CAL ENGINEERING DIVISION
E A -
CHEMICAL ENGINEERING EDUCATION
Chemical Engineering Division
American Society for Engineering Education
Some Phenomena From Fluid Mechanics,
by C. V. Sterling- - - - - 1
Advances in Heat and Mass Transfer,
by E. R. G. Eckert - - - - - 13
An Undergraduate Course in Analysis of Multistage
by D. N. Hanson - - - - - 24
John B. We
Chemical Engineering Division
American Society for Engineering Education
MArtin (Michigan) Vice C
at (Oklahoma State) Secret
CHEMICAL ENGINEERING EDUCATION (C), Journal of the Chemical
Engineering Division, American Society for Engineering Education.
Published Quarterly, in March, June, September and December, by
Albert H. Cooper,-Editor.
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SOME PHENOMENA FROM FLUID MECHANICS
C. V. Sterling
Shell Development Company
Fluid mechanics is a large and rapidly growing field. The first volume of
the Journal of Fluid Mechanics covered all of 1956 and contained about 400 pages.
Since 1960, in this journal alone, almost 2400 pages have been published each
year, about as many as are published in all the American chemical engineering
journals. Not only in bulk, but also in diversity has Fluid Mechanics been grow-
ing. Imagine a scientific meeting, where the latest advances in fluid mechanics
are to be presented, without regard for the professional interest of the author.
One would undoubtedly meet there a mathematician working on methods for solving
the non-linear flow equations, an aerodynamicist concerned with hypersonic flow,
a physicist formulating equations for the flow of plasmas, a meteorologist de-
vising models for the circulation in the atmosphere and another working on the
Impaction of rain particles, an oceanographer mapping and predicting the ocean
currents and the effects of waves, a geologist with improved methods for compu-
ting the flow of fluids through oil sands, and many mechanical and chemical en-
gineers expounding on a seemingly endless variety of processing problems. It is,
of course, impossible to describe adequately all the significant advances in fluid
mechanics in the space available. Instead, I will take a very restricted view and
ask what aid the science of fluid mechanics can give tb-the practicing chemical
engineer working mainly in the process industries. One might gain the impression,
from skimming recently published texts on fluid dynamics, that the subject has
reached maturity, that its laws are understood, and that solutions must follow
naturally on applying the deductive method. Let us see how far this is true with
regard to the questions in fluid flow that must be answered everyday by some chemi
cal engineer. Typical problems are: After the rupture of the bursting disk on
a vessel containing a vaporizing gas-liquid mixture what will be the rate of dis-
charge of liquid through the line? What part of the vessel contents will be
carried out through the line? How rapidly will the pressure in the vessel drop
to a safe level? If one wishes to protect the particles suspended in an agitated
tank from excessive degradation by collision with walls, impellers or other par-
ticles, what impeller speeds are tolerable? Will a jet of a gas-liquid mixture
entering a distillation column at high velocity splash on the wall opposite and
be partially entrained to the tray above? Will the downward deflected portion of
that jet stir the liquid in tae sump enough to cause troublesome entrainment or
vortexing of gas into the pump suction? How thick will be the head of froth on
top of a viscous liquid in a reactor and what will happen to its thickness when
the vapor velocity in the reactor is raised? How can one calculate the pressure
drop in a line carrying two phases and how does one know whether one of the phases
may settle out and block the pipe? Will a reactor whose heat evolution is removed
by a boiling of a liquid perform smoothly or will it act as a "geyser"? Can one
mix non-newtonian fluids in a tank by jetting a stream of liquid into them? How
will the dispersion point in a distillation tower using perforated plates be af-
fected by a drastic change in the properties of the fluids, such as a large in-
crease in the liquid viscosity and a large decrease in gas density? Will a con-
ventional pneumatic atomizing nozzle function satisfactorily on a gas of very
high density? Ought one to increase or decrease the velocity in a vibrating pipe-
line if one wishes to reduce the vibration?
In one respect, these problems are simple. For an answer often a simple yes
or no will suffice, or if it will not the quantitative answer need not be very ac-
curate. On the other hand, they are very complex in the sense that they have not
been formulated neatly like a textbook problem. Nothing warns the engineer to be-
ware of an instability or to beware of a change in regime of atomization. No one
gives him the hint that the affects of viscosity are negligible but that capillar-
ity must be considered. In all cases, the first step, and indeed, the greater
part of the solution lies in the proper formulation of the problem. To succeed
here, with a reasonable score, the engineer needs to know intimately the "things
that happen", the "phenomena" of fluid mechanics. Once these are recognized, he
can find expert help in the published literature with regard to simplifying con-
cepts, quantitative formulation, methods of solution, or perhaps he may even find
the solution to his immediate problems
With this in mind, we limit ourselves here to describing a few lesser-known
phenomena, selected I fear, somewhat capriciously, but with the intent of illus-
trating the great diversity, fascination and utility of fluid mechanics as viewed
by a chemical engineer.
Fluids flowing past solid, or fluid, bodies exert forces on that body drag
and lift. The calculation of these forces, which is probably the most technically
important problem in fluid mechanics is now fairly well understood. A few novel
studies, throwing light on the sources of the drag, are worth mentioning. Li and
Kusukawa (31) have shown that in the absence of viscosity, heat conduction, and
diffusion,-arag is yet induced by a finite rate of attaining chemical equilibrium.
2 CHEMCAL ENGINIEBING EDUCATION June 1963
Other dissipative processes normally increase drag. For example, complete solu-
tions for the very slow motion of a sphere in an electrically conducting and
m gnetically susceptible fluid are now available.
In calculating the drag on a body one can easily overlook the effects of ac-
celeration of deceleration. If a particle moving through a fluid has no "memory"
its kinematic state is determined by its velocity relative to the fluid, or its
Reynolds number in dimensionless terms. If it has a "memory" although this memory
is short, the drag is, in addition, a function of the derivatives of the velocity
with respect to time. One needs to consider, as the next higher approximation,
how the drag coefficient depends on the dimensionless group
Data on accelerational drag effects are rare. Ingebo's (234) data reproduced here
in Figure 1 show that drag coefficients for accelerating drops and solid particles
are sometimes vastly different from the drag coefficients for steady flow.
It has been shown by Sproull (43) that the addition of dust to the gas flow-
ing in a pipe can reduce the pressure drop. Two explanations of the effect have
been published. That of Sproull attributes the effect to a reduction of the mean
free path of the gas molecules because of collisions with the small solid particle
The other, advanced by SAffman (40) is that the particles act to change the stab-
ility characteristics of the flow and, by implication, the characteristics of the
fully developed turbulence in such a way as to reduce the pressure drop.
Kramer (29) has found that appreciable reductions in the drag that a fluid
exerts on a soTid body can be achieved by coating the surface with a special type
of compliant coating, the artificial "porpoise skin." Benjamin (k) has applied
linearized stability theory to the problem to test the assumption that the re-
duced drag is caused by a stabilization of the boundary layer with respect to the
onset of turbulence. He finds that there should be three modes of instability
with the flexible surface but only one with the rigid boundary. The possibility
of stabilizing the flow depends upon partially stabilizing the mode corresponding
to the rigid wall without "letting in" instabilities of the other types.
A surprising feature of the flow of suspensions of solids in liquids is the
behavior of turbidity currents in the oceans. It is difficult to reconcile the
great range and scouring power attributed by oceanographers to these currents with
the rather gentle slope of the terrain over which they are supposed to have occur-
red. Bagnold (2) has discussed this anomaly and another one related to it associ-
ated with the tFansport of very fine solids by a turbulent stream flowing under
gravity. Suspensions containing grains smaller than about 50 microns in diameter
are sometimes observed to flow with a concentration gradient which increases up-
wards, at least in part of the stream. The ordinary model which would attribute
the suspending action to eddy diffusion acting in opposition to the gravity set-
tling of the particles is hard pressed to explain this anomolous concentration
profile. Bagnold, by a simple semiquantitative argument, shows how gravity can
supply the energy to keep the current suspended in this configuration and also
keep it flowing over the great distances and small slopes observed in the case
of the oceanic turbidity currents. To start the flow, however, it is necessary
to postulate a very large scale avalanche.
At times the design engineer has overlooked the electrokinetic effects of
flow to his ultimate discomfiture. To quote from the text of Klinkenberg and
van der Minne (27): "Time and again there have been mysterious explosions whose
cause was subsequently traced to static electricity." The early theories of elec-
trokinetics, often the only ones finding their way into the textbooks, if that,
are inadequate to explain the ease with which intense potentials can be produced
by flow. Only by introducing modern concepts of turbulent diffusion has it be-
come possible to explain the rapid charging that can occur in the turbulent flow
of even very slightly conducting liquids. These interesting effects are due to
the fact that the electrical double layer lies partly outside of the laminar sub-
layer and hence is strongly disrupted by the turbulence. On the theoretical side
these effects are interesting because of the dominant role played by entrance af-
fects and because of the extreme sensitivity of the results to conditions very
near to the wall. Figure 2 shows how the charging occurs and contrasts the lami-
nar and turbulent flow cases.
As we have seen, a properly directed flow can separate electrical charges.
It can also separate energy. The Ranque-Hilech, or vortex tube, some years ago
popularly called a Maxwell's demon, easily separates a gas into hot and cold
streams without the use of moving parts. Thermodynamic efficiencies are surpris-
ingly high in a well designed tube. One commercial device can separate 4ofm of
air at 700F into a cold stream of .8 cfm at -400F and a hot stream at 90'F. The
tube 'is only 1/4 inch in diameter and 8 inches long. There is, of course, nothing
yoterious about such energy separation. Hartnett and Eckert (20) point out that
ordinaryy isothermal Poiseuille flow also causes n energy separation.
J*me 1963 CHEMICAL ENGINEERING EDUCATION 3
In the past sixteen years there have appeared over a hundred publications
4Lling with the vortex tube effect fVestley (41)7. The recent articles of Rey-
d de(37) present the most easily understood explanation of the various mechanism
leadinEgto the energy separation. Three effects are important. First is the
Knoernschild effect, which describes the heat exchange due to a compressible lump
of fluid moving rapidly through a pressure gradient and consequently undergoing
adiabatic expansion or contraction. Second, there is an Archimedean or buoyancy
effect, which makes the lighter clumps of fluid move inwards. Third, the turbu-
lent stresses furnish a means whereby one layer of fluid can do work on the ad-
jacent layers. All of these effects are also important in determining the energy
exchange taking place in the atmosphere.
Vibration effects flow in several ways. The streaming currents that flow
away from a dound transducer are well known. More spectacular are the inter-
actions of vibrations with a free surface. Figure 3 shows the multinodal columns
of fluid that have been observed to form on an ultrasonic transducer (38). A
simple laboratory experiment, illustrated in Figure 3, shown the almosTexplosive
nature of ultrasonic atomization. The spray is ejected as discrete bursts of
very finely atomized particles.
