I
STANLEY H. JURY and JACK M. ANDREWS Department of Chemical and Metallurgical Engineering, University of Tennessee, Knoxville, Tenn.
High Speed Memory Analog Computer Applications to Diffusional and Stagewise Operutions Recent developments in memory core and precision accounting techniques permit tackling problems previously considered too large for analog computers
CAPABILITY is usually not considered as part of the concept of analog computer design, the need for memory in analog computation was recognized as early as 1954 by Wade1 (47, 42). Such analog memory devices have been described by Kozak (25) and Bickart and Dooley (7). I n a recent report (22), the speed and memory capabilities of a high speed memory analog computer were described, along with the symbolism and nomenclature peculiar to this type of machine operation. [Standard analog nomenclature and symbols are outlined in a good text on the subject (24).] A number of possible applications were cited and the need for research into the means of exploiting the new machine capabilities was identified. I n a subsequent report ( 2 3 ) , one of the applications was the subject of a demonstration showing, through the medium of computer diagrams, how the machine may be used in solving a simple partial differential equation and its limiting conditions governing transient heat conduction in a slab. Since then, a number of other applications have been investigated. To extend the prior state of knowledge in the field, a series of problems has been selected which is considered a start a t an “across the board” sampling of chemical process industry and research problems the memory analog computer can cope with. The problems treated reveal a remarkable similarity of computer programs in spite of their mathematical diversity. This similarity might be attributed to the very use of memory, which in itself simplifies programming to the point where this observation is made possible. A more subtle conclusion, however, is concerned with the fact that the combination of memory and program simplicity permits a mental rearrangement of components. This suggests implied operations not considered before, as such, but which could lead to a generalized approach to
A L T H O U G H A MEMORY
Present address, KinOTrol, 8010 Glenview, No. 14, Houston 17, Tex.
programming, a matter which will be the subject of later reports. I n view of the broad scope and detailed nature of the work, which involved extensive mathematical exposes and numerous diagrams, only the highlights of these findings are reported in the limited space of a scientific journal. Complete details are available (see coupon below). Types of Problems
T h e major operations within a chemical process industry may be classed as diffusional, stagewise, or mechanical operations, depending on whether the operation involves transport phenomena, step by step treatment under equilibrium conditions, or purely mechanical manipulation, such as that involved in the separation of a solid from a fluid. In certain cases, chemical reaction may be associated with an operation. Heat conduction in a rod and the pseudo-isothermal catalytic reactor involve transport phenomena and are therefore diffusional operations. The stagewise countercurrent contactor, fractionation equipment, and the gaseous diffusion cascade fall into the stagewise operation class, although the cascade is not solely so because of barrier transport phenomena that are involved. Thus, the problems treated in this report are typical of situations wherein the memory analog may be applied in the chemical process industry and research.
The selection of problems is also significant from a mathematical viewpoint, in that some of the problems involve partial differential equations and others involve natural differential-finite difference equations or algebraic and difference equations as such. This is due, in part, to consideration given transient as well as steady phenomena. As shown in the complete report, all of the equations are handled on the memory analog in such a way that their computer diagrams in many respects show a remarkable similarity. This, in large measure, is attributable to the memory function of the computer. This fact is useful to the program because it helps to envision the diagram for new problems as they come along. I t also strongly suggests rules for memory analog programming, which has become an established fact for routine programming of a conventional analog computer. Steady
Heat Conduction in a Rod
T h e problem of heat conduction in a rod ( 8 ) is characterized by the governing differential equation:
and the limiting conditions T(y, 0 ) = 1
AVAILABLE FOR ONE DOLLAR The complete manuscript, including all mathematical exposes, computer diagrams, and additional text, by Jury and Andrews. After one year this material can be obtained from the AD1 Auxiliary Publications Project, Library of Congress, Washington 25, D. C., as Document No. 6850. The price will then be $3.75 for microfilm and $11.25 for photostat copies.
