Computer methods for solving dynamic separation problems (1983)

COMPUTER METHODS FOR SOLVING DYNAMIC SEPARATION PROBLEMS Charles D Holland Texas A&M University Athanasios I Liapis University of Missouri McGraw-Hill Book Company New York St Louis San Francisco Auckland Bogota Hamburg Johannesburg London Madrid Mexico Montreal New Delhi Panama Paris Sr?o Paulo Singapore Sydney Tokyo Toronto PREFACE Because of the availability of high-speed computers the time is fast approaching when the engineer will be expected to be as conversant with/the unsteady state solutions to process systems as was expected for the steady state solutions in the past In this book a combination of the principles of separation processes, process modeling, process control, and numerical methods is used to produce the dynamic behavior of separation processes That is, this book "puts it all together." The appropriate role of each area is clearly demonstrated by the use of large realistic systems The order of presentation of the material was selected to correspond to the order of the anticipated difficulty of the numerical methods Two-point methods for solving coupled differential and algebraic equations are applied in Part while multipoint methods are applied in Part 2, and selected methods for solving partial differential equations are applied in Part Also, the presentation of the material within each section is in the order of increasing difficulty This order of presentation is easily followed by the student o r practicing engineer who has had either no exposure or little exposure to the subject Techniques for developing the equations for the description of the models are presented, and the models for each process are developed in a careful way that is easily followed by one who is not familiar with the given separation process In general, the best possible models that are compatible with the data commonly available are presented for each of the separation processes The reliability of the proposed models is demonstrated by the use of experimental data and field tests For example, the dynamic behavior predicted by the model for the system of evaporators was compared with the observed behavior of the system of evaporators at the Freeport Demonstration Unit Experimental data as well as field tests on the Zollar G a s Plant for distillation columns, absorbers, and batch distillation columns were used for comparison purposes Experimental results were used to make the comparisons for adsorption and freeze-drying The development and testing of the models presented in this book required the combined efforts of many people to whom the authors are deeply indebted In particular, the authors appreciate the support, assistance, and encouragement given by J H Galloway and M F Clegg of Exxon; W E Vaughn, J W Thompson, J D Dyal, and J P Smith of Hunt Oil Company; D I Dystra and Charles Grua of the Office of Saline Water, U.S Department of Interior; J P Lennox, K S Campbell, and D L Williams of Stearns-Rogers Corporation Support of the research, upon which this book is based, by David L Rooke, Donald A Rikard, Holmes H McClure, and Bob A Weaver (all of Dow Chemical Company), and by the National Science Foundation is appreciated Also, for the support provided by the Center for Energy and Mineral Resources and the Texas Engineering Experiment Station, the authors are most thankful The authors acknowledge with appreciation the many contributions made by former and present graduate students, particularly those by A A Bassyoni, J W Burdett, J T Casey, An Feng, S E Gallun, A J Gonzalez, E A Klavetter, Ron McDaniel, Gerardo Mijares, P E Mommessin, and N J Tetlow The authors gratefully acknowledge the many helpful suggestions provided by Professors L D Durbin, T W Fogwell, and R E White of the Department of Chemical Engineering, Texas A&M University, and K Crosser, T W Johnson, and J M Marchello of the Department of Chemical Engineering, University of Missouri-Rolla