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 Permanent Link:
 http://ufdc.ufl.edu/AA00000383/00011
Material Information
 Title:
 Chemical engineering education
 Alternate Title:
 CEE
 Abbreviated Title:
 Chem. eng. educ.
 Creator:
 American Society for Engineering Education  Chemical Engineering Division
 Publisher:
 Chemical Engineering Division, American Society for Engineering Education
 Publication Date:
 June 1963
 Frequency:
 Quarterly[1962]
Annual[ FORMER 19601961]
 Language:
 English
 Physical Description:
 v. : ill. ; 2228 cm.
Subjects
 Subjects / Keywords:
 Chemical engineering  Study and teaching  Periodicals ( lcsh )
Notes
 Citation/Reference:
 Chemical abstracts
 Additional Physical Form:
 Also issued online.
 Dates or Sequential Designation:
 1960June 1964 ; v. 1, no. 1 (Oct. 1965)
 Numbering Peculiarities:
 Publication suspended briefly: issue designated v. 1, no. 4 (June 1966) published Nov. 1967.
 General Note:
 Title from cover.
 General Note:
 Place of publication varies: Rochester, N.Y., 19651967; Gainesville, Fla., 1968
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 Source Institution:
 University of Florida
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 All applicable rights reserved by the source institution and holding location.
 Resource Identifier:
 01151209 ( OCLC )
70013732 ( LCCN ) 00092479 ( ISSN )
 Classification:
 TP165 .C18 ( lcc )
660/.2/071 ( ddc )

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CHEMICAL
ENGINEERING
EDUCATION
CHE MI
THE AMERICAN
CAL ENGINEERING DIVISION
^ "
DUCTroN
e 1963
mml l
E A 
CHEMICAL ENGINEERING EDUCATION
June 1963
Chemical Engineering Division
American Society for Engineering Education
CONTENTS
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
Separation Operations,
by D. N. Hanson  24
Max Peters
Joseph J.
John B. We
Chemical Engineering Division
American Society for Engineering Education
Officers 196263
(Colorado) Chairs
MArtin (Michigan) Vice C
at (Oklahoma State) Secret
ian
chairman
aryTreasurer
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.
Publication Office: University of Connecticut
P.O. Box 445, Storrs, Connecticut
Subscription Price, $2.00 per year.
SOME PHENOMENA FROM FLUID MECHANICS
C. V. Sterling
Shell Development Company
Emeryville, California
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 nonlinear 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 tbthe 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 gasliquid 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 gasliquid 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 nonnewtonian 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 lesserknown
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.
1
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 RanqueHilech, 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 noncircular
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 uncanged 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 nonnewtonian
fluids.
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 nonlinear 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 eitherend 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
theory.
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). As7igure 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
initially outwards.
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 crossshaped 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, JebsenMarwedel
(25) has described the erosion of lTe walls of glass melting furnaces at a series
ofregularly 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 fluidfluid
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 noncirculating
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 flt 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
pseudocritical 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 formultion 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
tion.
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 gasliquid 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
ally confirmed.
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
transition.
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?
BIBLIOGRAPHY
1. Allan, R. S. and Mason, S. G., "Particle Behaviour in Shear and
Electric Fields", Proceedings of the Royal Society (London), A g6
Part I, 45, and Part II, 62, (1962).
2. Bagnold, R. A., "AutoSuspension of Transported Sediment; Turbidity
Currents", Proceedings of the Royal Society (London), A S5, 315 (19i*
3. Benjamin, T. B., "Wave Formation in laminar Flow Down an Inclined PIea
J. Fluid ?Moh. 2, 554 (1957).
4. Benjamin, T. B., "Effects of a Flexible Boundary an Bydrodynamio Stability
J. Fluid Mech. 513 (1960).
5. Bousmas, A. A., 'Streaming Currents in Turbulent Flows and Metal
Capillaries", Physica f3, 100755 (1957).
6. Bushmanov, V. K., 'qydrodynauic Stability of a Liquid layer on a
Vertical Wall", Soviet Physics JETP. 12, 873 (1961).
7. Caldwell, D. R. and Donnelly, R. J., "on the Reversibility of the
Transition Past Instability in Couette Flow", Proceedings of the
Royal Society (London) A g 197 (1962).
8. Christensen, H., "The Oil Film in a Closing Gap", Proceedings of the
Royal Society (London) A 266 312 (1962).
9. Christiansen, R. M. and Hixson, A. N., "Breakup of a Liquid Jet in a
Denser Liquid", Ind. Eng. Chem., !!, 1017 (1957).
10. Dombrowski, N., and Hooper, P. C., "The Effect of Ambient Density on
Drop Formation in Sprays", Chem. Bag. Sci. Z 291 (1962).
11. Durham, K., "Surface Activity and Detergency", aWMillan, London, 1961.
12. Eisenmenger, W., "Dynamic Properties of the Surface Tension of Water
and Aqueous Solutions of Surface Active Agents With Standing
Capillary Waves in the Frequency Range from 10 kc/s to 1.5 mo/s.
13. Fararoui, A., and Kintner, R. C., "Flow and Shape of Drops of on
Newtonian Fluids", Trans. of the Soc. of Rheology, J, 369 (1961).
14. Feldman, S., "On the Hydrodynamic Stability of Two Viscous Ineo a resib)*
Fluids in Parallel Uniform Shearing Motion", J. Fluid Meah. 2, (1957)
15. Forgacs, 0. L. and Mason, S. G., "The Flexibility of Wood Pulp Fibers",
TAPPI, 4_, 695 (1958).
16. Gaskina, F. H. and Philippoff, W., "The Behavior of Jet of Viteoelast,;
Fluids", Trans. of the Soo. of Rheology 181 (1959).
17. Goldsmith E. L. and maon, S. G., "Axial Migration orPartols in
poiseuillS Flow", Nature 19, 1095 (1961).
18. Oraebel, W. P., "'he Stability of a Stratified Flow", J. Fab rwb.
321(1960).
CHEMICAL ENGIIEERING EDUCATION June 1963
19. Haberman, W. L. and Sayre, R. M., "Motion of Rigid and fluid Spheres
in Stationary and moving Liquids Inside Cylindrical Tubes", U.S.
Navy Experimental MHdel Basin Report 1143 (1958).
20 Hartnett, J. P.,and Eckert, E. R. G., "Experimental Study of the Velocity
and Temperature Distribution in a HighVelocity Vortex Type Flow",
Trans. Amer. Soc. Mach. Eng. f2, 751 (1957).
21. Harris, 3., "Flow of ViscoElastic Liquids from Tubes", Nature 190.
993 (1961).
22. Hughes, R. R., Handlos, A. E., Evans, H. D. and Maycock, R. L., "The
Formation of Bubbles at Simple Orifices", Heat Transfer and Fluid
Mechanics Institute 143 (1955), Stanford University Press.
25. Hughes, R. R. and Gilliland, E. R., "Mess Transfer Inside Drops in a
Gas", Chem. Eng. Prog. Sym. Series 51, No. 16, 101 (1955).
24. Ingebo, R. D., "Drag Coefficients for Droplets and Solid Spheres in
Clouds Accelerating in Airstreams", NACA Technical Note 3762 (1956).
25. JebsenMHrwedel, H., Glastech. Berichte 2, 253 (1956).
26. Kapitza, P. L., Soviet Physics JETP 18 3, 19 (1948).
27. Klinkenberg, A. and van der Monne, J. L., "Electrostatics in the
Petroleum Industry", Elsevier, Amsterdam, (1958).
28. Knuth, E. L., "The Mechanics of Film Cooling", I Jet Propulsion 24
559 (1954). II, Jet Propulsion 25, 16 (1955).
29. Kramer, K 0., "Boundary Layer Stabilization by Distributed Damping",
Readers Forum, J. Aero/Space Sol. gz, 68 (1960).
30. Langmuir, I. and Langmuir, D. B., J. Phys. Chem. ji, 1719 (1927).
51. Li, T. Y. and Kusukawa, K., "Steady Subsonic Drag in nonEquilibrium
Flow of a Dissociating Gas", Proc. 1962 Heat Trans. and Fluid Mech.
Inst. Stanford University Press 1962.
32. Miles, J. W., "The Hydrodynamic Stability of a Thin Film of a Liquid in
Uniform Shearing Motion", J. Fluid Mech. 8, 593 (1960).
33. Miller, D. R. and Comings, E. W., "ForceMoomntun Fields in a Dual Jet
Flow", J. Fluid Mech. 7, 237 (1960).
34. yrsels, K. J., Shinoda, K., and Frankel, S., "Soap Films Studies of
Their Thinning", Pergamon Press New York 1959.
35. Oliver, D. R., "Influence of Particle Rotation on Radial Migration in the
Poiseuille Flow of Suspensions", Nature 94. 1269 (1962.
36. Orell, A., and Westwater, J. W., "Natural Convertion Cells Accompanying
LiquidLiquid Extraction", Chem. Eng. Sci. 1 127 (1961).
37. Reynolds, A. J., "Energy Flows in a Vortex Tube", J. Angew. Math. Phys.
12 543(1961).
38. Rosenberg, L. D. and Eknadiosyants, 0. K., "Kinetics of Ultrasonic Fog
Formation", Soviet Physics, Acoustics 6 3, 369 (1961).
39. Rumscheidt, F. D. and Mason, S. G., "Particle Motions in Sheared
Suspensions XII Deformation and Burst of Fluid Drops in Shear and
Hyperbolic Flow", J. Colloid Sci. 16 238 (1961).
40. Saffman, P. G., "On the Stability of Laminar Flow of a Dusty Gas",
J. Fluid Mech. h 120 (1962).
41. Sawyer, R. A., "The Flow Due to a TwoDimensional Jet Issuing Parallel to
a Flat Plate", J. Fluid Mach. 9, 543 (1960).
42. Scriven, L. E. and Sternling C. V., "The MArangoni Effects", Nature 187
186 (1960).
45. Sproull, W. T., "Viscosity of a Dusty Gas", Nature 190 976 (1961).
44. Tailby, S. R. and Portalsey S.,
Trans. Inst. Chem. Engrs. 3524 (1960).
45. Tailby, S. R. and Portalsey S., "Wave Inception on a Liquid Film Flowing
Down a Hydrodynanically Smooth Plate", Chem. Eng. Sci. l, 2835 1962).
46. van Duyne, R. J., "Measurement of Dynamic Surface Tension Changes in
FrothForming Aqueous Solutions", Thesis PhD University of Michigan
(1961).
47. Westley, R., "A Bibliography and Survey of the Vortex Tube", Note
Nr. 9 College of Aeronautics, Cranfield (1954).
48. Tib,
Proc. Second US Nat Cong of Appl. Hash. 623 (1954).
'urc 1 \CCELERATIONAL DRAG DATA OF INGEBO
a) How Charging Occurs
+ I
Concentration
+
+ I
1 Laminar Sublayer
b) How Turbulence Influences
the Process
Figure 2. ELECTRIC CHARGING INDUCED BY TURBULENT FLOW
After Klinkenberg and van der Minne
Water :,.