I
[
I I
I
Clip and mail coupon on reverse side VOL. 53, NO. 11
0
NOVEMBER 1961
883
4 u b s t . for 0 - M 84-6
Figure 1. A combination of integrator and memory loops calculates current values of T
-
R ;
(g)
v=1
(g) Y
= 7- (1, n )
=o
-0
where
T =
.T c TP
-
TP
y = r/R x =
z/L
Figure 2. Refined version of ACFG utilizes integrators instead of difference amplifiers and multipliers
The rod equations may be written in differential-finite difference form and diagrammed as shown (Figure 1). The major loops containing integrators solve the governing equation, while the Rev-M and RI memories are involved in satisfying the radiation boundary condition at y = 1. This calculation involves trial and error with the computer operating repetitively. The voltages u and u are compared in a circuit (not shown) which permits the computation to move to the next increment of x only when u = v.
which are dimensionless groups involving the temperatures T , r g , T,,, the radius variable r, and the longitudinal length variable z . Also h, k , R, and L are the parameters characterizing conduction into a rod from one end at T~ with radiation from all other surfaces into a medium at temperature T ~ . This conduction problem may be solved analytically ( 8 ) . I t is of educational value here, however, in that for programming purposes it represents a somewhat higher order of complexity than the slab problem treated previously ( 2 3 ) .
The voltage T(y, x) shown at the left of Figure 1 is, at the start, equal to the boundary value a t x = 0. At later values of x , both T ( y ,x ) and T(y,x 4-Ax) must be calculated from over-all heat balances, and here the need arises for rhe automatic continuous function generator (ACFG). An original version of this device was discussed in the previous report (22). The more recently refined ACFG is shown in Figure 2. Its design is dictated by the Kewton forward interpolation formula (2, 72) as before. Its refine-
EDITOR, I/EC 1155 Sixteenth Street, N.W. Washington 6, D. C. For complete manuscript by Jury and Andrews.
$1.00
Cash
Check 0
Money Order 0
(Make payable to American Chemical Society)
................................................ .....................................................
Nameand title Address
884
z = n
modify the ACFG then
f i ( x ) and i = l
i = n + 1
fi(x) are developed. I n the limit i = l
-
as f,, ~ ( x ) f * ( x ) the modified ACFG behaves as an integrator but with respect to the variable n, which is not machine time.
Pseudo-Isothermal Catalytic Converter
SEND ORDER COUPON TO:
Enclosed:
ment, hoLvever, stems from the use of integrators instead of difference amplifiers and multipliers, as was previously the case. The alternate connections may be used to eliminate summers 5 and 6 assuming that the 0 memories are required. I n many cases the a-b and c-d comparators are not required, and in this case the alternate connections are not used. Instead, the 0 memories are omitted, the dash-dot forward memory is connected, and summers 4 through 6 are replaced by summing junctions, assuming that the computer is so equipped. The g-j-k comparator switch is used for inserting the initial conditions at n+1=1. If the dotted circuitry is added to
INDUSTRIAL AND ENGINEERING CHEMISTRY
Heat transfer with chemical reaction in packed beds has been studied by a number of investigators (5, 77, 27, 30, 38, 45) because of its interest in the chemical process industry utilizing catalytic converters. The mathematical complexity of the cylindrical reactor problem was first demonstrated by Damkohler ( 73-76) in the form of vector equations which he did not solve. T h e form of these gener-
A N A L O G COMPUTER ally employed (38) in analytical work is as follows:
h,(rs -
T,)
= O
where
C
= heat capacity of the fluid
flowing through the bed, B.t.u./lb.-O F. CO,C1 = constants in R* equation, consistent dimensions = eddy thermal diffusivity, sq. ft./hr. = superficial mass velocity of fluid flow in axial direction, lb./ (hr.) (sq. ft.) = heat of reaction/lb. of key reactant, B.t.u./lb.; negative values indicate exothermic reaction = volumetric heat transfer coefficient from Darticle to fluid stream, B.t.u./(hr.cu. ft.) (" F.), = h,/a, where a = solid surface area per cu. ft. of reactor, sq. ft./cu. ft. = ~ k , CpE = effective fluid thermal conductivity, B.t.u./hr.-ft.-O F. = effective solid thermal conductivity, B.t.u./ (hr.) (ft.O F.) = fluid molecular thermal conductivity, B. t .u./ hr .-ft .-
,.
