LEON LAPIDUS
d Mathematics Different subject areas in the field of applied mathematics are presented and discussed
his review follows the format of previous years in that the subT.ject areas have been divided into specific sections. T h e section titles are not inclusive, however, and certain articles may be in sections that do not seem to coincide with the explicit titles. T h e articles listed cover roughly the period of July 1969 to July 1970. Three new journals were started in the past year which are important here. These are The Chemical Engineering Journal, SZAM Journal of Mathematical A n a b s i s , and JL-umerical Methods in Enzineering. I n particular, the first one is of direct interest to chemists and chemical engineers. Of further interest is the July 1969 issue of IBM Journal o j Research and Decelopment, which is devoted to a series of papers on various aspects of operations research, and the January 1970 issue of Proceedings ZEEE: devoted to process control in the chemical industry. Duriiig the past year there have been interesting articles placing chemical kinetics in a mathematical framework ( 2 ) , on general algorithms (5), on singular perturbation theory and geophysics ( 7 0 ) , on the use of Green functions (15,28), on the Monte Carlo technique ( 7 7 ) ,on the use of Lie algebra and groups (4, 291,on the solution of PDE’s in state space ( 7 3 ) , and on the computer (22). Two most interesting papers languages PDEL (8) and I ~ P L dealing with nonlinear superposition theory have also come out (25, 38). Two new books on numerical analysis have been published (9, 7 4 ) and onc book ( 4 7 ) and a papcr (24)dcaling with computeroriented design methods. One new definitive book on Laplace transforms has appeared (20) and a paper dealing with generalized integral transforms (48),with Fourier transforms ( 7 , 43), and with transform techniques in PDE’s (76)have been published. Hybrid computers have been the sourcc of many papers including .\DI procedures (7), solution of specific differential equations (33, 37, 4 6 ) , statistical correlations (do), and error propagation (78, 42). Finally, a series of new books have been published dealing with various aspects of system theory or nonlinear systems (6, 79, 27, 23, 26,32, 36, 44, 45).
Models and Analysis
There have been many publications dealing with system models and the subsequent analysis of the model equations. I n many cases the models are a priori known and the major feature is the simulation or solution of the differential equations. Thus there has been work on recycle systems ( 4 2 A ) ,on reservoir models (86A), on series of CSTR’s ( 3 9 A ) and continuous models for sedimentation (64A), on air and water pollution models ( 7 4 A ) , on topological models ( 8 5 A ) and networks ( 8 8 A ) , on matching asymptotic expansions to represent systems ( 8 7 A ) : on simplifying linear systems ( 2 7 A , 3 7 A ) and decoupling them ( 9 6 A ) , on the use of the superposition integral ( 5 2 A ) and the Laplace transform for distributed systems (67A), on time-constants 147A) and Laplace transform analysis of distillation columns (89A),on filtration 68
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analysis (35A), finally on all types of models and how to determine them (54-4-56A). More specifically in the area of mass transfer problems there have been papers on reverse osmosis ( / A , 59A, 64A), on membrane diffusion (25Aj, and on transfer with chemical reaction ( 7 A , 36A). I n the chemical reaction area there have been papers on a differential scaling technique ( Z Z A ) , on lumping the rate equations (34A, 7 9 A ) , and on Bode diagrams for reaction systems ( 6 8 A ) . There has also been a most interesting analysis of nonlinear phenomena in the transport equations ( Z A ) . ll’ork on the solution of the fixed bed adsorption (chromatographic) equations ( 6 A , 73A, 9 8 A ) have been reported as well as pulse input separation (74A),chromatography i77A), and driving forces (87A) in packed beds. Extensive details on packed bed catalytic reactors have also been published with a detailed review of the state-of-the-art of appropriate models heading the various publications (29A). Others on transport phenomena in such reactors (70A, ITA, 76A), on general features of models ( X A , 30A, 49A, 6 3 A ) , on selectivity ( 9 A ) , on deactivation ( 4 A ) ,on poisoning (24A), and on the Legendre transformation ( 6 2 A ) have also been detailed. Two other broad-gauged papers on flow through porous media (28A, 9 5 A ) have outlined the specific model equations and subsequent solution. Finally w-e mcntion some most interesting work on mulitiplicity of steady-state solutions (uniqueness) ( 4 5 A , 46A, 73.4) in various physical situations and a series of papers on effectiveness factors ( 7 0 A ) , equilibrium states ( 2 7 A ) , and multiplicity of solutions and gradients in catalyst particles (75A). Along the lines of porous catalyst particles, there has been excellent analysis of uniqueness of the solutions ( 7 6 A ) , of maximum temperature rise (38A)>of nonsymmetric profiles (33A), of equilibrium states ( 3 7 A ) , and of poisoning effects ( 7 9 A ) . Residence-lime and Dispersions
Publications continue in the area of residence-time or moment analysis as a vehicle for model building, o n dispersion effects in reactors, and in those areas which involve probabilistic or stochastic phenomena. As an illustration, residence-time analysis for parameter estimation (35B>36R, &B), for fluctuating throughput ( Z Q B ) ,for laminar flow reactors ( 5 7 B ) ,for general conditions ( 7 B ) using an analog computer ( 3 8 B ) , and for a large fluid bed ( 3 0 B ) have been presented. Analogous frequency response for flow in packed beds (26B), for a back flow cell model 132B), for stagewise processes ( 3 9 B ) , for extraction systems ( 4 3 B ) , and for nonlinear systems (44B)have been discussed. The use of moment analysis in connection with residence-time representation has been the subject of many publications in an effort to elucidate mathematical models. Thus, there has been work on parameter estimation via moments using delay-models ( Z B , 3 B ) , in CSTR configurations ( I B , 5B),in flow in gas-liquid bcds (74B, 5 0 B ) , in porous solids (73B), in recycle systems ( 2 0 B ) ,
in adsorption (4B, 42B), and in grinding (25B) and settling ( 4 7 B ) processes. The detailed connection between the residence-time and moment analysis has also been detailed (72B, 27B, 37B). Various aspects of dispersion phenomena in packed beds (76B78B, 4 0 B ) , in flow in tubes with adsorption ( 8 B ) , and in general porous media (75B)have been analyzed. Probability aspects of scale-up techniques ( 9 B ) , or surface-renewal effects ( 2 7 B ) , of suspensions ( 2 4 B ) ,of fluid bed behavior (28B, 34B), and of reactor analysis with stochastic mixing (47B, 5 3 B ) have all been reported. Periodic Systems
Inherently or forced periodic systems have been of continued interest during the past year. Reports on the equilibrium theory of parapumping (IC), on columns operating in cycling modes ( 6 C ) , on radiating gases (75C), on oscillating reactors (2C, 3C, 8C), on crystallizers ( 7 4 C ) , on gravity-oriented systems ( I Z C ) , on the response of viscoelastic materials to deformations (27C), and on pulsating flows (4C) in fluid media (QC, 70C, 76C, 79C) have been made. Further work on transport coupling leading to oscillations (5C), on residence-time behavior in oscillating reactors (73C), and on the general theory of periodicity (7C), as applied to Duffing’s equation ( I IC), to Mathieu’s equation (79C), and to Riccati-type equations (78C)have also been detailed. Optimization
This area continues to receive the attention of many research workers. Two new books ( 7 7 0 , 4 4 0 ) , an excellent general survey (450), and consideration of multiple objective functions ( 5 1 0 ) have been published. I n addition, generalized penalty function concepts have been reported (50,150, 2 6 0 ) , and the general use of inequalities ( 7 3 0 ) and Lagrange multipliers ( 7 6 0 ) as well as papers on general constrained and unconstrained minima ( 7 0 , 2 7 0 , 3 7 0 , 4 7 0 ) have all been reported. An interesting paper detailing a one-dimensional search procedure ( 7 7 0 ) has been presented as well as two general survey papers on gradient methods (230, 4 0 0 ) . Other papers on 280, 2 9 0 ) as well as gradient methods have been reported (40, one on convergence questions ( 6 7 0 ) and the use of a memory gradient ( 3 6 0 ) . The use of the Davidson-Fletcher-Powell algorithm for control ( 7 0 ) , a variable-metric generalization (780, 790, Z I D ) , and the use of Newton’s method ( 6 D )and its step size control ( 5 9 0 ) have been specified. Further results on conjugate gradients (520-550) have been reported as well as a general paper dealing with computational experience with a variety of second-order methods ( 3 8 0 ) . I n the applications area, work on the catalytic oxidation of SO2 ( 8 0 ) ,on graph theory and tearing ( l o o ) ,on product planning ( 7 4 0 , 3 0 0 ) , on multicycle steam simulation ( 2 5 0 ) , on catalyst makeup ( 3 7 0 ) , on minimum cost steel rolling ( 3 2 0 ) , on an adiabatic reactor sequence ( 3 9 0 ) , on batch distillation ( 4 8 0 ) , and on planning experiments (571)) have all been detailed within the framework of optimization problems. Dynamic Programming and Invariant Imbedding