One of the most intriguing uses of vibrations is in measurement. Eisen-
menger (12) has used an ultrasonically excited hydraulic tank to study surface
properties of fluids- He measured the surface viscosity of pure water, finding
a value of about 10" dyne sec/cm! Operating on a similar principle is the
oscillating jet. In principle one can obtain information about the dynamic sur-
face tension from the spacing of the nodes in a jet issuing from a non-circular
orifice. The recent painstaking efforts of van Duyne (46) were not entirely
successful in this regard however. He found the spacing between successive nodes
on a jet of such a mixture as .01% acetic acid in water to decrease with distance
from the orifice rather"than to increase as one might at first have expected.
Evidently, some factor is still missing from the elaborate theory of the vibrating
jet. This effect must be traced down before the vibrating jet can be exploited
fully as a measuring device. The attention given to the behavior of a jet of non-
newtonian fluids issuing from circular jets may throw some light on this problem
tarrls (21) 7. Such a jet does not always issue from the orifice with its dia-
meter unc-anged but, depending on conditions, may contract shortly after leaving
the tube or, more commonly, it may expand. Careful measurements of this effect
are potentially capable of giving information about the nature of non-newtonian
A rather puzzling effect in the realm of vibrations was found by Hughes
(23) in measuring uptake of carbon dioxide by water drops falling through a gas.
TE5 transfer rates correlate with a postulated "eddy" mixing inside the drops
supposedly caused by their vibration as they fall through the gas. For a linear-
ized model of a vibrating fluid sphere one would expect no gross mixing of con-
centric shells. One wonders whether non-linear effects can account for the degree
of mixing observed.
Resonance effects go hand in hand with vibration effects. The pronounced
effects of pressure resonance between a chamber and a growing and escaping bubble
were pointed out by Hughes et al (22). Perhaps more striking is the observation
by Christiansen and Hixson (9) thatone can make a dispersion of drops of almost
exactly equal sizes the adjusting the velocity with which a liquid jet issues be-
neath the surface of another liquid. This phenomenon is so precisely controllable
as to be useful as a means for measuring dynamic surface tensions.
The mechanics of particles, both fluid and solid, is a subject dear to the
heart of the chemical engineer. He wants to know how to make them (usually
neither too large not too small), how to suspend them, transport them, collect
and coalesce them. In these areas fluid mechanic is but slowly making the change
from art to science.
Dombrowski and Hooper (10) emphasize the rather surprising fact that the
average drop size formed in iEomization from fan sprays first decreases as the
ambient gas density is raised but later increases. This is in accord with pre-
dictions made from models of the instability of sheets of liquid moving relative
to a gas. Their photographs show clearly changes in the mode of breakup of the
sheets as a function of ambient gas density. As shown in Figure 4, the breakup
at very low densities, takes place almost exclusively from the edges of the sheet,
the drops formed there being relatively large. At moderate gas densities, the
leading edge of the sheet becomes unstable, breaks up into drops differing marked-
ly in size from those disrupted from the sides of the sheet.
Let us consider next the single fluid particle moving through another fluid.
When the surface tension is sufficient great as measured by the Laplace number.
D2g /gco, or a related number, the drop is very nearly spherical. When this
number is near unity or larger, however, the surface is no longer spherical but
is distorted in complex ways depending on flow conditions, drop size and on the
CHEMICAL ENGINEERING EDUCATION June 1963
properties of the fluids. Perhaps the least understood of the forms that appear
are those with sharp cusps. For drops falling under gravity a "tear drop" with
& inele cusp at the rear is observed /enraroui and Kintner (13)7, for a drop in
Couette flow a sigmoidd shape" is induced with cusps at either-end iWumscheidt
end Mason (39?17.-
As a problem in the statics branch of fluid mechanics, the theoretical and
experimental work of Allan and Mason (1) on the equilibrium shape and burst of
drops subjected to electric fields is Interesting. They predict that the drop wili
be deformed into a prolate spheroid and that it will burst when its eccentricity
would otherwise exceed /27 By assuming that each half of the burst drop retains
its individual charge they are able to estimate the charge separation caused by
the splitting. An experimental observation not explained is that some drops are
distorted to oblate spheroids rather than to prolate spheroids as required by the
The ways in which atomization can occur are almost bewildering in their com-
plexity. Rumscheidt and Mason (39) have summarized the classes of drop breakup
that they have observed in a Couette flow apparatus (concentric counterrotating
cylinders). As shown in Figure 5, adapted from their paper, there are three main
cases. At very low viscosity ratios, (inside/outside), the drop is progressively
stretched out as the shear rate is increased, finally forming a sigmoidall" drop
with two cusplike ends. From these ends continuous streams of very fine particles
are ejected. For viscosity ratios near unity, the drop develops a neck, which
then rapidly thins down and ultimately ruptures, leaving two almost equal sized
drops and several very much smaller satellite drops from the remnants of the neck.
A third type of breakup is also observed when the viscosity ratio is near one.
Then.the drop is drawn out into a long fine filament which breaks up into many
drops of almost equal size.
In the same apparatus, Forgacs and Mason (15) have observed the behavior of
small flexible fibers (mainly wood fibers). As-7igure 6 shows these fibers follow
quite complex orbits. The more flexible form single loops or multiple helixes.
The interactions of particles with one another and with the walls of the con-
tainer holding them are vital to an understanding of the flow of suspensions.
Oliver (35) has recently repeated in more detail the experiments of Segre and
SilberbeFg. It is observed that the particles in dilute, initially uniform, sus-
pension of spheres in Poiseuille flow are concentrated eventually into an annular
region about halfway between the axis and the wall. Previous theories, which pos-
tulate a lag between the particle and the neighboring flow, show that because of
induced rotation and the Magnus effect the particle ought to move always to the
center of the tube. Oliver's experiments show that particles initially near the
wall do indeed move inwards, but that also particles placed near the axis move
The contention of Goldsmith and Mason (17) that solid particles whether
spheres or elongated, stick close to the tube wall if initially placed there is
explained by Oliver as being due to the very small size of the particles used by
them. There is, however, no denying Goldsmith and Mason's observation that fluid
particles move away from the wall at velocities which greatly exceed that for a
similar solid particle. Related observations bearing on this point are those of
Rumscheidt and Mason (39). A fluid particle is placed in a hyperbolic flow ap-
paratus consisting of four cylinders rotating with theproper senses in a liquid
substrate. A particle of another fluid suspended in the center of the apparatus
exhibits circulation as shown in Figure 8. The circulation is made visible either
by fine suspended solids or by similar fine suspended liquid drops. only in the
case where the tracing particles are liquid is the cross-shaped zone indicated on
the figure swept free of tracer particles.
Lately, there has been a reawakening to the intriguing role played by surface
physics in the flow of fluids. The wide diversity of these phenomena is evident
from the review of Scriven et al (42). To cite only two examples, Jebsen-Marwedel
(25) has described the erosion of lTe walls of glass melting furnaces at a series
of-regularly spaced pits at the glass/gas interface and has attributed this un-
usual pattern to the flow induced by surface tension variations caused by the so-
lution of the wall material in the melt. Another example is the beautiful experi-
ments of the Langmuirs' (30) on the evaporation of ethyl ether out of water solu-
tion. Spontaneous agitation of the surface is very evident and the role of sur-
face active agents in arresting the agitation is demonstrated dramatically by the
quenching of the flame formed from burning ether vapors by the addition of a drop
of oleic acid. As a result of this reawakened interest there have been produced
several very interesting motion pictures /Urell and Weetwater (6)7 illustrating
interfaciall turbulence'. As yet unexplained is the very fast,-often almost ex-
plosive, growth of nascent convection cells that one often sees in these pictures.
The classical chemical engineering models for mass transfer at a fluid-fluid
interface have been somewhat discredited by the recognition of the wide occurenee
Sum 1963 CHEMICAL ENGINEERING EDUCATION 5
of Epontaneous agitation of the interface In the case of systems where mass is
bing transferred. That these models still have their uses can be attributed part-
ly to the equally dramatic effect of surface active materials in arresting the
spontaneous agitation. one indeed wonders whether most plant streams may not be
sufficiently contaminated to substantially suppress these bizarre effects. In any
ease, considerable doubt is thrown on the validity of inferences drawn from com-
parative tests on mass transfer equipment in which care is not taken to reproduce
the type and degree of contamination. An urgent requirement for the chemical en-
gineer is one or more reliable "meters" for characterizing the mechanical effects
of surface active materials. Any phenomena exhibiting these effects which are
sufficiently reproducible and accurately measurable are, therefore, well worth our
notice. The work of Haberman (19) on the rise of circulating and non-circulating
bubbles through tubes, the measurement of zeta potentials of falling jets (11),
and the many phenomena described by Mysels, Shinoda, and Frankel (34) Pre w4rth
mentioning in this regard.
The latter authors, in their Intriguing monograph on soap films, describe
many of the phenomena associated with the drainage and rupture of this films. In
certain cases, a soap film trapped on a vertical wire frame drains, not by a uni-
form thinning over the whole film, but rather by the formation of thin, "black"
spots at the bottom surface and the upward convection of these spots through the
film until they coalesce with a larger region of black film floating atop the
ordinary film. The spots, behave as a two dimensional analog of bubbles intro-
duced at the bottom of a vessel filled with liquid. They obey the two dimension-
al version of Stokes Law!
A determining step in the process of film breakage and of coalescence is the
drainage of fluids out of this films formed by the boundaries between two objects.
The usual approach taken in attempts at theoretical explanations has been unsatis-
fying since it is assumed that the shape of the bodies enclosing the thin film is
uninfluenced by the force generated by the approach. A quite similar problem
arises in the theory of hydrodynamic lubrication where it is found that the de-
formation of even solid bodies can be important in influencing the rate of drain-
age of the films. Christensen (8) shows how one may obtain solutions for the
flows and pressure induced in the gap between two solid cylinders as they approach
one another. Besides considering the case where the solids are perfectly rigid
and the viscosity is constant, he considers the effect of pressure on the viscosity
of the fluid filling the gap and also the effect of plastic deformation of the
solids. The shape pf the permanent set given to the solid bodies is very depend-
ent on the "lubrication" conditions. One might have here a model for calculating
film drainage in cases where the fluid viscosity depends on layer thickness.
The problem of predicting the conditions of flow on thin films flowing under
gravity has long interested chemical engineers. A review of the work on this
problem is instructive in showing both the power and the limitations of theoreti-
cal fluid mechanics at its present stage of development.
What happens when a liquid film flows down a fl-t surface under the influence
of gravity? At very low flow rates and near the top of the plate it flows with a
mirror smooth surface provided the liquid has been well distributed. At flows
above a certain critical rate, about 10 pounds per hour per foot of periphery for
water, rather regular waves develop on the surface. These waves grow rapidly as
the flow rate is raised. At much higher flows, say 1000 pounds per foot of per-
iphery, another change occurs. The appearance of the surface. is modified and the
trend of average film thickness as a function of flow rate shows a break. The
second transition is usually identified with the transition from laminar to turbu-
lent flow since the Reynolds number at which it is occurs is near 2000. Between
this transition and the lower one the flow is laminar but has pronounced waves on
the surface. The waves evidently will affect mass and heat transfer and are there-
fore of interest to chemical engineers.
