R. F. SWEENY R. S. DAVIS C. D. HENDRIX
ANNUAL REVIEW
Mathematics The solution o f dzferential equations l-y the jnite dzference methods is one o f the highlights of the reuiew o f the literature of applied mathematics in chemical engineering ne of the most important techniques of applied mathematics in chemical engineering is the solution of differential equations by the finite difference methods. A two-point pseudo Runge-Kutta method was presented ( 4 A ) and might be of use in some chemical engineering problems. Boundary value ordinary differential equations can be solved by finite difference approximations only (except for special cases such as linear equations) by some form of trial-and-error. A method for general nonlinear ordinary equations was given ( 9 A ) which first linearizes the equation by the Bellman and Kalaba procedure, solves the linearized equation; then from that solution a better linearization can be made, solved again, and so forth, until convergence is attained. I t has been suggested as especially useful in certain optimization problems using variational calculus; but should be quite generally useful. I n fact, the author found it could be used to help solve partial differential equations. Nonlinear partial differential equations solved by a simple forward finite difference tend to give unstable solutions. T h e instability can be overcome by putting the finite difference approximations in implicit form, such as the Crank-Nicholson form. This turns the problem into a trial-and-error, or iterative problem much like the boundary value ordinary equations. T h e linearization procedure is used, followed by a NewtonRaphson-type iteration. T h e results (8A) are especially good for the very common parabolic equations. I n
0
another paper (7.4) the instability of the Crank-Nicholson method when a derivative boundary is involved was discussed. Special problems arise when the location of a boundary moves with time, such as an ice melting situation. One approach to this problem was suggested in (75A). Whereas finite difference approximations are used to solve differential equations, many chemical engineering problems lead directly to true finite difference equations. Examples are plate distillation towers and successive extraction procedures. Two papers ( 6 4 72A) gave procedures for solving such problems. A number of books of special interest to chemical engineers were published during the past year. One was on general computational techniques ( 7 7A), another on systems engineering techniques (5A), mathematics and statistics ( 3 A ) , and a book on matrices and their application ( 2 4 ) . The papers of the joint London meeting symposium on applied mathematics were published ( I A ) in a package. Three rather novel applications defied classification. Vapor-liquid equilibria were calculated from solution vapor pressure measurements (70A), dispersed phase breakage was analyzed (73A), and equations were written and solved to describe the rate of discharge of electrically charged hydrocarbon liquids (74A). T h e many other applications have been classified into broad categories according to the type of application and are presented in Table I. Operations Research and Optimization
Only rarely does a text appear which offers broad perspective on and new insights into a field such as operations research. Such a work is “Today’s Information for Tomorrow’s Products-An Operations Research Approach” (4C) by George Chacko. This is a first-rate contribution to the operations research field VOL. 5 9
NO. 2 F E B R U A R Y 1 9 6 7 71
I
TABLE I. CLASSIFIED M A T H E M A T I C S APPLICATIONS
1
Simulation and Design
Ref.
Fluid Dynamics
Analytical solution for polycondensation i n C S T R , transient
Stokes resistance of a n arbitrary particle Vertical film flow i n entrance region
Stability of perforated plates Transients i n packed tubular reactors
Free convection of liquid metal from heated plate Entrance reyion flow
Unsteady-state operation of C S T R Transients in nuclear power reactor
Compressible and noncompressible fluid flow networks
Frequency response of plate absorber
Particle migration in shear fields
Short circuiting i n tubular reactors
Sedimentation of dilute suspensions
Transient convective transport, heat exchanger
Kumerical solution of boundary-layer problems
Filtration of dilute suspensions
Frictional h e a t in viscometers
Computer model for regenerative bed Clarification of suspensions
Particle a n d fluid diffusion i n homogeneous fluidization
Tutorial series on modeling
Wavc motion in falling film
Gas-solids flow-e.g.,
pneumatic conveying
Chemical reactor, analog
L a m i n a r flow of non-Newtonian Ellis fluid
Complex absorber
Viscoelastic flow i n inlet region of channel
Forced-flow h e a t exchanger dynamics
Confined wakes--Navier-Stokes
Batch grinding
T\vo-phase a n n u l a r flow
Equations Solved
Film stability Xvith heat and mass transfer
Residence T i m e Distribution ( A x i a l Dispersion)
Rheologically complex fluid i n helical flow
Packed absorption toxver
Flow in packed beds
Packed absorption to!ver
Multiphase systems--fliiiclization, precipitators, etc.