Dynamic programming continues to be of interest to many workers. Reduction of dimensionality problems (5E, 74E), the application to nonserial systems (ZE, 7 7 E ) , the connection between the Bellman and Krotov functions ( 7 E ) , and the concept of differential dynamic programming (8E, 9 E ) have all been detailed. Applications to the optimization of an ammonia plant ( I E ) , to terminal guidance ( 4 E ) , to stochastic service systems ( 6 E ) , to cluster analysis (IOE), and to the cold-shot adiabatic reactor ( 7 3 E ) have been presented. Invariant imbedding techniques have been used in filtering processes ( 7 F ) and connected to optimal control ( 2 F ) and the calculus of variations theory (3F). Computational aspects for countercurrent processes ( 4 F ) and for two-point boundary value problems (517)have been examined. Mathematical Programming
While the publications to be cited here might conveniently fit within the Optimit~tionarea above, we have chosen to cite these in a separate section. Thus, there have been papers dealing with stability questions in nonlinear programming (5G),with the use of mathematical programming for optimal control (22G, 23G), with the use of Lagrange multipliers ( 6 G ) , and two recent excellent books (76G, 25G) in the area. There have been papers on accelerating LP programs (8G), on
duality in LP (70G), on quadratic (3G), geometric (4G), and interval programming (79G), and on generalized Kuhn-Tucker conditions ( 7 7G). There have been publications on converting a constrained to a n unconstrained problem ( 7 G ) , on convexity and conjugate directions (73G), on the use of penalty functions ( I G ) , and on linearizing nonlinear constraints to form a LP problem (2G). In the area of linear constrained problems, there has been an application of Rosen’s gradient projection method for control (77G), an accelerating gradient projection method (ZUG), and some sufficiency conditions (26G). Applications to shortest path problems in networks (74G), to branch-and-bound algorithms ( 75G), to heuristic type programs (78G), and to plant location (27G) have also been reported. Stability
Here we are concerned with general concepts for determining stability or analyzing the behavior of special systems to perturbations in some variable. Thus there has been a n examination of Q use of spectral theory and linear system stability (ZH, 65H) V ~ the topology ( 7 7 H ) ,an examination of limit cycles ( 3 3 H ) uzu topology ( 3 4 H ) , a characterization of stability in a control context (48H, 52H, 66H, 72H, 73H), and the use of the comparison theorem (42H). Other papers have analyzed stability as related to transient bounds (50H)and in a purely computational mode ( 6 9 H ) . Liapunov’s method has received a considerable amount of attention this year with papers on Liapunov function generation ( 8 H ) and construction (IOH, 74H, 57H), on the calculation of closed loop eigenvalues ( 5 8 H ) and in adaptive systems (59H, 6 0 H ) , on the connection with Poisson-type stability ( 7 2 H ) , on the involvement of time lags ( 5 H ) ,on the generation of Liapunov functions for CSTR’s with chemical reaction ( 6 2 H ) , and as illustrated by the second part of a review article ( 6 7 H ) . Distributed type Liapunov functionals have been examined ( 3 H , 77H, 37H, 4 7 H ) as well as the comparison of many methods of stability calculation including Liapunov’s functional ( 3 6 H ) . An analogous energy principle has also been used for fluid stability
(SH, 20H). I n terms of specific systems, the packed bed has been analyzed in many ways for stability ( I H , 32H, 35H, 53H-55H, 6 8 H ) including the use of a simplified model (45H). Other systems have been those involving chemical reactions ( 2 6 H ) ,a heat exchanger system ( 4 H ) , inviscid fluids ( 7 H ) , catalytic wires (73H), stratified flow ( 7 5 H ) , oscillating or periodic systems (27H, 22H, 39H), the nonlinear Navier-Stokes equations ( 2 3 H ) , plane Couette flow (25H, 2 8 H ) , heated fluid layers (3UH), flow with a free surface ( 4 3 H ) , spinning a threadline (56H), and a n adiabatic reactor (7UH). Optimal Control
As in previous years, the number of mathematically oriented papers on optimal control exceeds those in any other area. I n particular, there have been some excellent review papers on control of lumped systems (7572, 1842), of distributed systems (82),and of stochastic systems (7871) plus some new books (672, 842) including one (7281) which presents computer programs for solving control problems. There has been a monumental and significant paper on the necessary conditions for optimality (7331), one on the discrete calculus of variations (232), a new book on the calculus of variations (421), two papers on adaptive control systems (7042, 7392),papers on assuring that discrete control systems approximate continuous systems (781, 321, 7402) (a most important area), a paper on the numerical solution of the state equations (381), on a differential-difference equation minimum principle ( 792), on integral equation control (7742), and a number of papers dealing with delay systems (271, 681, 872, 7452, 7791) of various types. There have also been publications on games (241,832, 981, 7552), on performance bounds for uncertain systems (786Z), o n including the cost of observation in the performance index (62), on free final time problems (7071), on decomposition principles ( 7 1 , 561) including multilevel con-epts (7302), on problems with incomplete state measurements (331, 342, 452, 7481),and on aspects of nonlinear controllability and observability ( 771,261). Specific constraints have been considered (41, 7721) including inequalities (472) and multiple constraints (7621) and the singular control problem (571) including necessary conditions (772),computational aspects (722, 731) and a conjugate-point condition (7561) has been detailed. Minimum time problems have also been investigated (641, 7781) including a discrete regulator (431), one with special constraints (802, 85Z), others with bounded control rates (7162, 7592), one which looks a t the switching times as a VOL. 6 2
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NOVEMBER 1970
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problem in root location (7772),and one which uses a time interval adjustment algorithm (7972). The linear regulator (linear-quadratic) problem has also been looked a t from many points of view. These include computational aspects (442), equivalence of different performance indices (962), stochastic questions (7002), the construction of the control (7052), with time-multiplied indices ( 7242), with hard terminal constraints (7492), and a special form which leads to a simplified version of the Riccati equation (7571). A number of publications have been concerned with the sensitivity (621, 792, 7461, 7531), or insensitivity of optimal control (721) to such features as parameter variations (7472). Such items as spectral factorization ( 751), generation of sensitivity coefficients ( 7821), performance sensitivity (881), minimum time sensitivity (952),and the introduction of sensitivity into a closed-loop control (942)have also been elucidated. I n the purely computational algorithm area, there have been papers of interest (351, 361, 571). I n particular, there were reports on a new algorithm (91, 691, 772), on the sweep or Newton-Raphson iteration in function space method (401), on contraction mapping (492),on one successive approximation approach (7351) and another iterating on the initial adjoint variables (742); on the use of Davidson’s method (731, 7701) and mathematical programming (7691), on a second variation algorithm (7702) with a n equivalent application to minimum fuel problems (7662), on the second order necessary conditions (7372), on the use of the conjugate gradient algorithm (7061),and a minimum energy algorithm
(7411). I n addition to all of the above publications, there has been work in two fast growing areas, namely That of suboptimal control and that of optimal control of distributed systems. I n the suboptimal area (31, 271, 342, 552, 7092, 7732, 7751, 7271, 7921), there have been details of a suboptimal regulator (ZOZ), of incomplete feedback (7232), of suboptimality with sensitivity and filtering (7271, 7671), and as applied to a reactor system (7382) and stochastic problems (7651). The distributed parameter control area (51,221, 372, 542, 581, 592, 871, 972, 1631) has seen a n examination of switching analysis (762), controllability and observability (281, 602, 662), ,periodic systems (462), successive approximation methods (821), suboptimality (891), analog simulation (971), minimum time control ( I I Z Z ) , various bounded inputs ( 7771, 7881, 1901), and application to stochastic (7742) and the vibrating plate or string (902,7222, 7892) problems. Finally, in the applications area we mention the publications on CSTR systems (702,7442, 7582),on the use of positive feedback for reactors (531), on polymerization reactors (671), on tubular (7081)and fixed bed (7682) reactors, on distillation systems ( 7 791, 7201), including minimum time batch distillation (736Z),and on a multibed deactivated catalyst system (7371). I n addition there have been reports on a questing controller (7802),on an optimal temperature problem (7752),and on a medical diagnostic process that is treated as a stochastic control process (7032). Identification
The papers dealing with the highly important area of model building or identification of systems have continued a t a furious rate. Thus there has been work reported on the use of Hermite polynomials ( 8 J , 7 6 J ) , on Volterra series ( 6 7 J ) ,and on piecewise continuous expansions ( 7 3 J ) as black-box representation of systems. Further papers on the use of the fast Fourier transform (75J),on general learning control systems ( 3 I J ) ,on the question of observers ( 6 7 J ) , and unmeasurable state variables ( 3 4 J ) , on linear system identification with long periods of observation ( 3 9 J ) , on converting observations to control signals ( 4 0 J ) , and on identification from input-output records (45J,4 9 5 ) have come out. Filtering techniques have been examined in a splendid article ( 7 3 J ) and a n excellent book ( 5 3 J ) , in terms of computational algorithms ( 6 5 J ) and as applied to distributed systems (76J, 7 7 J ) and reliability ( 7 7 J ) . Minimum realization algorithms ( 6 4 J ) have also been proposed, as well as connected to moment analysis (725). T h e greatest number of publications have dealt with parameter estimation. Here we have papers dealing with the determination of chemical rate constants in rate equations ( 4 J , 70J, 77J, 245,