Drop .
j Ned
a) Multinodal Liquid Pro
Ultrasonic Atomization
and Eknadi
I fezoelectric I
ejections Observed in Transducer
. After Rozenberg
osyants
b) Simple Experimental Setup to Demonstrate
Explosive Atomization
Figure 3. ULTRASONIC ATOMIZATION
I
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
Rigid
Turn
Springy
Turn
0 0 0 o.o
sSJ~
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) 00
c) 00
,,
a) Streamlines are Traced
by Solid Particles. No
Segregation Occurs
b) Streamlines are Traced
by Fluid Particles. Par
ticles Move away from
Boundaries
Solid Fluid
Object Object
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
SBinme
I I I I I I I I I I I I I I I I }
0 100
o(p
Figure 8. CRITICAL WEBER
1,000
FOR LIQUIDS FLOWI L
100
10,000
IvUivnen run ways runavi LLuI
10 Unstable with
10Respect to
Wave Formation
10 
Predicted
Curve of Miles
' 10
S10' Stable
10 Transition
Zone Observed
/byKnuth
0 0.1 0.2 0.3 0.4 0.5
Weber Number
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
by
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.
Heat Conduction
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 reentering 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 temperatureresistant 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.
Heat Convection
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
noncircular crosssections. 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 crosssection 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 crosssection 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 crosssection 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 crosssection 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
13
_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 reattaches 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 supercritical 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 ZehnderMach 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 nonexistent.
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.
Mass Transfer
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 twocomponent
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 twocomponent fluid. The
first term on the righthand 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 righthand 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 rewritten 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 righthand 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 Tot side of
the boundary layer could be adjusted by preheating or precooling 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 maybe 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 threecomponent 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
watercooled 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
plication.
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 radiatingab
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 nontransmitting 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 blackbody 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.
HOT
GAS
S V VELOCITY OF ABLATION
qc Vg VELOCITY OF ABLATED GAS
. q.c CONVECTIVE HEAT FLUX
Si q, RADIATIVE HEAT FLUX
FIG. I
CHEMICAL ERGIEERIUNG EDUCATION Juan 1963
References
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. 293306. 1961.
2) A. V. Likon, HBeat Transfer ibliograplh Russiaan Works,'
International Journal of Heat and Mes Transfer, vol. 5, pp. 571582.
192.
3) Recent Advances in Mast and tasg ransfegr J. P. Hartnett, Editor.
MUcGrawHill Book Company, New Tork. 1961.
4) T. R. Munson and R. J. Spindler, "Transient Thermal Behaviour of
Decomposing Materials," IAS Paper No. 6230. 1962.
5) E. R. G. Eckert and T. F. Irvlne, Jr., "Pressure Drop and Heat Transfer
in a Duct with Triangular CrosSectim,m Journal of Bansic Engineering,
v.l. 82, pp. 125138. 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. 208218.
1960.
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,
McGraHill Book mpany, New York. pp. 161222. 1961.
1i) J. R. Baron, "Thermodynaaic Coupling in Bondary Layers," American Rocket
society Snice Flight Report to the Nation 220661. 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. 261.
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. NorthHolland Publishing
Company, Amsterdam, Netherlands.
18) E. M. Sparrow, "A New and Simpler Formulation for Radlative Angle Factors,"
American Society of Mechanical Engineers, Paper No. 62HT17. 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. 425h. 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. 61NAl167.
1962.
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. 20828., American Society of Mechanical Engineers.
1961.
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. 6372. 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
Re
FIG. 2
30,000
LAYER
DlICIr,'G 3TREWM LINE
ATTACHED B01
FIG. 3
LAYER
Rew
FIG. 4
CONDUCTION
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
SECONDS
FIG. 5
300
200
100
80
60
'F
h
Bt
hrft
C
MASS AND HEAT TRANSFER EQUATIONS:
rhi= D12 Vwi + Pi V + COUPLED EFFECTS (1)
q=kVTpD2:hiVwiPVh+ (2)
= Vh(LeI)EhiVwiPVh+ + (3)
BOUNDARY LAYER EQUATIONS:
P(U + v )= (PD, i)+ Ki (4)
+ rPD, iL lh i] (5)
p(u "fi+vi) a (61
p (u s+vh s)= (s) (71
FIG. 6
I I I I I I I I I I I
800 r [Ibm/hr ft]
4.2 i ,
S2I0.2 HELIUM .
400 30.1
S* 62.7 AIR
X= 16
S*
(T,T.) F
FIG. 7
I I II
5 I= 150AMP, s=6mm
wev oy }SPLIT PLATE ANODE
4 TV *
C4 V ', ;}POINT PROBE ANODE
CONVECTIVE _
3 BY ELECTRON v' HEAT FLUX TO ANODE
GAS 1,
2 14 o0
ELECTRONS TO '',
CONVECTIVE
BY AT M GAS
0 I 2 3 4 5 6 7
r(mm)
FIG. 8
E, A,
FIG. 9
4(3) 4
2(3) 2
3: Specular
1,2,4: Diffuse
H3= BF3_+ B2F32+ B4F34
B4 = 4 T44 +p4H4
H4= BI(F4 + p3F41(3))
+B2 (F42 + P3 F4 2(3))
+* 3'T34F43
FIG. II
CONTINUUM
STATE
\ I \
N
V,T N
eb\
DIFFUSION
DIFFUSION
IL a Aluminum point
b Iron, scraped
30* c Iron, hot rolled
d Copperoxide
d 40*
50*
60*
70
80
0 90*
FIG. 10
MOLECULAR TRANSPORT
SLIP FLOW
TEMPERATURE
JUMP
L<'
SV,T ,
STRONGLY
ABSORBING
TRANSITION
STATE
SV,T
L
ABSORBING
FREE
MOLECULE
STATE
NONARSORRING
RADIATIVE TRANSPORT
FIG. 12
AN UNDERGRADUATE COURSE IN ANALYSIS
OF MULTISTAGE SEPARATION OPERATIONS
by
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 McCabeThiele 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 wellequipped 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
ques.
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 allinclusive 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 countercurrent 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 liquidliquid streams typical of liquid
liquid extraction. They may be gasgas 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 countercurrent 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 countercurrent 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
separation desired.
Figure 1
'General Multistage Separation Process
A
A+B A
A + BAB ( xB Phase 1
B ( Phase 2
CHEMICAL EIGIHEERINEERING EDUCATION
Figure 2.
Equations for Distillation Column
with Partial Condenser
June 1963
Component Material Balance around Each Stage
(xi)
Condenser V Y rxr DyD
VtlYt1 Ltxt Vtt r
t2Yt2 Ltixt Vtlt_1 Ltxt
n
[Vfy Lf+lxf+l V+lY C+1 Lf+2xf+2
Feed Stage Vflyf_1 Lfxf + FxF V y Lft+xf+1
Vf2yf_ Lf_1xf_1 Vflyfl Lfxf
m
VRYR Lx vly I
Reboiler .bxb VRYR L1x1
Energy Balance around
Each Stage
VtHt rhr = DHD +
VtlHt1 Lth = VtH rhr
Vt2Ht2 Lt ht1 v t1Htl Ltht
VfH Lf+lh f+1 f+lHf+1 Lf+2hf +
VfHf1 Lfhf + Fh = VfHf Lf+lhf+l
Vf2Hf2 Lf hf_1 VfHf_1 Lfhf
VRHR Ll V11 L2h2
Q hb = VRHR L1hl
Component Equilibrium
Relations
(xn)
YD Kcxr
Yt = Ktxt
ytI Kt1xt1
Yf+l = Kf+lxf+l
f fxf
Yf1 = Kflxfl
Y, Klxi
y1 1
RH "R
Defining Equations for
Component Equilibrium
Relations (xR)
Kc = O(TP)
Kt O(T,P)
Kt1 (TtP)
"K+1 (Tf+l1'P)
Kf = *(Tf,P)
Kf_1 = *(Tf_1'P)
S (T1,P)
O (TRP)
Defining Equations
for Molal Enthalpy
BD (Tc YD) i
Ht (Ttyt)
Ht_ (Tt1,Yti)
hr = Tc,Xr)
ht *(Tt,xt)
htl O(tlxtl)
Hf+1 = O(Tf+lYf+l) hf+l CTf+l,xf+I)
Hf = 0(Tf,yf) hf = *(Tf,xf)
H '(Tflyf) hf1 @(Tf_,hxfl)
H1 0(Ti,Y1) h1 O(T1,xl)
HR 4TRnyR) h, (TR',%)
ZyD 1
yt = 1I
t 1
Yf+l 1
f = 1
yf 1
fl 1
zx,
Ext 1
EXt1 
EXf+1 
EXf 1
Exf1
my, 1 ExI = 1
R 1 Etb 1
ZX 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 balancesaround 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 Kvalue.
The fourth set of equations defines the Kvalues. 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 wellunderstood 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
Figure 3
n n nn
ma
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
Figure 4
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 McCabeThiele 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
McCabeThiele 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 stagebystage analysis typified
by the McCabeThiele 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
(/A)d (/A)d
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 McCabeThiele diagram. The variables set are
(/i)d
(/j)d
n = o0
m oo
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
F 1
( j
The equation applies to any two components and hence, the distribution of all
other components can be backcalculated 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 stagebystage 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
tion.
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 timeconsuming 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
ally.
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
VRyR SR(bxb)
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
b SR
LV
E b E b
71 L1
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
and
Ed Er
6 b So b
are calculated.
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
+ 1
fb
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
converge rapidly.
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 welldeveloped 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
for $1.50.
The relaxation method is conceptully 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 massbalance solution to the Steadystate 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 flsh equation suffices. There are other and better ways to solve
liquidliquid 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.
NOMENCLATURE
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
lation column.
n number of stages in the rectifying section of a distil
lation column.
p pressure.
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
T temperature.
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
SUBSCRIPTS
A component A
B component B
b bottom product
C condenser
D vapor top product
d liquid top product or top product in general
F feed
f feed plate
I component i
j component j
L1 liquid flow leaving plate 1, etc.
R reboiler
r reflux
t top plate
VR reboller vapor, etc.
1 plate 1, numbering upward, etc.
GREEK
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.
function.
REFERENCES
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,
(1962).
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).
June 1963

Full Text 
PAGE 1
r ~ 1 A 5 r:: I v. E ~ E CHEMICAL ENGINEERING DIVISION THE AMERICAN SOCIEl"f FOR ENGINEERING EDUCATION Jtine 1963
PAGE 3
CHEMICAL EHGINE;ERIBG EDUCATION June 1963 Chemical Engineering D1v1a1on American Society for Engineering Education CONTENTS Some Phenomena From Fluid Mechanics, by C. V. Sterling 1 Advances in Heat and Mass Transfer, by E. R. O. Eckert 13 An Undergraduate Course in Analysis of Multistage Separation Operations, by D. N. Hanson 24 Chemical Engineering Division American Society for Engineering Education Max Peters Joseph J. M~rtin John B. West Officers 196263 (Colorado) (Michigan) (Oklahoma State) Chairman Vice Chairman SecretaryTreasurer CHEMICAL ENGINEERING EDUCATION @ Journal of the Chemical Engineering Division, American Society for Engineering Education. Published Quarterly, in March, June, September and December, by Albert H. Cooper,Editor. Publication Office: University of Connecticut P.O. Box 445, Storrs, Connecticut Subscription Price, $2.00 per year.