It would appear that the determination of the transition from smooth to wavy
flow would be amenable to theoretical analysis and indeed this was attempted as
early as 1924 by Kapitza (26). His analysis was apparently very successful since
it predicted the Reynolds nEumber for transition accurately and showed the effect
of surface tension as well (see Figure 8). The paper has not been well received
however because of the obscurity of the procedure used. In 1954, yih (48) made
the suggestion that this problem could be solved by the classical method of
linearized disturbances and that many of the mathematical difficulties that beset
this technique in other problems could be avoided since asymptotic methods ap-
propriate for small Reynolds numbers can be used. Using Yih's suggestion, Ben-
jamin (3) solved the problem and concluded, rather astonishingly, that the flow
is unstable at all Reynolds numbers, no matter how small. The theory, in this
form, apparently contradicts the experiments showing a fairly well defined tran-
sition. Benjamin noted, however, that the rate of growth of disturbances is very
small until the Reynolds number attains a value of about 10. He therefore made
the suggestion that the flow disturbances, although unstable at low Reynolds
6 CHEMICAL ENGINEERING EDUCATION June 1963
nImbers have not had sufficient time to develop to detectable size. Figure 8
shows the results of a quantitative development of this idea. One can define the
pseudo-critical Reynolds number as that at which the disturbances grow at a rate
which will double their amplitude during the time required for the film to move
one wavelength. It is seen that this model agrees with the data and also with
the formula of Kepitza obtained by a entirely independent route. One should note,
however, that the explanation of the transition given by Benjamin has not been
accepted by Tailby and portalsky (44) and (45) who have made the most recent and
extensive series of measurements or the transition.
One is justified, of course, in questioning the assumptions in Benjamin's
analysis. One of these, that the ambient phase exerts no drag on the film, has
been removed by Graebel (18) who considered the case of countercurrent flow of
two fluids of the same viscosity but of different densities. He finds, as did
Benjamin, that the flow on a vertical place is unstable at all Reynolds numbers.
However, if the channel is inclined as little as 1/2 degree off vertical the flow
is now stabilized below some definite Reynolds number. This sensitivity of the
analyses to small changes points the moral of not generalizing the results of
theoretical calculations too fast.
All these analyses assumed the interface to be perfectly compliant. But the
experiments by Tpilby and Portalsky and others show that the addition of surface
active agents suppresses waves formation at low Reynolds numbers. Although this
effect is understood qualitatively, a satisfying quantitative formul-tion has
never been made. As Gibbs has pointed out, in any multicomponent system, the
phase interface must exhibit an apparent elasticity or resistance to deformation.
Such an elasticity undoubtedly suppresses the waves.
As noted above, a falling film shows the classical transition to turbulence
at a Reynolds number of about 2000. It is curious that no one hrs investigated
the stability of this type of turbulent flow by the conventional techniques.
This configuration might facilitate comparative experiments on turbulent transi-
A flow closely related to that we have just been discussing is that of a
this film driven by friction exerted by another fluid. It occurs in transpiration
cooling and in the annular flow regime of gas-liquid pipe flow. Knuth (28)has
shown that waves are raised on the film at film Reynolds numbers larger Than about
200. The first attempt to explain this observation theoretically was made by
Feldman (14). He formulated the stability problem for the case where the driving
fluid behaves as if it were laminar near the interface. The results of the analy-
sis do not agree with the data of Knuth, the Reynolds number for transition being
about an order of magnitude higher than the experiments would lead one to believe.
Since in the experiments, the driving phase was turbulent, it is natural to
question and modify the boundary conditions expressing the tractions exerted by
the driving phase on the surface. Miles (32) has done this. At first glance,
his modification to the shear stress conditTon at the surface is surprising. He
assumes that the stress there has the same value in the disturbed flow as in the
undisturbed flow. While cogent arguments can be advanced for making this assump-
tion, they are not entirely convincing. The ultimate test must be comparison of
predictions with experiment. In this respect, Miles is moderately successful.
He finds that the film is stable provided either the Reynolds number or the Weber
number is small enough. As is shown in Figure 10, Knuth's results are substanti-
One may, I think, draw a moral from the history of these attempts to predict
conditions for the raising of waves. Theoretical analyses must be simplified
considerably in order to be tractible. It is possible by judicious juggling of
the assumptions to force the analyses to agree with data. Such analyses must at
present be regarded as secondary, though admittedly they are very powerful tools,
to experimental observations.
As a closing comment on this topic it is worth calling attention to the re-
cent paper of Bushmanov (6), who presents two analyses of the problem of the
stability of the falling film. By the first approach, substantially the same as
Benjamin, he gets the wrong answer! By the second, he improves the analysis of
Kapitza, by relaxing some of the assumptions made but in so doing destroys the
apparent good agreement of this analysis with the data which was shown in Figure
In view of the comments made by Bushmanov to reconcile his analyses of the
stability of thin films with experiment, the article by Caldwell and Donnelly (7)
is of great interest. They studied the hysteresis of transition to turbulence In
thepopular rotating concentric cylinder apparatus. Very precise measurements of
torque were made for contrasting series of runs in which the relative speed of
rotation was increased or decreased. Within the accuracy of the experiments,
which was very good, it appears that there is no hysteresis in this particular
June 1963 CHEMICAL ENGINEERING EDUCATION 7
Let us now return to the question posed at the beginning of the paper; To
what extent can the chemical engineer use the results of fluid mechanics in the
solution of his everyday problems? We must, I think, confess that there are a
great many problems where the aid furnished by the conventional literature is
slight; many of his problems must be solved by the engineer himself without the
aid of the professional fluid mechanician. The approach may be purely empirical
or the engineer himself may have to become an expert in fluid flow. Dpily, the
chemical engineer gives his opinion on problems similar to the ones listed in the
introduction. Often the answer is given tacitly, sometimes unfortunately, un-
knowingly, in the act of making of approving a design. Fortunately, luck is often
with him; his design works well without his understanding fully why. Most success-
ful, however, is the engineer who has formulated the relevant problems in his
mind, who has considered all the evidence that may bear on the answer, and has
designed into his product those features most likely to make it work and designed
out the sources of trouble. He has the honesty to recognize what he does not know
or cannot economically find out, and has the courage to assess realistically the
risks of uncertainty and to take these risks when Justified.
In some years, perhaps, the engineer will have more help in solving his flow
problems. To advance this day it is important for all of us to look sharply at
the "things that happen", examine them critically in the context of related phen-
omena and of theory and to point them out to one another. As a small contribution
I would call your attention to Figure 10, which shows an enlarged view of an air
bubble in a sparged vessel containing water. Does not the roiled surface of the
bubble strongly resemble the roiled surface of a quiet but turbulent river and
ought not the characteristics of the roiling have some bearing on the mass trans-
fer rates calculated on the model of surface renewal?
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Poiseuille Flow of Suspensions", Nature 94. 1269 (1962.
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Suspensions XII Deformation and Burst of Fluid Drops in Shear and
Hyperbolic Flow", J. Colloid Sci. 16 238 (1961).
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Down a Hydrodynanically Smooth Plate", Chem. Eng. Sci. l, 2835 1962).
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Nr. 9 College of Aeronautics, Cranfield (1954).
Proc. Second US Nat Cong of Appl. Hash. 623 (1954).
'urc 1 \CCELERATIONAL DRAG DATA OF INGEBO
a) How Charging Occurs
1 Laminar Sublayer
b) How Turbulence Influences
Figure 2. ELECTRIC CHARGING INDUCED BY TURBULENT FLOW
After Klinkenberg and van der Minne
a) Multinodal Liquid Pro
I fezoelectric I
ejections Observed in Transducer
. After Rozenberg
b) Simple Experimental Setup to Demonstrate
Figure 3. ULTRASONIC ATOMIZATION
b) Dominant B...reak.piat
Leading Edge when Gas
Density is Moderate
a) Breakup from Side Edges Only
Occurs at Low Gas Density
Figure 4. REGIMES OF ATOMIZATION OF FAN SPRAYS
According to Dombrowski and Hooper
0 0 0 o-.o
Increasing Shear Rate
Figure 5. BREAKUP OF DROPS IN COUETTE FLOW
According to Rumscheidt and Mason
Complex. C.) C- C4 C-" e:
Figure 6. ROTATION OF FIBERS OF VARIOUS FLEXImILITIES
According to Forgaes and Mason
a) Streamlines are Traced
by Solid Particles. No
b) Streamlines are Traced
by Fluid Particles. Par-
ticles Move away from
c) Contrasting Motion of Solid and Fluid
Particles Near Wall in Poiseuille Flow
Figure 7. MOTION OF SOLID AND FLUID PARTICLES
IN LAMINAR FLOW
After Rumscheldt and Mason
Kapitza s Equation
0 A --
-C "Data of
S Equaion Based on 0 Jackson
Benjamin's Model A Grimley
I I I I I I I I I I I I I I I I }
Figure 8. CRITICAL WEBER
FOR LIQUIDS FLOWI L
IvUivnen run ways runavi LLuI
10 Unstable with
Curve of Miles
0 0.1 0.2 0.3 0.4 0.5
Figure 9. STABILITY OF A THIN DRIVEN FILM
Figure 10. AIR BUBBLES IN WATER SHOWING ROILED SURFACES
Courtesy of G. D. Towell, Shell Development Co.
ADVANCES IN HEAT AND MASS TRANSFER
E. R. G. Eckert
University of Minnesota
Research in heat and mass transfer has received a strong impetus by new en-
gineering developments and has therefore grown considerably in recent years. This
is, for instance, evidenced by the fact that the last one of the yearly reviews
published in the International Journal of Heat and Mass Transfer contains approxi-
mately 500 references selected from more than double the number of published pap-
ers (ref. 1). A survey of recent papers and books from the Soviet Union lists 259
references (ref. 2). Consequently, the time available for this lecture permits
only to discuss in general terms some highlights in the recent research and in the
newer problems presented by engineering developments. The list of references at-
tached to this paper will be helpful for a more detailed information on the sub-
jects discussed. The availability of reference literature refss. 1 to 3) may also
be pointed out.
