( LECTURE 1: Introduction and Basic Concepts( Syllabus: Transfer
of Momentum, Heat Transfer (Conduction, Convection, Radiation),
Mass Transfer, Transport Phenomena Approach.
What is Transport Phenomena?Transport Phenomena is the study of
the transfer of momentum, heat and mass. These
three "modes" of transport are usually taught/grouped together
because: they have similar molecular origins
they yield similar governing equations/principles
they often occur simultaneously
they require similar mathematical/conceptual tools Transport
Phenomena and Unit Operations
The pioneering work Principles of Chemical Engineering was
published in 1923 under the authorship of Walker, Lewis, and
McAdams [1]. This book was the first to emphasize the concept of
unit operations as a fundamental approach to physical separations
such as distillation, evaporation, drying, etc.
This was the era when the profession of chemical engineering
matured into a separate area, no longer the province of the
industrial chemist. The study of unit operations such as
distillation is predicated on the idea that similarities in
equipment and fundamentals exist regardless of the process. In
other words, the principles of distillation apply equally to the
separation of liquid oxygen from liquid nitrogen as well as to the
thousands of other distillations routinely carried out in
industries around the world. The study of transport phenomena is
undertaken because this topic is the basis for most of the unit
operations. In simple terms, transport phenomena comprise three
topics: heat transfer, mass transfer, and momentum transfer (fluid
flow). In many of the unit operations (such as distillation), all
three transport phenomena (i.e., fluid flow, heat transfer, and
mass transfer) occur, often simultaneously. The concepts presented
in transport phenomena underlie the calculation procedures that are
used in the design of unit operations. Equilibrium and Rate
ProcessesMany problems can conveniently be divided into two
classifications: equilibrium and non-equilibrium. Under conditions
of non-equilibrium, one or more variables change with time. The
rates of these changes are of much interest, naturally. A typical
chemical engineer is involved with four types of rate processes:
rate of heat transfer, rate of mass transfer, rate of momentum
transfer, and rate of reaction. The first three of these are the
subject of this course. The fourth, rate of reaction, will not be
covered in any detail, except for the inclusion of the appropriate
terms in the general equations and in a few elementary
examples.
Transfer of momentum
In Fig. 1 a pair of large parallel plates is shown, each one
with area A, separated by a distance Y. In the space between them
is a fluid-either a gas or a liquid. This system is initially at
rest, but at time t = 0 the lower plate is set in motion in the
positive x direction at a constant velocity V. As time proceeds,
the fluid gains momentum, and ultimately the linear steady-state
velocity profile shown in the figure is established. We require
that the flow be laminar ("laminar" flow is the orderly type of
flow that one usually observes when syrup is poured, in contrast to
"turbulent" flow, which is the irregular, chaotic flow one sees in
a high-speed mixer). When the final state of steady motion has been
attained, a constant force F is required to maintain the motion of
the lower plate. Common sense suggests that this force may be
expressed as follows: (1)
That is, the force should be proportional to the area and to the
velocity, and inversely proportional to the distance between the
plates. The constant of proportionality is a property of the fluid,
defined to be the viscosity.
We now switch to the notation that will be used throughout the
book. First we replace F/A by the symbol yx, which is the force in
the x direction on a unit area perpendicular to the y direction. It
is understood that this is the force exerted by the fluid of lesser
y on the fluid of greater y. Furthermore, we replace V/Y by . Then,
in terms of these symbols, Eq. (1) becomes (2)
This equation, which states that the shearing force per unit
area is proportional to the negative of the velocity gradient, is
often called Newton's law of viscosity. (Actually we should not
refer Eq. (2) as a "law," since Newton suggested it as an
empiricism
Fig.1. The buildup to the steady, laminar velocity profile for a
fluid contained between two plates. The flow is called "laminar"
because the adjacent layers of fluid ("laminae") slide past one
another in an orderly fashion.
the simplest proposal that could be made for relating the stress
and the velocity gradient.