PAGE 4
'
PAGE 5
SOME PHENOMENA FROM FLUID MECHABICS C. V. Sterling Shell Development company Emeryville, California Fluid mechanics is a large and rapidly growing field. The first volume ot the Journal of Fluid Mechanics covered all of 1956 and contained about 400 pages. Since 1960, in this journal alone, almost 24,00 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 nonlinear 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 9tl sands, and many mechanical and chemical en gineers expounding on~ seemingly endless variety of processing problems. It la, 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 chemioal engineer working mainly in the process i ndustries. 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 b e answered everyday by some chemi cal engineer. Typical problems are: After the rupture of the bursting disk on a vessel containing a vap o rizin g gasliquid mixture what will be the rate of dis charge of liquid t h rough the line? What part of the vessel contents will be carried out througl1 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 exce ssive degradation by collision with walls, impellers or other par ticles, what impeller speeds are tolerable? Will a jet of a gasliquid mixture entering a distlllgtlon column at high veloctty 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 tne sump enough to cause troublesome entrainment or vortexlng of gas into t l1e pump suction? How thick will be the head of froth on top of a viscous liqulrl in a reactor and what will happen to its thickness when the vapor velocity in the re~ctor 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 b lock the pipe? Will a reactor whose heat evolution ls removed by a bolling of a liquid perform smoothly or will it act as a "geyser"? c an one mix nonnewtonian 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 increase in the liqui d visc osi ty and a large decrease in ga s density? Will a con ventional pneumatic atomizing nozzle function satisf acto ril y on a gas of very h i gh density? Ough t 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. O n the other hand, they are very complex ln t he sense that they have not been formulated ne at ly like a textbook problem. Nothing warns the enginee r to be ware of an instability or to beware of a chan ge in regime of atomization. No one gives him t ~e hint that t he affects of viscosity are negligible but that capillar ity must be considered. In all cases, the first step, and indeed, the gre ater t of the eolut ion lies in the proper formul~tion of the problem. To su cceed ~!~e, with a reasonable score, the engineer needs to know intimately t h e 11 thinga that happen", the "phenomena" of fluid mech anics. Once these are recognized, he can find expert help in the publis h ed literature with regard to simplify i ng con cepts, qusnt1tative formulation, methods of solution, or perh~ps he may e ven find the solution to his immediate probleml With thia in mind we limit ourselves here to describing a few lesse r lmown henomena selected I fear somewhat capriciously but wit h 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 d lift The calculation of these forces, whic h is probably the most technically t~portant problem in fluid mechanics is now fair l y well understood. A few novel studies throwing light on the sources of the drag, are worth mentioning. Li and Kusukaw; (31) have shown that in the absence of viscosity, heat conduction, and diffusion,arag is yet induced by a finite rate of attaining chemical aqu111brium.
PAGE 6
2 J'une 1963 Other dissipative proceaaea normally increase drag. Por example, complete solu tions for th e very slow motion of a sphere 1n an electrically conducting and m < gn et1call y suscep tible fluid are now available. CHEMICAL EliGllEERIBG EDtJCATIOB In calculating the drag on a body one can easily overlook the effects of ac c el eration 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 ita Reynolds number in dimensionless terms. If it has a "memory" although this memory is short, the drag is, in addlt16n, 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 D)p dv u2 dt Data on accelerational drag effects are rare. Ingebots (24) data ~eproduced here in Figure 1 show that drag coefficients for accelerating arops and solid particles are sometimes vastly different from the drag coefficients for steady flow. It has been shown by Sproull (.!!:J) 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, advanoed by Saffman (.!i:.Q) is that the particles act to change the stab1li ty <; haracteristics 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 ao!id body can be achieved by coating the surface with a special type of compliant coating, the artificial "porpoise skin." Benjamin(]) 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) hBS discussed this anomaly and another one related to it associ ated with the transport 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, of ten the only ones finding their way into the textboo ks, 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 RanqueHilsch, 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~ cfm of air at 70F into a cold stream of .8 cfm at .!+OF and a hot stream at 90F. The tubeis only 1/4 inch in diameter and 8 inches long. There is, of course, nothing .i yste rio us about such en ergy separation. Hartnett and Eckert (20) point out that v rdinary isothermal Poiseuille flo w also causes AD energy separi'Eion.
PAGE 7
.rwne 1963 CHEMICAL EllGIJEER,1llG EDUCATION 3 r; : ; ~. ln tne paat aixteen years there hav e appeared over a hundred publications '4taling with the vortex tub~ effect f;!estley (47}1. The recent articles of Rey nalda {37) present the most easily understood explanation of the various mechanism leading~o the energy separation. Three effects are important. First is the Knoernachild effect, which describes the heat exchange due to a compressible lump ot 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 ligh ter clumps of fluid move inwards. Third, the turbu lent stresses furnish a means whereby one layer of flu.id can do work on the ad jacent layers. All of these effects are also important in determining the energy exchange taking place in the a~mosphere. Vibration effects flow in several ways. The streaming currents that flow away from a sound transducer a re 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, shoWB 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 abo~t 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 noncircular orifice. The recent painstaking efforts of van Duyne {46) were not entirely successful in this regard however. He found the spacingbetween 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 Jiarris (21) 7. Such a jet does not always issue from the orifice with its dia meter uncnanged 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 nonnewtonian fluids. A rather puzzling effect in the realm of vibrations was found by Hughes (23) in measuring upta~e of carbon dioxide by water drops..falling through a gas. Tne transfer rates correlAte with a postulated "eddy" mixing inside the drops supposedly caused by their vibration as they f~~l 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 nonlinear 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 str1king is the observation by Christiansen and Hixson (9) tharone can make a dispersion of drops of almost exactl e ual sizes the adjusting the velocity with which a liquid jet issues be neath {heqsurface of another liquid. This phenomenon is so precisely controllable as to be useful as a means for meas uring dynamic surf ace tensions. The mechanics of particles, botl1 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 lar e not too small}, how to suspend them, transport them, collect and coalesce th:m. 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 average drop size formed in atomization from fan sprays first decrea~esi~~ =d it i raised but later increases. This is in accor w pr ~~!~~!sg!:dee~;omymo~els of the instability of sheets of liquid movi~ re}a~!ve to a as Their photographs show clearly changes in the mode of brea Po e :~e::!Y~o! ~:,:~~~~~s:rt~~;e~ia~:s.:~=~t!~c1~:1!~~;"r;~mF~~~=~~~::~Ie!~=::~e t the drops formed there being relativeiybi:Zg~~eat; differini marked leading edge of the sheet becomes uns a ly in size from those disrupted from the sides of the sheet. fl id ti 1 moving through another fluid. When~~ ~~~~~=i~=~sr~~t1:h:J~~~i:nt ~reata:s !e:sured b~e;f~ai~pl:~:nn:!:r o2g /gco, or a r~!ated ~~:=~' !~!e!:~P t~e v:~!r:~:ri; longer spherical but ., number is near un Y or a d, di ~n flow conditions drop size and on the is distorted in complex ways epen ng_
PAGE 8
4 CHEMICAL DGINEERING EDOCA!IOI .T~ 196~ P :o p ert ies ~f th e fluids. Perhaps the lea st understood of the forms that appear a e. th os e WJ. th ~ h arp cusps. For drops falling under grav1 ty a "tear drop" vi th & ~ i ngle c u sp a~ the rear is observed LJ'~raroui and Kintner (13) 7, for a drop in C ou ette flow a "sigmoid shape" is induced with cusps at eitherend IISumscheidt and M ae on ( 39)7 o '"~ As a problem in the statics branch of fluid mechanics, the theoret i cal and ex p erimental work of Allan and MAson (1) on the equilibrium shape and burst of dr o ps subjected to electric fields is Tnteresting. They predict that the drop wil be deformed into a prolate spheroid and that it will burst when its eccentricity would otherwise exceed ff. By assuming that each half of the burst drop reta ~ ns 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 theory. 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 s~ear rate is increased, finally forming a "sigmoidal" 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. rn the same apparatus, Forgacs and Mason (15) have observed the behavior of small flexible fibers (mainly wood fibers). AsPigure 6 shows these fibers follow quite complex orbits. The more flexible form single loops or multiple h elixes. 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 Silberberg. It i~ 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 a.xis 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 a.xis move initially outwards. The contention of Goldsmith and ?1eson ( 17) that solid particles whether spheres or elongated, stick close to the tubewall if initially placed there is explained by Oliver as being due t 0 0 the very small size of the particles used by them. There is, however, no denying Goldsmit ~ and Masons observation that fluid particles move away from the wall at yelocities 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 nyperbolic flow ap paratus consisting of four cylinders rotating with the proper 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 crossshaped zone indicated on the figure swept free of tracer particles. Lately t h ere has been a reawakening to the intriguing role played by surface h sics in the flow of fluids. The wide diversity of these phenomena is evident fr~m the review of scriven et al (4.2). To cite only two examples, JebsenMarwedel ( 2 5) has described the erosion of tlie walls of glass melting furnaces at a series d it t the ~lass/~as interface and has attributed this unof r!gul~~!~ns~~c!hepfl~wainduced by surface tension variations caused by the sousua P~f the wall material in the melt. Another example ls the beautiful experi lution f the Lan uirs ( 3 0) on the evaporation of ethyl ether out of water soluments o gm itation of the surface is very evident and the role of surtion. ~lontaneo~: arresting the agitation is demonstrated dramatically by the face ac ve agen d from ~urning ether vapors by the addition of a drop quenching of the flame fo~eof this reawakened interest there have been produced of oleic acid. As atresu tion pictures fcrrell and Westwater (36~/ illustrating several very interes mo 1 Ined is the very fast, o ten almost ex "interfacial turbulence t As yettfo!x~e~ls that one often sees in these pictures. plosive, growth of nascen convec i i models for mass t ransfer at a fluidtluid The classical chemichaltendgi neree~i by the recogni ti on of the wide occ~~n c ~ i nterface have been somew a sc ..