+
O
F.
= length
of the cylindrical reactor, ft. = rate of chemical reaction, lb./(hr.) (cu. ft.); e.g., R*AH = Co CIT,, B.t.u./(hr.) (cu. ft.) = radius of the cylindrical reactor, ft. = radial position, ft. = axial position, ft. = bed void fraction, dimensionless = fluid density, lb./cu. ft. = fluid temperature at location r and z, O F. = solid temperature at location r and z, O F. = wall temDerature of reactor at R, O'F. = initial fluid temperature, O F.
+
The equations are based on the assumption that fluid flows axially through the cylindrical reactor and the flow is uniform. The wall of the reactor is maintained at temperature T~ to produce pseudo-isothermal conditions in catalytic converters. This, of course, is not implied in the equations as such, because it is a boundary condition. Bernard and Wilhelm ( 5 ) have measured E for various conditions. A number of investigators (77, 27, 30, 45) have
measured h, for similar conditions. The values of K, have to be determined by experiment for the particular case in question. K, is the conductivity of a complex path involving conduction within the solid particle and from particle to particle through a thermal resistance which includes a path through the fluid fillet in the zone of contact and through the contact point itself. Singer and Wilhelm (38) attempted to solve the equations analytically, but because of mathematical complexity they neglected axial conduction. They also assumed that GO,C, K,, K,, and h, were constant. Various special cases were then solved wherein, for example, R* = 0, R*AH = Co C I T ~ . Various finite difference techniques have been tried in the solution to the equations. Perhaps the first of these was a calculation procedure of Wilhelm and others ( 4 3 ) . They didn't like their procedure, however, because they had assumed a linear dependence on temperature for the reaction rate. They also assumed constant radial gas temperature. Grossman (79) removed the necessity for the assumption of linear dependence on temperature in a graphical technique that he developed, but it had the new flaw that a complete set of gas and solid phase temperatures was required in a slice of bed before the calculations could be started. Wilhelm, Johnson, and others (44) built an electrical analog based on Johnson's development of a lumped parameter resistance network modeled after the finite difference equations. These investigators found that for a particular set of reactor conditions the network reduced computation time to 1 to 2 hours, not counting preliminary preparations required in setting up the analog for the particular set of conditions. They also neglected axial heat flow. The next logical step involves studying the possibilities of solving the equations using the memory analog computer, and this has been done. Although the diagrams are not included here, it is interesting that the basic solution of the problem has a diagram resembling very closely that of Figure 1 for the rod, except there are two major loops instead of one, as one would suspect from the simultaneous solution of two partial differential equations.
+
Satisfying limiting conditions is, as for the rod problem, a matter of satisfying over-all thermal balances which, as before, require use of ACFG capabilities. These diagrams are of particular interest in that the philosophy of their construction is quite different from that for the rod and they tend to simplify programming. We have been able to show that by simple modifications of the foregoing program, one can solve certain problems of heat conduction and heat convection in slug, laminar (78), and turbulent flow (27).
Stagewise Countercurrent Contactor A schematic of the stagewise countercurrent contactor is shown in Figure 3. Conventional fractional distillation nomenclature is used to permit direct comparison of the results for operations such as extraction and absorption with those for distillation. The first discussions (3, 9, 7 7 , 36, 39) of transient mass transfer appeared in the literature between 1940 and 1941 in connection with batch rectification. The concept and an analytical solution for a special case of transient behavior in stagewise operations were developed by Marshall and Pigford (37) in 1947. This special case was extended somewhat by Lapidus and Amundson (26) in 1950. In both cases the problem was amenable to analytical solution because it had been linearized in the sense that flow rates were assumed to be constant and the equilibrium relation had been assumed to be linear. The assumptions seriously restricted the usefulness of the solution, and the solutions are rather cumbersome to use in computations. The restriction of the linear equilibrium relation was removed in 1951 when Pigford, Tepe, and Garrahan (34) applied a mechanical analog computer to the problem of batch rectification. The same was true when in 1953 Acrivos and Amundson (7) solved the Lapidus and Amundson (26) version of the problem on a conventional electronic analog computer. I n 1955 and 1956, Rose and Williams (35, 37, 46) used an electronic analog computer to solve the problem of transients in continuous distillation but for the special case of constant relative volatility. It is interesting to note at this point that for constant flow rates and a linear
YN
.",
Xn+l, OntI
Figure 3.
