R. F. S W E E N Y R . S.
ANNUAL REVIEW
D A V I S
C. D. H E N D R I X
L. M . N A P H T A L I
MATHEMATICS IN CHEMICAL ENGINEERING T h i s year, optimization methods have been upplied, compared, combined, and extended-and have been debated heatedly and at length
he realm of mathematics and the chemical engineer
Tduring the past year has been largely characterized
by reflection upon the proper use of the many tools for optimization and optimal control made available recently. The methods, especially dynamic programming, maximum principle, and the direct method, have been applied and compared in application. They have been combined. And of course they have been extended to new applications. Perhaps a measure of the intense interest, even excitement, surrounding this field is the heated debates (some in print) between major advocates of the different methods. With only a few exceptions, this review covers May 1963 through May of 1964. General
During the past year, a new book (13A) was published, authored by two British chemical engineers; a it is also suitable textbook for applied mathematics. Points in favor of the book are that complex algebra and matrices are adequately treated, numerical methods and computers usage are emphasized, and the importance of optimization is recognized with a chapter. One section neglected was statistical treatment of data. Two CEP Symposium Series of special interest were published. The one on statistics ( 7 7A) also has some OR and computer solution content. The one on Process Systems Engineering (12A) is mostly process control, but has some good articles on simulation, frequency response, and optimization. Several papers appeared which presented either novel
mathematical solutions or particularly good explanations and examples of techniques in problem solving. Two numerical methods were compared (9A) and the problems of truncation studied. The accuracy of finite difference methods on convective transport problems was determined (16.4). I n another paper ( 1 8 4 ) three methods of approach to mass transfer (eigensolution, boundary layer and penetration theory) were applied to one problem, making a useful exposition of these methods. A new approach (3A) is given for solving that class of problems which involves the systematic change in size or weight of particles. An example is the growth of catalyst particles with deposits during reaction. I n another paper (2A), the differential equations of kinetics are related to nonassociative algebra. An interesting approach to essentially stochastic problems was given ( 7 4 4 ) . The usual continuum expressions were set up for flow through a packed bed and the random nature of the bed was expressed by introducing random perturbations using a Monte Carlo technique. Two authors presented rather similar approaches to simulation or optimization of nonlinear processes by imposing linear constraints. The one (6A) minimizes the free energy in multicomponent distillation, the other (77A) in chemical reaction. Finally, a method for inversion of Laplace transforms by linear programming was given (75A). Special computers are helping to solve problems. Adaptive optimization using a conditional probability computer was applied specifically to on-line process optimization in a highly recommended article (19A). Several other articles ( 7 4 4A, 5 4 7A, 84 70A) are concerned with various facets of hybrid or Rep-Op computers. Statistics
An excellent tutorial series (2B) was completed. A reprint of the collected series is available. An approach to the optimum design of screening tests is presented (6B). This method takes into account the VOL. 5 6
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Development of OR theory shows progress, as does application
values of the quantities at stake; the value and probability of a successful outcome, resources available, costs, etc. General discussions and methods in screening are presented (5B, 2OB) as is a group-screening method for catalyst selection (74B). Two examples of central composite design experiments appeared (9B, 19B). Low-arithmetic methods are reviewed ( I 1B). These methods are of particular value in devising procedures for laboratory and plant personnel. Although somewhat wasteful of the information contained in a set of data, these methods save substantial time in coniputation and interpretation. One of the hazards of multiple regression is described (/OB).Unfortunately, the author presents the problem only from the standpoint of conventional multiple regression. The problem is compounded in stepwise multiple regression, a tool in