PAGE 9
1963 CHEMIC.AL DOIIEERlllG EDOCATIOll S 0 r apont nasitation or the 1ntertace 1n the case of system wpere mass ill ~l~g tran ferred. That these models still have their uaea c,m be attributed part 7 to the equally dramatic effect of surface active materials in arresting the pontaneoua agitation. One :lndeed wonders whether most plant streams may not be :~t1c1ently contaminated to substantially suppress these bizarre effects. In any considerable doubt 1a thrown on the validity of inferences drawn from com tbarat1Te teats on mass transfer equipment in whioh care is not taken to reproduce type and degree of contam1nation. An urgent requirement for the chemical en gineer 11 one or more reliable "meters" for characterizing the mechanical effects ot aurtace active materials, Any phenomena exhibiting these effects which are au!'ticiently reproducible and accurately measurable are, therefore, well worth our notice. The work or Haberman (19) on the rise of circulating and noncirculating bubbles through tubea, the measurement of zeta potentials of falling jets (11), and ~he rnany phenomena described by Myaela, Shinoda, and Frankel (J!i) Pre worth mentioning in this regard. The latter authors, in their intriguing monograph on aoap 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" apota at the bottom surface and the upward convection of theae apota 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 veaael fil~ed 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 tmsatiafying since it la assumed that the shape of the bodies enclosing the thin film la 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 or drain age of the films. Christensen (8) shows how one may obtain aolutions 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 viscosi ty is constant, he considers the effect of pressure on t h e viscosi t3 of the fluid filling the g ap and also the effect of plastic deformation of the solids. The shape pf ttle permanent set given to the solid bodies ls very depend ent on the "lubrication" conditions. One might have here a model for calculatin g film drainage in cases where the fluid viscosity depends on layer thickness. The problem of predicting the condit i ons of flow on thin films flowin g under gravity has long interested chemical engineers. A rev i ew of the work on th i s problem is instructive in showing both the power and the limitations of theoreti cal fluid mechanics at its present stage of development WhRt happens when a liquid film flows dow~ a flat 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. ftt 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 g row rapidly as the flow rate is raised. At much hi gher flows, say 10 00 pounds per foot of per~ ipnery, another change occurs. The appearance of the surface is modi fied a nd t h e trend of average film th i ckness as a functio n of flow rate sh o ws a b reak. The second transition is usuall y identif i ed with the transition from laminar to turb u lent flow since the Re y nolds number at which it is occurs is nea r 200 0. B etween this transition and t h e lower one the flow is lam i n a r but ha s pro n o unced w aves on the surface. The waves evidently will affect mass a nd heat transf er a nd a r e 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 att~mpted as early as 1924 by Kapitza (26). His analysis was a pparently very successful since it predicted the Reynolds number for transition a c curately and showed the effect of surface tension as well (see Figure 8). The p a per 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 b e solved by the classical method.of linearized disturbances and that many of the mathematical ~ diff 1 cult l es t ha t beset this technique in other problems could be avoided since asymptotic met ho ds ~p propriate for small Reynolds numbers can be used. Using Yih 1 s suggest i on, Ben jamin (3) solved the problem and concluded, rather astonishingly, that the flow 1 8 unstable at all Reynolds numbers, no matter ho w small. The theory, in this form apparently contradicts the experiments show i ng a fairly well defined tran sit!~. Benjamin noted, however, that the rate of g rowth of disturbances is very small until the Reynolds number attains a value of a bout 10. He taerefore made the suggestion that the flow disturbances, altho~h u nstable at lo w Re ynol d s ...
PAGE 10
~ CHEMICAL EIGIIEERIIG EDOCA'tIOB June 1963 n,qubers have not had auff1c1ent time to develop to detectable aize Fi re 8 shows the results of a quantitative development of this idea. One.can ~~fine the pseudocritical Reynolds ~n:mher 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 1s seen t~at 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 B~njamin has not been accepted by Teilby and Portalsky (WJ.) and(~) who have made the most recent and extensive series of measurements orthe 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 fl~w 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 Reynold; numbers. However, if the channel ls inclined as little as 1/2 degree off vertical the flow ls now stabilized below some definite Reynolds number. This sensitivity of the analyses to small changes points the moral of not generalizing t h e 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~tjon has never been made. As Gibbs has pointed out, in any multicomponent system, the phase interf~ce must exhibit an apparent elasticity or resJstance 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. rt is curious that no one hF.s invest igP. ted the stability of this type of turbulent flow by the conventional techniques. This configuration might facilitate comparative experiments on turbulent transition. 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 gasliquid pipe flow. Knuth (28)has shown that waves are raised on the film at film Reynolds numbers lar g er man 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 drivin g fluid behaves as if it were laminar near the interface. The results of t h e 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 p l 1ase 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 condiUon 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 eit h er the Reynolds number or the Weber number is small enough. As is shown in Fi g ure 10, Knuth's results are substantially confirmed. One may, I think, draw a moral from t h e history of these attempts to pred i ct conditions for the raising of waves. Theoretical analyses must be simplified considerably in order to be tractible. It is possible by judicious jugglin g of the assumptions to force the analyses to agree wit~ 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 ?ilm. 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 destr~ys the apparent good agreement of this analysis wit h 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 transi tion.
PAGE 11
June 1963 CHEMIC.AL EIGIIEDIBG EDUCATION 7 Let us now return to the question posed at the beginnin g of the paper: To what extene can the chemical engineer use the r e sults 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 approech may be purely empirical or the engineer himself may have to become an expert in fluid flow. DPily, the ch e mical 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 successful, however, is the engineer who hP.S formulated the relevant proplems 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 lmow or cannot economically find out, and has the courage to assess realistically the rl k 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 or us to look sharply at the "things that happen 11 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 cont aining water. Doea not the roiled surface of the bubble strongly resemble the roil~d 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? BIBLIOGRAPHY 1. AJ J.en, R. s. and M:iaon, s. o., "Particle Behaviour :ln Shear am Electric Fields", froceed1ng,t ot tbe Royal Society (LaDc1on), A 867 Part I, 45, and Part II, 62, (1962). 2. Bagnold, R. A., "AutoSuspension ot Transported Sedinent; Turb!di'\y' Cmrents 11 Proceedings of tbe Royal Society ( IDDdon), A 265, '15 ( 1. 3. Benjamin, T. B., ''Wave Formation in Lem:lnar Fl.OW' Down an lncl1ne4 PJttn : J. Fluid M!ch. _g, 554 ( 1957) 4. Benjam:ln, T. B., "Ettects ot a Flexible Boundary on }lvlh'Cdyi J. Fluid M!ch. 2, 513 ( 1960) 5 Boumans, A. A., 11 Streeroing CUrrents 1n Turbulent Flows and Mttal Capillaries 11 Physica 100755 ( 1957) 6. Bl1sbmannv, V. K., 1 'l(ydrodynernin stability or a Liquid LB19r on a Vertical Wall", Sovie t Physics JETP. 12., 873 ( 1961) 7. Caldwell, D. R. and Donnelly, R. J,., 11 0n the .Reversibility ot the Transition Past Instability in Couette Flow"., Proceedings ot the Royal Society ( lol'.ldon) A 267, 197 ( 1962) 0.... 8. Christensen., H., "The 011 Film in a Closing Gap", Proceed1ngB or~ Royal Society ( london) A 266 312 ( 1962) 9. Christiansen, R. M. end Hixson, A. N., ''Breakup or a Liquid Jet 1n a Denser Liquid 11 Ind~ Eng. Chem. !t2., 1017 ( 1957) J 10. Dombrowski, N., and Hooper, P c., "The Effect or Ambient Density on Drop Formation in Sprays", Chem. Eng. Sci. .ll, 291 ( 1962) ll. Durham, K. 11 Surtaoe Activity end Detergency", MicM:l J Ian, London, 1961. 12. Eisenmenger, w., 11 Dynam:l.c Properties of the Surface Temiion or Water and Aqueous Solutions or Surface Active Agents With ~ing Capillary Waves in the Frequency Range from 10 kc/s to 1.5 mo/s. Fararoui, A., and Kintner, R. o., "Flow em Shape of Drops or Non Newtonian Fluids", Trans. or the Soo. or Rheology, 2, : 369 ( 1961). 14. Feldmn., s. "On the Hydrodynamic Stability of Two Viscous Incoq,reaai Fl.uids in Parallel Uniform Shearing tbtion"., J. Fluid M:!ch. g, ,., ( 19571 15. 16. Forgacs, o. L~ em tilson, S. G ., "'!be Flexibility of Wood Pulp Fi bera", TAPP I, 41, 695 ( 1958) Ga111dns, F. H. and Philippott, W., 11 1he Behavior ot Jet ot Viacoel.aetfA ~ Fluids", Trana. ot the Soc. of Rheology 2, 181 ( 1959). 17. Goldsudtb. 11 H. i.. and Miaon, S. G. 11 .ilial Mlgratian c:tr Ni_ '1~4le in PoiseuiJJe Flow", Nature 122, 1095 ( 1961) P., ,......,.. Stabil:i..,..: ot a sirat1t1ed Flaw" J t "lul 18. Graebel, W. ......., ..., 321 ( 1960)
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8 CHEMICAL DGIIEERIRG EDUCA'tIOI June 1963 19. Haberman, w. L. end Sayre, R. M.., "Motion or Rigid and Fluid Spheres in Stationary and 1bving Liquids Inside Cylindrical Tubes", U.S Navy Experimental 1bdel Basin Report ll43 ( 1958). 20 Har tnett, J. P end Eckert, E. R. G., "Experimental study ot the Velocity end Temperature Distribution in a HighVelocity Vortex Type Fl.ow" Trans. Amer. Soc. M!ch. Eng. 12, 751 ( 1957). 21. Harris, J., "Flow or ViscoElastic Liquids from Tubes", Nature 190, 993 (1961). 22. Hughes, R.R., Handlos, A. E., Evans, H. D. and Mlycock, R. L., 11 'lhe Formation ot Bubbles at Simple Orifices", Heat Tlsnarer end Fluid M!chanics Institute 143 ( 1953), Stanford University Press. 23. Hughes, R. R. and Gilliland, E. R., "Miss Transfer Inside Drops in a Gas", Chem. Eng. Prog. Sym. Series 2l, No. 16, 101 ( 1955). 24. Ingebo, R. D., "Drag Coefficients for Droplets am Solid Spheres in Clouds Accelerating in Airstreams", NACA Tachninal Note 3762 ( 1956). 25. JebsenMlrwedel, H., Glastech. Berichte .., 233 ( 1956). 26. Kapitza, P. L., Soviet Physics JETP 18, 3, 19 ( 1948). 27. 29. 30. 31. 32. 33. :,4. 35. 36. 37. ,a. 39. 40. Klinkenberg, A. and van der Minne, J. L., "Electrostati cs in the Petroleum Industry", Elsevier, Amsterdam, ( 1958). Knuth, E. L., "The ~chanios of Film Cooling", I Jet Propulsion 24, 359 (1954). II, Jet Propulsion E.2,, 16 (1955). Kramer, M. o., "Boundary Layer Stabilization by Distributed Damping", Readers Forum, J. Aero/Spac e Sci. gr, 68 ( 1960). Langmuir, I. and Langmuir, D. B. J. Phys. Chem. 21,, 1719 ( 1927) Li, T. Y. and Kusukawa, K., "Steady Subsonic Drag in no:rrF.quilibrium Flow of a Dissociating Gas", Proc. 1962 Heat Trans. and Fluid ~ch. Inst. Stanford University Press 1962. Miles, J. W., "The Hydrodynamic Stability of a Thin Film of a Liquid in Uniform Shearing Motion", J. Fluid Mech ., 593 ( 1960). Miller, D. R. and Comjngs, E. \./ ., 11 Fo rc c lk11rentun Fields in a llual Jet Flow", J Fluid ~ch. 1, 237 (196 0) Mysels, K. J., Shinoda, K., and Frankel, S., "Soap Films S tudies of Their Thinning", Pergamon Press New York 1959. Oliver, D. !., "Influence of Particle Rotation on Radial Migration in the Poiseuille Flow or suspensions", Nature 194, 1269 ( 1962_. Orell, A., and Westwa~r, J. W., "Natural Convertion Cells Accompanying LiquidLiquid Extraction", Chem. Eng. Sci. 16, 127 (1961). Reynolds, A. J., 11 Enerf?Y Flows in a Vortex Tube", J. Angew. Math. Phys. 12, 343 ( 1961). Rozenberg, L. D. and Eknadiosyants, O. K., "Kinetics o f 01 trasonic Fog Formation", Soviet Physics, Acoustics ., 3, 369 ( 1961). Rumscheidt, F. D. and Mason, S. G., "Particle rt:>tions in Sheared Suspensions XII Deformation and Burst of Fluid Drops in Shear and Hyperbolic Flow", J. Colloid Sci. 16, 238 ( 1961) Seffman, P. G., "On the Stability of Laminar Flow of a Dusty Gas", J. Fluid M!ch. .ll, 120 ( 1962} 41. Sawyer, R. A., "'lhe Flow Due to a 1\loDimensional Jet Issuing Parallel to a Flat Plate", J. Fluid M!ch. 2, 543 (1960) 42. Scriven, L. E. and Sternling C. V., "The ti!.rangoni Effects", Nature 187, 186 ( 1960). 46. 48. Sproull, W. T., ''Viscosity of a Dusty Gas", Nature 190, 976 ( 1961). Tailby, S. R. and Portalsky f3., Trans. Inst. Chem .Engrs. 324 ( 1960). Tailby, S. R. and Portalsey S., ''Wave Inc ept ion on a Liquid Film Flowing Down a Hydrodynami ~ally Sm::>oth Plate 11 Chem. Eng. Sci. 17, 283 l 1962) van Duyne, R. J., 11 M!asurement or Dynamic Surface Tension Changes in FrothForming Aqueous Solutions", Thesis PhD University of Michigan ( 1961). Westley, R. 11 A Bibliography and Survey or the Vortex Tube", Note Nr. 9 College of Aeronautics, Cranfield (1954). Yih, Proc. Second US Nat Cong of Appl. M!ch. 623 ( 1954).