A I Liapis thanks especially Professor D W T Rippin of E T H Zuuch, who encouraged his investigations in the field of separation processes, and stimulated his interest in the application of mathematics The senior author is deeply indebted to his staff assistant, Mrs Wanda Greer, who contributed to this book through her loyal service and assistance in the performance of departmental administrative responsibilities; to his daughter, Mrs Charlotte Jamieson, for typing this manuscript; and to his wife, Eleanore, for her understanding and many sacrifices that helped make this book a reality Charles D Holland Athanasios I Liapis COMPUTER METHODS FOR SOLVING DYNAMIC SEPARATION PROBLEMS CHAPTER ONE INTRODUCTIONMODELING AND NUMERICAL METHODS An in-depth treatment of both the modeling of dynamic separation processes and the numerical solution of the corresponding equations is presented in this book After the models which describe each of the separation processes at unsteady state operation have been formulated, the corresponding equations describing each of these models are solved by a variety of numerical methods, such as the two-point implicit method, Michelsen's semi-implicit Runge-Kutta method, Gear's method, collocation methods, finite-difference methods, and the method of characteristics The ability t o solve these equations permits the engineer to effect an integrated design of the process and of the instruments needed to control it The two-point implicit method (or simply implicit method) is applied in Part ; Michelsen's semi-implicit Runge-Kutta method and Gear's method in Part 2; and the collocation method, finite-difference methods, and the method of characteristics are applied in Part T o demonstrate the application of the numerical methods used in Parts and 2, the use of these methods is demonstrated in this chapter by the solution of some relatively simple numerical examples The methods used in Part are developed in Chap 10 and their application is also demonstrated by the solution of relatively simple numerical examples The techniques involved in the formulation of models of processes is best COMPUTER METHODS FOR SOLVING DYNAMIC SEPARATION PROBLEMS AND NUMERICAL METHODS INTRODUCTION-MODELING ] demonstrated by the consideration of p articular processes A wide variety of processes including evaporation, distillation, absorption, adsorption, and freezedrying are considered Both stagewise processes such as distillation columns equipped with plates and continuous processes such as adsorption processes are treated All of these models are based on the following fundamental principles: The contents are oerfecrlv mixed (/ !4 = moles of h o l d u ~ Conservation of mass or material balances Conservation of energy or energy balances Transfer of mass In order to demonstrate the techniques suggested for the formulation of the equations representing the mass and energy balances, several different types of systems at unsteady state operation are presented in Sec 1-1 These techniques are further demonstrated in subsequent 'chapters by the 'development of the equations for particular process models In order to solve the equations describing the model of a given process, a variety of numerical methods may be used Representative of these are the methods listed above An abbreviated presentation of selected methods and their characteristics are given in Sec 1-2 1-1 FORMULATION OF THE EQUATIONS FOR SELECTED MATERIAL AND ENERGY-BALANCE MODELS no -' *-, C&.O Material Balances Let the particular part of the universe under consideratiog be called the system and the remainder of the universe the surroundings A paterial balance for a system is based o n the law of conservation of mass For urposes of application, a convenient statement of this lau follows: Except for he conversion of mass to energy and conversely, mass can rle~rherbe created n o r destroyed Consequently, for a system in which the conversion of mass to energy and