COMPUTATIONAL ALGORITHMS for MULTICOMPONENT PHASE EQUILIBRIA and DISTILLATION by MATTHEW OCHUKOH OHANOMAH B.Sc.(Hons.), University of Ife, 1977 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of Chemical Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA November 1981 © Matthew Ochukoh Ohanomah, 1981 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree a t the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make it f r e e l y a v a i l a b l e f o r reference and study. I further agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by t h e head o f my department o r by h i s o r her r e p r e s e n t a t i v e s . understood t h a t copying o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my permission. Department o f CjrJuiY<*,C,oJL The U n i v e r s i t y o f B r i t i s h 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Da Itis £.<Wfrlrvgj^r w Columbia written ABSTRACT T h i s work has (1) To two major o b j e c t i v e s : develop r e l i a b l e , s t a b l e , easily-programmable and computational a l g o r i t h m s that would apply fast to process o p e r a t i o n and design. (2) To determine the best comparative The (1) algorithms following one liquid-liquid, solid — i n up to four phases — with emphasis on l i q u i d - s o l i d and S e n s i t i v i t y analysis i n vapour-liquid (3) Saturation-point (4) Adiabatic (5) Conventional equilibrium-stage The study d e a l s with multicomponent by one vapour, two vapour-liquid, v a p o u r - l i q u i d - l i q u i d systems. (2) constrained through areas: I s o t h e r m a l phase e q u i l i b r i a and operations studies. study covers the liquid for s p e c i f i c equilibria. c a l c u l a t i o n methods. vapour-liquid equilibria. l a c k of i n f o r m a t i o n , distillation-unit calculations. systems and, except where a l l n o n i d e a l i t i e s are rigorously accounted f o r . The p h a s e - e q u i l i b r i a s t u d i e s embody both mass-balance and e n e r g y - m i n i m i z a t i o n methods and sensitivity-analysis some of the a l g o r i t h m s substitution iteration, and a quadratic method have been developed and methods — saturation-points regula like study, based on geometric programming. p r o j e c t i o n methods, f o r a c c e l e r a t i n g m u l t i v a r i a t e The are, falsi and freethe Vector successive- form of Wegstein's p r o j e c t i o n successfully applied. study i n c l u d e s quadratic -ii- three i n t e r p o l a t i o n i n t e r p o l a t i o n s and a dynamic form of Lagrange i n t e r p o l a t i o n — , generally applied a quasi-Newton f o r m u l a t i o n and Newton and Richmond methods. The the r e s u l t s favour the i n t e r p o l a t i o n methods. For adiabatic-flash and distillation-unit calculations, a l g o r i t h m s employing two-dimensional Newton-Raphson, B P - p a r t i t i o n i n g SR-partitioning methods have been developed. equilibrium-ratio c a l c u l a t i o n to once per first and two classes Methods f o r i t e r a t i o n per for reducing enthalpy e v a l u a t i o n and reducing stage i n I n the the third c l a s s , have a l s o been developed. For every problem-type, r e l e v a n t a wide c l a s s of systems and excluded any applications on w i d e - b o i l i n g compared. on distillation. a l g o r i t h m s have been a p p l i e d However, the lack solid-liquid-liquid-vapour of to data equilibria and TABLE OF CONTENTS Page ABSTRACT i i LIST OF TABLES xii LIST OF FIGURES xv ACKNOWLEDGEMENT xlx CHAPTER ONE - INTRODUCTION 1-1 1-2 1 Preamble 1 1-1-1 1-1-2 5 7 Purpose and Scope of Study A Word on the R e s u l t s Isothermal Phase E q u i l i b r i a 1-2-1 1-2-2 1- 2-3 9 The Mass-Balance Approach The F r e e - E n e r g y - M i n i m i z a t i o n Approach The Geometric-Programming Method 10 14 17 1-3 S e n s i t i v i t y Analysis 18 1-4 Bubble- and Dew-Point Calculations 19 1-5 Adiabatic Vapour-Liquid 21 1-6 Multicomponent Multistage D i s t i l l a t i o n with Equilibria Equilibrium stages 22 1-7 Estimation of Physical Properties 24 1- 8 Structure of t h i s Work 26 CHAPTER TWO - ESTIMATION OF PHYSICAL PROPERTIES 29 2- 1 Introduction 29 2-2 Saturation Pressure 29 2-3 Liquid-Phase 30 2- 3-1 2-3-2 Reference Fugacity.... Condensable Components H y p o t h e t i c a l - L i q u i d and Components..... -iv- ' '< 30 Supercritical 32 Page 2-4 Solid-Phase Reference Fugacity 2-5 Vapour Molar Volume and Compressibility 2-5-1 2-5-2 33 Ratio The T r u n c a t e d V i r l a l E q u a t i o n of S t a t e The W i l s o n M o d i f i c a t i o n of the Redlich-Kwong EOS 35 35 38 2-6 L i q u i d Molar Volume 39 2-7 Vapour-Phase Fugacity C o e f f i c i e n t 42 2-8 Liquid-Phase A c t i v i t y C e o f f i c i e n t 43 2-8-1 2-8-2 43 44 2-9 2- 10 The W i l s o n E q u a t i o n The Non-Random-Two-Liquid Model 2-8-3 The U n i v e r s a l Q u a s i c h e m i c a l Model Vapour Molar Enthalpy 45 47 2-9-1 The Departure Function 47 2-9-2 The I d e a l - g a s E n t h a l p y 48 L i q u i d