Schematic diagram of general stagewise contactor
Conventional fractional distillation nomenclature i s used VOL. 53,
NO.
11
NOVEMBER 1961
885
equilibrium relation, the transient stagewise problem closely resembles the partial differential-finite difference form of the conduction (or diffusion) equation. Hartree (20) was the first to devise a machine solution of this equation using a mechanical differential analyzer. Machine solution actually involves simultaneous solution of a set of N equations, there being one such equation for each value of n ranging from 1 to N . This method is still used today (24) with conventional analog computers. In 1937, Beuken ( 6 ) suggested the use of a direct electrical analog for the solution of the diffusion equation. The direct analog involves flow of current along a lumped cable shown in Figure 4. The electrical equation (40) for the analog is:
g
=
'I
1 VB
Figure 5.
y+1
resistance per unit length of cable, ohms/ft. C = capacitance per unit length of cable, farads/ft. =
Paschkis (32) has described the actual construction of the direct electrical analog. Later, Paschkis and Hlinka (33) described an improvement in the analog using unity gain cathode followers (24). Liebmann (28, 29) replaced the time derivative in the voltage equation with its backward difference equivalent. H e then devised a resistance network analog which is described in detail by Karplus (24). Yesberg and Johnson (47) constructed a Liebmann-type electrical analog for the solution of the transient stagewise problem. They replaced the time derivative of concentration with its backward difference approximation. T h e analog was constructed for the special case of constant flows and a linear equilibrium relation. I n describing their results, the authors concluded that the Liebmann-type analog is no faster than programming a digital type computer to do the job. It also has the disadvantages that nonlinearities are difficult to introduce and the Liebmann analog is not suitable for investigating control dynamics due to truncation errors. The authors contended that had they built a Paschkis-
----""-1 C
---
886
Computer diagram for steady state stagewise countercurrent contactor Comparators are used to introduce boundary conditions
-(;;2+""-I)
where
R
I
type analog of the transient stagewise problem it would have been suitable for investigating control dynamics. However, the experimental difficulties would have been greater, and nonlinearites would be more difficult to introduce into the solution. From the standpoint of the memorytype computer, the steady state contactor problem is relatively easy to program, as is illustrated by the steady state diagram of Figure 5. This diagram is based on the conventional steady state mass balances and a nonlinear Langmuir distribution isotherm. Note that in the diagram comparators are used to introduce boundary conditions, and extensive use is made of stacked point memories for control of the sequential calculations involved, since only one stage-i e , the n stage-of the stages is actually patched according to the diagram. The computer and the memories do all the rest. 'The comparators are driven by n 1 which is obtained from a conventional accounting circuit The accounting circuit automatically repeats its accounting 1 = operation when jn$1 = y,,. or n N . Since 'V, an important design value, may be recorded, the effect is the design of an N stage contactor with but one plate patched up, which is a novel feature of the memory computer to say the least. T h e computer diagram
+
+
remains the same, independent of the number of stages being designed or investigated. Although the diagrams covering the investigation of transients and control dynamics are not included here, the one thing that distinguishes these from the one shown in Figure 5 is the necessity for remembering functions of time at each stage, and for that purpose ACFG's play an important role.