conversely is not involved, it follows that during the time period from t = t , to t = t , + At, / output of material Input of material from the system to the system (duu time "me $;;:II It accumulation of material within = he system during the time period A t The accumulation term is defined as follows: Accumulation of material within the system during the time period At amount of material in the system at 1( amount of - material i n t h e system at time t, In the analysis of systems at unsteady state, the statement of the material balance given above is more easily applied when restated in the following form: Figure 1-1 Sketch of a perfect mixer input of material per unit time - output of material (per unit time amount of material in the system YP v n XJ Qi(T) surface tension of a pure liquid surface and of a contaminated liquid surface, force per unit length = activity coefficients for component i in a nonideal mixture in the vapor phase and in the adsorbed phase, respectively = parameter in the kinetic model for multilayer adsorption = spreading pressure for the adsorbed phase, force per unit length = = functions (1 j n ) in the generalized equation of state for the adsorption of multicomponent mixtures in n layers = function defined by Eq (14-72) REFERENCES J H de Boer: The Dynanncnl Character of Adsorplion, Oxford University Press, New York, 1953 Kenneth Denbigh: The Pr~nciplesof Chemrcal Equ~lrhrium, Cambridge University Press, New York, 1955 A J Gonzalez: Ph.D d~ssertation,"Adsorption Equ~librlaof Multicomponent Mixtures," Texas A&M University, 1969 A J Gonzalez and C D Holland: "Adsorption of Multicomponent Mixtures by Solid Adsorbents," AIChE J., 16:718 (1970) A J Gonzalez and C D Holland: "Adsorption Equilibria of Light Hydrocarbon Gases on Activated Carbon and Silica Gel," AIChE J., 17:470 (1970) R J Grant, M Manes, and S B Smith: "Adsorption of Nornam Parafins and Sulfur Compounds on Activated Carbon," AIChE J., : (1962) E A Guggenheim: Thermodynamics, New York lnterscience Publishers Inc., 1949 E A Guggenheim: Mixtures, The Theory of the Equilihrrum Properties of Some Simple Classes of Mixtures Solutions and Alloys, Oxford University Press, New York, 1952 T L Hill: "Thermodynam~csand Heat of Adsorption," J Chem Phys., 17:520 (1949) 10 C D Holland: Fundamentals of Multicomponent Distillation, McGraw-Hill Book Company, New York, 1981 466 SOLUTION OF PROBLEMS INVOLVING CONTINUOUS-SEPARATION PRO( ES 11 A J Kidnay and A L Myers' " A Simplified Method for the Prediction of Multicomponent Adsorption Equilibria from Single Gas Isotherms," A I C h E J., 12:981 (1966) 12 I Langmuir: "Oil Lenses o n Water and the Nature of Nonomolecular Expanded Films," J Chem Phys., : 756 (1953) 13 A L Myers and J M Prausnitz: "Thermodynamics of Mixed-Gas Adsorption," AIChE J., 11: 121 (1965) 14 R K Schofield and E K Rideal: "The Kinetic Theory of Surface Films Part I-The Surfaces of Solutions," Proc R Soc London, A109: 57 (1925) 15 M Volmer: "Thermodynamische der Zustansgleichung fur Adsorbierte StotTe," Phys Chem., 115: 253 (1925) THERMODY( ,CS OF PHYSICAL ADSORPTION OF PURE GASES AND GAS MIXTURES 467 Hint: Make use of Eq (14-28) (b) By use of the results given by Eqs (A) and (B), show that 14-5 Show that for the general case of a nonideal adsorbed solution, 14-6 For one mole of a pure component in a closed system at constant temperature, Eq (14-11) reduces to PROBLEMS dGy = 14-1 Verify the results given by Eqs (14-29) and (14-30) v: dP The volume per mole of an actual gas is given by 14-2 Show that the equation of state obtained by substituting the expression for Cifor model I1 (Eq (11-28)) into Eq (14-104) is consistent with Gibbs' formula 14-3 Analogous to the expression for the volume of a liquid that forms a nonideal solution (Refs 7, 8, 10) the surface covered by a nonideal adsorbed solution may be expressed in terms of the partial molar areas as follows: where the compressibil~tyfactor Z has the property where the 