TECHNICAL REVIEW
Computer Simulations of Transport Processes Yatish T. Shah
YATISH T. SHAHis Assistant Professor in the Department of Chemical Engineering, University of Pittsburgh, Pittsburgh, P A . He received the BS degree from the University of Michigan and the M S and PhD degrees from Massachusetts Institute of Technology. Before joining the University of Pittsburgh, he was associated with Abcm Inc., Cambridge, M A . During the year 1970-71, he was a visitZng scholar at the University of Cambridge, Cambridge, U.K. H e presently teaches courses in reaction engineering and has research interests in reaction engineering and special types of transport phenomena.
R e c e n t developments in computer technology have created a significant impact on our theoretical as well as practical understandings of the transport processes. The problems, which appeared rather impossible to comprehend previously, are now evaluated easily with the help of a computer. Thus, a computer simulation has become a n intimate and important part of the study of transport phenomena. The advantages of a computer simulation are manyfold. It allows a rapid and relatively inexpensive evaluation of a complex process. It is almost essential for the identification, optimization, and control studies of a complex process. It can allow a direct and visual comparison between theory and experiments. Because of these and many other advantages, a computer simulation has become essential in the study of complex chemical process systems. I n recent years extensive literature on the subject of computer simulations of the transport processes has been scattered in a wide variety of scientific publications. A unified survey of this widely scattered literature should be useful for further research,both theoretical as well as experimental, on the study of transport phenomena. T h e survey presented here is a collection of the important reported literature on the computer simulations of twnsport processes. This collection covers the period from January 1, 1954, t o December 31, 1970, and includes the papers presented in the important scientific meetings along with those reported in the widely circulated scientific journals. For the convenience of the readers, the presentation of the references is divided into 22 separate categories: fluid-flow theory, fluid-flow equipment, beahtransfer theory, heattransfer equipment, mass-transfer general theory, heat and mass-transfer theory, distillation, absorption and stripping, extraction, Crystallization and growth processes, adsorption and chromatographic separations, filtration and membrane separations, equilibrium-stage calculations, multistage calculations, polymer processing, particlesize analysis, regenerators, miscellaneous systems, thermodynamic and physicalproperty calculations, mass transfer with chemical-change theory, mass transfer with chemical-change equipment (reactors), and lastly mathematics and c o m p u t e r s p r e v i o u s reviews. Some of the listed references in each of these categories can belong to more than one assigned category. T h e references on the simulation studies on all three types of existing comInd. Eng. Chem. Prod. Res. Develop., Vol. 11, No. 3, 1972
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Table 1. Simulation of Fluid-Flow Theory Subject
Reference numbers and types of simulations
Boundary-layer theory Convective flow Couette flow Flow exterior to submerged bodies Flow in pipes and ducts Flow through orifice, nozzle, jet high-speed flow Fluid networks Inviscid flow Laminar or viscous flow Miscellaneous types of flow
7A(D), Z9.4(D), 48A(D), 50A(D), 60A(D), 69A(D), 99A(D) 16A(D), ZRA(D), 23A(D), 26A(D), 65A(D), 80A(D), 84A(D) 6A (D), 63A (D) 2OA(D), 45A(D), 46A(D), 55A(D), 62A(D), 68A(D), 78A(D), 79A(D), 82.4(D), 87A(D), 92A(D), 9JA(D), 10$A(D) 4A(D), 33A(D), 36A(G), 37A(A), 40A(D), 56A(D), 58A(D), 86A(D) 4SA(D), 47A(D), 89A(D), lOOA(G)
Xavier-Stokes equations Plasma Pressure drop Steady-state flow Turbulent flow Two-phase flow Gnstability in flow Unsteady-state flow Water-hammer analysis Raves
1SA(D), 14A(D) SGA(D) S1A(D), 50A(D), 52A(D), 7lA(D), 86A(D), 88A(D), 91A(D), 9?A(D) 5A(D), 8 A ( D ) , 1ZA(D), S7A(D), 28A(D), 49A(D), 51A(D), 61A(D), 64A(D), 7OA(D), 76A(D), lOlA(D), 105A(D), 106A(D), 107A(D) 16A(D), 81A(D), 85A(D), 103A(D) 9 A ( D ) , 3Zz4(D),41A(D) 94A (D) 1OA(D), 19A(D), 20A(D), 53A(D), 57A(D), 62A(D), 6’i’A(D), 8IL4(D),92A(D) SA(D), 18A(D), SOA(D), 4ZA(D), 59A(D), 66A(D), 9SA(D), 98A(D)
$A (4 17il(D), Z l i i ( D ) , S4A(D), 54A(D), 58A(D), 60A(D), 96A(D) 1A(D), ZA(A), Z5A(A), S9A(D), 4ZA(D), 4 5 A ( D ) , 68il(D), 72A(D), 7SA(D), 77A(D) 16A (D,G) , 75A (A) , 104-4 (A) 9A(D), 1714(D), 94A(D), 44A(D), 74A(D), 9 0 A G )
Table II. Simulation of Fluid-Flow Equipment Subject
Blower, pump Calibration equipment Compressor Gas distribution-pipe line-feeder system Gas turbine Manometer Valves
Reference numbers and types of simulations
S B ( A ) ,15B(D) @(D)
SB(D), 14B(D,G) 5B(D), 6B(D), QB(D),IOB(D), 12B ( D ) , ISB(D) 7B(D), I I B ( A ) 1B(D),W D ) 16B(D)
puters-analog, digital, and hybrid-are included in the survey. It is worth noting several extensive reviews reported by Johnson et al. ( I V ) ,Lapidus (ZV), Sweeiiy (5V, 6V), Sweeny e t al. (/tV),and Killiams (7V-IOV) on the subject of mathematics and computers. These provide useful information on the basic fundamentals of mathematics and computers needed for the computer simulation of any type of transport process. Fluid-Flow Theory and Equipment
The complex theoretical analysis of fluid-flow processes can be suitably handled by a computer. This has been borne out by extensive literature reported on the use of a digital computer to study various types of fluid-flow processes. h summary of the computer simulation studies of fluid-flow theory is given in Table I. T h e letters in t h e brackets adjacent to each reference number in this and all subsequent tables denote the type of simulation carried out in t h a t particular study (A, analog; D, digital; H, hybrid; G, general). A substantial portion of the fluid-flox-, computer-simulated theoretical studies is reported in a single, special issue of The Physics of Fluids journal on computer applications in fluidflow studies. This special issue was published in 1969 as supplement 50.2 of The Physics of Fluids. The various fluid-flo\v equipment has also been simulated on the computers. These sirnulation studies are summarized 270 Ind. Eng. Chem. Prod. Res. Develop., Vol. 11, No. 3, 1972
in Table 11. The simulations of both fluid-flow theory and equipment have been carried out mostly on a digital computer. To the author’s knowledge, a hybrid computer has not as yet been used to simulate either fluid-flolv theory or fluid-flow equipment. Heat-Transfer Theory and Equipment
Like fluid-flow theory and equipment, heat-transfer theory and equipment have also been extensively simulated on the computers. These simulation studies are summarized in Tables I11 aiid IV, respectively. As one might expect, the most extensive computer-simulatioii study in this area has been done on the subject of the heat exchanger. Most of the literature on computer simulations of heat-transfer theory and equipment has been scattered in a wide variety of scientific publications. Unlike fluid-flow theory, both heat-transfer theory and equipment have been simulated quite significantly on a n analog computer. T h e hybrid computer has been used only by Carling (I ZD) for the simulation of the heat-exchanger partial-diff erential equations. Heat and Mass-Transfer Theory
Aicomputer simulation of the transport equations (generalheat and mass-balance equations) has been the subject of considerable research interest for the past several years. T h e various aspects of this subject have been studied by Fridender ( 4 F ) , Kazdoba (TF),and Kazdoba and Zagoruiko ( 8 F ) for analog simulation aiid by Xmes ( I F ) , Armstrong and Gloyna (RF),Beskov et al. ( S F ) , Garuer ( j F ) , Herron and Von Roseliberg ( 6 F ) , Kaphtali (9F), Ravicz and Kormaii ( l o p ) , Runchal et al. ( I I F ) ,Shinkarenko et al. (IRF),and Stone and Brian (13F) for digital simulation. T o the author’s knowledge, a general evaluation of heat aiid mass-balance equatioiis has not as yet been carried out on a hybrid computer. Mass-Transfer General Theory; Distillation
The survey of computer simulations of mass-transfer processes can be separated into tn o main divisions: mass-transfer general theory and separation processes conimonly knorw as
A survey of the studies on computer simulations of the transport processes is presented. This survey covers the period from January 1, 1954, to December 3 1, 1970, and includes the papers presented in the important scientific meeiings along with those reported in the widely circulated scientific iournals. Approximately 600 references provide the background and guidelines for evaluating the suitability of analog, digital, and hybrid computers for the simulations of a wide variety of transport processes. An analytical commentary i s presented on the question of selecting the appropriate computer species for a particular event.
