

magnetic forces caused by electric charge movement, propagation of electromag-

netic energy and waves, and other transfer phenomena, such as plasma physics and

magneto-hydrodynamics. The magneto-hydrodynamic dimensionless quantities

express the propagation of magneto-hydrodynamic waves, dynamic pressure, force

relations and energy conversion.

6.1 Electricity

A relatively large number of dimensionless quantities involve both electric circuits

and fields, and are as diverse as electrothermal, ionization and corona diffusion

processes; natural and forced electrothermal convection; heat transfer intensifica-

tion with dielectric fluid condensation; condensation in an electric field; condensate

film stabilization; mass transfer in electrochemical processes; the motion of

charged particles in flowing gas; and many other applications. Among them, the

dimensionless criteria for diverse boundary conditions and internal sources express

the inhomogeneity and non-linearity of the environment, charge density, electric

field intensity, corona phenomenon, earthing and puncture. Of the many dimension-

less numbers, the following are especially important: the electromechanical and

electronic numbers, the Fourier electrical number, the electric field intensity num-

ber, the Joule electrothermal number, the corona number, the Kronig number, the

Ohm number and many equivalents of basic quantities.



http://dx.doi.org/10.1016/B978-0-12-416013-2.00006-3

6.1.1 Biot Electrical Number Bie

Bie 5L

RαγS

L(m) � characteristic length; RαðΩÞ � electrical resistance equivalent to convective

thermal resistance; γ (S m21) � specific electrical conductance; S (m2) � surface

area in accordance with resistance Rα.

It characterizes the third-type boundary condition of an electric system. It is

analogous to the thermal Biot number Bi (p. 173).

Info: [A23].


6.1.2 Breakdown Coefficient N

N5Iρ

L2Ep

5U

LEp

I (A) � electric current; ρ (Ω m)� specific electrical resistance; L (m) � characteristic

length; Ep (V m21) � earth breakdown gradient; U (V) � spark zone surface voltage.

This coefficient expresses the dimensionless value of the earth breakdown gradi-

ent. Earthing.

Info: [A24].

6.1.3 Characteristic Electrical Number Nche

Nche 5 ρdεdT

� �L2ΔTE2

η2

ρ (kg m23) � density; ε (F m21) � permittivity; T (K) � temperature; L (m) �characteristic length; ΔT (K) � temperature difference; E (V m21) � electric field

intensity; η (Pa s) � dynamic viscosity.

It characterizes the effect of the temperature change of environmental dielectric

parameters with the electrostatic field in relation to the heat transfer from a heated

surface. Sometimes, it is incorrectly called the Kronig number Kr (p. 299).

Info: [A23].

6.1.4 Charge Density Number NQ

NQ 5wLρeH

w (m s21) � velocity; L (m) � characteristic length; ρe (C m23) � volume charge

density; H (A m21) � magnetic field intensity.


It expresses the ratio of the electric charge flow intensity to the magnetic field

intensity.

Info: [C173].

6.1.5 Clausius�Mossotti relation K

KðωÞ5 ε�1 2 ε�2ε�1 1 2ε�2

; where ε� 5 ε2jγω

ε�1; ε�2 (F m21) � permittivity of two different environments; γ (S m21) � specific

electrical conductance; ω (s21) � angular frequency of electrical field; j (2) �imaginary unit.

This relation is used in electrokinetics. The real part determines the coefficient

of the electromagnetic force acting on a particle. The imaginary part expresses the

electrorotating moment. The particle size and the electric field are other influences.

Electricity. Electrokinetics.

Info: [C20].

Rudolf Julius Emanuel Clausius (p. 175).

Ottaviano-Fabrizio Mossotti

6.1.6 Communication Coefficient N

N5ω ε0 β AR

ω (s21) � angular frequency; ε0 (F m21) � vacuum permittivity; β (m21) � wave

number (β5 2πλ21); A (m2) � cross section area of electron flux; R (Ω) �resistance.

It expresses the relative efficiency of the electron flow and the electric field

wave. Communications engineering.

Info: [A24].

6.1.7 Condensate Film Stability (1.) N

N5εE2L2

ρlν2l

ε (F m21) � permittivity; E (V m21) � electric field intensity; L (m) � characteristic

length; ρl (kg m23) � condensation layer density; νl (m2 s21) � condensation layer

kinematic viscosity.

It expresses the Coulomb-to-viscosity forces ratio. It characterizes the degree of

the stability damage of a condensate film and its deformation in an electric field.

This appears as formation of Coulombs and abruptly formed condensate drops on

287Electromagnetism

the surface due to longitudinally nonuniformly distributed pressure (εE2/2). With

sufficiently great E, the film is damaged and the condensate drops are sprayed,

with the electric force overcoming the viscosity force.

Info: [A23].

6.1.8 Condensate Film Stability (2.) N

N5εE2L

σ

ε (F m21) � permittivity; E (V m21) � electric field intensity; L (m) � characteris-

tic length; σ (N m21) � surface tension.

It expresses the ratio of the Coulomb force to the surface stress force. It charac-

terizes the stability degree of a condensate film and its deformation in an electric

field in relation to the surface stress force.

Info: [A23].

6.1.9 Condensation Electrostatic Number N

N5ðεgrad E2Þlδl

gρLð1Þ; N5

ðεE2ÞυgρL

ð2Þ

ε (F m21) � permittivity; E (V m21) � electric field intensity; δl (m) � condensate

film thickness; g (m s22) � gravitational acceleration; ρ (kg m23) � condensate

density; L (m) � characteristic length; index y: l � condensate, υ � vapour.

This number expresses the ratio of the gravity force to the ponderomotive force

of an electric field. It characterizes the heat transfer intensity with dielectric fluid

condensation on an outer tube surface (L5D/2) in an inhomogeneous electrostatic

field. In expression (1), it represents the volume ponderomotive force; in expres-

sion (2), it represents the surface force. The surface electrostatic force, acting on

the interface of phases, has an especially strong influence on heat transfer

intensification.

Info: [A23].

6.1.10 Corona Number NK

NK 5r0

ffiffiffiffiffiffiffiffiε0ω

pffiffiffiffiffiτk

p

r0 (m) � wire radius; ε0 (F m21) � vacuum permittivity; ω (s21) � angular fre-

quency; τ (A s m21) � linear density of electric charge; k (A s2 kg21) � ion

mobility.

It expresses the corona phenomenon.

Info: [A24].


6.1.11 Current Source Number NJ

NJ 5J0 R Δ x

U

J0 (A m21) � linear density of electric current; R (Ω) � resistance; Δx (m) �length of element; U (V) � voltage.

It concerns the current source related to 1 m of the conductor length. It deter-

mines, for example, the electric loss in conductors. Electric machines and devices.

Info: [A24].

6.1.12 Dielectric Constant εr

εr 5εsε0

ð1Þ; ε5 ε0εr ð2Þ

εs (F m21) � static material permittivity; ε0 (F m21) � vacuum permittivity

(ε05 8,8541873 10212 F m21); ε (F m21) � absolute permittivity.

In the case of a static field, it expresses the insulation properties of a dielectric

material. In the case of an alternating field or electromagnetic oscillation, it

expresses the relation between the vectors of electric induction and electric field

and depends on the frequency. Expression (2) represents the absolute permittivity.

The values of εr are: for air ðεr61Þ, for glass ðεr67:6Þ and for water ðεr680Þ.Electromagnetism. Electric fields. Physical properties.

Info: [C113].

6.1.13 Diffusion Corona Number N

N5D

LkE

D (m2 s21) � mass diffusion coefficient in electric field; L (m) � characteristic

length; k (A s2 kg21) � ion mobility; E (V m21) � electric field intensity.

It expresses the ratio of the diffusion mass transfer to rectified ion movement. It

characterizes the mass diffusion process in a corona discharge. The diffusion begins

to appear with N$ 1, approximately.

Info: [A23].

6.1.14 Dynamics of Inner Heat Source Number Dy

Dy5w2tδT 2w1t

w1t

; where w1t 5w1qV

cρ; w2t 5

w2qV

cρ

289Electromagnetism

δT (2) � thermogradient coefficient; w1, w2 (K s21) � warming velocity;

qV (W m23) � volume density of heat flux; c (J kg21 K21) � specific heat capacity;

ρ (kg m23) � density.

It characterizes the influence of an inner heat source on heat and mass transfers,

when power changes are affected by material temperature changes. When Dy, 0,

the power increase; when Dy. 0, it decreases. In dielectric heating, the internal

volume source qV is known. Induction and dielectric heating. Electric heat.

Info: [A33].

6.1.15 Electric Field Intensity N

N5E

wμHð1Þ; N5

EL

w

ffiffiffiγη

rð2Þ

E (V m21) � electric field intensity; w (m s21) � velocity; μ (H m21) � permeability;

H (A m21) � magnetic field intensity; L (m) � characteristic length; γ (S m21) �specific electrical conductance; η (Pa s)� dynamic viscosity.

It characterizes the ratio between the fed and induced electric field intensities in

moving electrically conductive fluids. In examining the outer electric field influ-

ence independently from the induced electric field � e.g. with a zero value for the

outer magnetic field � it is more suitable to use expression (2). Sometimes it is

called the criterion of loading characteristics MHD canal.

Info: [A23],[A33].

6.1.16 Electrical Circuit Number N

N5UτLI

5R2C

Lð1Þ; N5

τRC

ð2Þ

U (V) � voltage; τ (s) � time; L (H) � inductance; I (A) � electric current; R (Ω) �resistance; C (F) � capacitance.

Expression (1) describes the transient phenomena in an electric RLC circuit after

a sudden alternating voltage connection. The ratio of time to the electric circuit

time constant is expressed by (2).

Info: [A24].

6.1.17 Electrical Convection Number N

N5αη2

grad E2ρ

α (A2 s4 kg21) � molecule polarization coefficient; η (Pa s) � dynamic viscosity;

E (V m21) � electric field intensity; ρ (kg m23) � density.


It expresses the ratio of the molecular friction force to the electroconvective

one. It characterizes the dielectric fluids flow intensity influenced by the

electric field inhomogeneity. In natural convection, it appears as the Grashof heat

number. (p. 185)

Info: [A23],[A33].

6.1.18 Electrical Field Number NE

NE 5QNL4

τkU2

Q (C) � electric charge; N (m23 s21) � number of ion pairs; L (m) � length;

τ (C m21) � linear density of an electric charge; k (A s2 kg21) � anion mobility;

U (V) � voltage.

This number represents an electric field in ionized gas.

Info: [A12].

6.1.19 Electric Instability N2

N2 5βεdT

dxL

βε (K21) � thermal coefficient of permittivity; dT/dx (K m21) � temperature gradi-

ent; L (m) � characteristic length.

It characterizes the inhomogeneity grade of the electric environment permittivity

in solving problems of equilibrium fluid stability in a horizontal planar condenser.

Info: [A23].

6.1.20 Electric Relaxation Number Nrelax

Nrelax 5νεEJAL2

ν (m2 s21) � kinematic viscosity; ε (F m21) � permittivity; E (V m21) � electric

field intensity; JA (A m22) � surface density of electric current; L (m) � character-

istic length.

It expresses the ratio of electric phenomena relaxation in surroundings to the

mechanical relaxation time. It characterizes the relaxation effects related to electric

wind. When the ratio is small, electric relaxation occurs independently of mechani-

cal relaxation. Heat transfer in electrothermal systems.

291Electromagnetism

6.1.21 Electrocalorimetric Effect N

N5cpρΔT

εE2

cp (J kg21 K21) � specific heat capacity; ρ (kg m23) � density; ΔT (K) � tempera-

ture difference; ε (F m21) � permittivity; E (V m21) � electric field intensity.

This effect expresses the ratio of the heat obtained by transfer in the unit volume

to the energy density in an electric field. It characterizes the thermal effect of elec-

tric current passage through fluids without considering the fluid heating by friction,

polarization energy and energy of electrostatic charges. (K{1).

Info: [A12].

6.1.22 Electromechanical Number Es

Es5JAρL3

k1 η2

JA (A m22) � surface density of electric current at corona discharge in gas;

ρ (kg m23) � gas density; L (m) � characteristic length; k1 (A s2 kg21) � positive

charge carrier mobility; η (Pa s) � dynamic viscosity.

It characterizes the influence of the gas boundary layer damage, by corona elec-

tric discharge, on the intensity of the convective heat transfer between an electrode

and gas under conditions of natural or forced convection near the coronizing elec-

trode surface.

Info: [A23],[A33].

6.1.23 Electronic Number N

N5JAε0BLρ2

ð1Þ; N5Uε0L2ρ

ð2Þ;

N5JAL

Dρð3Þ; N5

Dε0BL2ρ

ð4Þ;

JA (A m22) � surface density of electric current; ε0 (F m21) � vacuum permittivity;

B (T) � magnetic induction; L (m) � characteristic length; ρ (C m23) � volume

density of electric charge; U (V) � voltage; D (m2 s21) � diffusion coefficient.

In expression (1), it expresses the current, limited by the volume charge, in real

dielectrics and semiconductors. In expression (2), it represents a voltage. In expres-

sion (3), it describes the current, limited by the volume charge, when considering

the diffusion of charge carriers in semiconductors and dielectrics with the diffusion


predominating over drift. In expression (4), the equivalent drift and diffusion influ-

ence is considered.

Info: [A24].

6.1.24 Electrothermal Convection Number Net

Net 5εβτE

2L3grad T

ρνa

ε (F m21) � permittivity; βτ 5 τ21r dτr=dTðK 21Þ � temperature coefficient of

relaxation time of electrical effects; E (V m21) � electric field intensity; T (K) �temperature; L (m) � characteristic length; ρ (kg m23) � density; ν (m2 s21) �kinematic viscosity; a (m2 s21) � thermal diffusivity; τr5 εγ21 (s) � relaxation

time of electrical effects.

It characterizes the electrothermal convection in fluids under the action of a

homogeneous or inhomogeneous field. It expresses also the forces of the electro-

thermal substance acting with the electrothermal convection. With no electric field

acting, it is Net5 0. The number Net expresses the electrothermal convection suit-

ably, especially with a homogenous field.

Info: [A23].

6.1.25 Electrothermal Convection Parameter N, r

N5εβτE

2

ρβgL5NetRa

212 5

εβτΔV2

ρβgL3

ε (F m21) � permittivity; βτ 5 τ21r dτr=dTðK 21Þ � temperature coefficient of elec-

trical effects on relaxation time; E (V m2 1) � electric field intensity; ρ (kg m23) �density; β (K21) � volume thermal expansion coefficient; g (m s22) � gravitational

acceleration; L (m) � characteristic length; V ðVÞ � electric potential; Net (2) �electrothermal convection number (p. 293); Ra2 (2) � Rayleigh number (2.)

(p. 187); τr (s) � relaxation time of electrical effects.

This parameter expresses the electrothermal to buoyancy forces ratio. It charac-

terizes the relation of electrothermal convection to free convection.

Info: [A23].

6.1.26 Electrothermal Conversion (1.) N

N5ρwhJAE L

293Electromagnetism

ρ (kg m23) � density; w (m s21) � flow velocity; h (J kg21) � specific enthalpy;

JA (A m22) � surface density of electric current; E (V m21) � electric field inten-

sity; L (m) � characteristic length.

It characterizes the process of electric energy conversion to thermal energy,

which proceeds in a gas arc. This criterion is among the fundamental ones of use,

for example, in work related to plasmatrones with heated gas.

Info: [A23].

6.1.27 Electrothermal Conversion (2.) N

N5ρw3

JAE L

ρ (kg m23) � density; w (m s21) � velocity; JA (A m22) � surface density of electric

current; E (V m21) � electric field intensity; L (m) � characteristic length.

It characterizes the process of electric energy conversion to kinetic energy of a

rectified flow. The sudden temperature increase of the heated gas and correspond-

ing density drop, with a given passage and cross section of a canal, are connected

to a remarkable velocity increase of the gradual gas flow. The criterion is of use,

for example, in a gas heater solution with supersonic flow in aerodynamic tunnels

if an electric arc is used in an accelerating nozzle.

Info: [A23].

6.1.28 Electrothermal Coupling Number NET

NET 5J2AL

2

γλΔT

JA (A m22) � surface density of electric current; L (m) � characteristic length;

γ (S m21) � specific electric conductance; λ (W m21 K21) � thermal conductivity;

ΔT (K) � temperature difference.

It expresses the coupling between electric current and heat conductions. With

this relation as defined by the number, further criteria expressing Joule heat action

can be obtained, for example, in flowing surroundings with physical and chemical

conversions.

Info: [A28].

6.1.29 Electrothermal Power N

N5γ2U4

gρ2c2T2d3βΔT5

γ2U4

ηλcT2

ηλgρ2cd3βΔT

5K1Gr21Pr21


γ (S m21) � specific electrical conductance; U (V) � voltage; g (m s22) � gravita-

tional acceleration; ρ (kg m23) � density; c (J kg21 K21) � specific heat capacity; T

(K) � local temperature in tank furnace; d (m) � electrode diameter; β (K21) � vol-

ume thermal expansion coefficient; ΔT (K) � temperature difference; η (Pa s) �dynamic viscosity; λ (W m21 K21) � thermal conductivity; K1 (2) � criterion K1;

Gr (2) � Grashof heat number (p. 185); Pr (2) � Prandtl number (p. 197).

It expresses the ratio of electric forces in molten material to mechanical forces

in free moving molten material with thermoelectric convection. It characterizes the

mutual relation among fundamental physical processes in an electric induction fur-

nace and their influence on heat transfer. The influence of the product (Gr Pr)

characterizes mechanical forces distributed in a tank that depends on slag para-

meters, temperature distribution and geometric dimensions. When Gr Pr. 23 107,

the process becomes automatic modelling. The criterion K1 depends on the charac-

ter of the electric field and inner source distribution.

Info: [A23].

6.1.30 Erlang Unit Eb, P

EbðN;AÞ5 AEbðN21;AÞN1AEbðN21;AÞ ; Ebð0;AÞ51 ð1Þ;

Pð. 0Þ5AN

N!N

N2APN21x5 0

Ax

x!AN

N!N

N2A

ð2Þ

N (2) � number of sources, for example, phones, servers or circuits; A (2) �operation size in dimensionless Erlang units.

The Erlang unit expresses the statistical degree of telecommunications opera-

tions used in telephony or internet connections, for example. One Erlang expresses,

for example, one source in uninterrupted operation or two sources at half of their

capacities. The Erlang unit is used to determine whether a telecommunications sys-

tem is overloaded or, on the contrary, underutilized � in other words, whether it

has many or very few utilized sources. It is applied to calculate the grade of service

(GoS) and the quality of service (QoS) as well. There are several diverse Erlang

equations, among which the Erlang B equation (1) and Erlang C equation (2) are

especially important.

Erlang relation B (1) expresses the probability Eb that a system is blocked. This

means that with a request not being realized immediately, it is blocked and can-

celled immediately. This relation is used for telephone systems which do not use

the so-called queues.

The Erlang relation C (2) expresses the probability P that a request remains in

the system queue for a certain length of time. In such a way, the quantity P(. 0)

expresses the probability that the response delay to the request is greater than zero,

meaning that the request is not realized immediately. This relation is used for tele-

phone ‘call centre’ systems, in which blocked requests remain in the queue as long

295Electromagnetism

as they can be realized. This system is utilized also in networks transmitting data

sets, for example, on the internet.

In addition to the Erlang unit, the Engset relation is applied to express the statis-

tical degree of a telecommunications operation where there is a narrow relation

between these quantities.

Info: [C52].

Agner Krarup Erlang (1.1.1878�3.2.1929), Danish math-

ematician, statistician and engineer.

He was the first to systematically deal with the problems

of telephone networks. In examining the activities of

municipal phone systems, he elaborated a dimensionless

expression which is known as the Erlang unit nowadays. In

particular, it represents the telecommunications criterion.

Even though the Erlang model is simple, the mathematical

foundation of present complex telecommunications net-

works is substantially based on this principle.

6.1.31 Fourier Electrical Number Foe

Foe 5τΔx2

RCL25

τRCn2

τ (s) � time; Δx (m) � geometric step; R (Ω) � resistance; C (F) � capacitance;

L (m) � characteristic length; n (2) � number of cells.

This is the dimensionless description related to time of electric current propaga-

tion through a conductor, e.g. through a geometrically discretized RC model.

Info: [A23].


6.1.32 Fourier Relaxation Number of Electrical Field Foe

Foe 5νeτeL2

5νeεL2γ

; where τe 5εγ

νe (m2 s21) � electric viscosity; τe (s) � relaxation time of electric field propaga-

tion; L (m) � characteristic length; ε (F m21) � permittivity; γ (S m21) � specific

electrical conductance.

This number expresses the charge relaxation time which influences the electric

and electromagnetic field distribution. Together with the Fourier magnetic number

Fom (p. 308), it expresses the Fourier electromagnetic number Foem (p. 308).

Electric fields.



6.1.33 Gain

Gain5 10 logP2

P1

ðdBÞ ð1Þ ; Gain5 20 logV2

V1

ðdBÞ ð2Þ

P1, P2 (W) � input and output power signal; V1, V2 (V) � input and output voltage

signal.

It is used in electronics especially and expresses the output-to-input power (1)

or voltage (2) signals ratio. It is used widely in amplifiers. Electronics,

Telecommunications. Automatic control.

Info: [C62].

6.1.34 Grashof Electrical Number Gre

Gre1 5F

ρν25

εE2L2

ρν2ð1Þ; Gre2 � Kr1 ð2Þ

F (N) � Coulomb force; ρ (kg m23) � density; ν (m2 s21) � kinematic viscosity;


length; Kr1 (2) � Kronig number (p. 299).

It expresses the electric-to-viscous forces under thermoelectric convection con-

ditions. It characterizes the Coulomb forces influencing the origin of the electric

convection and wind which occurs due to gradual environmental neutrality loss. In

gases, it is about a secondary phenomenon arising with corona discharge. The force

F expresses the force close to the coronizing electrode. However, the number Gredoes not express the field inhomogeneity as a necessary condition for electric con-

vection generation. The Grashof electrical number Gre2 is known as the Kronig

number Kr1 (p. 299) as well.

Info: [A23].


6.1.35 Grashof Electrical Relative Number Grrel

Grrel 5εE2

LgΔρ5Gre1Gr

21


length; g (m s22) � gravitational acceleration; Δρ � density difference; Gre1 (2) �Grashof electrical number (p. 297); Gr (2) � Grashof heat number (p. 185).

It expresses the ratio of a free electrothermic convection force to a free thermal

convection force. It characterizes the ponderomotive force influence on the heat

transfer under the action of electric convection.

Info: [A23].

Franz Grashof (see above).

297Electromagnetism

6.1.36 Ground Parameter N

N5UL

Iρð1Þ; N5

ρλΔT

U2ð2Þ

U (V) � voltage; L (m) � characteristic length; I (A) � electric current; ρ (Ω m) �specific electrical resistance; λ (W m21 K21) � thermal conductivity; ΔT (K) �ground warming around earthed system.

Expression (1) describes the voltage-to-grounding ratio recalculation, and

expression (2) represents the relation between the voltage and the heating of the

earth.

Info: [A24].

6.1.37 Joule Electrothermal Number Jo

Jo5λΔT

L2JAEð1Þ; Jo5

λTγU2

5λT

γE2L2ð2Þ

λ (W m21 K21) � thermal conductivity; ΔT (K) � temperature difference; L (m) �characteristic length; JA (A m22) � surface density of electric current; E (V m21) �electric field intensity; T (K) � temperature; γ (S m21) � specific electric

conductance; U (V) � voltage.

This number expresses the ratio of Joule heat to electric energy. Equation (1)

can be expressive for devices with low efficiency and low gradual motion velocity,

for example, the plasma in an electric arc. Equation (2) is called the Joule effect

criterion, which expresses the thermal to electric flows ratio in a fluid. With higher

values, the fluid Joule heating influence can be neglected in comparison to the

heating by simple heat conduction in the fluid.

Info: [A23].

James Prescott Joule (1818�1889), English physicist.

Above all, he is well known due to his experimental

determination of mechanical heat theory. This was con-

nected with his initial effort to determine the efficiency of

electric motors. He was one of the authors of the energy

conservation law, discoverer of the Joule heat unit and the

co-discoverer of the Joule�Thomson phenomenon. He

determined the mechanical equivalent of heat created by

friction.


6.1.38 Kirpichev Electric Number Kie

Kie 5JAL

γU


γ (S m21) � specific electrical conductance; U (V) � voltage.

It relates to the planar current density, planar source, surface loss and second-

type boundary condition. It is also called the Pomerantsev electrical number PoeA(p. 302).

Info: [A23].


6.1.39 Kronig Number Kr

Kr1 � Gre2 5L2βΔTE2Nk

Mν2α1

2p2

3kTN

� �ð1Þ;

Kr2 5εð3β 1 2βγÞE2L2ΔT

ρν25NetPr

21 ð2Þ

L (m) � characteristic length; β (K21) � volume thermal expansion coefficient;

ΔT5 TS2 TN (K) � temperature difference small against TN; TS, TN (K) � wall

and undisturbed fluid flow temperatures; E (V m21) � electric field intensity;

Nk (mol21) � Avogadro constant; M (kg mol21) � molar mass; ν (m2 s21) �kinematic viscosity; α (A2 s4 kg21) � gas molecule polarization coefficient;

p (C m) � electrical molecular dipole moment; k5RN 21k ðJ K21Þ � Boltzmann

constant; R (J mol21 K21) � molar gas constant; ε (F m21) � permittivity;

βε (K21) � thermal coefficient of permittivity; βγ (K21) � thermal coefficient of

specific electrical conductance; ρ (kg m23) � density; Gre2 (2) � Grashof electrical

number (p. 297); Net (2) � electrothermal convection number (p. 293); Pr (2) �Prandtl number (p. 197).

This number expresses the ratio of the product of the inertia and electrostatic

forces to the square of the viscous force. It characterizes the electrostatic field

effect on the heat transfer from a surface heated by convection. It expresses the

force relations caused by the inhomogeneity of electrophysical properties and vis-

cous friction in a fluid. Essentially, it is analogous to the Archimedes thermody-

namic number Ar (p. 184). In form (2), it is a modified Kronig electrothermal

convection number. The Kronig number Kr1 is also called the Grashof electrical

number Gre2 (p. 297).

Info: [A23].

Ralph De Laer Kronig (born 1904), German-American physicist.

299Electromagnetism

6.1.40 Kutateladze Electrothermal Number Kue

Kue 5JAEL

ρwΔh

JA (A m22) � surface density of electric current; E (V m21) � electric field inten-

sity; L (m) � characteristic length; ρ (kg m23) � density; w (m s21) � fluid velocity;

Δh (J kg21) � increment of specific enthalpy.

It characterizes the electrothermal process with an electric discharge in a flow-

ing fluid.

Info: [A23].

Samson Semenovich Kutateladze (18.7.1914�20.3.1986), Russian engineer

and physicist (p. 327).

6.1.41 Length Current Density K

K5γE2d2

cηT

γ (S m21) � specific electrical conductance; E (V m21) � electric field intensity;

d (m) � electrode diameter; c (J kg21 K21) � specific heat capacity; η (Pa s21) �dynamic viscosity; T (K) � temperature.

It expresses the ratio of longitudinal densities of electric and thermal flows. It

characterizes the electrothermal process in diverse electrothermal systems, for

example, in electric furnaces.

Info: [A23].

6.1.42 Motulevitch Number Mo

Mo5 ρL2E2

η2Δε; where Δε5

dεdT

� �ΔT

ρ (kg m23) � density; L (m) � characteristic length; E (V m21) � electric field

intensity; η (Pa s) � dynamic viscosity; ε (F m21) � permittivity; T (K) �temperature.

It expresses the ratio of the electric force caused by dielectric susceptibility

dependence on the environmental density to the viscosity force. It characterizes the

influence of the inhomogeneity, and the thermal system non-uniformity connected

with it, on the electric convection in dielectric fluids. It expresses the influence of

an electric field on the transfer phenomenon.

Info: [A23].


6.1.43 Nusselt Electrical Number Nue

Nue 5wL

D; where D5

1

2ðD11D2Þ

w (m s21) � flow velocity; L (m) � characteristic length; D (m2 s21) � final ion

diffusivity.

It expresses the ratio of the convective flow to the diffusion electric flow. It

characterizes the electrothermal convection in electrochemical processes.

Sometimes, it is called the Reynolds electrical number (2.) NuEL2 (p. 304).

Info: [A23].


6.1.44 Ohm Number Oh

Oh5γEJA

γ (S m21) � specific electrical conductance; E (V m21) � electric field intensity;

JA (A m22) � surface density of electric current.

In the dimensionless state, it characterizes the electric conductance of a system.

Info: [A23].

Georg Simon Ohm (16.3.1789�6.7.1854), German

physicist.

By experiment, he deduced the Ohm law and introduced

the electrical resistance concept. He was engaged in acous-

tics as well. He was influenced by French mathematicians

and physicists. On the base of the Fourier heat conduction

law, he modelled the electric current conduction in an elec-

tric network and described it mathematically.

6.1.45 Peclet Electrical Number Pee

Pee 5wLRC

Δx2

w (m s21) � velocity; L (m) � characteristic length; RC (s) � time constant of

electrical RC network; R (Ω) � resistance; C (F) � capacitance; Δx (m) � element

length dimension.

It expresses the relationship among motion velocity of a source, zone boundaries

and conduction in a geometrically discrete RC model.

Info: [A24].


301Electromagnetism

6.1.46 Pomerantsev Electrical Number Poe

PoeV 5JVL

2

γUr

ð1Þ; PoeA � Kie 5JAL

γUr

ð2Þ;

PoeL 5JL

γUr

ð3Þ; PoeB 5I

γUrLð4Þ

JV, JA, JL � volume density of electric current (A m23), surface density (A m22)

and linear density (A m21); L (m) � characteristic length; γ (S m21) � specific

electrical conductance; Ur (V) � reference voltage; I (A) � electric current; Kie(2) � Kirpichev electrical number (p. 299).

It expresses the ratio of the fed electric flow to that transferred by conduction in

a conductor. It characterizes volume (1), planar (2), longitudinal (3) or spot (4)

inner current sources. It represents thermal loss caused by electric current passage

through conductors.

Info: [A23].


6.1.47 Rayleigh Electrothermal Condensation Number Rael

Rael 5FehρlvlηλΔT

Feh (N) � equivalent excitation electrodynamic force acting on condensing film;

ρ (kg m23) � density; lvl (J kg21) � specific latent heat of condensation; η (Pa s) �

dynamic viscosity; λ (W m21 K21) � thermal conductivity; ΔT (K) � temperature

difference.

It expresses the ratio of the product of the buoyancy electrohydrodynamic force

and the inertia force to the square of the viscosity force. It characterizes the acting

forces influence on the condensation process in an electric field. Condensation in

the electric field.

Info: [A23].


6.1.48 Rayleigh Electrothermal Instability Number Rae1

Rae1 5β2εD

2ðrTÞ2L4ερνa

βε (K21) � thermal coefficient of permittivity for τr=τD{1; τr (s) � relaxation

time of temperature field; τD (s) � characteristic time of electric induction change;

D (C m21) � electric induction; T (K) � temperature; L (m) � characteristic


length; ε (F m21) � permittivity; ρ (kgτr m23) � density; ν (m2 s21) � kinematic

viscosity; a (m2 s21) � thermal diffusivity.