Leon Lajidus is Professor and Chairman of the Dejartment of Chemical Engineering, Princeton Universib, Princeton, N . J . 08540. AUTHOR
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Z S J , ZSJ, 44J, 50J, 57J, 7 0 J ) (for homogeneous and heterogeneous situations), the use of Bayesian techniques ( 5 4 J ) , the use of maximum likelihood techniques ( 3 J , 9J), the use of sequential estimation (78J, 22J, 58J),the determination of parameters in a , general concepts in nonlinear estimadispersion media ( 7 Q J ) and tion ( 6 8 5 )and identification (48J, 7 4 J ) . More specifically, there have been reports on many uses of quasilinearization ( 7 J , 74J, 23J, 47J, 6 0 J ) and a monumental paper testing almost all available techniques ( 5 4 . Least-square fitting has also been reported (33J,42J, 7 5 J ) . Finally, there have been papers dealing with identification of distributed systems in the face of a finite number of noisy measurement points ( 6 J , 30J, 5 6 J ) , oia finite differencing techniques (ZOJ), in the Laplace transform domain ( 8 7 J ) ,and for identifying Green’s functions ( 8 3 J ) . Numerical-Approximalion
Here we are concerned with some of the most fundamental problems in numerical analysis, namely, how to approximate functions or data, how to calculate derivatives and integrals, and how to solve integral equations. Papers on Chebyshev interpolation have appeared (3K, 6K, 2 6 K ) as well as on the newer spline function approximation ( 2 K ) . I n the latter category are the use of cubic splines (70K, 79K, 2 8 K ) for boundary-value problems (8K) and least-square fitting (23K),and the extension to two-dimensional splines ( 2 7 K ) . Methods for calculating derivatives ( 9 K ) and the convergence of discrete orthogonal polynomials (37K) have also been presented. Numerical details for inverting Laplace transforms ( 7 K , 2 Z K ) have been detailed and a handbook on the subject has been published ( 1 4 K ) . Intergral equations have been attacked (#K, 16K, 2 9 K ) with Chebyshev series ( 2 5 K ) and when singularities exist ( 7 K ) . I n addition, Filon ( 5 K ) and Clenshaw-Curtis ( 7 2 K ) quadrature methods have been analyzed along with adaptive trapezoidal ( 2 4 K ) and Newton-Cotes ( 7 3 K ) formulas. Repeated subdivision has been suggested ( 2 1 K ) as a means for evaluating a n integral, as well as analyzing the errors in the efficient Romberg method ( 7 7 K ) . Along these latter lines, one paper (30K) has shown how the Romberg method may be used to integrate the state equations in optimal control problems. Numerical-linear
Algebra
Since the manipulation of matrix equations is now common, papers have been written relating to many features of linear algebra. Thus there have been papers concerning optimally scaling matrices (ZL),symbol manipulation (.?L),generation of test matrices (72L, 38L), control systems ( 4 L ) , calculation of inverse matrices (39L),an interesting discussion on Gaussian elimination (42L), and some features of general matrix manipulation (46L). A number of papers have dealt with the pseudoinverse (7L, 22L) with special application to triangular ( 2 3 L ) and bordered (32L) matrices. Others have considered features of tridiagonal (8L, 33L) and band (74L) matrices and various aspects of eigenvalue and eigenvector evaluation (5L, 75L,25L). Authors have also been concerned with computation of the exponential or transition matrix in terms of Pade approximations ( 7 L ) or in general terms (77L, 78L), and the solution of the Riccati or steady-state Liapunov equation (24L, 34L, 4OL). Special reference to Davidson’s method (2OL) and the singular case (47L)have been made in the Liapunov equation solution. Applications in the area of multiple chemical rate equations (9L, 43L), of plate distillation and absorption ( 1 7 L ) columns, decoupling of distillation or other systems ( 7 9 L ) ,of binary distillation (29L), and of three-dimensional gas dynamics (37L) have all appeared. Numerical-Boundary-Value
Problems
Since multipoint (or two-point) boundary-value problems occur so frequently, there have been continued efforts on efficient numerical solution methods. Thus the Galerkin method ( 2 M ) has been proposed and applied to elliptic PDE’s ( 3 M ) ,the RayelighRitz method ( 7 8 M ) and cubic splines ( I M , 7 M ) outlined; the Cesaro summation (6,M), the use of particular solutions ( 7 0 M ) , and the Riccati transformarion (71.M) have been analyzed. One paper has shown how these problems evolve from chemical reactor systems ( 4 M ) , while another has analyzed a recycle reactor ( 1 4 M ) . Other works of interest have been the use of asymptotic
solutions ( I Z M ) , a level control method (5M), and the use of singular perturbations (73M, 75M). Numerical-Ordinary
Differential Equations
The publications in this important area have continued over the past year with some new formulas (ZN,7 7 N ) , a detailed analysis of the very efficient fifth-order Runge-Kutta forms ( 6 N ) and Runge-Kutta error estimates ( 7 3 N ) , with a means for automatic step change ( 7 N ) , with the use of spline functions (8N) and invariant imbedding ( 9 N ) , with the development of new hybrid correctors ( Z Z N ) , and with consistency and convergence of the numerical soultion ( 3 7 N ) . Implicit integration formulas have been further developed (32iV) with particular attention to the Runge-Kutta (74N) and the Rosebrock semi-implicit ( 7 5 N ) forms. Special formulas using Pade approximations ( 3 N ) , network combing ( 4 N ) , and implicit forms ( 3 7 N )have also been developed for the most important class of stzf problems (problems with widely separated timeconstants). Formulas with the feature of A-stability have been developed ( I N , 2 4 N ) , stability questions have been reviewed ( I O N ) , as well as new predictor-correctors ( 3 N , 2 8 N ) outlined, the extension of stability evaluation to variable coefficient problems made ( 7 9 N ) , and new formulas presented with maximum regions of both stability and accuracy (23N, 27N). Applications to problems with scale deposit in heat exchangers ( 7 6 N ) , to gas absorption ( 2 6 N ) ,and to multicomponent diffusion (3ON)have also been detailed. Numerical-Partial
Differential Equations
The solution of, and application to, many physical problems which involve PDE’s continues at a furious rate. I n the hyperbolic PDE area, there have been papers on iterative solutions ( I O ) , on the use of split difference equations ( 2 6 0 , 510), and on the solution for large times ( 3 0 0 ) . I n the elliptic PDE area, there have been papers on a n inverse problem ( 3 0 ) ,on the unusual case of a free boundary condition ( S O ) , on a review of available methods (ZOO), on boundary singularities (220 ), on high accuracy methods ( 3 2 0 ) ,and on the use of LP algorithms ( 5 3 0 ) . Parabolic PDE’s have received the most attention, with papers on the Ritz (20) and other variational methods (TOO), on a threedimensional mixed implicit-explicit method ( 2 4 0 ) , on three level forms (370),on rates of Convergence (330),on possible periodicity in the solution ( 3 5 0 ) , on asymptotic solutions ( 3 7 0 ) , on various AD1 methods ( 3 9 0 ) , 011 unbalanced forms ( 4 7 0 ) ,on an analog procedure ( 4 2 0 ) , on the Crank-Nicholson method (440),on techniques for refining the difference mesh ( 4 6 0 ) , on stability (620), and in particular the DuFort-Frankel method ( 6 7 0 ) , and on the use of a double-collocation method ( 6 3 0 ) . Other authors have also reported on curved meshes ( 6 0 ) , on AD1 or S O R methods (70, 7 7 0 ) , especially in reservoir flow ( 5 2 0 , 580), on implicit methods in two-phase flow (80, 770), on moving boundary problems ( E O ) , on the use of accelerated AD1 methods ( 2 7 0 ) ,on block-implicit methods ( 5 6 0 ) ,and on the effect of the use of wrong boundary conditions ( 4 5 0 ) . I n the application area, there have been papers dealing with PDE’s derived from low Reynolds number flow (do), boundary layer equations (go), cavity flow (720),the Navier-Stokes equations (130, 290, 6 8 0 ) , absorption ( 7 4 0 ) and mass transfer (750), three-dimensional flow ( 7 8 0 ) , diffusion ( 7 9 0 ) , pipe flow (280, 4OO), flow from an orifice ( 3 4 0 ) , multiphase flow ( 3 8 0 ) , filament generation ( 4 3 0 ) , catalyst regeneration ( 4 7 0 ) , oxygen transport in tissues (570, 6 7 0 ) , condensation polymerization ( 5 5 0 ) , gas piping systems (6OO), shock generation (650), and stochastic diffusion ( 6 6 0 ) . Numerical-Iteration
During the past year there have been general papers dealing with the solution of simultaneous nonlinear algebraic equations ( 3 P ) and errors in them ( 9 P ) , with the use of Newton’s method for PDE’s ( 4 P ) , with the question of round-off as it affects the convergence properties in an iteration ( 8 P ) , and with the use of Lehmer’s method ( 7 0 P ) for finding roots of polynomials. I n addition, there has been an excellent survey on the use of the root locus method ( 7 P ) . There have also been a number of interesting papers dealing with questions relating to solving large distillation systems (ZP, I I P , 7 2 ) . Further work in this area dealing with questions of convergence of the equations ( I P ) ,with the use of Newton’s method ( 5 P ) , and with an acceleration technique for speeding up convergence ( 6 P )have been reported.
