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Heat Transfer ME 431

Convection Heat Transfer

Dr. M.M. Abo Elazm

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Dimensionless Numbers

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Units & Dimensions

...???differencethe is WhatValues

DimensionsUnits

⇒

We always hear in heat transfer the names like Reynolds, Nusselt, and

prandtl, etc.

These are dimensionless numbers that are used to characterize and

classify the heat transfer problems.

So we need first to explain the meaning and significance of these

numbers.

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Many people aren't sure of the difference. Let's try and get

a set of definitions we can use. Consider:

10 grams of sodium

10 meters high

10 gallons of gasoline

Units & Dimensions

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We'll break them up this way:

Units & Dimensions

Value Unit Dimension

10 grams mass

10 meters length

10 gallons volume (length3)

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Units & Dimensions

1- Dimensions:

A "dimension" can be measured or

derived.

There are a "fundamental dimensions"

(length, time, mass, temperature,

amount) are distinct and sufficient to

define all the others.

There are also many derived dimensions

(velocity, volume, density, etc.) for

convenience.

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Units & Dimensions

2- Units:

"Units" can be counted or measured.

Measured units are specific values of

dimensions defined by law or

custom.

Many different units can be used for a

single dimension, as inches,

miles, and centimeters, are all

units used to measure the dimension

length.

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So what is a dimensionless number…????

When we say a quantity is dimensionless, we mean one of two

things:

First, it may just be a number like we get when counting. 10 Oranges,

5 circles (counted)

Or a ratio such as, [Pi or π] which is a dimensionless number

representing the ratio of the circumference of a circle to its diameter

Dimensionless Quantities

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Also numbers [of particular interest in engineering] are combinations of

variables when all the dimensions are "cancelled out" so the net term

has no dimension:

10 kg/2 kg = 5

These are often called "dimensionless numbers" and often have special

names and meanings.

Most of these have been found using techniques of "dimensional analysis"

[a way of examining physical phenomena by look ing at the dimensions that

occur in the problem w ithout considering any numbers].

Dimensionless Quantities

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So why do we need dimensionless numbers…???

These numbers do not exist in nature, or in other words The nature itself

does not have a clue about these numbers.

It is not like the air says to itself " boy, my Reynolds number is exceeding 2500

and I am in a pipe so I better switch to my turbulent mode or all the fluid

dynamics textbooks will be wrong".

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If we have come up with some formula for calculating the pressure drop

in a 2 inch pipe, can we use it for a 2 feet pipe ????

We have invented dimensionless numbers to be able to take our knowledge

from experimenting with one system to learn about another system with

different dimensions.

In a way, we are trying to get rid of dimensions in order to extend our

solution beyond its source.

Dimensionless Quantities

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Dimensionless numbers allow us to

experiment with models [cars, airplanes

and ships] and predict the behavior of

the original thing under actual

conditions.

We have to make sure that there is a

similarity between the model and the

actual thing.

Dimensionless Quantities

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Dimensionless Numbers In Heat Transfer

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The most common numbers used in heat transfer,

are:

Reynolds Number [Re]

Nusselt Number [Nu]

Prandtl Number [Pr]

Grashof Number [Gr]

Rayleigh Number [Ra]

Dimensionless Quantities

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These numbers are a combination of the physical properties of the

fluids….so

Before getting into the definitions of these numbers, we should define

and understand the meaning of some physical properties of

fluids since they show up all over the place in the dimensionless

numbers.

Dimensionless Quantities

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1- Density: [ρ]

It is defined as the “Mass of fluid contained in a unit volume”.

Its units are [kg/m3 or slugs/ft3]. Typical values are:

Water = 1000 kg/m3,

Mercury = 13546 kg/m3,

Air = 1.23 kg/m3,

Paraffin Oil = 800 kg/m3.

at Pressure of 1 bar or [1.013*105 Pascals] and Temperature=288.15K.

Dimensionless Quantities

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2- Viscosity: [μ]

As described before it is the property of a fluid, due

to interaction between molecules, which offers

resistance to shear deformation of the fluid.