Considerable attention has been directed in recent years to obtain new solu-
tions or to describe new methods of attach on conduction problems. This situation
has been created by the fact that new engineering developments required consider-
ation of new and more involved boundary conditions and also that the availability
of electronic computers made such solutions possible. The real challenging prob-
lems, however, are situations in which heat conduction is interrelated with con-
vection and possibly radiation. An example for such a situation may be discussed
with the help of Fig. 1, which is a schematic sketch of an ablation cooling pro-
cess. In this cooling method, a material which sublimes or decomposes under the
influence of heat is used to protect the surface of vehicles re-entering from
outer space through the atmosphere to the earth against the heat of friction
created in the boundary layer which surrounds the object. Some of the materials
used for ablation cooling are composed of a matrix of a temperature-resistant sub-
stance like asbestos or ceramics and an ablating substance, for instance, some
plastic. Under the influence of a convective heat flux qc and a radiative heat
flux qr into the surface, the plastic material decomposes and its surface recedes
with a velocity va, leaving the matrix through which the gas created by the abla-
tion process flows with the velocity v The low heat conductivity of the matrix
keeps the heat flow into the interior Small. Further cooling of the matrix and a
reduction of the convective flow into the surface is provided by the gases created
in the ablation process. It is easily seen that in this process conduction, heat
convection, and mass transfer processes are interrelated. Additionally, heat
sources or sinks are provided by the phase changes and the chemical reactions
occurring (ref. 4). In some materials, the radiative heat flux qr is only gradu-
ally absorbed while penetrating into the ablating material. This combination and
interrelation of various transfer processes is characteristic for many situations
in new applications.
Channel flows as well as boundary layer flows offer such a variety of boun-
dary conditions that they are far from being completely investigated. An area
in channel flow which has recently received attention is connected with ducts of
non-circular cross-sections. It was found that the flow and heat transfer char-
acteristics in such a duct are significantly different from those observed in
circular pipes, especially when the cross-section contains corners with small
opening angle. Fig. 2 shows as an example the results of measurements for fully
developed turbulent flow through a duct the cross-section of which has the shape
of an isosceles triangle with base to height ratio 1:5 (ref. 5). The measured
results indicated by the open circles and triangles are compared with the Nusselt
numbers Nu which would be predicted from measurements in a circular pipe of the
same hydraulic diameter. It can be recognized that the prediction would over-
estimate the heat transfer by a factor of two. The actual heat transfer in such
a cross-section depends strongly on the boundary condition around the periphery of
the duct, for instance, whether a constant wall temperature or a constant heat
flux from the duct wall into the fluid is prescribed. The boundary condition for
the results in Fig. 2 was between the two extreme cases which have just been
measured. Ducts of similar shapes have been used or are considered for coolant
passages in nuclear reactors and in gas turbines. The differences between non-
circular and circular ducts are less pronounced when the cross-section has only
large angle corners (ref. 6).
A situation in boundary layer flow which has found special attention only
recently is connected with heat transfer in regions where the flow has separated
from the surface. This occurs, for instance, at the downstream part of blunt
_4 CHEMICAL ENGINEERING EDUCATION Juno 1963
Objects or behindstefps in the surface contour. Fig. 3 sketches such a heat
transfer situation. The boundary layer which arrives in its downstream movement
at the corner of the atop separates from the surface and re-attaches again only
further downstream. The region between this boundary layer and the surface is
filled with a rotating body of fluid (dead water region). A second boundary
layer is created between this dead water and the surface. Heat transfer from the
main stream to the body surface behind the step has therefore to overcome two
resistances in series: one in the separated boundary layer and one in the at-
tached boundary layer. A large variety of flow situations may exist depending on
the Reynolds and Maoh number of the flow. The boundary layers may be laminar or
turbulent, or transition to turbulence may occur in the boundary layers. The body
of separated fluid may also be laminar or turbulent or in fluctuating unsteady
motion. This complicates an understanding and analysis of the heat transfer
process considerably. Nevertheless, analytical approaches for some of the flow
conditions were quite successful refss. 7, 8).
It is the opinion of this lecturer that the most significant advance in the
creation of a science of heat transfer was caused by the concept of a constant
property fluid introduced by Wilhelm Nusselt in 1916 and that teaching of heat
transfer has to be based extensively on the model of a constant property fluid.
New engineering developments, on the other hand, have created many situations
where large property variations exist. Such variations cause no principal diffi-
oulties in laminar flow but make the equations describing this situation much more
complicated. Only the advent of electronic computers made solutions to such prob-
lems tractable. For turbulent flow the question arises whether the turbulent
transport properties, for'instance the diffusivities for momentum and heat, are
changed in the presence of local property variations. From the results of analyse:
it appears that this is not the case even in the presence of strong property vari-
ations. Fig. 4 compares the results of experiments with theoretical predictions
for heat transfer connected with turbulent flow of carbon dioxide in the critical
and super-critical region through a tube (ref. 9). The properties involved in
the heat transfer process vary very strongly in the neighborhood of the critical
region. The parameter on the curves is a measure of the intensity of the
property variations and the good agreement between analysis and measurement sup-
ports strongly the assumption that the transfer properties were not influenced
by the property variations connected with these temperature differences. This
does not hold any more in the immediate neighborhood of the critical state. In-
vestigations on free convention heat transfer with a Zehnder-Mach interferometer,
on a vertical plate exposed to carbon dioxide within one degree to its critical
state and with temperature differences of order 0.010C, made it possible to cal-
culate the thermal conductivity (ref. 10). The remarkable conclusion has to be
drawn from the results that the thermal conductivity does not only depend on
temperature and pressure, but also on the intensity of the heat flux. Similar
observations had been made before but had been attributed to convection effects
which can be excluded in the present investigation. This is only one indication
that an understanding of the critical state is still almost non-existent.
Through many years, convective heat transfer had been studied almost exclu-
sively for steady state situations. This is justified by the fact that very rapid
changes of the boundary conditions are required to produce heat transfer coeffic-
ients which are significantly different from steady state values. Nevertheless,
new applications have raised the question on the limit for the use of steady state
values. Fig. 5 shows the results of another interferometric investigation of free
convection heat transfer to a vertical place under the condition that suddenly a
locally uniform heat flux from the surface into the surrounding water is started
(ref. 11). The measurements essentially indicate that for the initial period,
heat is transferred into the fluid by unsteady conduction and that the subsequent
transition period to steady state is quite short.
Heat transfer connected with boiling or condensation is an area which is still
understood only partially in spite of the intense research effort which has con-
tinuously been devoted to this process. The problems and research attempts in this
area are actually so large that they cannot even be sketched in this lecture. For
a discussion of the physical processes involved and of the analytic attempts which
have been published, the reader is referred to the attached references 12 and 13.
It is the feeling of this lecturer that the creation of concise and consistent
models is still lacking in the analytic investigations. Boiling and condensation
of liquid metals, for instance of mercury, and the influence of gravity on free
convection boiling are subjects which deserve special attention.
A discussion of mass transfer should certainly start with the analogy between
heat and mass transfer processes which has originally been pointed out by Wilhelm
Nusselt in 1916. This analogy permits to predict mass transfer situations solely
from information on an analogous heat transfer process especially when the mass
transfer rates and the temperature differences involved are small. Recently this
June 1963 CHEMICAL ENGINEERING EDUCATION 15
-nalog has been extended by Russian and American scientists to cover situations
With large mass transfer rates, large temperature differences chemical reactions
in the fluid, and also especially at high temperature levels (ref. 14). The ana-
logy includes processes in which heat and mass transfer are interrelated. They
are especially useful for gases and will be discussed on the example of mass
transfer from a surface into a gas flowing over it, thus creating a two-component
mixturein the laminar boundary layer. Eq. (1) in Pig. 6 describes a mass flux
vector mi for the component I in the mixture, that is, the fass flux per unit time
and unit area of a plane arbitrarily located within the two-component fluid. The
first term on the right-hand side of the equation describes the mass flux by dif-
fusion as a consequence of a gradient wi of the mass fraction of the component 1.
D12 is the mass diffusion coefficient and the mixture density. The second term
describes the convective transport of the component i with a density I as a con-
sequence of a movement of the mixture with the velocity V through the plane under
consideration. Mass transfer is also created by coupled effects like thermal dif-
fusion, pressure diffusion, or diffusion as a consequence of body forces. These
effects are generally small and are not spelled out in Eq. (1). A heat flux
vector q (heat flow per unit time through a unit area) will also exist and is de-
scribed by Eq. (2). The first term on the right-hand side of this equation des-
cribes heat transfer by conduction as a consequence of a temperature gradient T.
The second term describes the transport of enthalpy hi connected with the diffusion
mass flux of the components involved. The third term describes transport of the
enthalpy h of the mixture as a consequence of the mixture mass flow V through
the plane under consideration. Additional terms appear again as a consequence of
coupled effects described by irreversible thermodynamics like thermodiffusion.
Eq. (2) can be re-written in a different form when the gradient of the temperature
T is replaced by the gradient of the mixture enthalpy h (Eq. (3)). This equation
is especially useful for a fluid with a Lewis number Le equal to one, because the
second term drops out in this case. The Eqs. (1) to (3) have been written in a
form which is most useful for mass transfer situations where conduction and con-
vection occur simultaneously.
The boundary layer Eqs. (4) and (5) of Fig. 6 are obtained by a mass balance
of the individual components and by an energy balance on a volume element located
within the boundary layer. Chemical reactions occurring within the boundary layer
destroy or create one or the other component and appear, therefore, in the mass
balance of Eq. (4) as a source term Ki. The energy Eq. (5) has been written in
terms of the total or stagnation enthalpy hg (containing kinetic energy as well as
internal energy). The first term on the right-hand side of this equation describes
essentially heat transport by conduction; the second term, heat generation by
viscous dissipation which becomes important in high velocity flow; and the third
term is concerned with enthalpy transport by mass diffusion. From Eqa. (4) and
(5) one comes to the similarity considerations by two important steps: The first
one entails writing the mass balance for the chemical elements involved instead of
the two components; thus wi may indicate the mass fraction of the element i in the
mixture regardless whether the element appears as such or in the chemical compound.
No chemical element is created or destroyed in a chemical process, and as a conse-
quence the source term Ki vanishes in Eq. (4) when it is written for the chemical
element. The second step assumes that, for the fluid mixture under consideration,
the Prandtl number Pr as well as the Lewis number Le are both equal to one. Eqs.
(4) and (5) simplify then to Eqs. (6) and (7) which can be recognized to be com-
pletely similar. As a consequence, analysis of mass transfer processes and of
combined mass and heat transfer processes becomes much simpler because the most
difficult elements in such an analysis can often be taken over from known solu-
tions of an analogous heat transfer situation. Proper boundary conditions have
of course to be considered in such an analysis which may include chemical processes
occurring at the surface. It should be mentioned that the mass and energy conser-
vation equations alone do not describe the transfer problem completely. A momen-
tum equation and equations for the thermodynamic and transfer properties have to
be added. The analogy, however, holds independent of these. It has become es-
pecially useful in an analysis of problems like combustion or heat transfer to re-
entering vehicles as mentioned in the section on Conduction. Approximate rela-
tions have also been developed which extend the analysis to situations with Prandtl
and Lewis numbers different from one (ref. 14).