It expresses the ratio of the product of the electric buoyancy force and the inertia

force to the square of the viscosity force. It characterizes the electrothermal con-

vection in ideal liquid dielectrics. It occurs in the stability description of a balanced

planar horizontal layer in a vertical electric field.

Info: [A23].


6.1.49 Rayleigh Electrothermal Number Rae

Rae 5 nεβεL

3E3rTρνa

n5 0, 1, 2; ε (Fτr m21) � permittivity; βε (K

21) � thermal coefficient of permittivity;

L (m) � characteristic length; E (Vτr m21) � electric field intensity; T (K) �

temperature; ρ (kg τ m23) � density; ν (m2 s21) � kinematic viscosity; a (m2 s21) �thermal diffusivity.

This number characterizes the electrothermal convection originating in a fluid

due to field inhomogeneities which are influenced by the geometric shape of

electrodes.

Info: [A23].


6.1.50 Reynolds Electrical Number (1.) ReEL1

ReEL1 5wL

νe5wL

γL2

ε

� �21

5wεγL

ð1Þ;

ReEL1 5wερkL

5w

kEð2Þ

w (m s21) � flow velocity; L (m) � characteristic length; νe (m2 s21) � electrical

kinematic viscosity; γ (S m21) � specific electrical conductance; ε (F m21) �permittivity; k (A s2 kg21) � charge carrier mobility; E5 ρL /ε (V m21) � electric

field intensity; ρ (C m23) � volume density of electric charge.

This number expresses the ratio of the fluid inertia force to the friction electric one

(of atoms, electrons or ions). It characterizes the influence of the inertia force and

that of the electric viscosity with forced electrothermal convection. In equation (2), it

expresses the volume density of charges. Magneto-hydrodynamics.

Info: [A23].


303Electromagnetism

6.1.51 Reynolds Electrical Number (2.) ReEL2

See the Nusselt electrical number Nue (p. 301).


6.1.52 Schmidt Electrodiffusion Number Sc3

Sc3 5νεγL2

ν (m2 s21) � kinematic viscosity; ε (F m21) � permittivity; γ (S m21) � specific

electrical conductance; L (m) � characteristic length.

It expresses the ratio of the swirl diffusivity of a system to the mass diffusivity

of ions. It characterizes the mass transfer in electrochemical processes.

Info: [A23].

Ernst Schmidt (p. 264).

6.1.53 Senftleben Number Se

Se5NkE

2

LMgα1

2

3

p2

kTN

� �5KrGr21

Nk (mol21) � Avogadro constant; E (V m21) � electric field intensity; L (m) �characteristic length; M (kg mol21) � molar mass; g (m s22) � gravitational accel-

eration; α (A2 s4 kg21) � gas molecule polarization coefficient; p (C m) � electric

molecular dipole moment; k (J K21) � Boltzmann constant; TN (K) � temperature

of undisturbed fluid flow; Kr (2) � Kronig number (p. 299); Gr (2) � Grashof

heat number (p. 185).

This number characterizes the electric field influence on the heat transfer of

paraelectric fluids (with constant dipole moment). Essentially, it is another expres-

sion of the Kronig number Kr (p. 299).

Info: [A23].

Hermann Senftleben (born 1890), German physicist.

6.1.54 Source of Electrical Heat N

N5J2AL

γρwΔh5

u

Δh

JA (A m22) � surface density of electric current; L (m) � characteristic length; γ(S m21) � specific electrical conductance; ρ (kg m23) � density; w (m s21) �motion velocity; Δh (J kg21) � specific enthalpy change; u (J kg21) � specific

inner energy.


It expresses the ratio of the fed specific inner energy to the specific enthalpy

change of moving surroundings. It characterizes the thermal source caused by elec-

tric current passage through moving surroundings.

Info: [A23].

6.1.55 Stefan Electrical Number Sfe

Sf e 5σγU3L

σ (A m22 V24) � electric equivalent of surface immisivity or emissivity; γ (S m21) �specific electrical conductance; U (V) � voltage; L (m) � characteristic length.

This number expresses the ratio of the electric energy transmitted by a surface

under a non-linear boundary condition to that transmitted by conduction in a sys-

tem. It characterizes a non-linear boundary condition, expressed by a power func-

tion, in an electric system which is analogous to a thermal one. It is of use, for

example, in the electrothermal modelling of heat transfer by radiation. It is the

physical analogue to the Stefan number Sf (p. 213).

Info: [A23].


6.1.56 Temperature Non-Uniformity N

N5εβ2

εL2E2ðrTÞ2ρνa

ε (F m21) � permittivity; βε (K21) � thermal coefficient of permittivity; L (m) �

characteristic length; E (V m21) � electric field intensity; T (K) � temperature;

ρ (kg m23) � density; ν (m2 s21) � kinematic viscosity; a (m2 s21) � thermal

diffusivity.

It characterizes the influence of environmental thermal non-uniformity on an

inhomogeneous electric field and the phenomena it causes. Especially with high

field intensities, it has influence but does not have any substantial influence on

thermal process intensification except in cases of thin fluid layers.

Info: [A23].

6.1.57 Viscous Electrical Number E

E5

ffiffiffiffiffiffiffiffiffiffiρ

2πε0

rρL2Qηm

305Electromagnetism

ρ (kg m23) � density; ε0 (F m21) � permittivity; L (m) � characteristic length;

Q (C) � electric charge; η (Pa s) � dynamic viscosity; m (A m2) � magnetic

moment.

This number expresses the relation between electrostatic and hydrodynamic

quantities or, alternatively, the motion of charged particles in flowing gas.

Info: [A24].

6.2 Magnetism

In magnetism, the dimensionless quantities are related to magnetic processes and

forces caused by the electric charge motion. Some of them express steady-state

magnetic fields or unsteady electromagnetic ones, propagation of electromagnetic

energy and magnetic waves, thermomagnetic convection and transfer phenomena

in plasma and magnetic turbulence. Among the important numbers are the Ampere,

magnetic field, Joule magnetic, Kubo, Lorentz, Maxwell and Stuart, and other num-

bers. A great number of magnetic dimensionless quantities are related to magneto-

hydrodynamics.

6.2.1 Ampere Number Am

Am5JAL

H


H (A m21) � magnetic field intensity.

This number expresses the ratio of flow density to magnetic field intensity. It

characterizes the mutual action of the current and the magnetic field in magneto-

hydrodynamic processes.

Info: [A23].

Andre-Marie Ampere (20.1.1775�10.6.1836), French


He contributed markedly to the electricity and magne-

tism theory which became the base of the nineteenth cen-

tury scientific development. He executed many experiments

and, in addition to other things, he demonstrated the origin

of a magnetic field around a conductor through which elec-

tric current was flowing. The unit of electric current was

named in his honour.

6.2.2 Electromagnetic Coupling Constant α

α5e2

hcB

1

137:03599 . . .


e (C) � elementary charge; h (J s) � Dirac constant; c (m s21) � speed of light in

vacuum.

This constant characterizes an electromagnetic field. It can be utilized to demon-

strate the probability that an electron emits or absorbs a photon. It is a very impor-

tant phenomenon in modern physics and in quantum electrodynamics especially. It

is a non-linear expression. With extremely high energy levels, the effective value

of the electric charge changes the coupling constant value. Electromagnetic fields.

Quantum physics. Astrophysics.

Info: [C23].

6.2.3 Electromagnetic Effect K

K5μH2

εE2

μ (H m21) � permeability; H (A m21) � magnetic field intensity; ε (F m21) �permittivity; E (V m21) � electric field intensity.

This effect expresses the ratio of the magnetic field energy density to that of the

electric field. It characterizes the degree of the hydrodynamic action of electric and

magnetic fields in fluids. It measures the degree of magnetic effect on electric

convection.

Info: [A23].

6.2.4 Electrothermal Time Nτ

Nτ 5c

wA

� �2

5τrmτ

5 γμc2τrm;

where

τrm 5ρ

γμ2H2; τ5

1

γμc2

c (m s21) � speed of light; wA (m s21) � speed of Alfven waves; τrm (s) � relaxa-

tion time of magnetic field; τ (s) � conversion time of electromagnetic field energy

into Joule heat; γ (S m21) � specific electrical conductance; μ (H m21) � perme-

ability; ρ (kg m23) � fluid density; H (A m21) � magnetic field intensity.

It expresses the ratio of the lines-of-force relaxation time in an electrically con-

ductive fluid to the conversion time of the electromagnetic field energy to the heat

in Joules. It characterizes time relations involved in the conversion of the electro-

magnetic field to heat in Joules.

Info: [A23].

307Electromagnetism

6.2.5 Fourier Electromagnetic Number Foem

Foem 5νmτemL2

; where τem 5ffiffiffiffiffiffiffiffiffiffiτeτm

p; τe 5

εγ; τm 5μγL2

νm (m2 s21) � magnetic viscosity; τem (s) � propagation time of electromagnetic

wave; L (m) � characteristic length; τe (s) � relaxation time of electrical field

propagation; τm (s) � time of magnetic diffusion; ε (F m21) � permittivity;

γ (S m21) � specific electrical conductivity; μ (H m21) � permeability.

In the dimensionless expression, it characterizes the magnetic field propagation

time, for example, in an electrically conductive fluid to the depth L.

Info: [A23],[A33].


6.2.6 Fourier Magnetic Number Fom

Fom 5νmτmL2

5 νmμγ; where τm 5μγL22

νm (m2 s21) � magnetic viscosity; τm (s) � time of magnetic diffusion; L (m) �characteristic length; γ (H m21) � magnetic permeability; γ (S m21) � specific

electrical conductivity.

It expresses magnetic diffusion time. It points to the electromagnetic effects

influence on electromagnetic wave propagation. Together with the Fourier relaxa-

tion number of electrical field Foe (p. 296), it influences the characteristics of an

electromagnetic field. Magnetism.

Info: [A24].

6.2.7 Grashof Magnetic Number Grm

Grm 5μ0

@M

@T

rHL3βρν2

ð1Þ;

Grm 5μ0βMMðTS 2 TNÞL3

ρν2dH

dxð2Þ;

Grm 5 4πγμνGr ð3Þ

μ0 (H m21) � vacuum permeability; M (A m21) � magnetization; T (K) �temperature; H (A m21) � magnetic field intensity; L (m) � characteristic length;

β (K21) � volume thermal expansion coefficient of fluid; ρ (kg m23) � density;

ν (m2 s21) � kinematic viscosity; βM (K21) � volume thermal expansion of

magnetization; TS, TN (K) � wall and undisturbed fluid flow temperature;


x (m) � length, coordinate; γ (S m21) � specific electrical conductance;

μ (H m21) � permeability; Gr (2) � Grashof heat number (p. 185).

This number characterizes the heat transfer by free thermomagnetic convection

in paramagnetic fluids with a gravity acceleration which depends on the space

coordinates. It is of use under non-isothermic conditions with the action of thermo-

magnetophoresis transfer of particles.

Info: [A23],[A29].


6.2.8 Interaction Parameter Ninter

See the Stuart number (1.) (magnetic force) St (p. 329).

6.2.9 Joule Magnetic Number Jom

Jom 52ρcpΔT

μH2

ρ (kg m23) � density; cp (J kg21 K21) � specific heat capacity; T (K) � tempera-

ture; μ (H m21) � permeability; H (A m21) � magnetic field intensity.

It expresses the ratio of the heat in Joules to the total magnetic field energy. It

characterizes the heat transfer in Joules when acted upon by a magnetic field, for

example, in magneto-hydrodynamics and inductive heating.

Info: [A23].

James Prescott Joule (p. 298).

6.2.10 Kubo Number Kub, R

Kub5δBB0

Lpar

Lper

δB (T) � mean fluctuation of magnetic field; B0 (T) � mean value of magnetic

induction; Lpar, Lper (m) � parallel and perpendicular correlation lengths in regard

to B0.

The Kubo number is used to classify the transfer phenomena in turbulent mag-

netic and other systems. With a small Kubo number, the quasi-linear diffuse coeffi-

cient is of use, as is the filtration diffusion coefficient with a large Kubo number.

The Kubo number expresses the level of a magnetic flux lines chaos. For Kub{1,

a weak chaos is valid and for Kub. 0.3, the extent of the chaos damages the sur-

faces which had been closed before. With Kubc1, a complete stochastic behaviour

occurs in which the filtration diffusion coefficient is of use. Transient phenomena

in plasma. Magnetic turbulence. Diffusion.

Info: [B130].

309Electromagnetism

6.2.11 Lorentz�Lorenz Number n

n �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi11

3Ap

R T

r

A (m3 mol21) � molar index of refraction; p (Pa) � gas pressure; R (J mol21 K21) �molar gas constant; T (K) � absolute temperature.

This number represents the refraction index of rarefied gas. It expresses its

dependence on absolute temperature, pressure and the refraction index.

Electrodynamics. Magnetism. Electromagnetic radiation.

Info: [C109].

Hendrik Antoon Lorentz (p. 323).

Ludvig Valentine Lorenz.

6.2.12 Magnetic Dimensionless Frequency Nω

Nω 5τrxwref

L

τrx (s) � magnetic relaxation time of particles; wref (m s21) � reference velocity of

particles; L (m) � characteristic length.

It expresses the relation between the relaxation time of particles and the convec-

tion time. Usually, Nω{1 holds, which means that the influence of the angular

velocity of fluid particles is not as effective as the convective influence. Magneto-

hydrodynamics. Magnetic fields. Technology of magnetic separation, emulsifi-

cation, connection, etc.

Info: [B27].

6.2.13 Magnetic Field Number N

N5JAL

H5

I

LHð1Þ; N5 L2f 2με ð2Þ;

N5 L2fμγ5L2μγτ

ð3Þ; N5U

I

ffiffiffiεμ

rð4Þ


H (A m21) � magnetic field intensity; I (A) � electric current; f (s21) � frequency;

μ (H m21) � permeability; ε (F m21) � permittivity; γ (S m21) � specific electri-

cal conductance; τ (s) � time; U (V) � voltage.

In equation (1), it expresses a steady-state magnetic field. In expression (4), it

relates to an unsteady electromagnetic field. Further, expression (2) represents


electromagnetic energy propagation and expression (3) represents it electromag-

netic waves propagation.

Info: [A24].

6.2.14 Magnetic Force Parameter NmgF

See Stuart number (1.) (magnetic force) St (p. 329).

6.2.15 Maxwell Number (1.) Fundamental Mx

Mx5ε E2

μ H

ε (F m21) � permittivity; E (V m21) � electric field intensity; μ (H m21) �magnetic permeability; H (A m21) � magnetic field intensity.

It expresses the electric-to-magnetic field intensity ratio. It is the basic criterion

in the set of dimensionless Maxwell equations. Electromagnetic fields.

Info: [C87].

James Clerk Maxwell (13.6.1831�5.11.1879), Scottish

theoretical physicist.

He made revolutionary works in electrical engineering,

magnetism and the kinetic theory of gases. He predicted the

existence of electromagnetic waves, which propagate at

the same velocity as light. He created the electromagnetic

theory describing the phenomena of electrodynamics, optics

and thermal radiation. The substance of the Maxwell theory

is expressed by four Maxwell equations describing the time

and space dependence of the electromagnetic field.

6.2.16 Maxwell Number (2.) Electrodynamical Mx

Mx5τ2

εμL25Fom

τ (s) � time; ε (F m21) � permittivity; μ (H m21) � magnetic permeability; L (m) �characteristic length; Fom (2) Fourier magnetic number (p. 308).

This number expresses the dynamics of the electromagnetic field. It occurs in

the system of dimensionless Maxwell equations. Electromagnetic fields.

Info: [C87].

James Clerk Maxwell (see above).

311Electromagnetism

6.2.17 Maxwell Number (3.) Time Mx

Mx5ρτμL2

5Foem

ρ (Ω m) � specific electrical resistance; τ (s) � time; μ (H m21) � magnetic per-

meability; L (m) � characteristic length; Foem (2) � Fourier electromagnetic num-

ber (p. 308).

Essentially, it expresses the dimensionless time of an electromagnetic process. It

is also called the Fourier electromagnetic number (p. 308). Electromagnetic

processes.

Info: [A23],[C87].

James Clerk Maxwell (see above).

6.2.18 Ohmic Heating Oht

Oht 5λ γ ΔT

J2A L2

λ (W m21 K21) � thermal conductivity; γ (S m21) � specific electrical conduc-

tance; ΔT5 T2 TS (K) � temperature difference, difference of mean plasma tem-

perature and temperature of channel wall surface; JA (A m22) � surface density of

electric current; L (m) � characteristic length.

It expresses the ratio of the heat transferred by conduction to the electric energy

transferred by plasma. It characterizes the influence of the heat led away by con-

duction on the electric arc motion in a transversal electric field. It expresses the

heat transfer intensity from a plasma into a canal wall with laminar equilibrium

and fully developed flow in a canal spot heated with an arc plasma generator.

Info: [A23].

6.2.19 Peclet Magnetic Rotational Number Pemg

Pemg 5τBτrx

5L2

Dmgτrx

τB (s) � Brown relaxation time; τrx (s) � magnetic relaxation time of particles;

L (m) � characteristic length; Dmg (m2 s21) � magnetic diffusivity.

It expresses the relation between the Brown time and the magnetizing relaxation

time and, alternatively, it defines how fast the magnetic particles can be oriented

according to the local magnetic field. Magnetic field. Magnetic emulsification.

Magnetic separation. Magnetic spraying. Destructive methods.

Info: [B27].



6.2.20 Pomerantsev Electromagnetic Number Pomg

Pomg 5Lεqmg

λT

L (m) � characteristic length of solid particle; qmg (W m22) � surface density of

electromagnetic flux; λ (W m21 K21) � thermal conductivity; T (K) � source tem-

perature; ε (2) � emissivity (absorptivity) (p. 206).

This number characterizes the heating of solid particles in an electromagnetic

field. It is applied, for example, to discover the thermal properties of microscopic

particles by means of the impulse laser heating method. It is analogous to the

Pomerantsev heat number Po (p. 181).


6.2.21 Prandtl Magnetic Number Prm

Prm5El215νμγ5ννm

5RemRe21

ν (m2 s21) � kinematic viscosity; μ (H m21) � permeability; γ (S m21) � specific

electrical conductance; νm (m2 s21) � magnetic viscosity; El (2) � Elsasser num-

ber (p. 318); Rem (2) � Reynolds magnetic number (p. 328); Re (2) � Reynolds

number (p. 81).

It expresses the ratio of viscosity to magnetic diffusion. It characterizes the simi-

larity between the magnetic and the velocity fields of an electrically conductive

fluid flowing in the magnetic field. It is used to express the exciting effect on the

induced magnetic field current.

Info: [A23],[B20].


6.2.22 Rayleigh Magnetic Number Ram

Ram 5 γ2μ2L3gβΔT 5GrPr2m

γ (S m21) � specific electrical conductance; μ (H m21) � permeability; L (m) �characteristic length; g (m s22) � gravitational acceleration; β (K21) � volume

thermal expansion coefficient; ΔT (K) � temperature difference; Gr (2) �Grashof heat number (p. 185); Prm (2) � Prandtl magnetic number (p. 313); Ra2(2) � Rayleigh number (2.) (p. 187).

It characterizes the magnetic convection influence on heat transfer intensifica-

tion. The influence of the Ram number appears from Ram. 150 Ra2 approximately.

With RamcRa2, only the thermomagnetic convention is of use.

Info: [A23].


313Electromagnetism

6.3 Magneto-hydrodynamics

Magneto-hydrodynamics is focused on the hydrodynamics of electrically conduc-

tive fluids on plasma and liquid metals, especially. In this field, the range of dimen-

sionless quantities used is extensive, which corresponds to the significance of

dimensionless quantities of this field and to the wide range of applications in

applied physics, engineering branches, geophysics, astrophysics and other spheres.

Individual dimensionless quantities are focused on dynamic velocity, pressure and

force conditions in MHD, on the continuity of flow of electrically conductive fluid

or plasma and on the level of non-stationarity. The Alfven, Cowling, Einstein,

Elsasser, Hartman, Karman, Mach magnetic, Reynolds and Strouhal numbers are

among the basic dimensionless quantities used within this area.

6.3.1 Alfven Number Al

Al5Co12 5M21

m 5wA

w5

B

wffiffiffiffiffiffiμρ

p ð1Þ; where wA5H

ffiffiffiμρ

r;

Al5Eu12m 5

H

w

ffiffiffiμρ

rð2Þ

wA (m s21) � Alfven’s wave velocity; w (m s21) � velocity; B (T) � magnetic

induction; μ (H m21) � permeability; ρ (kg m23) � density; H (A m21) �magnetic field intensity; Co (2) � Cowling number (p. 316); Mm (2) � Mach

magnetic number (p. 324); Eum (2) � Euler magnetic number (p. 319).

This number expresses the magnetic-to-inertia forces ratio or the ratio of the

Alfven wave propagation rate to the fluid flow propagation rate. It characterizes

the propagation of magneto-hydrodynamic waves (Alfven waves) along the mag-

netic field flux lines in a flowing electrically conductive fluid flow (the plasma). It

corresponds to the Euler magnetic number Eum (p. 319) and sometimes is used in

equation (2) to express the magnetic-to-kinetic energy ratio in unit volume. Its

inverse value is called the Mach magnetic number Mm (p. 324) or the Karman num-

ber (2.) Ka (p. 321). Magneto-hydrodynamics.

Info: [A23],[B20].

Hannes Olof Gosta Alfven (30.5.1908�2.4.1995),

Swedish astronomer. Nobel Prize in Physics, 1970.

He was engaged in research in electrical engineering,

electronics and astrophysics, and is one of the founders of

modern plasma physics. He published the theory of mag-

netic storms and of polar glares and laid the foundation of

the Earth magnetosphere theory. His principal works con-

cern theoretical research of magneto-hydrodynamics, for

which he received the Nobel Prize.


6.3.2 Batchelor Number Bt

Bt5wγLc2ε

w (m s21) � velocity; γ (S m21) � specific electrical conductance; L (m) � charac-

teristic length; c (m s21) � speed of light; ε (F m21) � permittivity.

It characterizes the dynamic relations in magneto-hydrodynamics.

Info: [A23],[A35]

George Keith Batchelor (8.3.1920�30.3.2000), Australian


He worked in hydrodynamics and magneto-hydrodynam-

ics. The focus of his work was primarily in turbulence

research. Of his monographs, Homogenous Turbulence

(1953), Turbulent Diffusion (1956) and An Introduction to

Fluid Dynamics (1957) are the most important.

6.3.3 Brinkmann Modified Number Brmod

Brmod 5BrLffiffiffiζ

p 5EcPrLffiffiffiζ

p

L (m) � characteristic length (width) of channel; ζ (m2) � permeability of porous

material; Br (2) � Brinkmann number (p. 174); Ec (2) � Eckert number (p. 191);

Pr (2) � Prandtl number (p. 197).

It expresses the ratio of the heat caused by viscous friction of the fluid in micro-

scopic canals of a porous material to the heat transferred by molecular conduction.

Two-phase fluid mechanics. Magneto-hydrodynamics.

Info: [B70]

Henri Coenraad Brinkmann, German physicist.

6.3.4 Chandrasekhar Number Ch

Ch5B2L2

ρνρe

B (T) � magnetic induction; L (m) � characteristic length; ρ (kg m23) � fluid density;

ν (m2 s21) � kinematic viscosity; ρe (Ω m) � specific electrical resistance.

It expresses the magnetic-to-dispersion forces ratio. Magneto-hydrodynamics.

315Electromagnetism

Subrahmanyan Chandrasekhar (19.10.1910�21.8.1995),

Indian astrophysicist. Nobel Prize in Physics, 1983.

He was engaged in research in various areas of physics,

in astrophysics especially. Besides the theory of the structure

and dynamic stability of stars, he devoted himself to radia-

tion, the mathematical theory of black holes and general rel-

ativity. His name has been given to the Chandrasekhar limit.

According to this, a cooling star, with 1.5 times the Sun’s

mass, cannot resist its own gravitation. He received the

Nobel Prize for his theoretical work about the gravitational

collapse of stars.

6.3.5 Cowling Number, Magnetic Force Number Co

Co5Al2 5wA

w

� �2

5B2

μρw25

BH

ρw2

wA (m s21) � Alfven’s wave velocity; w (m s21) � velocity; B (T) � magnetic

induction; μ (H m21) � magnetic permeability; ρ (kg m23) � density; H (A m21) �magnetic field intensity; Al (2) � Alfven number defined by equation (1) (p. 314).

This number expresses the magnetic-to-inertia forces ratio. It is also called the

magnetic force number. It expresses the second power of the ratio of the Alfven

wave velocity to the fluid velocity. Magneto-hydrodynamics.

Info: [A29],[A43],[B17]

Thomas George Cowling (17.6.1996�16.6.1990), English


He contributed significantly to modern research on stel-

lar energy, especially that of the sun. He played an impor-

tant role in finding the convective nucleus in stars. Due to

this convection, the sun can behave as a giant dynamo, the

rotation, internal circulation and flow of which generate

electric currents and magnetic fields connected with solar

spots.

6.3.6 Eckert Magnetic Number Ecm

Ecm 5B2

μρcpΔT5

w2A

cpðT 2 TSÞ

B (T) � magnetic induction; μ (H m21) � permeability; ρ (kg m23) � density;

cp (J kg21 K21) � specific heat capacity; ΔT5 T2 TS (K) � fluid and wall tem-

peratures difference; wA (m s21) � Alfven’s wave velocity.


It characterizes heat transfer under the conditions of a laminar magneto-hydro-

dynamic boundary layer caused by a magnetic field of induced currents. The value

of Ecm. 0 corresponds to outer heating and the Ecm, 0 represents fluid flow

cooling.

Info: [A23].

Ernst Rudolf Georg Eckert (p. 192).

6.3.7 Einstein Number Ei

Ei5w

cð1Þ; Ei � R5

w2

c25

w

c2γμLð2Þ

w (m s21) � flow velocity; c (m s21) � speed of light; γ (S m21) � specific electri-

cal conductance; μ (H m21) � permeability; L (m) � characteristic length; R (2) �relativistic parameter (p. 327).

This number expresses the ratio of the local flow rate of electrically conductive

fluid to the conversion rate of electromagnetic field energy to heat in Joules. In the

form of equation (2), it is also called the velocity criterion or the relativistic param-

eter R (p. 327).

Info: [A23].

Albert Einstein (14.3.1879�18.4.1955), American physi-

cist of German origin. Nobel Prize in Physics, 1921.

He is the ingenious physicist who brought to physics the

modern vision of physical reality, particularly with his spe-

cial and general relativity theories. He also explained

the concept of the photoelectric phenomenon. He received

the Nobel Prize for his photoelectric phenomenon explana-

tion, but not for the theory of general relativity, which was

his main contribution. He spent the last years of his life try-

ing to create a unified theory which could explain all

known forces in nature.

6.3.8 Electric Field Number RE

RE 5E

wμH

E (V m21) � electric field intensity; w (m s21) � flow velocity; μ (H m21) �permeability; H (A m21) � magnetic field intensity.

It characterizes the mutual action of hydrodynamic and electromagnetic fields in

magneto-hydrodynamics.

Info: [A23],[A33].

317Electromagnetism

6.3.9 Electric Field Parameter NE

NE5E

wμH5

E

Bw

E (V m21) � electric field intensity; w (m s21) � velocity; μ (H m21) � permeability;

H (A m21) � magnetic field intensity; B (T) � magnetic induction.

It expresses the ratio of the active electrodynamic force to the induced electro-

dynamic force. Magneto-hydrodynamics.

Info: [A23].

6.3.10 Elsasser Number El

El5ReRe21m 5Pr21m 5

ρηγμ

ρ (kg m23) � density; η (Pa s) � dynamic viscosity; γ (S m21) � specific electrical

conductance; μ (H m21) � permeability; Re (2) � Reynolds number (p. 81);

Rem (2) � Reynolds magnetic number (p. 328); Prm (2) � Prandtl magnetic num-

ber (p. 313).

Under magneto-hydrodynamic conditions, it expresses the hydrodynamic-to-

magnetic field propagation ratio. It characterizes the relation between hydrody-

namic and magnetic quantities in the magneto-hydrodynamics field.

Info: [A23],[B20].

Walter Maurice Elsasser (20.3.1904�14.10.1991),

American theoretical physicist of German origin.

He occupied himself with fundamental problems of

atomic physics, geophysics, radiation heat transfer in the

atmosphere, terrestrial magnetism and � in the last years of

his life � the theory of organisms. In 1939, he proposed

that the Earth’s rotation generates turbulent currents in both

its liquid nucleus and in its terrestrial magnetism.

6.3.11 Entropy Generation Number NS

NS 5 SgL2 T0

λ ðΔTÞ2

Sg (W m23 K21) � entropy generation rate; λ (W m21 K21) � thermal conductiv-

ity; ΔT (K) � temperature difference; L (m) � characteristic length (width) of

channel; T0 (K) � initial temperature.

This number expresses the ratio of the generated entropy rate change to the

characteristic entropy transfer. In magneto-hydrodynamics, it is used to express the


entropy changes when acted upon by mixed convection and radiation in vertical

porous canals. Magneto-hydrodynamics.

Info: [B70].

6.3.12 Euler Magnetic Number Eum

Eum 5μH2

ρw25

B2

μρw2

μ (H m21) � permeability; H (A m21) � magnetic field intensity; ρ (kg m23) �density; w (m s21) � velocity; B (T) � magnetic induction.

In a fluid moving in a magnetic field, it expresses the magnetic-to-dynamic pres-

sures ratio. It characterizes the voltage acting along the magnetic lines of force in

an electrically conductive fluid which moves in a magnetic field. Essentially, it is

the degree of the magnetic-to-kinetic energies ratio for a unit fluid volume.

Sometimes, it is also called the magnetic pressure N (p. 325).

Info: [A23].


6.3.13 Fourier Magnetic Relaxation Number Fomr

Fomr 5τrmνmL2

5ReRe21m Ha22 5Eu21

m Re22m 5 St21Re21

m ;

where

τrm 5ρ

γμ2H2; νm 5

1

γμ

τrm (s) � magnetic relaxation time; νm (m2 s21) � magnetic viscosity; L (m) �characteristic length; ρ (kg m23) � density; γ (S m21) � specific electrical

conductance; μ (H m21) � permeability; H (A m21) � magnetic field intensity;

Re (2) � Reynolds number (p. 81); Rem (2) � Reynolds magnetic number

(p. 328); Ha (2) � Hartmann number (p. 320); Eum (2) � Euler magnetic number

(p. 319); St (2) � Stuart number (1.) (magnetic force) (p. 329).

In an electrically conducting fluid, moving at the characteristic velocity

we (γμL)21, it expresses the ratio of lines of force relaxation (attenuation) time to

the lines of force passage time through a magnetic field and the conversion of its

energy to electric heat.

Info: [A23].


319Electromagnetism

6.3.14 Hall Coefficient CH

CH 5λrL

λ (m) � mean free path; rL (m) � Larmor radius. This coefficient expresses the

ratio of the gyro-frequency to the frequency of collisions. Magneto-hydrodynamics.