Fractional Distillation T h e diagrams for binary distillation in many respects resemble those for countercurrent contacting. T h e distinguishing feature: of course, is the switching arrangements through comparators necessary for introducing feed and reflux in the column shown in Figure 3. Although one plate is diagrammed, as was the case for the contactor, one can read out of the computer not only the cooling load and total number of plates in the column but also the feed plate location which is made possible through a comparison of x , and x R to avoid mixing on the feed plate. Gaseous Diffusion Cascade
Gaseous diffusion cascades for the separation of isotopes have been the subject of considerable investigation during and since World War 11. Perhaps the best known operation of this type is the one operated for the government by the Union Carbide Nuclear Co. at Oak Ridge, Tenn. This cascade separates uranium-235 and uranium-238. Because of the C OW separation factor and the extremely dilute feed, thousands of stages are required, and this leads to extended periods of time lost in start-up. I t can also produce transient problems which are difficult if not impossible to tackle analytically. Benedict and Pigford (4) have given
""""-----T+;--"" _--
C
-
INDUSTRIAL AND ENGINEERING CHEMISTRY
-
C
- --
A N A L O G COMPUTER a detailed account of steady state cascade theory, particularly for small values of the separation factor. Cohen (70)presented a little different approach to the problem and derived the socalled “fundamental equations of isotope separations.” These equations are presented in partial differential equation form based on the assumption that the separation factor is so small and so many stages are involved that changes from stage to stage may be represented by derivatives instead of finite differences. For operations other than those involving a small separation factor, this approach has serious limitations. Cohen attempted to treat the cascade control problem, identifying two parts to the problem. T h e hydrodynamic part is the one which involves controlling temperatures, pressures, flows near their design values, assuring no leakage in or out of the cascade, and preventing small disturbances from building into large ones. The second part of the control problem deals not with average conditions but with the effect of fluctuations from the average. A slight change in operating conditions in general causes mixing, and mixing reduces separation. Cohen limited his treatment of transients to those arising from fluctuations, because for certain types of cascades these are susceptible to treatment through perturbation theory. The results are applicable to small deviations from steady state operatiig conditions. Gross upsets and phenomena associated with the approach to equilibrium were excluded from consideration. The interesting thing about cascade theory in its present state of development is that it leads to a paradox. The transient equations are inherently nonlinear finite difference-differential equations. If, for analytical purposes, they are reduced to differential equations, truncation errors are introduced. O n the other hand, if the equations are reduced to finite difference equations so that they may be programmed for a digital computer, then truncation errors are also involved. Even though truncation errors are avoided on analog computers, it is hopeless to consider a conventional analog computer, because, in general, the total number of stages and the feed plate location are not known at the start. For many of the cascade problems, knowing these answers wouldn’t simplify the situation because of the phenomenal amplifier requirement plus the nonlinear equipment requirements and associated problems. The memory analog computer changes this situation. A schematic of the gaseous diffusion cascade is shown in Figure 6. Here, conventional fractional distillation nomenclature is used :
xi = mole fraction of valuable con-
stituent in downflow from stage i or an external stream y i = mole fraction of valuable constituent in upflow from stage i Oi= total molar flow rate of downflow from stage i , moles/hr. Vi= total molar flow rate of upflow from stage i, moles/hr. F , P, VV = molar flow rates of feed, product, and waste, respectively, moles/ hr . 0 1 ~= separation factor, dimensionless T h e cascade consists of stages ranging from 1 to N . Each stage is physically divided into two compartments by means of a porous barrier through which “effusion” may take place because of a pressure differential maintained across the barrier by the turbocompressors shown in Figure 6 . Here, we define as follows :
C,
= compressor capacity at stage
i,
cu. ft./hr. Po, = high pressure side gas pressure in stage i, lb./sq. ft. P,, = low pressure side gas pressure in stage i, lb./sq. ft. h, = moles of gas at pressure Po,in stage z H, = moles of gas at pressure P,, in stage i NA, = molar flow rate of valuable constituent through barrier of stage i, moles/hr. iVBI = molar flow rate of less valuable constituent through barrier of stage i, moles/hr.
Figure 6.