Acknowledgmenl
As in previous years, this Review would not have been possible without the help of a number of graduate students a t Princeton University. This year these students included: W. P. Burgess, V. J. Corbo, T. F. Edgar, T. L. Englert, V. P. Leung, T. K. Pho, A. V. Prabhu, R. H. Rossen, E. J. Schlossmacher, J. G. Vermeychuk, S. D. Weinrich, and Y. K. Yang.
REFER ENCES GENERAL (1) Amba-Rao, C. L., “Fourier Transform Methods in Elasticity Problems and a n Application,” J . Franklin Inst., 287, 241 (1969). (2) Aris, R., “Chemical Kinetics and the Ecology of Mathematics,” Bmer. Sci., 58,419 (1970). (3) Banerii, R . E., “The State Space in Systems Theory,” Inform. Sci., 1, 335 (1969). (4) Belinfante, J. G., and Kolman, B., “An Introduction to Lie Groups and Lie Algebra,”SZAM Reo., 11, 510 (1969). (5) Bellman R . “A Function is a Mapping-Plus a Class of Algorithms,” Inform. Sci., 1, 257’ (19k9). (6) Bellman, R., “Methods of Nonlinear Analysis,” Academic Press, New York, N . Y., 1970. (7) Bishop, K . A,, and Green, D. W “ H brid Computer Implementation of the AD1 Procedure for the Solution of $wo-gimensional Parabolic PDE,” AZChE J., 16, 139 (1970). (8) Cardenas, A . F., and Karplus, W. J., “PDEL-A Language for P.D.E.’s,” Comm. ACIM, 13,184 (1970). (9) ,CarnEhan, B., Luther, H . A., and Wilkes, J. D., “Applied Numerical Techniques, . . John Wiley, New York, N. Y., 1969. (10) Carrier, G . F., “Singular Perturbation Theory and Geophysics,” SZAA4 Rev., 12, 175 (1970). (11) Chan, S . K., “ T h e Serial Solution of the Diffusion Equation Using Nonstandard Hybrid Tecliniques,” ZEEE Trans. Corn,/mt., 18, 786 (1969). (12) s u e n o d , M., and Durling, A , , “A Discrete-Time Approach for System Analysis, AcadernicPress. New York, N. Y., 1969. (13) Dodd, C . W., and Gupta, S. C., “Solution of Some Partial Differential Equations using State Space Techniques,” J . Franklin Inst., 287, 305 (1 969). (14) Fox, L., and Mayers, D. F., “Computing Methods for Scientists and Engineers,’’ Oxford University Press, London, 1968. (15) Gallof, S . , “Numerical Solution and Utilization of Greens Functions,” Numcr. AlethodsEng., 1, 169 (1969). (1 6 ) Gutterman, M., “An Operational Method in Partial Diff erential Equations,” SIAM J . Abbl. Math.., 17., 468 119691. , (17) Halton, J. H., “ A Retrospective and Prospective Survey of the Monte Carlo Method,” S I A M Rev., 12, 1, 1970. (18) Hammond, J. L., and Alford, C. O., “Sampling Errors in Closed-Loop Hybrid Computer Programs,” Simulation, 13, 307 (1969). (1 9 ) Himmelblau, D. M . , “Process Analysis by Statistical Methods,” John Wiley, New York, N. Y., 1970. (20) Jaeger, J. C., and Newstead, G., “An Introduction to the Laplace Transformation: With Engineering Applications,” Barnes and Noble, New York, h’.Y., 1969. (21) Kalman, R . E., Falb, P. L., and Arbib, M. A,, “Topics in Mathematical System Theory,” McGraw-Hill, New York, N. Y , 1970. (22) Kolsky, H. G., “Problem FormulationJJsing APL,” Z R M Syst. J.,8,204 (1 9 6 9 ) . (23) Kupperman, R . H., and Smith, H . A,, ”Mathematical Foundations of Systems Analysis,” Vol. 1, Addison-Wesley, Reading, Mass., 1969. (24) Lee, K. F., Masso, A. H . , and Rudd, D . F., “Branch and Bound Synthesis of Integrated Process Dcsigns,” IND. CNO.Cmhi., FUNDAM., 9, 48 (1970). (25) Levin, S. A,, “Principle of Nonlinear Superposition,” J M A A , 30, 197 (1970). (26) Lewis, L. J., Reynolds, D . K., Bergseth, F . R., and Alexandro, F. J., Jr., “Linear Systems for Engineers,” McGraw-Hill, New York, N . Y . , 1970. (27) Little, J. D. C., and Lodish, L. M., “A Media Planning Calculus,” Oper. Res., 17, 1 (1969). (28) Loud, M’.S “Some Examples of Generalized Green’s Functions and Green’s Matrices,” S I A i 4 Rev., 12, 194 (1970). (29) Miller, W., “Lie Theory and Difference Equations,” J M A A , 28,383 (1969). (30) Morse, P . M., and Elston, C., “A Probabilistic Model for Obsolescence,” Ofier. Res., 17, 36 (1969). (31) Nash, R. T . , “Event Classification,” ibid., p 70 (32) Perkins, W . R., and Cruz, J. B., “Engineering of Dynamic Systems,” John Wilev, New York, N. Y., 1969. (33) Poulsen, PIT.J., “Distributed-parameter Problem Solved o n a Hybrid Computer bv a Modified Function Storaee Technioue.” Simulation., 13., 187 11969). , . (34) Prater. S. V., “Numerical Analysis in a Ph.D. Computer Science Program,” Comm. A C M , 12,661 (1969). (35) Sicgel, A. I., and Wolf, J. J., “Man-Machine Simulation Models,” WileyInterscience, New York, N. Y., 1969. (36) Siliak, D., “Nonlinear Systems-Their Parameter Analysis and Design,” John Wiley, New York, N. Y., 1969. (37) Silvey, T. I., and Barker, E. J. R., “Hybrid Computing Techniques for Solving Parabolic and Hvperbolic PDE’s,” Cornput. J.,13. 164 (1970). .. (38) Spijkcr, M . N., “Superposition in Linear and Nonlinear O.D.E.,” J M A A , 30, 206 (1970). (39) Swainston, J. B., “Hybrid Computation in the Process Industries,” Brit. Chem. Enp .. 15.661 11970). , (40) Tacker, E . C., and Linton, T. D., “Hybrid Computation of .4utocorrelation Functions,” Simulation, 14, 137 (1970). (41) Tribus, M., “Rational Descriptions, Decisions, and Designs,” Pergamon Press, Elmsford, N . Y., 1969. (42) Vensteenkiste, G., “Study yf the Dynamic Errors Introduced by the Delays Inherent in Hybrid Computers, AZCA, 11,226 (1969). (43) Warmbrod, G . K., “ T h e Distributional Finite Fourier Transform,” SZAM J . AppC. Math., 17, 930 (1969). (44) Ward, J. R., and Strum, R . D., “State Variable Analysis,” Prentice-Hall, Englewood Cliffs, N. J., 1970. (45) White, H. J., and Tauber, S., “Systems Analysis,” W. B. Saunders Co., Philadelphia, Pa., 1969. (46) Wozny, M,, J., Raytell!, P. :., and Bailey, R . E., “Hybrid Computer Simulation of Zenon Spatial Oscillations, Simulation, 14, 239 (1970). I
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(47) Ying, C. C., “A Model of Adaptive Team Decision,” Oper. R e f , , 17, No. 5 (1969). (48) Zemanian, A. H., “Generalized Integral Transformations,” Wiley-Interscience, New York, N . Y., 1968.