However, it has been found that the resistance to flow of all
gases and all liquids with molecular weight of less than about 5000
is described by Eq. (2), and such fluids are referred to as
Newtonian fluids. Polymeric liquids, suspensions, pastes, slurries,
and other complex fluids are not described by Eq. (2) and are
referred to as non-Newtonian fluids. Equation (2) may be
interpreted in another fashion. In the neighborhood of the moving
solid surface at y = 0 the fluid acquires a certain amount of
x-momentum. This fluid, in turn, imparts momentum to the adjacent
layer of liquid, causing it to remain in motion in the x direction.
Hence x-momentum is being transmitted through the fluid in the
positive y direction. Therefore yx may also be interpreted as the
flux of x-momentum in the positive y direction, where the term
"flux" means "flow per unit area." This interpretation is
consistent with the molecular picture of momentum transport and the
kinetic theories of gases and liquids. It also is in harmony with
the analogous treatment given later for heat and mass transport.The
above idea may be paraphrased by saying that momentum goes
"downhill" from a region of high velocity to a region of low
velocity-just as a sled goes downhill from a region of high
elevation to a region of low elevation, or the way heat flows from
a region of high temperature to a region of low temperature. The
velocity gradient can therefore be thought of as a "driving force"
for momentum transport.
In what follows we shall sometimes refer to Newton's law in Eq.
(2) in terms of forces (which emphasizes the mechanical nature of
the subject) and sometimes in terms of momentum transport (which
emphasizes the analogies with heat and mass transport).
This dual viewpoint should prove helpful in physical
interpretations.
Often the symbol represents the viscosity divided by the density
(mass per unit volume) of the fluid, thus: (3)This quantity is
called the kinematic viscosity. The appropriate SI units for and
are as follows: [] = Pas , [] = m2s-1. Heat Transfer Internal
energy may be viewed as the sum of the kinetic and potential
energies of the molecules. The portion of the internal energy of a
system associated with the kinetic energy of the molecules is
called sensible energy or sensible heat. The internal energy is
also associated with the intermolecular forcers between the
molecules of a system. These are the forcers that bind the
molecules to each other; they are strongest in solids and weakest
in gases. If sufficient energy is added to the molecules of a solid
or liquid, they will overcome these molecular forcers and simply
break away, turning the system into a gas. This is a phase change
process and because of this added energy, a system in the gas phase
is at a higher internal energy level than it is in the solid or the
liquid phase. The internal energy associated with the phase of a
system is called latent energy or latent heat. The sensible and
latent forms of internal energy can be transferred from one medium
to another as a result of a temperature difference, and are
referred to as heat or thermal energy. Thus a heat transfer is the
exchange of the sensible and latent forms of internal energy
between two mediums as a result of a temperature difference. The
amount of heat transferred per unit time is called heat transfer
rate, [W]. The rate of heat transfer per unit area is called heat
flux, [W m-2]The First Law of Thermodynamics manifests the
principle of the conservation of energy:
(4)In heat transfer analysis, we are usually interested only in
the form of energy that can be transferred as a result of a
temperature difference, that is, heat or thermal energy. In such
cases it is convenient to write a heat balance and to treat the
conversion of nuclear, chemical, mechanical, and electrical
energies into thermal energy as heat generation. The energy balance
in that case can be expressed as (5)
When a stationary closed system involves heat transfer only and
no work interactions across its boundary, the energy balance
relation reduces to
(6)Under steady conditions and in the absence of any work
interactions, the conservation of energy relation for a control
volume with one inlet and one exit with negligible changes in
kinetic and potential energies can be expressed as
(7)where [kgs-1] is the mass flow rate.
Heat can be transferred in three different modes: conduction,
convection, and radiation. Conduction is the transfer of heat from
the more energetic particles of a substance to the adjacent less
energetic ones as a result of interactions between the particles,
and is expressed by Fouriers law of heat conduction as , (8)
or, in general case (8a)Convection is the mode of heat transfer
between a solid surface and the adjacent liquid or gas that is in
motion, and involves the combined effects of conduction and fluid
motion. The rate of convection heat transfer is expressed by
Newtons law of cooling as (9)
where h [Wm-2K-1] is the convection heat transfer coefficient.