Info: [A29].

Edwin Herbert Hall (see below).

6.3.15 Hall Parameter, Hall Coefficient H, β

H5ωτ5eBτm

5KnLr21

ω (s21) � resonant frequency of electron; τ (s) � mean time of electron relaxation;

e (C) � electron charge; B (T) � magnetic induction; m (kg) � electron mass;

Kn (2) � Knudsen number (1.) (p. 69); Lr (2) � Larmor number (p. 322).

It represents the ratio of the gyro-frequency to the frequency of collisions. In the

magneto-hydrodynamics, it expresses the ratio of the mean time between collisions

to the time of their circulation around a magnetic flux line.

Info: [B20].

Edwin Herbert Hall (7.11.1855�20.11.1938), American

physicist.

In the year 1879, he discovered the Hall effect, the sub-

stance of which is the transversal electric voltage in a cur-

rent loaded conductor or in a semiconductor located in an

outer magnetic field. In two-dimensional electron systems,

the quantum Hall effect appears in strong magnetic fields

and under temperatures close to those of liquid helium.

Together with superconductivity, this effect is one of the

most significant manifestations of quantum theory.

6.3.16 Hartmann Number Ha

Ha5μHLffiffiffiγη

r5BL

ffiffiffiγη

r5

ffiffiffiffiffiffiffiffiffiStRe

p5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiReEumRem

p

μ (H m21) � permeability; H (A m21) � magnetic field intensity; L (m) � charac-

teristic length; γ (S m21) � specific electrical conductance; η (Pa s) � dynamic vis-

cosity; B (T) � magnetic induction; St (2) � Stuart number (1.) (magnetic force)

(p. 329); Re (2) � Reynolds number (p. 81); Eum (2) � Euler magnetic number

(p. 319); Rem (2) � Reynolds magnetic number (p. 328).


It is an important criterion of magneto-hydrodynamics. It expresses the ratio of

the induced electrodynamic (magnetic) force to the hydrodynamic force of the vis-

cosity or, alternatively, the ratio of the ponderomotive force (the electromagnetic

volume force by means of which the magnetic field acts on a conductor through

which electric current flows, which causes magnetic pressure) to the molecular fric-

tion force. It characterizes the magnetic field influence on the flow of viscous, elec-

trically conducting fluid. With small Ha values, the motion proceeds as if no

magnetic field were acting. With great Ha values, the viscosity forces act only on a

thin layer of the electrically conducting fluid (ionized gas) which adheres closely

to a by-passed wall surface. In other cases, the motion resistance does depend on

the viscosity and is determined completely by electromagnetic volume forces which

are acting on the fluid. With high velocities and turbulent flow, it is more

suitable to use the Stuart number (2.) St (p. 330), expressing the mutual magneto-

hydrodynamic action, instead of the Ha number.

Info: [A23],[A43],[B20].

Julius Frederik Georg Poul Hartmann (1881�1951), Danish physicist.

6.3.17 Hydromagnetic Velocity Number Nhm

Nhm 5w

wmg

w (m s21) � fluid flow velocity; wmg (m s21) � velocity of magnetic field

propagation.

It expresses the ratio of the fluid flow rate to the magnetic field propagation

rate. It corresponds to the Reynolds magnetic number Rem (p. 328). Magneto-

hydrodynamics.


Ka5w

wA

5Al21 5w

ffiffiffiffiffiffiμρ

pB

; where wA 5H

ffiffiffiμρ

r

w (m s21) � local fluid velocity; wA (m s21) � velocity propagation of Alfven’s

magnetic wave; μ (H m21) � permeability; ρ (kg m23) � fluid density; B (T) �magnetic induction; H (A m21) � magnetic field intensity; Al (2) � Alfven num-

ber (p. 314).

It characterizes the influence of the mutual action of hydrodynamic and mag-

netic fields in magneto-hydrodynamics. It expresses the ratio of the fluid flow rate

to that of the Alfven waves. It expresses also the kinetic-to-magnetic energies ratio

in a unit volume. Magneto-hydrodynamics.

Info: [A35].


321Electromagnetism

6.3.19 Larmor Number Lr

Lr5rL

L

rL (m) � Larmor radius; L (m) � characteristic length.

In the magnetic field, it expresses the ratio of the Larmor rotation radius of a

charged mass particle to a characteristic longitudinal dimension. It characterizes

the continuity of the electrically conductive flow. With the Lr negligibly small, the

fluid or plasma flow can be described as a continuous environment exposed to

magnetic field influence. For a large Lr value and the Knudsen number (1.) (p. 69)

Knc1, the fluid flow must be described as the flow of mutually acting discrete

charged particles, exposed to magnetic field influence. Magneto-hydrodynamics.

Info: [A23].

Joseph Larmor (11.7.1857�19.5.1942), Irish physicist.

He was engaged in researching electricity, dynamics and

thermodynamics. He suggested the physical theory of light,

according to which ether can represent a homogenous fluid

which would be both perfectly incompressible and perfectly

elastic. He developed a theory according to which the spec-

tra could be explained by oscillations of electrons. He

determined the energy released with radiation of an acceler-

ated charge and explained the spectral lines splitting in a

magnetic field. The Larmor radius represents that of the

charged particle movement in a magnetic field.

6.3.20 Lorentz Factor, Lorentz Term γ, Lof

γ5dτmdτ

51ffiffiffiffiffiffiffiffiffiffiffiffiffi

12 β2p ð1Þ; γ5

cffiffiffiffiffiffiffiffiffiffiffiffiffiffic2 2 u2

p ð2Þ;

where

β5u

c

τm (s) � measured time; τ (s) � supposed time; u (m s21) � velocity at reference

point where time τm is measured; c (m s21) � speed of light.

It is used in the special relativity theory to simplify the recording of equations

and, especially, to express the related time dilation, length contraction and rest-mass

conversion to the relativistic mass. Physics. Electromagnetism. Electrodynamics.

Magneto-hydrodynamics.

Info: [C79].


Hendrik Antoon Lorentz (18.7.1853�4.2.1928), Dutch


He was engaged in optics, electrodynamics and atomis-

tics. He contributed significantly to the general relativity

theory, elaborated by Einstein, specifically by experimen-

tally verifying the special relativity theory (1919). He is

known for his work on the electric and magnetic field influ-

ence on electromagnetic radiation. He laid the foundation

for the electron theory of the electric conductivity of sub-

stances. He received the Nobel Prize for his work on elec-

tromagnetic radiation.

6.3.21 Lundquist Number Ld

Ld5γHμ

32L

ρ12

5HaRe12mRe

212

γ (S m21) � specific electrical conductance; H (A m21) � magnetic field intensity;

μ (H m21) � permeability; L (m) � characteristic length (thickness of conductive

fluid); ρ (kg m23) � density; Ha (2) � Hartmann number (p. 320); Rem (2) �Reynolds magnetic number (p. 328); Re (2) � Reynolds number (p. 81).

This number characterizes the size of the attached magnetic field influence on

the distribution of the induced magnetic field in the flow of an electrically conduc-

tive fluid. It has been used in magneto-hydrodynamics to express unidirectional

Alfven waves or, alternatively, the waves caused by the magnetic field in the flow

of the electrically conductive fluid.

Info: [A23].

6.3.22 Lykoudis Number Ly

Ly5γμ2H2L

12

ðgβΔTÞ12ρ5Ha2Gr2

12

γ (S m21) � specific electrical conductance; μ (H m21) � permeability; H (A m21) �magnetic field intensity; L (m) � characteristic length; g (m s22) � gravitational

acceleration; β (K21) � volume thermal expansion coefficient; T (K) � temperature;

ρ (kg m23) � density; Ha (2) � Hartmann number (p. 320); Gr (2) � Grashof heat

number (p. 185).

It expresses the ratio of the ponderomotive force to the product of buoyancy and

inertia forces. It characterizes the magnetic field influence on the convection of an

electrically conductive fluid in magneto-hydrodynamics.

Info: [A23].

Paul S. Lykoudis (born 1926), American engineer of Greek origin.

323Electromagnetism

6.3.23 Mach Electromagnetic Number Mae

Mae 5E

Bw5ReEL1Rem 5

we

w

E (V m21) � electric field intensity; B (T) � magnetic induction; w (m s21) � fluid

flow velocity; we (m s21) � drift velocity; ReEL1 (2) � Reynolds electrical

number (1.) (p. 303); Rem (2) � Reynolds magnetic number (p. 328).

It expresses the drift-to-flow rates ratio. Magneto-hydrodynamics.

Info: [A24].

Ernst Mach (p. 73).

6.3.24 Mach Magnetic Number Mm

Mm 5Al21 5w

wA

5w

ffiffiffiffiffiffiρμ

pB

ð1Þ; Mm 5w

ða2 1w2AÞ

12

ð2Þ;

Mm 5w

a1wA

ð3Þ

w (m s21) � velocity; wA (m s21) � Alfven wave velocity; ρ (kg m23) � density;

μ (H m21) � permeability; B (T) � magnetic induction; a (m s21) � sound speed;

Al (2) � Alfven number (p. 314).

This number expresses the ratio of the local flow rate of an electrically conduc-

tive fluid to the propagation rate of the Alfven (magneto-hydrodynamic) wave.

Like the Alfven number (p. 314) or the Euler magnetic number Eum (p. 319), it

characterizes the flow rate in magneto-hydrodynamics. Sometimes, expression

(2) or (3) is used. It is also called the Alfven�Mach number.

Info: [A23].

Ernst Mach (p. 73).

6.3.25 Magnetic Fluidization Number Nmgf

Nmgf 5μsH

2x

ρsu2f

μs (H m21) � powder permeability; H (A m21) � magnetic field intensity; x (2) �volume fraction of magnetic powder; ρs (kg m23) � mean density of fluidized bed;

uf (m s21) � surface fluid velocity.

It characterizes the magnetic fluidization of powders in a flowing fluid. It

expresses the magnetic-to-kinetic energy ratio for particles. It is a modification of

the Euler magnetic number Eum (p. 319). Magneto-hydrodynamics.

Info: [B129].


6.3.26 Magnetic Force Parameter Nmg,F

Nmg;F 5μHγLρw

μ (H m21) � permeability; H (A m21) � magnetic field intensity; γ (S m21) �specific electrical conductance; L (m) � characteristic length; ρ (kg m23) � density

of environment; w (m s21) � flow velocity.

It expresses the magnetic-to-dynamic forces ratio. It is analogous to the mag-

netic number Nmg (p. 325). Magneto-hydrodynamics.

Info: [A29],[B17].

6.3.27 Magnetic Interaction Nmg

Nmg 5μ0H

2r

2σ

μ0 (H m21) � vacuum permeability; H (A m21) � magnetic field intensity; r (m) �radius; σ (kg s21) � liquid surface tension.

It relates to ferrous fluids dynamics. Magneto-hydrodynamics.

Info: [A24],[B20].

6.3.28 Magnetic Number Nmg

Nmg 5B

ffiffiffiffiffiffiγLρw

s5N

12mgF

B (T) � magnetic induction; γ (S m21) � specific electrical conductance; L (m) �characteristic length; ρ (kg m23) � density; w (m s21) � fluid flow velocity;

NmgF (2) � magnetic force parameter (p. 311).

It expresses the magnetic-to-inertia forces ratio. It is analogous to the magnetic

force parameter NmgF (p. 311). Magneto-hydrodynamics.

Info: [C84].

6.3.29 Magnetic Pressure N, S

N5p

pm5 nkT

2μB2

ð1Þ; S5μH2

ρw2ð2Þ

p (Pa) � gas pressure; pm (Pa) � magnetic pressure; n (m23) � numerical

density of particles; k (J K21) � Boltzmann constant; T (K) � temperature;

325Electromagnetism

μ (H m21) � permeability; B (T) � magnetic induction; H (A m21) � magnetic

field intensity; ρ (kg m23) � density; w (m s21) � velocity.

It expresses the ratio of the magnetic pressure to that of the gas. It characterizes

the pressure in magneto-hydrodynamics. pm5B2/2μ holds for magnetic pressure

and p5 nkT for gas pressure. It is also called the Euler magnetic number Eum(p. 319).

Info: [A23],[B20].

6.3.30 Magnetic�Dynamic Number Nmgd

Nmgd 5γwB2L

ρw2

γ (S m21) � specific electrical conductivity; w (m s21) � fluid flow velocity;

B (T) � magnetic induction; L (m) � characteristic length; ρ (kg m23) � fluid

density.

It expresses the magnetic-to-dynamic pressures ratio of the fluid. Magneto-

hydrodynamics.

Info: [A24],[B20].

6.3.31 Magnetization Parameter Ω

Ω5ωef

νef

ωef (s21) � angular frequency; νef (s

21) � effective electron collision.

It represents the local electron magnetization parameter. Magneto-hydrodynamics.

Flow in accelerator canals.

6.3.32 Morozov-Hall Parameter Mor

Mor5ΩRem

Ω (2) � magnetization parameter (p. 329); Rem (2) � Reynolds magnetic number

(p. 328).

It characterizes the magneto-hydrodynamic processes for flow in accelerator

canals. Magneto-hydrodynamics.

Info: [C96].

V.A. Morozov, Russian physicist.

Edwin Herbert Hall (p. 320).


6.3.33 Naze Number Na

Na5wA

a5MAl

wA (m s21) � Alfven’s wave velocity; a (m s21) � sound speed; M (2) � Mach

number (p. 73); Al (2) � Alfven number (p. 314).

In an electrically conductive fluid, it expresses the ratio of the propagation rate

of magneto-hydrodynamic waves (the Alfven waves) to the sound propagation rate

in the fluid. It characterizes the velocity relations in electro-magneto-

hydrodynamics.

Info: [A23].

Jacqueline Naze (born 1935), French mathematician.

6.3.34 Oseen Magnetic Number Os

Os51

2ð12Al2ÞNmg

Al (2) � Alfven number (p. 314); Nmg (2) � magnetic number (p. 325), or the

Stuart number (1.) (magnetic force) (p. 329).

It expresses the magnetic-to-inertia forces ratio. It characterizes the magneto-

hydrodynamic relations in electrically conductive fluid flow in the magnetic field.

Info: [A23].

6.3.35 Porous Media Inertia Coefficient Γ, Npmi

Γ 5Ckε2Lffiffiffi

ζp

Ck (2) � Forchheimer coefficient; ε (2) � porosity (p. 24); ζ (m2) � permeability

of porous material; L (m) � characteristic length (width of channel).

It characterizes the porous environment behaviour with convective and mixed

convective radiation flow and interaction in vertical porous canals. It occurs, for

example, in magneto-hydrodynamics and is accompanied by entropic changes.

Two-phase flow. Magneto-hydrodynamics.

Info: [B70].

6.3.36 Relativistic Parameter R

See the Einstein number Ei defined by relation (2) (p. 317).

327Electromagnetism

6.3.37 Reynolds Magnetic Number Rem

Rem 5RePrm 5wL

νm5 γμwL5

w

wm

ð1Þ;

Rem 5γμELB

5Re�mM21m ð2Þ

w (m s21) � velocity; L (m) � characteristic length; νm5 (μmσ)21 (m2 s21) �

magnetic viscosity or magnetic diffusivity; γ (S m21) � specific electrical conduc-

tance; μ (H m21) � permeability; wm5 (γμL)21 (m s21) � characteristic velocity;

E (V m21) � electric field intensity; B (T) � magnetic induction; Re (2) � Reynolds

number (p. 81); Prm (2) � Prandtl magnetic number (p. 313); Re�m(2) � Reynolds

magnetic number defined by the equation (1); Mm (2) � Mach magnetic number

(p. 324).

This number expresses the ratio of a movable induced magnetic field to an intro-

duced outer magnetic field. It characterizes the influence of the electrically conduc-

tive fluid flow rate on the magnetic field distribution in the fluid. Sometimes, it is

also called the velocity number Rw.

It can be expressed as the ratio of the fluid flow rate to the characteristic velocity

wm5 (γμL)21 of a magnetic field moving over a conductor, or as the ratio of the

flow dimension L to the characteristic length of a magnetic field Lm5 (γμwm)21

which moves through a conductor. It is the degree of outer magnetic field excita-

tion. For Rem5 0, the fluid flow does not influence the magnetic flux lines of the

field in which the fluid is moving. The magnetic field of induced currents does not

originate in the fluid. For Rem{1, or w{wm, L{Lm alternatively, the magnetic

field influence can be ignored and it can be assumed that the flow is due to outer

magnetic field influence only, for example, without fluid motion influence on the

magnetic field. The plasma moving in an electromagnetic field or liquid metal in a

pipeline, in canals of magnetic pumps, or in MHD generators, are examples. For

Rem�1, the induced magnetic field can influence the outer magnetic field action

considerably. It causes some entraining of a resulting magnetic field by the fluid,

this entraining growing greater with the increasing Rem. For Remc1, the fluid flow

becomes a good electric conductor and acts on a magnetic field strongly. It is as if

the magnetic flux lines are frozen in the fluid flow and entrained by it. These ques-

tions are the subject of magneto-hydrodynamics research in astrophysics.

Info: [A23],[A29],[B20].


6.3.38 Roberts Number Rob

Rob5a

νm5RemPe

21

a (m2 s21) � thermal diffusivity; νm (m2 s21) � magnetic viscosity; Rem (2) �Reynolds magnetic number (p. 328); Pe (2) � Peclet heat number (p. 180).


It characterizes the magnetic convection of an electrically conducting fluid in

fast rotating systems under action of a toroidal magnetic field. It expresses the ratio

of induced and outer magnetic fields to convective and molecular heat transfers in

the flowing fluid. It is a very important criterion in examination of the convection

around planets. Magneto-hydrodynamics. Astrophysics.

Info: [B125].

6.3.39 Strouhal Electromagnetic Number Shm

Shm 5Eτ

BL5

weT

Lð1Þ; Shm 5

wB

Eð2Þ;

Shm 5τ

γμL2ð3Þ

E (V m21) � electric field intensity; τ (s) � time; B (T) � magnetic induction;

L (m) � characteristic length; we5E/B (m s21) � drift velocity; w (m s21) �velocity; γ (S m21) � specific electrical conductance; μ (H m21) � permeability.

It expresses the ratio of the local inertia force to the convective electromagnetic

force. It characterizes the dynamic relations in non-stationary electromagnetic pro-

cesses. It expresses the degree of the movement non-stationarity in a system.

Info: [A23].

Vincenc Strouhal (p. 87).

6.3.40 Stuart Number (1.) (Magnetic Force) St

St5γμ2H2L

ρw5

B2Lγρw

5Ha2Re21 5EumRem

γ (S m21) � specific electrical conductance; μ (H m21) � permeability; H (A m21) �magnetic field intensity; L (m) � characteristic length; ρ (kg m23) � density;

w (m s21) � velocity; B (T) � magnetic induction; Ha (2) � Hartmann number

(p. 320); Re (2) � Reynolds number (p. 81); Eum (2) � Euler magnetic number

(p. 319); Rem (2) � Reynolds magnetic number (p. 328).

This number expresses the ratio of the ponderomotive force caused by induced

currents to the fluid inertia force. It characterizes the electrically conductive fluid

flow in a magnetic field. This criterion can also be held as the ratio of the persis-

tence time of the fluid in a magnetic field to the relaxation time of magnetic field

flux lines. Sometimes, it is called the magnetic parameter or the parameter of mag-

neto-hydrodynamic mutual interaction (interaction parameter). For St. 1 with

Rem{1, the magnetic forces strongly influence the flow. Magneto-hydrodynamics.

Info: [A23],[A33].

John Trevor Stuart (born 1929), English mathematician.

329Electromagnetism

6.3.41 Stuart Number (2.) (Charge Volume Density) St

St5εE2

ρw25

ρeELρw2

ε (F m21) � permittivity; E (V m21) � electric field intensity; ρ (kg m23) � fluid

density; w (m s21) � flow velocity; ρe (C m23) � volume density of an electric

charge; L (m) � characteristic length.

It expresses the electrostatic-to-dynamic pressures ratio for a fluid. It is also

called the electrodynamic number Nmgd. It characterizes the electrodynamic process

in magneto-hydrodynamics.

Info: [A23].

John Trevor Stuart (see above).


7 Physical Technology

There is a Plenty of Room at the BottomRichard Phillips Feynman (1918�1988)

7.1 Micro- and Nanotechnology

In micro- and nanotechnology, the dimensionless quantities differ substantially

from those in macroscopically oriented physics and mechanics. Here, specific phys-

ical problems of these technologies appear as adhesion, tension, deformation, plas-

ticity, peeling and processes of flow in microchannels and nanopores. The

microfluidic phenomena � the flow of micro- and nanoscopic particles, such as

carbon soot, dust and tissue cells � are connected with these technologies. The

Knudsen, Mason, microfluidic, Tabor and Zhang-Zhan numbers are among the

most popular dimensionless quantities in micro- and nanotechnology.

Richard Phillips Feynman (11.5.1918�15.2.1988),

American theoretical physicist. Nobel Prize in Physics, 1965.

The main area of his research was in the field of quan-

tum physics, in which he participated in the origin of quan-

tum electrodynamics. He contributed to defining the

mechanisms of interactions between elementary particles

and elaborated on the technique to describe them, namely

the eponymous Feynman diagrams. He predicted the exis-

tence of the internal structures of the proton and the neu-

tron. His works contributed significantly to describing the

microworld we now know. He played an important role in

creating a new sphere: nanotechnology.

7.1.1 Adhesion Parameter Nadh, Θ

Nadh 5Eh3=2

r1=2a Δσ

5Eh

Wadh

f 1=2r ; where Δσ5σ1 1σ2 2 σ1;2

E (Pa) � modulus of elasticity; h (m) � standard deviation of the peak heights of

roughness; ra (m) � radius of curvature of an asperity; Δσ (N m21) � Dupre



http://dx.doi.org/10.1016/B978-0-12-416013-2.00007-5

adhesion, adhesion surface tension; σ1, σ2, σ1,2 (N m21) � surface tension between

two spheres; Wadh (J) � adhesion work; fr (2) � roughness factor.

It expresses the ratio of the elastic energy to the adhesion work, provided there

is a contact. For Nadhc1, there is only a partial contact, where the elastic material

has contact only on peaks of the highest roughness. Full contact occurs with

Nadh{1 only. The parameter is based on the Gauss transversal roughness distribu-

tion on an elastic material surface. Micro- and nanotribology.

Info: [B128].

7.1.2 Benard�Marangoni Instability Number Mrcrit

Mrcrit 5σ0ΔTh

ηa5 80; where σ0 5

dσdT

σ0 (N m21 K21) � rate of change of surface tension with temperature; ΔAT (K) �temperature difference between lower level and free liquid level; h (m) � depth of

fluid layer; η (Pa s) � dynamic viscosity; a (m2 s21) � thermal diffusivity;

σ (N m21) � surface tension; T (K) � temperature.

This number is expressed by the value of the critical Marangoni number Mr

(p. 186), with which instability occurs in a fluid surface layer. This instability man-

ifests itself by loss of contact with the bottom wall, of which the temperature must

be higher than the air temperature above the fluid layer surface. The

Benard�Marangoni set of hexagonal cells builds a system in which the fluid flows

upwards through the centre of the cells, and downwards near the walls.

Benard�Marangoni instability arises from the temperature nonuniformity in the

fluid and is a certain modification of the classic Rayleigh�Benard instability num-

ber, which dates from the beginning of the 20th century. It relates to hydrothermal

flow and surface waves under the action of an intensive heat source (i.e. laser or

electron beam) on the molten material surface. Thermocapillary phenomena.

Microgravitational applications. Microtechnology.

Info: [C7].

Carlo Guiseppe Matteo Marangoni (1840�1925), Italian physicist.

7.1.3 Darcy Number (3.) Porous Da

Da5h2

ξ

h (m) � thickness of porous layer; ξ (m2) � permeability of porous material.

It characterizes the permeability in porous material and in microchannels. It is

analogous to the Darcy granulation number (2.) Dc (p. 98). Two-phase flow.

Micro- and nanotechnologies. Thin layers.

Info: [B95].



7.1.4 Deborah Number (2.) De

De5 τrxγ52τrxwmed

kð1Þ;

De5ReNelast ð2Þ

τrx (s) � fluid relaxation time; γ (s21) � shear rate; wmed (m s21) � mean fluid

flow velocity; k (m) � contraction width of the channel; Re (2) � Reynolds num-

ber (p. 81); Nelast (2) � elasticity number (4.) (p. 333).

It characterizes the low viscosity elastic fluid flow in industrially made micro-

channels with various degrees of narrowing when the fluid contracts and expands.

It is a modification of Deborah number (1.) De (p. 120). It relates to rheologic fluid

flow in technological microchannels with diverse geometric contraction.

Info: [B97].

7.1.5 Dispersion Number Ndis

Ndis 52kd

σwR2h

kd (N) � dispersion coefficient; σw (N m21) � fluid surface tension; Rh (m) �hydraulic diameter.

It represents the quantity of the dispersion force in the thin surface layer of a

microchannel. Two-phase flow in microchannels. Micro- and nanotechnology. Thin

layers.

Info: [B95].

7.1.6 Elasticity Number (4.) Nelast , El

Nelast 5τrxηRbDh

5DeRe21; where Dh 52bh

b1 h

τrx (s) � fluid relaxation time; η (Pa s) � dynamic viscosity; R (kg m23) � fluid

density; b (m) � width of contraction; Dh (m) � hydraulic diameter; h (m) �height of channel; De (2) � Deborah number (1.) (p. 120); Re (2) � Reynolds

number (p. 81).

It characterizes the non-Newtonian flow of the low viscosity elastic fluid

flow in industrially made microchannels in which the concentration and expansion

processes of the fluid flow occur. It is a modification of elasticity number (1.) El

(p. 121). It relates to the flow of polyethylene and similar rheologic fluids through

technological channels with diverse geometric contraction.

Info: [B97].

333Physical Technology

7.1.7 Evaporation Momentum Force (1.) Nv

Nv 5q

qmlv

� �2 RlRv

q (W m22) � heat flux; qm (kg m22 s21) � mass flux density; lv (J kg21) � latent

heat of vapourization; Rl, Rv (kg m23) � density of liquid and vapour, respectively.

It expresses the ratio of the momentum force, during fluid evaporation, to the

inertia force in microchannels. Primarily, it involves the boiling number Boi

(p. 215) and fluid to vapour densities ratio. Boiling. Heat transfer in microchannels.

Microtechnology.

7.1.8 Evaporation Momentum Force (2.) Nv

Nv 5q

lv

� �2L

Rvσ

q (W m22) � heat flux; lv (J kg21) � latent heat of vapourization; L (m) � charac-

teristic length (bubble diameter); Rv (kg m23) � vapour density; σ (N m21) �surface tension.

It expresses the ratio of the momentum force, in fluid evaporation, to the surface

strain force with fluid passage and evaporation in microchannels. Heat transfer.

Boiling. Microtechnology.

7.1.9 Force Flow Boiling Number in Microchannels (1.) NF1

NF1 5qA

qmlv

� �2 RlRv

qA (W m22) � surface heat flux; qm (kg m22 s21) � mass flux density; lv (J kg21) �

specific latent heat of vapourization; Rl, Rv (kg m23) � density of liquid and vapour.

It expresses the ratio of the evaporating momentum force to the inertia force in

fluid flow and evaporation in microchannels. For these relations, low values of the

hydraulic diameter and Reynolds numbers ReAh(100; 1000i (p. 81) are characteris-

tic. Two-phase flow. Boiling. Nonequilibrium processes. Micro- and

nanotechnologies.

Info: [B57].


7.1.10 Force Flow Boiling Number in Microchannels (2.) NF2

NF2 5qA

lv

� �2L

Rvσ

qA (W m22) � surface heat flux; lv (J kg21) � specific latent heat of vapourization;

L (m) � characteristic length, bubble diameter; Rv (kg m23) � vapour density;


This number expresses the ratio of the evaporating momentum force to the sur-

face strain force in fluid flow and evaporation in microchannels. The expression

does not involve the influence of the contact angle. Two-phase flow. Boiling.

Nonequilibrium processes. Micro- and nanotechnologies.

Info: [B57].

7.1.11 Heat Removal Nrem

Nrem 5λlðTsat 2 TwÞ

Rlνlllv

λl (W m21 K21) � fluid thermal conductivity; Tsat (K) � saturation temperature;

Tw (K) � wall temperature; Rl (kg m23) � liquid density; νl (m2 s21) � kinematic

viscosity; llv (J kg21) � latent heat of evaporation.

It expresses the ratio of the heat transfer rate, in terms of the channel length unit,

to the transfer of fluid properties. Two-phase flow. Micro- and nanotechnology.

Info: [B95].

7.1.12 Kandlikar Number (1.) Kan

Kan5q

Gr

� �2 RlRg

q (W m22) � heat flux; G (kg m22 s21) � mass flux density; r (J kg21) � specific

heat of evaporation; Rl, Rg (kg m23) � fluid and vapour density.

This number expresses the ratio of the evaporation momentum force to the iner-

tia force. Comparing to the boiling number Boi (p. 215) � which does not, itself,

represent the real evaporation momentum effect � the Kandlikar number involves

the ratio of fluid and vapour densities, too. Among other things, the number Kan

describes the heat transfer mechanism of a boiling fluid flowing through micro-

channels. Two-phase flow.

Info: [B56].


7.1.13 Kandlikar Number (2.) Kan

Kan5q

r

� �2 L

Rgσ

q (W m22) � heat flux; r (J kg21) � specific heat of evaporation; L�D (m) �characteristic number, hydraulic diameter D; Rg (kg m23) � vapour density;


It expresses the ratio of the evaporation momentum force to the surface stress

force. It describes the heat transfer mechanism of a boiling fluid flowing through

microchannels. Two-phase flow.

Info: [B56].

7.1.14 Knudsen Micro- and Nanometric Number Kn

Kn5

ffiffiffiffiffiπk2

rM

Re5

ffiffiffiffiffiffiffiffiffiπ

2RT

rηRL

R (J mol21 K21) � molar gas constant; T (K) � absolute fluid temperature; η (Pa s) �dynamic viscosity; R (kg m23) � fluid density; L (m) � characteristic length

(microchannels and nanopores dimension); K (2) � specific heat ratio (p. 28);

M (2) � Mach number (p. 73); Re (2) � Reynolds number (p. 81).

It characterizes the flow in microchannels and nanopores. For KnAh0; 1023i, itdescribes continuous flow; for KnAh1023; 1021i it describes shear flow; for

KnAh1021; 101i it describes transient flow; and for KnAh101; Ni it describes freemolecular flow. Micro- and nanomechanics.

Info: [B99].


7.1.15 Mason Dielectrophoretic Number Masdf

Masdf 5Mf1

Mfdf15

8ηwε0εfβ2dE2

η (Pa s) � fluid dynamic viscosity; w (m s21) � characteristic velocity; ε0,εf (F m21) � vacuum and fluid permittivity, respectively (ε05 8.8542310212 F m21); β (2) � Clausius�Mossotti factor; d (m) � particle diameter;

E (V m21) � electric field intensity; Mf1 (2) � microfluidic number (p. 338);

Mfdf1 (2) � microfluidic dielectrophoretic number (1.) (p. 337).