Enthalpy 49 2-10-1 2-10-2 2-10-3 50 51 51 Departure from S a t u r a t i o n Value E n t h a l p y of V a p o r i z a t i o n E n t h a l p y of M i x i n g CHAPTER THREE - ISOTHERMAL VAPOUR-LIQUID FLASH 3- 1 3-2 53 Introduction 53 3-1-1 3-1-2 53 Nomenclature. Systems Employed and a Measure of Nonideality Double-Loop Univariate 3-2-1 3-2-2 3-2-3 3-2-4 3-2-5 3-2-6 3-2-7 56 Methods 63 T h e o r e t i c a l Background A G e n e r a l Form of the A l g o r i t h m A Comparative Study of the D i f f e r e n t Formulations E x p l o r i n g Other P o s s i b i l i t i e s Investigating Different I n i t i a l i z a t i o n Schemes Applications Deductions 64 65 -v- 68 76 85 89 91 Page 3-3 Free-Energy-Minimlzation Methods 3-3-1 3-3-2 3-3-3 3-3-4 3-3-5 3-3-6 3-3-7 3-3-8 3-3-9 3-3-10 3-3-11 3-4 4-3 108 109 I l l 121 122 123 3-4-1 3-4-2 3-4-3 3-4-4 128 129 130 How Richmond-Accelerated Methods Methods U t i l i z i n g the Mean-Value Theorem W e g s t e i n - P r o j e c t e d Methods A Q u a d r a t i c Form of Wegstein's A c c e l e r a t i o n Method A p p l i c a t i o n s and Deductions the D i f f e r e n t Methods Compare A p p l i c a t i o n s and D e d u c t i o n s V a r y i n g the Frequency of K-Computation Conclusions CHAPTER FOUR - SENSITIVITY ANALYSIS IN VAPOUR-LIQUID EQUILIBRIA 4-2 105 128 3-5-1 3-5-2 3- 5-3 4- 1 93 94 96 100 Single-Loop Univariate Methods 3-4-5 3- 5 G e n e r a l Theory The Rand Method A M o d i f i e d Rand Method A Geometric-Programming F o r m u l a t i o n Seeking a S o l u t i o n Method f o r the Geometric-Programming Problem (GPP) Various Successive-Substitution Arrangements of the GPP A c c e l e r a t i n g the GPP by Hyperplane Linearization A c c e l e r a t i n g the GPP by V e c t o r P r o j e c t i o n A Method Based on S e n s i t i v i t y A n a l y s i s I n i t i a l i z a t i o n Schemes A p p l i c a t i o n s and Deductions 92 139 146 148 148 149 153 154 Introduction 154 4- 1-1 154 Nomenclature The Vapour-Liquid Equilibrium Formulation 155 4-2-1 4-2-2 156 158 Eliminating Matrix Inversion A p p l i c a t i o n s and Outcome A Quadratic Taylor-Approximation -vi- 159 Page 4-4 A Mean-Value-Theorem Approach 162 4-4-1 4-4-2 4-4-3 4-4-4 163 167 168 168 Obtaining Algorithm Algorithm Algorithm a Best a A B C 4-5 Error-Tracking 4-6 Predictor-Corrector 4- 7 176 Approach 179 4-6-1 Method 1 179 4-6-2 4-6-3 4- 6-4 Method 2 Method 3 Applications 179 180 180 and Deductions Conclusions 181 CHAPTER FIVE - ISOTHERMAL LIQUID-LIQUID AND LIQUID-SOLID FLASH CALCULATION 5-1 5-2 Introduction 184 5- 1-1 184 Nomenclature Liquid-Liquid E q u i l i b r i a 5-2-1 T h e o r e t i c a l Background 5-2-2 5-2-3 5-3 5-4 184 185 187 O u t l i n e of A l g o r i t h m s NRTL Model and the Problem of M u l t i p l e Solutions 5-2-4 I n i t i a l i z a t i o n Schemes 5-2-5 Applications 5-2-6 Deductions Liquid-Solid E q u i l i b r i a 190 192 203 207 207 5-3-1 5-3-2 5-3-3 5-3-4 5-3-5 207 211 211 214 214 T h e o r e t i c a l Background Choice of A l g o r i t h m s I n i t i a l i z a t i o n Schemes Applications O b s e r v a t i o n s and Deductions Conclusions 187 216 -vii- Page CHAPTER SIX - MULTIPHASE EQUILIBRIA 6—1 6-2 6-3 Introduction. 217 6-1-1 218 Nomenclature The Phase-Fraction Approach 219 6-2-1 Problem F o r m u l a t i o n 220 6-2-2 6-2-3 6-2-4 S o l v i n g By a Newton-Raphson Approach Employing a Quasi-Newton Approach P a r t i t i o n i n g Method w i t h MVT and Tetrahedral Projection 222 223 224 The Geometric-Programming Formulation 225 6-3-1 6-3-2 226 6-3-3 6-4 217 The Three-phase L i q u i d - L i q u i d - V a p o u r Case.... The G e n e r a l i z e d S o l i d - L i q u i d - L i q u i d Vapour Problem S o l u t i o n Methods 230 231 The S e n s i t i v i t y Approach 234 6-4-1 6-4-2 6-4-3 235 237 6-4-4 A G e n e r a l Two-phase Problem A G e n e r a l Three-phase Problem The G e n e r a l i z e d S o l i d - L i q u i d - L i q u i d Vapour Problem The G e n e r a l A l g o r i t h m 239 242 6-5 Some Alternative Formulations 243 6-6 I n i t i a l i z a t i o n Schemes 245 6-7 Applications 249 6- 8 Conclusions 254 CHAPTER SEVEN - BUBBLE- AND DEW-POINT CALCULATIONS 7- 1 255 Introduction 255 7-1-1 256 Nomenclature 7-2 Theoretical Background 257 7-3 A Regula-Falsi Interpolation Method 261 7-4 A Quadratic Interpolation Method 261 -viii- Page 7-5 The Dynamic Lagrange Interpolation Method 262 7-6 Newton's Method 264 7-7 The Third-order Richmond Method 266 7-8 A Quasi-Newton Approach 269 7-9 I n i t i a l i z a t i o n Schemes 276 7-10 Applications 279 7-10-1 7- 10-2 283 284 7-11 Double-Loop A l g o r i t h m s Fixed-Inner-Loop A l g o r i t h m s Conclusions 290 CHAPTER EIGHT - ADIABATIC VAPOUR-LIQUID FLASH CALCULATION 8- 1 293 Introduction.. 