MODELS AND ANALYSIS (1A) Agrawal, J. P., and Sourirajan, S.; “Reverse Osmosis,” I N D .ENG. CIiEhl., 61 ( l l ) , 62 (1969). (2A) Ames, W.!., “Recent Developmpnts in the Nonlinear Equations of Transport Processes, IND.ENC.CH-LM., FUNDAM., 8, 522 (1969). (3A) Barak, A,, and Dagan, G., “An Analytical Investigation of the Flow in the Saturated Zone ofIce Counterbashers,” A I C h E J . , 16, 9 (1970). (4A) Bischoff, K. B., “General Solution of Equations Representing Effects of Catalyst DeactiQation in Fixed-Bed Reactors,” I K D .ENC. CHEM.,FUNDAM., 8, 665 (1969). (SA) Bogomoiov, A . M . , and Tverdokhlebov, V. A . , “Diagnosability of Complex Systems,” Automat. Remote Contr. (USSR). . .. 1969., 1519. (6A) Bowen, J. H., and Lacey, C. T., “A Single Pellet Prediction of Fixed Bed Behavior,” Chem. Eng.Sct., 24, 965 (1969). (7A) Brunson, R . J., and Wellek, R . M . “Determination of the Enhancement Factor for Mass Transfer with an Instantkeously Fast Chemical Reaction,” ibid., 25, 904 (1970). (8.4) Burnett, J. C., and Himmelblau, D. M., “ T h e Effect of Surface Active Agents on Interphase iMassTransfer,” AIChE J . , 16, 185 (1970). (9A) Campbell, D. R., and Wojciechowski B. W. “Selectivity of Aging Catalvst in Static, Moving and rluidized Bed Reactdrs,” Ca;. J . Chem. Eng., 48, 224 (1970). (10A) Carberry, J. J., “Heat and Mass Diffusional Intrusions in Catalytic Reactor Behavior,” Catal. K E U..3. . , 61 (1969). (11A) Cafberry, I. J., and White, D., “ O n the Role of Transport Phenomena in Catalytic Reactor Behavior,” I N D .ENC. CHEM.,61 (7), 27 (1 969). (12A) Chen, F. C., Griskey, R . G., and Beyer, G . H., “Thermally Induced Solid State Polycondensation,” AIChE J . , 15, 860 (1969). (13A) Chi, C. W., and IYasan, D. T., “Fixed Red .4dsorption Drying,” itzd., 16, 23 (1970). (14A) Conder, J. R., and Purnell, I. H., “Separation and Throughput in Production and Preparative Chromatography,” Chem. Eng. Sci., 25,353 (1970). (15A) Copelowitz, I., and Aris, R., “Communications on the Theory of Diffusion and Reaction. V I . T h e Effectiveness of Spherical Catalyst Particles in Steep External Gradients. V. Findings and Conjectures Concerning the Multiplicity of Solutions,” ibid., pp 885, 906. (16.4) Cyesswell, D. L., “ O n the Uniqueness of Stea Involving both Intraphase and Interphase Transfer (17A) Deans, H. A . , Horn, F. J. bi., and Klauscr, G . “Perturbation Chromatography in Chemically Reactive Systems,” AIChE J . , 16, 426 (1970). (18A) de Santiago, M., and Farina, I. H., “Mass Transfer with Second Order Reaction. Numerical Solution,’’ Chem. Eng. Sci.. 25, 744 (1970). (1914) Dougharty, N. A . , “Self-Poisoning Reactions in the Single-Pellet Catalytic Reactor,” ibid., p 489. (20A) Feick, J., and Quon, D., “Mathematical Sfodels for the Transient Behavior of a Packed Bed Reactor,” Can. J . Chem. Eng., 48,205 (1970). (21A). Fossard, A . , “ O n a Method for Simplifying Linear Dynamic Systems,” I E b E Aulnmat. Conlr., 15, 261 (1970). (22A) Fuller, 0. M., “Differential Scaling: A Method of Chemical Reactor Modelling,” Can. J. Chem. Eng., 48, 119 (1970). (23A) 9 1 1 , W. N., “Axial Dispersion with Time Variable Flow in Multiphase Systems, AIChE J . , 15,745 (1969). (24A) Gioia F . Gibilaro, L. G., and Greco G . “ A Model far the Independent Reversiblk Pojsoning of Porous Catalyst b; Feid Stream Impurities,” Chern. Eng. J., 1, 9 (1970). (25.4) Goddard, J. D., Schultz, .I. S., and Bassett, R. J. “ O n Membr.ine Diffusion with Near-Equilibrium Reaction,” Chem. Eng. Sci., 25, k65 (1970). (26A) Hartland, S., “ T h e Optimum Operation of an Existing Forward and Back Extractor,” ibid., 24, 1075 (1969). (27A) Hatfield, B., and Aris, R., “Communications on the Theory of Diffusion and Reaction. IV,” ibid., p 1213. (28A) Herzig, J. P., LeClerc, D . M., and LeGoff, P “Flow of Suspensions through , (5), 8 (1970):’ PorousMedid,” IND.ENC.C H E M .62 129A) Hlavacek, V., “Aspects of Design of Packed Catalytic Reactors,” ibid., (7), p 9 (1970). (30A) Hlavacek, V., and Hofmann, H., “Modeling of Chemical Reactors.” Chem. Eng. Sci., 25, 173, 187 (1970). (31.4) Hlavacek, V., Kubicek, h l . , and Marek M . ”Analysis of Nonstationary Heat and MassTransfer in a Porous Catalyst P;rticle,” J.Cotal., 15, 17, 31 (1969). (32A) Hochmuth, R. and Sutera, S. P.: “Spherical Caps in Low ReynoldsNumber Tube Flow, Chem. E n f . Sci., 25, 593 (1970). (33A) Horn, F. J. M . , Jackson, R., Martel, E., and Patel, C., “ O n Asymmetric Temperature and Composition Profiles in Catalyst Particles,” Chem. Enq. J . , 1, 79 (1970). (34A) Hutchinson, P., and Luss, D., “Lumping of Mixtures with lhany Parallel First Order Reactions,” ibid., p 129. (35A) Kozicki, W., Rao, A . R . K . , and Tiu, C;,, “Correction for Transient Flow in the Initial Stage of Constant-Rare Filtration, INn. ENG.CHEM.,F U N D A M . , 9, 261 (1970). (36A) Kuo, C., and Huang, C., “Liquid Phase Mass Transfer with Complex Chemical Reaction,” AIChE J . , 16, 493 (1970). (37A),,Kuppurajulu, A , , and Elangovan, S., “System Analysis by Simplified Models, IEEEAutomnt. Contr., 15, 234 (1970). (38A) Le:, J. C. M . , and Luss, D., “Maximum Temperature Rise Inside Catalytic Pellets, IND. E N D .CHEM.,FUNDAM.: 8, 596 (1969). (39A) Lelli,,,U., Gatta, A . , and Pasquali, G., “Heat Transfer in Multistage .Mixer Columns, Chem. Eng.Sci., 24, 1203 (1969). (40A) Leon, J., and Herrero, E. E,, “Determination of Higher-Order Transfer Functions by Analog Computation,” Szmulaiion, 14, 123 (1970). (41A) Levy, R . E., Foss, A . S . , and Grens, E. A , , “Response Modes of a Binary Distillation Column,” I N D ENG. . CHEM.,F U N D A M8,. ,765 (1969). (42A) Liddle, C. J., “Modelling and Analysis of Recycle Systems,” Brit. Chem. Eng., 15, 64 (1970). (43A) Liljenzin, J., and Reinhardt, H., ”Theoretical Models for Behavior of Drops i n a Centrifuge,” I N D .EKG.C H E M .FUNDAM., , 9, 248 (1970). (44A) Lim, H . C., “ O n Transient Heat Transfer in a Porous Medium,” AIChE J., 16, 153 (1970). (45A) Lindberg, R . C . , and Schmitz, R . A , , “ O n the Multiplicity of Steady States