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'"' ...., ; c., ..... ._ ::) U ':.L C: "' J (J(J r:~ I O I 0 l I A cc el c> rating Spheres N e gative Accelerations IO Solid Spheres at Constant V e lo c ity 100 R ey nolds Number, R e l~:il_ ur< ~ :. ..:' \CCE:LER A TIONAL DRAG l)ATA OF ING.EBO ~. .. .... ,. I V .. a) How Charging Occurs + + Concentration + of Negative I ons + + + Concentration + of Positive Ions + + + + Laminar Sublayer + + + + r T urb ulent Zone + I + + b) H ow Turbulence Infl uences the Process 1000 Figure Z. ELECTRIC CHARGING INDUCED BY TURBULENT FLOW After Klinkenber g and van der Minne Water Drop ,. 1 ...., ...... .. ... .. .: Erlenmeyer Fl as k with Neck Drawn Out a) Mu ltino dal Liquid Proje ctio ns Ob served in Ultrasonic Atomization After R ozenberg and Eknadi osyants Piezoelectric Transducer b) Simple Experimental Setup t o Dem on strat e Explosive Atom izat ion Figure 3. ULTRA SON IC ATOMIZATION
PAGE 14
a) b ) c) . .. I a) Br eakup from Side Edges Only Occurs at Low Gas Dens ity . .. :~~=:.~~ .. .. .. .. .. .. :,;._ .. o .. '. .,, 0 D 0 4:.:a .o b) Dominant Breakup is at Leading Edge when Gas Density is Moderate Figure 4. REGIMES OF ATOMIZATION OF FAN SPRAYS A ccordin g to Dombrowski an d Ho oper I In creasi ng Shear Rate Figure 5. BREAKUP OF DROPS IN COUE TTE FLOW A cc ording to R umsche id t and Mason Rigid Turn Springy Turn Snare T urn Turn Complex ( ,,. c '(, Sf C C r .) Fisure 6. ROTATION OF FIBERS OF VARIOUS FLEXIBILITIES Acc o rding to Forgacs and Mason
PAGE 15
' ;.. ro ::I ..c; "f' II ::t h ...,. II h CJ QJ ~ a) Streamlines are Traced by Solid Particles No Segregation Occurs Solid b) Streamlines are Traced by Fluid Particles. Par ticles Move a w ay from Boundaries F l uid Objec t Object I ......._ C) 9 '1 ~ ,, c) Contrasting Motion of Solid and Fluid Particles Near Wall in Poiseuille Flow F igure 7. MOT I ON OF SOLI D AND F L UID P A R T I CLES IN LAMINAR FLOW After Rumscheid t and Mason 100 .~ 10 10 0 0 Equation Based on Benjamin's Model 100 Kapitza I s Equation Data of 0 Jackson 6. Grimley Binnie 1,000 \.. ;;;.... 00 Figure 8 CRITICAL WEBE R NUM B E R FOR WAVE FORMATION FO R LIQUIDS FLO W IN G D O W N A VERTICAL WALL 10,000
PAGE 16
S.. ..c 9 z CD 0 .... 0 cP p:: .... al u ... ... J.. u 10 7 r~ 10 5 10 10 3 0 Unstable with Respect to Wave Formation Transition / Zone Observed by Knuth 0 .1 O.Z Predicted Curve of Miles Stable 0.3 0.4 Weber Number 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.
PAGE 17
ADVANCES IN HEAT AND MASS TRANSFER by 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. Thia 1s, 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 app~oxi mately 500 references selected from more than double the ntimber 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 (refs. 1 to 3) may also be pointed out. Heat conduction 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 reentering 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 temperatureresistant sub stance like asbestos or ceramics and an ablating substance, for instance, some plastic. Under the influence of a convective heat flux q 0 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 Vg 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 procQss conduction, heat convection, and mass transfer processes are interrelated. Ad~itionally, heat sources or sinks are provided by the phase changes and the chemical reactions oocurrlne (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. Heat Convection 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 connect ed with ducts of noncircular crosssections. 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 crosssection contains corners with small openin g angle. Fig. 2 shows as an example the results of m easuremen ts for fully developed turbulent flow through a duct the cros sse ction of which has the shape of an isosceles triangle with base to height rati o 1:5 (ref. 5) T he measured results indicated by the open circles and trian g les 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 pred ic tion would over estimate the heat transfer by a factor of two. The actual heat transfer in such a crosssection 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 crosssection 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 reg ions where the flow has separated from the surface. This occurs, for instance, at the downstream part of blunt 13
PAGE 18
~ ~ CHEMICAL DGIBEERil'G EDUCATIOB June 1963 ._jeot's op : ,ii1ict ~ teja 1n the aurface contour. F 1g. 3 sketches suoh a heat i P aaater situation. !he boundary layer which arr1vea in its downstream movement at the corner of the atep aeparatea from the surface and reattaches again only further downstream. The region between this boundary layer and the surface is f illed vith a rotating body of flu.id (dead water region). A second boundary l a yer 1a ~reated batveen thia dead water and the surface. Heat transfer from the wain stream to the bodj surface behind the step hes therefore to overcome two Ns1s~anoes in aerie : one 1n the separated boundary layer and one in the at tached bounda17 layer. A large variety of flow situations may exist depending on the Reynold and Mach ~umber of the flow. The boundary layers may be laminar or turbulent. or transition to turbulence may occur in the boundary layers. The body of eparated fluid may also be laminar or turbulent or in fluctuating unsteady motion. Thi complicates an understanding and analysis of the heat transfer process considerably. Nevertheless, analytical approaches for some of the flow oond1t1ona were quite successful (refs 7, 8). It ia the opinion of this lecturer that the most significant advance in the creation ot a science of heat transfer was caused by the concept of a constant property fluid introduced by Wilhelm Busselt in 1916 and that teaching of heat tranater haa to be baaed extensively on the model of a constant property fluid. Bev engineering developments, on the other hand, have created many situations vhere large property variations exist. such variations cause no principal diffi oultiea 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, tor'instance the d1ffusiv1ties for mementum and heat, are ohanged in the presence of local property variations. From the results of analyse! 1t 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 tor heat transfer connected with turbulent flow of carbon dioxide in the critical and supercritical 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 irnm~diate neighborhood of the critical st a te. I n vestigations on free convection heat transfer with a ZehnderMach interferometer, on a vertical_plate exposed to carbon dioxide within 0 one degree to i ts crit i cal atate and with temperature differences of order 0 .01 C made it poss i ble to c a culate the thermal conductivity (ref. 10). The rem a rk ab le conclusion has to be drawn from the results that the thermal conductivity does not only depend or temperature and pressure, but also on the intensity of the heat flux. S imi ar observations had been made before but had been attri b uted to convection ef f ects which can be excluded in the present investigation. This is only o ne i ndic r. t i on that an underatand1ng of the critical state is still almost nonex j stent. Through many years, convective heat transfer had been studied almost exclu aivel for steady state situations. This is justified by the fact that very rapi d chang!s of the boundary conditions are required to produce heat transfer coeffic ients which are significantly different from steady state values. Nevert h eless, new a plications have raised the question on the limit for the use of stead y state 1 p Fig S shows the results of another interferometric investigat i on o f free !~n::~tion h;at transfer to a vertical place under t h e condition that suddenly a locally uniform heat flux from the surface into the surroundin g water is starte d (~ f 11) The measurements essentially indicate that f o r t h e initial peri o d, he:tis t;ansferred into the fluid b y unsteady c o nduct i on and t h at the su b sequent transition period to steady state is quite sh o rt. H t transfer connected with boiling or condensation is an area whic h is still unders~~od only partially in spite of the intense rese a rch effort w h ic h has con tinuously been devoted to this process. The problems an d research attempts in this area are actually 80 large that they cannot even be sketched in this lecture. For a discussion of the physical processes involved and of the analytic attempts wh~c h have been published, the reader is referred to the attac h ed references 12 and 1 It is the feeling of this lectur~r that the creation of concise and consistent models is still lacking in the analytic investigations. Boilin g and conden;ation of liquid metals, for instance of mercury, and the influence of gravity on ree convection boiling are subjects which deserve special attention. Mass Transfer A discussion of mass transfer should certainly start with th~ an~l~gyw~!~w~n heat and mass transfer processes which has originally been pointe ou y lel Nusselt in 1916. This analogy permits to predict mass transfer situations so e Y from information on an analogous heat transfer process especially when the massi transfer rates and the temperature differences involved are small. Recently th s
PAGE 19
June ~?63 CHEMICAL EllGIBEERI!IG EDUCATIOB 1$ :~~ 0 !7 has been extended by Russian and American acienti1ts to oover ituation in th ~ge mass transfer rates, large temperature differences chemical reaction 1 luid, and also especially at high temperature levels (ref 14) The ana a~:Y ncludes processes in which heat and mass transfer are inter;elat;d. They t e;pecially useful for gases and will be discussed on the example ot ma11 nJ~ 8 er from a surface into a gas flowing over it, thus creating a twocomponent x ure!in the laminar boundary layer. Eq. (1) in Fig. 6 describes a ma11 tlu vector mi for the component 1 in the mixture, that is, the tass flux per unit time and unit area of a plane arbitrarily located within the twocomponent fluid. The ~ir st term on the righthand side of the equation describes the mass flux by dif uaion as a consequence of a gradient wi of the mass fraction of the component 1. ~lf is the mass diffusion coefficient and the mixture density. The second tem 8 cribes the convective tr~sport ot the component i with a density as a consequence of a movement of the mixture with the velocity v through the 1 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 1n 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 righthand aide ot 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 dittuaion mass flux of the components involved. The third term describes transport or 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 thermodiffuaion. Eq. (2) can be ~ewritten in a different form. when the gradient of the temperature Tis replaced by the gradient of the mixture enthalpy h (Eq. (3)). Thia equation is especially useful tor a fluid with a Lewis number Le equal to one, because the second term drops out in this case. The Egs. (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 occuring within the boundary layer destroy or create one or the other component and appear, therefore, in the maaa balance of Eq. (4) as a source term Ki. The energy Eq. (5) has been written in terms of the total or stagnation enthalpy hs (containing kinetic energy as well aa internal energy}. The first term on the righthand 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 w may indicate the mass fraction of the element i in the mixture regardless whether 1 the element appears as such or in the chemical compound. No chemical element is created or destroyed in a chemical process, and aa a conse quence the source term K 1 vanishes 1n Eq. (4) when it ls 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 moat 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 oonser vation equations alone do not describe the transfer problem completely. A momen tum equation and e~uations 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 transf e r 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 n11mbars dlff erent 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 (refs. 15, 16), however, have demonstrated that one has to be careful in this respect. Thia 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 Tg:r,, outside of the boundary layer could be adjusted by preheating or precooling or the helium. Fig 7 presents the heat flux q through the porous surface as a function of the difference between the wall pur1ace temperature Tw and the air temperature T00