2,'s are the partial molar areas, which are defined by Equations (A) and ( )may be combined to give dGy = Z,RTd In P (at constant T) By use of Eqs (C), (D), (14-43), show that (a) By use of Eqs (B), (14-62), (14-63), and (14-86), show that af Ilm = I (at constant T) P-0 (b) Use Eq (C) to verify the result given by Eq (14-91) (c) Show that when the temperature T and all of the nTs are held fixed, Eq (C) may be integrated to give In = I , - a,) dn 14-4 Partial differentiation of both sides of Eq (14-20) with respect to n: at constant n and T gives where (a) Show that 14-7 Show that the result given by Eq (14-78) is also obtained for the general case of n adsorbed layers for both models I and [I 14-8 Show that the expression given for OXT) by Eq (14-81) is also obtained for the general case of n layers for both models I and 11 INDEX Absorbers, 217, 235-247, 253-268 field tests, 258-260 fractional response, 265 packed absorbers, 253-258 Acrivos, A., 356 Activity coefficients, 43, 128, 129 Adiabatic adsorbers, 404-414 Adsorbers: breakthrough curves, 378, 392, 393 countercurrent operation, 415, 416 fixed-beds, 374-384, 389-414 periodic operation, 414-416 Adsorption: chemical adsorption, 363, 364 physical adsorption, 363-372 thermodynamics of physical adsorption, 439-463 Adsorption isotherms: of mixtures, 369-372, 398 of pure components, 364-369, 427 Allen, R H., 19 Anzelius, A., 378 Arnold, J R., 369 Balzli, M W., 392 Barb, D K., 183, 207 Bassyoni, A A., 254-255, 262, 264 Batch-distillation, 177-207 comparison of model predictions with experimental results, 199-202 cyclic operation, 195-197 optimization of, 202-207 Bennett, J M., 67, 167 Benton, A,, 369, 371 Bernouli's theorem, 270, 271 BET isotherm, 367-369 Bolles, W L., 273, 274 Bonilla, C F., 73 Breakthrough curves (see Adsorbers) Broyden, C G., 71, 167 Broyden-Bennett algorithm, 162 Broyden's method, 63-67, 162 Brunauer, S., 364, 366, 369 Bullington, L A, 272 Burdett, J W., 71, 73, 81, 86, 92 Buron, A G., 274 Butcher, J C., 19 Caillaud, J B., 19, 217, 218, 304 Calahan, D A., 19 Carnahan, B., 336 Carslaw, H S., 82 Characteristics (see Method of characteristics) Chemical potential, 442, 445 Chilton, C H., 103 Chua, L O., 309 Churchill, R V., 84, 114 Clenshaw, C W., 341 Conte, S D., 31, 337 Control valves, 279, 282-285 Controllers : for distillation columns, 279-281, 282-285 proportional-integral controller, 281 proportional-integral-rate controller, 281 INDEX Convective mass transport, 372-373 Cooke, C E., Jr., 369 Crank, J., 348 Crank Nicolson method, 348, 356, 432, 433 Crosser, K., 408 Countercurrent operation (see Adsorbers) Coupled differential and algebraic equations, 218-235,321-326 Gear's kth-order algorithms, 222-229 Michelsen's algorithms, 218-222 semi-implicit Runge-Kutta algorithms, 222-229 Dahlquist G., 31 Damkohler, G., 382, 384 D'Arcy equation, 427 de Boer, J H., 439, 440 dc Boor, C., 31, 337 Deming, L S., 364 Deming, W E., 364 Denbigh, K., 42, 442, 456 Departure functions, 127 Desalination plant, 73-77, 88-1 11 comparisons of model predictions with experimental data, 109, 110 equipment parameters, 103-105 of Freeport, 73, 8 11 I Diffusion: axial diffusion, 394 coefficients, 380, 382 Knudsen diffusion, 427 in pores, 379-384, 395, 427 solid diffusion, 395 surface ditTusion 382 Distillation: batch distillation (see Batch distillation) continuous distillation, 123-164, 269-285 examples, 138 - 143, 153, 285-292 control of distillation columns, 279-281, 282-285 equilibrium relationships, 128, 129 Duhring lines, 43 Dukler, A E., 73 Dusty-gas model, 427 Dykstra, D I., 77 Dynamics of sieve trays, 270-276 Eckert, C A,, 286 Emmett, P H., 367, 369 Energy balances, 5-12 Enthalpy, Evans, R B., 427 Evaporation, 37-45 Evaporators: boiling point elevation, 42, 45 of multiple-effect, 41, 68, 69, 72 of single-effect, 40, 41, 46-57 control of mass holdup, 101, 102 steam consumption, 42 Swenson type, 38-40 Euler's method, 13, 14 Euler's theorem, 34, 444 Feng, An, 248, 268 