Based on thermodynamics, streams of different concentrations should not be mixed; that is, at stage n 1 we must satisfy the criterion that :
+
Xn+2
= yn
A cascade satisfying this criterion is said to be an “ideal cascade,” and for o u r illustration this case is considered. However, because of construction practices, it is cheaper even in a cascade to build certain groups of stages of the same size so that A , is constant over a given group. As a consequence, in practice the reflux ratio is changed in steps rather than continuously from stage to stage down the cascade. Within a given group there is “constant molar over-flow’’ rather than x,+% = Yn. Cohen designated such a cascade as a “squared-off” cascade. This is a refinement that can be easily accommodated on the computer, but it adds unnecessary complications to the demonstration at hand. The plate area in a distillation column is related to vapor velocity, which in turn is related to operating pressure within the column. So it is in the steady state cascade. In the steady state design of a cascade, the principal contributors to its size, and consequently its cost, are compressor capacity, barrier area, and power consumption. It would be desirable to hold all of these to their minimum values, which of course cannot be done.
Schematic diagram of gaseous diffusion cascade
Conventional fractional distillation nomenclature is used to identify variables
VOL. 53, NO. 1 1
NOVEMBER 1961
887
I/Dp
t-””
“ A
I
It 2
Figure 7. Computer diagram for steady state ideal cascade i s based on conventional mass balances and barrier effusion
We must either compromise or settle for one of these that is reasonable for illustrative purposes. I n the case illustrated, minimum total compressor capacity is selected, and for this steadv state case Benedict has shown that: Po = 1.207 P, P , = 0.5 P,
wherr
and p = gas viscosity, lb./(ft.)(hr).
MB
atomic mass of the less valuable isotope, lb. u, b = parameters characterizing the barrier, as a n effusion medium, consistent dimen=
sions T h e computer diagram based on conventional mass balances and barrier effusion for the steady state ideal cascade is shown in Figure 7 . The diagram is simpler than some of those developed previously because of the ideal cascade restrictions, even though o(0 has not been restricted to small values. In a sense, this is quite remarkable because normally on a n analog computer the number of stages would primarily determine the amplifier requirement, and this would tend to cover u p any advantages arising from the ideal cascade restrictions. Perfect mixing has been assumed at each stage, and amplifiers 1, 2, and 3 with the associated divider in Figure 7 solve the effusion relation. All other amplifiers are involved in mass balances. For simplicity, pot settings greater than 1 are shown. I n practice, one would
888
decrease the setting by a factor of 10 and then feed to a subsequent amplifier on a factor of 10. A high precision accounting circuit not shown would be used to generate n 1, and from it one would also determine the feed stage location and the total number of stages in a given cascade. This is independent of the size of cyo and the size of the cascade. With the circuit of Figure 7, one can investigate a 10,000stage cascade just as readily as a 10stage cascade. T h e computer diagram for studying transients, start-up, and temporal failures in a cascade are somewhat more involved than that of the steady state situation. This, of course, is attributable to the need for remembering functions of time at each stage and the consequent use of ACFG‘s.
+
Literature Cited (1) Acrivos, A., Amundson, N. R., IXD. ENG.CHEM.45, 467 (1953). (2) Andrews, J. M., Znrtr. Control Systems 33, 1540 (1960). (3) Bardeen, J., Phys. Reu. 57, 35 (1940). (4) Benedict, M., Pigford. T. H., “Nuclear Chemical Engineering,” McGraw-Hill, New York, 1957. (5) Bernard, R. A,, Wilhelm, R. H., Chem. Eng. Progr. 46, 233 (1950). (6) Beuken, C. L., Econ. Tech. Tydschr. No. 1, l(1937). (7) Bickart, T. A., Dooley, R. P., Electronics 33, 71 (December 1960). (8) Carslaw, H. S., Jaeger, J. C., “Conduction of Heat in Solids,” Oxford Univ. Press, London 1947. (9) Cohen, K., J . Chem. Phys. 8, 588 (1941). (IO) Cohen, K., “The Theory of Isotope Separation,” McGraw-Hill, New York, 1951. (11) -Colburn, A. P., Sterns, R. F., Trans. Am. Inst. Chem. Engrc.. 37, 291 (1941).