unit operations. X summary of simulation studies of general mass-transfer theory is given in Table V. As shown in this table, t h e most extensive study in this area has been done on the subject of molecular-diffusion theory. In t h e comput'er studies of the unit operations, distillation has been the most widely examined process because of t'he common and frequent use of this process in chemical industries. Furthermore, the design of a distillation 'column for a multicomponent mixture almost inevitably requires a computer. Like fluid-flow theory, most of the distillation calculations have been simulated on a digital comput,er. Only the simulation study of Ruszky and Mitchell (ti3G) has been carried out on a hybrid computer. h summary of simulation studies of the distillation process is given in Table VI. As noted, this table indicates t h a t multicomponent distillation calculations have been simulated extensively on a digital computer. Absorption and Stripping
Even though absorption and stripping are almost as widely used unit operations as distillation, i t is somewhat surprising t h a t there are few specific studies reported o n t h e computer simulation of the absorption and stripping calculations. This may be owing to the fact that' absorption and stripping calculations are similar to the distillation calculations. T h e reported studies on this subject have used only analog ( 3 H ) and digital ( I H , Z H , 4H,5 H , 6H)computers. Extraction
There are only nine important reported studies on t h e computer simulation of the extraction calculations. These are either devoted to t h e design of various types of extraction columns ( S I , 4 I , 51, 7.1, 91) or to t h e general evaluation of the extraction process (11, 21, 61, 81). All these studies have used a digital computer as a simulator. Crystallization and Growth Processes
Crystallization and gron.th processes are common occurrences in nature. T h e chemical engineers have not only made use of this process as a separat'ion means but have also used it as a tool for the production of various types of chemical compounds. Even though the majority of the research in these areas has so far been esperimental, few studies on t h e computer simulation of these processes have been reported. 9 JI ) , and T h e two types of computers, analog ( 2 J , S J , & digital ( I J . 4.J, 6 J , Y J , 8 J , fOJ, I I J ) , have been used to simulate either a n opei~it'ionof a crystallizer or the dynamics of n growth process. Adsorption; Chromatographic Separation; Filtration; Membrane Separation
Two most recent and novel separation processes are chromatographic and membrane separations. The former uses t h e basic fundamentals of adsorption phenomenon, whereas
the latter is in many respects a filtration-like process. All the studies escept one on the computer simulation of a fixed-bed adsorption column ( I K , ZK, 5 K , 7 K , IOK) or a chromatographic column ( S K , 6 K , 8 K , 9 K ) have used a digital computer. Eteson and Zlviebel ( 4 K ) have recently used a hybrid computer for the simulation of a fixed-bed adsorption column. Like adsorption and chromatographic-separation processes, both filtration and membrane-separation processes have been mainly simulated on a digital computer. T h e simulation studies of Gul'ko and Sovosel'tseva (3L) of a n unsteadystate filtration problem and t h a t of Jlaier et al. (4L) for a trickling filter process have been carried out on a digital computer. I n t h e membrane-separation process, either by reverse osmosis or by ultrafiltration, the problem of solute concentration buildup a t t'he membrane surface has drawn quite a bit of attent'ion. T h e important digital simulat'ions of this problem have been reported by Brian ( I L ) for t h e laminar f l o ~and by Srinirasan and Tien (5L) and Winograd and Solan (6L)for t h e turbulent flow. An analysis of this problem in t h e case of concentration-dependent viscosity has been carried out by Ginette and Merson (2L) with the help of an analog simulation. Equilibrium and Multistage Calculations
;\lost' of the unit operations-distillation, absorptioii and stripping, estraction, filtration-encountered in t,he chemical industry are stage operations. There has been considerable theoretical work on general evaluation of the stage operations. T h e computer simulations of this type of analysis can be divided into two main categories: equilibrium-stage calculations and multistage calculations. T h e reported studies of Friday and Smith ( I X ) , Tierney and Bruno @Ai), and Tomich (SJI) on t h e computer simulation of equilibrium-stage calculations have been carried out on a digital computer. Except for the study of Franks ( I S ) , all the simulation studies on multistage calculations have been carried out on a digital computer. Franks ( I S ) has analyzed t h e dynamics of a multistage system on a hybrid computer. Polymer Processing; Regenerators; Particle-Size Analysis
I t may be difficult to say whether the computer-simulation studies on the transport processes associated with polymer processing and t h e regenerators and the studies on the particle-size analysis should be treated separately. Ho\vever, because of t h e grolying importance of these subjects, these studies are listed in categories 0 , Q, and P, respectively. The reported studies on estrusion ( 2 0 , 50)' injection ( I O ) , and drying (SO, 40)of polymeric materials have all used a digital computer as a process simulator. On the ot'her hand, the simulations of three basic types of regenerators, fixed-bed ( I Q , 2Q, 4Q, 5Q), moving-bed (SQ, SQ), and thermal reInd. Eng. Chem. Prod. Res. Develop., Vol. 1 1 , No. 3, 1972
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Table 111. Simulation of Heat-Transfer Theory Subject
Reference numbers and types of simulations
Conduction Change of phase Forced convection Free convection General I n miscellaneous systems and processes Mixed mechanism (e.g., free -t forced convection) Radiation Unsteady heat transfer
Table IV. Simulation of Heat-Transfer Equipment Subject
Boiler, steam generator, and vaporizers Condensers Driers Evaporators Furnaces and heaters H e a t exchangers
Reference numbers and types of simulations
l O D ( A ) , I I D ( A ) , 2 6 D ( X ) , 2 4 D ( A ) , ZSD(A), 38D(A),39D(D), 44D(A) 8D(D), 1 5 D ( A ) , 27D(D), 1 9 D ( A ) ,S l D ( D ) 4D(D), 5 D ( D ) , SOD(A), 34D(A),55D(G),4OD(A),47D(D) 2D(D), 25D(D), 26D(D), 5 2 D ( A ) , 41D(D), 4 Z D ( D ) , 45D(D) 6 D (-A), 5 3 0 (A) SD(D), 7 D ( D ) ,9 D ( D ) , 1 2 D ( H ) , 1SD(D), 1 4 D ( D ) , 1 8 D ( h ) , ZOD(.i), 21D(G), 22D(*k),23D(X), 27D(D), 29D(G), 33D(.l), 3 6 D ( D ) ,3 7 D ( A ) , 43D(G), 4 6 D ( A ) , 49D(D), 5OD(G), 51D(G),
52D(D),5 5 D ( D ) , 56D(G), 57D(G) Temperature measurements
I D @ ) , 48D(A), 54D(D)
~
Table V. Simulation of Mass-Transfer General Theory Subject
Reference numbers and types of simulations
I E (Al,H), 2E(A), 6 E ( D ) , 9 E ( A , H ) , 1 I E ( D ), 12E ( D,G) 14E (A), I S E ( A ) , 27E(A), 2OE(D), 23E (D), 26E(D), 38E(A),29E(D) Convective mas5 transfer I E ( A , H ) ,25E(D) Mass transfer in bioS E ( H ) , IOE(D,G),21E(D) logical systems Mass transfer in droplets 3OE(.I) Material balances 18E(D),24E(D) Mising 7 E ( A ) ,8 E ( D ) , 25E(A),22E(D) Multicompoiieiit mass 27E(D), 3 1 E ( D ) transfer Particulate mixing 4E(D) Permeation 23E(D) Sorption 5E(D), 19E(D) Diffusion
~
these properties are readily obtainable and can be fed into the computer. I n more compley situations, however, the knowledge of thermodynamic and physical properties may involve complex calculations best handled by a computer. Thus, under these circumstaiices, a computer simulation of thermodynamic and physical-property calculations should be useful. -111 the reported studies on this kind of calculation have used a digital computer. The simulation studies have been reported on the calvapor-liquid culations of vapor pressure (IS,I d s , equilibrium curves, or ' K ' values (2S,5S, 11S, 138,15S, 18S), enthalpy and Gibbs free energy (28, 4S, 158, IYS),specific heat (SA'), and compressibility factors (168). There are several general (or package) studies reported on thermodynamic as well as physical property (SS,78, SS), olily thermodynamic (IOS), and only physical-property (19s) calculations.
ICs),
Miscellaneous Systems
generators (7Q) have been carried out on analog (I&) and digital (2Q, 4Q,5Q, ?Q) as well as hybrid ( S Q , SQ) computers. The knowledge of particle-size distribution is of significant demand in evaluations of many types of transport processes. The calculations involved in obtaining this kind of knowledge have been simulated only on analog ( I P ) and digital (2P, 3P, 4P) computers. Thermodynamic and Physical-Property Calculations
The simulations of transport processes on a computer nil1 almost inevitably require the kiion ledge of thermodynamic and physical properties, such as density, vapor pressure, specific heat, thermal conductivity. In simple instances 272
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Several reported studies on the computer simulation of transport processes can either belong to several categories described above or may not belong to any. These studies (Miscellaneous Systems) include the digital simulations of a continuous-process analysis ( I R ) , a recycle problem (ZR),a cascade aiialysis (5R),a neutralization process (GR),air and water pollution dynamics (7R, 12R, 14R), a distributed parameter-process analysis (9R), a n analysis of microwavemeasurement process (IOR),a tower-vibration problem ( I S R ) , a transient-shock wave problem (IbR),the analog simulations of a distributed parameter-system analysis (8R),a calculation of tliermolumiiiescence glow curves ( I I R ) ,t h e hybrid simulations of a countercurrent process ( S R ) ,and various chemical engineering-system analyses (4R).