Fick's law, 379 Film resistance and diffusion modcl, 394404 Finite-difference methods, 348- 356 explicit methods, 351-354 implicit methods, 354-356 Finlayson, B A,, 341, 344 Fluid flow in pipes, 6-12 Foam factor, 274 Fowler, R H., 366 Franke, F R., 153 Freeze-drying, 420-435 adsorbed water, 426,427 models, 421-429 model solutions, 429-435 Freundlich isotherm, 365 Friction factor, 275 Friedly, J., 356 Fritz, W., 372 Fritz-Schluender isotherm, 372 Fugacity, 42, 43, 448, 451- 454 Furnas, C C., 378 Gallun, S E., 248, 281, 285 Gear, C W., 30, 248, 285, 309 Gear's integration algorithm, 276 279 Gear's method of integration, 22-24, 269, 285, 315-326 Generalized theorem of integral calculus, 33, 79 Gentzler, G L., 421 Gerlack, A., 45 Gibb's adsorption formula, 447, 448 Gill, S., 17, 304 Glueckauf, E., 375, 376, 387 Glueckauf model, 375-379, 389-393 Gonzalez, A J., 369, 372, 451, 460 Greenfield, P G., 431 Groves, D M., 268 Guggenheim, E A,, 439, 442, 456 Gunn, R D., 427,430 Hamming R W., 337 Hanson, D T., 407, 408,413,414 Harper, J C., 422 Harwell, J H., 407, 408, 413, 414 Heat, Heat transfer models, 77-88 errors in the predictions, 85, 86 Henrici, P., 30 Henry's law, 365 Hildeband, F B., 334 Hill, T L., 369, 439, 450 Hlavatek, V., 293 Holland, C D., 67, 71, 76, 81, 84, 86, 91, 92, 126, 167, 208,248, 264,291, 369,448, 456, 463 Holmes, M J., 300 Holmes, R E., 378, 379 Horvay, G., 341 Hougen, A., 378 Householder, A S., 293 Ifuang C J., 73 Huckaba, C E., 153, 208 Hugmark, G A,, 275 Hutchinson, M H., 274 Hwang, M., 32 Hydraulic gradient, 271, 275 Hydraulic radius, 275, 276 Ideal adsorbed solution, 455-457 Integration of differential equations (rve Numerical methods of ~ntegration) Interface (see Phase interface) Interfacial area, 373, 426 Internal energy, Isothernis for adsorption 364-372, 398 427, 439 ~ Itahara S 73 Jacobian matrix, 19, 53 54 Jaeger J C., 82 K , method, 136 Kelley, R E., 272 Kinetic energy, King, C J., 42 1, 426, 430 Kirkpatr~ck,S D., 103 Krylov, V I., 361 Kubitek, M., 293 KubiEek's algorithm for matrices, 291, 293, 294 Lam, W K., 426, 433 Lanczos, C., 341 Langmulr, I., 364, 366, 440, 458 Langmuir isotherm, 365 Lapidus, L., 32, 167 Lee, H M., 73 Leibson, I., 272 Leland, T W., Jr., 378, 379 473 Liapis, A I., 392, 396, 402, 404, 408, 414, 421, 427,431, 435 Lin, Pen-Min, 309 Liquid surface, 440 Litchfeld, R J., 396, 407, 414, 421, 430 Lord, R C., 279 Lugin, V V., 361 Luther, tl A,, 327, 336 McBain, J W., 364 McCabe, W L., 44 McDanicl, R., 248, 258, 262, 264 Markham, E C 369, 371 Markham-Benton isothcrms, 369 371 Marshall, W R., 167, 377, 378 Mason, E A., 427 Mason, J., 369 Mass transfer coelticicnth, 373, 374, 426 Material balances, 2-5 May, R B., 153 Mean-valuc theorem of diffcrential calculus, 4, 5, 33 Meo, 11 430 Mcthod ofcharacteristics 356 360, 390, 408 Method of weighted residuals, 340, 341 Michelsen, M L., 19, 218, 248, 308, 341, 342, 344 Michelscn's method of integration, 18, 308, 398, 403 Mickley, H W., 377 Mijarcc, G., 164 Miller, B P 274 Milne, W E., 32 M ~ n t o n P , E., 279 Modeling: fundanientals of 12 cncrgy balances I2 material balance\, rate expressions, 373 374, 379, 382, 394,405 407, 427 MoRcrt, H T., 42 Moving-boundary problem, 429 solution of, 430, 431 Mult~co~nponent adsorhcrs, 389-416 Multistep intcgrat~onmethods, 308 of Adams Bashforth, 31 1, 312 of Adams Moulton, 12 Gear's, 312 326 Myers, A L., 439, 442 Newton -Raphson method, 53-55, 102, 134, 346, 348, 409,412 the 2N, 160-164, 192195 Nicolson, P., 348 474 INDEX Nonlinear algebraic equations: solution of (see Newton-Raphson method) Nordsieck vector, 22, 317-319 Norton, H T., 341 Numerical methods of integration: of ordinary differential equations, 13-24, 308 -326 Euler's method, 13 Gear's methods, 22-24, 269, 276-279, 285, 315-326 Michelsen's method, 18, 308, 398, 403 multistep methods, 308, 398, 403 point-slope predictor, 15, 16 Runge-Kutta methods (see Runge-Kutta methods) trapezoidal corrector, 19 two-point implicit method, 21, 