Coupled effects have been neglected in the discussion up to now because they
are unimportant in many mass transfer situations. Recent studies refss. 15, 16),
however, have demonstrated that one has to be careful in this respect. This will
be discussed with the help of Fig. 7 which presents the results of the following
experiments (ref. 16). A cylinder with porous surface was exposed on its outside
to a flow of air in axial direction at a Reynolds number which created a turbulent
boundary layer. Helium was injected from the inside of the cylinder through the
porous surface into this boundary layer. A difference in the temperature with
which the helium was fed into the cylinder and the air temperature T-ot side of
the boundary layer could be adjusted by pre-heating or pre-cooling oothe helium.
Fig. 7 presents the heat flux q through the porous surface as a function of the
difference between the wall surface temperature Tw and the air temperature Too
16 CHEMICAL ENGINEERING EDUCATION June 1963
wilh the specific mass flow m of the helium as parameter. One series of measure-
me-,ts made with air instead of helium injection is also entered as dashed line.
The heat flux qw is defined as the sum of the first two terms in Eq. (2). The
striking feature in this figure is the observation that the heat flux becomes zero
at a finite temperature difference Tw Too and it is believed that this is a con-
sequence of the fact that concentration differences within the boundary layer ap-
pear through thermodiffusion as driving force in addition to conduction. Another
consequence of this interplay of driving potentials is the fact that a finite dif-
ference between the wall surface temperature T and the air temperature T exists
when the helium is admitted into the porous cylinder at a temperature To equal to
the temperature in the outside air flow. This situation is marked in the figure
by crosses and it can be recognized that the wall temperature may-be up to almost
30 degrees higher than both the helium and the air temperatures, depending on the
injection rate of the helium. Similar effects have also been observed in laminar
boundary layers for forced and free convection.
An area in which investigations have recently started are transfer processes
in a gas plasma. In such a plasma, the temperature is so high that dissociation
and ionization occur. Transport processes in such a situation are therefore most
involved because mass transfer processes are interrelated with heat transfer,
chemical reactions occur, at least three-component mixtures of neutral atoms, ions,
and electrons are involved, and electric as well as magnetic body forces influence
the flow. An example of a recent experimental investigation in this area is shown
in Fig. 8 which presents and analyzes the local heat flux distribution into a
water-cooled anode of an electric arc burning in argon (ref. 17). It can be recog-
nizdd that only a small portion of the specific heat flux q into the anode surface
is caused by convection of the atom gas. Convection of the electron gas, which
is generated at the cathode and absorbed by the anode, contributes approximately
an equal fraction and the rest is due to energy released when the electrons enter
the anode material (similar to a heat of condensation). The heat flux q indi-
cates the electrical energy which is converted into heat within the current tube
ending at the anode location under consideration. It may be recognized that the
majority of the electrically generated heat enters the anode surface. This is
the reason for the many burnouts occurring in electrically heated plasma genera-
tors at the anode surface. The heat fluxes q occurring at the anode surface on
spots at which the arc strikes are among the largest known in any engineering ap-
Radiative Heat Transfer
An important tool in all radiative heat transfer calculations is the shape
or angle factor. Graphical, mechanical, and optical means have been described,
in addition to analytical methods for its calculation. The analysis can, in many
cases, be considerably simplified by converting an area integral describing the
shape factor into a line integral (ref. 18).
The network method for the solution of radiative heat transfer problems in
enclosures, which lumps emitted radiation together with the reflected parts, is
of such advantage that it has been introduced into practically all recent books
on heat transfer. The analogy to electric circuits, illustrated in Fig. 9, gives
in many situations without analysis a feeling for the heat flux distributions oc-
curring in such a transfer process. The network method is applicable to enclosures
the surfaces of which emit radiation according to Lambert's cosine law and which
reflect diffusely. Many engineering materials, on the other hand, have surfaces,
the reflection of which comes closer to a specular than to a diffuse character.
Some measured directional distribution curves of reflected radiation are shown in
Fig. 10. In enclosures with specularly reflecting surfaces, the analytical ap-
proach has to be different and has to consist of summation of the first, second,
third, and so on reflections. The analysis is simplified when one introduces the
optical images as shown in Fig. 11 in which 1 (3), 2 (3), and 4 (3)denote the
optical images of the diffuse surfaces 1, 2, and 3 created by the specularly re-
flecting surface 3 (ref. 19). This image method can be combined with the network
method for enclosures consisting partially of diffusely and partially of specular-
ly reflecting surfaces (ref. 20). The network method is actually an approximation
to the integral equations which in principle describe radiative transfer processes.
It is important to obtain exact solutions to the integral equations for a few
simple situations in order to get a feeling for the errors which may be introduced
in the network method. Several recent papers have started to formulate radiative
heat processes in this form including the scattering mechanism in a radiating-ab-
sorbing medium filling the enclosure (ref. 21). Engineering analyses usually
attack problems in which radiative energy occurs simultaneously with other trans-
fer mechanisms like conduction or convection in such a way that the various con-
tributicns are calculated separately and that the total energy transfer is ob-
trined by a summation of the individual parts. In reality, situations are en-
countered in which the various transfer processes interact. Such interactions
jave been studied for a few oases in the recent past (ref. 22).
June 1963 CHEMICAL ENGINEERING EDUCATION 17
At the end of our discussion we will return to the molecular and convective
transport processes with a brief review of the similarity between radiative and
molecular transport (ref. 23). We can consider radiative transport as caused by
the movement of photons in a similar way as energy, mass, or momentum transport
is caused by the movement of molecules. Fig. 12 illustrates this similarity by
considering Couette flow or heat conduction in a rarefied gas between two parallel
walls on one hand, and radiative energy transfer in a radiating and absorbing
medium between two parallel walls which are non-transmitting on the other hand.
The temperature or velocity variation between the two walls follow a straight line,
as indicated in Fig. 12a, as long as the ratio of the mean free path length is
very small compared to the distance L of the two wells. With increasing path
length, temperature or velocity still exhibits the linear variation within the gas;
however, a slip of the velocity or a temperature jump can be observed in the im-
mediate neighborhood of both wall surfaces (fig. 12b). For situations, on the
other hand, in which the mean free path length is large compared to the distance
L, velocity or temperature in the gas is uniform (Fig. 12d). The terms for the
corresponding regimes are indicated above the figures. Completely analogous situ-
ations exist for the radiative transfer process. The black-body emissive power
eb has now to be considered instead of the velocity or temperature. This emissive
power drops linearly in the absorbing and radiating medium as long as the free
path length of the photons is small compared to the distance L. Jumps near the
surface of the two walls occur with increasing photon path length. The variation
in the absorbing medium itself decreases towards zero when the photon path length
vets larger and larger. The terms for these regimes are listed below the figures.
This similarity is very helpful in unifying the concepts for transfer processes
and such a unification and interrelation of the concepts I would consider as one
of the most essential' requirements of a good course in transfer processes.
S V VELOCITY OF ABLATION
qc Vg VELOCITY OF ABLATED GAS
.- q.c CONVECTIVE HEAT FLUX
Si q, RADIATIVE HEAT FLUX
CHEMICAL ERGIEERIUNG EDUCATION Juan 1963
1) E. R. 0. ,ehart, T. F. Irvine, Jr., I. N. Sparrow, and V. E. Thele,
"Heat Transfer, A Review of Current Literature,' International Journal
of Heat and Mas Transfer. vol. 3, pp. 293-306. 1961.
2) A. V. Likon, HBeat Transfer ibliograplh Russiaan Works,'
International Journal of Heat and Mes Transfer, vol. 5, pp. 571-582.
3) Recent Advances in Mast and tasg ransfegr J. P. Hartnett, Editor.
MUcGraw-Hill Book Company, New Tork. 1961.
4) T. R. Munson and R. J. Spindler, "Transient Thermal Behaviour of
Decomposing Materials," IAS Paper No. 62-30. 1962.
5) E. R. G. Eckert and T. F. Irvlne, Jr., "Pressure Drop and Heat Transfer
in a Duct with Triangular Cros-Sectim,m Journal of Bansic Engineering,
v.l. 82, pp. 125-138. 1960.
6) w. H. Lodaermilk, V. F. Weiland, Jr., and J. N. B. Livingood,
"Measurement of Heat Transfer and Friction Coefficients for Flow of Air
in Nonecircular Duets at High Surface Temperatures," NACA Research
Memc. L53J07. 195k.
7) D. R. Chapman, "A Theoretical Analyaia of Heat Transfer in Regions of
Separated Flow,'- ICA TN 3792. 1956.
8) W. 0. Carlson, 'Heat Transfer in l minar Separated and Wake Flow Regions,"
1959 Heat Transfer and Fluid Mechanics Instituta, Stanford University
ress, Stanford, California.
9) R. G. Deissler, "Convective Heat Transfer and Friction in Flow of Liquids,"
iAh Speed Aerodynaics and Jet Propulsion, vgl. V, Princeton University
Paes, Princeton, H. J. 1959.
10) Harold A. Simo, "An Interferostric Inreatigation of Laminar Free
Convection in Carbon Dioxide Near Its Critical Point,' Ph.D. thesis,
University of Minnesota. 1962.
11) R. J. Goldstein and E. R. 0. Eckert, "The Steady and Transient Free
Convention Boundary Layer on a Uniformly Heated Vertical Plate,'
International Journal of Heat and Mass Transfer, vol. 1, pp. 208-218.
12) R. B. Sabersky, "Survey of Problems in Boiling Heat Transfer,"
High Speed Aerodynamica and Jet Propulsion, vol. V, Princeton Univereity
Press, Princeton, N.J. 1959.
13) W. M. Robshnow and H. Y. Choi, "Heat, Mass, and Momentum Transfer,"
Prentice Hall, New York. 1961.
14) L. Lees, "Convective Heat Transfer with Mass Addition and Chenicl
Reaction,." Recent Advances in Heat and Mass Transfer, editor J. P. Hsrtnett,
McGra-Hill Book mpany, New York. pp. 161-222. 1961.
1i) J. R. Baron, "Thermodynaaic Coupling in Bondary Layers," American Rocket
society Snice Flight Report to the Nation 2206-61. 1961.
16) 0. E. Tewftk, F. R. G. Eckert, and C. J. Shirtliffe, "Thermal Diffusion
Effects on Energy Transfer in a Turbulent Boundary Layer with Helium
Injection." Proceedings 1962 Heat Transfer and Fluid Mechanice Institute,
Stanford University Prees, Stenford, California. pp. 2-61.
17) P. Schoeck and E. R. 0, Eckert, "An Investigation of Anode Heat Transfer
in High Intensity Ares," Proceedings of the 5th InteonmU onal Conference
on Tonloation Phenomena in Gases, Munih 1961. North-Holland Publishing
Company, Amsterdam, Netherlands.
18) E. M. Sparrow, "A New and Simpler Formulation for Radlative Angle Factors,"
American Society of Mechanical Engineers, Paper No. 62-HT-17. 1962.