293 8- 1-1 293 Nomenclature 8-2 Theoretical Background 295 8-3 Newton-Raphson Methods 298 8-3-1 8-3-2 8-3-3 300 301 303 The C o n v e n t i o n a l F o r m u l a t i o n The B a r n e s - F l o r e s Form A M o d i f i e d Approach 8-4 A Double-Iteration Regula-Falsi Method 309 8-5 Sum-Rates P a r t i t i o n i n g Methods 311 8-5-1 8-5-2 8-5-3 8-6 A S u c c e s s i v e - S u b s t i t u t i o n Method with Vector-Projection Absorption-Factor Formulation Vector-Projection Absorption-Factor Formulation Acceleration 311 with 314 without 315 Bubble-point P a r t i t i o n i n g Methods 315 8-6-1 8-6-2 8-6-3 8-6-4 316 317 322 The C o n v e n t i o n a l Form E l i m i n a t i n g Bubble-point C a l c u l a t i o n Reducing K Computation E l i m i n a t i n g D i r e c t K-T D e r i v a t i v e Calculation -ix- 324 Page 8-7 8- 8 Applications 325 8-7-1 8- 7-2 326 328 Convergence C r i t e r i a R e s u l t s and O b s e r v a t i o n s Conclusions 334 CHAPTER NINE - DISTILLATION-UNIT CALCULATION 9-1 338 Introduction 338 9- 1-1 9-1-2 339 Nomenclature Problem S p e c i f i c a t i o n and Working Equations 342 9-2 BP Methods — A General Algorithm 347 9-3 BP Methods — Updating Total-Flow P r o f i l e s 348 9-4 BP Methods — Updating Temperature P r o f i l e 348 9-4-1 9-4-2 9-4-3 9-5 9-5-2 9-7 b BP Methods — 9-5-1 9-6 A Bubble-point-Temperature Approach The K Method The One-Step N e w t o n - I t e r a t i o n Approach Updating the Composition P r o f i l e s . . . . M o d i f i e d Thomas A l g o r i t h m w i t h the 6 Method of Convergence An A l t e r n a t i v e Composition U p d a t i n g Scheme 349 349 353 359 359 362 SR-type Solution Methods 372 9-6-1 9-6-2 9-6-3 372 372 A General Algorithm Updating the Temperature P r o f i l e U p d a t i n g the Flow and C o m p o s i t i o n Profiles Boiling-Range-Unlimited 9-7-1 9-7-2 Methods Tomich's 2N Newton-Raphson Method A Stage-wise Two-dimensional NewtonRaphson Approach -x- 376 378 379 383 Page 9-8 Applications 9-8-1 9-8-2 9-9 I n i t i a l i z a t i o n Schemes and Convergence Criteria Results Conclusions 389 389 397 400 CHAPTER TEN - GENERAL CONCLUSIONS AND RECOMMENDATIONS 401 BIBLIOGRAPHY 406 APPENDICES 411 A G e n e r a l Notes on Programming B Availability 411 of Programs and Some S p e c i f i c Program D e t a i l s 415 C The Geometric Programming Concept 427 D The S o l i d - p h a s e Reference F u g a c i t y 430 E Sources of Data 438 LIST OF TABLES Table Page 3-1 Vital 3-2 Computation times (CPU seconds) f o r doubleloop u n i v a r i a t e a l g o r i t h m s 90 Computation times (CPU seconds) f o r f i r s t phase of a p p l i c a t i o n of f r e e - e n e r g y m i n i m i zation algorithms 125 Computation times (CPU seconds) f o r the f i n a l comparison of f r e e - e n e r g y m i n i m i zation algorithms 127 Computation times (CPU seconds) f o r s i n g l e loop u n i v a r i a t e a l g o r i t h m s 147 Absolute errors frequencies 151 3-3 3-4 3-5 3-6 3- 7 4- 1 4-2 4-3 4-4 4-5 4- 6 i n f o r m a t i o n on v a p o u r - l i q u i d systems 57 (E) f o r v a r i o u s K-computation F i n a l comparison of v a p o u r - l i q u i d Computation times (CPU seconds) algorithms: 152 E r r o r s f o r s e n s i t i v i t y a n a l y s i s ( o r i g i n a l GP v e r s i o n ) f o r system VA, w i t h T* = 336.5K 160 Errors f o r s e n s i t i v i t y - a n a l y s i s Algorithms A and B f o r system VA, w i t h T* = 336.5K 169 Errors f o r s e n s i t i v i t y - a n a l y s i s Algorithm C f o r system VA w i t h T* = 336.5K 174 E r r o r s f o r a more comprehensive a p p l i c a t i o n of s e n s i t i v i t y - a n a l y s i s A l g o r i t h m C 175 E r r o r - t r a c k i n g r e s u l t s (system VA w i t h T* = 336.5K and AT = 2K) 178 Computation times (CPU seconds) f o r P r e d i c t o r - C o r r e c t o r algorithms i n f o r m a t i o n on l i q u i d - l i q u i d 182 5- 1 Vital systems 5-2 8, and i t e r a t i o n count (I.C.) f o r d i f f e r e n t 186 Li initialization schemes -xii- 205 Table r 3-J 0 Computation times (CPU seconds) f o r l i q u i d l i q u i d systems Vital 5- 5 Computation times (CPU seconds) f o r l i q u i d s o l i d systems 215 I t e r a t i o n counts o b t a i n e d from a p p l y i n g the two GP arrangements to l i q u i d - l i q u i d - v a p o u r systems 233 I t e r a t i o n counts f o r d i f f e r e n t i n i t i a l i z a t i o n schemes, a p p l i e d to l i q u i d - l i q u i d - v a p o u r systems.... 248 I t e r a t i o n counts f o r the v a r i o u s m u l t i p h a s e a l g o r i t h m s , based on assumption of the r i g h t number of phases 251 I t e r a t i o n counts f o r the v a r i o u s m u l t i p h a s e a l g o r i t h m s , based on assumption of redundant phases. 253 Computation times (CPU seconds) f o r b u b b l e - p o i n t c a l c u l a t i o n based on the d o u b l e - i t e r a t i o n algorithm 285 Computation times (CPU seconds) f o r dew-point c a l c u l a t i o n based on the d o u b l e - i t e r a t i o n algorithms 286 Computation times (CPU seconds) f o r bubblep o i n t c a l c u l a t i o n based on a l g o r i t h m s with i n n e r i t e r a t i o n f i x e d at 2 288 Computation times (CPU seconds) f o r dew-point c a l c u l a t i o n based on a l g o r i t h m s w i t h i n n e r i t e r a t i o n f i x e d at 2 289 Computation times (CPU seconds) f o r b u b b l e - p o i n t c a l c u l a t i o n based on a l g o r i t h m s with i n n e r i t e r a t i o n bounded at 4 291 Computation times (CPU seconds) f o r dew-point c a l c u l a t i o n based on a l g o r i t h m s w i t h i n n e r i t e r a t i o n bounded a t 4 292 Vital 294 6-2 6-3 6- 4 7- 1 7-2 7-3 7-4 7-5 7-6 8-1 systems. 