Radiation is the energy emitted by matter in the form of
electromagnetic waves (or photons) as a result of the changes in
the electronic configurations of the atoms and molecules. The
maximum rate of radiation that can be emitted from a surface at a
thermodynamic temperature Ts is given by the Stefan-Boltzmann law
as
(10)where = 5.6710-8 Wm-2K-4 is the Stefan-Boltzmann
constant.When a surface of emissivity and surface area As at a
temperature Ts is completely enclosed by a much larger (or black)
surface at temperature Tsurr separated by a gas (such as air) that
does not intervene with radiation, the net rate of radiation heat
transfer between these two surfaces is given by (11)Ii this
(limited) case, the emissivity () and the surface area of the
surrounding surface do not have any effect on the net radiation
heat transfer.
The rate at which a surface absorbs radiation is determined
from
(12)Where is absorptivity of the surface.Radiation is usually
considered to be a surface phenomenon for solids that are opaque to
thermal radiation such as metals, wood, and rocks since the
radiation emitted by interior region of such material can never
reach the surface, and the radiation incident on such bodies is
usually absorbed within a few microns from the surface.
The heat transfer problems encountered in practice can be
considered in two groups: (1) rating and (2) sizing problems. The
rating problems deal with the determination of the heat transfer
rate for an existing system at a specified temperature difference.
The sizing problems deal with the determination of the size of a
system in order to translate heat at a specified rate for a
specified temperature difference. Mass Transfer When a system
contains two or more components whose concentrations vary from
point to point, there is a natural tendency for mass to be
transferred, minimizing the concentration differences within the
system and moving it towards equilibrium.
The transport of one component from a region of higher
concentration to that of a lower concentration is called mass
transfer. Distinction should be made between mass transfer and the
bulk fluid flow that occurs on a macroscopic level as a fluid is
transported from one location to another. Many of our daily
experiences involve mass-transfer phenomena. The invigorating aroma
of a cup of freshly brewed coffee and the sensuous scent of a
delicate perfume both reach our nostrils from the source by
diffusion through air. A lump of sugar added to the cup of coffee
eventually dissolves and then diffuses uniformly throughout the
beverage. Laundry hanging under the sun during a breezy day dries
fast because the moisture evaporates and diffuses easily into the
relatively dry moving air.
Mass transfer plays an important role in many industrial
processes. A group of operations for separating the components of
mixtures is based on the transfer of material from one homogeneous
phase to another. These methods covered by the term mass-transfer
operations- include such techniques as distillation, gas
absorption, humidification, liquid extraction, adsorption, membrane
separations, and others.
The driving force for transfer in these operations is a
concentration gradient, much as a temperature gradient provides the
driving force for heat transfer. (N.B. Although it should be
remembered that a rigorous analysis employs the gradient of
chemical potential as the driving force for mass transfer.)
Returning to the lump of sugar added to the cup of coffee, it is
evident that the time required for the sugar to distribute
uniformly depends upon whether the liquid is quiescent or whether
it is mechanically agitated by a spoon. In general, the mechanism
of mass transfer depends upon the dynamics of the system in which
it occurs. Mass can be transferred by random molecular motion in
quiescent fluids, or it can be transferred from a surface into a
moving fluid, aided by the dynamic characteristics of the flow.
These two distinct modes of transport, molecular mass transfer and
convective mass transfer, are analogous to conduction heat transfer
and convective heat transfer. Each of these modes of mass transfer
will be described and analyzed. The two mechanisms often act
simultaneously. Frequently, when this happens, one mechanism can
dominate quantitatively so that approximate solutions involving
only the dominant mode can be used.The description of diffusion
involves a mathematical model based on a fundamental hypothesis or
law. Interestingly, there are two common choices for such a law.