Multicomponent fixed-bed sorption
Couette flow i n a soap film
I o n exchange column, resistance i n particles
Transient flow around h o t wire
Nonlinear chromatography
Frictional heating i n cone a n d plate viscometer
Packed absorption tower
Couette flow between rotating cylinders
Cascade of mixed vessels
Equations of motion in porous media Falling film stability \ i i t h mass transfer
Reactors and Kinetics Neat-Mass Transfer
Absorption of ethylene in aqueous chlorine, mass transfer Book-analysis of chemical reactors Cascaded mixed stages with back flow between stages
Forced ancl free convection heat transfer
Mass a n d heat transport effect o n catalytic behavior
Porous gas diffusion electrodes
M o n t e Carlo simulation of reaction kinetics
Concentration polarization in reverse osmosis
C h a i n growth type problem for vinyl polymerization
Bubble grou th by dissolution
Long chain approximation i n free radical systems
Unsteady heat transfcr to slug flou;r
Analysis of chemical reactor stability a n d control-polymerization models
bVa11 heat coefficients in packed beds Multicomponent gas absorption
L a m i n a r dispersion in capillaries
( 1 771) ( 7211) ( 7.373)
H e a t a n d mass transfer effect on reaction i n catalyst Combustion \vith recirculation treated as a reactor Mass transfer Tvith chemical reaction i n tubular reactor Diffusion with consecutive heteroseneous reactions
(74H) ( 7 577) (TOHI
H e a t and mass transfer effect on reaction in catalyst Effect of linearization i n dynamic simulation of nonlinear reactor Noncatalytic gas-solid reaction with diffusion through product layer (ore-roasting)
( 7 711)
Free diffusion in binary systems Numerical solution of diKusion i n infinite m e d i u m H e a t a n d mass transfcr o n moving plate with suction or injection Effect of particle size distribution on mass transfer i n dispersions Fuel cell battery H e a t a n d mass transfer i n floti systems Diffusion in laminar boundary layer Ivith variable density
Step a n d pulse response analysis of reactor
(78H)
H e a t transfer i n tube with sinusoidal heat flux
Reactor stability by Liapunov direct method
(7SH) (20H) (2711) (2211) (2373) (24W (2517, (2671) (2771)
Transfer i n laminar flow through annulus
Averaging technique for stability analysis Performance of fouled catalyst pellets Steady-state stability of tubular reactor Lvith recycle Effectiveness factors for porous catalysts Effect of convective diffusion on surface catalytic reactions H e a t a n d mass transfer with reaction i n laminar flow systems Effect of internal diffusion on catalytic reactions
Viscous single d r o p extraction Laminar countercurrcnt double-pipe exchanger Forced convection of variable viscosity fluids M o m e n t u m , heat, mass transfer in turbulent non-Sewtonian boundary layers Freezing outside infinite isothermal cylinder Uncoupled multicomponent diffusion equations T h e r m a l history of a granule i n a rotary cooler
Noncatalytic gas-solid reaction with diffusion through product layer (ore-roasting)
( 2 W
Evaporation from falling saline water film Thermodynamics a n d heat transport-viscoelastic
Reaction mechanism in underground coal gasification
(ZgH) (3011)
N a t u r a l convection in a rectangular enclosure
D u a l diffusion-reaction coupling i n multicomponent systems
Gas flow reactors-hydrocarbon
(3717)
pyrolysis
(3211, (33H)
Reaction stability with h e a t a n d mass transfer i n particleLiapunov methods Batch reactor polymerization models
,
(34H) ‘
Robert F. Sweeny is Associate Professor at the Chemical Engineering Department of Villanova University; Robert S. Davis is President of Realtime Systems, Inc.; AUTHORS
72
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
I
Charles D . Hendrix is Engineer and Statistician, Union Carbide Chemicals Co.