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in Boundary Layer Problems with Surface Reaction,” Chem. En!. Sci., 24, 1113 (1969). (46.4) Lindberg, R . C., and Schmitz, R . A,, “Multiplicity of States with Surface Reaction on a Blunt Object in a Convective System,” ibid., 25, 901 (1970). and Toor, H. L., ‘’A Diffusion Model for Reactions with T u r (47A) Mao, K . W-., bulent Mixing,” AIChE J., 16, 49 (1970). (48A) .Matovich, M. A . , and Pearson, J. R . A., “Spinning a Molten Threadline,” I N D END. . CHEM.,FUNDAM., 8,512 (1969). (49A) McGreavy, C., and Cresswell, D. L., “A Lumped Parameter Approximation to a General Model for Catalytic Reactors,” Can. J.Chem. Eng., 47, 583 (1969). (50A) Mecklenburgh, J. C . , and Hartland, S., “Design of Differential Countercurrent Extractors with Backmixing,” ibid., p 453. (51A) Mecklenburgh, J. C., and Hartland, S., “Design Methods for Countercurrent Flow with Backmixing. 11,” Chem. E n g . Sci., 24, 1063 (1969). (52A) Mihail, R . , “ 4 Superposition Integral Equation for Catalytic External Surfaces,” ibid., 25, 463 (1970). “A Liquid Phase Adsorption Study,” (53A) Miller, C. 0. M., and Clump, C. W., A I C h E J . , 16, 169 (1970). (54A) Murrill P . W . Pike R . W.,and Smith, C. L., “Modelsfor a Process Plant,” Chem. Eng., i s , 97 (i969).’ (55.4) Murrill P. W Pike, R . W., and Smith, C. L., “Models for Process Equipment,” itid.,’, 139.” (56A) Murrill, P . W., Pike, R . I%‘.,and Smith, C. L . , “Algorithm for Computing Fourier Integrals Rapidly,” ibid., p 125. (57A) Nunge, R . J., and Gill, W . N., “Mechanisms Affecting Dispersion and Miscible Displacement,” I N D ENC. . CHEM.,61 (9), 33 (1969). (58.4) Ohki, Y., and Inoue, H . , “Studies on Yield in Gas-Liquid Contacting Reactors, Can. J . Chem. Eng., 47, 576 (1969). (59A) Ohya, H . , and Sourirajan, S., “Effect of Longitudinal Diffusion in Reverse Osmosis,” AIChE J., 15, 780 (1969). (60A) Ohya, H., and Sourirajan, S., “Some General Equations for Reverse Osmosis Process Design,” ibid., p 829. (61.4) Oldenburger, R . , “Theory of Distributed Systems,’’ J. Basic Eng., 92, 1 (1970). (62A) Ozawa, Y . , “Application of Legendre Transformation to One-Dimensional Packed Bed Model,” Chem. Eng. Sci.,25, 529 (1970). (63A) Paris, J. R., and Stevens, 1%‘. F., “Mathematical Xfodels for a Packed-Bed Chemical Reactor,” Can. J . Cham. Eng., 48, 100 (1970). (64A) Phillips: C. R., and Smith, T . N., “Random Three-Dimensional Continuum Models for Two-Species Sedimentation,“ Chem. E n g . Sci.,24, 1321 (1969). (65A) Placheo, F . P.: and Krasuk, J. H.?“Solid-Liquid Countercurrent Extractors,” I K D .ENG.CHEM.,PROCESS DES.DEVELOP., 9,419 (1970). (66A) Pollack, G . G., and Johnson, A . I., “ T h e Dynamics of Extraction,” Can. J . Chem. Eng., 47,469 (1969). (67A) Ramadadhan, K., and Gill, W .N . , “Combined Forced and Free Convective Diffusion,” A K h E J . , 15, 872 (1969). (68A)RdO, A. M., and Corrigan, T. E., “Bode Diagrams for Some Chemical Reactors,’’ Brit.Chem. Eng., 14, 1406 (1969). (69A) Rao, K . B., and Doraiswamy, L. K . , “Combined Reactors. Formulation of Criteria and Operation of a Mixed Tubular Semifluidized Reactor,” .4IChB J . , 16, 273 (1970). (70A) Rester, S., Jouven, J., and Aris, R., “Communications on the Theory of Diffusion and Reaction. 111,” Chcrn. Eng. Sci., 24, 1019 (1969). (7111) Rieger, F., “Viscous Heating ofLiquid in Torsional Flow,” ibid,, p 1017. (72A) Robinson, E. R., and Goldman, M. R., “Multicomponent Batch Distillation Simulation on an IBM 1130 Computer,”Simulntion, 13, 289 (1969). (7324.1 Root, R. B., and Schmitz, R . A .Multiplicity in a Loop Reactor,” .4IC (74A) Ross, L. W., “Sirnulation of Air and Water Pollution Dynamics,” Simulation, 14, 165 (1970). (75A) Ruckenstein, E . , “ T h e Effectiveness of Diluted Porous Catalysts,” AIChE J . , 16, 151 (1970). (76A) Schertz, W. W., and Bischoff, K . B., “Thermal and Material Transport in Nonisothermal Packed Beds,” zbid., 15, 597 (1969). (77A). Schneider, P., and Mitschka, P., “Intraparticle Diffusion and Apparent Activarion Energy,” Chem. Eng. Sa.,24, 1725 ( 1 9 6 9 ) . (78A) Scrivner, N . C., and Manning, F. S., “Reduction Kinetics of Swelling LVustite Particles,” A Z C h E J . , 16, 32G (1970). (79.4) Shapifo, N . Z., “A Generalized Techniquc for Eliminating Species in Complex Chemical Equilibrium Concentrations,”,SIAMJ.Aflpl. Math., 17, 960 (1969). (80A) Simmons, P . J.,and Spinner, I. H., “Cyclic Steady-State Diffusion,” AIChE J . , 15, 489 (1969). (81A) Sleicher, C. A , , ATotter, R . H . , and Crippen, M. D., “A Solution to the Turbulent Graetz Problem by Matched Asymptotic Expansions,” Chem. Eng. Sci., 25, 845 (1970). (82A) Sohn, H . Y.:Merrill, R . P.: and Petersen, E. E., “Effectiveness Factor for Surface Diffusion and Reaction on Catalyst Surface,” ibid., p 399. (83A) Tommasi, G., and Rice, P., “Dynamics of Packed Tower Distillation,” Cirshr., PROCESS DES.DEVELOP., 9, 234 (1970). I N D ENG. . (84.4) Treybal, R . E.: “Adiabatic Gas Absorption and Stripping in Packed Towers,” I N D ENC. . CHEM.,61 (7), 36 (1969). (85.4) Vaclavek, V., “Studies in System Engineering-111,” Chem. Eng. Sci., 24, 947 (1969). (86Aj van Poolcn, H. K., Breitenbach, E. A , , and Thurnau, D.’H., “Treatment of Individual Wells and Grids in Reservoir Modelling,” Soc. Petrol. E n g . J., 8, 341 (1968). (87A) Vermeulen, T., and Quilici, R . E., “Analytic Driving-Force Relation for Pore-Diffusion Kinetics in Fixed-Bed Adsorption,” IKD,END.CHEM.,FUNDAX., 9, 179 (1970). (88A) Vlach: J., “h-umerical Method for Transient Responses of Linear Networks with Lumped, Distributed or Mixed Parameters,” J . Franklin Inst., 288, 99 (1969). (89A) Wahl. E. F., and Harriott, P., “Understanding and Prediction of the Dynamic Behavior of Distiilation Columns,” TNU. ENC. Crieu., PnocEss Des. DEVELOP., 9, 396 (1970). (90A) Wajc, S. J., “Analytical Solutions for Adiabatic Reactor Problems,” I N D . ENG.CHEM., FUKDAM., 8, 593 (1969). (91A) Wang, S. C., “Diffusion-Controlled Rate Mechanisms in Gas-Solid Reaction Systems,” .4ZChE J., 15, 624 (1969). (92.4) Ward, W. J,, “Analytical and Experimental Studies of Facilitated Transport:” ibid., 16, 405 (1970). (93A) Wasen, D . T . , and Ahluwalia, M. S., “Consecutive Film and Surface Renewal Mechanism for Heat or Mass Transfer from a Wall,” Ciiem. Eng. Sci., 24, 1535 (1969).