The more fundamental, Ficks law of diffusion, uses a diffusion
coefficient. The second, which has no formal name, involves a mass
transfer coefficient. Now we want to illustrate the two basic ways
in which diffusion can be described. To do this, we first imagine
two large bulbs connected by a long thin capillary (Fig.2). The
bulbs are at constant temperature and pressure and are of equal
volumes. However, one bulb contains carbon dioxide, and other is
filled with nitrogen. To find out how fast these two gases will
mix, we measure the concentration of carbon dioxide in the bulb
that initially contains nitrogen. We make these measurements when
only a trace of carbon dioxide has been transferred, and we find
that the concentration of carbon dioxide varies linearly with time.
From this, we know the amount transferred per unit time.
Fig. 2. A simple diffusion experiment. Two bulbs initially
containing different gases are connected with a long thin
capillary. The change of concentration in each bulb is a measure of
diffusion and can be analyzed in two different ways.There are two
basic ways in which diffusion (mass transfer) can be described.
Firstly, imagine two large bulbs connected by a long capillary
(Fig.2). The bulbs are at constant temperature and are of equal
volumes. However, one bulb contains carbon dioxide, and the other
is filled with nitrogen. To find how fast these two gases will mix,
we measure the concentration of carbon dioxide in the bulb that
initially contains nitrogen. We make these measurements when only a
trace of carbon dioxide has been transferred, and we find that the
concentration of carbon dioxide varies linearly with time. From
this, we know the amount transferred per unit time. We want to
analyze this amount transferred to determine physical properties
that will be applicable not only to this experiment but also in
other experiments. To do this, we first define the flux: (13)In
other words, if we double the cross-sectional area, we expect the
amount transported in double. Defining the flux in this way is a
first step in removing the influences of our particular apparatus
and making our results more general. We next assume that the flux
is proportional to the gas concentration:
(14)
The proportionality constant k is called a mass transfer
coefficient. Its introduction signals one of the two basic models
of diffusion. Alternatively, we can recognize that increasing the
capillarys length will decrease the flux, and we can assume that
(15)The new proportionality constant D is the diffusion
coefficient. Its introduction implies the other model for
diffusion, the model often called Ficks law.
These assumptions may seem arbitrary, but they are similar to
those made in many other branches of science. For example, they are
similar to those used in developing Ohms law, which states that
(16)Thus, the mass transfer coefficient k is analogous to the
reciprocal of the resistance. An alternative form of Ohms law
is
(17)The diffusion coefficient D is analogous to the reciprocal
of resistivity.Neither the equation using the mass transfer
coefficient k nor that using the diffusion coefficient D is always
successful. This is because of the assumptions made in their
development. For example, the flux may not be proportional to the
concentration difference if the capillary is very thin or if the
two gases react. In the same way, Ohms law is not always valid at
very high voltages. But these cases are exceptions; both diffusion
equations work well in most practical situations, just as Ohms law
does.
The parallels with Ohms law also provide a clue about how the
choice between diffusion models is made. The mass transfer
coefficient in Eq. (14) and the resistance in Eq. (16) are simpler,
best used for practical situations and rough measurements. The
diffusion coefficient in Eq. (15) and the resistivity in Eq. (17)
are more fundamental, involving physical properties like those
found in handbooks. How these differences guide the choice between
the two models?The choice between the two models represents a
compromise between ambition and experimental resources. Obviously,
we would like to express our results in the most general and
fundamental ways possible. This suggests working with diffusion
coefficients. However, in many cases our experimental measurements
will dictate a more approximate and phenomenological approach. Such
approximations often imply mass transfer coefficients, but they
usually still permit us to reach our research goals.
This choice and the resulting approximations are best
illustrated by two examples. In the first, we consider hydrogen
diffusion in metals. This diffusion substantially reduces a metals
ductility, so much so that parts made from the embrittled metal
frequently fracture. To study this embrittlement, we might expose
the metal to hydrogen under a variety of conditions and measure the
degree of embrittlement versus these conditions. Such empiricism
would be a reasonable first approximation, but it would quickly
flood us with uncorrelated information that would be difficult to
use effectively.