widespread use in the chemical industry. Residuals contain information about the variables not included in an analysis of variance or a regression analysis. These variables are often time trends, weekday-weekend patterns, operator variations, etc. Procedures for the analysis of residuals are described ( 3 B ) . The properties of transformations are presented ( 7 B ) . I n spite of (or because of) the advent of digital computers and powerful estimation methods, the extraction of reaction rate constants remains a tricky problem full of pitfalls for the amateur. One attempt at the extraction of rate constants from existing data is presented ( I 7B). Another approach ( l 5 B ) is more satisfactory. I t is often convenient to have “running estimates” of the magnitudes of effects while executing factorial designs. 4 procedure for rapidly estimating such effects is presented (73B). The estimation of plant yields by data adjustmentleast squares is presented ( 4 B ) , A review of the properties of statistical distributions appeared ( 1 6 3 ) . Although the numerous references to Soviet standard methods make one book (77B) sometimes difficult to read, the basic statistical principles still apply. Not written for the amateur, this book contains a diversity of information of interest to serious workers in analytical chemistry who will recognize this text as unique. A thorough description of nomographic methods is presented ( 7 B ) . This text elevates nomography from an art to a science. Several interesting articles on statistical and numerical methods appeared in a series (72%). Of particular interest are the articles on random numbers (pp. 8-13), reaction rate estimation (pp. 14-27), optimum operating conditions found from arbitrary preference ratings (pp. 34-38), investment risk analysis (pp. 55-64). application of queuing theory (pp. 74-77), and simultaneous regression equations (pp. 84-89). 58
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
An example of regression analysis in chemical engineering is fully described (8B). Basic Operations Research
,4 new journal appeared which the serious worker in O R should find most helpful. I t is the “Journal of the Canadian Operational Research Society,” published monthly in Ontario, Canada. This section will cover the development of OR thzory. Industrial operations research which is concerned with industrial applications and the practice of O R is covered in the next section, Bellman (3C) has developed a new application of dynamic programming as an algorithm for the numerical treatment of multistage decision processes. This approach gives some interesting results in multistage decision processes. The subject of linear programming was covered in two new books (40,7 0 ) . Multistage decision models and techniques for developing them were discussed in another book ( 9 0 ) . A method of solving very large linear programming problems by breaking them into smaller subsets was mentioned (78C). The use of an analog computer in solving multistage decision problems was presented (76C). The use of partition programming to solve nonlinear problems was covered (ZOC). Sequential decision models and their use in queuing and replacement problems appeared (22C) The combination of classical critical path techniques with statistics and stochastic programming to introduce probability considerations into this type of problem was presented (6C). A series of review volumes which will serve as basic reference works to O R workers has been begun (50). T h e books are a n inventory of mathematical techniques and research methods available. I
Industrial Operations Research
Many excellent general texts and articles on O R and its applications appeared this past year. ’4 good introduction for executives appears in Fortune ( 4 C ) . X survey of industrial operations research and its areas of application appeared ( I 4 C ) . Several books of a general nature appeared ( 7 0 , 60, 700, and 7 7 0 ) . Dun and Bradstreet (71C) published an excellent monograph on the gains in efficiency and the payout that can be achieved from applying OR techniques. Some excellent articles appeared on the use of OR in selecting and making decisions in research and development projects (7C,SC, 27C). .4n excellent article by J. W. Forrester (73C) gives a rather broad view of the field of systems management and how the concepts involved are influencing the approaches to undergraduate education. The article is highly recommended to both managers and practicing chemical engineers, as is his in[eresting text, “Industrial Dynamics,” published by the
TABLE 1.