(94A) Wasserman M. L. and Slattery .I. C., “Cree ing Flow Past a Fluid Globule When a Trace oiSurfaciant is Presen;,” AIChE J.,f5, 533 (1969). (95A) Whitaker S., “Advances i n Theory of Fluid Motion in Porous Media,” IND.END.CHLhi., 61 (12), 14 (1969). (96A) Wonham, W. M . , and Morse, A . S., “Decoupling and Pole Assignment in Linear Multivariable Systems: A Geometric Approach,”. SIAM .I. Cunlr., 8, 1 (1970). (97A) Yang, J. Y., Thomas, C. C., and Cullinan, H. T . , “Radiation-Initiated Side-Chain Chlorination of Toluene,” I N D ENG. . CHEM.,PROCESS DES.DEVELOP., 9, 214 (1970). (98A) Zwiebel, f;, “Fixed Bed Adsorption with Variable Gas Velocity Due to PresSureDrop, IND. ENG.C H E M . , FUNDAM., 8, 803 (1969).
RESIDENCE-TIME AND DISPERSION (1B) Buffham, B. A , , “Impulse Responsc of Infinite and Semi-Infinite Sequence of Identical Stirred Tanks with Backflow,” I N n . ENG. CHmf., FUNDAM., 8, 428 (1969). (28) Buffham, B. A , , and Gibilaro, L. G . “A Unified Time Delay Model for Dispersion in Flowing Media,” Chem. Eng. J:, 1,31 (1970). (3B) Buffham, B. A , , Gibilaro, L. G., and Rathor M M . “A Probabilistic Time Delay Description ofFlow in Packed Beds,” I I I C i E i , 16,’218 (1970). (4B) Germ, R . L., and Smith, J. M . “Effects of Heat Release and Nonlinear Equilibrium on Transient Adsorption,” ~ N D ENC. . CHEM.,FUNDAM., 8, 797 (1969). (5B) Ccrro, R . L., and Parera, J. M . , “1)ynamic Similarity in CSTR’s,” ibid., 9, 181 (1970). (6B) Chio, S., Fan, L., and Akins, R . G . , “Experimental Study of the Ice-Making Operation in the Inversion Desalination Freezing Process,” INn. ENC.CHEM., PROCESS DES.DEVELOP., 8, 347 (1 969). (7B) Cokelet, G . R., and Shair, F. H., “Concerning the Calculation of Residence Time Distribution Functions for Systems in which Diffusion is Negligible,” AIChE J . , 15, 939 (1969). (8B) Dayan, J., and Levenspiel, O., “Dispersion in Smooth Pipes with Adsorbing Walls,” I N D .ENG.CHEM.,FUNDAM., 8, 840 (1969). (9B) Evangelista, J. J. Katz S. and Shinnar, R., “Scale-up Criteria for Stirred Tank Reactors,” A d E J., i5, b43 (1969). (10B) Evans G . V . “Kote on the Dispersion of a Tracer in Turbulent Pipe Flow,” Chem. Eng.’Sci., 24, 1736 (1969). (1 1B) Fedotkin, I. M., “ T h c Residence Time of a Two-Phase Current in the T u b e Element of a Continuous-Action Apparatus,” Theor. Found. Chem. Eng. (USSR), 2, 632 (1969). (12B) Fri$ly J. C., “On the Use of ’I’ransfrr Function in Dynamic Process Models, C h m . Eng. Sci., 25, 119 (1970). (13B) Gibilaro, 1. G., Gioia, F., and Grrco, G., “Unsteady-State Diffusion in a Porous Solid Containing Dead-Ended Porcs,” Chem. Eng. J., 1, 85 (1970). (14B) Gilath C . Naphtali, L. M., and Resnick, W., “Transicnt Response of a Packed Cciumh to Changes in Liquid and Gas Flow Rate,” IND.ENC.CHEM., PROCESS DES.DEVELOP., 8, 324 (1969). (1511) Greenkorn, R . A., and Kessler, D . P. “Dispersion in Heterogeneous Non, C l l E M . , 61 (P), 14 (1969). uniform Anisotropic Porous Media,” I N D Ehc. (16B) G u n n ’ 1). .J. “Theory of Axial and Radial Dispersion in Packed Beds,” 7 ’ r a n s . I n r t ~ C h e m . ~ n g . , 4 7 , T 3 5(1969). 1 (17B) Gunn, D. J., “ T h e Transient and Frequcncy- Response of Particles a n a Beds of Particles,” Chem. Eng. Sci.,25, 53 (1970). (18B) Gunn, D . J., and Pryce, C., “Dispersion in Packed Beds,” T m n r . Inst. Cllem. Eng., 47, T341 (1969). (19B) Han, C . D., “Steady-State Behavior of Continuous Granulators-An Elementary Mathematical Analysis,” Chem. Eng. Sci., 2 5 , 875 (1970). (20B) Hochman, J. M . , and McCord, J. R., “Residmce-Time Distribution in Recycle Streams with Crossmixing,” ibid., p 97. (21B) Hopkins, M. J,, Shcppard, A. J., ,and Cisenklam, P., “ T h e Use of Transfer Functions in Evaluating Residence Time Distribution Curves,” ibid., 24, 1131 (1969). (228) K a f x o v , V. V., and Klipinitser V. A. “Stochastic Model of a Cascade of 2, 761 (1968). Non-ldeal Mixers,” I‘heor. Found. Ch:hb6.Rng.’(USSR), (23B) Kafarov, V. V., Klipinitser, V. A., and Dudowv, A. A , , “Stochastic Model of a Kon-Idral Mixer.” ibid.. D 671. (24B) Krssler, r).P., and York, J. L., “Characteristics of Inclusions in the Dispersed Phase of Liquid-Liquid Suspensions,” AlChE J . , 16, 369 (1970). I
(40B) Patel, R . D., and Greenkorn, R . A “ O n Dispersion in Laminar Flow Through PorousMedia,” AIChE J., 16,33211970). (41B) Pell, T. M., and Aris R “Problems in Chemical Reactor Analysis with M., 9,15 (1970). Stochastic Features,” IND.~ N C : ’ C H E FUNDAM., (42B) Petho, A . , “Distributions in Time and Space, Respectively, in the Case of Continuous-Flow Combined with Sorption in a Fixed Bed,” Chem. Eng. Sci., 25, 769 (1970). (43B) Pollock G. G and Johnson A . I., “ T h e Dynamics of Extraction Processes,” Can. J. Ched. Eng.,’)47, 565 (1969)). (44B) Ritter, A. B., and Douglas, J. M . “Fre uency Response of Nonlinear Systems,” I N D . ENC.C H E h i . , FUNDAM., 9, 211 (19707. (45B) Sagara, M., Schneider, P., and Smith, J. M., “ T h e Determination of HeatTransfer Parame.ters for Flow in Packed Beds Using Pulse Testing and Chromatography Theory,” Chem. Eng. J., 1, 47 (1970). (46B) Sastry, K . V. S., and Fuerstenau, D. W., “Size Distribution of Agglomerates in Coalescing Dispersed Phase Systems,” IND. ENO. CHEM.,FUNDAM., 9, 145 (1970). (47B) Sl;!veston, P. L., “Design of Settling Basins with Allowance for Residence Time, Can. J . Chstn. Eng., 47, 521 (1969). (48B) Smith, J. V., and Perona, J. J “Axial Mixing of Fluids in Turbulent Flow Through Concentric Annuli,” IND.~ N G CHEM., . FUNDAM., 8, 621 ( 1 9 6 9 ) . (49B) Szabo, T. T., and Nauman, E. B., “Copolymerization and Terpolymerization in Continuous Nonideal Reactors,’’ AIChE J., 15, 575 (1969). (50B) van Swaaij W. P . M . Charpentier, J. C., and Villermaux, J., “Residence Time Distribution in the i i uid Phase of Trickle Flow i n Packed Columns,” Chem. Eng. Sci.,24, 1083 ( 1 9 6 4 . (51B) Wen, C., and Ziegler, E “On the Axial Dispersion Approximation for Laminar Flow Reactors,” ibid., 25, 723 (1970). (528) Yoshida K . Kunii D and Levenspiel, 0. “Axial Dispersion of Gas in Bubbling F1:idiz;d Beds,’” I&. ENG.CHHDI., FUNDLM., 8,402 (1969). (53B) Zoularian, A,, and Villermaux, J., “An Experimental Investigation of the State of .Mixing of the Fluid in a Chemical Reactor,” Chem. Eng. J., 1,76 (1970).