Fig. 3. Hydrogen diffusion into a metal. This process can be
described with either a mass transfer coefficient k or a diffusion
coefficient D. The description with a diffusion coefficient
correctly predicts the variation of concentration with position and
time, and so is superior. As an improvement, we can undertake two
sets of experiments. First, we can saturate metal samples with
hydrogen and determine their degrees of embrittlement. Thus we know
metal properties versus hydrogen concentration. Second, we can
measure hydrogen uptake versus time, as suggested in Fig. 3, and
correlate our measurements as mass transfer coefficients. Thus we
know average hydrogen concentration versus time. To our dismay, the
mass transfer coefficients in this case will be difficult to
interpret. They are anything but constant. At zero time, they
approach infinity; at large time, they approach zero. At all times,
they vary with the hydrogen concentration in the gas surrounding
the metal. They are inconvenient way to summarize our results.
Moreover, the mass transfer coefficients give only the average
hydrogen concentration in the metal. They ignore the fact that the
hydrogen concentration very near metals surface will reach
saturation but the concentration deep within the bar will remain
zero. As a result, the metal near the surface may be very brittle
but that within may be essentially unchanged.We can include these
details in the diffusion model that assumes that
(18)
or, symbolically,
(19)Where the subscript 1 symbolizes the diffusing species. In
this equations, the distance l is that over which diffusion occurs.
In the case described in Fig.2, the length of the capillary was
appropriately this distance; but in this case, it seems uncertain
what distance should be taken into consideration. If we assume that
it is very small, (20)We can use this relation and the techniques
developed later in this course to correlate our experiments with
only one parameter, the diffusion coefficient D. We then can
correctly predict the hydrogen uptake versus time and the hydrogen
concentration in the gas. Moreover, we get the hydrogen
concentration at all positions and times within the metal.
Thus the model based on the diffusion coefficient gives results
of more fundamental value than the model based on mass transfer
coefficients. In mathematical terms, the diffusion model is said to
have distributed parameters, for the dependent variable (the
concentration) is allowed to vary with all independent variables
(like position and time). In contrast, the mass transfer model is
said to have lumped parameters (like the average hydrogen
concentration in metal).
Fig.4. Rates of drug dissolution. In this case, describing the
system with a mass transfer coefficient k is best because it easily
correlates the solutions concentration versus time. Describing the
system with a diffusion coefficient D gives a similar correlation
but introduces an unnecessary parameter, the film thickness l.
These results would appear to imply that the diffusion model is
superior to the mass transfer model and so should always be used.
However, in many interesting cases the models are equivalent. To
illustrate this, imagine that we are studying the dissolution of a
solid drug suspended in water, as schematically suggested by Fig.
4. The dissolution of this drug is known to be controlled by the
diffusion of the dissolved drug away from the solid surface of the
undissolved material. We measure the drug concentration versus time
as shown, and we want to correlate these results in terms of as few
parameters as possible.
One way to correlate the dissolution results is to use a mass
transfer coefficient. To do this, we write a mass transfer balance
on the solution: ( (21)where V is the volume of solution, A is the
total area of the drug particles, c1(sat) is the drug concentration
at saturation and at the solids surface, and c1 is the
concentration in the bulk solution. Integrating this equation
allows quantitatively fitting our results with one parameter, the
mass transfer coefficient k. This quantity is independent of drug
solubility, drug area, and solution volume, but it does vary with
physical properties like stirring rate and solution viscosity.
Correlating the effects of these properties turns out to be
straightforward. The alternative to mass transfer is diffusion
theory, for which the mass balance is (22)in which l is an unknown
parameter, equal to the average distance across which diffusion
occurs. This unknown, called a film or unstirred layer thickness,
is a function not only of flow and viscosity but also of the
diffusion coefficient itself.
Equations (21) and (22) are equivalent, and they share the same
successes and shortcomings. In the former, we must determine the
mass transfer coefficient experimentally; in the latter, we
determine instead the thickness l. The choice between the mass
transfer and diffusion models is often a question of taste rather
than precision. The diffusion model is more fundamental and is
appropriate when concentrations are measured or needed versus both
position and time. The mass transfer model is simpler and more
approximate and is especially useful when only average
concentrations are involved. The additional examples given below
should help us decide which model is appropriate for our purposes.