and has material in it which has not appeared any place else. It deals with operations research rather than with an assortment of mathematical techniques. I t is more concerned with the strategies of policy-making rather than with the techniques of suboptimization. Illustrations and case histories are given from actual industrial situations. The book develops an integrated way of looking at widely varying entities that enter into decision-making with respect to tomorrow’s products, such as consumer products, durable goods, weapons systems, space technology and hardware, and services of all types. Chacko has used a unique approach for predicting the characteristics which tomorrow’s products must have in order to satisfy the demands which will occur. H e uses a concept similar to that of the periodic table of elements which has enabled us to predict the existence of elements which had not previously been discovered, but which we have subsequently found. The book bears careful study by all who are concerned with the planning function and who are taking a systems approach to corporate planning problems. Theory and practice are expertly blended in this lucid work by an international expert in the operations research (OR) field. An excellent review of optimization theory (75B) appeared this past year by Prof. Wilde of Stanford University. I t surveys recent developments in optimization techniques and presents a unified approach to the operational problems of a hypothetical chemical plant. I t shows the use of the classical approach to this type of problem followed by using a stationary point technique, then goes into geometric programming, optimum seeking methods, the use of Fibonacci search, linear programming and nonlinear programming, as well as geometric programming. This is an excellent article from which to get an overall view of the current state of the art of applied optimization in the chemical and petroleum fields. Table I1 lists other optimization articles. An article appeared (74B) on using mathematical schemes to help determine whether to sell a product immediately or to reprocess it for more attractive future marketing potential. This is an interesting concept and an unusual application of OR techniques.
I
TABLE I I .
T h e use of linear programming (LP) to solve the problem of optimal short-term financing was presented in (73B). There are many mechanisms for obtaining short-term cash to cover seasonal business needs. For example, one can use lines of credit, delay accounts payable, obtain term loads, or finance receivables. All of these strategies are compared for a given situation and the optimum one is determined using an LP technique. The use of dimensional analysis in operations research was discussed in ( I I B ) . The basic dimensions in OR are time, quantity, and money. Naddor describes the dimensions of quantities used in probability distributions, inventory models, LP, and queueing theory and gives examples of the use of dimensional analysis in solving these kinds of problems. The use of OR techniques in plant investment decisions was discussed in (8B). T h e present value and expected rate of return criteria in plant investment decisions were covered, and the parameters affecting the cash flow estimates using probability techniques were discussed. An interesting multiproduct, multifacility production and inventory model appeared in (76B). T h e article expands the single production and inventory model to the multiproduct plant model. A useful text on systems and simulation appeared (5C). A system is defined as a group of interdependent elements acting together to accomplish a predetermined task. T h e text covers industrial operations, inventory control, tools for data generation, the presentation of management data in mathematical form, program evaluation and review technique (PERT), market evaluation, traffic and cargo problems, steel simulation, and the simulation of industrial processes and manufacturing. I t is a highly recommended text on industrial systems. One of the classic texts on operations research appeared with a new revision this past year (2C). There is a new chapter added on linear programming and the theory of production. Management Texts
There is a growing need for clear and concise texts useful to managers to help them grasp an understanding
OPTIMIZATION ARTICLES AND BOOK
I
Ref. Recycle reaction example, m a x i m u m principle Ammonia synthesis, vector calculus Recycle reactor example, m a x i m u m principle Several methods compared, constrained problems O p t i m a l on-off control Simplex EVOP Recycle example Batch distillation T u b u l a r reactor temperature profiles, m a x i m u m principle Box Wilson-type EVOP Green’s functions, weak m a x i m u m principle Distributed parameter systems Tutorial review of variational methods Application of similitude integrals Book on continuous m a x i m u m principle CSTR’s with recycle-maximum principle