19) E. R. 0. Eckert and E. M. Sparrow, "Radiative Heat Exchange between
Surfaces with Specular Reflection," International Journal of Heat and
Mass Tranafar, vol. 3, pp. 42-5h. 1961.
20) E. M. Sparrow, E. R. 0. Eckert, and V. K. Joneson, "An Enclosure Theory
for Radiative Exchange between Specularly and Diffusely Reflecting
Surfaces," American Society of Mechanical Enginsers, Paper No. 61-NAl-167.
21) R. Viskanta and R. J. Groshb, "Heat Transfer in a Thermal Radiation
Absorbing and Scattering Medint," International Developments in Heat
Transfer, part IV, pp. 20-828., American Society of Mechanical Engineers.
22) R. Viskanta and R. J. Grosh, "Heat Transfer by Simaltaneo Condnctlmo
and Radiation in an Absorbing Madium," Journal of Heat Transfer,
vol. 84, pp. 63-72. 1962.
23) E. R. G. Eckert, "Similarities between Energy Transport in Rarefied Gases
and by Thermal Radiation." Modern Developments in Heat Transfer,
Academic Press, New York. 1962.
I I I I I I I I I I
1000 3000 10,000
DlICIr,'G 3TREWM LINE
v "STEADY STATE)
o x 3.82 in. 00r 03.--
a 2.11 In. 0
A 1.03 In.
v 0.32 In.
10 100 12
MASS AND HEAT TRANSFER EQUATIONS:
rhi= D12 Vwi + Pi V + COUPLED EFFECTS (1)
=- Vh-(Le-I)EhiVwi-PVh+ + (3)
BOUNDARY LAYER EQUATIONS:
P(U + v- )= (PD, i)+ Ki (4)
+ rPD, i--L lh i] (5)
p(u "fi+vi)- a (61
p (u s+vh s)= (s) (71
I I I I I I I I I I I
800- r [Ibm/hr ft]
4.2 i ,
S2I0.2 HELIUM -.
S* 62.7 AIR
I I II
5 I= 150AMP, s=6mm
wev oy }SPLIT PLATE ANODE
4 TV *
C4 V ', ;}POINT PROBE ANODE
3 -BY ELECTRON v' HEAT FLUX TO ANODE
2 14 o0
ELECTRONS TO '',
BY AT M GAS
0 I 2 3 4 5 6 7
H3= BF3_+ B2F3-2+ B4F3-4
B4 = 4 T44 +p4H4
H4= BI(F4 + p3F41(3))
+B2 (F4-2 + P3 F4 2(3))
\ I \
IL a Aluminum point
b Iron, scraped
30* c Iron, hot rolled
AN UNDERGRADUATE COURSE IN ANALYSIS
OF MULTISTAGE SEPARATION OPERATIONS
D. N. Hanson
Department of Chemical Engineering
University of California, Berkeley
Multistage separation operations have been a part of Chemical Engineering
ever since the beginning of Chemical Engineering as a field. The unit operation
of distillation appeared in the earliest textbooks on unit operations and has been
studied by every undergraduate chemical engineer in the United States. Other sepa-
ration operations such as absorption and extraction have had less emphasis. Pro-
cesses such as reboiled absorption and refluxed stripping, as typified by crude
oil columns, and processes which exist but are unnamed, have had essentially no
attention in undergraduate chemical engineering courses. In addition, the treat-
ment of the whole of these operations has been limited in the usual undergraduate
course. If we take distillation as an example, the usual undergraduate course
discusses principally the McCabe-Thiele diagram or the Ponchon diagram for the
analysis of binary distillation problems. Extraction is limited to problems solve
on a triangular diagram or a JInecke diagram.
The student certainly gains an understanding of separation processes from
these problems and he is also well-equipped to solve additional problems of the
same type. Unfortunately the chances of his encountering a binary system in
later work are very small while his chances of encountering a multicomponent sys-
tem are very large. Also, many of the processes he meets will not be simple dis-
tillation or extraction, and he will need methods of analysis which are general
enough to be extended to any problem.
At most schools, this more general material is presently taught in graduate
courses. However, there are increasing numbers of B.S. graduates doing the pro-
cess design work of industry while the M.S. and Ph.D. graduates are increasingly
doing less of it, and it is thus the B.S. graduate who needs to know the techni-
Even without the justification of the need of technique, the undergraduate
courses should certainly treat as many areas of the subject matter as possible in
as general or all-inclusive a fashion as can be done. All parts of the field are,
of course, not susceptible to generalization. Calculational procedures or analy-
ses have been worked out which should be taught simply because the single problem
for which they are applicable is a highly important problem. However, the funda- .
mental requirements which must be met to create a multistage separation process
can be generalized, and the first basic analysis of a given process in terms of
what independent or arbitrary numbers can and must be set can be generalized. In
ordinary distillation, as soon as the number of components is three or four, many
methods of attack on problems are of a general character for any number of compo-
nents. In addition, methods of calculation which are capable of either a very
high degree of generalization or complete generalization have become useable over
the past few years because of the routine availability of computers, and these
methods should certainly be taught.
I would like to outline here the content of a course we have been offering
for the last few years to the first semester seniors in approximately twenty
lectures, and which we intend to offer in the future to second semester juniors
in approximately thirty lectures. The course assumes that the student already
has a knowledge of equilibrium constants and their use in the calculation of
simple bubble points, dew points, or flashes. Plate efficiency and the design
and capacity of equipment are omitted to be treated in later mass transfer and
plant design courses.
The course thus concentrates completely on the calculational analysis of
multistage processes, and even within this limitation it is necessarily a col-
lection of a few methods and approaches out of the tremendous bulk of methods
which has been published. The choice of the particular methods could easily be
different since so much is available. Even the areas of coverage could be dif-
ferent and in the future undoubtedly will be.
If we refer to Figure 1, the course starts out by illustrating a typical
multistage separation process in which the separation desired on the mixture of
A and B fed to the column of stages is not good enough in a single stage and is
in effect, multiplied by a succession of stages to obtain the separation desired.
The process shown assumes counter-current flows linking the stages and shows that
in the separation between A and B, a section of stages is needed above the feed
point in order to produce a purified product of A rnd a section of stages is
needed below the feed point to produce a purified product of B- The individual
stage is described as a mixing and separating device in which various streams
June 1963 CHEMICAL ENGINEERIIG EDUCATION 25
may be entered to be mixed, and then separated into new streams which leave the
stage. What the stage might be or do is discussed.'
The only two requirements which must be met to create a workable multistage
separation process are then discussed; namely, 1) that it be possible to separate
the mixture of input streams into two phases which can be transported to the ad-
joining stages, and 2) that the components to be separated must appear in these
streams in different ratios of their mole fractions.
There is no requirement as to how the phases are produced as long as the
materials to be separated appear in both phases. It is pointed out that the
streams may well be vapor and liquid, typical of distillation, absorption, and
stripping operations, or they may be liquid-liquid streams typical of liquid-
liquid extraction. They may be gas-gas streams in barrier diffusion, or gas-
solid streams in adsorption processes. Any mechanism by which two different
phases can be produced from the input mixture which can be handled so that they
can flow from stage to stage will produce a separation process. As shown in
Figure 1, the stages are linked in a counter-current fashion. As far as I know,
there is no way to prove that this is the optimum fashion for linkage of stages.
But by example, one can easily show that counter-current flow of the phases is
better than parallel flow, which is, of course, no good at all, and is better
than a linkage scheme as typified by such processes as extraction in which a sol-
vent might be split into parts, each part being introduced in order to further
extract the other phase flowing through the stage. An attempt is made to impart
an understanding of the purpose of reflux and stripping vapor in a distillation
column, but the results are sometimes doubtful.
The second general requirement of the multistage process is that the con-
centration ratios of the two components to be separated must be different in
the two phases produced in the stage. This quantity is usually labeled AB as
shown in Figure 1. It is inevitably called relative volatility in distillation,
but is more generally defined as the separation factor. As long as this separ-
ation factor remains on the same side of unity throughout the concentration range
which is to exist in the separation process, a process is feasible to produce the
'General Multistage Separation Process
A + BAB ( xB Phase 1
B ( Phase 2
CHEMICAL EIGIHEERINEERING EDUCATION
Equations for Distillation Column
with Partial Condenser
Component Material Balance around Each Stage
Condenser V Y rxr DyD
Vt-lYt-1 Ltxt Vtt r
t-2Yt-2 Lt-ixt Vtlt_-1 Ltxt
[Vfy- Lf+lxf+l V+lY C+1 Lf+2xf+2
Feed Stage Vflyf_1 Lfxf + FxF V- y Lft+xf+1
Vf-2yf_- Lf_1xf_1 Vflyfl Lfxf
VRYR Lx vly I
Reboiler .-bxb VRYR L1x1
Energy Balance around
VtHt rhr = DHD +
Vt-lHt-1 Lth = VtH rhr
Vt-2Ht-2 Lt- ht1 v t-1Ht-l Ltht
VfH Lf+lh f+1 f+lHf+1 Lf+2hf +
Vf-Hf1- Lfhf + Fh = VfHf Lf+lhf+l
Vf-2Hf-2 Lf- hf_1 VfHf_1 Lfhf
VRHR Ll V11 L2h2
Q hb = VRHR L1hl
Yt = Ktxt
Yf+l = Kf+lxf+l
Yf-1 = Kf-lxf-l
Defining Equations for
Kc = O(TP)
Kf = *(Tf,P)
Kf_1 = *(Tf_1'P)
for Molal Enthalpy
BD (Tc YD) i
hr = Tc,Xr)
Hf+1 = O(Tf+lYf+l) hf+l CTf+l,xf+I)
Hf = 0(Tf,yf) hf = *(Tf,xf)
H- '(Tflyf) hf-1 @(Tf_,hxfl)
H1 0(Ti,Y1) h1 O(T1,xl)
HR 4TRnyR) h, (TR',%)
yt = 1I
f = 1
my, 1 ExI = 1
R 1 Etb 1
June 1963 CHEMICAL ENGINEERING EDUCATION 27
After discussing the general features of multistage separation processes, the
course continues with a general discussion of the variables associated with the
process. I believe this is one of the most important steps in convincing the stu-
dent that he can deal with any process. He determines with very little effort what
can be done with the process and what possibilities he has to alter or control
the separation. In addition, he discovers that the analysis of the process is
simply the solution of a rather lengthy and complicated set of equations, for
which he must find a method of attack.
For example, in Figure 2, all of the equations which define a distillation
column with a partial condenser have been written down. The first set of equations
is a set of mass balances for each component around each stage. The second set
of equations is a set of energy balances-around each stage. The third set of
equations defines the relation between mole fractions in the vapor and the liquid
for each component in each stage in terms of an equilibrium constant or K-value.