(This choice is often difficult, a junction where many have
trouble).
Example 1: Ammonia scrubbing. Ammonia, the major material for
fertilizer, is made by reacting nitrogen and hydrogen under
pressure. The product gas can be washed with water to dissolve the
ammonia and separate it from unreacted gases. How can you correlate
the dissolution rate of ammonia during washing?Concept of solution:
The easiest way is to use mass transfer coefficients. If you use
diffusion coefficients, you must somehow specify the distance
across which diffusion occurs. This distance is unknown unless the
detailed flows of gases and the water are known; they rarely
are.
Example 2: Reaction in porous catalysts. Many industrial
reactions use catalysts containing small amount of noble metals
dispersed in a porous inert material like silica. The reaction on
such a catalyst are sometimes slower in large pellets than in small
ones. This is because the reagents take longer to diffuse into the
pellet than they do to react. How should you model this
effect?Concept of solution: You should use the diffusion
coefficients to describe the simultaneous diffusion and reaction in
the pores in the catalyst. You should not use mass transfer
coefficients because you cannot easily include effect of
reaction.Example 3: Corrosion of marble. Industrial pollutants in
urban areas like Venice, Athens etc. cause significant corrosion of
marble statues. You want to study how these pollutants penetrate
marble. Which diffusion model should you use?
Concept of solution: The model using diffusion coefficients is
the only one that will allow you to predict concentration versus
position in the marble. The model using mass transfer coefficients
will only correlate how much pollutant enters the statue, not what
happens to the pollutant.Example 4: Protein size in solution. You
are studying a variety of proteins that you hope to purify and use
as food supplements. You want to characterize the size of the
proteins in solution. How can you use diffusion to do this?
Concept of solution: Your aim is determining the molecular size
of the protein molecules. You are not interested in the protein
mass transfer except as a route to these molecular properties. As a
result, you should measure the proteins diffusion coefficient, not
its mass transfer coefficient. The proteins diffusion coefficient
will turn out to be proportional to its radius in solution.
Example 5: Antibiotic production. Many drugs are made by
fermentations in which microorganisms are grown in a huge stirred
what of a dilute nutrient solution or beer. Many of these
fermentations are aerobic, so the nutrient solution requires
aeration. How should you model oxygen uptake in the type of
solution?
Concept of solution: Practical models use mass transfer
coefficients. The complexities of the problem, including changes in
air bubble size, flow effects of the non-Newtonian solution, and
foam caused by biological surfactants, all inhibit more careful
study.
Example 6: Facilitated transport across membranes. Some
membranes contain a mobile carrier, a reactive species that reacts
with diffusing solutes, facilitating their transport across the
membrane. Such membranes are used to concentrate copper ions from
industrial waste and to remove carbon dioxide from coal gas.
Similar membranes are believed to exist in the human intestine and
liver. Diffusion across these membranes does not vary linearly with
the concentration difference across them. The diffusion can be
highly selective, but it is often easily poisoned. Should this
diffusion be described with mass transfer coefficients or with
diffusion coefficients?
Concept of solution: This system includes not only diffusion but
also chemical reaction. Diffusion and reaction couple in a
nonlinear way to give the unusual behavior observed. Understanding
such behavior will certainly require the more fundamental model of
diffusion coefficients.
Example 7: Flavor retention. When food products are spray-dried,
they lose a lot of flavor. However, they lose less than would be
expected on the basis of the relative vapor pressures of water and
the flavor compounds. The reason apparently is that the drying food
often forms a tight gel-like skin across which diffusion of the
flavor compounds is inhibited. What diffusion model should you use
to study this effect?Concept of solution: Because spray drying is a
complex, industrial-scale process, it is usually modeled using mass
transfer coefficients. However, in this case you are interested in
the inhibition of diffusion. Such inhibition will involve the sizes
of pores in the food and of molecules of the flavor compounds. Thus
you should use the more basic diffusion model, which includes these
molecular factors.