Fluid with a high viscosity such as syrup,

deforms more slowly than fluid with a low

viscosity such as water.

[Units: N.sec/m2, or kg/m.sec]

Typical values: Water =1.14x10-3 , Air =1.78x10-5 ,

Mercury =1.552 , Paraffin Oil =1.9 kg/m.sec.

Dimensionless Quantities

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3- Kinematic Viscosity [ν]

Defined as the ratio of dynamic viscosity to mass density, [ν= μ/ ρ]

Units: m2/sec, (often expressed in Stokes [St], where St = 10-4 m2/sec .)

Typical values: Water =1.14x10-6 , Air =1.46x10-5 ,

Mercury =1.145x10-4 , Paraffin Oil =2.375x10-3 m2/sec.

Dimensionless Quantities

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4- Thermal conductivity: [k]

It is a measure of the ability of a material (Solid, Liquid or Gas) to

conduct heat. It is defined using the Fourier's law of conduction

which, relates the rate of heat transfer by conduction to the

temperature gradient.

Dimensionless Quantities

CmW

Cm

mWk

dxdTkAQ

oo .2 ==

−=

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Thermal conductivity of most liquids increase with decreasing

temperature. “Water is, however, an exception to this rule”.

According to the kinetic theory of gases, the thermal conductivity of gases

is proportional to the square root of temperature and inversely

proportional to the square root of the molecular mass.

It is obvious that the thermal conductivity of a gas increases with increasing

temperature.

Dimensionless Quantities

mkandTk ~1αα

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5- Specific heat: Cp

Specific heat is the amount of heat that is required to raise the

temperature of a unit mass of a substance by one degree. In a

constant pressure process

Q = m . Cp . Δ T

The units for the specific heat are kJ/kg.K (or oC).

Dimensionless Quantities

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6- Thermal Diffusivity a

As we know, the heat travels from the high temperature region to

the low temperature.

A measure of how heat propagates through a medium may be defined

as “the ratio of the heat conducted through the material to the

heat stored in the material per unit volume”.

Dimensionless Quantities

pC ρkα

fluidthe in stored heatfluidthe through conducted heatα

=

=

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7- Coefficient of Thermal Expansion: [β]

It is defined as the change in the density of a substance as a

function of temperature at constant pressure. It can be

approximated as:

Dimensionless Quantities

TβρRTP

ρβΔTΔρsoΔTΔρ

ρβ

1:Gases IdealFor

1

=∴⇒=

−≈−≈

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Important Dimensionless numbers

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1- Reynolds Number

After exhaustive experiments in the 1880s,

Osborn Reynolds discovered that the flow

regime depends mainly on the ratio of the

inertia forces to viscous forces in the

fluid. This ratio is called the Reynolds

number, which is a dimensionless

quantity, and is expressed as:

FUNDAMENTALS OF CONVECTION

νµρ

µ

ρ

ULULL

UL

UForceViscousForceInertia

==

=

=

2

2

Re

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At large Reynolds numbers, the inertia forces (density

and the velocity) of the fluid, are large relative to the

viscous forces, which cannot prevent the random and

rapid fluctuations of the fluid, Thus the flow is

turbulent.

At small Reynolds numbers, however, the viscous

forces are large enough to overcome the inertia

forces and to keep the fluid “in line.” Thus the flow is

laminar.

FUNDAMENTALS OF CONVECTION

νµρ

µ

ρ

ULULL

UL

UForceViscousForceInertia

==

=

=

2

2

Re

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Any fluid flow contains small disturbances that will grow given enough

opportunities. as long as the viscous forces dominate, these disturbances

are under control.

As the inertia forces get bigger, the viscosity can no longer maintain order

and these disturbances grow into trouble makers and then transition to

turbulent flow.

FUNDAMENTALS OF CONVECTION

νµρ ULUL

==Re

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Critical Reynolds number

The value of Reynolds number beyond which the flow is no longer considered

laminar is called the critical Reynolds number.