206 5-4 6- 1 i n f o r m a t i o n on l i q u i d - s o l i d Page i n f o r m a t i o n on a d i a b a t i c - f l a s h systems. -xiii- 208 Table 8-2 Page I t e r a t i o n counts f o r d i f f e r e n t c h e c k - f u n c t i o n f o r m u l a t i o n s f o r one-step N e w t o n - i t e r a t i o n update of temperature i n BP a l g o r i t h m s 323 Reference names and d e s c r i p t i o n s of a d i a b a t i c f l a s h algorithms 327 Computation times (CPU seconds) f o r a d i a b a t i c f l a s h algorithms 329 8-5 Comparison 331 8-6 Computation times (CPU seconds) f o r a d i a b a t i c f l a s h a l g o r i t h m s f o r n a r r o w - b o i l i n g systems 333 I t e r a t i o n counts f o r the two v e r s i o n s of 'SR a l g o r i t h m 3' without temperature damping 335 8-3 8-4 8-7 8- 8 Computation of i t e r a t i o n counts f o r BP a l g o r i t h m s times (CPU seconds) f o r a d i a b a t i c - f l a s h a l g o r i t h m s f o r w i d e - b o i l i n g systems 336 9- 1 Vital 341 9-2 I t e r a t i o n counts f o r one-step N e w t o n - i t e r a t i o n temperature-updating schemes 358 I t e r a t i o n counts f o r a l t e r n a t i v e - c o m p o s i t i o n u p d a t i n g convergence schemes 369 Code names and d e s c r i p t i o n s of the d i s t i l l a t i o n u n i t computational a l g o r i t h m s 390 X v a l u e s f o r the d i f f e r e n t i n i t i a l i z a t i o n schemes 394 9-3 9-4 9-5 9-6 9-7 i n f o r m a t i o n on d i s t i l l a t i o n T systems temperature- I t e r a t i o n counts f o r d i f f e r e n t i n i t i a l i z a t i o n schemes temperature396 Computation times (CPU seconds) and i t e r a t i o n counts ( i n p a r e n t h e s i s ) f o r n a r r o w - b o i l i n g systems 398 B-l I n f o r m a t i o n on a c c e s s i b l e programs 416 E-l Source r e f e r e n c e s f o r data employed -xiv- i n study 438 LIST OF FIGURES Figure 3-1 3-2 3-3 Page Two-phase r e g i o n temperature p r o f i l e f o r narrowb o i l i n g systems 59 Two-phase r e g i o n temperature p r o f i l e f o r wideb o i l i n g systems 60 K 61 V. 9 r 3-4 3-5 3-6 3-7 3-8 3-9 3-10 3-11 3-12 f o r the d i f f e r e n t v systems at 9* « 0.5 v P l o t of n o n i d e a l i t y parameter v e r s u s p o i n t vapour f r a c t i o n solution62 Standard form c h e c k - f u n c t i o n p r o f i l e s f o r system VA 69 Standard l o g a r i t h m i c c h e c k - f u n c t i o n f o r system VA 70 profiles Rachford-Rice check-function p r o f i l e s f o r system VA 71 Barnes-Flores check-function p r o f i l e s f o r system VA 72 Comparing at 340K 74 the f o u r f o r m u l a t i o n s f o r system VA Comparing the f o u r f o r m u l a t i o n s f o r system VA at 342.5K 75 Iteration units vapour f r a c t i o n method.) 77 (on a s c a l e of 0 t o 100) v . f o r system VC (based on Newton's I t e r a t i o n u n i t s v. vapour f r a c t i o n : comparison between Newton and Richmond f o r system VB 78 f, v. 9 f o r d i f f e r e n t v a l u e s of n f 6 r system VA a t 341.66K 81 3-14 < | > v. 8^ f o r d i f f e r e n t 83 3-15 f. v. 9 f o r d i f f e r e n t <p v VA a t 341.66K 3-13 -xv.- forms of < | > forms of <b f o r system 84 Figure 3-16 Page o * 9 v . 9 f o r systems VA and VP based on v v z a t i o n Schemes 3 and 4 initiali88 3-17 Solution-space triangle 114 3-18 Disparity-space triangle 114 3-19 Disparity-space 117 3-20 D i s p a r i t y - s p a c e t e t r a h e d r a l base 119 3-21 G o l d e n - s e c t i o n s e a r c h f o r o p t i m a l MVT parameter 131 3-22 9 v. v g 8 tetrahedron for different v at T corresponding 3-23 3-24 9 v v. 9 v v. s s 9 9 v a l u e s of t f o r system VA g to 9^ « 0.35 134 f o r t = 1 and t = 20 f o r system VA 135 f o r t= 1 and t = 20 f o r system VP 136 v v 3-25 t p r o f i l e s f o r systems VA and VP 3-26 9 v . 9 f o r the R i c h m o n d - a c c e l e r a t e d s t a n d a r d v v formulation 3-27 3-28 3-29 138 g g 8 v . 8 f o r the N e w t o n - a c c e l e r a t e d v v formulation 140 standard 8 v . 8 f o r the R i c h m o n d - a c c e l e r a t e d R a c h f o r d v v R i c e form 141 s 142 g 8 v . 8 f o r the N e w t o n - a c c e l e r a t e d R a c h f o r d v v R i c e form 143 3-30 8 v . 