52, 94, 129, 160, 179 of partial differential equations, 348 360, 390-393,400-404,408-413,431-433 finite-d~tTerenccmethods (see Finite-difference methods) method of characteristics (see Method of characteristics) orthogonal collocation method (see Orthogonal collocation) O'Connell, H E., 275 O'Connell, J P., 286 Open-boundary system, 11, 12 Orthogonal collocation: method of, 331-348, 398, 400-403 applications, 341 348, 400-403 Orthogonal polynomials, 332 336, 402, 403 Chebyshev, 333 Hermite, 333, 334 Jacobi, 334-336 Laguerre, 332, 333 Legendre, 332 Orye, R V., 286 Padmanabhan, L., 19, 217, 304 Partial molar enthalpies, 127 Peck, R E., 421 Percolation processes, 362 Perfect gases, 449-451 three-dimensional gases, 449, 450 two-dimensional gases, 450, 451 Perfect mixer, 3-5, 11, 12 Periodic operation (see Adsorbcrs) Perry, R H., 103 Peters, W A., Jr., 260 Phase interface, 373 Phase rule, 446, 447 INDEX Pigford, R L., 167, 377 Point-slope predictor, 15, 16 Pore diffusion (see Diffusion) Potential energy, Prausnitz, J M., 286, 439, 442 Prochaska, F., 293 Quadratures, 336-340 gaussian quadrature, 336, 337 Gauss-Jacobi quadrature, 337-340 Rayleigh, Lord, 208 Reed, C E., 377 Reid, R C., 300 Residuals (see Method of weighted residuals) Reynolds number, 275 Richtmyer, R D., 353, 356 Rippin, D W T., 392, 402, 403, 404, 415, 416 Rosenhrock, H H., 19, 308 Rungc-Kutta methods, 17 -19, 301-308 explicit, 17, 302-304 of fourth-order, 17 Michelsen's, 18 of Runge-Kutta-Gill, 17 semi-implicit, 18, 19, 218-229, 265, 304 -308, 393 Sandall, C., 426, 430, 433 Scaling procedures, 57-63 column scaling, 62, 63 row scaling, 60 63 variable scaling, 60-62 Schlucnder, E U., 372 Schmidt, I.' W., 421 Secrest, D., 335 Seinfeld, J H., 32 Semi-implicit Runge-Kutta methods, 18, 19, 218-229, 265, 304-308, 393 generalized algorithm, 222 229 Separation by sublimation (see Frcczc-drying) Shapiro, A H., 356 Sheng, T R., 421 Sherwood, T K., 300, 377 Sieve trays (see Dynamics of sieve trays) Simultaneous differential and algebraic equations (see Coupled differential and algebraic equations) Single-component adsorbers, 374, 384 Sips I1 isotherm, 427 Slusser, R P., 279 Smith, B D., 274, 275 Solid diffusion (see Diffusion) Spiess, F N., 341 Spreading, 439 Spreading pressure, 441 Stability of numerical integration methods: for ordinary differential equations, 25-30 for partial differential equations, 353-356 explicit methods, 353, 354 implicit methods, 355, 356 for stiff differential equations, 29, 30 Stewart, W E., 341 Stiel, L I., 73 Stiff ordinary differential equations, 29, 30 Stroud A H., 335 Sublimation interface, 422, 428, 429 Surface dilfusion (see Diffusion) Surtace tension, 441 System with open boundary, 11, 12 Taylor's theorem, 33 Teller E., 364, 369 Temperature: bubble-point, 129 dewpoint, 129 Tetlow, N J., 268 Tewarson, R P., 248 475 Theta method, 124, 132, 137, 156-159, 183-188 exact solution, 157-159 modified, 156, 157 Thomas algorithm, 131, 132 Trapezoidal corrector, 20 Trapezoidal rule, 359, 360, 409, 410 Van Winkle, J., 300 Van Winkle, M., 274 Vichnevetsky, R., 344 Villadsen, J., 341, 342, 344 Viscous flow, 427 Volmer, M., 366, 439, 459 Waggoner, R C., 153, 208 Watson, G M., 427 Wetting, 439, 440 Wicke, E., 384 Wilke, C R., 430 Wilkes, J O., 327, 336 Wr~ght,K., 341 Yanovich, L A., 361 Zoller gas plant, 254, 255, 258 ... Athanasios I Liapis COMPUTER METHODS FOR SOLVING DYNAMIC SEPARATION PROBLEMS CHAPTER ONE INTRODUCTIONMODELING AND NUMERICAL METHODS An in-depth treatment of both the modeling of dynamic separation processes... INTRODUCTION-MODELING AND NUMERICAL METHODS 13 COMPUTER METHODS FOR SOLVING DYNAMIC SEPARATION PROBLEMS method are used to solve a single differential equation To explain the behavior of these methods, a stability... compute y,,,, which may be ,, 20 INTRODUCTION-MODELING COMPUTER METHODS FOR SOLVING DYNAMIC SEPARATION PROBLEMS AND NUMERICAL METHODS 21 Calculations for the next time step are carried out in the following