PERIODIC SYSTEMS (1C) Ark, R., “Equilibrium Theory of the Parametric Pump,” I N D .E N G . CHEY., FUNDAM., 8, 603 (1969). (2C) Baccaro, G . P., Gaitonde, K. Y., and Douglas, .I.M., “ A n Experimental Study of Oscillating Reactors,” AIChE J., 16, 249 (1970). (3C) Bailey, J. E., Horn, F. J. M., Bankofl, S. G., and Chang, K . S., “Oscillatory Operation of Jacketed Tubular Reactors,” IND.Erc. CHEM.,FUNDAK, 9, 299 (1970). (4C) Fagela-Alabastro, E. B. and Hellums J. D “Diffusion inpulsating Flow in a Distensible Conduit,” A d E J., 15, 803 i1969j: (5C) Frazier, G. C., Jr., and Ulanowicz, R . E. “Film Transfer with Non-Equilibrium Chemical Reaction,” Chem. E n g . Sci., 24, 549 (1970). (6C) Gerster J. A. and Scull H. M. “Performance of Tray Columns Operated in the Cyclin$ Modi,” A I C h E j . , 16, 168 (1970). (7C) Grafton, R . B., “A Periodicity Theorem for Autonomous Functional Differcntial Equations,” J . Dif.Eqnr. 6 , 87 (1969). (8C) Lannus, A , , and Kershenbaum, L. S., “ O n the Cyclic Operation of Tubular Reactors,” AlChEJ., 16, 329 (1970). (9Cj McGonaghy, G . A , , and Finlayson, 73. A. “Surface Tension Driven Oscillatory Instability in a Rotatin2 Fluid Layer,” .J. )Fluid Mech., 39,49 (1969). (1OC) McEwan, A . I>., “Inertial Oscillations in a Rotating Fluid Cylinder,” zbid., 40, 603 (1970). (11C) Meissner, L. P. “High Order Resonance for Duffing’s Differential Equation,”SlAM J . Appl.’Math., 17, 240 (1969). (12C) Modi, V. J., “Periodic Solutions Associated with the Gravity-GradientOriented System,” A I A A ~ J .7, , 1217 (1969). (1 3C) Nduman E. B. “Residencp-Time Distribution Theory for Unsteady Stirred Tank React&” Chdm. Eng. Sa., 24, 1461 (1969). tilt J. and Mullin J. W., “ T h e Periodic Behavior of Continuous Crystali5, 131 (i97oj. (‘2r;’ibii., (15C) Olfe D. R. and DePlomh E. P., “ T h e Decay of Perturbations in a Radiating Gas,” J . hltiid hech., 40, 127 (i970). (16C) ,Pekeris, C . L., and Shkoller, B., “Stability of Plane Poiseuille Flow to Periodic Disturbances of Finite Amplitude,” ibid.,39, 61 l (1969). (17C) Perko, L. M., “Higher Ordcr Averaging and Related Methods for Per-’ turbed Periodic and Quasi-Periodic Systems,” S I A M J . Appl. Moth., 17, 698 (1969). (18C) Sanchez, D . A , , “A Note on Pcriodic Solutions of Riccati-Type Equations,” ibid., p 957. (19C) Subramanyam, S . V., “ A Notc o n the Damping and Oscillations of a Fluid Drop Moving in Another Fluid,” J . Fluid Mech., 37, 715 (1969). (2OC) Taylor, J . , H . ; , a n d Narcndra K. S. “Stability Regions for the Damped Mathieu Equation, SZAM J . A/@.’Math., i 7 , 343 (1969). (21C) Yerushalmi, J., Katz, S., and Shinnar, R., “ T h e Response of Viscoelastic Materials to Small Deformations,” Chem. Eng. Sei., 24, 1371 (1969).
OPTIMIZATION (1D) hlfriend, K . T., and Kashiwagi, Y.. “Minimum Time Orbital Rendezvous Between Neighboring Elliptic Orbits,” J . Opt. Theory Appl., 4, 260 (1969). (2D) Anderson, N., Arthurs, A. M., and Robinson, P. D.. “Pairs of Complementary Variational Principles,” J . Inst. Math. Itr Appl., 5 . 422 (1969). (3D) Aubin, J. P., “Characterization of the Sets of Constraints for which the Necessary Conditions for Optimization Problems Hold,” SZAM J . Contr., 8, 148 (1970). (4D) Bach, H., “ O n the Downhill Method,” Comm. A C M , 12, 675 (1969). (5D) Bellmore, M., Greenberg, 13. J., and Garvis, J . J., ”Generalized Penalty-Function Concepts in Mathematical Optimization,” Oper. Res., 18, 229 (1970). (6D) Brown K . M . “A Quadratically Convergent Newton-Like Method Based Upon Gadssian E l i h n a t i o n , ” SIAM J . Niimer. Anal., 6 , 560 (1969). (7D) Br,o den;, C. G., “The Convergence of a Class of Double-Rank Minimization Algoritxms, J . Inst. Math. Zts Appl., 6 , 76 (1 970). (XD) Chartrand, G., and Crowe, C. M., “ T h e Optimization of the Catalytic Oxidation of Sulphur Dioxide in a Multi-Bed Adiabatic Reactor,” Can. J . Chem. Eng., 47, 296 (1969). (OD) Chazan, D., and Miranker, W. L., “A Nongradient and Parallel Algorithm for Unconstrained Minimization,” SIAM J . Cunlr., 8, 207 (1970). (10D) Christensen, J. H., “ T h e Structuring of Process Optimization,” AICfiE J . , 16,177 (1970). (11D) Conversr, A. O., “Optimization,” Holt, Rinehart and Winston, New York, N . Y., 1970.
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(12D) Demenchuk, V. A l . , “Flow Interpretations and an Algorithm for Solving the Problem of Time-Optimal Allocation of Limited Resources,” Automat. Remote Contr. ( U S S R ) , 1969, 1697. (13D) Eben, C . D., and Ferron, J. R., “Inequality Methods for Computation of Optimal Systems,” IND.E N G CHEM,, . FCNDAM., 8, 749 (1969). (14D) Fan, L. T., Huang, C. L., and Tillman, F. A., “A Sequential Simplex Pattern Search Solution to Production Planning Problems,’’ A I E E Trans., 1, 267 (1 969). (15D) Fiacco A . V. and Jones A . P. “Generalized Penalty Methods in Topological Spades,” Sd.14 J.Appl. h a t h . , i 7 , 996 (1969). (16D), Fox, B. L., and Landi, D. ?vi.>“Searching for the Multiplier in One-Constraint Optimization Problems,” Oper. Res., 18, 253 (1970). “A New One Dimensional Search Technique,” Chern. Eng. (17D) Gentry, J. W., Sci., 25, 425 (1970). (1 8D) Goldfarb, D., “Extension of Davidson’s Variable Metric Method to Maximization under Linear Inequality and Equality Constraints,” S I A M J . Appl. Math., 17, 739 (1969). (19D) Goldfarb, n., “A Family of Variable-hletric Methods Derived by Variational Means,” Mnth. Comp,: 24, 23 (1970). (20D) Gorelik, N. G . , and Kaplinskii, A . I., “Investigation of Extrema1 Systems with Variable Optimizer-Adjustment Parameters, ”Automat. Remote Contr. ( U S S R ) , 1969, 1790. (21D) Greenstadt, J.: “Variations on Variable-Metric Methods,” ,Math. Comp., 24, 1 (1970). (22D) Grigoriadis M . D . and Ritter K . “A Parametric Method for Semidefinite Quadratic Prog;ams,” 2IA.M J . C o n k , 559 (1969). (23D) Hestenes, M . R . “Survr Paper: Multiplier and Gradient Methods,” J . Opt. Theory Appl. 4, i 0 3 (19697. (24D) Hix, A. H., “Status of Process Control Computers in the Chemical Industry,” Proc. ZEEE, 58, 4 (1970). (25D) Hang, K. C., and Jensen, R . B.: “Optimization of Multicycle Steam Simulation,”Soc. Petrol. Eng. .l.,9, 357 (1969), (26D) Julio S . D. “Numerical Solution of Dynamical Optimization Problems,” S I A M J . kontr., 135 (1970). (27D). Kowalik, J., Osborne, A