It expresses the ratio of the viscosity force to the electrostatic interaction force

between particles. It characterizes the microfluidic phenomena in the flow of

micro- and nanoscopic particles (i.e. carbon black, glass dust or tissue cells).

Info: [B55].


7.1.16 Mass Flux Number Nqm

Nqm 522 c

2c

λv

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRðTw 2 TlÞ3

qσwllv

ffiffiffi2

pπ

c (2) � constant, usually c5 1; λv (W m21 K21) � vapour thermal conductivity;

R (J kg21 K21) � gas constant; Tw (K) � wall temperature; Tl (K) � fluid tempera-

ture; σw (N m21) � fluid surface tension at the wall; llv (J kg21) � latent heat of

vapourization.

It characterizes the mass flow on the interface between phases. It is implied

from the kinetic theory of fluid flow in thin layers and microchannels under phase

conversion conditions.

Info: [B95].

7.1.17 Microchannel Flow Boiling Number Boim, Co

Boim 5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiσgðRl 2 RgÞD2

h

r

σ (N m21) � surface tension; g (m s22) � gravitational acceleration; Rl, Rg(kg m23) � density of liquid, gas or vapour; Dh (m) � hydraulic diameter.

When boiling fluid passes through microchannels, it expresses the ratio of the

surface strain force to the buoyancy force acting on a vapour bubble.

Thermomechanics. Boiling. Micro- and nanotechnologies.

Info: [B56].

7.1.18 Microfluidic Dielectrophoretic Number (1.) Mfdf1

Mfdf1 53πε0β2d3E2

4mw2

ε0 (F m21) � vacuum permittivity (ε05 8.85423 10212 F m21); β (2) � positive

or negative direction of dielectrophoretic force; d (m) � particle diameter;

E (V m21) � electric field intensity; m (kg) � particle mass; w (m s21) � charac-

teristic velocity of particle.

It expresses the ratio of the electrostatic interaction force, between particles, to

the inertia force. It relates to microfluidic phenomena in the flow of micro- and

nanoscopic particles in an electrical field.

Info: [B55].



Mfdf2 54πε0εfβd4E2

mw2L; where β5

εp 2 εfεp 1 2εf

ε0 (F m21) � vacuum permittivity (ε05 8.85423 10212 F m21); εf (F m21) � fluid

permittivity; β (2) � Clausius�Mossotti factor; d (m) � particle diameter;

E (V m21) � electric field intensity; m (kg) � particle mass; w (m s21) � particle

flow velocity; L (m) � characteristic length (electrodes distance).

It expresses the dielectrophoresis and inertia forces ratio in the flow of particles

in an electric field. Microfluidic phenomena.

Info: [B55].


Mfdf3 5Mfdf1

Mfdf25

3βL16d

β (2) � Clausius�Mossotti factor; L (m) � characteristic length (electrodes dis-

tance); d (m) � particle diameter; Mfdf1 (2) � microfluidic dielectrophoretic num-

ber (1.) (p. 337); Mfdf2 (2) � microfluidic dielectrophoretic number (2.) (p. 338).

When the particles flow in an electric field, it characterizes the force relations.

It always applies in cases of important influence of mutual interaction of particles

and the dielectrophoresis force. Microfluidic phenomena.

Info: [B55].

7.1.21 Microfluidic Number Mf1

Mf1 56πηd2

mw

η (Pa s) � dynamic viscosity of fluid; d (m) � particle diameter; m (kg) � particle

mass; w (m s21) � characteristic velocity.

It characterizes the behaviour of micro- and nanoscopic particles in a flowing

fluid, for example, that of live cells which are suspended in the fluid. It expresses

the viscous to inertia forces ratio.

Info: [B55].

7.1.22 Mismatch Parameter δ

δ5 100sff 2 sss

sss; where sfs 5

sff 1 sss

2


sff (m) � atom geometric parameter in thin layer; sss (m) � atom geometric param-

eter in substrate; sfs (m) � atom geometric parameter on the boundary line of thin

layer and substrate.

It characterizes the growth of thin layers, with deposition processes, and the

structure originating on the interface between a thin layer and the substrate mate-

rial. It is applied, for example, in the simulation of deposition processes to create

interatomic potentials. Micro- and nanotechnology. Thin layers.

Info: [B84].

7.1.23 Peel Number Np

Np53Es3h2

2L4Wadh

E (Pa) � modulus of elasticity; s (m) � layer thickness; h (m) � distance of

unstuck layer from substrate; L (m) � characteristic length of crack; Wadh (J) �adhesion work.

With the deposited layer deviated from the substrate, it characterizes the rela-

tions of adhesion. It expresses the ratio of the elastic deformation energy, stored in

the deformed microstructure memory, to the adhesion work between the micro-

structure and the substrate. When Np. 1, the elastic deformation stored in the

memory is greater than the work of adhesion and the microstructure will not adhere

to the substrate. When Np# 1, the deformed layer does not have enough energy to

overpower the adhesion.

Info: [B128]

7.1.24 Plastic Adhesion Index Npl.ad.

Npl:ad: 5π2H4raσ8E2

ekvW2a

H (N) � material hardness; ra (m) � curvature radius of the roughness peaks,

in point as a sphere; σ (m) � standard deviation of the roughness height distribu-

tion; Eekv (Pa) � equivalent modulus of elasticity (see Tabor number Tab)

(p. 343); Wa (J) � Dupre adhesion work.

It characterizes the quantitative point of view of the relation between adhesion,

plastic deformation and other corresponding phenomena. Micro- and nanotechnology.

Info: [B100].

7.1.25 Plasticity Index Nplast

Nplast 5Eekv

H

ffiffiffiffiσra

r5

Eekv

H

ffiffiffif

p


Eekv (Pa) � equivalent modulus of elasticity (see Tabor number Tab) (p. 343);

H (N) � material hardness; σ (m) � standard deviation of the roughness height dis-

tribution; ra (m) � curvature radius of the roughness peaks, in point as a sphere;

f (2) � relative surface roughness.

It is the plastic deformation criterion of a microscopic contact. For elastic rough-

ness it is Nplast, 0.6; for plastic roughness it is Nplast. 1; and for elastoplastic

roughness it is 0.6,Nplast, 1. Micro- and nanotechnology.

Info: [B128].

7.1.26 Prediction Number of Adhesion Npred

Npred 5 24

3πEσ2

WadhL

� �2=3

E (Pa) � equivalent modulus of elasticity; σ2 (m2) � quadratic deviation of the

roughness height distribution; Wa (J) � adhesion work; L (m) � characteristic cor-

relation length.

It characterizes the quantitative predictions of adhesion between deformable

fractal surfaces. It expresses the ratio between the standard deviation of the rough-

ness heights and the maximum height of the roughness peaks before transforming.

Micro- and nanotechnologies.

Info: [B128].

7.1.27 Pulse Electromagnetic Heating of Microparticles Nheat , β

Nhead 5dεP2λTref

d (m) � microparticle diameter; P (W m22) � power density of the electromag-

netic field; λ (W m21 K21) � thermal conductivity of microparticles; Tref (K) �reference temperature; ε (2) � emissivity (p. 206).

It characterizes the pulse heating of microscopic particles with a laser, as an

example. Among other things, it is used to determine the thermal, physical proper-

ties of the particles. Physical technology. Microtechnology. Thermomechanics.

Electromagnetism.

7.1.28 Pumping Intensity Npi

Npi 5ðpl 2 pvÞL2

ν2l Rl


pl (Pa) � fluid pressure; pv (Pa) � vapour pressure; L (m) � characteristic pressure

length; νl (m2 s21) � kinematic viscosity of liquid; Rl (kg m23) � liquid density.

It represents the ratio of the maximum power � which can be obtained with the

pressure difference between a fluid and a vapour � to the transfer parameters.

Porous flow through microchannels. Two-phase flow. Micro- and nanotechnology.

Thin layers.

Info: [B95].

7.1.29 Pumping Resistance Npr

Npr 5pRh

4σ

p (Pa) � fluid pressure; Rh (m) � hydraulic diameter; σ (N m21) � fluid surface

tension on the channel wall.

It characterizes the resistance of the two-phase miniature microchannel flow in

porous layers which serve, for example, to cool down the electronic circuits or as

miniature heat exchangers or pumping microcircuits. Two-phase flow. Micro- and

nanotechnology. Thin layers.

Info: [B95].

7.1.30 Response Number Rn

Rn5Rw2

ini

τL

H

� �25Dn

L

H

� �2ð1Þ;

Rn5I2

RτH2

L

H

� �2ð2Þ

R (kg m23) � material density; wini (m s21) � initial velocity of particles; τ (Pa) �shear stress of material; L (m) � characteristic length; H (m) � thickness of beam

or slab; I (Pa s21) � pressure impulse velocity of rectangular shape; Dn (2) �Johnson’s damage number.

This number characterizes the dynamic plastic response in a material exposed to

impulse or stroke pressure loading. It especially relates to diverse beams or plates.

However, for the dynamic load of generally shaped bodies, expression (2) is valid.

Info: [B127].

7.1.31 Reynolds Rheological Number (2.) Rerh

Rerh 5wmedDh

ν05

2Q

ðb1 hÞν0;

where wmed 5Qbh; Dh 5

2bhb1 h


wmed (m s21) � average fluid velocity; Dh (m) � hydraulic diameter; ν0 (m2 s21) �

kinematic viscosity at zero shear speed; Q (m3 s21) � volume flow rate; b (m) �width of contraction; h (m) � height of the channel.

It characterizes the forced flow of a low viscosity elastic fluid through micro-

channels with a range of technological cross section narrowing when the fluid is

contracted and expanded. Rheology. Micro- and nanotechnologies.


7.1.32 Strain Peel Number Npε

Npε 5128Es3h2

8L4Wadh

114σrL

2

21Es21

256

2205

h

s

� �2" #ð1Þ;

Npε 5186Es3h2

ð12 ν2Þb4Wadh

1127ð12 ν2Þσrb

2

310Es21

12

31

h

s

� �2" #ð2Þ;

Npε 540Es3h2

3ð12 ν2ÞR4Wadh

1151ð12 ν2ÞσrR

2

160Es2

� �ð3Þ

E (Pa) � modulus of elasticity; s (m) � layer thickness; h (m) � distance of

unstuck layer from substrate; L (m) � length of double-sided, close-set beam;

Wadh (J) � adhesion work; σr (Pa) � residual stress; ν (2) � Poisson’s ratio

(p. 143); b (m) � width of beam; R (m) � radius of circular plate.

In a dimensionless shape, the number Npε expresses the residual strain in a

beam fixed at both ends (1), in a square plate (2) or in a round plate (3). From these

equations, the maximum dimensions in the microstructure can be determined

(the beam length or fixation, the width or the plate radius), with which the layer

will not be connected with the substrate. It is valid provided the condition Np5 1

holds, with the inner energy of the elastic deformation corresponding to the

adhesion.

Info: [B127].

7.1.33 Subcooling Number (2.) Nsc

Nsc 52ηvλvðTw 2 TlÞRhRvπllvσw

ηv (Pa) � vapour dynamic viscosity; λv (W m21 K21) � vapour thermal conductivity;

Tw (K) � wall temperature; Tl (K) � fluid temperature; Rh (m) � hydraulic diameter;

Rv (kg m23) � vapour density; llv (J kg

21) � latent heat of vapourization; σw (N m21) �fluid surface tension at the wall.


In porous material microchannels, this number expresses the condensation

process and its influence on the heat dissipation with transfer zone reduction and

very high dissipation capability. Heat transfer in microchannels. Condensation and

controlled cooling. Microtechnology.

Info: [B95].

7.1.34 Tabor Number Tab, μ

Tab5RW2

a

E2ekvε2

� �1=3; h5

RW2a

E2ekv

� �1=3

where E21ekv 5

12 ν21

E11

12 ν22

E2; R (m) � equivalent radius of curvature; Wa (J) �

Dupre adhesion work; Eekv (Pa) � equivalent modulus of elasticity; E1, E2 (Pa) �modulus of elasticity of two hemispheres; ν1, ν2 (2) � Poisson’s constant of two

hemispheres; ε (m) intermolecular distance; h (m) � depth of penetration.

The Tabor number is the degree of elastic deformation, which depends on the

surface forces’ size. It characterizes the nonequilibrium adhesion mechanism of a

contact in micromechanical systems. Alternatively, it expresses the ratio of the

impression height to the intermolecular distance. Bodies in contact characterized

by a small number, Tab , 0.1, have slight influence on the elastic deformation

and adhere one to the other more easily. For Tab. 5, it is the opposite. With both

materials and quantities Wadh, Eekv and ε being equal, TabBR1=3 holds. Micro- and

nanotribology.

Info: [B128].

7.1.35 Transport Heat Number Nheat

Nheat 5Rlν2lσL

Rl (kg m23) � liquid density; νl (m

2 s21) � liquid kinematic viscosity; σ (N m21) �surface tension; L (m) � characteristic length of channel.

It characterizes the condition for the maximum heat transfer on the microchan-

nel length unit. Two-phase flow. Micro- and nanotechnologies.

Info: [B95].

7.1.36 Zhang-Zhao Number (1.) Zh1

Zh1 5Wa

σH

Wa (J) � Dupre adhesion work; σ (m) � standard deviation of the roughness height

distribution; H (N) � hardness of material.


It expresses the ratio of the adhesion work to the product of the standard deviation

of the roughness peak heights and the material hardness. The greater this number is,

the more easily the plastic deformation occurs. Micro- and nanotechnologies.

Info: [B90].

7.1.37 Zhang-Zhao Number (2.) Zh2

Zh2 5d

σ

d (m) � difference between mean surface and glare surface during loading; σ (m) �standard deviation of the roughness heights.

It expresses the ratio of the difference between the central surface height and the

smooth surface during loading. It affects the plastic deformation adhesion.

7.2 Plasma Physics and Technology

The dissimilarity of plasma � in the form of ionized gas � from the solid, liquid

or gaseous phases manifests itself in dimensionless quantities as well. The quanti-

ties relate to gas particle behaviour; wavelength; frequency and thermal equilibrium

of the plasma; magnetic field influence on the plasma; and plasma pressure obser-

vation with regard to magnetic pressure, to magnetohydrodynamic load and to other

quantities. Plasma is associated with various technologies utilizing intensive

sources such as the laser, an electric arc in arc welding, etching of dielectric layers

in electronics with integrated circuits manufacture, thermal loading of the front sur-

face of space bodies in descent into the atmosphere, nuclear fusion research and

other areas of physics and technology. The de Broglie and Larmor numbers, Hall’s

parameter and many other numbers are well known dimensionless quantities in

plasma physics. Systematic using of the dimensionless quantities for plasma tech-

nologies has been lacking up to now.

7.2.1 Anisothermal Plasma Number Nta,τ

Nta 5Te

Ti

Te, Ti (K) � kinetic temperature of electrons and ions.

This number expresses a dimensionless parameter characterizing thermal plasma

anisotropy. It appears in plasma density determination and discovering out the ion

collision influence in measurement with the Langmuir cylindrical probe. The


anisothermal plasma number and the Knudsen number for electrons (ions) Kne,i(p. 349) represent two parameters which presume the Maxwell distribution of elec-

tron energy. Plasma physics and technology.

Info: [B88].

Irving Langmuir (31.1.1881�16.8.1957), American chem-

ist and physicist. Nobel Prize in Chemistry, 1932.

He devoted himself to studying light and contributed sig-

nificantly to vacuum engineering improvement, which led

to the idea of a high vacuum tube. He was engaged in

research related to glowing filament behaviour in a vacuum

and in the thermal emission problem. He was one of the

first scientists who worked with plasma, which he called

ionized gas. He introduced the concept of electron tempera-

ture and the diagnostic method to measure vacuum temper-

ature and density with the Langmuir probe. He devoted also

to atmospheric sciences and meteorology.

7.2.2 Attraction Number Na

Na 5A

12πηr2pw

A (J) � Hamaker constant; η (Pa s) � dynamic viscosity; rp (m) � particle radius;

w (m s21) � approximate fluid velocity.

It represents the combined influence of the van der Waals attraction forces and

the fluid rate on the deposition velocity with particle capture. Deposition processes.

Physical and chemical filtration in saturated porous materials.

Info: [B48].

7.2.3 de Broglie Thermal Wavelength Λ

Λ5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffih2

2πmkT

rð1Þ;

V

nΛ3# 1 ð2Þ;

V

nΛ3c1 ð3Þ;

Λ5ch

2kT3ffiffiffiπ

p ð4Þ


h (J s) � Planck constant; m (kg) � mass of gas particles; k (J K21) � Boltzmann

constant; T (K) � absolute temperature; c (m s21) � speed of light; V (m3) � vol-

ume; η (m23) � number of particles in volume V.

This number represents roughly the average of the de Broglie wavelength of ideal

gas particles with specific density. It can be held as a mean distance between the par-

ticles in the gas, approximatelyffiffiffiffiffiffiffiffiffiffiffiVn213

p: When the thermal wavelength is much less

than the distance between the particles, the gas behaves as a classic

Maxwell�Boltzmann gas. However, when the thermal de Broglie wavelength is

greater � by one or more orders � than the distance between the particles, the quan-

tum effects become dominant and the gas must be considered a Fermi gas or a Bose

gas, depending on the substance of gas particles. In this type of situation, expression

(2) is valid, and for the preceding situation, the Maxwell�Boltzmann expression (3)

holds. For a massless particle, the thermal wavelength can be expressed by expres-

sion (4). Plasma physics.

Info: [C133].

Louis-Victor Pierre Raymond, duc de Broglie(15.8.1892�19.3.1987), French mathematician and physi-

cist. Nobel Prize in Physics, 1929.

He studied quantum theory and the wave theory of elec-

trons based on Einstein’s and Planck’s works. This led to

the proposal of the wave particle duality theory. According

to it, a substance has properties both of particles and of

waves. The results of de Broglie’s theory of electromag-

netic waves later served Schrodinger, Dirac and other scien-

tists in the creation of wave mechanics.

7.2.4 de Broglie Wavelength λ

λ5h

pð1Þ;

λ5h

mvð2Þ

h (J s) � Planck constant; p (kg m s21) � momentum of particle; m (kg) � mass of

particle; v (m s21) � speed of particle.

It represents a wavelength as the ratio of the kinetic energy of a particle to its

momentum. In physics, de Broglie’s hypothesis primarily expresses the fact that all

materials have properties of waves. Plasma physics.

Info: [C31].

Louis-Victor Pierre Raymond, duc de Broglie (see previous entry).


7.2.5 de Broglie Wave Number k

k52πph

5p

h

p (kg m s21) � momentum; h (kg m s21) � Planck constant; h (kg m s21) � Planck

constant (h-bar).

This number expresses the ratio of a particle’s momentum to its kinetic energy.

Plasma physics.

Info: [C32].

Louis-Victor Pierre Raymond, duc de Broglie (see previous entry).

7.2.6 Debye Number ND

ND 54

3πneλ2

D ð1Þ;

λD 5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiε0kq22

e

neT 21e 1

Pij j

2nijT21i

sð2Þ;

λD 5

ffiffiffiffiffiffiffiffiffiffiffiε0kTeneq2e

sð3Þ

ne (m23) � electron density; λd (m) � Debye length ε0 (F m21) � vacuum permittiv-

ity; k (J K21) � Boltzmann constant; qe (C) � electron charge; Te, Ti (K) � electron

and ion temperatures, respectively; nij (m23) � density of atom particles i with posi-

tive ion charge jqe.

This number represents the relation between the Debye length and the probe

radius. It expresses the state of the thermal mode and the density during a plasma’s

existence. For ND5 1, the state occurs which is denoted as the ratio of the collec-

tive behaviour to the independent behaviour of particles. When NDc1, it repre-

sents the collective behaviour mode with which the dynamic properties of the

plasma are changing. When ND, 1, it is called the mode of independent particles,

in which the collective phenomenon has slight influence in the plasma. Expressions

(2) and (3) represent the Debye length, which expresses the shielding diameter in

the plasma. Expression (3) is valid only if the ions are much colder than the elec-

trons. The size of λD changes substantially from 10211, for the plasma in the Sun’s

nucleus (ne5 1032), over 1024 for a gas discharge (ne5 1016), to values of 105

(ne5 1) for the intergalactic environment. Plasma physics and technology.

Tokamak. Astrophysics.

Info: [A24], [C33].


Peter Josephus Wilhelmus Debye (24.3.1884�2.11.1966),

Dutch-American electrical engineer, physicist, chemist.

Nobel Prize in Chemistry, 1936.

He formulated the specific heat theory (1911) and

formed the theory of electrolytic solutions. He engaged in

work related to dipole moments of molecules. He had

remarkable knowledge of the arrangement of atoms and

molecules and of the distances between atoms. Together

with P. Scherrer, he developed (1916) the method to deter-

mine the atomic structure of crystals by means of X-rays.

This is called the Debye�Scherrer method.

7.2.7 Hall Parameter, Hall Coefficient H, β

H5ωτ5eBτm

5KnLr1

ω (s21) � resonant frequency of electron; τ (s) � average relaxation time of elec-

tron; e (C) � electron charge; B (T) � magnetic induction; m (kg) � electron

mass; Kn (�) � Knudsen number (1.) (p. 69); Lr (�) � Larmor number (p. 322).

In magnetohydrodynamics, it expresses the ratio of the mean time between colli-

sions of particles to the time of their circulation around a magnetic flux line. It is

an important criterion for examining three plasma acceleration modes in a magnetic

field. There is no dispersion for βc1 , but it occurs for β5 1 and is aperiodic for

β{1 . Plasma physics.

Info: [A29].

Edwin Herbert Hall (p. 320).

7.2.8 Ionization Plasma Number Ni

Ni 5ni

nn� 33 1027

T32

niexp 2

Ui

kT

� �

ni (m23) � number of ionized atoms per unit volume; nn (m

23) � number of neu-

tral atoms per unit volume; T (K) � absolute temperature; Ui (J) � ionizing energy;

k (J K21) � Boltzmann constant.

It expresses the general approximation of the thermal plasma equilibrium. It pre-

dicts which ionization degree can be expected in a gas with thermal equilibrium.

Plasma physics.

Info: [C103].


7.2.9 Knudsen Number for Electrons (Ions) Kne,i

Kne;i 5λe;i

rp

λe,i (m) � electron (ion) mean free path; rp (m) � probe radius.

It is the characteristic number for the theory of collisions, as it describes the

mean free path of electrons or ions in relation to the dimension of the applied

probe. For collisionless conditions, Kne,i - N. In other words, with a probe, the

collisions act on the charged particles more intensely as the probe radius increases,

especially under experimental conditions. Plasma physics and technology.

Info: [B88].


7.2.10 Larmor Plasma Number Lrpl

Lrpl 5rB

rt5

v

rtωB

; where ωB 5 gB

rB (m) � Larmor radius; rt (m) � toroid radius; v (m s21) � particle velocity per-

pendicular to magnetic field; ωB (s21) � Larmor frequency; g (s21 T21) � gyro-

magnetic radius; B (T) � magnetic induction.

It expresses the ratio of the Larmor radius to that of a toroid. Alternatively, it is

the ratio of the particle velocity to the magnetic field intensity. Plasma and atomic

physics.

Info: [B86],[C131],[C104].

Joseph Larmor (p. 322).

7.2.11 Magnetization Parameter δ

δ5RL

ð1Þ

R (m) � Larmor radius, gyroradius; L (m) � characteristic length.

It characterizes the magnetic field influence on a plasma. The plasma process or

system is magnetized if its characteristic length is much greater than the gyrora-

dius. When R{L, the charged particles follow the magnetic field lines of trajecto-

ries directly. In the opposite case, when RcL; the charged particles strike against a

chamber wall before they can be accelerated due to the magnetic field influence.

Plasma physics.

Info: [C131].


7.2.12 Controlled Thermonuclear Fusion number

β52μ0p

B2

μ0 (H m21) � vacuum permeability; p (Pa) � plasma pressure; B (T) � magnetic

induction.

It plays a key role in controlled thermonuclear fusion. It expresses the plasma to

magnetic pressures ratio. To maintain the plasma, the value must be β, 1 and the

magnetic flow must be higher than the plasma flow. Nevertheless, in reality it is

often β# 0.2, which limits, for example, the pressure p# 107 Pa with BB10 T.

Info: [C131].

7.2.13 Plasma Collision Frequency Nfpl ,γ

Nfpl 5Nπd2w

ω5Nπd2

ffiffiffiffiffiffiffiffiffi8πTmπ

r1

ω

N (m23) � number of particles per unit volume m3; d (m) � atom diameter;

w (m s21) � average atom velocity; ω (s21) � characteristic frequency of plasma;

k (J K21) � Boltzmann constant; T (K) � absolute temperature; m (kg) � atom

mass.

This expression represents an estimate � based on the gas kinetic theory � of

the plasma collision frequency. When Nfpl{ω; then Nfpl5 0.01. Plasma physics.

Info: [B86],[C104].

7.2.14 Plasma Parameter Λ

Λ5 4πnλ3D

n (m23) � number of particles per unit volume; λD (m) � Debye length.

It is among the basic plasma parameters. Usually, it expresses the number of

particles contained in a Debye sphere. When Λ{1, the Debye zone is rarely occu-

pied, which corresponds to a weakly coupled plasma. When Λc1; in the Debye

sphere there is a great concentration of particles to which strongly coupled plasma

corresponds. For strongly coupled plasma, lower temperatures and great densities

of particles are typical, whereas weakly coupled plasma is diffusive with low con-

centrations of particles and high temperatures. Plasma with a laser ablation and

plasma with an arc discharge are examples of strongly coupled plasma. Some

examples of typical values of a weakly coupled plasma are Λ5 53 108 for nuclear

fusion and Λ5 43 104 for the interstellar environment. Λ5 23 103 is valid for a

chromosphere and Λ5 33 102 for a glow discharge. Plasma physics.

Info: [C131].


7.2.15 Pressure Plasma Number Nβ , β

Nβ 52μ0p

B2; where p5 nkT

μ0 (H m21) � vacuum permeability; p (Pa) � plasma pressure; B (T) � magnetic

induction; n (m23) � number of particles per unit volume; k (J K21) � Boltzmann

constant; T (K) � absolute temperature.

It expresses the ratio of plasma to magnetic pressures or, alternatively, that of

thermal energy density to magnetic energy density. It characterizes the conditions

under which certain limitations are valid to maintain the plasma for a magnetic

field. The value must be less than unity (Nβ, 1). As a matter of fact, the most

suitable value is NβB0.2. The pressure plasma number plays a key role, for exam-

ple, in controlled thermonuclear fusion. Plasma physics and technology.

Info: [B86],[C131],[C104].

7.2.16 Thrust Coefficient Ct

Ct 54πμ0

FT

I2

μ0 (H m21) � vacuum permeability; FT (N) � thrust force; I (A) � total current

between electrodes.

It characterizes the magnetoplasmadynamic load by the total current.

Magnetodynamics. Electromagnetic plasma accelerators.

Info: [B23].


8 Technology and MechanicalEngineering

The intuition is the only valuable thing indeed.Albert Einstein (1879�1955)

8.1 Technology and Material

In this section, quantities are presented which are applied only in specific areas

of a technology or materials processing. Others are not considered here because

their main application is in fundamental fields, and they are presented, therefore,

in other chapters of this book. In this section, technologies such as heating in

diverse combustion devices, welding, melting, surface treatment and solidification

processes, and other technologies, are discussed. The following numbers are

among the known quantities: the Boltzmann and Craya�Curtet numbers for fire-

places and furnaces, the Christensen, Rosenthal and Rykalin numbers for welding,

the Tikhonov number for thermochemical materials processing, and the Chvorinov

number for solidification time.

8.1.1 Agglomerate Number Ag

Ag5 Rw2

ffiffiffiffiffiffiffiffiffiffiffiD5E2

Γ 5

3

s

R (kg m23) � particle density; w (m s21) � particle velocity; D (m) � particle

diameter; E (Pa) � modulus of elasticity; Γ (J m22) � surface energy.

It expresses the degree of contact damage during agglomeration due to the

impact rate, the adhesion energy, and the properties and shape of particles. It is

based on the assumption that the energy used to erode the material is proportional

to the incidental kinetic energy of the agglomerate. The number is a modification

of the Weber agglomerate number Weagl (p. 363).



http://dx.doi.org/10.1016/B978-0-12-416013-2.00008-7

8.1.2 Boltzmann Number of Combustion Chamber Bo

Bo5k1Qmcp

σk2ST3sp

k1 (2) � heat hold coefficient in combustion chamber, comprising heat losses to

ambient; Qm (kg s21) � fuel mass flux calculated; cp (J kg21 K21) � specific heat

capacity; σ (W m22 K24) � absorptance (absorption capacity, radiating capacity);

k2 (2) � contamination coefficient of wall surface; S (m2) � surface area of

absorption walls; Tsp (K) � theoretical combustion temperature.

It expresses the ratio of the heat transferred to a fireplace surface to the maxi-

mum heat which can be transferred by radiation. It characterizes the heat transfer

in boiler fireplaces. Their inner space is considered as an absolutely black body.

Info: [A23].

Ludwig Boltzmann (p. 205).

8.1.3 Christensen Number Chr

Chr5w2A

a2

w (m s21) � motion velocity; A (m2) � surface; L (m) � characteristic length,

weld width; a (m2 s21) � thermal diffusivity.

It is often called the dimensionless weld width. It expresses the coupling

between the weld dimension and the dynamic thermal process in electric arc weld-

ing. It is valid for both 3D and 2D solutions.

Info: [B37].

N. Christensen.

8.1.4 Chvorinov Cast Mould System Number Ch

Ch5τL2

RclΔTcfRf

Vc

Vf

λf

λc

� �12

5FoPVP12λ

τ (s) � time; L (m) � characteristic length L5Vc

S

� �; Rc, Rf (kg m23) � density of

casting and mould; l (J kg21) � specific latent heat of solidification; ΔT (K) �superheating of melting; cf (J kg21 K21) � specific heat capacity of a mould;

Vc (m3) � casting volume; Vf (m

3) � mould volume; S (m2) � surface area of cast-

ing; λf, λc (W m21 K21) � thermal conductivity of mould and casting, respectively;

Fo (2) � Fourier number (p. 175); PV (2) � volume parameter; Pλ (2) �thermal conductivity parameter.


In the case of a contact task, it expresses the mutual dynamic coupling between

thermomechanical quantities, during the process of casting solidification in a mould

and the cooling time of the casting. Foundry technology.

Nikolai Chvorinov (15.11.1903�7.11.1987), Czech engi-

neer of Russian origin.

He was an engineer, but a talented painter too, enabling

him to introduce abstraction into his foundry work, which

was only empirical before. With extensive experiments

researching the solidification and crystallization of steel

and diversely formed castings, he deduced that solidifica-

tion time is proportional to the second power of the vol-

ume-to-surface ratio. His work formed the basis for

subsequent theoretical research on foundry processes.

8.1.5 Chvorinov Shrink Formation Number Chsh

Chsh 5grad Tffiffiffiffiffiffi

wTp grad Tffiffiffiffiffiffi

wTp

� �21ref

grad T (K m21) � temperature gradient in solidifying casting;WT (K s21) � solidifi-

cation rate of casting; subscript ref � limiting value for shrink hole formation.