OTHER APPLICATIONS OF M A T H E M A T I C S
Optimization, frequency response, etc. ( 7 J ) Response to random inputs ( 2 J ) Effect of mixing in reactor ( 3 5 ) Dynamic programming in reactor ( 4 5 ) Dynamic programming in multistage separation ( 5 5 ) Gradient method in CSTR ( 6 J ) Katz maximum principle in multistage sequence ( 7 5 ) Maximum principle in cross-current extraction
Simulation a n d design ( 7 K ) Molecular weight distribution from polymer reaction ( 2 K ) Analog simulation of hydrodealkylation (naphthalene) ( 3 K ) Flow rates in isotope separation-material balances ( 4 K ) Analog simulation of batch distillation ( 5 K ) Effect of horizontal barriers in thermal diffusion ( 6 K ) Kinetics and control of microorganisms ( 7 K ) Method for Underwoods’ minimum reflux on computer ( 8 K ) Concentration response in intermittent irradiated reaction ( 7OK) Piping network analysis-Hardy-Gross method on digital computer ( 7 I K , 7 2 K ) Yield studies in packed tubular reactors ( 73K) Extraction with longitudinal mixing ( 7 4 K ) Catalyst regeneration in fixed beds ( 75K) Continuous, transient extraction with agitation ( 7 6 K ) Dynamics of pulsed-plate extractor Fluid dynamics ( 7L) Non-Newtonian fluids in entry to pipe ( 2 L ) Laminar-turbulent transaction from semifluids ( 3 L ) Profile relaxation in Newtonian jets ( 4 L ) Eddy formation in film flow ( 5 L ) Initial in stability viscous interface ( 6 L ) Heterogeneous flow ( 7 L ) Forced convection in 3-D flow ( 8 L ) Longitudinal dispersion in two phase flow ( 9 L ) Viscous heating in cone-and-plate viscometer
M I T press. Two other general articles are also of interest (ZC, 5C). Scheduling applications of O R continued to receive emphasis. A general system of plant wide PERT control was described (25C). Manpower scheduling and allocation were covered ( 9 C ) . Methods of multiple resource scheduling and allocation were discussed ( 2 3 C ) . Use of O R in construction planning was presented (77C). Scheduling by means of O R in a data processing installation was covered (24C). Critical path scheduling was developed from the point of view of network flow theory ( 7 9 C ) . The problem of scheduling jobs through a fixed number of machines was considered (70C). In the areas of decision theory, production, and cost control, some interesting work was reported. The use of PERT for cost and profit control was covered in a new book ( 8 0 ) . The modeling of production and operations management was also considered (30). Applications AUTHOR Robert F. Sweeny is Assistant Professor of Chemical Engineering at Villanova University. Robert S. Davis is Assistant Vice President-Director of Development for Halcon hternational, Inc. Charles D . Hendrix is Engineer and Statistical at LJnion Carbide Chemicals Co. Leonard M . Naphtali is Associate Professor of Ct,emical Engineering at Polytechnic Institute of Brook4yn.
( IOL) ( I7L) (7215) ( 73L)
Laminar flow at inlet to parallel plates Drag coefficient of sphere in viscoelastic fluid Effect of surface active agents on falling film Oscillatory behavior in viscoelastic fluids
Fluid flow-heat transfer ( I M ) Free and forced convection in vertical and horizontal conduits ( 2 M ) Radiative and convective transfer in slug flow ( 3 M ) Flow and transfer in rarefied gas ( 4 M ) Numerical method for steady turbulent transfer Heat a n d mass transfer ( I N ) Condensation of vapors from fixed gases ( 2 N ) Heat and mass transfer in falling film ( 3 N ) Evaporation from spheres (4N) Pure diffusional mass transfer in zone melting Heat transfer i n reacting systems ( 7P) Pipe flow heat transfer with reaction ( 2 )Effect of kinetics on thermal conductivity Diffusion a n d reaction ( 7 Q ) Absorption with reaction ( Z Q ) Density effects and axial dispersion in reactors ( 3 Q ) Diffusion controlled reaction ( 4 4 ) Diffusion in catalyst pellets (5Q, 6Q) Effect of diffusion on surface reactions ( 7 Q ) Interfacial turbulence in diffusion with reaction ( 8 Q ) Diffusion and reaction in porous catalysts Diffusion
(7R) (ZR, 3 R ) (4R, 4R) (6R,7 R ) (8R) (9R) (70R)
Pseudo steady state for moving boundary Multicomponent restricted diffusion Thermal diffusion separation from dilute solution Laminar boundary layer mass transfer Transient mass transfer in fixed bed Transfer in laminar jet Transient diffusion with large flux
of O R and computers to marketing were described ( 2 0 ) . An interesting article discussing the use of O R in deciding whether to rebuild or replace industrial equipment appeared ( 7 C ) . An excellent article on applying OR to decisions concerning new plant investment for manufacturing a new product was published (75C), which attacks the problem of deciding whether a plant should be built or not. The application of various OR techniques to the economic planning of production activities was discussed ( 72C).