PAGE 20
.. f . 16 CHEMICAL EIGl,EERlBG EDUCATIOB June 1963 wi th the specific ma~a fiow m of the helium as parameter. One aeries or measure me ., :1. t s made vi th air instead or helium injection la also entered as dashed line. ; Th e heat flux qv is defined as the sum or the first two te~s 1n Eq. (2). The striking feature in this figure ia the observation that the heat flux bec omes zero at a finite temperature dift"erenc e T TOQ. and 1 t is believed that this is a con.. sequence of the fact that concentrat!on d1rt"erences within the boundary layer ap. pear through thermodiffusion as driving force in addition to condotion. Another consequence of this interplay of driving potentials is the fact that a finite dif. terence between the wall surface tempe rature Tlil and the air temperature Tcx, exists when the helium is admitted into the po~ous cyiinder at a temperature T00 equal to the temperature 1n the outside air fl ow. This situation is marked in the .figure by croaaes and it can be recognized that the wall temperature may be \lP to almost 30 degrees high.er than both the helium end the air temperature~, depending on the injection rate of the helium. Similar effects have also been observed in laminar boundary layers for forced and free oonvect16n. An area in which investigations have recently started are transfer processes 1n a gas plasma. In such a plasma, the temperature 1 s 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 threecomponent mixtures of neutral atoms~ ions and electrons are involved, and electric as well as magnetic body toroea in.flue~c~ the flow. An example or a recent experimental investigation in this area is shown in Fig. 8 which presents and analyzes the local heat flux di~tribution into a watercooled anode ot an electric arc burning in argon (ref. 17). It can be ~ecog nized that only a small portion of the specific heat flux q into the anode surfa ce la caused by convection of the atom gas. Convection of the e lectron gas which .is generated at the cathoqe and absorbed by the anode, contributes approximately an equal fraction and the rest is due to energy released when the electrons en~er the anode material (similar to a heat of condensation). The heat flux q ind!cates the electrical energy which is converted into heat within the currint tube ending at the anode location Wlder consideration. It may be recognized that the majority o.f the electrically generated heat enters the anode surface. This is the reason f o~ 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 kr;l.own in any engineering ap plication. Radiative Heat Transfer An important tool in all radi ative heat transfer calculations is the shap.e or angle factor. Graphical, mechanical, and optical means have been described, in addition to analytical methoqs 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 show:n 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 optic~l images as shown in Fig. 11 in which l (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 proc~sses. It is important to obtain exact solutions to the integral equ~tions 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 radiatingab sorbing medium filling the enclosure (ref. 21). Engineering analyses usually attack problems in which radiative energy occurs simultaneously with ot~er transfer mechanisms like conduction or convection in such a way that the various con tributions are calculated separately and that the total ener~y t~ansfer is ob~ t i d by a 8 ,1mmati on of the individual p9.l'ts. In reality, s1tuat1ona are en c ~~~ered in ;hich the various transfer processes interact. Such interaotions n ave been studied for a few cases in the rece~t past (ref. 22).
PAGE 21
. I . June 1963 . CHEMICAL ENGINEERING EDUCATIOi .. .. 17 At the end of our discussion we will return to the molecular and convective trans port processe s with a brief revie w of the similarity between radiative and molecular transport (ref. 23 ). We can consider radiative transport a~ caused by the movement of photons in a stmi lar way as energy, mas s or momentum tran sport is caused by the movement of molec~les. Fig. 12 illus trates this similarity by considering Couette flow or h~a.t conduct~ori in a r.are.f;'iEtd gas betwe.en two parallel walls on one hand, and radiative energy transf'.er in a radiating and abs orb ing medium between: two parallel wal:). .' s which are nontransmittin g on the other hand .. The temperature or velocity varl ation be tween the two wal ls : f~llow a stra ight line as i~dicated in Fig. 12a, as long as the ratio of the me an free path length is v ery small com'Pared to the di stance L of tpe ~wo wtlls. Wi tp ~nc~asing path length, temperature or velocity still exhibits the linear vari ation within the gas; however, a slip of the velocity or a temperature jump can be obser~ed 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 (F ig. 12d). The terms for the correspond i ng regimes are indicated above the figures. c ompletely analogous ai tuations exist for the radiative transfer process. The black:body emissive power eb has now to be considered instead of the velocity or temper atur~. T~1s emisai ve p o wer d rops linearly in the absorbing and radiating medium as long as the free pe t h len g t h of the photons i s small compared t'o the dist~ce L. Jump~ near the surface of the two walls occur with increas ing photon path length. The variation in the absorbing medium 1 tself d e cr eases toward~, zero when the photon path 1en g. t h gets l e rger and larger. Tl1e terms t:or the.se regim~ s are 11 st ed belo w tne figures. This ~imilarity is very helpful in unifying tbe conc,pts for transfer processes a n c l such a unification and interrelation of the c9ncepts I w9uld conside r as one of tl 1 e most essential' requirements of Ei. good co~ae in transfer processes HOT GAS Va . FIG. I ... . I V 0 VELOCITY OF ABLATION Vg VELOCITY OF ABLATED GAS qc CONVECTIVE HEAT FLUX qr RADIATIVE HEAT FLUX .. . ..
PAGE 22
18 CHEMICAL BIGilkBRllG EDUCATIOB l'1M 1963 l) I. R. G. lcllllrt, T. P. Irn.ne, Jr., I. Ke Sparrow, and W. I. Ibele Heat 'lnnster, A ReT1ev of C\4"lent Literature, International Jo~aklll ot Heat and Maaa T.ran,ter, 'fol. 3, pp. 293306. 1961. 2) .l V. tu1 klff, "Beat Tnoter Bibliography Ruaeian Varn, Intematf anal Journal ot Beat and Mau 'l'lai!ater, 'fol. S, pp. S71S82. 1962. 3) Recent Advaucea 111 Beat am Maas Transfer, J. P. Bartnett, !ditar. Rearavm1 Book c0111pa~, Wew fork. 1961. 4) T. R. Munson and R. J. Spindler, "Transient 'l'hermal Beba"Yiour of Dec0111poaing Materials, ns Paper No. 6230. 1962. S) E. R. G. Eckert and T. F. Irvine, Jr., "Pzieaaure Drop and Heat Trimafer in a Duet with '&iangular Croaasectton., Jum,1l of Basic 'lngimering Tol. 82, pp. 1251.38. 1960. 6) w. H. Lowdermilk, w. r. Weiland., Jr., and J. 1. B. Livingood, "Measurement ot FIBat Tranater and Friction Coetticimits tor Fl.ow of Air in NCl'lcircular Duct.a at High Surface TemperatUl'8a, HA.CA Research Mem o L5)J07. 1954. 7) D. R. Chapun, "A 'l'heoret.ical .lna~a of Heat Transfer 1n Regions ot Separated Flow, aCA Tl 3792. 1956. 8) W. o. Carlson, Heat 'lranater in tendnar Separated and Wake Flow Regi.0118 1 9 1959 Heat Transfer and Fluid Mechanics Inatitute 1 Stanford Um.versityPress, Stanford, California. I 9) R. G. Deiasler, "ConTect1n Beat 'transfer and Friction 1n nov of Liquids," BJ:b Speed Aerodz1r:f "11 and Jet Propulsion, Yal. V, Princeton Uni:versity Pr as, ~eton, J. 1959. 10) Harold A. Simcn, "An Interteromtric Imestigation ot Lamnal' Free Convection in Carbon Dioxide Near Its Critical Point," Ph.D. thesis, Uni"Yersity ot Minnesota. 1962. 11) R. J. Goldstein and E, R. O. Ickert, The Stead1' and Transient Free Conection Boundary Layer on a Un:U'Oflllly' Heated Vertical Pl.ate, International Journal. at Bat 1nd 'fa.as Transfer., vol. 1, pp. 208218. I96o. 12) R. R. Sa~raky. "Surve:y ot Problems 1n Boiling Heat 'l'ranster," High Speed Aerodynam1 ea and Jet Propulsion, vol. V, Prince ten Um. verei ty Press, Prin ~ eton, N.J. 1959, 13) V. M, Roheenov and H. Y. Choi, "Heat, Mass, and Momentum Transfer, 11 Prentice Hall, Nev York. 1961. lh) L. lees, "C o n..,.ecUve Heat Transfer with Mas:i Addition and Chendc::il 16) 17) 18) 19) 20) 21) 22) 2.3) Ra~ct.ion~." Recent Advances 1n Heat c1nd ?1a i;s Transfer, editor J. P. Hartnett, McGTavHill Book Caupany, Rew York. pp.lol.222. 1961. J. R. &ron, "ThemodynaJ1ic Couplin11: in Bo, : nrlary Layers," American Rocket 6~:i ety Sp ice Flight Report to the Nation 220661. 1961. o. r. Tewfik, F. R. G Eckert, and r. J. Shirtlifte, "Ttu,1nal Ditfusim Eff ec ts cm Energy Transter in a Turbulent Boundary Layer with Helium Injection." Proce~s 1962 Heat banster and Fluid Heebani..cs Ineti1nte 1 Stanford Uni versi cy es., Stanford, California. PP 42ol. P. Schoeck and E. R. o. Eckert, "An Investigatim of Anode Heat Transter in High Intensiey Arcs," Proceed! s or the th IntemaU onal Cmterence on Joni1 ation Pbe110111Bna in ases, ..,a__.. eaap&ny, Amsterdam, Netherlands. !. M. Sparrow, "A New and Simpler Formulation tor Rad:fatiTe Angle Factors, American Society of Mechanical Engineers, Paper lo. 62HT17. 1962. !. R. o. !ckert and E. M. Sparrow, "Radiative Heat Excha~e between Surtacea with Speculllr Renection," International Joumal ot Heat and Maae Transfer, 'Yol. 3, PP 42Sh. 1961. E. M. Sparrow, E. R. o. Eckert, and V. K. Jonason, "An !ncloeure TheoJ'1 tor Radiative Exchange between Specularly and Dif.f\lse~ Renecting Surfaces," American Sociev ot Mechanical ltngin eers, Paper !lo. 61WA167 1962. R. Viskanta and R. J Oroeh, "Heat Transter in a Thermal Radie tion Absorbin~ and Scattering Medimn," International DeveieOJ!!!nts i~ Heat Tranater, part IV., PP 820828. American scid.eti ot chanical Engineers. 1961. R. Viakant.a and R. J. Grosh, "Heat Transfer b7 S1nm1 tlnellllil Conduct1cm and Radiatioo 1n an Abaorbing Medium, Journal ot Beat b'&neter, vol. 84, PP 6.372. 1962. E. R. o. Eckert, 1tS:'nd.lal'i ties between Energ1 Transport 111 Rarefied Oases and by' Thermal Rad1ation. Modern Dnel(!Plllnts in Heat Transfer, Academic Press., Hew York. 1962.