Table VI. Simulation of Distillation Reference numbers and types of simulotions
Subject
Batch distillation Binary distillation Distillation control Distillation design Distillation dynamics Distillation optimization bfulticomponerit distillritioii Packed-tower distillation Simulation technique Uiisteady-state distillat ion
ISG(A), 18G(D), 19G(D), 22G(D), 26G(d,D), 27G(D), S2G(D), 43G(D), 56G(D), 57G(D) 25G(A), 35G(d), 5OG(D), 55G(D,G), 67G(D,G) 6G(D,G), 16G(D), 24G(D,G), SSG(A), SQG(D),4OG(D), 47G(D), GIG@), 69G(D), 76G(A) 8G(D), I I G ( D ) , 14G(D), dSG(D), SIG(D), 41G(D), 42G(D), 45G(D), 52G(D), 59G(D), 64G(D), 66G(D), 72G(D,G), 7SG(D) SG(D), 6G(D,G), I7G(D), BOG(A), S4G(D), 37G(-4), 44G(D), 6fG(-4), 65G(G), 7OG(D), 77G(D) W D ) , 46G(D), 74G(D,G) 1G(D), 2G(D), 7G(D,G), 9G(D), IOG(D), 21G(A), 22G(D), 27G(D), 28G(D), 29G(D), SOG(D,G), SSG(D), 4SG(D), 48G(.4), 5SG(D), 66G(D), 57G(D), 58G(D), 64G(D) l5G(A) 12G(D), 14G(D), 36G(D), 49G(D), GSG(H), 76G(D) 25G(A), SSG(D), S5G(h), 54G(D), 6OG(D), 62G(A,D), 68G(D), 71G(D)
Mass Transfer with Chemical-Change Theory and Equipment
T h e subject, of maw transfer with chemical change is commonly treated separately from the transport processes under a title of reaction engineering. Since the entire field of reaction engineering is basically a distinct type of transport process, the computer-simulation studies reported in this area are also included in this survey. T h e summaries of t'he simulation studies on t h e reaction processes and the reactors are given in Tables VI1 and. V I I I , respectively. These studies have made significant use of both analog and digital computers. Some studies are reported on the hybrid simulation (56C, 67C, 71 V) of t h e react'ors. I n summary, a few general remarks regarding t h e computer simulations of the transport processes can be made from this survey. h s one might expect, the computers have been quite extensively used, particularly over the past decade, to simulate both theoriez and equipment eiicountered in the study of transport phenomena. Out of three types of existing computers, analog, digital, and hybrid, the digital computer has so far been most rvidely used for t h e simulation purpose. However, t h e analog computer has been used extensively as a simulator when a visual display of the simulated process is required (e.g., heat-transfer equipment, reaction processes). Because of its rersatility, in the future one may find substantial use of the hybrid computer as a process simulator. Finally, this survey should provide the background and guidelines for evaluating t h e suitability of a given computer for simulations of a wide variety of transport processes. Analytical Commentary on Preference in Computer Selection
Oiie question t h a t would inevitably be raised from this survey is, " K h a t is t h e most suitable computer to solve a given transport-process problem: analog, digital, or hybrid?" T h e purpose of this commentary is to briefly analyze this quest'ion. One of the most imporbant criteria for making a choice between ail analog and a digital computer in solving a given transport-process prolslem has been the size and t h e complexity of the mathematical model. If t h e model iiivoh-es a large number of variables aiid equat'ions, 3 digital computer is probably a n inevita,ble choice. Let us briefly evaluate this statement. A solution of a large number of equations on a n analog coniputer would require a large number of amplifiers. This means t h a t t h e phyi;ical size and the cost of the analog computer required to solve t8heproblem may be large. Konlinearities in the ma~:liematical models kill have t h e same
Table VII. Simulations of Mass Transfer with Chemical-Change Theory Subject
Catalytic gas-solid reactions Gas-liquid reactions
Reference numbers and types of simulations
9T(A4), 28T(D), 3 4 T ( A ) ,37T(A)
Kinetic calculations
4T(D), 5 T ( D ) , 14T(D), 15T(D), 25T (D), 267'(D), ZgT(D) 3 T ( A ) , 7T(D), I S T ( G ) , 1 7 T ( D ) , 1 9 T (*%,D),bOT(.%), 24T(G),27T(d), SOT(D), 31 T ( X ) I T ( D ) , 6 T ( D ) , IOT(D), 11 T ( D ) , 12T
Simulation technique
(A), f?IT(A),22T(D), 2ST(G), SST(G) WT(D),8 T ( D ) , 1 6 T ( h ) ,32T(d), 352' (A), 3 6 T ( A ) ,S8T(G)
Industrial reactions
considerations, since the solution of nonlinear problems on a n analog computer may require a large number of multipliers. Because multipliers are expensive parts of a n analog computer, t h e solution of a highly noillinear equation on a n analog computer may thus become expensive. T h e required manpower may vary for solving a given problem on a n analog or a digital computer. T h e time required to design and patch the analog-circuit boards for a complex problem may be more difficult t h a n w i t i n g a computer program to do t h e same job 011 a digital computer. This point' is: of course, debatable. T h e above brief indicates t h a t for a solution of the KavierStokes equation for two or more dimensions or for the solutions of theoretical aiid equipment design problems in heat and mass transfer which involve more than one-dimensional equations or a large number of equations, a digital computer would probably be a better choice than a n analog computer. T h e complex iioiiliiiear multidimeiisional problems commonly eiicountered in the mathematical simulations of polymer processing and in the theory of mass transfer with a chemical change n-ould be similarly better solved by a digital computer. -1theoretical evaluation of coniplex multicomponent distillatioii is also bett,er suited for a digital computer. T h e reported literature indicates this to be t h e case. I n general, the use of a n analog computer has been avoided to solve mult idimensioiial problems. The incremental time step required for a stable digital simulation would be dependent upon the time constant of the system. Furthermore, if there is more than one time coiistaiit iiivolved in t h e problem, t h e largest incremental time step which would give a stable and efficient digital solution Ind. Eng. Chem. Prod. Res. Develop., Vol. 1 1 , No. 3, 1972
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Table VIII. Simulations of Mass Transfer with Chemical-Change Equipment (Reactors) Subiect
Autorefrigerated reactor Batch reactor Cracking furnace Continuous stirred-tank reactor Fixed-bed reactor Fluidized-bed reactor Industrial reactors Packed-bed reactor Reactor design Reactor dynamics Reactor optimization and control Reactor response Reactor start-up Residence-time distribution Spray-column reactor Trickle-bed reactor Tubular reactor
Reference numbers and types of simulations
48U(D)
i 5 U ( A ) ,S i U(D), 35U(D), 51 U(D) 47U(D) 2OC(A),34 U(h) , 39 C(D,G), 40 U(A), 49 C(G), 58 I;(A), 60U(D) 13C(D), 1 8 U ( D ) , 19U(D), 67L’(A), 70C(D) 11 U ( A ) ,45U(A,D), 51U(D) 1C(D,G), 3 C ( D ) , 4 C ( D ) , 9U(D), 14L7(D),25C(A), 47U(A), 53C(D), 59C(D), 66L7(D), 72C(A), 74C(A) 1 6 U ( D ) ,SOU(D), SSC(D), 42U(D), 46C(D) SC(A), 7U(G), 1‘7C(A),24r(.4), 3 1 C ( D ) , 55C(A), 57U(A), 65U(D) ~ c ( D , G ) 41 , G(G), ~ u ( A ) 5, 4 ~ ( ~WU(H) ) , 8U(A,D), 22U(A), dSG(D), 28U(A)
4SC(A),64U(D), 69U(A) lSU(A) 26U(D), %‘U(D), 56U(H) 36C(D) 33U(D)
27r(D), 32U(D), 44C(D), 62U(D), 6SC(D), 71 C(A,D,H)
would be dependent upon the magnitude of the smallest time constant. On the other hand, the analog simulation gives results as a continuous function of time so that incremental time-step considerations are not required for stable solutions. Thus, problems associated n ith design of simple unsteady-state heat exchangers and reactors and simple unsteady-state theoretical heat and mass-transfer problems are often more suitable for an analog computer. T h e survey of the reported literature in this area indicates this to be the case. 111the hydrodynamical analysis of turbulent flow where the mathematical model is usually complex, a digital computer has been widely used for mapping t h e complex turbuleiice pattern. This type of complex mapping problem is handled more appropriately by a digital computer than a n analog computer. One reason for t h e suitability of a digital computer to solve large problems is t h a t in digital simulation a large problem can be divided into several smaller ones Ivhich can be solved separately and stored as subroutines. These subroutines can be easily called upon when required for a complete solution of the large problem. Since this type of “division of labor” can be carried out more easily on a digital computer than on a n analog computer, the design calculations of distillation, absorption, and stripping processes have been more applicable to digital computations. T h e ease of storage and duplication of the computer solution is also another important consideration for evaluating the suitability of a computer for a given problem. The programs for a digital computer can be easily stored 011 magnetic memory or cards, both of which are available for immediate use when desired and occupy less storage space and financial commitment than maintaining or storing prewired circuit patchboards for an analog computer. Duplication of programs or tapes for a digital computer is certainly easier than the duplication of circuit patchboards for a n analog computer. For these reasons, t h e physical property calculations as indicated by the literature are usually carried out on a digital computer. I n a n analog simulation of a problem, the “scaling” of the problem variables is important. T h e “scale factor” for a variable requires a priori laowledge of the range of Its magnitudes in the simulated model. -1variation in “scale 274
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factor” may require a time-consuming review and revision of t h e analog-circuit board. On the other hand, in digital simulation, except for extremely large or small values of the variables, the scaling of a variable is not a n important consideration. As a result, in the computer simulation of any problem of transport processes in which the variables are required to be changed over a wide range of magnitudes, digital simulation may be a more favorable choice than a n analog simulation. An analog simulation is distinctly more advantageous than a digital simulation when a n immediate interaction and visual display of the computer results are required. The interaction and visual display, normally achieved by use of a n oscilloscope, can be useful for quick on-the-spot decisions for problems such as the determination of the best rate expression for kinetic data or for a n evaluation of the best curve through a set of experimental data. The visual display is also useful for the simulations of equipment designs of simple heat exchangers or reactors where the effects of a small change in variables (such as flow rate and inlet concentration) on the final results, such as temperature a t the heat exchanger outlet or conversion a t the reactor outlet, may be obtained immediately. This is one of the main reasons why several studies on the analog simulation of heat exchangers and reactors are reported in the literature. An immediate visual display of the results in a n analog simulation also a l l o w a n easy evaluation of some simple boundary value problems. A solution to a one-dimensional heat-conduction problem or the solutions to other siniilar problems which should satisfy a particular set of boundary conditions can be easily evaluated with t h e visual display of a n analog simulation. An analog computer will handle the sharp discontinuity in problem variables more suitably than a digital computer. I n the evaluations of problems such as the gas-liquid or gassolid interface temperature rise owing to the heat of solution or heat of adsorption or the neutral stability plot of a twophase gas-liquid reactor, the discontinuities or sharp changes in the problem variables are suitably handled by a n analog computer. T h e use of a digital computer in these situations would be less accurate and efficient because of its inherent discrete operation.