Optimization
A book (75E) appeared for chemical engineers on dynamic programming. I t has been recommended as a textbook, and includes many example applications as well as computational methods. The same author published a paper (73E) on the application of Lagrangian multipliers in dynamic programming and another (74E) in which the method was applied to control of pyrolysis in a tubular reactor. Dynamic programming was used to determine optimal operation of a batch still (4E) and then compared with the methods of classical calculus of variations as well as the maximum principle method. This paper gives the development of a coinVOL. 5 6
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puter algorithm for a numerical solution. T w o separate papers by the same authors used the methods of dynamic programming ( 7 E ) and the Katz version of maximum principle (ZE)in the optimum design of a grain drier. The dynamic programming was combined with maximum principle for optimum design and operation in another paper ( 9 E ) . The same author used random search methods combined with the maximum principle in conjunction with an analog computer (TOE). For those interested in the maximum principle, a translation ( 7 IE) of Pontryagin’s book is now available. The problem of design and optimization for complex or recycle types of problems was treated in a series of papers and notes ( 5 4 7E, 8E) using modifications of the inaxinium principle. Two other papers (12E, T6E) which are not truly optimization but are related concerned the recycle or “chasing one’s tail” problem. Finally, two papers used the statistical search type approach. The first is primarily mathematical (6E) and the other ( 3 E ) is an application of fairly conventional response surface designs to a reaction. A book (77E) also appeared. Process Stability
A series of papers on chemical reactor stability has appeared (3F-7F. 9F, 701;). The first four deal with various two-phase systems and the last two have started on the use of Lyapunov’s method. These papers are r-ery complete. Another paper ( 3 F ) on Lyapunov’s method is a particularly readable piece (for such a topic) Two other papers use Lyapunov’s method. The first ( I F ) is concerned with stability with no control, the second (21;)introduces different modes of ferdback control. Finally, stability of adiabatic packed-bed reactors with axial mixing is treated (BF)in a paper which gives rather complete computational details. I
Residence-Time Distributions
A review of residence-tirnr paraineters appeared (2G) showing the relationships betu een them. ,4 very good analysis of mixing on a valve-tray and in downcoiners appeared ( 3 G ) . iVhat appears to be a novel (and perhaps very useful) approach was used in one article (7G). The method was based on an intensity function, actually a transform of the usual density function. Arquiiients for its advantages are given.
describes the simplification of boundary and initial condition problems by the introduction of dimensionless quantities. Other Application of Mathematics
It is impractical to critically review all of the applications of mathematics. Table I lists many papers which used mathematics in a n interesting, though not necessarily novel, way. They are considered worthwhile froin the point of view of the particular application and so are classified in that way. Many of them offer computer algorithms for useful applications.
BIBLIOGRAPHY General