The fourth set of equations defines the K-values. The fifth and sixth set define
the molal enthalpies, H and h of the vapor and liquid phases leaving each stage.
Lastly, the seventh and eighth sets state that the mole fractions in all phases
must sum to unity. It might be noted that the pressure has been assumed constant
for all stages; it need not be, but would simply require more equations and would
leave the basic conclusions unchanged. Also, the equations could have been written
differently but with no change in the conclusions.
It is apparent to every student that in order to solve this set of equations
one must have as many equations as unknowns. Hence, the number of independent
variables which must be assigned values in order to set up a solvable problem on
the process can be obtained by counting the number of equations and the number
of unknowns. The number of unknowns inevitably exceeds the number of equations
and the difference in these numbers is thus the number of independent variables
to be assigned values.
If one does this for the process of Figure 2, where R components are fed, n
stages exist above the feed stage, and m stages exist below the feed stage, he
finds the number of equations is
9(nem+37 + gn+m+317 + tL(ntm+3L7 4 /B(n+m3)_7 +
nem37 fMn.m + fn4.*m+*7 L`meg
= (3R + 5) (n4m+3) + 1
Similarly the number of variables is
/S(2) (n+m+3) + R + 2 (n+m43) + 1_7 4 2(n4m43) 37J 0
ZR(n+m+3)7 + J/n+m+47
= (3R + 5) (n+m+3) + R 4 5
For the column shown in Figure 2, then it appears that R 4 variables must
be assigned values in order to describe a meaningful problem. Actually, in
writing the set of equations, two other variables, n and m, must also be assigned
arbitrary values, so that in defining problems for this column, R + 6 variables
must be set. One is, of course, fre to set any R + 6 variables he wishes so long
as they are independent, and he is then posed with the problem of solving the set
of equations for the remaining dependent variables.
This method of analysis can be extended to any process, but if the process
is reasonably complicated, it is quite easy to write the wrong number of equations
and hence get a false answer. The student is then shown a far easier way to count
the number of independent variables by what we have called the Description Rule.
The Description Rule simply says that one must set a number of variables equal to
the number of independent choices he can make in construction plus the number of
independent choices he can make during operation of the column. It is obvious
that during construction one does have independent choices to make which consti-
tute independent variables. After the column is built, he has certain valves
and other features on the column which allow him to make arbitrary adjustments.
The number of these arbitrary adjustments must also constitute a number of indepeno
ent variables. Thus, if one simply draws the column of stages and examines it for
these independent choices, he can determine in seconds how many independent vari-
ables he must set in order to describe a problem on the process. As an example
of this, consider in Figure 3 the same distillation column for which the equations
were all written down in Figure 2. In drawing the column, one can arbitrarily say
there are n stages above the feed stage and m stages below, thus generating two
independent variables. It is then apparent that one can feed as much as he wants
of any component to the system; so that if he has R components, there are R inde-
pendent variables in the component feed amounts. He can also_arbitrarily set the
CO CHEMICAL ENGINEERING EDUCATION June 1963
enthalpy of thi feed, adding one more independent variable, hP. He can, within
lir ;, opera, at any pressure he chooses, generating an independent variable, 1
Wit uhe column in operation, he can, within limits, change the steam valve and
put an arbitrary setting on the reboiler load, generating one more independent
variable, QR. He can, again within limits, arbitrarily set the cooling water to
the condenser and hence the condenser load, creating one last independent variable
Qc" If one suns these variables, as shown in Figure 3, one obtains R 6 vari-
ables, which is identical with the result obtained by consideration of the set of
equations shown in Figure 2.
As a further illustration, if one has a total condenser on the distillation
column instead of a partial condenser, it is apparent that he can put a valve in
the reflux line as shown in Figure 3 and split the flow of liquid from the con-
denser in any proportion he wishes. This constitutes one more independent vari-
able, and if one considers the equations for this process, he will find that
there is indeed one more independent variable. I usually tell the students, with
an explanation, that this total condenser which contains two independent variables,
rather than the one of a partial condenser, is best treated by always setting the
condenser temperature. One can arbitrarily say that the reflux coming from the
condenser will be saturated reflux, which sets one variable, or alternatively, he
can choose an arbitrary temperature.
To illustrate still further what would be a reasonably complicated process,
a refluxed stripper with a side stripper which might be considered to be a very
basic model of a crude oil column is shown in Figure 3. Again, one can count the
variables very simply as the list there shows. No matter how complicated the pro-
cess, the counting of the variables is quite simple.
I believe the use of the Description Rule for the counting of variables,
because of its simplicity, is of real benefit to the student. He is able to set
up a correctly defined problem on any complicated process with real assurance.
He has counted the number of independent variables to be set. He replaces as
many of those which he counted as he wishes with others for which he wants to
specify values in his particular problem. The only requirement left is to find
a way to solve the problem for the remaining dependent variables. If he cannot
find a way to solve the particular problem he has set up, he can at least set up
a series of others which he can solve, and then find the particular one of these
which corresponds to the original problem.
Again, I believe the usual undergraduate course dwells too much on one par-
ticular type of problem, although there are many problems which can be of interest .
in any process. If we take distillation as an example, almost inevitably the
student is asked to solve a problem in which he sets the separation specifications
on two components, sets the reflux and then arbitrarily sets the feed plate lo-
cation during the course of his calculation. This is the standard design problem,
and if the column happens to have a partial condenser, the variables set are shown
in the second list of Figure 4. This solution of this problem has a distinct
method of attack which is well-understood by students if they read any textbook.
However, there can be many other problems of interest on the same distillation
column; and in Figure 4, I have added a few which might be of interest and which,
incidentally, can also be reasonably easily solved.
Almost every problem encountered sets certain variables. For example, the
amount of each component in the feed is always set. The enthalpy of the feed is
almost always set. The pressure under which the column is to operate is always
set. These variables I have shown in Figure 4 listed above the line drawn in
each list, and the remaining variables below. The first list of Figure 4 is the
problem described by counting the variables. It would not be easily solved and
luckily is not often of interest. In the second list, which is the typical de-
sign problem, the recovery fractions in the top and bottom product for two compo-
nents to be separated have been set, the reflux has been set, and the feed plate
location has been set. in the third list, a problem is shown for a column which
might already be in existence. The number of stages in the column above the feed
point and the number of stages in the column below the feed point are set. As
the last two independent variables, one could set the bulk split and the reflux,
obtaining then as an answer the separation which will be achieved. This problem,
although not in itself often of interest, is the stepping stone to the solution
of many problems. Through it one could easily get the solution to the problem of
the fourth list in which the last two variables set are the recovery and purity
of a component in the top product. Also, through it one could even more easily
get the solution to the problem of the fifth list in which the octane number of
the bottom product were set and the problem is to find the amount which could be
produced under the set reflux. The only requirement would be that it be possible
to calculate the octane number from a knowledge of the bottom product composition.
All of these problems and many others could be solved, although the last two
would have to be solved by a parametric solution. But it is helpful to point out
to the student that many problems exist other than the straightforward design prob
June 1963 CHEMICAL ENGINEERING EDUCATION 29
n n nn
n n n QC
m m n' Amount of SI
R of Fxp R of FxF m H of S1
hF hF ma Amount of S3
P P R of FXp H of S2
QC __ P
r r R+ 13
R+6 R + 7
Set Variables for
Various Problems in Distillation
All FXF All FXF All FxF All FxF All FX
hF hp hF hF hp
P P P P P
n (/i)d n n n
m (/J)d m m m
QR r r (/i)d r
QC Fd Plate Loo. d (xi)d Octane no. of b
lem which he normally encounters in the textbooks. A new problem does not present
him witn a requirement for developing a new understanding. He simply must search
for a way in which to solve it. I might say also, that while I have dwelt at some
length on the subject of variables here, the discussion in the course occupies
only about two lectures, which is easily sufficient for it.
The next two sections of the course consider binary distillation, starting
with McCabe-Thiele diagram, which I think is very Instructive. The student sees
grapnically what the column looks like and he obtains an insight into the op-
eration of a typical distillation column by looking at the result of increasing
the reflux, or increasing the number of stages, or for example, tightening the
separation between the two components. However, I do not intend to discuss the
McCabe-Thiele diagram here. Every teacher uses it, I am sure, or uses the Ponchon
diagram as a substitute for it. I might point out that the "assumption" of con-
stant molal overflow requires justification. If one has been very careful to ex-
@3 B EU0lMERING BDUOATION June 1963
B611F t0 tt@ *t4a*t that there are only a certain number of variables which can
S.got. h4 t.hg fIa ; the ipgtruotor setting all liquid flows in the rectifying
QWi 1 t"kq gal to the reflux and all liquid flows in the stripping seo-
l 9gua4 to the reflux plus whatever liquid flow might have been gen-
PCl A a e8 t he introduction of the feed, he is naturally puzzled. The instruc-
t B8R M@ttlW4 a new number of independent variables equal to the number of
069g,* iR t4o oatem. This is easily explained to him by simply postulating that
fWH "t Itroduced a heat exchanger on each atage, and it is possible for you to
B 8P p0 i the stage in an arbitrary amount thus creating the necessary new set
@t i44Wpedent variables equal to the number of stages. Thus, you can then con-
404 t he liquid flow off the stage at the quantity you wish. Since the column is
Rover built with this set of heat exchangers on each stage, the answer calculated
fpr the problem cannot correspond to reality. You can then only explain that in
many Oases the result is close enough to the truth to satisfy you, and if it is
not, it is possible to use a neat balance to correct the calculation.
The following section of the course deals with a set of "group" methods,
still applied first to binary systems. It is useful to the student to know that
there are analytic methods by which he can solve directly for the number of plates
required in distillation columns provided he make certain assumptions, namely that
the flows through any section of the column are constant and that the relative
Volatility is constant through the column or at least that an average relative
volatility will suffice. He can then avoid the stage-by-stage analysis typified
by the McCabe-Thiele diagram and calculate his problem by means of a set of equa-
tions such as Underwood's equations (9). I have used Underwood's equations be-
cause their derivation is simple and extends to multicomponent systems by simply
adding terms, because the equation by which he defines his parameters is easily
shown to be simply an infinite section equation, and also, because through them
the minimum reflux calculation of Underwood can be proved. The equations are
solved for the standard design problem, the second list of Figure 4, and the only
problem for which they allow an easy solution.
The concepts of minimum reflux and minimum plates are next developed, but
again it is emphasized that they are merely different problems in which the vari-
ables set below the line are respectively
n 0o r oo
m co0 Arbitrary feed plate location
At this point tne course extends naturally into consideration of multicompo-
nent systems. A general equation for minimum reflux in multicomponent systems
can be easily developed from infinite section equations under the assumption that
the infinite sections meet at the feed plate, a concept the student is already
familiar with from the McCabe-Thiele diagram. The variables set are
n = o0
where i and j are any two components of a multicomponent mixture. If the feed is
a saturated liquid feed, the equation is exceedingly simple.