Finally, some general advantages of a digital simulation over a n analog simulation should be pointed out. I n a digit’al solution there are some valuable premritten performance packages, such as statistical data analysis, at hand for use. Furthermore, the evolu.tion of simulation languages such a s MIDAS and CSMP in recent years allows the application of analog circuits and their inherent advant’ages directly t o a digital computer. In the previous discussion the hybrid computer has not been evaluated. I n theory the hybrid computer has the combined advant’ages of both analog and digital computers for simulation purposes. Hence, it should be preferred over the independent use of a n analog or digital computer. However, not enough work on hybrid simulation of transport processes has been reported t o critically evaluate justification of its use a t the present time. I n summar?, it can be deduced from the present analysis t h a t even though ailslog simulation has several specific advantages over digital simulation, the available literat’ure indicates that’a digital sirnulation is, in general, a more powerful and convenient research tool than a n analog simulat’ion i n the st,udy of complex transport processes. At present, hybrid simulation for such use is rare. However, its apparent superiorit’y over both analog and digital simulation may in the f u k e make it the most popular and u-idely used Computer for the solution of tramport-phenomena problems. Ac knowledgmenl
Various discussions and comments of J. K, Tierney and H. Beisel are appreciat,ed. References
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(53D) Wade, H. L., llcCann, 11.J., “?\Iodelling the Dynamics of a lIonomer Preheater for a Supercritical Fluid,” paper presented at AIChE Meeting, Houston, TX, February 20, 1967. (54D) Wagner, R. E., “Dynamic Study of Temperature Transducers by Use of an Optical Method,” J . Basic Eng. (ASME), 287 (June 1967). (55D) Whitley, D,; L., Ludwig, E. E., “Rate Exchangers This Computer Way, Petrol. Refiner, 40 ( l ) ,147-56 (January 1961). (56D) Williams, T. J., Morris, H. J., “A Survey of the Literature on Heat Exchanger Dynamics and Control,’’ Chem. Eng. Progr., S y m p . Ser., 57 20-33 (1961). (57D) Willmott, A. J., The Regenerative Heat Exchanger Computer Representation,’’ Int. J . Heat X a s s Transfer, 12, 997-1014 (1969). Mass-Transfer General Theory (1E) Bailey, G. IT., Slater, I. W.,Eisenklam, P., “Dynamic Equations and Solutions for Particles Gndergoing Mass Transfer,” Brit.Chem. Eng., 15 ( i )912-16 , (July 1970). (2E) Bingulac, S.,Radanovic, I., IIattisek, 11.. Azarevic, B. L., “Elimination of Vndesired Sensitivity in the Analog Solution of Diffusion Equations,” paper presented a t Fourth International Analogue Computation Meetings, Brighton, England, New Compendium, September 14-18, 1964; also Proceedings of the Meetings, p 146,JIay 1966. (3E) Bruley, D . F., Knisley, 11. H., “Hybrid SimulationOxygen Transport in the llicrocirculation,” Chem. Eng. Progr., Symp. Ser., 66 (99):,22-32 (1970). (4E) Cahen, I). S., Simulation of Diffusional Mixing of Particulate Solids by Monte Carlo Techniques,” Diss. Abstr., 27B, 4 (1966). (5E) Colwell, C. J., Dranoff, J. S., “Xonlinear Eqailibriiim and Asia1 llixing Effects in Intraparticle Diffusion Controlled Sorption by Ion Exchange Resin Beds; Computer Analysis,’’ Ind. Eng. Chem. Fzlndana., 8 (2), 193-98 ( J l a y 1969). (BE) Duda, J. L., Vrentas, J. d., “Siinierical Technique for Solut,ion of the Diffusion Equation in an Infinite Medium,” ibid., 5 ( l ) ,69-74humerical Solui.ion of the Elliptic Equations for Transport of T’orticity, Heat and hIatter in Two-Dimensional Flow,” Phgs. Fluids, Suppl. 11,21-27 (1969). (12F) Shinkarenko:‘A. V.,Zhivotnikov, V. I., Shevyreva, L. I., Filippov, G. G., A Method of Accelerat,ing the Convergence of Balance Calculations,” Theor. Found. Chem. Eng., 3, 130-32 (1969). (13F) Stone, H. L., Brian, P. L. T., ”Numerical Solution of Convective Transport Problems,” AIChE J . , 9 ( 5 ) , 681-88 (September 1963). Distillation (1G) Amundson, N. R., Pontinen, A. J., Tierney, J . W., Multi-Component Distillation of a Large Computer I1 Generalization with Side-Stream Stripping,” AIChE J . , 5 ( 3 ) ,29-5-300 (September 1959). (2G) qmundson, 5 . R., Pontinen, A. J., “Multi-Component Distillation Calculations on a Large Digital Compnter,” Ind. Eng. Chem., 50 ( 5 ) ,730-36 (May 1958). ( 3 G ) Anisimov, I. V.,Vlasov, -4.E., Pokrovskii, V. B., “Computer Calculations of Dynamics of Plate Rectification Columns,ji Theor. Found. Chem. Eng., 1, 211-17 (1967). (4G) Anisimov, I. V.>Bodrov, V. I., Lebedev, K. V., Koshcheev, N . S . ,“Simplified Algorithm of Search for Optimum Control of a Fractionating Coliimn,” ibid., p 591-95. 280
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(5G) Anisimov, I. V., Bodrov, V. I., Koshcheev, N. X., Pokrovskii, V. B., “llaximation of the Yield of Distillate During Rectification in Plate Columns,” zbzd., 2, 259-66 (1968). (6G) Archer, D. H., Rothfus, R. R., “The Dynamics and Control of, Distillation Units and Other Mass Transfer Equipment, Chem. Eng. Progr., Symp. Ser.?!57 (36), 2-19 (1961). (7G) Aristovich, V. Yu., Levin, A. I., Method for Calculating Rectifiytion of Multi-Component Mixtures on Digital Computers, Int. Chem. Eng., 8 ( l ) ,1-5 (Janutry 1968). (8G) Armstrong, AI., Schofield, A. E., The Design,, of Air Separation Distillation Columns Using a Computer, Chem. Eng. (London), 184-89ijAIay 1969). (9G) Billingsley, D. S., On the Numerical Solution of Problems in llulticomponent Distillation at the Steady State,” A I C h E J . , 12 (6), 1134-41 (September 1966). (10G) Billingsley, I>. S.,“On the Numerical Solution of Problems in Multi-Component Distillation at the Steady State,” ibzd., 16 (3), 441 (Mfty 1970). (11G) Bond, P. D., Distillation Column with One Feed and One Sidestream, Chem. Eng. (London), 5 (January/February 1965). (12G) Bonner, J. S., “An Integrated System for the Automatic Solution of Distillation