(1A) A n d i e w . J. hl., Aloore, C. E., Jiinnecrh: C . F., Munsan, J . K., Chem. E n g . Prog. 60 (h’u. 3 ) , 57-64 (1964). (2.2) Aris, Rutherford, IND. Esc. CizEM. F U N ~ A U I : N ? A3,I .28-37 S (1964). (3.4) Behnken, D. W., Horowitz. J., Katz, S., Ibtd., 2,212-16 (1963). (4A) Brown. E. C., VonRosenberg, D. V., Ciiem. En?. Prog. 59, No. 10; 75-80 (1963). 15.4) Dahlin, E. B., hTelson,J. M., I b i d . , 60, S o . 3. 49-56 (Mar. 1964). (6A) Doland, E. C., Wolf. h4. B., INII. 3, 100-6 (1964). (7.4) F r m k , .4., Lapidus, L., Ciiem. Eng. P, (8A) f r a n t , R . G. E., Ihd., No. 3, 65-9 (1 964). (9A) Gonzalez, 2 . O., Spencer, E. H.. Ckem. Eng. Sci. 1 8 , 753-66 (1963). (10.4) Gruet, I. C . , Cliem. EfiE. Prog. 60, No. 3, 70-4 11964). (11.4) Hriny, R. L.,ed., Ckem. Eng. Prog. Simp. Ser. 59,No. 42 (1963). (12.4) Huckaba, C. E., Aionet, G. P., ed., Clirm. Eqg.Prog. S y n p . 59, No.46, (13.4) Jenson, V. G., JeHreys, G. V.. “Mcithernaricnl Method3 in Chemical Engineering,“ Academic, h-ew York, 1963. 114~1) Lamb, D. E., TVilhelm, R. H.: I N D .Euo, CIIIM. F u z n 4 ~ 1 : s . r2 ,~ 173-82 ~~ (19G3). (15.A) Pearson, J. R. A,, Richards, G. A t . : Skorz)-las, H., Cirem. Eng. Sn. 19, 82-5 (1964). (1GA) Stone, A. L.:Rrain, P.L.T., A.I.Ch.E. .J. 9, 681-8 (1963). ( 1 7 h ) Storey, S. H., Zeggeren, F. Van, Can. .I. Cham. E n g . 42, No. 2, 54--5 (1964). (182%) T a n g , Y . P., Himmelblau, D. M., A.I.C/i.E. .J. (19:1) Tinkler. E.R.,Nikifoi.uk, 1’. K., Cnn. .J. Chem. Statistics
(1B) Allcock. H. J., Jones, J . R., Michcl, J. G. I . . , ‘ T h e Nomogr,irn.” P i t m a n , New Y o r k (1963). (2B) Anderson, I>,Bryce, Cireni. I:ng. 70, h-0. 10. 173-8 (1963’1. (3B: Anscornbe, r. J., Tukey, J. W.. l’echnonie/riis 5 , 141-59 ( M a y 1963). (4R) Clementson, A. T.: Brit. Chem. Eng. 8, 564-5 ( A u s . 1963). ENG.CHEM. 5 5 , No. 5. 45-8 (1963). (5B) Daniel, Cuthrbert, IXD. (6B) Davies, 0 . L., 7’echnamelrirs 5 , 481-9 ( N a v . 1963). (7B) Dolby, James L.. Ibzd.,pp. 317-25. (8B) Dukler: A. E., tVicks, M,, Cleveland, R. G., II.I.C/I.E.J . 10, 38-43 (1946). (9B) Finch, R. N., V a n IYinkle, M., Iso. Frxc. C I I ~ W Prtoci~ss , DFSIGNI~EVFLGP. 3,106-16 (1964). (1OB) Freuncl, R. J., dm.Stolislicran, 13-15 (Dec. 1963). (1lB) Hays. G. E., Poska, F. L., Ciiem. E n g . Pmg. 60, 61-5 (Jan. 1964). (12B) Heiny, R. L., ed., Ciiem. Eng. Prog. Jimp. Srr. 59, No. 42, (1963). (13B) Hunter, J . S.. lbchnonie1,ics 6 , 41-55 (Feb. 1964). (14B) Hunrer, T\’. G., Mezaki, K., ISD. E N G CEIEhf. . 5 6 , 38-40 ( I f a r c h 1964). (15B) Hunter, TV. G., Mezaki,R., A.I.Ch.E.J . 10, 315-22 (bfay 1964). rii. Ciiem. E’ng. 8, 478-84 (July 1963). T h e Appliculion of Mathematical Statistics t o Chemical s k y Puhl. Co., Re‘iding, I’d. 1863. (188) R a t k o w s k y , D. A,: Brit. Chem. 2:‘ng. 9, 305-30 (May 1964). (19B) Smith, A , Rose, A , , ISD.EXG. CHrnr. 5 5 , No. 7, 25-R (1963). (ZOB) TVillianis, K. B.:Ibid., No. 6. 29-32.
Similarity Operations research-articles
A good example and application to flow in porous media by similarity and dimensional analysis was given (27). I t makes a good tutorial article. Another significant analysis of the use of similarity was presented (6H).The relationship between scaling and numerical analysis was discussed. Dimensional analysis was used in drag experiments ( I N ) and some pitfalls are illustrated. Similarity transformation was used in moving boundary problems with heat transfer in three papers (3H-5H) by the same authors. Another paper ( 7 H ) also 60
INDUSTRIAL A N D ENGINEERING CHEMISTRY
Disman, S.: Chem. Icng. Pro