Example 8: The smell of marijuana. Recently, a large shipment of
marijuana was seized in the Minneapolis-St. Paul airport. The
police said their dog smelled it. The owners claimed that it was
too well wrapped in plastic to smell and that the police had
conducted an illegal search without a search warrant. How could you
tell who was right?
Concept of solution: In this case, you are concerned with the
diffusion of odor across the thin plastic film. The diffusion rate
is well described by either mass transfer or diffusion
coefficients, However, the diffusion model explicitly isolates the
effect of the solubility of the smell in the film, which dominates
the transport. This solubility is the dominant variable. In this
case, the search was illegal.
Example 9: Scale-up of wet scrubbers. You want to use a wet
scrubber to remove sulfur oxides from the flue gas of a large
power. A wet scrubber is essentially a large piece of pipe set on
its end and filled with inert ceramic material. You pump the flue
gas up from the bottom of the pipe and pour a lime slurry down from
the top. In the scrubber, there are various reactions, such as: CaO
+ SO2 CaSO3The time reacts with the sulfur oxides to make in
insoluble precipitate, which is discarded. You have been studying a
small unit and want to use these results to predict the behavior of
a larger unit. Such an increase in size is called a scale-up.
Should you make these predictions using a model based on diffusion
or mass transfer coefficients?
Concept of solution: This situation is complex because of the
chemical reactions and the irregular flows within the scrubber.
Your first try at correlating your data should be a simple model
based on mass transfer coefficients. Should these correlations
prove unreliable, you may be forced to use the more difficult
diffusion model.
When facing the mass transfer operation problem, keep both
models in mind. People involved in basic research tend to be
overcommitted to diffusion coefficients, whereas those with broader
objectives tend to emphasize mass transfer coefficients. Each group
should recognize that the other has a complimentary approach that
may be helpful for the case in hand.The purpose of the equipment
used for mass-transfer operations is to provide intimate contact of
the immiscible phases in order to permit interphase diffusion of
the constituents. The rate of mass transfer is directly dependent
upon the interfacial area exposed between the phases, and the
nature and degree of dispersion of one phase into the other are
therefore of prime importance.
The operations which include humidification and
dehumidification, gas absorption and desorption, and distillation
all have in common the requirement that a gas and a liquid phase be
brought into contact for the purpose of diffusional interchange
between them. The equipment for gas-liquid contact can be broadly
classified according to whether its principal action is to disperse
the gas or the liquid, although in many devices both phases become
dispersed. Transport Phenomena Approach (Transport Phenomena-II)If
we introduce a general variable the conservative form for all fluid
flow equations, including equations for scalar quantities such as
temperature and species concentration etc., can usually be written
in the following form:
(23)
In words,
Eqn. (23) is the so-called transport equation for property . It
clearly highlights the various transport processes: the rate of
change term and the convective term on the LHS and the diffusive
term ( = diffusion coefficient) and the source term respectively in
the RHS.
The integration of Eqn. (23) over a three-dimensional control
volume (CV) yields:
(24)
In words:
The following is the energy equation:
(25)
or
Here we are interested in a stationary volume element, fixed in
space, through which a fluid is flowing (control volume). Both
kinetic energy and internal energy may be entering and leaving the
system by convective transport. Heat may enter and leave the system
by heat conduction as well. Heat conduction is fundamentally a
molecular process. Work may be done on the moving fluid by the
stresses, and this, too, is a molecular process. This term includes
the work done by pressure forces and by viscous forces. In
addition, work may be done on the system by virtue of the external
forces, such as gravity.
References:
1. Walker, W. H., W. K. Lewis, and W. H. McAdams: Principles of
Chemical Engineering,McGraw-Hill, New York, 1923.
2. Y. A. engel, A. J. Ghajar . Heat and Mass Transfer.
Fundamentals and Applications. 4th Ed., McGraw-Hill.
3. E.L. Cussler. Diffusion Mass Transfer in Fluid Systems. 2nd
Ed., Cambridge University Press.
4. R.B. Bird, W.E. Stewart, E.N. Lightfoot. Transport Phenomena.
2nd Ed., John Wiley & Sons. _1418124557.unknown
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