Problems,” Chem. Eng. Progr., Symp. Sei-., 55 (21))87-92 (1959). (13G) Bowman, W. H., Clark, J. B., “Analog Simulation of Batch Distillation,” Chem. Eng. Progr., 59 ( 5 ) , 54-58 (May 1963). (14G) Bovnton, G. W., “Iteration Solves Distillation,” Hydrocarbon Process., 49 ( l ) ,153 (January 1970). (15G) Brown, E. C., Von Rosenberg, D. U., “Anziig Computation of Packed Tower Distillation Problems, Chem. Ens. Progr., 59 (IO), 75-80 (October 1963). (16G) Buster, A. A4.,“Application of Controlling Computers to Fractionating Units,” Chem. Eng. Progr., Symp. Ser., 57 (34), 85-94 (1961). (17G) Ceaglske, N. H., “Dynamics of Fractional Distillation,” AIChE J . , 7 (4), (353-57 (December 1961). (18G) “Computer Program for 11ulti-Component Batch Di+ tillation (loo),” Chem. Eng. Progr., 58 (9), 96 (September 1962). il9Gi Converse. A. 0.. Gross. G. D.. “ODtimal Distillate-Rate Policy i n Bat‘ch Di. G., “Plate to Plate Binary Distillation Calculations,” Chem. Eng. (London),280 (November 1965). (SlG) Peiser, A. M.,Grover, S. S., ”Dynamic Simulation of a Distillation Tower,” Chem. Eng. Progr., 58 (9), 65-70 (September 1962). (52G) Petlyuk, F. B., Yampol’skaya, 11. Kh., Platonov, T’. AI., “1Iethod of Calcu1ai;ing the Rectification of Mixtures with a Continuous Composition,” Theor. Found. Chem. Eng., 3, 157-63 (1969). (33G) Petryschuk, W. F., Johnson, A. I., “Simulation and Parametric Study of Four Existing AIulti-Component Distillation Columns,” Can. J . Chena. Eng., 44, 241-51 (October 1966). (54G) Platonov, V. , AIonko, Ya. D., Bergo, B. G., “The ‘Ural’ Digital Computer as an Aid in Calculating Unsteady State Rectification,” Int. Chem. Eng., 8 (l),1-5 (January 1968). ( 3 3 G ) Pohjola, V. J., Norden, H. V., “Process Dynamics of Binary Distillation,” Chem. Eng. Sci., 24, 1t3!7-98 (1969). (.i6G) Robinson, E. R.,Goldman, 11. R., Multi-Component Batch Distillation Simulation of an IBM 1130 Computer,” Simr~lation,13, 289 (1969). (57G) Robinson, E. R., Goldman, 31. It., “The Simulation of Multi-Component Eiatch Distillation Processes on a Small Digital Computer,” Brit. Chem. Eng., 14 (G), 318 (June 1969). (.58G) Rose, A,, Williams, T. J., Stillman, R. E., Carlson, IT. C., “Automatic Computer Procedure for Calculating Plates Required for Sonideal Ternary Continuous Distillation,” Chem. Eng. Progr., Symp. Ser., 55 (21), 79-85 (195?(). (d9G) Rose, A, Sweeny, R. F., Schrodt, V. S., Continuous Distillation Calculations by Relaxation Method,” Ind. Eng. Chcm., 50 (s), 737-40 (hlay 1958). (60G) Itosenbrock, H. H., “An Investigation of the Transient Response of a Distillation Column: Part 1, Solution of the Equations,” Trans. Inst. Chem. Eng., 35, 347-51 (1957). (61G) Itosenbrock, €I H., “The Control of Distillation Columns,” zbzd., 40,35-53 (1962). (62G) Itosenbrock, H. H., “The Transient Behavior of Distillation Cohimns and Heat Exchangeri; an Historical and Critical Review.” zbzd.. D 37ti-84. (63G) Itikzky, it:, iIitchell, E. E. L., “Hybrid Simulation of a Iteaction Distillation Column,” paper presented at SJCC Meeting, Boston, hL4, April 28, 1966. (64G) Sargent, R. W. H., hlurtagh, B. A., “The Design of Plate Distillation Columns for Multi-Component Mixtures,” Trans. Inst. Chenr. Eng., 4 7 , 8 5 9 3 11969). (65G) Hchrodt, V. N., Sonimerfeld, J. T., Martin, 0. R., Parisot, P. E., Chien, H. H., “Plate-Scale Study of Controlled Cyclic I>istillation,” Chcm. Eng. Sci., 22, 759-67 (1967).
(66G) Shelton,, R. J., “A Computer Program for Calculating Flash Equilibrium Characteristics and Heat Contents of Hydrocarbon Systems,” Chem. Eng. (London), 385-98 (November 1968). (67G) Shunta, J. P., Luyben, W.L., “Comparison of ‘Stripping’ and General Complex Matrix Inversion Techniques in Calculating the Frequency Response of Binary Distillation Columns,” Znd. Eng. Chem. Fundam., 8 (4), 838-40 (Xovember 1969). (68G) Srygley, J. M., Holland, C. D., “Optimum Design of Conventional and Complex Distillation Columns at Unsteady State Operation,” AZChE J., 11 (l), 112-20 (January 1965). (69G) Stainthorp, F. P., “The Computer Controlled Fractionating Columns at U.hl.I.S.T.,” Brit. Chem. Eng., 15 (6), 794 (June 1970). (7OG) Tetlow, N. J., Groves, D. &I., Holland, C. D., “A Generalized ?lode1 for the Dynamic Behavior of a Distillation Column, AZChE J., 13 (3), 476-85 (May 1967). (71G) Waggoner, It. C., Holland, C. D., “Solution of Problems Involving Conventional and Complex Distillation Columns at Unsteady StateOperation,” ibid., 11 (l),112-20 (January 1965). (72G) Wajc, S. J., “Une Methode de Calcul Peu Iterative des Etates d t , Regime des Colonnes de Distillation Continue a Plateaux, Chem. Eng. Sci., 23, 211-20 (1968). (73G) Waterman, W. nT.,Frazier, J. P., “This Distillation Program Generates Its Own Data,” Hydrocarbon Process., 44 (9), 155-60 (1965). (74G) Waterman, W.W., Frazier, J . P., Brown, G. M.,“Compute Best Distillation Feed Point,,” ibid., 47 (6), 155-60 (June 1968). (75G) Waterman, W.W., Frazier, J. P., “Computer Simulation of Distillation Columns,” AIChE J., 4,598 (1961). (76G) Williams, T. J., Harnett, R. T., “Automatic Control in Continuous Distillation,” Chena. Eng. Progr., 53 ( 3 ) , 220-25 (May 1957). (77G) ’Winteringham, R., Haselden, G. G., “An Approach t o Thermodynamic Reversibility in the Fractionation of Liquid Air,” Trans. Inst. Chem. Eng., 44,55 (1966). Absorption and Stripping (1H) Coggan, G. C., Bourne, J. R., “The Design of Gas Absorbers with Heat Effects:,Part I : A General Program for Adiabatic Plate Absorbers, Trans. Inst. Cheni. Eng., 47, 96-106 (1969). (2H) Davis, P. C., Sobel, B. A., “Absorber-Stripper Calculations with a Digital Computer,’’ Chem. Eng. Progr., Synip. Ser., 59 (42), 95-98 (1963). (3H) Kafarov, V. V., Perov, V. L., Rysin, G. Sh., “Modelling of Yon-Stationary Operations of the -4bsorption Process in a Packed Column on Analog Computers,” Theor. Found. Chem. Eng., 3,95-104 (1969). (4H) hIcNee;e, C. R., “Gas Absorber Solution by Digital Computer, Chem. Eng. B o g r . , Symp. Ser., 58 ( 3 7 ) , 43-53 (1962). (5H) Petryschuk, W. F., Johnson, A. I., “Simulation