PAGE 23
100 kw b 24 Nu 50 10 5 I 1000 3kDh ~'v
PAGE 24
1poo 800 0 0 9 60 0 Nuw 0 400 o /3=0 011 2 0.0 14 3 o 0.00 7 00 0078 v 0 00520.0067 100 .___L...,11;..L._.....L..L..l.....L'llL~~ 20,000 40,00 0 6 0 000 1 00 000 200 000 5 00 000 Rew FIG. 4 300r.,..,~,.,,,...._h, ONDUCTION 80 t+++~ 60t++++4STEADY STATE Btu v v v v v hr ft 1 F 40 t==::~++++1U: o X = 3.82 in. c 2.11 In. 1.03 In. 0 c9 v 0.32 In. 20 r.tr+++4+++_:_ _J 10 I 10 100 12C SECONDS FIG. 5
PAGE 25
MASS AND HEAT TRANSFER EQUATIONS : mi= P D,2 Vwi + Pi V + COUPLED EFFECTS (I) q=k VT p D 1 2 Ihi Vwi pVh + 11 11 (2) r =!_ Vh(Lel)IhVw pVh+ 11 11 (3) Cp I I BOUNDARY LAYER EQUATIONS: p (u 0~ + V OWj ) = a (PD oWj) + K (4) ax 0 y O y ,2 a y P (u ohs+ v ohs)= a (.l!::._ ohs)+ o f.(i1) a (..Y.) 2 ] ox iJy iJy Pr iJy iJy Pr iJy 2 iJ [ ( I ) iJ Wj] + oy PD,2 '~Ce Ihi ay (5) ~ ~ ~ p (u iJWi + V iJWi) = iJ (. OWj) ax ay ay ay (6) p (u ohs + v ohs)= o f. ohs) ax ay ay~ oy (7) FIG. 6 800 m (lbm /hr ft 2 ] 0 4.2 A 10.1 HELIUM a 20.2 400 I 30.1 ., y ... 62.7 AIR .. X= 16 11 :, 0 ,, m ., I O" 400 800 60 40 20 0 20 40 60 80 100 FIG. 7
PAGE 26
qO s 5LS' lb o 0 999999, 00 9 99 o 4~ V9 ~Oo I= 150 AMP s = 6 mm kW '99 9 o cm2 CONVECTIVE,:L.'~; 0 ~~~31BY ELECTRON 9 0 0 0 11, : } SPLIT PLATE ANODE : }POINT PROBE ANODE HEAT FLUX TO ANODE GAS 1, o 2. WORK FUNCTIO ELECTRONS TO COPPER ,9 o, ..,.......qo 99f 00 ...,........, q 9 io 0 "' 0 .o 0 #19 o !lvvo ""'x : o'! : 9 9' 9 9 9 9 9 0~~~~~o~o~~'r'..:.''''::! 0 I 2 3 4 5 6 7 CONVECTIVE BY AT M GAS s. 3 4( 3) r, 4 I / I (3) l /. H .. I/ 3,~ .., r 0 3 L ____ .,_ __ __. 2(3) 2 3: Specular I, 2 ,4: Diffuse r(mm) FIG. 8 2 q=O I Az Fzs(IEzs,ca> ec 1f, f, A 1 1,1, e, I FIG.9 I 4 83 = E3crT3 + P3 H3 H 3 = 0,F31 + Bz F32 + B 4F34 84 = E4 trT44 + p4 H4 H4 = 81 ( F41 + P3 F41(3)) FIG. II + 82 ( F42 + P3 F42(3l) + E3crT34F43
PAGE 27
r4, r f/,W CONTINUUM STATE ~< C Iron, hot rolled I I d Copper oxide 40 50 60 70 80 0 90 FIG 10 MOLECULAR TRANSPORT SLIP FLOW TEMPERATURE JUMP TRANSITION STATE FREE MOLECULE ~< I L ~> I L Vw, V,T A r>> I orO NONABSORBING J
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AN UNDERGRADUATE COURSE IN ANALYSIS OF MULTISTAGE SEPARATION OPERATIONS by 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 tmit operation of distillation appeared in the earliest textbooks on tmit 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 rebo~led 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 McCabeThiele diagram or the Ponchon diagram for the analysis of binary distillation problems. Extraction is limited to problems solver on a triangular diagram or a Janecka diagram. The student certainly gains an understanding of sep~ration processes from these problems and he is also wellequipped to solve additional problems of the same type. Unfortunately the chances of his encounterin g a binary system in later work are very small while his chances of encountering a multicomponent sys tem are very lPrge. 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 1s thus the B.S. graduate who needs to lmow the techni ques. Even without the justification of the need of tec h nique, the undergraduate courses should certainly treat as many areas of the subject matter as possible in as general or allinclusive a fashi on as can be done. All parts of the field are, of course, not susceptible to g eneralization. Calculational procedures or analy ses have been worked out which shou ld be taught simply because the sin g le 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 t~ese methods should certainly be taught. I would like to outline here the content of a course we he ve been offering for the last few yeP.rs to the first semester seniors in approximately twenty lectures, and which we intend to offer in the future to second semester jun io rs in approximately thirty lectures. The course assumes th ~ t the st 1.1 dent already has a knowledge of equilibrium constants and their use in the calculstion 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 appropches 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 sep~ration desired on the mixture of A and B fed to the column of stages is not good enough tn a single stage and is in effect, multiplied by a succession of stages to obtain the separation desired. The process shown assumes countercurrent flows linking the stages and shows that in the separation between A end B, a section of stages is needed above the feed point in order to produce a purified product of A P.nd a section of stages is need ed below the feed point to produce a purified product of BThe individual stage is described as a mixing and separating device in which various streams 24
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l une 1963 CHEMICAL EIGIIEERIG EDUCATI05 as may be entered to be mixed, and then separated into stage. What the stage might be or do is discussed. new streams which leaYe t~e 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 sep~rated appear in both phases. It 1s pointed out that the streams may well be vapor and liquid, typical of distillation, absorption, and stripping operations, or they may be liquidliquid streams typical of liquid liquid extraction. They may be gasg as 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 countercurrent fashion~ As far as I know, there is no wa y to prove that this ls the optimum fashion for linkage of stages. But by example, one can easily show that count~rcurrent flow of the phases is better than parallel flow, which is, of course, no good at all, and is better than a linkage s cheme 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 strlpping vapor in a distillation col11m11, 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 t h e stage. This quantit y 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 separation desired. i A+ B Figure 1 ~General Multistage Separation Process .=,,: A ,. ' ~ ._ ___ .. B .. XA XB C>( AB Phase 1 = XA XB Phase 2
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16 CHEMICAL EIGllEIRilkRRIBG lsDUCATIOB June 1963 I Figure 2. Equations for Distillation Column with Partial Condenser .a .... ~ Component Material Ba.lance around Each Stage (xR) _, Condenser Vt Yt rxr a DyD VtlYt1 Ltxt a VtYt rxr Vt2Yt2 Ltlxt1., VtlYt1 Ltxt n Feed Stage Vflyfl Lfxf + FXp., Vfyf Lf+lxf+l Vr2Yr2 Lr1xr1 Vr1Yr1 Lrxr m Reboiler bxb VRYR L 1 x 1 Energy Ba.lance around Component Equilibrium Each Stage Relations (xR) V tHt rhr = Dffo + Qc Yn ""Kcxr y t "" Ktxt 1 tl.,. Ktlxt1 vt_lHt1 Ltht = VtHt rhr Vt2Ht2 Ltlht1 a VtlHt1 Ltht I VfHf Lf+lhf+l a Vf+lHf+l Lf+2hf+2 vflHf1 Lrhr + F~ = VrHr Lf+lhf+l Vr2Hr2 Lrlhf1 = vf_lHf1 Lrhr Defining Equations for Component Equilibrium Relations {xR) K 0 = (Tc,xr) ht= q,(Tt,xt) ht1 ... (Ttl'xt1) I:yt1 = l Kf+l = ct>(Tf+l'P) Kf = 4>(Tf,P) Kf1 = 4>(Tfl'P) K 1 = (Ti,P) .t = Hr= ct>(Tr,Yf) Hf1 = q,(Tf1' 1 fl) Hl = cp(T1,Y1) HR= 4>(TR,yR) hf+l.,. cp(Tf+l'xf+l) hr= (T 1 ,x 1 ) ct>(TR.,~) ~f+l = 1 !yr a 1 l:yf1 Cl l l:yl 1 l:yR 1 r.x l r r.xt .,. l LX.t1 r.xf+l Cl I:xr= 1 l:xf1 = r.xl a 1 ~b .. 1 1
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June 1963 CHEMICAL EBGIIE.tm.l.BG EDUCATIOB 27 After discussing the general features or multista e oourae continues with a general discussion or the varigblseparation prooeaaes, the process. I believe this is one of the most important !t es associated with the dent that he can deal with any process He determin ff~ in convincing the stu can be done with the process and what ;oasibillties ~: s tveritlittle effort what the separation. In addition, he discovers that the a o a er or control shimlphlyhthe solution of a rather lengthy and oomplica:~1!:iso~fetq~:ti~~:s;o;s w c e must find a method of attack. For example, in Figure 2, all of the equations which define a distillation ~olumn with a partial condenser have been written down. The first set of equations s a set of mass balances for each component around each stage. The second set of equations ls 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 Kvalue. The fourth set of equations defines the Kvalues. The fifth and sixth set define the molal ent h alpies, H e.nd h of the vapor and liquid phases leaving each stage. Lastl y 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 unlmowns. 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 b e 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 Lff(n+m+)l7 + Lfn+m+Jl7 .L'ff(ntm+3l7 + fEi(n+m~)l7 + f.n+m+l7 f.n+m+1J f [.n+m+J7 + ffi+m+!J c ( 3R + 5) ( n +m+ 3) 1 Similarly the number of variables is Lff(2) (n~m+3} + R + 2 (n+m43) + 17 + f2(n+m~J) + I/ Lff(n+m+3l7 + Lfn+m+417 = ( 3R + 5) ( n.f.m 13) + R 5 For the colunm shown in Figure 2, then it appears that R 4 varia b les must be assigned values in order to describe a meaningful problem. Act u all y 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 varia b les 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 t he set of equations for the remaining dependent variables. This method of anal y sis can be extended to an y process, b u t if the p ro ces s is reasonably complicated, it is quite eas y to wr i te the wro ng numb e r o f equ a t i one and hence get a false answer. T h e student l s then shown a fa r e a sier wa y to count the number of independent variables by what we h ave called t h e D e sc r i ption Rule. The Description Rule simply says that one must set a number of var i ables equal to the number of independent choices he can make in construction plus the number of independent choices he can make du.ring operati on o f the col11mn. It is obvious that during construction one does have independe n t 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 adjustm e nts. The n11mber of these arbitrary adjustments roust also constitute a number of indepencl ent variables. Th u s, if one simply draws the column of stages and examines it for these independent choices, he can determine in se c onds how many in dependent vari ables he must set in order to describe a problem o n the process. As an example of this, consider in Figure 3 the same distillation col 1 1 :rnn for which the equations were all written down in Figure 2. In drawing t h e column, one can arbitrarily say there are n stages above the feed stage and m sta g es 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 ~f he has R components, there are R 1nde pendent variables in the compo ne nt r~ eg ~ ~~~! .! !! ~ '1 + .!9 2.__!~~~ ~~~1 ~:1 se t the I
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28 CHBMICAL ERGIREERDlG EDUCATIOI June 1963 enth alpy of t h o t e ed, adding one more independent variable hF He can, within 1 1 :? ope:ra t.,:j a t any pressure he choose a, generating an !ndependent variable, l W 11' :. o he c o l l lDIX l in operation, he can, within limi ta, change the ateam valve and pu t an ar b i t rary setting on the reboiler load, generating one more independent vari a ble, 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 sums 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 conden~er, 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 te~l 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 o an choose an arbitrary temperature. To illustrate still further what would be a rea~onably complicated process, a refluxed strippe r with a side stripper which might be considered to be a very basic model of a crude oil col11mn is shown in Figure ). 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, h e 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 proc ess. If we take distillation as an example, almost inevitably the student is asked to solve a problem in Which he sets the separation apecificetions 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 varia b les se~ are shown in t h e second list of Figure 4. This solution of this problem has a dist;nct method of attack which is wellunderstood by students if they read any textbook. However, there can be many other problems of interest on the same distillation col11mn; and 1n 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. T l 1e 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 ;eparated have been set, the reflux has been set, and the feed plate location has been set. in the third list, a problem is s h own 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 ac h ieved. 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 pro?lem of the fourth list in which the last two variables set are the recovery and purity of a component in the top product. Also, thro~gh it one could even more easily get the solution to the prpblem 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~~ roduced under the set reflux. The only requirement would be that it be possi 8 ~o calculate the octane number from a lmowledge of the bottom product composition. All of these problems and many others could be solved, although the last o uld h ave to be solved by a parametric solution. But it is helpful to poin to the student that many problems exist other than the straightforward design two out prob.J ... ~.