This criterion is based on Chvorinov’s idea about the influence of the simulta-

neous action of the geometric and time temperature gradients on a shrink hole in a

solidifying casting. The solution process of the casting solidification dynamics

starts by modeling the temperature field and � from it � the module and argument

fields of the geometric and time temperature gradients. From these, the distribution

field of the criterion for the shrink hole arising in a casting is determined. In the

casting, the zones having the value of Chsh, 1 point to the place of possible shrink

hole origin. For Chsh$ 1, no shrink hole appears. It is analogous to the Niyam

shrink formation number Φ (p. 358).

Nikolai Chvorinov (see above).

8.1.6 Chvorinov Time Number Ch, Chf

Ch5a1τL21

5λ1ΔTl

llR1

τL21

ð1Þ;

where l1 5 lls 1ΔTlc1 5lls

ΔTl1 c1

� �ΔTl 5 cekvΔTl;

355Technology and Mechanical Engineering

Chf 5a2τL22

5λ2

ðcRÞ2τL22

ð2Þ

a1, a2 (m2 s21) � thermal diffusivity; τ (s) � time; L1, L2 (m) � characteristic

length L1 5V1

S

� �; L2 5

V2

S

� �; V1, V2 (m

3) � volume; S (m2) � surface area of cast-

ing; λ1, λ2 (W m21 K21) � thermal conductivity; ΔTl (K) � superheating of melt-

ing; l1, lls (J kg21) � specific latent heat of solidification; R1, R2 (kg m23) �density; c1, c2 (J kg

21 K21) � specific heat capacity; cekv (J kg21 K21) � equivalent

specific heat capacity; subscripts: 1 � casting, 2 � mould, ls � liquidus�solidus.

This number expresses the dimensionless solidification time of a casting (1) and

a mould (2). It is a modification of the Fourier number Fo (p. 175).

Nikolai Chvorinov

8.1.7 Craya�Curtet Number Ct

Ct5wkffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

w2d 2

12w2k

q ;

where wk 5Qm1 1Qm2

RA; wd 5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiH1 1

12H2

RA

s

wk (m s21) � mean kinematic velocity; wd (m s21) � mean dynamic velocity; Qm1,

Qm2 (kg s21) � mass flux relative to burner and to induced air stream; R (kg m23) �gas density; A (m2) � cross section of furnace; H1, H2 (kg m s21) � momentum rela-

tive to burner and to induced air stream.

It characterizes the radiation flow and the heat transfer in industrial furnaces by

means of a passing mass and the momentum of basic and induced currents. For

Ct-0, the axial flow has great momentum and recirculation. When Ct. 0.74, the

return flow disappears.

Info: [A23].

Antoine Joseph Edouard Craya (1911�1976), French engineer.

Roger Michel Curtet (born 1923), French engineer.

8.1.8 Heat Diffusivity B

B5a

aref; b5

ffiffiffiffiffiffiffiffiRcλ

p

a (m2 s21) � thermal diffusivity; aref (m2 s21) � reference value of thermal diffu-

sivity; R (kg m23) � material density; c (J kg21 K21) � specific heat capacity;

λ (W m21 K21) � thermal conductivity.


Most frequently, it characterizes the diffusivity in a cast-mould system with the

relative value corresponding to the casting diffusivity and the value of b corre-

sponding to the mould. Foundry work technology. Solidification and cooling of

castings.

Info: [B81].

8.1.9 Latent Heat NL

NL 5lsl

cðTl 2 TiniÞð1Þ;

NL 5lls

cðTl 2 TsÞð2Þ

lsl (J kg21) � specific latent heat of melting; lls (J kg

21) � specific latent heat of

solidification; c (J kg21 K21) � specific heat capacity; Tl, Ts, Tini (K) � tempera-

ture of liquidus and solidus, initial mould temperature.

It expresses the latent heat of melting (1) and solidification (2) of a casting in a

mould. Foundry work technology.

Info: [B81].

8.1.10 Loss Dissipation Coefficient η, Ndis

η5ΔU

2πU; where U5

ðσmax

0

dε1

2

σ2

E

U (J m23) � elastic energy in material per volume unit; ΔU (J m23) � difference

of dissipation energy between loading and unloading; σmax (Pa) � maximal stress

at elastic loading; E (Pa) � modulus of elasticity; ε (2) � relative elongation

(p. 144).

It expresses the degree which is reached by a material due to vibration energy

dissipation. The cycle can be expressed in various ways, either as fast or slow.

Usually, the coefficient value depends on a timescale or a cycling frequency.

Info: [C65].

8.1.11 Melting Efficiency ηW

ηW 5wAQV

Q

w (m s21) � motion velocity; A (m2) � weld surface;QV (J m23) � heat per unit

volume; Q (W) � heat flux.


It expresses the melting efficiency as a ratio of the power necessary to melt the

welding zone to the net power absorbed in a part. It represents an important aspect

of the welding process. The theoretical maximum value is 0.48 for the 2D heat

propagation process. For the 3D process, it is 0.37.

Info: [B37].

8.1.12 Niyam Shrink Formation Number Φ

Φ5rΘ N21η

rΘ (2) � temperature gradient (p. 361); Nη (2) � solidification rate constant (p. 360).

It is an approximate criterion of the shrink hole originating in a casting. It is

based on the assumption that geometric and time gradients influence shrink hole

formation in a solidifying casting. It is analogous to the Chvorinov shrink formation

number Chsh (p. 355). Foundry work technology. Solidification of castings.

Info: [B81].

8.1.13 Part of Liquid Phase β

β5T 2 Ts

Tl 2 Tsfor Ts , T, Tl ð1Þ;

β5 0 for T , Ts ð2Þ;

where h5 href

ðTTref

cp dT ; ΔH5βL ð3Þ;

H5 h1ΔH ð4Þ;

where h5 href

ðTTref

cp dT ; ΔH5βL

T (K) � temperature of melting; Ts (K) � solidus temperature; Tl (K) � liquidus

temperature; H (J) � enthalpy of material; h (J) � enthalpy of heating up; href (J) �reference enthalpy; Tref (K) � reference temperature; cp (J kg2 1 K2 1) �specific heat capacity; ΔH (J) � addition of enthalpy; L (J) � latent heat of

melting.

In equation (1), it expresses the liquid phase portion of material melting. The

enthalpy increment can be determined from the latent heat of the material and is in

the range of h0; Li for the solidus and liquidus. For the enthalpy of the material,

expression (4) holds.


8.1.14 Power of Smeltery N

N5P

cηTdð1Þ;

N5Pz

cηTdð2Þ

P (W) � furnace power per electrode; c (J kg2 1 K2 1) � specific heat capacity;

η (Pa s) � dynamic viscosity; T (K) � temperature; d (m) � electrode diameter;

Pz (W) � power dissipation per electrode.

It expresses the electrical-to-thermal-power ratio. In the dimensionless state, it

characterizes the electrical power (1) or the power dissipation (2) of an electric

induction furnace.

Info: [A23].

8.1.15 Pressure in Smeltery N

N5p

g R β ΔT d

p (Pa) � pressure in a molten material in smelting furnace; g (m s22) � gravita-

tional acceleration; R (kg m22) � density; β (K21) � volume thermal expansion

coefficient; ΔT (K) � temperature difference; d (m) � electrode diameter.

It expresses the relationship between the pressure in a molten material and the

pressure caused by the temperature non-uniformity in the molten material. It char-

acterizes the pressure relations in a melting pan of an electric smelting furnace

with free convection of the molten material.

Info: [A23].

8.1.16 Rockwell Scale HR

HR A fHRA; HRB; HRC;?g

It expresses the hardness of a material based on the depth of a body impression

in a material sample compared to that in a reference material. The HR hardness is a

dimensionless quantity which is characterized by its simplicity and wide applica-

tion in mechanical engineering and � most especially � in metallurgy. Several

hardness scales are used, mostly the B and C scales, in which the B scale applies to

soft material (aluminium, bronze and mild steel). In the B scale, hard steel balls

and 100 kg loads are used in the measurement. The measured value is denoted as

HRB. With the C scale, a diamond cone and 150 kg loads are used. For very hard

steel, HRC A h55; 62i. For the steel for spades, axes and chisels, for example,


HRC A h40; 45i. The values below HRC5 20 are considered unreliable, as are

those over HRC5 100. Many other scales, including the HRA, have specific

applications.

Info: [C117].

8.1.17 Rosenthal Number Ro

Ro5Q

LλΔT

Q (W) � heat flux; L (m) � characteristic length, width of weld; λ (W m21 K21) �thermal conductivity; ΔT (K) � temperature difference.

Together with the Rykalin number Ry (p. 360), it is used for the 3D and 2D ther-

mal process analyses of electric arc welding. It expresses the influence of a charac-

teristic weld dimension in welding. It is analogous to the Pomerantsev heat number

Po (p. 181). Welding.

Info: [B37].

D. Rosenthal.

8.1.18 Rykalin Number Ry

Ry5Qw

a2QV

Q (W) � heat flux; w (m s21) � motion velocity; a (m2 s21) � thermal diffusivity;

QV (J m23) � heat per unit volume.

This number expresses the dynamic thermal process in electric arc welding. It is

used to analyse the heat conduction in melting down. It can be used for 3D and 2D

thermal processes.

Info: [B37].

N. N. Rykalin, Russian engineer.

8.1.19 Solidification Rate Constant Nη

Nη 5η

2ffiffiffia

p

η (m s21/2) � solidification rate; a (m2 s21) � thermal diffusivity.

It expresses the solidification rate of a casting in a mould. Together with the

temperature gradient rΘ (p. 361), it influences the shrink hole formation in a cast-

ing. Foundry work technology. Solidification of castings.

Info: [B81].


8.1.20 Solidification Shrinkage β, Nsh

β5vl 2 vs

vs

vl, vs (m3 kg21) � specific volume of liquidus and solidus.

It expresses the volume changes with different molten materials and casting

shrinkage during solidification. Foundry work technology. Solidification of

castings.

Info: [B81].

8.1.21 Specific Heat of Melting Nmelt

Nmelt 5ΔTc

lsl5

ðT 2 TminÞclsl

T (K) � temperature; c (J kg21 K21) � specific heat capacity; lsl (J kg21) �specific latent heat of melting.

It characterizes the specific heat which is necessary to melt the material in a

furnace.

Info: [A23].

8.1.22 Temperature Gradient rΘ

rΘ5

ffiffiffiffiffiaτ

p rTrefTl 2 Tini

a (m2 s21) � thermal diffusivity; τ (s) � time; rTref (K m21) � reference tempera-

ture gradient; Tl (K) � liquidus temperature; Tini (K) � initial mould temperature.

It expresses the temperature gradient in a solidifying casting during time

τ, related to the temperature difference between the molten material and a

mould. Together with the solidification rate constant Nη (p. 360), it is the main

quantity acting during shrink hole formation. Foundry work technology.

Solidification of castings.

Info: [B81].

8.1.23 Temperature Range of Solidification Θsl

Θsl 5Tl 2 Ts

Tl 2 Tini

Tl, Ts, Tini (K) � temperature of solidus and liquidus, initial mould temperature.


It expresses the temperature range during casting solidification in a mould.

Foundry work technology. Solidification of castings.

Info: [B64].

8.1.24 Tikhonov Number (1.) Ti

Ti5w

ffiffiffiffiffiτnD

r

w (m s21) � reaction rate; τn (s) � surface saturation time; D (m2 s21) �diffusivity.

This number expresses the ratio of the velocity of transferring an element from

the surroundings on the surface of an object to that of the element propagation in a

body surface layer. It characterizes the process of diffusion saturation of a rigid

body surface by various atoms from the gas or vapour phase, for example, in

cementation, oxidation or chrome plating of steel or cast iron parts. It is the crite-

rion of the thermochemical treatment of materials.

Info: [A33].

Andrey Nikolayevich Tikhonov (30.10.1906�8.11.1993),

Russian mathematician and physicist.

He was engaged in research in topology, functional anal-

ysis and differential equations. He focused especially on

mathematical problems in geophysics, the physics of

plasma, gas dynamics and electrodynamics, and solved

many mathematical and physical problems in the area of

thermomechanics, for example, in heating, cooling and

solidification. He formulated and proved the Tikhonov the-

orem in the field of topology.

8.1.25 Tikhonov Number (2.) Ti

Ti5αchar

b

ffiffiffiτ

p ð1Þ;

Ti5αchar

b

ffiffiffiffiffiτ1

p αmed

αchar

ð2Þ;

where αmed 5

ðτ10

α dτ

α (W m22 K21) � heat transfer coefficient at the melt�mould interface; αchar

(W m22 K21) � characteristic heat transfer coefficient; αmed (W m22 K21) � mean

heat transfer coefficient from the initial time of mould heating-up to the time

τ1; b ðkg s232 K21Þ � mould capability of heat accumulation; τ (s) � heating time

of monitored mould section; τ1 (s) � heating time.


It characterizes the mould heating time by considering the heat transfer on the

boundary between molten material and a mould, and the thermal accumulating

capabilities of the mould, whereas the Chvorinov cast mould system number Ch

(p. 354) solves the heat transfer between a casting and a mould as a contact prob-

lem (a fourth-type boundary condition).

Info: [B33].

Andrey Nikolayevich Tikhonov (see above).

8.1.26 Weber Agglomerate Number Weagl

Weagl 5RDw2

Γð1Þ;

Weagl 5RDðw2w0Þ2

Γð2Þ

R (kg m23) � particle density; D (m) � particle diameter; w (m s21) � particle

velocity; w0 (m s21) � limiting velocity at which no agglomeration of particles

occurs; Γ (J m22) � surface energy.

It expresses the ratio of the dynamic agglomeration energy to the surface energy.

Expression (2) is a modification of expression (1), a modification taking into con-

sideration the limiting velocity at which no agglomeration occurs.

Ernst Heinrich Weber (p. 92).Wilhelm Eduard Weber

8.1.27 Weld Size Parameter PW

PW 5wA

La

w (m s21) � motion velocity; A (m2) � weld surface; L (m) � characteristic weld

length; a (m2 s21) � thermal diffusivity.

It is often called the dimensionless weld width. It characterizes the weld width in

considering 3D and 2D thermal processes.

Info: [B37].

8.1.28 Weld Width Nww

Nww 5wA

La


w (m s21) � motion velocity; A (m2) � weld surface; L (m) � characteristic length,

weld width; a (m2 s21) � thermal diffusivity.

It expresses the weld width in 2D analysis of an electric arc welding process.

Together with the Rykalin number Ry (p. 360) and the Rosenthal number Ro

(p. 360), it is among the fundamental welding criteria. Welding technology.

Info: [B37].

8.2 Mechanical Engineering

In mechanical engineering, many dimensionless quantities are applied in all aspects

of the design and construction of machines or parts thereof. This includes, for

example, the design and construction of water pumps, compressors, water and

steam turbines, fans and mixers [A29].

These quantities relate to rotating parts dynamics in moving blades, airscrews,

propellers and worm conveyers. Other dimensionless quantities relate to the design

of bearings, their loading and lubrication, dimensionless cryogenic cooling of

machines and devices, the observation of water drops in steam, including the flow

through part of the last stages of steam turbines, and the dimensioning of combus-

tion chambers, rocket engines and other machines and devices. Of the applied

dimensionless quantities, the Harrison and Addison numbers are used, for example,

to design a water turbine or a rotary pump. Similar to the Harrison and Ocvirk

numbers for bearings, the Leroux and Thoma numbers can be used for the cavita-

tion phenomena in water machines, and the Tomson number can be used for fuel

consumption depending on environmental resistance in aircraft and water

machines. Many of the numbers are modifications of applied dimensionless

quantities.

8.2.1 Addison Shape Number Ad

Ad5 103Kn; where Kn is specific velocity

Kn 5nP

12

R12ðghÞ54

for water turbines;

Kn 5nQ

12

ðghÞ34for rotary pumps

n (2) � revolutions per time interval; P (W) � power; R (kg m23) � density;

g (m s22) � gravitational acceleration; h (m) � hydrostatic head of machine;

Q (m3 s21) � volume flow.


This number is a multiple of the specific velocity. It is used with water

machines. It is an empirically obtained dimensionless number.

Info: [A29].

Herbert Addison (born 1889), English engineer.

8.2.2 Advance Ratio J

J5w

nD

w (m s21) � motion velocity; n (s21) � revolutions per second; D (m) � outer

diameter of screw or propeller, impeller.

It characterizes the relation between the velocity and the geometric parameters

of an airscrew or a ship propeller. With the ultrafiltration of mixing chambers,

it serves to predict the permeability and clogging degree thereof. It enables

dimensioning of a drive system.

Info: [A29].

8.2.3 Bearing Damping Coefficient Ndam, D0

Ndam 5DωRFηðTÞ

D (m kg2 s22) � general damping coefficient; ω (s21) � angular frequency; R (m) �radius of bearing journal; F (N) � bearing loading; η (Pa s) � dynamic viscosity

T (K) � temperature.

This coefficient expresses the plain bearing damping capabilities. Mechanical

engineering. Tribology. Bearings.

8.2.4 Bearing Modulus Nmod

Nmod 5ηnp

η (Pa s) � dynamic viscosity; n (s21) � revolutions per second; p (Pa) � bearing

pressure.

It is an important bearing characteristic, according to which three bands of the

bearing friction coefficients can be distinguished (thin film friction, transient

boundary and hydrodynamic friction). Mechanical engineering. Tribology. Plain

bearings.

Info: [C77].


8.2.5 Bearing Number Nbea, S0

Nbea 5F

ηðTÞRωL

F (N) � loading force; η (Pa s) � dynamic viscosity; T (K) � temperature; R (m) �radius of bearing journal; ω (s21) � angular frequency; L (m) � characteristic length

of bearing.

It expresses the plain bearing load size. It is a modification of the Sommerfeld

number Sm (p. 385).

8.2.6 Bearing Stiffness Nst

Nst 5kst

EðTÞtp

kst (N m21) � stiffness; E(T ) (Pa) � modulus of elasticity; tp (m) � thickness of

lubricating film.

It expresses the stiffness of various kinds of fluid and gas plain bearings.

Mechanical engineering. Plain bearings.

8.2.7 Betz Number Be

Be5ðωrÞ2Lw2r

ω (s21) � angular frequency; r (m) � radius; L (m) � characteristic length, chord

of profile; ω (m s21) � flow velocity.

It expresses the flow in rotating channels of flow machines.

Info: [A24].

8.2.8 Cavitation Number Ncav, σ

Ncav 5p2 pv

Rw2ð1Þ;

Ncav � Th5ha 2 hs 2 hv

hð2Þ


p (Pa) � total static pressure; pv (Pa) � partial vapour pressure; R (kg m23) � den-

sity; w (m s21) � mean flow velocity; h, ha, hs, hv (m) � total, atmospheric, suck-

ing and water vapour pressure heads; Th (2) � Thoma number (1.) (p. 90).

It characterizes the formation of cavitation in a flowing fluid. Above all, it

depends on the flow rate, the kind and purity of the fluid, and the inlet flow turbu-

lence. In equation (1), it expresses the ratio of the static pressure drop, between the

fluid and vapour, to the fluid dynamic pressure. It characterizes the thermohydrody-

namic process of the cavitation origin in flow through parts of water machines. It is

a modification of the Euler number (1.) Eu (p. 61). In equation (2), it is called the

Thoma number (1.) Th (p. 90), and it expresses the cavity relations in water pumps.

See the Leroux number Lx (p. 374). Hydrodynamics. Hydraulics. Water machines.

Info: [A7],[A26].

8.2.9 Critical Number of Brake Disc Ndisc, crit

Ndisc; crit 5η20wsu

R2σ2max

ð1Þ;

Pe5uΔx

a. 2 ð2Þ

η0 (Pa s) � dynamic viscosity of oil at the temperature at input to contact area;

ws (m s21) � shear speed; u (m s21) � surface speed; R (m) � disc radius; σmax (Pa) �maximum Hertz stress under different loading, not leading to seizing; Δx (m) � grid

space; a (m2 s21) � thermal diffusivity.

The number size expresses the brake disc critical temperature and the possibility

of it subsequently seizing up. Usually, a high disc velocity and arising superficial

thermal instabilities cause brake disc oscillations. The Peclet heat number Pe

(p. 180), expressing heat transfer by convection and conduction, is the characteris-

tic dimensionless number for formation of motion thermal field. In the case of disc

brakes, the value of this number is Pe� 10.5 With the numerical solution using the

finite elements method (FEM) application, expression (2) is valid for the selection

of the grid space.

Info: [B16].

8.2.10 Cryogenic Cooling of Superconductor Nsupra

Nsupra 5λqV1OαqV1A

5Bi21PoisoPo21supra

λ (W m21 K21) � thermal conductivity of isolation; qV1, qV2 (W m23) � volume

density of heat flux in isolation and superconductor; O (m) � helium cooled


perimeter; α (W m22 K21) � effective heat transfer coefficient; A (m2) � cross sec-

tion area of supraconductor; Bi (2) � Biot number (p. 173); Poiso (2) �Pomerantsev number for isolation; Posupra (2) � Pomerantsev number for super-

conductor, see Pomerantsev heat number Po (p. 181).

It characterizes the affect of insulation on superconductor cooling. Cryogenic

cooling.

Info: [B118].

8.2.11 Damping Coefficient Ndb

Ndb 5hωkdpabL

h (m) � bearing clearance; ω (s21) � angular frequency of bearing journal;

kd (kg m21 s21) � damping coefficient of bearing; pa (Pa) � ambient pressure;

b (m) � bearing width; L (m) � bearing length.

It characterizes the plain bearing damping. It depends on the geometric proper-

ties, revolving speed and damping properties of the bearing. Mechanical engineer-

ing. Aerostatic bearings.

8.2.12 Delivery Number ND

ND 5Q

Au

Q (m3 s21) � volume flow; A (m2) � impeller surface A5 14πd2

� �; u (m s21) �

peripheral speed.

It characterizes the volume throughflow in rotary machines. It is used in

throughflow blade machines.

Info: [A35].

8.2.13 Diameter Group ND

ND 51

4πD2

� �12

ð2ghÞ14w12

D (m) � diameter of screw propeller, impeller; g (m s22) � gravitational accelera-

tion; h (m) � hydrostatic head; w (m s21) � flow velocity.

It serves to dimension water and turbojet machines.


8.2.14 Discharge Number NΦ

See the flow coefficient φ (p. 370).

8.2.15 Droplets Formation in Jet N

N5ηlrmin

RlL2ffiffiffiffiffiffiffirT1

p ð1Þ;

N5λrmin

clRlL2ffiffiffiffiffiffiffirT1

p ð2Þ;

N5σRlLη2l

ð3Þ;

N5L

rmin

ð4Þ;

N5 5cl

rð5Þ;

N5rT1Rlp1

ð6Þ;

N5LRl

ffiffiffiffiffiffiffirT1

p

ηlð7Þ

ηl (Pa s) � dynamic viscosity; rmin (m) � radius of minimal jet cross section;

Rl (kg m23) � drop density; L (m) � characteristic length, initial drop diameter;

r (J kg21 K21) � specific gas constant; T1 (K) � temperature in input cross section; p1(Pa) � pressure in input cross section; λ (W m21 K21) � thermal conductivity; cl(J kg21 K21) � drop specific heat capacity; σ (N m21) � surface tension.

The set of criteria (1)�(7) describes gas flow influence on the laws of droplets

crumbling in nozzles without considering evaporation and condensation. The frac-

tional composition of droplets is presumed according to the law of normal logarith-

mic distribution. Temperature dependence of physical properties is neglected.

Info: [B119]

8.2.16 Eccentricity of the Bearing ε

ε5e

c

e (m) � eccentricity (distance between centre of journal and bearing); c (m) �interspace between journal and bearing.


It expresses the bearing journal deviation which occurs in the hydrodynamic

mode. Sliding bearings, lubrication.

Info: [B31].

8.2.17 Ekman Number (2.) Ek

Ek5w

ωL2

w (m s21) � motion velocity; ω (s21) � angular frequency; L (m) � characteristic

length.

It characterizes, for example, the inertia to centrifugal forces ratio during

rotation of a vessel with the fluid in motion. It appears in cases of heat and mass

transfer solutions in rotary systems.

Info: [A5].

8.2.18 Elastic Component of Indentation Energy ηIT

ηIT 5Welast

Wcelk

100; where Wcelk 5Welast 1Wplast

Welast, Wplast, Wcelk (J) � elastic, plastic and total indentation energy, respectively.

It characterizes the hardness measurement. It expresses the portion of the elastic

energy in measurement of macro-, micro-, and nanohardness by means of diverse

kinds of indenters. The elastic energy component is given with the test conditions,

of which the first is the loading force in newtons (N) and the second is the loading

time in seconds (s), for example, ηIT (0, 5/10)5 36, 5%.

Info: [B111].

8.2.19 Fineness Coefficient, Waterplane Coefficient Ψ

Ψ5 LW13

L (m) � characteristic length; W (m3) � volume draft.

It is applied in ship modelling and design.

8.2.20 Flow Coefficient φ

φ5Q

nD3ð1Þ;


φ5Q

12πD2u2

ð2Þ;

φ5Q

14πð12D2

r ÞD2u2ð3Þ;

φ5v

u2ð4Þ

Q (m3 s21) � volume flow; n (s21) � rotational frequency; D (m) � diameter of

flow part; u2 (m s21) � peripheral flow velocity; Dr (2) � ratio of inner to outer

diameter; v (m s21) � mean flow velocity.

It expresses the throughflow through turbojet rotary machines. In equation (1), it

is often called the outlet number. Together with modifications (2) through (4), it is

used in the hydrodynamics and aerodynamics of rotary machines, fans and blowers.

Info: [B20].

8.2.21 Froude Number (3.) Rotation Fr3

Fr3 5Dn2

g

D (m) � diameter of impeller; n (s21) � revolutions; g (m s22) � gravitational

acceleration.

It expresses the influence of rotation on the flow through a pump running wheel.

Hydrodynamics. Water machines. Mixing.

Info: [A29].

William Froude (p. 63)

8.2.22 Geometric Parameter of Water Jet A

A5Rzrd

Nr21

Rz (m) � curvature radius of liquid flow in sprayer chamber; rd (m) � radius of

jet; N (2) � number of canals; r1 (m) � radius of input canals.

It characterizes the geometric parameter of a fluid sprayer. It is applied in the

heat transfer with condensation of the sprayed fluid.

Info: [B39].

8.2.23 Gravitational to Centrifugal Acceleration Ratio Ngc

Ngc 5q

ω2R


g (m s22) � gravitational acceleration; ω (s21) � angular frequency; R (m) �radius.

It characterizes the dynamic force relations, for example, in rotary systems.

Info: [A5].

8.2.24 Head Coefficient CH, Ψ

CH 5H

n2d2

H (J kg21) � specific pressure energy; n (s21) � rotational frequency; d (m) �impeller diameter of pump, ventilator, etc.

It characterizes the flow in flow machines. It serves to dimension rotary pumps

and ventilators.

Info: [A4],[A29].

8.2.25 Head Coefficient, Loading Coefficient, Energy TransferCoefficient Ψ

Ψ5ΔP

Rn2D2ð1Þ;

Ψ5ΔP

Rω3D5ð2Þ;

Ψ5gh

n2D2ð3Þ;

Ψ5gh

u2ð4Þ

ΔP (Pa) � pressure difference; R (kg m23) � density; n (s21) � rotational fre-

quency; D (m) � diameter of flow part; ω (s21) � angular frequency; g (m s22) �gravitational acceleration; h (m) � hydrostatic head; u (m s21) � peripheral speed.

It is used in rotary machine dynamics. Expressions (1) and (2) are used to design

fans and rotary blowers. Expressions (3) and (4) are applied in the designing of

water pumps and turbines.

Info: [A29].

8.2.26 Homochronicity Number Ho

Ho5 nτ

n (s21) � revolutions; τ (s) � mixing time.


It is used for time comparison of mixing, whisking and other processes in time.

Info: [A29].

8.2.27 Homochronous Number Ho

Ho5wτL

w (m s21) � motion velocity; τ (s) � transfer time to the distance L; L (m) �distance.

It serves to choose the timescale for similar processes.

Info: [A29].

8.2.28 Kennedy Number Ω

Ω5τdapfRb

Tda (s) � deaerating time; pf (m3 kg21 s21) � constant of material permeability

(specific volume per second); Rb (kg m23) � material density.

It expresses the ratio of the deaerating time to the delivery time of the unit mate-

rial amount in pneumatic transport. Mechanical engineering. Pneumatic transport.


Lg2 5P

ηL3n25RerotNp; where Np 5

P

L5Rn3

P (W) � power input of impeller; η (Pa s) � dynamic viscosity; L (m) � character-

istic length; n (Hz) � revolutions per second; R (kg m23) � density; Rerot (2) �Reynolds number (p. 81); Np (2) � power input criterion.

It characterizes the energy loss from mixing viscous fluids in vessels, depending

on the physical properties of the fluid and the dimensions and speed of the mixer.

Info: [A14],[A24].

Joseph-Louis Lagrange (p. 70).

8.2.30 Lautrec Number Lau

Lau5L

δsð1Þ;


Lau5R

δkð2Þ

L (m) � characteristic length of the wall; R (m) � characteristic length (gap thick-

ness) between two walls; δs (m) � penetration depth in solid wall; δk (m) � pene-

tration depth in liquid.

It characterizes the thermoacoustic phenomenon of the thermal and acoustic

energy transformations in a wall (1) or in a gap between two walls (2) in thermoa-

coustic devices such as thermal pumps and cooling devices. It appears together

with the Mach number M (p. 73) and the Prandtl number Pr (p. 197), especially in

observing the interaction between porous surroundings and a sound field in pipes

and pipelines.

Info: [C128].

8.2.31 Leroux Number Lx, σc

Lx5p2 pv12Rlw2

p (Pa) � local static pressure; pv (Pa) � vapour pressure; Rl (kg m23) � liquid den-

sity; w (m s21) � velocity.

This number expresses the ratio of the difference between the local static pres-

sure head and the vapour head to the velocity head. In water machines it charac-

terizes the cavitation phenomenon with water flow through. See the cavitation

number Ncav (p. 366).

Info: [A35].

8.2.32 Load Parameter of Journal Bearing NB

NB 5ηur2

FL2

η (Pa s) � dynamic viscosity; u (m s21) � peripheral speed; r (m) � radius; F (N) �force; L (m) � characteristic length.

It is another variant of the Sommerfeld number Sm (p. 385). Mechanical engi-

neering. Tribology.

8.2.33 Loading Number of Bearing NM, M

NM 5 qα2

ηusR2

� �13


q (N m21) � length loading; α (m2 N21) � coefficient of piezoviscosity; η (Pa s) �input dynamic viscosity; us (m s21) � sum of tangential surface velocities: R (m) �effective radius curvature.

It expresses the ratio of the plain bearing load to surface friction forces.

Elastohydrodynamic lubrication. Bearings.

Info: [B77].