of the Steady-State Behavior of a Jlulti-Component, biultifeed Reboiled-Absorber,” Can. J . Chem. Eng., 43 (4), 209-16 (August 1965). (6H) Tomio, U., “Optimum Design of an Absorber-Stripper System: Application of the Complex Method,” Ind. Eng. Chem. Process Des. Develop., 8 ( 3 ) , 308-17 (July 1969). Extraction (11) Allen,, P., Kropholler, H. W., Spikins, 11. J., “Use of a Ferranti Argus Computer for Control of a Liquid Extraction Process,” Chem. Eng. (London), 182-88 (September 1966). (21) Belter, P. A., “Simulation of Fractional Liquid-Liquid Extraction Process,” Ind. Eng. Chem., 59 (3), 14-21 (March 1967). (31) Biery, J., C., Boylan, D. R., “Dynamic Simulation of Liquid-Liquid Extraction Column,” Ind. Eng. Chem. F w d a m . , 2 (l),44-50 (February 1963). (41) Diliddo, B. A., Walsh, T. J., “Computer Simulation of Pulse Columns,” Ind. Eng. Chem., 53 ( l o ) , 801-04 (October 1961). (51) Doninger, J. E., Stevens, W. F., “Th; Dynamic Behavior of a Packed Liquid Extraction Column, AIChE J . , 14 (4), 591-99 (July 196?). (61) Lowe, J . T., Calculation of the Transient Behavior of Solvent Extraction Process,” Ind. Eng. Chcm. Process Des. Develop., 7 (3), 362-66 ( J u ~1968?., v (71) Sebenik, It. F., Smut< AI., Optimization of a Solvent Extraction Plant to Process Monazite Rare Earth Xitrates,” ibid.,8 (2), 225-31 (April 1969). (81) Sleicher, Jr., C. A , , “Axial Mixing and Extraction Efficiency,” AZChE’J., 5 (2), 145-49 (June 1959). (91) Wilburn, N. P., “~IathematicalDetermination of Concentration Profiles in Two-Phase Continuous Counter Current Extractors,” Znd. Eng. Chem. Fzindam., 3 ( 3 ) , 189-95 (August 1964). Ind. Eng. Chem. Prod. Res. Develop., Vol. 1 1 , No. 3, 1972
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Crystallization a n d Growth Processes ( I J ) Aiba, S., Nagai, S., Endo, I., Nishizawa, Y., “A Dynamic Analysis of Microbial Growth,” AZChE J., 15 (4), 624-26 (July 1969). (25) Godfrey, W. L., Benham, R. D., “Analog Simulation of Counter-Current Crystallization Process," Simulation, 4 (I), 26 (January 196*5). (35) Han, C. D., Determination of Crystal Growth Rate by Analog Computer Simulation,” Chem. Eng. Sci., 22, 611-18 (April 1967). (45) Han, C. D., “A Control Study on Isothermal Mixed Crystallizers,” Znd. Eng. Chem. Process Des. Develop., 8 (2), 150-58 (April 1969). (5J) AIarique, L. ii., Houghton, G., “Analog Computer Solution of the Modified Rayleigh Equation and Parameter Affecting Cavitation,” Can. J . Chem. Eng., 40 ( 3 ) , 122-26 (June 1962). (6J) Murray, 1).C., Larson, M.A., “Size Distiibution Dynamics in a Salting Out Crystallizer,” AZChE J . , 11 (4), 728-33 ( J ~ l y 1965). (75) Orcutt, J. C., Carey, T. P., “Simulation of Liquid Mixing in a Freezer-Crystallizer Vessel, Znd. Eng. Chem. Process Des. Develop., 9 (l),58 (January 1970);t (85) Randolph, A. D., Larson, AI. A., Transient and Steady State Size Distributions in Continuous Suspension Crystallizers,” AZChE J., 8 ( 5 ) , 639-45 (Kovember 1962). (9J) Randolph, A. D., Larson, 11. A., “Analog Simulation of Dynamic Behavior in a Mixed Crystal Suspension,” Chem. Eng. Progr., Symp. Ser., 61 ( 5 5 ) , !?7-64 (1965). (1OJ) Schneider, K., Adoutte, IZ., Simulation and Control of a Crystallization Process,” Brzt. Chem. Eng., 11 (lo), 1217-19 (October 1966). ( l l J ) Sherwin, AI. B., Shinnar, R., Katz, S., “Dynamic Behavior of the Well-Mixed Isothermal Crystallizer,” AZChE ,J., 13 (6), 1141-54 (November 1967). Adsorption a n d chromatographic Separations (1K) Carter, J. W., “ A Numerical Method for Prediction of Adiabatic Adsorption in Fixed Beds,” Trans. Znst. Chem. Eng., 44,253 (1966). (2K) Chen, J. Ti-., Buege, J. .4.,Cunningham, F. L., Northam, J. I., “Scale Up of a Column Adsorption Process by Computer Simulation,” Ind. Eng. Chem. Process Des., Develop., 7 (I), 26-31 (January 1968). (3K) Dnnckhorst, F. T., Houghton, G., “Digital Computer Simulation of Yon-Linear Equilibrium Chromatography with Axial Dispersion,” Ind. Eng. Chem. Fundam., 5 (l), 93-98 (February 1966). (4K) Eteson, D. C., Zwiebel, I., “Hybrid Computer Solution of the Simple Fixed Bed Adsorption Model,” AZChE J., 15 (l), 124-26 (Januarr 19691. (5K) Fair, G. 11.;Geminell, S., “ A Mathematical Model for Coagulation,” J . Colloid ~SCZ.,1 9 , 360 (1964). (6K) GOUW,T. H., Hinkins, It. L., Jentoft, R. E., “Simulated Distillation by Gas Chromatography,” J . Chromatogr., 219 (June 1967). ( 7 K ) Hall, K. R., Eagleton, L. C., Acrivos, -4., Vermeulen, T., Pore and Solid Diffusion Kinetics in Fixed-Bed Adsorption under Constant-Pattern Conditions,” Znd. Eng. Chem. Fundam., 5 @), 212-23 (May 1966). ( 8 K ) Lai, Cheng-Liang, Roth, J. A., “Dynamic Simulation of Gas Chromatographic Column,” Chcm. Eng. Sei., 22, 1299-304 (1967). (9K) Mears, F. C., “Computer Controls Chromatographs,” Hydrocarbon Process., 46 (12), 105-08 (December 1967). (10K) Pollock, A. W., Brown, M.F., Dempsey, C. W., “Machine Solution of a Boundary Value Problem for a Continuous Arosorb Process,” Ind. Eng. Chem., 50 (6), 72j-29 (May 1958). Filtration a n d M e m b r a n e Separations (1L) Brian, P. L. T., “Concentration Polarization in Reverse Osmosis Desalination with Variable Flux and Incomplete Salt Rejection,” Ind. Eng. Chem. Fundam., 4, 439-43 (1965). (2L) Ginette, L. F., Merson, R. L., “Maximum Permeation Rate in Reverse Osmosis Concentration of Viscous Materials,” paper presented at 63rd National AIChE Meeting, St. Louis, 310, February 1968. (3L) Gul’ko, F. B., Xovosel’tseva, Zh. A., “Solution of Unsteady State Problems of Filtration and Prediction by Means of Simulation Methods,” Automat. Remote Contr. (4), 645 (.4pril 1966). (4Lj Maier, W. J., Behn, V. C., Gates, C. D., “Simulation of the Trickling Filter Process,” J . Sanit. Eng. Diu., Amer. Soc. Civil Eng., 91 (August 1967). ( j L ) Srinivasan, S., Tien, Chi, “A Finite Difference Solution for Reverse Osmosis in Turbulent Flow,” Desalination, 7 (l), 51-74 (1969-70 j. (6L) Wino rad, Y., Solan, A., “Concentration Build-Up in Reverse 8smosis in Turbulent Flow,” ibicl., p 97-109.