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June 1963 n m 'n m R of Fxp n m hF p R+ 6 ( I "I ( ',J I I CHEMICAL ENGINEERIJIG EDUCATION Figure 3 n m n m R of FxF hF p r R + 7 Figure 4 Set Variables for n n m n n' m mB R or FxF p r L B ~ Amount Hor s 1 or s 1 Amount of s2 Hor s 2 Ls R + 13 Various Problems in Distillation (/i)d (/j)d r Fd Plate Loe, n m r d n m n m r Octane no of b lem which he normall y encounters in the textbooks. A new problem does not present him wit h 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 ls easily suf'ficient for it. The next two sections of the course consider b inary distillation, starting with McCabeThiele diagram, which I think is very instructive. The student sees grepnically w ha t t he column looks like and he obtains an insight into the op eration of a typical distillation column by lookin g 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 McCabeThiele diagram here. Every teacher uses it, I em 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
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" "I"'" .. .. ) June 196.3 Sl t!R \@ \fl @ ~\Y@@H\@ \~,~ tq,~ are ~nly a c~rtain number of variables which~~ It ,i fl@ ~ ~@ R f!R@' ~he !~~~~uc~or ~~~ing all liquid flows in the rectifying I @~ '. ~ f .~ ~ 7 1 : i \l@ @s,'4,~ tb,e reflux and all liquid flows in the str1 pping sec \ s.. t 6 @9~l tg the reflux plua whate~er liquid flow might have been gen I ff t
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June 1963 CHEMICAL EIGIBEERIBO EDUCATION The whole pro b lem of stagebystage calculations for multicomponent systems is discussed as a logical extension of binary systems. I believe the student is made well awe.re of the fact that the only new feature is si1nply 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, A 1 1so, algebraic calculation replaces grap h ical methods, and the students calcuate design problems for short columns. 31 The use of Underwood's equations for design problems on multicomponent sys tems is also discussed alt~ough the exact solution is avoided. Instead, time is devoted to showing how the concentrations of components ot~er 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 Underwoodts equations for multicomponent systems wit h the approximate feed plate composition and the exact solutions, the errors are of the order of 1%. The student is t h us able to use Underwood's equations for multicomponent systems wit h out the extremely tedious solution of t h e determinants necessary in the exact solu tion, The last part of the course is devoted t o t h e iteration and relaxat i on schemes w h ic h h ave come into use in recent years. Students are asked to solve simple pro b lems, b ut the timeconsumin g nature of the methods would allow solution of lon g pro b lems only if a computer were used and we have n ot yet used one in the course. However, the ver y fact t ~ at computers are widely available in industry makes t h e methods important since they are capable of extension and generalization to a de gree t h e otl1er metl1od 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 ally. The basic iteration scheme was proposed by Thiele and Geddes in 1932( 8 ) 2 Many workers have elab9rated and changed the sche~e,since, nota b ly Edmister< ) Admundson and Pontinenll), and Lyster and Hollandl~, and t h e choice of exact procedure is today quite broad. The particular scheme I teach the students is one I ~ ave worked on a considerable amount and I h ave chosen it to a large degree b ecause of this. All iteration schemes break the s y stem of equations for the column into groups and solve the groups separately. As an example, consider the simple dis tillation colunm of Figure 2. The mass balance equations will be solved first, and tl1e set of equations for each component must b e separately solved. If t h e equations for the first component are written down it is apparent t h at nothin g 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 arbitraril y choose x, and will solve for all values of x. The technique of solving the set of equations I use was first Sm1thl 7 )in essentiall y t l 1e form I use it. Since nothing is Jmown, value of bxb for the component, the amount of the component in the The amount ls wrong b y some unknown error, b' w h ere b.xb : {b) true + b publis l 1ed b ;r one assumes a bottom product A typical stage by stage calculation of the component can now eas i l y be run up the column. Thus VRYR 3R{bxb) L1 xl : VRyR + bxb V1Y1: 8 1 (L1x1) etc. to the top of the column where rxr Vtyt + b F~ DyD = Sc(rXz.) Dyn is, of course, wrong since b is wrong.
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32 CHEMICAL DGlNEER!BG BDUCATIOB June 1963 However, we lmow the error in bx. is some uantit la t h en S R b 3Il d the error in L x iso g Y, b The error in VRYR ca ) 1Gcd di r c tly but all of tlh 11 tViR &b None of these errors can be EvR b v R b er ra os can. Thus + 1 ect., on up the column. The feed has no not enter the error ratio calculation. error associated with it and hence, doeE At the top of the column the ratios tr v t ,. 1 b Eb and d Er Sc b are calculated. Then one can write and, sine e ( Dy ) + D true (bxt,} true = F~ a slight rearrangement yields d b + 1 The amount of the component in every flow can then be corrected to the true value through the lmowledge of eb and all of the error ratios. This procedure is followed for each component (the reverse calculation down the column must be employed for lig h t 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 belance 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 converge rapidly. 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 somew ha t harder problem. correction methods are welldeveloped at the present time for many common processes and the iteratior method has been extensively used in industr y The student is impressed when he is told that columns with 40 plates and 20 components can be solved on a computer for $1.50. The relaxation method is conceptuPlly the most simple method which one can 'q g{n e, and becau~e of t hi s it 1s also completely general. The basic idea was ,, d i j rat by Rose ( 6) for the massbalance solution t~ the steadystate by cal c 1,1l a tion th rou gh the un ste ad y s ta te period of a column startup. However, if the a' i st a te i s t h e desire d a nswer, no consideration need be given th~ 4 time be '\'" tl1e me ih od can t.f; s implif led and at the same time extended( ) The
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June 196.3 CHEMICAL DGIKEERIG EDUCATIOX 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: '~f I bring in one, two, three, or more streams into the stage of a given amount and composition and with a given ener gy, what will be composition, amount and energy or the p ha ses 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 conver~e as:vmptotically 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, alt h ough if one could locate the stage of largest error, 1\ 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 di stribution of t h e compo nents be between the phRses le aving the stage. Again, for eac h relPxat i on 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 ot h er and be tter wa ss to solve liquidliqu id extraction systems, but t h e relaxation method works well and s ho ws the complete generality of the method. AS I said initially, t he course we are now teaching is a collection of methods which we lmow particularly well and which cover t he gene ral field of multist age separation operations. T he re is suc h a tremendous literature in this field that one could easily pick a totally different set of met hod s. The course will undoubtedly c h ange in the future, hopefully in t h e 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 a mount of computer time effect! vely, however and tt1e use of t h e computer will not in itself add to tt1e understanding the student ma y have gained in the field. The advantage of the computer may well be that it will allow tt1e 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 D F H h K L Lmin m n p QR R r s T X y (/i)d (/i)b NOMENCLATURE moles of bottom product. moles of vapor top product. moles of feed. molal enthalpy of a vapor leaving a stage. molal enthPlpy of a liquid leavin& a stage. component equilibrium constant, y/x. moles of liquid flow leaving a stage. mlnimum ljquid flow in rectifying section for a distil lation column, usually taken as minimum external reflux. number of stages in the stripping section of a distil lation column. number of stages in t h e rectifying section of a distil lation column. pressure. condenser load. rebeller load. number of components. moles of reflux. stripping factor for a component in a stage, VKi. temperature. L mole fraction of a component in a liquid. mole fraction of a component in a vapor. fraction of 1 fed which is removed in the top product. fraction or 1 fed which is removed in the bottom product.
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1. 2. 3. 4. 5. 6. 1. 8. 9. 10. A B b C D d F r i j CAL BIGI ... .... 1. ,~, BG BUucnIOB SUBSCRIPTS component A component B bottom product condenser vapor top product liquid top product or top product in fe.ed feed plate component i component j June 1963 general L1 liquid flow leaving plate 1, etc. R reboiler r reflux t top plate VR reboller vapor, etc. 1 plate 1, numbering upward, etc. GREEK separation factor between components A And Bin phases leaving a stage. AB XA XB Phase l XA Xe Phase 2 separation factor or relative volatility of based on some unnamed reference component. based on same reference component. component 1 All 1 are error in the calculated amount of a component in a flow stream leaving a stage. function. REFERENCES Amundson, N. R., And Pontinen, A. J., Ind. Eng. Chem., 50, 130 (1958). Edmister, W. c., A.I.Ch.E. Journal, 3~ 165 (1957). Fenske, M. R., Ind. Eng. Chem., 24, (1932). Hanson, D. N., Duffin, J. H., anaSomerville, G. F., Computation of Multistage Separation Operations, Reinhold Publishing Corp., New York, ( 1962). 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, 1'2'I (1959), ibid, 38, No. 10, U9 (1959). Rose, A.~ Sweeney, R. F., and Schrodt, V. N., Ind. Eng. Chem., 50, 737 (195~). Smith, B. D., and Brinkley, W. K., A.I.Ch.E. Journal, 6i 451 (1960). Thiele, E.W. and Geddes, R. L., Ind. Eng. Chem., 24, (1933). Underwood, A. J. v., Chem. Eng. Prog., 44, 603 (19zrB). Underwood ;A. J. V., Jour. Inst. Petroleum (London), 32, 614 (1946).
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