(/i)d C (/J)d
The equation applies to any two components and hence, the distribution of all
other components can be back-calculated once the value of Lmi is known for a
particular system and a particular sytemanda particular of (/i)and J)d
The students quickly find that by backcaCul n of the distribution of te
components other than i and J, impossible results are often obtained, e.g. more
of a light component will be calculated in the top product than enters in the
feed, and it is apparent that the assumption that the infinite sections meet at
the feed stage has broken down. Also, with infinite stages, it is possible to
reduce the amount of a component in the product to zero.
The next logical step is the de ivation of Underwood's equation for minimum
reflux for multicomponent systems(10 This equation is derived, along with all
o ,ier equations in the course so that no method is discussed which is not defiyed
from basic principles. Fenske's equation for minimum stages at total reflux is
also derived for multicomponent systems.
June 1963 CHEMICAL ENGINEERING EDUCATION 31
The whole problem of stage-by-stage calculations for multicomponent systems
is discussed as a logical extension of binary systems. I believe the student is
made well aware of the fact that the only new feature is simply that his calcu-
lations must now include estimates of the distribution of all the components whose
separation he cannot set because of the limited number of independent variables.
Also, algebraic calculation replaces graphical methods, and the students calcu-
late design problems for short columns.
The use of Underwood's equations for design problems on multicomponent sys-
tems is also discussed although the exact solution is avoided. Instead, time is
devoted to showing how the concentrations of components other than the two compo-
nents whose separation has been set, can be estimated on the feed plate, thus
yielding an approximate feed plate composition. In comparisons between solutions
of Underwood's equations for multicomponent systems with the approximate feed
plate composition and the exact solutions, the errors are of the order of 1%. The
student is thus able to use Underwood's equations for multicomponent systems with-
out the extremely tedious solution of the determinants necessary in the exact solu-
The last part of the course is devoted to the iteration and relaxation schemes
which have come into use in recent years. Students are asked to solve simple prob-
lems, but the time-consuming nature of the methods would allow solution of long
problems only if a computer were used and we have not yet used one in the course.
However, the very fact that computers are widely available in industry makes the
methods important since they are capable of extension and generalization to a de-
gree the other method cannot achieve.
Iteration methods directly solve only problems in which the numbers of stages
in all sections of the column are fixed, plus a sufficient number of other inde-
pendent variables to completely define the problem. The "other" variables are the
bulk split and a flow such as the reflux; direct solution for problems described
by other variables is at present not worked out, but must be obtained parametric-
The basic iteration scheme was proposed by Thiele and Geddes in 1932(8)
Many workers have elaborated and changed the schema since, notably Edmister( ),
Admundson and Pontinen(1l), and Lyster and Hollandt-), and the choice of exact
procedure is today quite broad. The particular scheme I teach the students is
one I have worked on a considerable amount and I have chosen it to a large degree
because of this.
All iteration schemes break'the system of equations for the column into
groups and solve the groups separately. As an example, consider the simple dis-
tillation column of Figure 2. The mass balance equations will be solved first,
and the set of equations for each component must be separately solved. If the
equations for the first component are written down it is apparent that nothing,
or almost nothing, is known about them. However, we could solve the equations
for the mole fraction of the component on each stage and in the products if all
unknown flows and temperatures were assumed. The set of equations is then a
linear set of equations in either x or y, we arbitrarily choose x, and will solve
for all values of x.
The technique of solving the set of equations I use was first published by
Smitht7)in essentially the form I use it. Since nothing is known, one assumes a
value of bxb for the component, the amount of the component in the bottom product
The amount is wrong by some unknown error, b, where
bxb = (bxb)true + 6b
A typical stage by stage calculation of the component can now easily be run up
the column. Thus
L1x1 = RR + bxb
VlYl = S1 (LlxI)
etc. to the top of the column where
rxr a Vtt + bxb Fx
DyD = S,(rxr )
DyD is, of course, wrong since bxb is wrong.
32 CHEMICAL ENGINEERING EDUCATION June 1963
However, we know the error in bxb is some quantity,E bh The error in VYR
is then SR b nd the error In Llx1 is CV .b None of these errors can be
ca d directly but all of their ratios can. Thus
E b E b
Lb b 1
eat., on up the column. The feed has no error associated with it and hence, does
not enter the error ratio calculation.
At the top of the column the ratios
rb b t 1
6 b So b
Then one can write
DyD + bxb (DYD)true + E (bxb) true + Cb
and, since (DYD)true (bxb)true = FxF
a slight rearrangement yields
b = DYD + bxb -FXF
The amount of the component in every flow can then be corrected to the true
value through the knowledge of 6b and all of the error ratios.
This procedure is followed for each component (the reverse calculation down
the column must be employed for light components) and a complete set of mole frac-
tions is obtained for the column. All of these mole fractions should sum to
unity, and if they do not, the assumed temperatures or flows must be corrected to
better values. Either could be corrected at this point, but most procedures cor-
rect the temperatures. I use a simple bubble point calculation for this purpose,
after which the mass balance equations are solved again.
The heat balance equations are used to correct the flows, either at the end
of each temperature correction or after several temperature corrections. If the
correction methods for the temperatures and flows are effective, the calculations
The students understandably shy from the tedious work of the method, but
they certainly have no trouble understanding it, and it is apparent that with
suitable correction methods, the iteration scheme could be applied to essentially
any process. Actually, multiple feeds present no problem, side streams only a
slight problem, and side strippers a somewhat harder problem. Correction methods
are well-developed at the present time for many common processes and the iteratior
method has been extensively used in industry. The student is impressed when he
is told that columns with 40 plates and 20 components can be solved on a computer
The relaxation method is conceptu-lly the most simple method which one can
Sge, and because of this it is also completely general. The basic idea was
d Ljrst by Rose(6) for the mass-balance solution to the Steady-state by cal-
c .acion through the unsteady state period of a column startup. However, if the
-tqte is the desired answer, no consideration need be given the time be-
the -&rhd can % simplified and at the same time extended~ The
June 1963 CHEMICAL ENGINEERING EDUCATION 33
relaxation scheme solves the same type problem which the iteration methods solve,
more slowly in every case, but with great reliability.
One simply takes each stage in a process and answers the single, simple
question: "If I bring in one, two, three, or more streams into the stage of a
given amount and composition and with a given energy, what will be composition,
amount and energy of the phases leaving the stage?" In most processes which one
calculates in the present day, one assumes that the phases leaving the stage are
in thermodynamic equilibrium. In a distillation calculation, for example then,
an isenthalpic flash will answer the question. If one then loads his column with
any material and simply starts to calculate from stage to stage, answering the
question in turn for each stage, the calculation will converge asymptotically to
the steady state condition of the process. It is not necessary to completely
solve the isenthalpic flash on each stage. Commonly two isothermal flashes are
used to predict the results of the isenthalpic flash and the calculation moves on
to the next stage. There does not appear to be any particular benefit in any
order of consideration of the stages, although if one could locate the stage of
largest error, i would undoubtedly be beneficial to relax this one and then re-
lax the one which had again the largest error, etc. Such calculations are slow,
even on a computer, but they inevitably result in solution. Many fairly compli-
cated processes which have been unsolvable by the present iteration schemes can
be very readily solved by the relaxation technique.
This simple concept obviously can be extended to any process. In extraction
processes, all one needs to consider is what will the distribution of the compo-
nents be between the phases leaving the stage. Again, for each relaxation of a
stage, it is not necessary to calculate exactly what this distribution will be.
Simple prediction of the distribution from the results of one cycle in an iteratior
solution of the fl-sh equation suffices. There are other and better ways to solve
liquid-liquid extraction systems, but the relaxation method works well and shows
the complete generality of the method.
As I said initially, the course we are now teaching is a collection of
methods which we know particularly well and which cover the general field of
multistage separation operations. There is such a tremendous literature in this
field that one could easily pick a totally different set of methods. The course
will undoubtedly change in the future, hopefully in the direction of increasing
generality, and will also perhaps have the benefit of having some computer time
available. It is not a simple matter to use a limited amount of computer time
effectively, however and the use of the computer will not in itself add to the
understanding the student may have gained in the field. The advantage of the
computer may well be that it will allow the student to invent a new separation
process himself and do a sufficient amount of calculation to test it and thereby
gain a little more understanding.
b moles of bottom product.
D moles of vapor top product.
F moles of feed.
H molal enthalpy of a vapor leaving a stage.
h molal enthplpy of a liquid leaving a stage.
K component equilibrium constant, y/x.
L moles of liquid flow leaving a stage.
Lmin minimum liquid flow in rectifying section for a distil-
lation column, usually taken as minimum external reflux.
m number of stages in the stripping section of a distil-
n number of stages in the rectifying section of a distil-
QC condenser load.
QR reboiler load.
R number of components.
r moles of reflux.
S stripping factor for a component in a stage, 1 I
x mole fraction of a component in a liquid.
y mole fraction of a component in a vapor.
(/A)d fraction of I fed which is removed in the top product.
(/i)b fraction of i fed which is removed in the bottom product.
CHEMICAL NGIUEERING EDUCATION
A component A
B component B
b bottom product
D vapor top product
d liquid top product or top product in general
f feed plate
I component i
j component j
L1 liquid flow leaving plate 1, etc.
t top plate
VR reboller vapor, etc.
1 plate 1, numbering upward, etc.
ccAB separation factor between components A and B In phases
leaving a stage.
AB = -
7B/ Phase 1
8 )/Phase 2
qO I separation factor or relative volatility of component 1
based on some unnamed reference component. All i are
based on same reference component.
g error in the calculated amount of a component in a flow
stream leaving a stage.
1. Amundson, N. R., And Pontinen, A. J., Ind. Eng. Chem., 50, 730 (1958).
2. Edmister, W. C., A.I.Ch.E. Journal, 3, 165 (1957).
3. Fenske, M. R., Ind. Eng. Chem., 24, 582 (1932).
4. Hanson, D. N., Duffin, J. H., and Somerville, G. F., Computation of
Multistage Separation Operations, Reinhold Publishing Corp., New York,
5. Lyster, W. N., Sullivan, S. L. Jr., Billingsley, D. S., and Holland,
C. D., Petroleum Refiner, 38, No. 6, 221 (1959), ibid, 38, No. 7, 151
(1959), ibid, 38, No. 8, 121 (1959), ibid, 38, No. 10, T39 (1959).
6. Rose, A., Sweeney, R. F., and Schrodt, V. N. Ind. Eng. Chem., 50,
7. Smith, B. D., and Brinkley, W. K., A.I.Ch.E. Journal, 6, 451 (1960).
8. Thiele, E. W. and Geddes, R. L., Ind. Eng. Chem., 24, 289 (1933).
9. Underwood, A. J. V., Chem. Eng. Prog., 44, 603 (191b).
10. Underwood, A. J. V., Jour. Inst. Petroleum (London), 32, 614 (1946).