For flow over a flat plate, the critical Reynolds number is observed to vary

between [1x105 to 3x106] depending on the turbulence level in the

free stream and the roughness of the surface.

We normally use 5x105 as the critical Reynolds number for flow over flat plates.

We normally use 2400 to 4000 as the critical Reynolds number for flow inside

tubes.

FUNDAMENTALS OF CONVECTION

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2- Nusselt number:

Is the ratio of convective to conductive heat

transfer across the boundary. It is a

dimensionless number named after Wilhelm

Nusselt.

The conductive component is measured under the

same conditions as the heat convection but with a

stagnant (or motionless) fluid.

FUNDAMENTALS OF CONVECTION

Nuk

hLLTk

Thqq

LTkq

Thq

cond

conv

cond

conv

==∆∆

=

∆=

∆=

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So, the Nusselt number may be viewed as the

ratio of convection to conduction for a

layer of fluid.

Higher values of Nusselt mean that the

heat transfer is enhanced by convection.

FUNDAMENTALS OF CONVECTION

Nuk

hLLTk

Thqq

LTkq

Thq

cond

conv

cond

conv

==∆∆

=

∆=

∆=

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3- Prandtl Number:

The ratio between the rate of viscous diffusion to the rate

of thermal diffusion within a fluid flow, is best described

by the dimensionless parameter Prandtl number, It is

named after Ludwig Prandtl, who introduced the concept

of boundary layer in 1904 and made significant contributions

to boundary layer theory.

FUNDAMENTALS OF CONVECTION

kc

ratediffusionThermalratediffusionViscous pµ

αν===Pr

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The Prandtl numbers of gases are about 1,

which indicates that both momentum and

heat dissipate through the fluid at about

the same rate.

FUNDAMENTALS OF CONVECTION

Heat diffuses very quickly in liquid metals (Pr<<1) and very slowly

in oils (Pr>>1) relative to momentum.

Consequently the thermal boundary layer is much thicker for liquid metals

and much thinner for oils relative to the velocity boundary layer.

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Prandtl number can also be related to the thickness of the thermal and velocity

boundary layers. It is actually the ratio of velocity boundary layer

thickness to the thermal boundary layer thickness.

FUNDAMENTALS OF CONVECTION

th

vel

thicknesslayerBoundaryThermalthicknesslayerBoundaryVelocity

δδ

==Pr

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4- Grashof Number: Gr

You see this number and you should think of natural or free convection. The

Grashof number is the ratio of buoyancy forces to the viscous forces.

Grashof number In natural convection plays the same role that is played by

Reynolds number in forced convection.

FUNDAMENTALS OF CONVECTION

2ForcesViscousForcesBoyancyGr

vvg

ρρ∆

==

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The buoyant forces are fighting with viscous forces and at some point

they overcome the viscous forces and the flow is no longer nice and

laminar.

For a vertical plate, the flow transitions to turbulent around a Grashof

number of Gr=109.

FUNDAMENTALS OF CONVECTION

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5- Rayleigh Number: Ra

The Rayleigh number is the product of Grashof and Prandtl numbers.

Ra = Gr * Pr

It turns out that in natural convection the Nusselt number scales with

Rayleigh rather than just Grashof.

Most correlations in natural convection are of the form:

FUNDAMENTALS OF CONVECTION

[ ]( ) Pr......... 2

3

vxTTgPrGrRaWhere

RaCNu

s

n

∞−=×=

=

β

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6- Richardson number [Ri]

Is named after Lewis Fry Richardson (1881 – 1953).

It is the dimensionless number that expresses the ratio of potential to

kinetic energy or importance of natural convection relative to the forced

convection. The Richardson number in this context is defined as:

FUNDAMENTALS OF CONVECTION

22ReGr

EnergyKineticEnergyPotential

Convection ForcedConvection Natural

UTLgRior

Ri

∆==

==

β

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Heat Transfer by Cengel 2nd Ed See Examples … to …

Thank You…

CONDUCTION