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Equilibrium-Stage Calculations (111) Friday, J. R., Smith, B. D., “An Analysis of the Equilibrium Stage Separations Problem-Formulation and Convergence,” AZChE J., 10 (5), 698 (September 1964). (231) Tierney, J. W.,Bruno, J. A., “Equilibrium Stage Calculations,” ibid., 13 (3), 556 (+y 1967). (3M) Tomich, J . F., “A New Simulation Method for Equilibrium Stage Processes,” ibid., 16 (a), 229 (March 1970). Multistage Calculations ( I N ) Franks, G. E., “Hybrid Simulation of Multi-Stage System Dynamics,” Chem. Eng. Progr., 60 (3), 65 (March 1964). (2N) Koenig, D. AI., “Invariant Imbedding and Concurrent Multi-Stage Operations,” Znd. Eng. Chem. Fundam., 8 ( 3 ) , 537 (iiugust 19B9). (3N) Lee, E,. S., Invariant Imbedding, Iteratiye Linearization, and Multi-Stage Countercurrent Processes, AIChE J., 16 (4), 679 (July 1970). (4N) Nah, R. S.H., Michaelson, S., Sargent, R. W., “Dynamic Behavior of Multi-Component Multi-Stage Systems; Xumerical Xethods for the Solution,“ Chem. Eng. Sei., 17, 619 (August 1962). (5s)Miirdoch, P. G., “Finite-Difference Transforms for Applications to Stage by Stage Processes,” AZChE J., 7 ( 3 ) , 526-29 (September 1961). (6Nj Sargent, R. W. H., “The Dynamic Behavior of RlultiStage Systems: Further Improvement in the Xumerical Solution,” Trans. Inst. Chem. Eng., 41, 01 (1963). (7N) Wells, J. F., “Automatic Control in a Dye-Works,” Brit. Chem. Eng., 15 ( 5 ) , 655 (May 1970). Polymer Processing (10)Harry, D. H., Parrott, R. G., “Numerical Simulation of Injection llold Filling,” Polym. Eng. Sei., 10 (4), 209-14 (J~ily1970). ( 2 0 ) Klein, I., Tadmore, Z., “The Simulation of the Plasticating Screw Extrusion Pr0ce.s with a Computer Programmed Theoretical Model,” ibicl., 9 (l), 11-21 (January 196;). (30) Riek, It. F., AIcAvoy, T. J., Chappelear, D. C., Diffusion in Polymer-Solvent Systems; a Study of Numerical Methods of Simulation,” J . Polym. Sci., Part A-2, 6 , 1863 (1968). (40) Robinson, D. E., Higinbotham, A. E., Wankat, P. C., Accelerated Polymer Film Drying at Elevated Pressures,” Ind. Eng. Chem. Process Des. Develop., 8 (4), 502-08 (October 1969). (j0) Tadmor, Z., Duvdevani, I. J., Klein, I., “Xelting in Plasticating Extruders-Theory and Experiments,” Polym. Eng. Sei., 7, 198-217 (July 1967). Particle-Size Analvsis (1Pj Cole, H..if;, “An Analog Computer Model for Particle Size .4nalysis, Report No. AERE-R-5096 (G.B.) XASA N66-35297, 1967. (2P) Cole, H,. “ A Special-Purpose Computer foi Particle Size Analysir, Radio Electron. Eng., 325 (May 1967). (3P) Collins, S. B., Knudsen, J. G., “Drop-Size Distributions Produced by Turbulent Pipe Flow of Immiscible Liquids, ’’ AZChE J., 16 (6), 1072 (November 1970). (4Pj Suzuki, A., Ho, S . F. H., Higuchi, W.I., “Predictions of the Particle Size Distribution Chances in Emulsions and Susuensions by Digital Cornputation,”-J. Colloid Interface Sci.,’ 2 9 , 552 (1969). Regenerators (I&) Duhne, C., Garcia, H., “ A Hydraulic Analog Computer for Regenerator Calculations,” Brit. Chem. Eng., 7 (l),39-41 (January 1962). (2Q) Gonzalez, L. O., Spencer, E. H., “Studies in the It’umerical Solution of a Model Simulating Fixed Bed Regeneration,” Chem. Eng. Sei., 18, 763-66 (December 1963). 1 3 0 ) Harter. 11. 0.. JIarr. G. R.. Weekman. T’. W..“Hvbrid Computer’ Simulafion of ’a Moving Bed Regenerator,’’ paper presented at Fall Joint Computer Conference, San Francisco, C4, November 10, 1966. (4Q) Leung, P. K., Quon, D., “ A Computer Model for Regenerative Bed,” Can. J . Chem. Eng., 44,26-31 (February 1966). ( 5 Q ) Sculman, B. L., “Building a Mathematical Model for Catalyst Regeneration in Fixed Bed,” Znd. Eng. Chem., 55 (12), 44-49 (December 1963). (6Q) Weekman, Jr., V. IT., “Hlbrid Computer Simulation of a Moving Bed Catalyst Regenerator,’’ zbicl., 59 (l), 84-91 (January 1967). (7Q) Willmott, A. J., “Digital Computer Simulation of a Thermal Regenerator,” Znl. J . Heat lllass Transfer, 7, 1291-302 (1964). Miscellaneous Systems (1R) Box, G. E. P., Chanmugam, J., “Adaptive Optimization of Continuous Processes,” Znd. Eng. Chem. Fundam., 1 (l), 2 (February 1962).
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(2R) Chien, H. H. Y., “Optimization of Recycle Problem,” ibid., 5 (l),66 (February 199;). (3R) Frank, A., Lapidus, f;., Hybrid Computations Applied to Chem. Eng. Progr., 60 (4),61 (April , Countercurrent Process, 1964). (4R) Frank, A,, Lapidus, L., “Hybrid Simulation of Chemical Engineering Systems,” ibid., 62 (6), 66 (June 1966). (5R) Franks, R. G. E., Worley, C. W.,“Quantitative Analysis of Cascade Control,” I$. Eng. Chem., 48 (6), 1074 (June 1956). (6R) Geerlings, 11. W., Dynamic Behavior of pH-Glass Electrodes and Neutraliz,ation Process,” Proceedings of Plant and Process Dynamic Characteristics, The Society of Instrument Technology, p 101, Elutterworths, London, England, 1957. (7R) Jaske, R. T., “The Use of Digital Systems Modelling in the Evaluation of Regional Water Quality Involving Single or llultiple Releases,” Chem. Eng. Progr., Symp. Ser., 64 (go), I ( 1968). (8R) Kummel, M.,“Analog Simulation and Control of a Distributed Parameter System,” Chem. Eng. Sci., 24, 1035 ( 1969). (91t) Paraskos, J. A., IIcXvoy, T. J., “Feedforward Computer Control of a Class of Distributed Parameter Processes,” AIChE J., 16 ( 5 ) , 7.54 (September 197p). (1011) Pieterse, J. D., T’ersnel, W., Numerical Method for Obtaining the Scattering 3Iatrix of a Micro-Wave Two-Port from Standing Wave Measurements,” A p p l . Sci. Res., 21, 13 (1969). ( I l R ) Razdan, K. K . , Brennan, R. D., Grosswei, L. I., “Analog Computer Calculation of Thermoluminescence Glow Curves,” J . Appl. Phys., 41 (K2), 832 (1970). (I2R) Ross, L. W., “Simulation of Air and Water Pollution Dynamics-a Surve:v-,” Simulation, 14 (+), 165-70 (April 1970). (13R) “Tower Vibration Program (091), Chcm. Eng. Progr., 86 (April 1962). (14R) i’enn Leeds, Jr., J., “Dynamic Curve Fitting, Identification Problem of System Theory as Applied to Pollution I>ynamics,” Chem. Eng. Progr., Symp. Ser., 64 (?O), 92 (1968). (ltjR) Watts, J. W.,Yon Rosenberg, D. E., A Kumerical Solution of Transient Shock M-ave Problem,” Chem. Eng. Sei., 24, 49 (1969). Thermodynamic and Physical-Property Calculations (IS) Acciarri, J. *4., Gerlach, R. H., “Automatic Computation of hntoine Equation Constants,’’ Ind. Eng. Chem., 51 (3), 239 (llarch 19.59). ( 2 s ) Bond, P. I]., “Preparation of Equilibrium and Enthalpy Data,” Chem.. Eng. (London), 6 ,(!anuary/February 1965). ( 3 5 ; ) Bracie, W.C., Liou, D. W., Chemical Structure Coding,” Chcna. Eng. Progr., 61 (.j),102 (May 1965). (48) “Computer Program for Enthalpy Tables (090),” ibid., 58, 96 (February 1962). (.is)Goodwill, AI. J., Gordon, E., Paylor, J . IT7., “ AIachine Computation of K Values,” Chem. Eng. Progr., Symp. Ser., 55 (21), l ( l 9 5 9 ) . (6s) Heitman, R. E., Harris, G. H., “Estimation of Physical Properties by Minimum Error Analysis,” Ind. Eng. Chenz., 60 ( 2 ) , ,51 (February 1968). 17s) Henlev. E. J.. I3eirute. R. XI.. “A Thermodvnamic and Physical ”Properties Package (TAP),” Chem. En;. J . , 1 (4), 291-9.; (October 1970). (8s) Meadows. E. L.. “The A.1.Ch.E. Svstem.” Chem. Enor. Progr., 61 (3j, 93 ( l l a y 196.5). (9s) Morris, J. R., “Heat Capacity of Gases at Constant Pressure,” Chem. Eng. (London), ,57 (March 1965). (10s) Myers, I). B., Scott, R . “Thermodynamic Functions for Sonelectrolyte Solutions, Ind. Eng. Chem., 55 (7), 43 (July 1963). (11s) Peacock, D. G., “ Vapor-Liquid Equilibrium; Polynomial Representation of Non-Ideal Systems,” Chem. Eng. (London), 280 (November 1965). (12s) Peacozk, 11. G , “Fitting and Testing of Yapor-Pressure Equation, zbzd., p 281. (13s) Price, A. R.’,-Leland, T. IT., Kobayashi, R., “Evaluation of Benedict-Webb Rubin Equation for Prediction of Phase Equilibrium of Light Hydrocarbon AIixtures at Low Temperatures,” Chcm. Eng. Progr., S y m p . Ser., 55 (21), 13 (1939). (14s) Rose, A , , .4cciarri, J. A,, Johnson, R. C., Sanders, W.W., “Automatic Computation of Antoine Equation ConstantsCaproic and Caprylic -4cids and 11ethyl Esters,” Ind. Eng. Chem., 49 ( 2 ) , 104 (February 1937). (1.53) Shelton, R. J., “K-Value Prediction, Flash and Enthalpy Calculations,” Chem. Eng. (London), 281 (November 1965). (16s) Sonimerfeld, J. T., Perry, G. L., “Compressibility Factors by Computers,” Hydrocarbon Process., 47 (IO), 109 (October 1968). (17s) Wilcox, D. E., Bromley, L. A., “Computer Estimation of Heat and Free Energy of Formations for Simple Inorganic Compounds,” Ind. Eng. Chem., 55 (i), 32 ( J d y 1963).