Effect of noise Decomposition of large systems Multicomponent distillation design Experimental design for plant testing Reactor startup, m a x i m u m principle Note concerning discrete maximum principle Note on d y n a m i c programming Mixed and tubular reactor-dynamic programming Time-optimal control with dead time-maximum principle Time-optimal control of discrete time systems Objective function for design T i m e optimal batch reactions Multibed reactor, d y n a m i c programming M i n i m u m time batch processing T i m e d o m a i n solution of steepest descent Dynamic optimization
VOL. 5 9
NO. 2
FEBRUARY 1967
73
of operations research techniques. We have attempted to group some of the better texts appearing this past year in this category. An excellent basic introduction and background text appeared ( Q C ) . The book has many excellent examples and can be readily followed with a minimum of math background. Industrial applications of operations research are presented in (7C). I t presents the significance of the analytical approach to decisionmaking and helps one become proficient in the application of quantitative models in operational systems. I t provides the math background required and presents models of production processes, equipment replacement, interest, depreciation, and the control of operations by use of procurement and inventory models. Waiting line operations, linear programming, and dynamic programming are also discussed. Teichroew (73C) presents the use of mathematical models in business problems using operations research techniques and weaves together a good balance between mathematical and business applications. The book discusses math models and mathematics including use of series, polynomials, exponentials, differential and integral equations, matrices, and linear programming. Applications presented are interest-calculation models, investment growth and advertising, determining optimum product mix, personnel assignment problems, transportation, and production and employment scheduling. T h e book has many excellent exercises and examples. The use of quantitative methods in management is described in (70C). The presentation is clear and understandable without a mathematical background, yet it is a sound and respectable one. I t covcrs the scientific method, break-even analysis, probability and statistics, inventory models, linear programming, matrix algebra, games theory, Markov analysis, and queueing. I t is clear and carries many good examples. PERT and CPM
The use of P E R T and critical path method (CPM) in project planning and scheduling is almost a classical O R application at this time. Significant savings in time and money have been realized by using these techniques. An excellent survey article appeared entitled “Resource Allocation in Project Network Models-A Survey’’ (5B). Three main categories of resource allocation problems are discussed : (1) scheduling of limited resources to meet the project schedule; ( 2 ) leveling of resources and usage-i.e., the commitment of resources at a constant rate; and (3) time/cost tradeoff, or the reduction of scheduled completion time through allocation of more resources to critical activities. As is proper in a survey, the author does not attempt to present the computational approach for each method discussed in detail; instead the basic concept and approach are examined. The bibliography is excellent and quite exhaustive u p to the time of writing. The article should serve as required reading for all those concerned with any phase of the resource allocation problem and the application of PERT/CPM network models. 74
INDUSTRIAL A N D ENGINEERING CHEMISTRY
An excellent book on construction scheduling and control was published (6C). Use of critical path methods in construction practice was discussed (1C). T h e book describes the techniques and gives examples of resource leveling and the use of computer US. manual techniques in pipeline and other construction projects. A do-ityourself text appeared on the critical path mcthod (72C). No background in linear programming is required and it uses words rather than formulas. I t gives examples and provides practice problems as well. T h e petroleum industry was the source of a CPM application to multiproject drafting (7OB) and to shutdown scheduling (ZB). A book entitled “Control and Management of Capital Projects” (8C) was written by an author who has had many years of experience in all aspects of project management at The M. \Vi. Kellogg Company. The text includes estimating, control, and management by owner corporations of cost, time, and resources on engineering-construction projects. I t gives the basic principles and techniques of project control and management. I t presents information on capital cost estimating and control, cost accounting, the use of PERT/CPSII and other techniques in project work. I t is highly recommended for anyone involved in plant construction projects. Resources Allocation