8.2.34 Loading Number of Sliding bearing NB

NB 5ηuR2

Fh25 2Gu

η (Pa s) � dynamic viscosity; u (m s21) � peripheral speed; R (m) � radius of

bearing journal; F (N m21) � length force loading of bearing; h (m) � thickness of

round gap; Gu (2) � Gumbel number (1.) (p. 164).

It expresses the ratio of the dynamic friction force to the loading force of a bear-

ing. It is another variant of the Sommerfeld number Sm (p. 385). Tribology.

Lubrication. Bearings.

8.2.35 Lock Number (1.) Lk

Lk5RR4la

I

R (kg m23) � liquid density; R (m) � rotor radius; l (m) � blade chord;

a (m2 kg21) � steepness of lifetime curve of rotor; I (m4) � moment of inertia of

blade at hangings.

It expresses the dynamics of rotor blades. It is an empirical dimensionless

number.

Info: [A35].

Christopher Noel Hunter Lock (1894�1949), English mathematician.

8.2.36 Lock Number (2.) Lk

Lk5dCL

dαRlR4

I

dCL

dα(2) � angular coefficient of rotor lifting curve; CL, (2) � lifting coefficient

of rotor; α (rad) � angle of pitch of copter rotor; R (kg m23) � liquid density;

l (m) � blade chord; R (m) � rotor radius; I (m2 kg) � inertia moment of rotary

rotor blades.


It expresses the buoyancy to inertia forces ratio in helicopters.

Info: [29].

Christopher Noel Hunter Lock (see above).

8.2.37 Mach Blowing Number Mb

Mb 5wb

a

Wb (m s21) � inject velocity at surface of porous boundary; a (m s21) � mean stag-

nation sound speed.

It is the basic criterion of combustion chambers. The effect of fuel viscous dissi-

pation is important with a low Mb value occurring with small fuel injection, high

frequency and kinematic viscosity, which is possible in hot gases. With a very

small Mach number (p. 73) corresponding to slight fuel injection, the solution

approaches the injectionless Stokes solution. Jet engines. Combustion chambers.

Info: [B119].

Ernst Mach (p. 73).

8.2.38 Mach Propeller Blade Tip Number Ma

Ma5πnDa

n (s21) � revolutions per second; D (m) � diameter; a (m s21) � sound speed.

It characterizes the velocity of a propeller blade tip or of a pump propeller run-

ning wheel, as a couple of examples.


8.2.39 Mechanical Load Group NM

NM 5ασHZ 5αE

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiF

2πLER

r

α (m2 N21) � pressure coefficient of viscosity; σHZ (Pa) � maximal dry (Hertzian)

stress; E (Pa) � reduced modulus of elasticity; F (N) � force loading of bearing;

L (m) � characteristic length (width of bearing); R (m) � reduced radius of

curvature.

It expresses the mechanical contact load influence on the oil film thickness and

pressure in a bearing. Electrohydrodynamic lubrication. Tribology. Lubrication.

Plain bearings.

Info: [B117].


8.2.40 Mishken Number Mi

Mi5DefL

q

Def (m2 s21) � effective diffusivity; L (m) � characteristic length (endless screw

diameter); q (m3 s21) � volume flow.

It describes the mixing process in worm conveyors from the kinematic point of

view.

8.2.41 Modified Power Number NPM

NPM 5Np

D

L

� �Δs0:5

ðnbnsÞ0:67

D (m) � effective diameter of blade wheel; L (m) � effective wheel length;

Δs (2) � wall closeness factor; nb, ns (2) � blade number of impeller or stator;

Np (2) power coefficient, power number (p. 379).

It is used to recalculate the power output of rotary mixers and pumps after

changing the rotating wheel parameters.

8.2.42 Muramtsev Number Mo

Mo5Raλbccp

ηd

R (kg m23) � fluid density; λ (W m21 K21) � thermal conductivity; cp(J kg21 K21) � specific heat capacity; η (Pa s) � dynamic viscosity; a, b, c, d (2) �dimensionless constants.

It is the criterion used to choose the heat transferring material with the heat

transfer properties in a limited range of working temperatures, specifically from the

point of view of heat transfer intensity under the equal velocity condition. It can be

used to design energy generating plants. It is an empirical dimensionless number.

Info: [A33].

8.2.43 Nusselt Modified Number Numod

Numod 5αλcrit

λv

; where λcrit 5 2πffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

σgðRl 2 RvÞ

s

α (W m22 K21) � heat transfer coefficient; λcrit (m) � critical wavelength;

λv (W m21 K21) � vapour thermal conductivity; σ (N m21) � surface tension;


g (m s22) � gravitational acceleration; Rl, Rv (kg m23) � density of liquid and

vapour, respectively.

It expresses the ratio of the load due to convection heat transition on a corre-

sponding transfer surface to the nucleon film boiling, to the thermal load by con-

duction in vapour. Cryogenic cooling of machines and plants.


8.2.44 Ocvirk Number Oc

Oc5q

ηwΔs

r

2r

L

� �2

q (kg s22) � length loading; η (Pa s) � dynamic viscosity; w (m s21) � surface

speed of bearing journal; Δs (m) � bearing clearance; r (m) � bearing radius;

L (m) � bearing length.

This number expresses the ratio of the bearing load to the friction force.

Tribology. See also Hersey number He (p. 165) and Sommerfeld number Sm

(p. 385).

Info: [A29].

Frederick A. Ocvirk (1913�1967), American engineer.

8.2.45 Orifice Design Parameter Nλ, λ

Nλ 5

ffiffiffi23

pcdD

20ηLffiffiffiffiffiffi

Rpp

e3D

cd (2) � orifice outflow coefficient; D0 (m) � nozzle diameter; η (Pa s) �dynamic viscosity; L (m) � characteristic bearing length; R (kg m23) � lubricant

density p (Pa) � pressure; e (m) � eccentricity clearing; D (m) � bearing journal

diameter.

Mechanical engineering. Tribology. Bearings.

8.2.46 Peclet Brake Number Pebrake

Pebrake 5uL

a. 2; where u5ωr

u (m s21) � sliding speed; L (m) � characteristic length; a (m2 s21) � thermal dif-

fusivity; ω (s21) � angular speed; r (m) � radius of disc.

It characterizes the convectively diffusive heat transfer in disc brakes from the

point of view of the finite elements method solution. Incorrect size of the brake


disc causes often inadmissible oscillations and instability. This is one o- the main

problems of brakes, in which the typical size of the number Pebrake is approxi-

mately 105.

Info: [B123].


8.2.47 Penetration Number Npen, Sp

Npen 5w3z

ω2νL5M3

bN22ω Re21

wz (m s21) � inject velocity at surface of porous boundary; ω (s21) � angular fre-

quency; v (m2 s21) � kinematic viscosity; L (m) � characteristic length, radius

Mb (2) � Mach blowing number (p. 376); Nω (2) � wave number, angular speed

(p. 389); Re (2) � Reynolds number based on sound speed (p. 383).

This number governs the depth and structure of the unsteady boundary layer,

e.g. in jet engines, particularly the shape and dimension of the outer casing Jet

engines. Combustion chambers.

Info: [B71].

8.2.48 Petrov Number Pt

Pt5Δp1 1Δp2

Δp3

Δp1 (Pa) � pressure difference on water part of boiler; Δp2 (Pa) � pressure differ-

ence on economizer; Δp3 (Pa) � pressure difference on vapour�liquid part and

superheating part.

It characterizes the stability of the flow parameters of a heat transferring

medium which moves in a steam boiler heating system. For steam boilers, Pt . 1

is valid. Generally, it is Pt A h0.1; 10i.Info: [A33].

P. A. Petrov.

8.2.49 Power Coefficient, Power Number NP

NP 5M

Rn2D5ð1Þ;

NP 5Mω

Rn3D5ð2Þ;


NP 5P

Rn3D5ð3Þ;

where P5 2πnM

M (m2 kg s21) � force moment; R (kg m23) � fluid density; n (s21) � rotational

frequency; D (m) � rotor diameter; ω (rad s21) � angular speed; P (W) � shaft

power.

This coefficient relates to the dynamic criteria for rotary machines. For example,

it represents the hydrodynamic resistance on the mixer blade against the inertia

force. It expresses the energy loss in mixer vessels. In the shapes (1) and (2), these

numbers express the ratio of the shaft moment to the inertia force. In the shape (3),

it is the ratio of the thrust force (resistance) to the inertia force. It characterizes the

power output of some rotary machines, such as pumps, turbines, fans, mixers and

other devices. It is among the widely used dimensionless numbers in mechanical

engineering. Mechanical engineering.

Info: [A29],[A20],[A35],[B20].

8.2.50 Power Pressure Number NP

NP 5Δp

RD2Sn

2ð1Þ;

NP 5P

12RAw3

ð2Þ;

Δp (Pa) � pressure difference; R (kg m23) � fluid density; Ds (m) � impeller

diameter; n (s21) � rotation of mixer, etc.; P (W) � impeller power; A (m2) �effective surface; w (m s21) � wind speed.

This number expresses the ratio of the pressure gradient to the dynamic pres-

sure. It serves to design and dimension rotary mixers (1) and wind energy

plants (2).

Info: [B19].

8.2.51 Pressure Number (2.) NP2

NP2 5 e1

2u2

� �21

e (m2 s22) � specific energy; u (m s21) � peripheral speed.

It describes the flow through turbines, pumps and other flow machines.

Info: [17].


8.2.52 Propeller Torque Coefficient Mt

Mt 5MF

Rn2D5

MF (N m) � torque of propeller or impeller; R (kg m23) � density; n (s21) � revo-

lutions per second; D (m) � diameter of propeller or impeller.

It expresses the torque caused by the force acting on a rotating propeller or a

rotating wheel in rotary machines. It is not derived strictly from similarity theory

but from empirical experience.

Info: [A29].

8.2.53 Pulsation Number Pu

Pu5fdRqm

f (s21) � frequency of pulsation; d (m) � equivalent pipe or channel diameter;

R (kg m23) � fluid density; qm (kg m22 s21) � mass flux density.

This number characterizes the pulsation transport of fluids.

8.2.54 Rayleigh Centrifugal Number Rac

Rac 5ω2TβΔCR3

wD5Re2ScRo

ω (s21) � angular frequency; T (K) � temperature; β (K21) � volume thermal coef-

ficient of expansion; ΔC (2) � difference of ion concentration; R (m) � character-

istic length, vessel diameter with fluid; w (m s21) � travel speed; D (m2 s21) �diffusivity; Re (2) � Reynolds number (p. 81); Sc (2) � Schmidt number (p. 263);

Ro (2) � Rossby number (p. 406).

It expresses the influence of temperature, mass diffusion and motion in rotating

systems, such as volume conveyors and containers, as some examples. Mechanical

engineering.


8.2.55 Rayleigh Modified Number Ramod

Ramod 5gλ3

cr

νvavRl 2 RvRv

ð1Þ;


Ramod 5PrGrmod ð2Þ;

where λcrit 5 2πffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

σgðRl 2 RvÞ

r

g (m s22) � gravitational acceleration; λcrit (m) � critical wavelength; νv (m2 s21) �

vapour kinematic viscosity; av (m2 s21) � vapour thermal diffusivity; Rl, Rv (kg m

23) �liquid and vapour density; σ (N m21) � surface tension; Pr (2) � Prandtl number

(p. 197); Grmod (2) � Grashof modified number (p. 186).

In expression (1), it is a modification of the Rayleigh number (2.) (heat instability)

(p. 187). It is used in thermal calculations of cryogenic cooling systems such as cryo-

genic turboalternators. Expression (2) is a general equation expressing the natural

convective heat transfer from a heated or cooled wall.

Info: [B60].


8.2.56 Reynolds Acoustic Number Rea

Rea 5ωL2

ν5NωRe

ω (s21) � angular frequency; L (m) � characteristic length, radius; ν (m2 s21) �kinematic viscosity; Nω (2) � wave number, angular speed (p. 389); Re (2) �Reynolds number based on sound speed (p. 383).

In comparison to the Reynolds number based on sound speed (p. 383), it

involves the influence of the angular velocity of combustion products. Rocket

machines. Combustion chambers.

Info: [B71].


8.2.57 Reynolds Mixing Number Remix

Remix 5D2nRmηm

D (m) � propeller diameter; n (s21) � propeller rotational frequency; Rm (kg m23) �mixture density; ηm (Pa s)� mixture dynamic viscosity.

It characterizes the ratio of dynamic to viscous forces which act on an airscrew

with mixing. It is used to design and dimension rotary mixers.

Info: [C3].



8.2.58 Reynolds Number Based on Sound Speed Re

Re5aL

ν

a (m s21) � mean stagnation sound speed; L (m) � characteristic length, radius;

ν (m2 s21) � kinematic viscosity.

It expresses the ratio of the mean dynamic force of acting combustion products

to the viscous force. Rocket engines. Combustion chambers.

Info: [B71].


8.2.59 Reynolds Oscillatory Number Re

Re52πDfA

ν

D (m) � column diameter of oscillatory reactor; f (s21) � oscillatory frequency;

A (m) � oscillatory amplitude; ν (m2 s21) � kinematic viscosity.

It characterizes the intensity of mixing and flow through a circulation reactor.

The fluid (e.g. denatured protein) is delivered into the mixing chamber where it is

vibrated with an oscillating piston. Mixing of fluids. Physical chemistry. Chemical

mechanical engineering.

Info: [C130].


8.2.60 Reynolds Rotation Number Rerot

Rerot 5L2nRη

ð1Þ;

Rerot 5ωr2

νð2Þ

L�D (m) � characteristic length (diameter); n (s21) � rotational frequency;

R (kg m23) � fluid density; η (Pa s) � dynamic viscosity; ω (s21) � angular fre-

quency; r (m) � characteristic radius; ν (m2 s21) � kinematic viscosity.

This number characterizes the rotating viscous fluid flow. It expresses the

dynamic similarity of turbojet rotary machines (e.g. fluid mixers).

Info: [A29],[B17].



8.2.61 Richardson Gradient Number of Wind Turbine Rig

Rig 5g

Tmed

ΔT

Δz

Δw

Δz

� �22

g (m s22) � gravitational acceleration; Tmed (K) � mean air temperature; w (m s21) �wind speed; z (m) � height above ground.

This number expresses the ratio of the generated turbulence, with vertical tem-

perature (density) difference or buoyancy, to the wind velocity change according to

height. A negative value of Rig corresponds to an unstable condition of the flow in

a layer between the surface and the turbine rotor height. On the contrary, its posi-

tive value corresponds to a stable condition. The zero value of Rig represents a

neutral condition under which the buoyancy does not have any influence and the

turbulence is generated only by wind action. Mechanical engineering. Wind tur-

bines. Aerodynamics.

Lewis Fry Richardson (p. 403).

8.2.62 Size Number NSN

NSN 5DðghÞ14Q

12

D (m) � diameter of reactor or propeller; g (m s22) � gravitational acceleration;

H (m) � pressure head difference across turbine; Q (m3 s21) � volume flow.

It is also called the specific diameter. It serves to dimension water turbines and

pumps. It is an empirical dimensionless number.

Info: [A29].

8.2.63 Slosh Time Nτ

Nτ 5

ffiffiffiffiffiffiffiffiσRR3

rτ

σ (N m21) � surface tension; R (kg m23) � density; R (m) � tank radius; τ (s) �time.

It expresses the time of non-linear free oscillations of the interface of immiscible

fluids in containers under zero friction and zero gravity conditions. Tanking of fluid

in a low gravitation field. Astronautics. Space mechanical engineering.

Info: [B8].


8.2.64 Sommerfeld Number Sm

Sm5ηnp

D

h

� �2

ð1Þ;

Sm5ηnp

R

h

� �2

ð2Þ;

Sm5F

ηuh

R

� �2

ð3Þ;

Sm5F

2ηnh

R

� �2

ð4Þ

η (Pa s) � dynamic viscosity; n (s21) � rotational frequency; p (Pa) � mean bear-

ing pressure; D (m) � shaft diameter; h (m) � gap thickness; R (m) � shaft radius;

F (N m21) � length loading of bearing; u (m s21) � peripheral speed of shaft.

In equations (1) and (2), this number expresses the viscosity to loading forces

ratio. In equations (3) and (4), it represents the inverse value. It is analogous to the

Gumbel number (1.) Gu (p. 164), and in equation (4), it is identical to it. It charac-

terizes the relation of forces in a bearing with lubrication.

For small Sommerfeld numbers ðSm{1Þ; the lubrication pressure represents a

small part of the increase in the loading capability of a plain bearing. Two surface

sides are contacted by means of the roughness. For larger numbers Smc1; thelubrication pressure wholly supports the bearing load and the friction strain corre-

sponds to the viscous tension in the wholly developed lubrication mode. For

Sm�1, a mixed mode exists, with which the friction is influenced both by the vis-

cous resistance and by the roughness of contacts.

Info: [A29],[B20],[C77].

Arnold Johannes Wilhelm Sommerfeld (5.12.1868�26.4.1951), German physicist.

He was engaged in research in atomic physics, electro-

dynamics, mechanics and partial differential equations. He

modified the Niels Bohr atomic theory by considering

the elliptical path of electrons. Similarly, he also modified

the special relativity theory by formulating the Sommerfeld

model for energy in which the energetic levels are distrib-

uted into numerous parts. He was the first to come up with

an exact solution to the diffraction problem by making use

of the Maxwell equations.


8.2.65 Speed of the Bearing Ns

Ns 56ηω

paR2e22

η (Pa s) � dynamic viscosity; ω (s21) � angular frequency; pa (Pa) � ambient pres-

sure; R (m) � radius of bearing journal; e (m) � bearing gap.

It characterizes the lubrication rate of a plain bearing. Tribology. Bearings.

8.2.66 Speed Specific Number N3

N3 5nQ

12

ðghSÞ34

5N12DN

234

P2

n (s21) � rotational frequency; Q (m3 s21) � volume flow; g (m s22) � gravita-

tional acceleration; hS (m) � hydrostatic head; ND (2) � delivery number

(p. 368); NP2 (2) � pressure number (2.) (p. 380).

It expresses the dimensionless velocity in rotary flow machines, such as pumps,

water turbines and compressors.

Info: [A29],[A35],[B17].

8.2.67 Squeeze Number NS

NS 512ηL2ωpah2

η (Pa s) � dynamic viscosity; L (m) � characteristic length (length of plate);

ω (s21) � oscillation frequency; pa (Pa) � ambient pressure; h (m) � gap height.

It characterizes the squeezing of a thin isothermic film between two flat plates

of which the bottom one is steady and the top one is oscillating. Among other

things, it is used to damp dynamic systems and microsystems.

Info: [B20].

8.2.68 Stability of Cryogenic Cooling Nstab

Nstab 5qmax

LffiffiffiffiffiRv

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiσgðRl 2 RvÞ4

pqmax (W m22) � maximal surface heat flux; L (J kg21) � specific latent heat of

evaporation; Rv, Rl (kg m23) � vapour and liquid density, respectively; σ (N m21) �surface tension; g (m s22) � gravitational acceleration.


It expresses the ratio of the maximum planar thermal surface load, correspond-

ing to the transfer with nucleon film boiling, to the load caused by the surface

strain force and the force of gravity. It characterizes the stability of the cryogenic

cooling process in machines and plants.

8.2.69 Storage Energy Number Nen

Nen 5Ef

p

E (J) � accessible energy; P (W) � nominal actual output; f (s21) � synchroniza-

tion frequency.

It expresses the number of basic cycles which should be chosen for accessible

energy to reach the power corresponding to the nominal energy. When

P5 100 MW, f5 50 Hz, E5 20 MW, the number of cycles is Nen5 10.

Info: [C47].

8.2.70 Strouhal Boundary Layer Number St

St5ωLwb

5NωM21b

ω (s21) � angular frequency; L (m) � characteristic length, radius, ratio of volume

to half porous part; wb (m s21) � inject velocity at surface of porous boundary;

Nω (2) � wave number, angular speed (p. 389); Mb (2) � Mach blowing number

(p. 376).

It is a factor in the wavelength control of the unsteady swirl velocity of the

oscillatory flow in a cylindrical combustion space with side injection. It provides

precise observation of the amplitude and phase between non-stationary velocity

and pressure. Rocket engines.

Info: [B71].

Vincenc Strouhal (p. 87).

8.2.71 Structural Utilization Number N

N5gRLE

g (m s22) � gravitational acceleration; R (kg m23) � density; L (m) � characteris-

tic length; E (Pa) � modulus of elasticity.

It expresses the ratio of the weight of a structural part to its rigidity.


8.2.72 Subcooling Number (1.) Nsubc

Nsubc 5cpðTs 2 TlÞ

l

cp (J kg21 K21) � specific heat capacity of liquid; TS (K) � temperature of satura-

tion; Tl (K) � temperature of liquid; l (J K21) � specific latent heat of evaporation.

It expresses the heat transfer in subcooled liquid cryogenic materials such as

helium, hydrogen and nitrogen. Cryogenic cooling of machines and plants.

8.2.73 Thoma Number (2.) Tm

Tm5pH 2 pV

p2 2 p1

pH (Pa) � local total hydrostatic pressure; pV (Pa) � vapour pressure under given

temperature; p2 (Pa) � absolute input pressure; p1 (Pa) � absolute output pressure.

It expresses the ratio of the pressure gradient in cavitation to the pressure

increase in a pump. It characterizes the cavitation in water pumps.

Info: [A29].

Dieter Thoma (1881�1942), German hydraulic engineer.

8.2.74 Thrust Coefficient CDP

CDP 5FT

Rn2D4

FT (N) � drag force of propeller or impeller; R (kg m23) � fluid density; n (s21) �rotational frequency; D (m) � propeller diameter.

It characterizes the ratio of thrust to dynamic forces which act on a propeller.

Info: [A29].

8.2.75 Tomson Number To

To5gQ

Rw3L

g (m s22) � gravitational acceleration; Q (kg s21) � fuel usage; R (kg m23) � envi-

ronment density; w (m s21) � motion velocity; L (m) � characteristic length.

It expresses the ratio of fuel time unit consumption to environmental resistance.

Flying or floating machines. Aeronautics. Ships.

Info: [B20].


8.2.76 Torque Coefficient N

N5Mk

Rv2D3ð1Þ;

N5Mk

Rn2D5ð2Þ

Mk (N m) � torsion moment; R (kg m23) � density; v (m s21) � peripheral speed;

D (m) � impeller diameter; n (s21) � revolutions per second.

It is used to design propellers and running wheels. It is an empirically deter-

mined dimensionless number.

Info: [A29],[A35].

8.2.77 Wave Number, Angular Speed Nω

Nω 5ωLam

ω (s21) � angular frequency; L (m) � characteristic length, radius; am (m s21) �mean stagnation sound speed.

It expresses the ratio of the dynamic force of the combustion products flow to

the dynamic force corresponding to the stagnation sound velocity. Rocket engines.

Combustion chambers.

Info: [B71].

8.2.78 Weaver Flame Speed Number Wea

Wea5w

wH

100

w (m s21) � laminar combustion rate of gas; wH (m s21) � laminar combustion

rate of hydrogen.

It expresses the ratio of the laminar combustion rate of the considered gas to that

of hydrogen gas (Wea5 100). It characterizes the tendency of the gas to react. A

lower value of this number represents a lower flame rate. The value of a high com-

bustion rate of the gas is Wea A h32; 45i; of a medium rate, it is Wea A h25; 32i;and of a low rate, it is Wea A h13; 25i.

Info: [C37].

8.2.79 Weber Number of Damp Steam Wev

Wev 5Rvðwv 2w2ÞL

σ


Rv (kg m23) � vapour density; wv (m s21) � absolute vapour speed; w (m s21) �absolute speed of water drop; L (m) � characteristic length, maximal length of sta-

bilized water drop dimension; σ (N m21) � surface tension.

It characterizes the process of the erosive influence of water droplets in damp

steam on the throughflow parts, for example, of the last steam turbine stages.

Wev A h21; 24i.Info: [C37].



8.2.80 Wobbe Number, Wobbe Index Wob

Wob5λV1

ffiffiffiffiffiR2

pλV2

ffiffiffiffiffiR1

p ð1Þ;

Wob5λV

pð2Þ

λV1, λV2 (J m23) � volume density of energy in gas 1 and 2; R1, R2 (kg m23) den-

sity of gas 1 and 2; λV (J m23) � gas total volume energy density; p (Pa) � gas

pressure.

In equation (1), this number enables a comparison of the energies of changing

gases, provided the burner is the same (having an equal cross section and pressure

drop). In equation (2), it expresses the ratio of the energy density and pressure in

the same gas. Wob A h17.8; 35.8i is valid for a coke oven gas, Wob A h35.8; 71.5iholds for natural gas, and Wob A h71.5; 87.2i is for liquefied gases (LPG).

Info: [C63].


9 Geophysics and Ecology

To withdraw from Nature would mean to languish physically and a total divorce

would be death.Jan Neruda (1834�1891)

9.1 Geophysics, Meteorology and Astrophysics

In recent years, in geophysics, meteorology and astrophysics, the number of dimen-

sionless quantities give evidence of the increasing importance of these scientific

disciplines. Geophysics is about geodynamic, geothermic, geomagnetic, gravimetric

and seismologic processes, to which natural catastrophes are closely related, for

example, eruptions of volcanoes, tsunamis and earthquakes. Meteorology, as atmo-

spheric physics, deals especially with climatological conditions and accompanying

phenomena. In geophysics, the Ekman, Hatta, Rossby, Argand and Love numbers,

as well as the tsunami velocities and the Shida number, are very important. In

meteorology, the Richardson number and the Burger, Davies and Rossby atmo-

spheric numbers are among the best known. In astrophysics, the redshift number

and the Eddington, Golitsyn, Hubble, and Wesson numbers are among the most

significant.

9.1.1 Albedo Number A

A5 124σ0T

4

εC

σ0 (W m22 K24) � Stefan-Boltzmann constant; T (K) � absolute temperature;

ε (2) � emissivity (p. 206) of the atmosphere, around the height of about 5.5 km

where long-wave radiation is emitted; C (W m22) � solar constant.

It expresses the reflectance of surfaces and bodies. It characterizes the climatic

conditions on the Earth which are produced by the solar radiation reflected from

the Earth. The global value of the Albedo number can be determined by measuring

the amount of solar radiation, reflected from the Earth, and back to the Earth during

rotation from the dark surface of the Moon. Meteorology. Climatology. Astronomy.

Info: [B73],[C42].



http://dx.doi.org/10.1016/B978-0-12-416013-2.00009-9

9.1.2 Argand Number Arg

Arg5gLRcðRm 2 RcÞ

BRm

ffiffiffiffiffiL

wc

n

r

g (m s22) � gravitational acceleration; L (m) � characteristic length, thickness of

lithosphere; Rc (kg m23) � crustal density; Rm (kg m23) � mantle density;

B (N s m22) � constant characterizing vertical average of compositional and ther-

mal influences on the lithosphere rheology; wc (m s21) � imposed collision veloc-

ity; n (2) � stress�strain exponent.

It expresses the deformation in the terrestrial crust as the ratio of the buoyancy

force to the boundary force acting on the terrestrial plate. Geophysics.

Info: [B73],[C42].

9.1.3 Argand Number (1.) Arg

Arg5ΔRghη_ε

ΔR (kg m23) � density difference of the oceanic and continental lithospheres;

g (m s22) � gravitational acceleration; h (m) � thickness; η (Pa s) � lithosphere

dynamic viscosity; _ε (s21) � strain rate.

This number expresses the ratio of the tension, induced by the density difference

on the bottom part of the lithosphere, to the ductile deformation of the lithosphere.

With a certain simplification (the Newtonian fluid is considered instead of the

material creep law), it describes the lithosphere rheology. The criterion Arg charac-

terizes the interface area between the ocean and the mainland. For Arg. 1, the lith-

osphere deformation force cannot lead to lower energetic potential so that the

ocean lithosphere will act on the mainland lithosphere and the interface angle

decreases. Geophysics. Terrestrial crust geology. Rheology.

Info: [B73].


Arg5Rcghc

C1μcRchc

Rc (kg m23) � density of continental lithosphere; g (m s22) � gravitational acceler-

ation; hc (m) � thickness of the continental brittle lithosphere, which is unsup-

ported by a juxtaposed oceanic lithosphere; C (kg m21 s22) � cohesion;

μc (kg m21 s22) � constant dependent on friction coefficient (usually 0.5).

It expresses the ratio of the tension, on the bottom part of the non-hardened

mainland brittle lithosphere, generated with various increases of the mainland and


oceanic lithospheres, to the brittle tension of the lithosphere. The Arg reproduces

the development of the destruction in the upper lithosphere along the interface of

the mainland plates. Geophysics. Terrestrial crust geology. Rheology.

Info: [B73].


Arg5τ

μσn

τ (Pa) � shear stress; μ (2) � coefficient of friction between the lithospheres;

σn (Pa) � normal stress.

It expresses the shear strain-to-resistance ratio along the interface between the

ocean and the mainland. Therefore, with Arg. 1, the slip of the lithosphere and

its shift down into the asthenosphere can be expected. The shear resistance can be

used to represent the main impediment force in the subduction process (subshifting).

Geophysics. Terrestrial crust geology. Rheology.

Info: [B73].

9.1.6 Beaufort Number B

B5w

0:836

� �23

for w½ �Bm s21 ð1Þ;

B5w

1:87

� �23

for w½ �Bmile hour21 ð2Þ

w (m s21) � wind speed.

It characterizes the wind velocity on the sea. The value is in the range

BAh0; 15i.Info: [A43].

Francis Beaufort (7.5.1774�17.12.1857), Irish oceanographer.

He formulated the Beaufort scale indicating wind force.

His digital scale extended from zero (calm) to 12 (hurricane)

and determined the wind force and possible thunderstorm

activity. Later, the scale was extended to 17 (tropical

cyclones) and supplemented by a formula to determine the

Beaufort number conversion to wind velocity. He was a

significant contributor to the development of the fields of

geography, astronomy, oceanography and meteorology.

393Geophysics and Ecology

9.1.7 Buoyancy Wave Number NB

NB 5

ffiffiffiffif 3

ε

r

f (s21) � buoyancy frequency; ε (s21) � dissipation of turbulent kinetic energy.

This number characterizes the buoyancy vertical oscillations originating in a

stable environment as considered in meteorology. This is different to gravitational

waves arising in a gravitation field. On the turbulent kinetic energy spectrum, the

wave number separates the buoyancy band from the inertia band. The measure-

ments show a relatively small portion of this number on the spectrum.

Info: [C13].

9.1.8 Businger�Dyer Relationship NBD

NBD 5wmed

wf

51

klnz2 d

z01Ψ

z2 d

L

� �

wmed (m s21) � mean wind velocity; wf (m s21) � friction velocity; k (2) �Karman constant; z (m) � height above the ground; d (m) � distance of transloca-

tion; z0 (m) � roughness length; Ψ (2) � empirical correction; L (m) �Monin�Obukhov length.

It characterizes the similarity relation between the surface flow of a variable

quantity and the mean profile value of this variable. Micrometeorology of surface

layer. Boundary conditions.

Info: [C15].

9.1.9 Centripetal Acceleration Ac

Ac 5ac

g; where ac 5ω2R

ac (m s22) � centrifugal acceleration; g (m s22) � gravitational acceleration;

ω (s21) � angular frequency of body (planet) rotation; R (m) � body radius.