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(18s) Williams, R. A., Henley, E. J., “ A Comprehensivy,VaporLiquid Equilibrium Computer Program-KVALUE, Chem. Eng. J., 1 (a), 145 (April 1970). (19s) Yen, L. C., Cantwell, K. R., Giles, B. L., “A General Purpose Physical Data System for Computer Process Calculations,” Ind. Eng. Chem., 60 ( 2 ) , 70 (February 1968). M a s s Transfer with Chemical-Change Theory ( I T ) Arbesman, R. W., Kim, Y. G., “Generalized Relaxation Method in Chemical Kinetics,” Ind. Eng. Chem. Fundam., 8 ( 2 ) , 216-21 ( l l a y 1969). (2T) Bacher, S., Kaufman, A., “Computer-Controlled Batch Chemical Reactions,” Ind. Enq. Chem., 62 (l),53-61 (January 1970). (3T) Ballard, B. E., Goyan, J. E., “Application of Analog Computer Techniques to In-Vivo Drug Kinetic Studies,” M e d . Bzoi. Eng., 483 (September 1966). (4T) Brian, P. L. T., Gas Absorption Accompanied by an Irreversible Reaction of General Order,” AIChE J., 10 (I), 5-10 (1964). /ST) Brian. P. L. T.. Beaverstock. AI. C.. “Gas Absorution &companied by a Two-step Chemical Reaction,” Chem.-Eng. SCZ.,20, 47-*56 (1965). (6T) Burkhard, C. A . , “Computer Program for Frost-Schwerner Kinetic Equations and Derived Tables,” Ind. Eng. Chem., 52 (8), 678-80 (August 1960). (7T) Cam, Y. L., Somers, A. E., “Computer Ctjlization in a Kinetic Study of n-Butane Isomerization,” Can. J . Chem. Eng., 48, 456-62 ( - 4 ~ g ~1970). st (8T) Cohe:: G. D., “Monte Carlo Simulation of Reaction Kinetics, Ind. Eng. Chem. Fundam., 4 (4), 471-73 (November 146.5)
(STj-Fahidy, T. J., Perlmutter, D. D., “Dynamics of a SurfaceCatalyzed Chemical System,” Can. J . Chem. Eng., 44, 95-99 (April 1966). (10T) Giese, C., “Determination of Best Kinetic Coefficients of a Dynamic Chemical Process by On Line Digital Simulation,” Simulatzon, 8 (3), 141 (March 1967). ( I l T ) Griswold, R., Haugh, J . F., “.4nalog Computer Simulation, an Experiment in Chemical Kinetics,” J . Chem. Educ., 576 (Seotember 1968). (12Tf Hyman, 31. H., “Simulate Methane Reformer Reactions,” Hydrocarbon Process., 47 (7), 131-37 (July 1968). (13T) Janata, J., Schmidt, O., “Application of Analog Computer to the Study of Reactions Involving Electrolytically Generated Reagent,” J . Electround. Chem., 224 (March 1966). (14T) Johnson, A. I., Akehata, T., “Reaction Accompanied Mass Transfer froni Fluid and Solid Spheres at Low Reynolds Xumbers,” Can. J . Chem. Eng., 43, 10-15 (February 196*?). (l5T) Johnson, A. I., Hamielec, A. E., Houghton, W. T., Mass Transfer with Chemical Reaction from Single Gas Bubbles,” AIChE J . , 13 (2), 379-83 (March 1967). (16T) Kafarov, 1;. V., Lutsenko, A. A., “The Bnalysis of Typical Chemical Reactions with Analog Computers,” Int. Chewi. Eng., 5 (4), 623-32 (October 1963). (17T) Xartin, R. A,, Hoy, K. L., Peterson, R. H., “Computer Simulation of the Tolylene Diisocyanate-Polyol Reaction,” Ind. Eng. Chem. Prod. Res. Deaeiop., 6 (4),218-22 (December 1967). (18T) Matthews, T., “Simulating Continuous Reaction Processes,” Chcm. Eng., 93-96 (August 3, 1964). (19T) Naxon, W. D., Chen, J. W., Hanson, F. R., “Simulation of a Steroid Bioconversion with a 1Iathematical llodel,” Ind. Eng. Chem. Process Des. Develop., 5 ( 3 ) , 285-89 (July 1966). (20T) LIeeks, 31. R., “An Analog Computer Study of Polymerization Rates in Vinyl Chloride Suspensions,” Polym. Eng. Sei., 9 (a),141-31 (March 1969). (21T) AIumme, K. I., “Analysis of the Control Potential of the Copolymerization Equation by Analog Simulation,” Ind. Eng. Chem., 59 (4), 64 (March 1967). (22T) Sesbit, R. A, Engel, R. D., “.4n Example Program for the Determination of Chemical Rate Coefficients from Experimental Data,” Simulation, 8 (3), 133 (March 1967). (23T) Kilsen, P. H., “Cse of Computers in Kinetic Calculations,” AIChE J., 1 (4), 561-62 (December 1955). (24T) O’Dom, G., Fernando, Q., “Kinetics of Bromination of Certain Substituted 8-Quinolinols,” Anal. Chem., 38 ( i ) , 844 (June 1866). (25T) Onda, K., Sada, E., Kobayashi, T., Fujine, AI.> Gas Absorption iiccompanied by Complex Chemical Reactions, Part I and 11,” Chern. Eng. Sci., 25,753-68 (1970). (26T) Onda, K., Sada, E., Kobayashi, T., Fujine, lI., “Gas Absorption Accompanied by Complex Chemical Reactions, Part 111,Parallel Chemical Reactions,” ibid., p 1023-31. (27T) Parker, W.A., Prados, J. R., “ilnalog Computer Design of an Ethylene Glycol System,” Chem. Eng. Progr., 60 (6), 74-78 (June 1964). Ind. Eng. Chem. Prod. Res. Develop., Vol. 1 1 , No. 3, 1972
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(67U) Sinai, J., Foss, A. S., “Experimental and Computational Studies of the Dynamics of a Fixed Bed Chemical Reactor,” AIChE J., 16 (4), 658-69 (July 1970). (68U) Vichnevetsky, R., “High Speed Computing for the Hybrid Simulation of Reactors and Heat Exchangers,” paper presented at Eastern Simulation Council Meeting, Kew York, XY, December 13, 1967. (69U) Wen, C. Y., Chung, S. F., “Dynamic Response Equations for Various Reactor Models,” Can. J . Chem. Eng., 43 (3), 101-09 (June 1965). (70U) Wheeler, J . PI.,Middleman, S., “Machine Computation of Transients in Fixed Beds with Intraparticle Diffusion and Non-Linear Kinetics,” Tnd. Eng. Chem. Fundam., 9, 624-27 (1970). (71U) Wright, B. S., “Computer Simulation of a Plug Flow Reactor for a Complex Reaction,” Brit. Chem. Eng., 12 ( l l ) , 1750-52 (November 1967). (72U) Yablonskii, G. S., Kamenko, B. L., Gel’Bshtein, A. I., Slin’ko, A I . G., “Modelling the Hydrochlorination of Acetylene in a Fixed Catalyst Bed,” Znt. Chem. Eng., 8 ( l ) , 6-13 (January 1968). (73U) Zasmolin, A. V., Yablonskii, G. S., Beskov, V. S., “Calculation of Temperature and Concentration Fields in a Catalyst Bed for a Process with Variable Flow Properties,” Theor. Found. Chem. Eng., 2,201-08 (1968). (74U) Zayarnyi, N. S., “Computer Model for Ammonia Synthesis Columns,” Int. Chem. Eng., 2 (3), 378-83 (July 1962). Mathematics and Computers-Previous Reviews (1V) Johnson, R. C., Ball, W. E., Burggrabe, W. F., Heinyj R. L., Russell, J . L., Sweeny, R. F., “LIathematice, Computers, Operations Research, and Statistics: Fundamentals of Chemical Engineering,” Ind. Eng. Chem., 52 (4), 359-67 (April 1960). (21‘) Lapidus, L., “Annual Review of Applied Alathematics,” ibid., 60 (12), 42-52 (December 1968); 61 (la), 43-54 (December 1969). (37.’) “Survey Report: Computer Applications in Chemical Engineering,” Chem. Eng. (London), 222, 343-51 (October 1968). (4V)Swee;y, R. F., Davis, R. S.,Lapidus, L., Roberts, S. AI., Sax, E., Mathematics, Computers, Operations Research, and Statistics: Chemical Engineering Fundamentals Review,” Ind. Eng. Chem., 53 (4), 329-36 (April 1961). ( 5 v ) Sweeny, R. F., “LIathematics in Chemical Engineering,” zbid., 56 ( l l ) ,57-61 (November 1964). (61’) Sweeny, R. F., “Annual Review of llathematics,” ibid., 57 (12), 72-77 (December 1965); 59 (2)) 71-76 (February 1967). (7V) Williams, T. J., “Annual Review of Process Control,” ibid., 57 (12), 33-43 (December 1965). ( 8 V ) Williams, T. J., “Computers, Automation and Process Control,” ibid., 56 ( l l ) , 47-56 (November 1964). (9V) Williams, T. J., “Chemical Engineers and Computers,” Can. J . Chem. Eng., 39 (5), 219-26 (October 1961). (1OV) Williams, T. J., “Computers and Process Control,” Znd. Eng.Chem., 58 ( l j , 33 (January 1966); 58 (la), 55-70 (December 1966); 61 ( l ) ,76-89 (January 1069); 62 (2)) 28-40 (February 1970j.
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