9 f o r the M V T - a c c e l e r a t e d R a c h f o r d - R i c e v v form. 144 4-1 F i t of m e a n - v a l u e - t h e o r e m p a r a m e t e r , a , f o r system VA 166 S -xv i - Figure Page 4- 2 A n K - v e r s u s - T p a r t i t i o n i n g scheme 5- 1 f ( 9 L ) v e r s u s e f L o r t h 171 system LA at 298K e (based on NRTL) 5-2 F ^ L^ * 6 v 9 L' u s i n 191 § t h e UNIQUAC model, with K ' s c o r r e c t e d f o r c o m p o s i t i o n dependence 5 3 L F ( 9 ) ' L» V L 6 u s i n § t h e UNIQUAC model, 193 with K ' s u n c o r r e c t e d f o r c o m p o s i t i o n dependence 5-4 f(8 ) v. Li 9 Li 194 u s i n g the NRTL model, w i t h K ' s c o r r e c t e d f o r c o m p o s i t i o n dependence 195 f(8 ) v. 9 u s i n g the NRTL model, w i t h L L K ' s u n c o r r e c t e d f o r c o m p o s i t i o n dependence 196 5-6 T e m p e r a t u r e - s o l i d - f r a c t i o n p r o f i l e f o r the s o l i d systems 209 5-7 f ( 9 ) v e r s u s 9 f o r the system SA s s 212 5-8 f ( 9 ) v. s 213 7-1 The B u b b l e - p o i n t check f u n c t i o n f o r system VA at P = 1.0 Atm 259 7-2 The Dew-point check f u n c t i o n f o r system VA at P = 1.0 Atm 260 7-3 P e r c e n t A b s o l u t e d e v i a t i o n of approximate K v a l u e s [ E q . ( 7 - 1 9 ) ] from exact v a l u e s as f u n c t i o n s of Temp, f o r system VD, u s i n g 370K as base t e m p e r a t u r e . . 267 7-4 t r v e r s u s r f o r p o l y n o m i a l s of d i f f e r e n t 272 7-5 I t e r a t i o n - u n i t s - v e r s u s - t p r o f i l e r e s u l t i n g from the golden s e c t i o n o p t i m i z a t i o n scheme 7-6 V a r i a t i o n of 5-5 8 f o r system SB s T. . - T , w i t h system p r e s s u r e | init sol| 3 f o r system VC 7-7 V a r i a t i o n of f o r system VA degrees T. . I init T v \ w i t h system p r e s s u r e soil -xvii- 275 280 n 281 Figure 7- 8 Page V a r i a t i o n of J T for I N I T - T G O L | w i t h system p r e s s u r e system VP 282 8- 1 An a d i a b a t i c f l a s h drum 8-2 V a r i a t i o n of E n t h a l p i e s w i t h temperature c o n s t a n t composition 8- 3 V a r i a t i o n of system average K w i t h at constant c o m p o s i t i o n 296 Distillation 9-2 Model assumed f o r f e e d - p l a t e behaviour 9-3 Normalized r e l a t i v e v o l a t i l i t y versus t u r e f o r system DA 9-5 D-l D-2 306 temperature 9- 1 9-4 at 307 column c o n f i g u r a t i o n 343 344 tempera352 P r o f i l e s of check f u n c t i o n r e p r e s e n t e d by E q u a t i o n ( 9 - 4 1 ) at f i r s t i t e r a t i o n 367 V a r i a t i o n of 8 w i t h i t e r a t i o n f o r Convergence Schemes 1 and 2 368 Comparison of e r r o r s i n f°/f° f o r t h r e e systems J-i s included i n a-correlation f i t 436 Comparison of e r r o r s i n f°/f° f o r t h r e e systems lj s not i n c l u d e d i n a - c o r r e l a t i o n f i t 437 -xviii- ACKNOWLEDGEMENT Any report of t h i s nature i s the product of the e f f o r t s of many more people than j u s t i t s acknowledged author. This report exception. I would l i k e to express my g r a t i t u d e contributed i n some p o s i t i v e way, no matter how s m a l l , i s no to everyone who has towards i t s making. I am e s p e c i a l l y i n d e b t e d t o Dr. D.W. Thompson, who t h i s work, f o r h i s unremitted moral support, frequent s u g g e s t i o n s and g e n e r a l u n a l l o y e d by co-operation. supervised helpful The c o n t r i b u t i o n the members of my t h e s i s committee i n the process of g i v i n g report a f i n i s h i n g touch i s a l s o The made this appreciated. s t a f f of the Department of Chemical E n g i n e e r i n g , The U n i v e r s i t y L i b r a r y and the Computing Centre ( a l l o f the U n i v e r s i t y of B r i t i s h Columbia) have been e s p e c i a l l y c o - o p e r a t i v e and I am t h a n k f u l t o them. S p e c i a l acknowledgement a l s o goes to the Canadian Government f o r providing me w i t h f i n a n c i a l sustenance (through the Canadian Commonwealth S c h o l a r s h i p Association), and t o the U n i v e r s i t y Columbia f o r supplementary f i n a n c i a l s u p p o r t . - xix - of B r i t i s h CHAPTER ONE INTRODUCTION 1-1 Preamble The i n d u s t r i a l world has over the l a s t (1) a realignment (2) the advent The advocates few first and decades. The experienced a significant two main agents of change are: i n s o c i a l p e r s p e c t i v e , and of the computer. f a c t o r i s the product environmental groups, who of the a c t i v i t i e s of consumer have generated increased public awareness, which has i n t u r n p r e s s u r e d l e g i s l a t o r s i n t o i n c r e a s i n g l y more s t r i n g e n t quality r e g u l a t i o n s on environmental imposing and product- control. The cherished based reorientation industrialists are thus b e g i n n i n g to r e a l i z e t h a t the much- ' s a f e t y - f a c t o r approach' to process d e s i g n , whereby designs on h i g h l y - c r u d e methods are c e r t i f i e d good through of a l a r g e ' s a f e t y f a c t o r ' , are l e s s than s a f e . t h e i r a c t i v i t i e s w i t h i n the l i m i t s that ensure the i m p o s i t i o n I f they must that n e i t h e r human h e a l t h , nor the n a t u r a l elements w i t h which the e f f l u e n t s from