INDUSTRIAL AND ENGINEERING CHEMISTRY
(12) Computer Systems, Inc., Monmouth Junction, N. J., “Applications of Dystac Memory,” Form 80-105,1959, (13) Damkohler, G., Chern. Fabrik. 12, 469 (1939). (14) Damkohler, G , 2. Elektrochem. 42, 846 (1936). (15) Zbid., 43,8 (1937). (16) Zbid., 44,240 (1938). Thodos, G.: Hougen, (17) Gamson, B. W., 0 . A,, Tt-ans. Am. Inst. Chem. Engrs. 39, 1 (1943). (18) Gractz, L., Ann. P h y . 2 5 , 337 (1885). (19) Grossman, L. M., Trans. Am. Znst. Chem. Engrs. 42, 535 (1946). (20) Hartree, D. R., “Calculating Instruments and Machines,” Univ. Illinois Press, Urbana, Ill., 1949. (21) Hurt, D. M., IND. EX. CHEM.35, 522 (1943). (22) Jury, S. H., Zbid.?53, 173 (1961). (23) Ibid., p. 177. (24) K,yplus, W. J., “Analog Simulation. McGraw-Hill. New York. 1958. (25) Kozak, -W.S., Can. Eledronics Eng. 2, 38 (1958). (26) Lapidus, L., Amundson, N. K . , IND. E N G . CHEM. 42, 1071 (1950). (27) Latzko, H., 2. angew. M a t h . u. Mech. 1, 268 (1921). (28) Liebmann, G., Brit. J . A$$. Phys. 6, 129 (1956). 129) Liebmann. G., Trans. Am. Soc. Mech. Engrs. 78, 655 (1956). (30) McCune, C. H.: Wilhelm, R. H., IND.ENG.CHEM.41, 1124 (1949). (31) Marshall, W. R., Pigford, R. L., “The Applications of Differential Equations t o Chemical Engineering Problems,” Univ. Delaware Press, Newark, Del., 1947. (32) Paschkis, V., Baker, H. D., Trans. Am. SOC.Mech. Engrs. 64, 105 (1942). (33) Paschkis, V., Hlinka, J. W.,Trans. N . Y . h a d . Sci. Ser. 11, 19, 714 (1957). (34) Pigford, R. L.: Tepe, J. B., Garrahan, C. J., IND.ENG.CHEM.43, 2592 (1951). (35) Rose, A., Johnson, C. L., Williams, T. J., Zbid., 48, 1173 (1956). (36) Rose, A., Welshans, L. M., Ibzd., 32, 668 (1940). Williams, T. J., Zbid., 47, (37) Rose, *4., 2284 (1955). (38) Singer, E., Wilhelm, R. H., Chem. Eng. Progr. 46, 347 (1950). (39) Smoker, E. H., Rose, A.: Trans. Am. Znst. Chem. Engrs. 36, 285, 675 (1940). (40) Sokolnikoff, I. S., Redheffer, R. M., “Mathematics of Physics and Engineering,” McGraw-Hill, New York, 1955. (41) Wadel, L. B., J . dssoc. Computing Machinery 1, 128 (July 1954). (42) Wadel, L. B., Miestern Electronics Show and Convention, Los Angeles, Calif.! August 1954. (43) Wilhelm, R. H., Johnson, W. C., Acton. F. S., IND.ENG.CHEM.35, 562 (1943). (44) Wilhelm, R. H., Johnson, W. C., Wynkoop. R., Collier, D. W., Chem. Eng.Progr. 44, 105 (1948). (45) Wilke, C. R., Hougen, 0. A., Trans. Am. Znst. Chem. Engrs. 41,445 (1945). (46) Williams. T. J., Petrol. Refiner 35, No. 4: 211 (1956). (47) Yesburq, D., Johnson, A. I., Can. J.Chem. Eng. 38, 49 (April 1960). ~
RECEIVED for review January 25, 1961 ACCEPTED June 23, 1961 Research sponsored by Computer Systems, Inc., Monmouth Junction, N. J. After November 1962 the complete article can be obtained from the AD1 Auxiliary Publications Project, Library of Congress, Washington 25, D. C., as Document No. 6850, at $3.75 for microfilm and $11.25 for photostat copies.