The use of OR techniques in situations in which individuals are faced with multiple activities among which they must allocate efforts was discussed in (3B). Optimal allocation processes are found for four different motivating structures, one involving profit maximization and the others involving performance goals. The location of warehouses under continuous economies of scale appears in (7B). T h e work was performed at Esso Research and Engineering and gives some cxcellent examples on nonconvex programming techniques. Two examples are given on warehouse location in an article well worth study by anyone concerned with the problems involved. A discussion, “Plant Location,” appeared (6B) on the use of integer programming techniques for solving a special class of discrete programming problems. T h e method has been used to solve practical plant location problems with u p to 50 plants. T h e plant location problem is basically a transportation problem with no constraints on the amount shipped from any source but there is a cost associated with each source and the plant cost does not vary linearly with the amount shipped from the plant. This is a fine article with good examples on plant location problems. Mathematical Programming
An introductory book on dynamic programming was published (77C). An article also appeared (72B) reporting on the use of dynamic programming for optimizing multibed adiabatic reactors in sequence and gave an example of the technique applied to the design of water-gas shift reaction systems. The optimal design
of multiple-effect evaporators using dynamic programming was described in (9B). The solution involves selecting the evaporator system to give a minimum of initial cost based on the minimum area criteria. Another interesting application of dynamic programming appeared in (4B). A model for combined production, distribution, and storage, as well as the warehouse location problem, is discussed. An actual example is given involving the production, distribution, and storage of fertilizer over a 52-week period. T h e subject of quadratic programming was presented in (3C). Quadratic programming is basically the problem of maximizing a concave quadratic function subject to linear and equality constraints. Examples are given such as portfolio selection, a constrained least squares regression technique, and applications to resource allocation. Integer programming was discussed in (7B). The article covers the methods used and computational techniques involved. Statistics
A self-instructing probability and statistics text for engineers has been published (75E),and a tutorial series was started on statistical design of experiments ( 7 E ) . Although not a book for beginners, (6E) is a strong contribution to probability. A recently announced statistical numerical systems computer program (9E) should be of general interest. Design of experiments as usual held the attention of several authors. A new class of factorial plans (4E) has been considered. I t is not always convenient to use a regular 2 P - 9 factorial plan in 8, 16, 32,. . . .experiments. An irregular fraction can be devised in many instances (ZOE). Other schemes include the PlackettBurman plans (77E). Finally a good example of response surface designs was given in a cellulose vinylation problem (ZE). Another fruitful application of statistics for chemical engineers is the general problem of fitting a model or equations to data. Five applications of least squares analysis of nonlinear models appeared (7OE-74E). One of these (72E) contrasts several approaches to nonlinear estimation. I n the absence of a “model,” one usually begins fitting a multivariable response surface with coefficients of linear terms, then quadratics, and so forth. Inverse polynomials may be a more profitable approach (76E). The response to an experiment may be expressed in numerous ways. For example, the response might be the shape of a stress-strain curve. A method is discussed (3E) for extracting the orthogonal principal components of a response curve and for relating these components to the independent variables. The analysis of covariance is considered in (8E). There is a rich, but bewildering, variety of opinion on multiple regression analysis. Some order is, however, being evolved (5E, 7E). I n other papers of interest, one author discussed the statistical properties of blends (79E) and considered confidence intervals on an economic basis (78E).
B I BL IOG RAPH Y General a n d Miscellaneous (1A) A.1.Ch.E.-1.Chem.E. Symposium, Series 4, “Application of Mathematical Models in Chemical Engineering Research, Design, and Production,” The Institution of Chemical Engineers, 16 Belgrave Square, London, S.W.1, England (1965). (2A) Amundson, N. R., “Mathematical Methods in Chemical EngineeringMatrices and their Applications,” Prentice-Hall, Inc., New Jersey, 1966. (3A) Brookes, C. J., et at. “Mathematics and Statistics for Students of Chemistry, Chemical Engineering, Chemical Technology, and Allied Subjects,” Wiley, New York, 1966. (4A) Byrne, G. D., Lambert, R. J., J.A.C.M. 13, (1) 114-23 (1766). (5A) Chestnut, Harold “Systems Engineering Tools,” Wiley, New York, 1965. (6A) Davison, R. R., A.Z.Ch.E. J . 11, 743-5 (1965). (7A) Keast, P., Mitchell, A. R., Computer J . 9 (l), 110-14 (1966). (8A) Lee, E. S., Chem. Eng. Sci. 21, 143-57 (1966). (9A) Zbid., pp. 183-94. (10A) Mixon F. 0. Gumowski, B., Carpenter, B. H., IND. ENG. CHEM. F U N D A M E N ~4,A4