p r o c e s s p l a n t s u l t i m a t e l y blend, are unduly is the need to adopt c o n t r o l of the v a r i o u s process T h i s b r i n g s us t o the second with i t a gradual s h i f t factor. The units. computer has i n p r o c e s s - d e s i g n methods — T h i s trend i s s i g n i f i c a n t - 1- i n more ways than brought from simple one: there guarantee approximate g r a p h i c a l methods to complex but f a r more r e l i a b l e methods. the i n t e r f e r e d w i t h , then more s o p h i s t i c a t e d d e s i g n methods that v e r y e f f e c t i v e o p e r a t i o n and control but computer (1) I t constitutes an agreeable response to the dictates of the new s o c i a l order (of environmental groups and consumer advocates). (2) I t makes f o r large-scale and more economically f e a s i b l e designs. (3) In terms of design e f f o r t as w e l l as process operation, the drudgery i s s h i f t e d onto the machine; by so doing, many man-hours are saved. I t w i l l not be far-fetched to conjecture that i n the not-too- d i s t a n t future, we s h a l l a r r i v e at a point where any process design that i s less than optimal w i l l be regarded with displeasure. P o s i t i v e i n d i cators i n t h i s d i r e c t i o n are provided by: (1) The tremendous amount of e f f o r t that has been devoted, over the past twenty years, to the development of improved mathematical models f o r the p r e d i c t i o n of the properties of substances. (2) The sustained i n t e r e s t i n the development of computational algorithms f o r handling the design and operation of d i f f e r e n t types of (3) process u n i t s . The recent and increasing devotion to the study of techniques f o r the synthesis of optimal chemical processes (see, f o r example, Thompson and King, 1972; Hendry and Hughes, 1972; Fernando et a l . , 1975; Westerberg and Stephanopoulos, 1975; Nishida and Powers, 1978; O s t r o v s k i i and Shevchenko, 1979). A chemical process plant i s a c o l l e c t i o n of reactors, separation u n i t s and a n c i l l a r y equipment ( f o r mass, momentum and energy t r a n s p o r t ) , j u d i c i o u s l y combined to e f f e c t , at reasonable cost, the elevation of the utility-', of some material by converting i t from i t s 'raw' state to a desirable ' f i n i s h e d ' product. Any serious attempt at the design or - 3 - analysis of any of the often-multitudinous process units that constitute the process plant involves the solution of a large number of highly nonlinear mathematical relationships. Because of their multivariate and nonlinear c h a r a c t e r i s t i c s , their solution i s necessarily of an i t e r a t i v e nature. algorithm. This c a l l s for an e f f i c i e n t solution path or To be considered e f f i c i e n t , one would expect a computer-oriented algorithm to s a t i s f y the following c r i t e r i a : (1) I t should be r e l i a b l e — (2) It should be stable — oscillatory (3) converging to the right solution. preferably yielding smooth non- convergence. I t should be applicable to a reasonably large number of chemical systems. (4) I t should be easily programmable. (5) I t should be able to locate the solution i n a reasonable period of time. The t h i r d condition arises from the fact that while the o v e r a l l configuration of a plant depends on the p a r t i c u l a r chemical system being processed, the building blocks — process units — are not unique to the plant, with the result that algorithms are developed not with any p a r t i cular chemical system i n mind, but on the basis of the momentum-, massand heat-transfer principles underlying the operation of the process unit. The fourth condition i s s t r i c t l y for user convenience and need not be a constraining determinant, since an algorithm that shows superiority over a l l competing algorithms except i n complexity can easily be converted by a programming expert into a readily-accessible computer package. -4- An for algorithm w i l l , of course, have to s a t i s f y a g i v e n system f o r i t to be a p p l i c a b l e at a l l . standing i n r e s p e c t of the f i r s t and the f i f t h An satisfies the f i r s t a computer t i m e - s c a l e , d i f f e r e n c e of a decisecond, nay a centisecond, (1) Any plant design tations — optimal (2) The considers on system, only i s the computation time. on a high-speed time digital i n the comparison of a l g o r i t h m s small-scale process-operation comes apparent when one For any a c e n t r a l - p r o c e s s i n g - u n i t (CPU) computer c o u l d be q u i t e s i g n i f i c a n t handling i s dependent c o n d i t i o n , then the l o g i c a l q u a n t i t a t i v e measure of i t s e f f i c i e n c y On criterion algorithm's criteria the degree to which i t meets the second c o n d i t i o n . g i v e n that an a l g o r i t h m the f i r s t problems. for T h i s s i g n i f i c a n c e be- that: i n v o l v e s a very l a r g e number