It expresses the ratio of the centrifugal acceleration, with uniform circular

motion of a body, to the gravitational acceleration. It is used for bodies in approxi-

mate hydrostatic equilibrium. Geophysics. Astrophysics.

Info: [C18].

9.1.10 Dalton Number Dal

Dal5wv

wðks 2 kvÞ5CD;evSc

223


wv (m s21) � resultant evaporation velocity; w (m s21) � wind speed above water

level; ks (2) � saturation vapour concentration; kv (2) � vapour concentration;

CD,ev (2) � drag evaporation coefficient (p. 228); Sc (2) � Schmidt number

(p. 263).

It expresses the local evaporation rate of a substance from the free surface into

the atmosphere. It characterizes the atmospheric diffusion in a thin subsurface layer

between the surface and the surrounding atmosphere. Meteorology. Geophysics.

Info: [B48].

9.1.11 Davies Number Dav

Dav5CDRe2

CD (2) � drag coefficient (p. 60); Re (2) � Reynolds number (p. 81).

It is used in meteorology to calculate the final velocity of water droplets falling

from the atmosphere to the earth. It is also called the Best number. Meteorology.

Info: [C30].

9.1.12 Dissipation Number Dn

Dn5αgLcp

α (K21) � linear thermal expansion coefficient; g (m s22) � gravitational accelera-

tion; L (m) � characteristic length (depth); cp (J kg21 K21) � specific heat

capacity.

It expresses viscous dissipation with rapidly rotating thermal convection. It fol-

lows the dynamics of the terrestrial crust, but without the influence of the magnetic

field. Geology. Geophysics.

9.1.13 Ekman Number (1.) Ek

Ek5

ffiffiffiffiffiffiffiffiv

ωL2

r5

ffiffiffiffiffiffiRo

Re

rð1Þ;

Ek5

ffiffiffiffiffiffiffiffiffiffiffiν

2ωL2

rð2Þ;

Ek5ν

2L2ω sin ϕð3Þ

ν (m2 s21) � kinematic viscosity; ω (s21) � angular frequency of planetary rota-

tion; L (m) � characteristic length (usually vertical); ϕ (2) � latitude;


2w sin ϕ (s21) � Coriolis frequency; Ro (2) � Rossby number (p. 406); Re (2) �Reynolds number (p. 81).

This number expresses the ratio of the viscosity force to the Coriolis force. It

characterizes the flow in rotating channels. In expression (2) Ek, as defined by rela-

tion (1), it is called the velocity pressure criterion. Sometimes, equation (2) is

applied. In magnetohydrodynamic processes, it expresses the viscous fluid friction

influence and the effect of a secondary carrying motion, for example, the rotation of

the Earth. In equation (3), it describes geophysical phenomena in oceans and in the

atmosphere. With a small Ek, the arising disturbances can propagate due to a weak-

ening friction effect. Magnetohydrodynamics. Geophysics of viscous friction forces.

Info: [C45].

Vagn Walfrid Ekman (3.5.1874�9.3.1954), Swedish

oceanographer and physicist.

He is among the most famous oceanographers of his

time. He observed the movement of icebergs and found that

they did not move in the wind direction, but had a deviation

of 20�40�. He published his theory of the Ekman spiral, in

which this phenomenon was explained as the equilibrium

between friction effects and the fictive forces due to the

Earth’s rotation. He examined how the steady-state winds

formed a thin layer on the ocean surface, the eponymous

Ekman layer, and defined its velocity and thickness.

9.1.14 Froude Biomechanical Number Fr

Fr5w2

gLl

w (m s21) � forward speed of animal; g (m s22) � gravitational acceleration;

Ll (m) � leg length, usually measured from coxa.

It expresses the classic Froude number (1.) Fr (p. 62), but with parameters which

correspond to the motion of a person or animal. It is an important similarity criterion

in modelling the motion of a person under conditions of reduced gravitation, for

example, of the astronauts on other planets. Biomechanics. Astrophysics.

Astronautics.

Info: [B29].


9.1.15 Geometry Erosion Parameter Pg

Pg 5

ffiffiffih

L

r

h (m) � depth of surface water; L (m) � slope surface size.


It characterizes the geometric parameters of the hydrodynamic and kinematic

processes of soil erosion. The parameter value is very small, among other things,

with respect to physical conditions connected to a channel acting with surface

water flow.

Info: [B65].

9.1.16 Golitsyn Energy Balance Number Goenergy, E

Goenergy 5qabs 1 qint

qabs5

Tef

Tekv

� �4

qabs (W m22) � absolute planet radiation; qint (W m22) � internal planet radiation;

Tef (K) � effective temperature; Tekv (K) � reference temperature.

This number expresses the thermal equilibrium between an internal heat source

and planet radiation. When Goenergy. 1.27, the internal heat source can be

expected to play an important role in controlling atmospheric dynamics.

Geophysics and astrophysics. Dynamics of the atmosphere.

Info: [B111].

Georgy Sergeyevich Golitsyn (see below).

9.1.17 Golitsyn Number Go

Go5τdynτrad

; where τdyn 5Rp

a; τrad 5

cppef

σ0gT3ef

τdyn (h) � dynamic time constant; τrad (day) � radiative time constant; Rp (m) �planetary radius; a (m s21) � sound speed; cp (J kg

21 K21) � specific heat capacity;

pef (Pa) � effective pressure characterizing the condition when hydrogen

optical depth is 1; σ0 (W m22 K24) � Stefan�Boltzmann constant; g (m s22) �gravitational acceleration; Tef (K) � effective temperature.

It characterizes the thermal inertia of the atmosphere or its structure, which is

important from an atmospheric radiative heating or cooling point of view, as com-

pared to the dynamics thereof. With Go{1, the atmosphere would be in equilib-

rium from a radiative point of view. In this case, the thermal mode plays a

controlling role over atmospheric dynamics and leads to low temperature gradients.

For Go$ 1, the global circulation requires longer time periods for the transference

of heat. The atmosphere is closed for local radiation equilibrium originating and

the dynamics is set on this thermal mode. The Golitsyn number Go is the time mea-

suring number Nτ (p. 30) for the atmosphere. Geophysics and astrophysics.

Dynamics of the atmosphere.

Info: [B111].

Georgy Sergeyevich Golitsyn, Russian astrophysicist.


9.1.18 Grouch Number Gro

Gro5wvωg

5Fr Sh21

wv (m s21) � vertical landing velocity; ω (s21) � natural frequency of mass

dynamical system during impact (incidence); g (m s21) � gravitational accelera-

tion; Fr (2) � Froude number (1.) (p. 62); Sh (2) � Strouhal number (p. 87).

It expresses the ratio of the vertical Froude number (1.) Fr (p. 62) and the verti-

cal Strouhal number Sh (p. 87). In dimensionless form, it represents the landing

velocity of a person or animal under various gravitational conditions. Astrophysics.

Astronautics.

Info: [B29].

9.1.19 Hubble Number Hu, z

Hu5δλλ

5H

cr

δλ (m) � wavelength shift of spectral lines; λ (m) � wavelength of spectral line;

H (s21) � Hubble constant; c (m s21) � speed of light; r (m) � distance of each

nebula from the Earth.

Essentially, it expresses Hubble’s law. According to it, the electromagnetic radi-

ation propagating from the Universe to the Earth has shifted towards longer

wavelengths (the so-called redshift in the visible light band) and is proportional to

the distance the radiation travelled. Together with the Doppler law, it confirms the

expansion of the Universe.

Info: [C44],[C71].

Edwin Powell Hubble (20.11.1889�28.9.1953), American

astronomer.

He was a prestigious astronomer because of his discovery

of galaxies behind the Milky Way and his cosmological sub-

stantiation of the wavelength redshift. He was the first to

argue that this shift was due to the Doppler effect, induced

by the expansion of the Universe. He was one of the leading

astronomers of his time who laid the foundation for physical

cosmology. Shortly before his death, the Hale telescope was

finished and Hubble was the first to use it.

9.1.20 Jeffreys Phase Φ

Φ52π

ðFo=2Þ13Ai

X2Fo

ðFo=2Þ13

0@

1A; where Fo5 τ

ffiffiffig

h

r; X5

x

h


τ (s) � time; g (m s22) � gravitational acceleration; h (m) � depth of ocean; x (m) �distance; Fo (2) � Fourier number characterizing tsunami propagation; X (2) �geometrical coordinates (p. 15).

It characterizes the phase shift of tsunamis. Geophysics. Oceanography.

9.1.21 Kibel Number Ki

See the Rossby number Ro (p. 406).

9.1.22 Kirpichev Number of Solar Radiation KiS

KiS 5εSLλT

ε (2) � relative absorption of solar radiation; S (W m22) � solar constant, density

of solar radiation; L (m) � characteristic length; λ (W m21 K21) � thermal conduc-

tivity of absorption environment; T (K) � temperature.

It expresses the ratio of the specific solar heat flow, transferred by the solar

radiation to the system, to the specific heat flow transferred by conduction in the

system. It characterizes the solar radiation heat flow onto the surface of a body. It

is the modified Kirpichev heat number Ki. (p. 176)

Info: [A23].


9.1.23 Kolmogorov Parameter Kol

Kol5Rgwffc

w2m

@w

@z

� �21

where R5Rw 2 R

R; wm 5

ffiffiffiτR

r; 0#Kol, 1

R (2) � relative density; g (m s22) � gravitational acceleration; wff (m s21) � free

fall velocity of droplets in air; c (2) � droplet concentration in thin layer;

wm (m s21) � shear stress velocity; @w@z ðs21Þ � gradient of flow velocity in the

z axis; w (m s21) � mean flow velocity at water level; Rw (kg m23) � water den-

sity; R (kg m23) � air density; τ (Pa) � shear stress.

For the suspension of droplets, it expresses the ratio of the energy dissipation

rate to the turbulent energy generation rate. The acceleration rate and the creation

of water spray connected with it on the surface of oceans is an example. The

Coriolis effect and the effect of cooling act with this, causing water droplets to be


formed and to evaporate. It is a specific analogy of the Kolmogorov number Kol

(p. 104). Geophysics. The physics of water spray generation in ocean surf.

Info: [C75].

Andrey Nikolayevich Kolmogorov (p. 104).

9.1.24 Langmuir Turbulent Number Lat

Lat 5

ffiffiffiffiffiuf

uS

r

uf (m s21) � water friction velocity; uS (m s21) � Stoke’s drift velocity.

It expresses the ratio of the friction force to the Stokes drift velocity. It charac-

terizes the turbulent process in the ocean surface layer with this process causing

mixing in a layer which becomes deeper due to Langmuir circulation. The

Langmuir turbulence has various characteristics, depending on the wind acting at

the surface. Geophysics.

Info: [C35].

Irving Langmuir (p. 345).

9.1.25 Lochtin Number Lo

Lo5Q

d2ffiffiffiffiffiffiffiffigdL

p

Q (m3 s21) 2 volume flow; d (m) � size of particles; g (m s22) � gravitational

acceleration; L (m) � characteristic length.

It characterizes the steady-state natural river flow which depends on the size of

particles of sediments, forming the river bed, and on the bottom inclination and

flow. It determines the width and depth. It is analogous to the Froude river inflow

number Frin (p. 417).

Info: [B52].

V.M. Lochtin.

9.1.26 Love Number Lov, k

Lov5Vred

Vtide

r

R

� �3

ð1Þ; Lov5ΔVred;2

Vtide

r

R

� �3

ð2Þ;

Lov53

211

19w2s

2gr

� �21ð3Þ; Lov � 3

19

gr

w2s

ð4Þ;


Vred (m2 s22) � deformation potential due to influence of tidal mass redistribution;

Vtide (m2 s22)2 tidal potential; ΔVred,2 (m2 s22) � increase of deformation poten-

tial at excitation8; r (m) � local radius of planet; R (m) � outer radius of planet;

ws (m s21) � shear speed dependent up to tidal excitation; g (m s22) � gravitational

acceleration.

The Love numbers characterize the influence of the Moon’s rotation around the

Earth and its periodic action on oceans (ebb and flow) and on the atmosphere and

rigid surface of the Earth. Due to the tide, this surface is deformed alternatively

and changes the gravitational field. As a result, the tide potential is created, which

differs on opposite sides of the Earth and causes two different shifts. The Love

number expresses the ratio of the vertical shift, leading to surface heightening and

to the tide potential. The expressions (1)�(4) are valid for homogeneous and

incompressible planets. For the Earth, it is Lov, 10 and for the Mars Lov. 10.

The expression (4) gives an approximate value. Geophysics. Mineralogy.

Info: [C81].

Augustus Edward Hough Love (17.4.1863�5.6.1940),

English mathematician.

He became famous by his work on the mathematical

elasticity theory. He was also engaged in the wave propaga-

tion and elaborated the mathematical model of terrestrial

surface waves which are known as the Love waves. The

book Some Problems in Geodynamics is among his most

popular works. Also important is his two-volume work, A

Treatise of the Mathematical Theory of Elasticity.

9.1.27 Moment Magnitude Scale Mw, Me, Mn

Mw 52

3log10

M0

ðN mÞ 29:1

� �ð1Þ;

Me 52

3log10

Es

ðN mÞ 22:9 ð2Þ;

where Es 5 1024:8M0 5 1:63 1025M0;

Mn 52

3log10

mtnt

ðkgÞ ð3Þ pro mtnt½ �5 kg;

Mn 52

3log10

mtnt

ðktÞ ð4Þ pro mtnt½ �5 kt;

Mn 52

3log10

mtnt

ðMtÞ ð5Þ pro mtnt½ �5Mt;


M0 (N m) � seismic moment of the total amount of energy that is transformed dur-

ing an earthquake; Me (N m) � small fraction of the moment M0 converted into

radiation seismic energy; Mn (N m) � comparative nuclear detonation moment;

Es (J) � radiation seismic energy; mtnt (kg, kt, Mt) � mass of explosive TNT serv-

ing for comparison.

The expression (1) represents the successor of the well-known Richter’s scale. It

serves to compare the energy loosened during the earthquake. At this logarithmic

scale, the rise by 1 degree corresponds to the loosened energy increase multiplied

by 101.55 31.65. The degree 2 corresponds to 1035 1000 times greater energy.

The expression (2) shows a part of the energy, released during the earthquake, in

the form of the radiation seismic energy. Expressions (3) and (4) present the com-

parison with nuclear detonations of the trinitrotoluene in kg, kilotons and

megatons.

Info: [C92].

9.1.28 Pratt Number Pra

Pra5Ls

Ll5

τwLl

Ls (m) � step length; Ll (m) � leg length, usually measured from the coxa; τ (s) �time duration of step; w (m s21) � forward speed of animal motion.

Together with the Froude biomechanical number Fr (p. 396), it expresses the

fundamental similarity criteria in modelling human motion under reduced gravity

conditions, such as in the motion of astronauts on other planets. Biomechanics.

Astrophysics. Astronautics.

9.1.29 Redshift Number Z

See the Hubble Number Hu (p. 398).

9.1.30 Richardson Bulk Number Rib

Rib 5gT 21

V ΔTVΔz

ðΔw1Þ2 1 ðΔw2Þ2ð1Þ; Rib 5

gΔRh

R0ðΔwÞ2 ð2Þ

g (m s22) � gravitational acceleration; TV (K) � absolute virtual temperature;

ΔTV (K) � virtual potential temperature difference across a layer of thickness Δz;

h�Δz (m) � layer thickness; Δw1, Δw2 (m s21) � changes in horizontal wind

components across the same layer; ΔR (kg m23) � change of density across a

layer; R0 (kg m23) � initial density; Δw (m s21) � change of resulting horizontal

wind component across a layer.


In meteorology, it represents the relation between the vertical flow stability and

the vertical slip. A high value indicates the instability and a low one shows stabil-

ity. The value RibAh50; 100i represents advantageous development. It is the differ-

ence analogy of the Richardson gradient number Rig (p. 384) and expresses the

approximation of local gradients by finite differences across a layer. Both numbers

characterize a slip instability model which expresses mixing. It is used to simulate

daily fluctuations in the depth of an instable layer and to determine the flow pro-

files, for example from subtropical regions. A complete mathematical model also

involves the influence of thermal flows and wind tension on layer instability. The

number Rib itself expresses the approximation of local gradients by finite differ-

ences across a layer. It is used in expression (1) or (2). The mathematical difference

model works, for example with the so-called surface box, and with the slip being

greater than the stratification density necessary to support it (Rib. 0.55), the prop-

erties of both adhering boxes are averaged because both have been stirred. This

process drops continuously up to Rib. 0.65. Meteorology.

Info: [C12].

Lewis Fry Richardson (11.10.1881�30.9.1953), English

mathematician, physicist, meteorologist, psychologist and

pacifist.

His interest was concentrated on meteorology and, above

all, on the design of a mathematical model, using differen-

tial equations (1922), to predict the weather, which is still

used today. He was also engaged in atmospheric turbulence

and executed many terrestrial experiments. The dimension-

less parameters of turbulence theory are named after him.

9.1.31 Richardson Flux Number Rif

Rif 5gT 21

VP wTV

uwðrUÞ1 vwðrVÞ

g (m s22) � gravitational acceleration; TVP (K) � virtual potential

temperature (theoretic temperature of dry air having the same density as wet air);

TV (K) � virtual temperature (coming from state equation of dry air for wet air

except substitution of T for TV); U, V (m s21) � horizontal Cartesian wind compo-

nents from the west and from the east; rU, rV (s21) � vertical wind velocity gra-

dients @U=@z; @V=@z; u, v, w (m s21) � axial components of wind velocity.

It expresses the ratio of the buoyancy force limit of turbulent kinetic energy to

the negative slip limit. It is the measure of the dynamic instability describing the

flow capability to stay turbulent in case of a slip wind and certain static stability.

In the expression, the denominator is usually negative. Therefore, with a positive

numerator, the Rif is negative for statically instable flow. For Rif, 1, the flow is

dynamically instable and turbulent. For Rif. 1, the flow is dynamically stable, the


turbulence tends to decrease and, according to the number Rif, the flow cannot be

defined. Meteorology.

Info: [C12].

Lewis Fry Richardson (see above).

9.1.32 Richardson Gradient Number Rig

Rig 5gT 21

A rTPðrw1Þ2 1 ðrw2Þ2

ð1Þ; Rig 5 g@R@z

R210

@w

@z

� �22

ð2Þ

g (m s22) � gravitational acceleration; TP (K) � virtual potential temperature;

TA (K) � absolute virtual temperature; w1, w2 (m s21) � wind components towards

the east and north; R (kg m23) � air density; R0 (kg m23) � initial density; z (m) �height of the layer.

This number is used in expression (1) or (2). It expresses the capability of gener-

ation or dissipation of the turbulence, divided according to direction, in the slip

development thereof. It is used to determine the dynamic stability and formation of

turbulence. For Rig less than the Richardson critical number (approximately

Rig# 0.25), the air becomes dynamically instable and turbulent. The origin of hys-

teresis can be considered too, with which the laminar flow must sink under the

above-mentioned critical value in order to pass to a turbulent flow. However, turbu-

lent flow can exist up to Rig5 1.0, provided it was primarily laminar before.

Meteorology.

Info: [C12].


9.1.33 Richardson Meteorology Number (1.) Rim

Rim 5gαrTðrwÞ2

g (m s22) � gravitational acceleration; α (K21) � linear thermal expansion coeffi-

cient; rT 5 @T=@z (K m21) � temperature gradient expressing vertical stability;

rw5 @w=@z (m s22) � velocity gradient expressing characteristic vertical wind

shear.

It expresses the buoyancy-to-inertia forces ratio or, alternatively, the capability

to suppress turbulence in the formation of the turbulence slip. It is used as a stabil-

ity measure to determine whether turbulence arises. Meteorology.

Info: [C12].



9.1.34 Richardson Meteorology Number (2.) Ri

Ri5gLΔRRw2

5Gr Re22 ð1Þ; Ri5gαΔTL

w2ð2Þ

g (m s22) � gravitational acceleration; L (m) � characteristic length; ΔR (kg m23)

� density difference; R (kg m23) � density; w (m s21) � flow velocity; α (K21) �linear thermal expansion coefficient; ΔT (K) � temperature difference; Gr (�) �Grashof heat number (p. 185); Re (�) � Reynolds number (p. 81).

It expresses the ratio of buoyancy effects to vertical slip effects. With Ri{1, the

buoyancy influences the turbulence with forced convection. For natural convection,

it is Ric1. At Ri�1, it expresses mixed convection. The value of the number

Ri5 1 is known as the Monin�Obukhov length L (p. 21). Mechanics of fluids.

Meteorology.

Info: [C12].


9.1.35 Richardson Overall Number Ri

Ri5g0Lw2

g0 (m s22) � reduced gravitational acceleration; L (m) � characteristic length;

w (m s21) � velocity imposed by the boundary condition of the problem.

In the above-mentioned expression, this number presents a basic view of its

physical meaning only. A more detailed description is in the following Richardson

numbers.


9.1.36 Rossby Atmospheric Number (1.) Roatm

Roatm 5τrotτdyn

Ba

Rpω

τrot (h) � time period of rotation; τdyn (h) � dynamic time constant; a (m s21) �sound speed; Rp (m) � planet radius; ω (s21) � angular frequency of planet

rotation.

It expresses the importance of planet rotation velocity (the Coriolis force) to cre-

ate atmospheric motion. Alternatively, it expresses the ratio of the rotation period

to the dynamic time scale. With Roatm, 1, the Coriolis forces play an important

role in atmospheric motion control.

Carl Gustaf Arvid Rossby (see below).


9.1.37 Rossby Atmospheric Number (2.) Ro

Ro5τrotτconv

5w

ωH; where τconv 5

H

w; τrot 5

1

ω

τrot (h) � time period of rotation; τconv (h) � time period of convection; w (m s21) �vertical velocity in the convective layer; ω (s21) � angular frequency of planet

rotation; H (m) � atmospheric height at static equilibrium.

This number characterizes the rotation acting on the vertical motion which

causes buoyancy. Therefore, it is also called the Rossby buoyancy number. With

Ro, 1, the planetary rotation causes the deviation of moving objects.


9.1.38 Rossby Concentration Number RoC

RoC 5 βCΔC

βC (M21) � volume ion concentration coefficient11; ΔC (M) � ion concentration

difference between two electrodes12.

It expresses the change of the ion concentration in the atmosphere or other sys-

tems. Geophysics. Meteorology.


9.1.39 Rossby Heat Number RoT

RoT 5βΔT

β (K21) � volume thermal expansion coefficient; ΔT (K) � temperature difference

between surfaces of warm and cold systems.

It expresses the thermal dilatability of solid, liquid or gaseous materials.

Mechanics. Geophysics. Meteorology.


9.1.40 Rossby Number Ro

Ro5w

2ωL sin αð1Þ; Ro5

w

ωLð2Þ

w (m s21) � fluid flow velocity; ω (s21) � angular frequency; L (m) � characteris-

tic length; α (�) � angle between the axis of rotation and the direction of fluid

motion.


It expresses the ratio of the fluid inertia force to the Coriolis force in the atmo-

sphere or in an ocean, based on the planetary rotation. It characterizes the influence

of the rotation of the Earth on the flow in channels and pipelines. It describes the

geophysical phenomena in oceans and in the atmosphere. With a large number Ro,

usually Ro. 1, for example in the tropics or lower geographical latitudes, the plan-

etary rotational influence is insubstantial and can be neglected. If the Ro number is

small (Ro, 1), this influence cannot be neglected. It is also used in expression (2).

It is also known as the Kibel number (p. 399).

Info: [A21],[B20].

Carl Gustaf Arvid Rossby (28.12.1898�19.8.1957),

Swedish-American meteorologist and oceanographer.

He was the first to clarify the wide movements of the ter-

restrial atmosphere by using fluid mechanics. He introduced

a mathematical description (model) for these movements.

He was engaged in atmospheric thermodynamics, in mixing

and turbulence, and in the ocean�atmosphere interaction.

He identified and characterized nozzle flowing and Rossby

waves in the atmosphere. After the Second World War, he

modified his previous mathematical model of atmospheric

dynamics for the computer and weather predictions.

9.1.41 Rossby Temporal Number Roτ

Roτ 52πτω

τ (s) � time, time of the one rotation period; ω (s21) � angular frequency of the

Earth’s rotation.

It compares the local velocity changes in the atmosphere to the Coriolis force.

Geophysics. Meteorology.

Info: [C48].

Carl Gustaf Arvid Rossby (see above).

9.1.42 Seismic Efficiency η

η5Erad

Etot

5Δσ2σ

; where σ5σ1 1Δσ2

Erad (J) � radiated seismic energy; Etot (J) � total loading energy; Δσ (Pa) �difference of stresses before and after dislocation; σ (Pa) � average stress during

dislocation; σ1 (Pa) � stress after dislocation.

It expresses the ratio of the radiated seismic wave energy during earthquakes to

the total energy. Alternatively, it also expresses the ratio of the tension increment

to the mean tension value. Geophysics. Earthquakes.

Info: [C41].


9.1.43 Shida Number Shi, l

Shi5uλg sin λR gradλV

ð1Þ; Shi5uϑg

R gradϑVð2Þ

uλ, uϑ (m) � components of horizontal shift; g (m s22) � gravitational accelera-

tion; λ, ϑ (�) � shift angles; R (m) � outer Earth radius; V (m2 s22) � tide

potential.

Like the Love number, it characterizes the influence of the moon’s rotation

around the Earth, which causes the action of the tide on the surface deformation

thereof. The number expresses the ratio of the angles of horizontal shifts with two

components of the tide potential gradient.

9.1.44 Smith Erosion Number Sm

Sm5εwL

ε (�) � scale factor, usually ε{1; w (m s21) � mean effective velocity of rain,

incident to Earth surface; τ (s) � time; L (m) � characteristic length of a surface,

vertical or horizontal.

It characterizes the erosion time during which measurable changes occur on the

soil surface. Especially, it is used with physical modelling and analysis of land-

scape terrain changes. Geophysics. Morphology. Ecology.

Info: [B65].

9.1.45 Strouhal Biomechanical Number St

St5wvL

21l

wn

ð1Þ; St5whL

21l

ws

ð2Þ

wv (m s21) � vertical velocity of landing; Ll (m) � length of leg, usually measured

from coxa; ωn (s21) � natural frequency; wh (m s21) � horizontal speed at jumping

motion; ωs (s21) � jumping frequency at horizontal motion.

It is a modification of the basic Strouhal number Sh (p. 87) for the dynamics of

vertical motion with landing (1) or with horizontal spring motion (2). Astrophysics.

Astronautics.

Info: [B29].

Vincenc Strouhal (p. 87)


9.1.46 Tidal Horizontal Displacement Number Nhor

Nhor 5uϑ

R5

Shi

g

@V

@ϑð1Þ;

Nhor 5uλ

R5

Shi

g sin λ@V

@λð2Þ;

uϑ, uλ (�) � components of horizontal displacement; R (m) � outer Earth radius;

g (m s22) � gravitational acceleration; V (m2 s22) � tide potential; ϑ, λ (�) �angles of deviation; Shi (�) � Shida number (p. 408).

It expresses the horizontal displacement components caused by the tidal redistri-

bution of the terrestrial mass.

Info: [C81].

9.1.47 Tidal Radial Displacement Number Nrad

Nrad 5ur

R5

Lov2Vtide

g

ur (�) � radial displacement; R (m) � outer Earth radius; Vtide (m2 s22) � tide poten-

tial; g (m s22) � gravitational acceleration; Lov2 (�) � Love number.

It expresses the radial displacement caused by the tidal redistribution of the

mass of the Earth.

Info: [C81].

9.1.48 Tide Height Ratio H, h

H5 Lov5 Shi5 0 ð1Þ; H5 Shi5 1 ð2Þ;Hn 5 11 Lovn ð3Þ

Lov (�) � Love number (p. 400); Shi (�) � Shida number (p. 408); Lovn (�) �Love number (p. 400) at excitation potential of nth degree (here n5 2).

It expresses the ratio of the main tide height to the static height of the sea tide.

The expression (1) holds for a fully rigid body, the expression (2) for a liquid body

and the expression (3) for real body. If they have equal density, then Lov2 5 3=2and H2 5 5=4 are valid. Geophysics.

Info: [C81].

9.1.49 Tsunami Frequency Number Ω

Ω5ω

ffiffiffih

g

sð1Þ; Ω2 5K tanh K ð2Þ; where K5 kh


ω (s21) � angular frequency; g (m s22) � gravitational acceleration; h (m) � char-

acteristic length, depth of flat ocean; k (m21) � dimension wave number; K (�) �wave number (p. 33).

In expression (1), it represents the tsunami propagation frequency on the ocean

surface. In expression (2), it represents the dispersion relation for created gravita-

tional waves. Geophysics. Oceanography.

Info: [B18].

9.1.50 Tsunami Velocities C, W

C5cffiffiffiffiffigh

p 5 12K2

6ð1Þ;

W 5wffiffiffiffiffigh

p 5 12K2

2ð2Þ

c, w (m s21) � velocity propagation of water wave; g (m s22) � gravitational accel-

eration; h (m) � sea depth; K (�) � wave number (p. 33).

It expresses the approximate velocity, provided the sea depth is constant and the

velocity and phase approximation is quadratic. In addition, it is valid for long-wave

tsunamis (K{1). Geophysics. Oceanography.

9.1.51 Wesson Coupling Constant β

β5G

pcB33 1023Bα

G (m3 kg21 s22) � Newtonian constant of gravitation; p (kg21 m2 s21) � Wesson’s

empirical parameter expressing the ability of planets, stars and galaxies to super-cluster-

ing; c (m s2 1) � speed of light; α (�) � electromagnetic coupling constant (p. 306).

It is an analogy to the gyromagnetic ratio, but is more suitable from the general

relativity point of view. Astronomy. Astrophysics.

9.2 Ecology and Biology

Above all, ecology, as a multidisciplinary field, is based on biology and other

branches such as physics, chemistry, meteorology, geology and other technical as

well as non-technical fields. The increasing significance of ecology is demonstrated

in a lot of dimensionless quantities. They involve general ecology as well as


industrial ecology, ecology of seas, landscapes and forests, and global ecology.

Among the dimensionless quantities, the Brown number for atmospheric pollution,

the Burger number for the influence of the earth’s rotation on the water movement

in water systems, and the Imberger number to express the tendency of a surface

water layer to be mixed are presented. Several dimensionless quantities relate to

exergy and industrial ecology. As for biology, the following numbers are presented:

the sperm number for viscous and elastic tensions in microbiologic fibres, the radi-

ation weighting factor and the breakup wavenumber expressing the influence of

biological heterogeneity on intercellular relations in wave propagation in cardiac

tissue. A survey of fundamental dimensionless quantities for biological systems,

with more detailed comments on their formulation is in [B95]. In addition to the

dimensionless quantities in biology, microbiology, and agrobiology, these quanti-

ties are presented for pulmonology and botany as well.

9.2.1 Absorbed Agrobiological Dose Ndose

Ndose 5RedE2γτ

where edBE2γτR

for ω{ωcr; edBE2εR

for ωcωcr

R (kg m23) � density of soil; ed (J kg21) � specific energy; E (V m21) � electric

field intensity; γ (S m21) � specific electrical conductance; τ (s) � time; ω (s21) �frequency of supply; ωcr (s

21) � critical frequency; ε (F m21) � permittivity.

Together with other dimensionless quantities, it characterizes both the capability

of soil to absorb electric field energy and the efficiency of the delivered energy. It

is used in agricultural soil refining technology, based on the electric field method,

enabling inactivation of various harmful microorganisms in the soil. Agrobiology.

Info: [B113].

9.2.2 Association Constant kA

kA 5konC0

koff

kon (m3 mol21 s21) � kinetic speed constant of direct reaction; C0 (mol m23) �volume concentration of the target DNA in solution; koff (s21) � kinetic speed

constant of reversed reaction.

It expresses the ratio of the maximum direct biochemical reaction rate to the

maximum reversed reaction rate. It characterizes the biochemical processes, for

example, in DNA microorganisms in the hybridizing process and diffusion and


convective motions. In kA{1; it corresponds to the equilibrium relative molar sur-

face concentration of the formation of hybrid pairs. Microbiology. Microorganisms.

Info: [B86].

9.2.3 Bond Natural Number Bo

Bo5RghL2σ

R (kg m23) � sap density; g (m s22) � gravitational acceleration; h (m) � height in

capillary; L (m) � characteristic length, capillary radius; σ (N m21) � surface

tension.

It expresses the ratio of the gravitational force to that of the surface tension

force in the sap movement in plants. It characterizes this movement originating

from water loss as a result of evaporation from leaves. In the trunk or stem of a

plant, the radius of capillary tubes equals several tens of micrometres, and the

radius of leaves equals several tenths of micrometres. Biomechanics. Botany.

Info: [A49].

Wilfrid Noel Bond (1897�1937), English physicist.

9.2.4 Breakup Wave Number NS

NS 5 τr

ffiffiffiffiffiffiffiffiSRD

τ

r

τr (s) � enhanced time of output phase; SR (m22) � density downtrend per unit

area; D (m2 s21) � diffusion coefficient; τ (s) � transfer time rising between

inhomogeneities.

This number expresses the heterogeneity of a biological system and the proper-

ties of a medium capable of being excited. The isolated heterogeneity causes trans-

fer wave decomposition. Microbiology. The influence of heterogeneity and

intercellular coupling on wave propagation in the heart tissue.

Info: [B104],[B108].

9.2.5 Brown Number Bro

Bro5

ffiffiffiffiffiffiffiffiwp

2

wf2

s5

wp0

wx0

�� ð1Þ;

where wf0�� 5

ffiffiffiffiffiffiffiffiffiffi8kT

πmf

;

r


w51

2cfλ ð2Þ;

where λ51ffiffiffi

2p

πnd2m5

kTffiffiffi2

pπd2mp

wp ;wf (m s21) � mean velocity of particles (p) and fluid (f); wp0;wf

0 (m s21) � mean

temperature fluctuation of the velocity of particles (p) and fluid (f); k (J K21) �Boltzmann constant; T (K) � absolute temperature; mf (kg) � mass of fluid;

w (m s21) � velocity of fluid; cf (m s21) � sound velocity in fluid; λ (m) � mean free

path of molecules in gas; n (cm23) � numerical density of molecules in gas; dm (m) �diameter of gas molecule; p (Pa)� pressure in fluid.

Most frequently, it characterizes aerosols as suspensions of solid or liquid parti-

cles in a fluid and in the air. Dust, smoke, smog and mist are common aerosols. It

describes a colloid system in which the dispersed phase is represented by sprayed

solid or liquid particles. Usually, the radius of aerosol particles is in the range of

0.01�10 μm. The Brown number represents an important ecological criterion

expressing atmospheric pollution conditions, among other things. Ecology.

Meteorology.

Info: [A0].

Robert Brown (21.12.1773�10.6.1858), English botanist.

Above all, he dedicated himself to botany. In Australia,

his research over many years led him to discover roughly

2000 hitherto unknown plants. He edited a whole range of

works about the results of his Australian flora research. He

became most famous for his microscopic observations of

the random movement of dust pollen flecks and for formu-

lating the law of this movement. To honour him, this move-

ment has been named Brown movement. He also

discovered and described vegetal cells’ nuclei.

9.2.6 Burger Number, Si Number Bur

Bur5fBLV

ωLh5

Ro

Fr

� �2

5Rr

L

� �2

fB (s21) � buoyancy frequency; LV (m) � vertical depth of flow; ω (s21) � inertial

angular frequency of the Earth; Lh (m) � characteristic horizontal length of flow;

Rr (m) � Rossby deformation radius; L (m) � characteristic length; Ro (2) �Rossby number (p. 406); Fr (2) � Froude number (1.) (p. 62).

It is also called the Si number. It expresses the influence of the earth’s rotation

on the motion in a lake or other large water basin. With its value less than the criti-

cal one (Bur 5 1), the waves have the character of Kelvin or topographic waves.


For Bur , 1, the earth’s rotation influences the dynamics and the waves have an

oscillating character with most of their energy in the form of kinetic energy. The

Burger number characterizes the atmospheric or oceanographic flow expressed by

the ratio of the vertical stratification density to the horizontal terrestrial rotation.

For Bur-0, the rotational flow predominates, whereas the vertical stratification

does for Burc0: Geophysics. Ecology. Meteorology.

Info: [B52],[C14].

Alewyn Burger, mathematician and meteorologist.

9.2.7 Crop Coefficient KC

KC 5ETC

ET0

ETC (mm day21) � crop evapotranspiration; ET0 (mm day21) � reference crop

evapotranspiration.

It expresses the ratio of the real evaporation rate from plants and the soil under

them to the relative rate, generally determined by calculation according to the

Penman�Monteith equation for grass. The coefficient value is used to determine

the amount of water in soil irrigation. The evaporation coefficient value is usually

within the range KCAh0.1; 1.5i. Ecology. Botany. Agriculture.Info: [C26].

9.2.8 Damkohler Biological Number Dabiol

Dabiol 5konHmaxh

Dmol

kon (m3 mol21 s21) � constant of kinetic speed of direct reaction; Hmax (mol m22) �

molar surface concentration freely connecting the position of specific places on the

probe; h (m) � height of liquid layer; Dmol (m2 s21) � molecular diffusion

coefficient.

It expresses the ratio of the maximum direct reaction rate to the maximum nor-

mal diffusion rate. It characterizes biochemical processes, for example, those in

DNA microorganisms related to the hybridizing process and diffusion and convec-

tive motions. In the case of convective hybridization ðDabiol{1Þ; the hybridizing

time dependencies (in the probe covering place) are almost identical to the average

value of the relative molar superficial concentration of created hybrid pairs on the

probe surface. For Dabiol$ 1, pure diffusive hybridization occurs. Microbiology.

Microorganisms.

Info: [B86].



9.2.9 Deep Parameter Pd

Pd 5

ffiffiffih

L

r

h (m) � depth of surface water; L (m) � characteristic length of surface gradient.

It expresses the relative depth of the surface water on an observed surface sec-

tion. Geophysics and geomorphology. Ecology.

Info: [B65].

9.2.10 Depletion Number Dp

Dp 5εdεl

5DpðΨ ;Φ;ΩÞ

εd (kW, MJ year21) � exergy depletion rate; εl (kW, MJ year21) � exergy loss rate,

exergy destruction; Ψ (2) � exergy cycling fraction number (p. 415); Φ (2) � exergy

efficiency number (p. 416); Ω (2) � renewed exergy fraction number (p. 422).

It expresses a dimensionless indicator of exergy consumption for the loss exergy

unit. The depletion number Dp is the function of three dimensionless indicators:

the exergy cycling fraction number Ψ (p. 415), the exergy efficiency number Φ(p. 416) and the renewed exergy fraction number Ω (p. 422).

Info: [B25].

Catherine Preston Koshland (p. 416)

9.2.11 Exergy Cycling Fraction Number Ψ

Ψ 5 rm rq 5mRΔε1mCΔε2

; where rm 5mR

mC

; rq 5Δε1Δε2

rm (2) � mass recovery factor; rq (2) � quality recovery factor; mR (kg s21) � mass

flux of resources recovered from output of consumption process; mC (kg s21) �mass flux of consumption process; Δε1 (kJ kg21) � specific exergy difference of

recovered resources; Δε2 (kJ kg21) � specific exergy difference of consumption

process.

It expresses the ratio of the velocity of the exergy return to sources to that trans-

ferred from the sources. Ecology.

Info: [B25].

Catherine Preston Koshland (see page 416).


9.2.12 Exergy Efficiency Number Φ

Φ5ε1 2 ε2

ε1

ε1 (kW, MJ year21) � exergy removal rate; ε2 (kW, MJ year21) � exergy loss rate.

It expresses the ratio of the difference between transferred and lost exergies and

converted exergy or, alternatively, the ratio of the exergy, distributed into other

sources, to the transferred one. Ecology.

Info: [B25].

Catherine Preston Koshland (born 11.5.1950), American

engineer and ecologist.

She concentrates mainly on the problems of combustion

and arising emissions. Among other things, she is engaged

in research related to reducing the influence of temperature,

oxygen and other substances on the environment, especially

in relation to older industrial facilities, such as incinerators

and incineration furnaces, among others. She is engaged in

work related to increasing plant efficiency and to reducing

atmospheric contamination. She investigates the areas that

are the origins of toxic industrial by-products. Above all,

she tries to optimize industrial systems to achieve signifi-

cant reductions of emissions.

9.2.13 Foppl�von Karman Number NFK, γ

NFK 5EL2

k

E (Pa) � modulus of elasticity; L (m) � characteristic length (radius of roundel,

virus etc.); k (Pa m2) � flexural rigidity.

In the case of Newtonian fluids, it characterizes, for example, the behaviour of

the cytoplasm of erythrocytes flowing through a circular cross section of a capillary

tube. It also characterizes changes in spherical swirl walls, among other things. Its

value is NFKAh103; 108i for elastic walls. Medicine. Biology.

Info: [B82],[B68].


9.2.14 Froude Dam-Breach Number Frdb

Frdb 5qdffiffiffiffiffiffigV

p


qd (m2 s21) � maximum outflow rate for unit width of breach; g (m s22) � gravita-

tional acceleration; V (m3) � volume outflow from hydrograph.

It characterizes the outflow wave propagation from a breached dam.

Geophysics. Ecology. Floods.

Info: [B89].


9.2.15 Froude River Inflow Number Frin

Frin 5wiffiffiffiffiffiffiffig0R

p 5Qiffiffiffiffiffiffiffig0R

pAi

; where g0 5ΔRiR0

g

wi (m s21) � inflow velocity; g0 (m s22) � corrected gravitational acceleration;

R (m) � hydraulic radius of the underflow; Qi (m3 s21) � inflow volume rate;

Ai (m2) � inflow area; ΔRi (kg m23) � density difference between inflowing river

water and surface lake water; R0 (kg m23) � average density of water in the lake;

g (m s22) � gravitational acceleration.

It characterizes the river water inflow into a lake or similar water system. It is

an important hydrodynamic criterion in describing solid particle sedimentation. The

critical value is Frin5 1. For Frin, 1, inflow strokes of discharge occur.

Geophysics. Hydrodynamics. Ecology.

Info: [B52].


9.2.16 Froude River Outflow Number Frout

Frout 5QoutffiffiffiffiffiffiffiffiffigH5

pQout (m

3 s21) � outflow volume rate; g (m s22) � gravitational acceleration; H (m) �total depth of lake.

It characterizes the lake water outflow into a river or other water system. It is

often applicable as an important hydrodynamic criterion in describing solid particle

sedimentation and dispersion. The critical value is Frout5 1. Geophysics.

Hydrodynamics. Ecology.

Info: [B52].


9.2.17 Heating Agrobiological Temperature Qheat

Qheat 5cpTτa


cp (J kg21 K21) � specific heat capacity; T (K) � temperature; τ (s) � time;

a (m2 s21) � thermal diffusivity of ground.

It characterizes temperature penetration into soil which depends on specific ther-

mal capacity and diffusivity, and on action time. It is used in the cultivation of

agricultural soil using electrical field technology.

Info: [B113].

9.2.18 Human Inhalation Number Ninh

Ninh 5DQ

m5

D

Dexp

D (kg m23) � personal dose or exposition to concentration; Q (m3 s21) � volume

air flux in input; m (kg s21) � mass flux intensity of polluted gas; Dexp (kg m23) �

fully mixed gas concentration.

It expresses the ratio of the inhaled concentration dose to the fully mixed gas

concentration. Pulmonology. Ecology.

Info: [B51].

9.2.19 Imberger Number Im

Im5h

L012

1

Rk

ð12 e2Rk Þ�

1h

L1; where Rk 5 kh

h (m) � reference depth (e.g. daily thermocline); L0, L1 (2) � Monin-Obukhov

length for solar wave radiation and for remaining heat fluxes at the lake

surface (p. 21); Rk (2) � function of surface water mixing; k (s21) � damping

coefficient.

On a water surface, it expresses the surface water tendency to mix. The critical

value is Im5 0.2. With Im5 0.3, the tendency appears to cause stratification and

mixing at the surface. Geophysics. Ecology.

Info: [B52].

Jorg Imberger (born 10.9.1942), Australian engineer and

limnologist.

He is engaged in research on fluid dynamics, especially

in mixing and streaming in lakes, at river mouths and at the

sea coast. His work consists of designing computer models

to investigate the movement of water with biochemical

activity. It concerns interactions between physical, biologi-

cal and chemical processes. With the range of his work and

the standard of its results, he has contributed markedly

to the improvement of the environment in many countries

of the world.


9.2.20 Internal Mixing in the Metalimnion RS

RS 5τsτi

5h2

h2 2 h1Lak; where h1 , h2

h1 (m) � depth of the upper layer; h2 (m) � depth of the lower layer; τs (s) � time

of internal wave period for reserved water system (e.g. lake or pond); τi (s) � time

of free wave period; Ts (K) � temperature of surface water layer; Ti (K) � temper-

ature of lower water layer; Lak (2) � lake number (p. 420).

It expresses the time of one strong wave period. With h2. (h2�h1) and

Ts/Ti, 1, shear instability may be expected with increased mixing in the top part

of the fluid surface. With h2/(h1�h2). 1 and Ts/Ti, 1,26 vehement oscillation

occurs (Lak, 1), which is connected to bottom water rising to the surface. The

critical value is RS5 1. Geophysics. Ecology.

Info: [B52].

9.2.21 Joule Agrobiological Number Joargo

Joagro 5L2E2γλT

L (m) � characteristic length; E (V m21) � electric field intensity; γ (S m21) �specific electric conductance; λ (W m21 K21) � thermal conductivity; T (K) �temperature; ω, ωcr (s

21) � supply and critical frequency of electric field.

It expresses the ratio of the total energy of a low frequency ðω{ωcrÞ electricfield to the generated heat. It describes the amount of energy dispersed during the

process of soil cultivation using the electric field method, which deactivates various

detrimental microorganisms in the soil. Agrobiology.

Info: [B113].

James Prescott Joule (p. 298).

9.2.22 Knudsen Dispersion Number Kn

Kn52λd

54Ma

Re

λ (m) � mean free path; d (m) � particle diameter; Ma (2) � Mach dispersion

number (p. 421); Re (2) � Reynolds dispersion number (p. 423).

This number expresses the ratio of the mean free path of molecules to the char-

acteristic length (i.e. a particle diameter). It is a modification of the Knudsen num-

ber (1.) Kn (p. 69) for colloidal systems with dispersed solid or liquid particles.

Ecology. Meteorology.

Info: [A0].


Martin Hans Christian Knudsen (15.2.1871�27.5.1949),

Danish physicist and oceanographer.

He was engaged especially in research related to statisti-

cal mechanics, continuum mechanics and fluid mechanics.

Above all, he is known because of his studies in molecular

gas flow and for the development of the Knudsen cham-

bers, which is the basic component of molecular epitaxial

systems. The problems connected with the Knudsen number

are related, for example, to dust particle movement in the

atmosphere, to satellite movement and to the aerodynamics

of aircraft wings.

9.2.23 Lake Number Lak

Lak5SSchðH2 htÞ

w2A32LðH2 hV Þ

SSch (m5 s22) � Schmidt stability; H (m) � total depth of the lake; ht (m) � depth

of diurnal thermocline; w (m s21) � water shear velocity due to wind; AL (m2) sur-

face area of the lake; hV (m) � height of water centre.

It expresses the ratio of the acting forces prohibiting mixing and the forces

delivering the energy for system mixing. In the expression, the first number (SSch)

expresses the stability, which depends on the water system size. With Lak. 1,

there is no flow of a lower water layer to the surface. With Lak, 1, the cold and

the depth cause a flow towards the surface. The critical value is reached with

Lak5 1. Geophysics. Ecology.

Info: [B52]

9.2.24 Littoral Water Exchange Nlit, CV

Nlit 5τfτc

5Vh

13

w tan ϕL73τc

;where w5

ffiffiffiffiffiffiffiffiffiffiffiffiffiαqAghcR

3

s

τf (s) � flushing time; τc (s) � time over which night cooling acts; V (m3) � vol-

ume of the lake; h (m) � total depth of the lake; w (m s21) � convective flow

velocity; ϕ (2) � bottom slope angle in littoral area; L (m) � characteristic

length (equivalent radius of the lake bottom); α (K21) � thermal expansion coeffi-

cient of water; qA (W m22) � surface heat flux; g (m s22) � gravitational accelera-

tion; c (J kg21 K21) � specific heat capacity of water; R (kg m23) � water density.

It characterizes the influence of the night cooling of shallow littoral waters on the

exchange of this colder water with the warmer water from superficial layers of the

water system in its deeper parts by free convection. The critical value is Nlit5 1.


When Nlit, 1, in the shallow littoral zone the cooled water will flow into deeper

waters of the lake because of natural (gravitational) flow, and the warmer water takes

its place from the superficial layer at the deeper areas of the lake. When Nlit. 1 in

the littoral zone, the night cooling destroys the thermal equilibrium of the water, but

this water exchange is not influenced significantly. Geophysics. Ecology.

Info: [B52]

9.2.25 Mach Dispersion Number Ma

Ma5wp 2wf

�� cf

wp, wf (m s21) � velocity of particles (p) and fluid (f); cf (m s21) � sound speed.

It expresses the ratio of the difference between particle and fluid velocities to

the sound velocity under equal conditions of temperature and pressure. Ecology.

Meteorology.

Info: [A0].

Ernst Mach (p. 73).

9.2.26 Manning Roughness Coefficient Nn, n

Nn 5A53S

12wini

QO23

A (m2) � flow cross section; S (m m21) � inclination of water level; wini (m s21) �inlet velocity; Q (m3 m21 s21) � water volume flux on 1 m ground width; O (m) �wetted perimeter.

It characterizes the irrigation of furrowed clayey soil at diverse inflow velocities

and surface inclinations with various kinds of grain crops. Geophysics. Biosystems.

Info: [B104]

9.2.27 Peclet Solar Number Pesol

Pesol 5H2

aτ

H (m) � water height in solar reservoir; a (m2 s21) � thermal diffusivity; τ (s) �daytime duration of solar radiation.

It expresses the convective-to-conductive heat transfer ratio as water flows into

a solar reservoir. Ecology.

Info: [B48].



9.2.28 Radiation Weighting Factor wrad

wrad 5H

D

H (Sv5 keV μm21) � equivalent dose in sievert (Sv); D (Sv) � absorbed dose by

the biological tissue.

The radiation weighting factor is used because some types of radiation are more

harmful biologically than others, provided the absorbed dose is equal. It charac-

terizes the influence of various kinds of radiation on biological damage to body tis-

sue. By means of the wrad, the dose absorbed by biological tissue can be

recalculated to an equivalent radiation dose by distinguishing the harmful effects of

diverse kinds of absorbed radiation. wrad5 1 holds for photons and electrons,

wradAh5; 20i for neutrons, wrad5 5 for protons, and wrad5 20 for alpha particles.

Info: [C50].

9.2.29 Ratio of Spot Radius R

R5r

h

r (m) � spot radius; h (m) � height of liquid layer.

It expresses the ratio of the spot radius to the fluid layer height. It represents a

geometric parameter, of which a large value corresponds to a large target, and vice

versa. Microbiology. Microorganisms.28

Info: [B39],[B86].

9.2.30 Renewed Exergy Fraction Number Ω

Ω5εren

εren 1 εnon

εren (kW, MJ year21) � exergy from renewed resources; εnon (kW, MJ year21) �exergy from nonrenewable resources.

This number expresses the exergy from renewable sources which is multiplied

by the inverse value of the sum of the exergies from both renewable and nonrenew-

able sources. It influences the depletion number Dp (p. 415) by expressing supplies

to the system which have been derived from renewable sources. The renewable

resources in question are, for example, solar radiation, wind or tide energy, and the

supply sources include biomass, among other sources.

Info: [B25].

Catherine Preston Koshland (p. 416).


9.2.31 Resuspension Number Nres, NRS

Nres 5ws

wb

5d2g0pLH

18νffiffiffiffiffiffiCD

phlwi

where ws 5d2g0p18ν

; wb 5

ffiffiffiffifficD

pwihl

LH; g0p 5

ΔRpR

g

ws (m s21) � sedimentation rate at the bottom; wb (m s21) � shear velocity at the

bottom; d (m) � effective sedimentation size; g0p (m s22) � corrected gravitational

acceleration; L (m) � characteristic length; H (m) � total depth of the lake;

ν (m2 s21) � kinematic viscosity; hl (m) � depth of upper water layer; wi (m s21) �phase speed of the internal wave; ΔRp (kg m23) � density difference between the

particles and water; R (kg m23) � water density; g (m s22) � gravitational accelera-

tion; CD (2) � drag coefficient (p. 60).

This number is an important ecological criterion determining whether a sedimen-

tation will settle on the bottom or be dispersed during the activity of waves in a lake

or other similar water systems. In compliance with the data in [C36], the value

Nres, 1 holds for the particle diameter d5 0.1 mm, as does the value Nres, 1024

for the diameter d5 1 μm. The critical value is Nres5 1. Geophysics. Ecology.

Info: [A30],[B25].

9.2.32 Reynolds Dispersion Number Re

Re5wp 2wf

�� dν

54Ma

Kn

wp, wf (m s21) � velocity of particles (p) and fluid (f); d (m) � particle diameter;

ν (m2 s21) � kinematic viscosity; Ma (2) � Mach dispersion number (p. 421);

Kn (2) � Knudsen number (1.) (p. 69).

It is a modification of the Reynolds number Re (p. 81) for colloidal systems with

dispersed solid or liquid particles. Ecology. Meteorology.

Info: [A0].


9.2.33 Schmidt Dispersion Number Sc

Sc5νD

5nfλd4

4

ν (m2 s21) � kinematic viscosity; D (m2 s21) � diffusivity; nf (m23) � numerical

gas density; λ (m) � mean free path; d (m) � particle diameter.

It is a modification of the Schmidt number Sc (p. 263) for colloidal systems with

dispersed solid or liquid microparticles. Ecology. Meteorology.

Info: [A0].

Ernst Schmidt (p. 264).


9.2.34 Sediment Number Eis

Eib 5qbffiffiffiffiffiffiffiffiffiRgD

pD

ð1Þ;

Eit 5qtffiffiffiffiffiffiffiffiffiRgD

pD

ð2Þ;

where qt 5 qb 1 qs; R5RsR21

qb (m2 s21) � volume bedload transport rate per unit width of cross-section diame-

ter; qt (m2 s21) � total volume bed material transport rate per unit width;

qs (m2 s21) � volume suspended load transport rate per unit width; R (2) � sedi-

ment submerged specific density; g (m s22) � gravitational acceleration; D (m) �characteristic sediment size; Rs (kg m23) � sediment density; R (kg m23) � water

density.

It expresses the sedimentation process with water flow in rivers, channels and

other water courses. It is also called the Einstein sedimentation number.

Geophysics. Ecology. Morphodynamics.

9.2.35 Smith Number Smi

Smi5εwτL

ε (2) � scale factor, its value ε{1; w (m s21) � effective rain velocity; τ (s) �time connected with measurable surface changes; L (m) � characteristic length.

It characterizes the influence of rain in changing the character of a land surface.

Geophysics and geomorphology. Ecology.

Info: [B65].

9.2.36 Sperm Number Sp

Sp5 Lηωk

� �14

L (m) � characteristic fibre length; η (Pa s) � dynamic viscosity; ω (s21) � angular

frequency of motion; k (N m21) � linear stiffness.

This number expresses the viscous-to-elastic tensions ratio on microbiological

fibres, such as bacteria and many eukaryotic cells moving by means of a hair

structure (flagellum). For the so-called simple swimmers, internal elasticity domi-

nance holds with a low value of Sp. With high values of Sp, the influence of vis-

cous friction predominates. Microbiology. Microphysics.

Info: [B32].


9.2.37 Tank Solar Number Ntank

Ntank 5RcpVSατ

R (kg m23) � water density; cp (J kg21 K21) � specific heat capacity of water;

V (m23) � water volume in reservoir; S (m2) � surface area of reservoir;

α (W m22 K21) � heat transfer coefficient; τ (s) � daytime duration of solar

radiation.

It expresses the ratio of the amount of water in an exchanger to the absorbing

area. It is an important quality in designing of solar collector equipment utilizing

natural water convection. Ecology.

Info: [B48].

9.2.38 Timescale Parameter Pτ

Pτ 5L

wΔτ

L (m) � declivity length of surface; w (m s21) � mean declivity velocity of surface

water; Δτ (s) � time interval of surface change.

It characterizes the development changes caused by hydrodynamic action of sur-

face water on a landscape. Geophysics and geomorphology. Ecology.

Info: [B65].

9.2.39 Volumetric Concentration of Target DNA C 00

C00 5

h

Hmax

C0

h (m) � height of liquid layer; Hmax (mol m22) � molar additional value of volume

concentration of free bonded parts in spot place on the probe; C0 (mol m23) �initial volume concentration on target DNA in the solution.

It characterizes the ratio of the number of DNA targets, represented over the

spots, to the number of tied up sides in a spot in the instant τ5 0. Explicitly, it

expresses the parameter determining the need to demarcate a target from outside

of the spot surface. Its value acts on the relative molar surface concentration of

formed hybrid pairs. Microbiology. Microorganisms.

Info: [B86].


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http://en.wikipedia.org/wiki/Damping

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http://physics.mercer.edu/hpage/damping.htm

http://www.answers.com/Darcy%20friction%20factor

http://amsglossary.allenpress.com/glossary/search?p=1&query=Davies+number







http://scienceworld.wolfram.com/physics/deBroglieWavelength.html

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http://scienceworld.wolfram.com/physics/deBroglieWavenumber.html

http://scienceworld.wolfram.com/physics/deBroglieWavenumber.html

http://en.wikipedia.org/wiki/Debye_length

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[C38] Dimensionless number. http://en.wikipedia.org/wiki/Dimensionless_number [cit. 2010-

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[C41] Earthquake source parameters. http://www.earth.northwestern.edu/people/seth/324/

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[C42] Earthshine measurements of global atmospheric properties. http://www.bbso.njit.edu/

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[C43] Eddington number. http://mathworld.wolfram.com/EddingtonNumber.html [cit. 2007-

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[C44] Edwin Powell Hubble. http://en.wikipedia.org/wiki/Edwin_Hubble [cit. 2007-09-21].

[C45] Ekman number. http://en.wikipedia.org/wiki/Ekman_number [cit. 2007-09-21].

[C46] Energy efficiency. http://en.wikipedia.org/wiki/Energy_efficiency [cit. 2007-09-21].

[C47] Energy storage. http://en.wikipedia.org/wiki/Energy_storage [cit. 2007-09-21].

[C48] Equations governing geophysical flows. http://engineering.dartmouth.edu/BBenoit R

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[C49] Equations of motion and energy in Cartesian coordinates. http://www.owlnet.rice.

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[C50] Equivalent dose and radiation weighting factor. http://trshare.triumf.ca/Bsafety/EHS/

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[C51] Ericksen number. http://en.wikipedia.org/wiki/Ericksen_number [cit. 2010-05-30].

[C52] Erlang unit. http://en.wikipedia.org/wiki/Erlang_unit [cit. 2007-09-21].

[C53] f-number. http://en.wikipedia.org/wiki/F-stop [cit. 2007-09-21].

[C54] Fermi paradox. http://www.en.wikipedia.org/wiki/Fermi_paradox [cit. 2007-09-21].

[C55] Fluent 6.3 User’s Guide � Nomenclature. http://my.fit.edu/itresources/manuals/

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[C57] Force of friction. http://faculty.wwu.edu/vawter/PhysicsNet/Topics/Dynamics/Forces/

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[C58] Fraction mechanics. http://encyclopedia.thefreedictionary.com/crack1 propagation

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[C59] Fresnel number. http://en.wikipedia.org/wiki/Fresnel_number [cit. 2007-09-21].

[C60] Friction. http://en.wikipedia.org/wiki/Friction [cit. 2007-09-21].

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[C62] Gain. http://www.reference.com/browse/wiki/Gain [cit. 2007-09-21].

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[C70] Hagen number. http://en.wikipedia.org/wiki/Hagen_number [cit. 2007-09-21].

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[C73] Internal Froude number. http://scienceworld.wolfram.com/physics/InternalFroude

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[C74] Karlovitz number. http://www.cfd-online.com/Wiki/Karlovitz_number [cit. 2010-05-30].

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[C76] Laplace number. http://en.wikipedia.org/wiki/Laplace_number [cit. 2007-09-21].

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[C78] Lockhart-Martinelli parameter. http://en.wikipedia.org/wiki/Lockhart-Martinelli parameter

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[C81] Love number. http://scienceworld.wolfram.com/physics/LoveNumber.html [cit. 2007-

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[C91] Molecular mass. http://en.wikipedia.org/wiki/Molecular_mass [cit. 2007-09-21].

[C92] Moment magnitude scale. http://en.wikipedia.org/wiki/Moment_magnitude_scale

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[C93] Moment of inertia ratio. http://scienceworld.wolfram.com/physics/MomentofInertia

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[C94] Monin�Obukhov length. http://scienceworld.wolfram.com/physics/Monin-Obukhov

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[C112] Rejection coefficient. http://www.msu.edu/course/fsc/229/Notes/Lecture%208.htm

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[C113] Relative permittivity. http://en.wikipedia.org/wiki/Dielectric_constant [cit. 2010-05-30].

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[C117] Rockwell scale. http://en.wikipedia.org/wiki/Rockwell_scale [cit. 2007-09-21].

[C118] Roshko number. http://en.wikipedia.org/wiki/Roshko_number [cit. 2007-09-21].

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[C120] Sedimentation. http://en.wikipedia.org/wiki/Sedimentation [cit. 2010-10-19].

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[C124] Stratification parameter. http://scienceworld.wolfram.com/physics/Stratification

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[C125] Strehl ratio. http://scienceworld.wolfram.com/physics/StrehlRatio.html [cit. 2007-09-21].

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http://www.efunda.com/formulae/fluids/calc_pipe_friction.cfm

http://www.efunda.com/formulae/fluids/calc_pipe_friction.cfm

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