of such compu- more so i f the u l t i m a t e g o a l i s to achieve design. f i n a n c i a l l y - c o n s t r a i n e d d e s i g n e r who i s not b l e s s e d with a high-speed computing u n i t c o u l d i n c u r on h i s modest computer a time m u l t i p l i c a t i v e - f a c t o r of the order The l i t e r a t u r e i s r e p l e t e with a l g o r i t h m s given process-design it necessary subjected of one that any calculation. new to a comparative t e s t . f o r performing This m u l t i p l i c i t y algorithms, The before million. they of a l g o r i t h m s are absence of any any makes 'marketed', reliable be compara- t i v e s t u d i e s c o n s t i t u t e s a problem t o : (1) The a l g o r i t h m user, who would have to p a i n s t a k i n g l y go mountains of p u b l i s h e d work i n order more o f t e n than not, (2) The ending up w i t h through to s e l e c t an a l g o r i t h m the wrong c h o i c e . i n d u s t r y , which i s robbed of v i t a l man-hours due consequent u n d e r u t i l i z a t i o n of i t s employees. to the — - 5 - The l i t e r a t u r e reveals that most of the comparative s t u d i e s have been undertaken on computational a l g o r i t h m s that have been founded on those same assumptions of i d e a l i t y or s e m i - i d e a l i t y that render the g r a p h i c a l methods u n s u i t a b l e ions f o r h i g h - q u a l i t y d e s i g n work. that d e r i v e from such s t u d i e s cannot but be viewed w i t h because the r e l a t i v e performance of the a l g o r i t h m s d i f f e r e n t when they are a p p l i e d to r e a l 1-1-1 parative c o u l d be q u i t e systems. study has been d i r e c t e d towards the f o r m u l a t i o n study of computational a l g o r i t h m s the p r i n c i p a l o b j e c t i v e s (1) and a n a l y s t s To f u r n i s h the d e s i g n e r s algorithms and t h a t , i n a d d i t i o n to being reliable, stable, v e r s a t i l e f a s t as to save on costs. To e l i m i n a t e much of the c o n f u s i o n or misguidance plagues the d e c i s i o n on the c h o i c e of a l g o r i t h m specific The being: of process u n i t s w i t h easily-programmable, a r e a l s o s u f f i c i e n t l y computer and the com- that a r e a p p l i c a b l e t o a number of process o p e r a t i o n s , (1) skepticism Purpose and Scope o f Study This (2) The c o n c l u s - that f o r performing operations. study covers the f o l l o w i n g areas: A thorough i n v e s t i g a t i o n of the i s o t h e r m a l vapour-liquid f l a s h problem, w i t h new methods developed based on the theory of geometric programming, and w i t h new techniques i n t r o d u c e d f o r accelerating multivariate successive-substitution (2) An e x p l o r a t i o n of the u t i l i t y iterations. o f the s e n s i t i v i t y - a n a l y s i s method of geometric programming i n so f a r as i t a p p l i e s to the isothermal vapour-liquid f l a s h problem. - 6 - (3) A study of isothermal l i q u i d - l i q u i d and l i q u i d - s o l i d equili- b r i a , drawing on the experience gained from the vapour-liquid case. (4) An extension of the methods studied for two-phase e q u i l i b r i a to v a p o u r - l i q u i d - l i q u i d - s o l i d e q u i l i b r i a , with the algorithms formulated to be able to also handle any two- and three-phase system, so long as i t contains at least one l i q u i d phase. (5) A quite thorough investigation of the adiabatic vapour-liquid f l a s h problem, with the introduction of a number of time-saving techniques. (6) A comparative study of different methods — newly introduced i n this study — some of them of handling bubble- and dew-point calculations. (7) An investigation of computational methods for handling the conventional distillation-column operation problem, with applications to the s p e c i f i c case of one with a p a r t i a l condenser and with d i s t i l l a t e rate and reflux r a t i o s p e c i f i e d . For each of the investigations itemized above, the following experimental path was adopted: (a) New methods are developed and implemented. (b) The best of the existing methods i s implemented as documented i n the l i t e r a t u r e . Where no comparative studies exist or where the outcomes of such studies are considered inconclusive, the most promising or the most widely-applied methods are implemented. (c) The methods of Step b are studied c r i t i c a l l y and, where possible, modifications aimed at improving them are made. The modified versions are then implemented. - (d) 7 - The methods of Steps a through c are a p p l i e d to a number of systems w i t h a view to d e t e r m i n i n g the best s o l u t i o n method. 1-1-2 A Word on t h e R e s u l t s The CPU time consumed i n the e x e c u t i o n of an a l g o r i t h m ( w i t h the time f o r d a t a i n p u t excluded) ciency, given that i t was chosen as the measure of i t s i s r e l i a b l e , s t a b l e and v e r s a t i l e . employed i s AMDAHL-47,0.) Unfortunately, i t was o b s e r v e d , (The efficomputer at a l a t e stage i n the s t u d y , t h a t the PL/1 (F - v e r s i o n 5 . 5 ) c o m p i l e r , which was used i n t h i s s t u d y , c o n t a i n e d a bug w h i c h , where i t o c c u r r e d , had a way of a p p r o x i m a t e l y d o u b l i n g the e x e c u t i o n t i m e . Every e f f o r t was made, c o l l a b o r a t i o n w i t h the s t a f f of the u n i v e r s i t y computing c e n t r e , e l i m i n a t e the problem - w i t h o u t t h a t the bug, where i t o c c u r r e d , success. However, i t was where i t did a f f e c t a l l three o p t i o n s , of cases to e l i m i n a t e i t program. It to discovered d i d not always a f f e c t a l l o p t i o n s of the c o m p i l e r t h a t are a v a i l a b l e . in three was a l s o found that i t was p o s s i b l e i n a good number from at l e a s t one of them by r e s t r u c t u r i n g The approach adopted i n every one of the comparisons the thus involved: (a) C o m p i l i n g each program u s i n g a l l t h r e e a v a i l a b l e c o m p i l e r options. (b) A p p l y i n g a l l t h r e e c o m p i l a t i o n s to one system and p r i n t i n g the e x e c u t i o n times as w e l l as the number of iterations required. (c) Determining, through a comparison of the e x e c u t i o n times as w e l l as the i t e r a t i o n s f o r a l l a l g o r i t h m s of the same c l a s s and by a rough a n a l y s i s of the computation e f f o r t per i t e r a t i o n r e q u i r e d by each of them, whether the bug has i n t e r f e r e d or not. - 8 - (d) R e s t r u c t u r i n g those all three c o m p i l a t i o n l e a s t one (e) programs f o r which the bug options u n t i l algorithm, the minimum e x e c u t i o n a p p l i c a t i o n of the T h i s approach was and, than 200 f o r these, time. For f l a s h algorithms, lines. sented. For time and found to work q u i t e w e l l f o r the f a i r l y programming l i n e s f o r the i t e r a t i o n to f i n d , T h i s was i n every the case w i t h execution quite d i f f e r e n t , be- a program s t r u c t u r e that the comparison of m u l t i p h a s e the comparison on d i s t i l l a t i o n - u n i t not 340 calculation, involving l e n g t h ( r a n g i n g from 200 c e r t a i n i f the bug was to e l i m i n a t e d i n every i n terms of both e x e c u t i o n time 360 case, and and count. All the a l g o r i t h m s implemented i n t h i s work have been based r i g o r o u s methods that i n v o l v e no a l l n o n i d e a l i t i e s , using property-estimation as one s i m p l i f y i n g assumptions and The methods should emphasis attached account f o r not to the c h o i c e be misconstrued the p r e d i c t i v e c o r r e l a t i o n s . Far from i t . borne of the b e l i e f , This of to mean that of I t s o b j e c t i v e s a ' g o o d n e s s - o f - f i t ' study w o r l d s ' approach was on the best methods c u r r e n t l y known f o r e s t i m a t i n g the p r o p e r t i e s of substances. work has routine) only the i t e r a t i o n counts have been pre- the r e s u l t s have been presented iteration case, small where the i t e r a t i o n r o u t i n e s ranged i n l e n g t h from I n t h i s case, i t was gives algorithm. i t e r a t i o n r o u t i n e s of i n t e r m e d i a t e lines), o p t i o n that u s i n g that i n every l a r g e programs, the s t o r y was impossible e l i m i n a t e d the bug. to 450 the compiler the comparisons have been based s o l e l y on the very cause i t was i t i s e l i m i n a t e d from at option. Choosing, f o r any programs ( l e s s has a f f e c t e d of any this of 'best-of-all-known- expressed e a r l i e r on, that any -9- computational it a l g o r i t h m cannot be s a i d i s s u b j e c t e d to the same treatment hands of the u l t i m a t e 1-2 'consumer': the to have been t r u l y tested unless as i t would experience at designer. Isothermal Phase E q u i l i b r i a Phase-equilibrium calculation constitutes a s i g n i f i c a n t of the work i n v o l v e d i n chemical process tion. the I t f e a t u r e s prominently d e s i g n , m o d e l l i n g and simula- i n g e n e r a l m a t e r i a l - t r a n s p o r t con- s i d e r a t i o n s where the f o r m a t i o n of more than one the flow l i n e c a l l s component phase anywhere along f o r a knowledge of the r e l a t i v e d i s t r i b u t i o n of the components i n the d i f f e r e n t phases, without which knowledge a d e c i s i o n cannot be made as to the best mode of t r a n s p o r t a t i o n and of m a t e r i a l treatment. It i s even more important where the o b j e c t i v e c o u l d be ture i n t o a l i g h t drums and the s e p a r a t i o n of a vapour or l i q u i d mix- (vapour) phase and a heavy ( l i q u i d ) phase (as i n f l a s h d i s t i l l a t i o n columns); or the r e d i s t r i b u t i o n of the components of a mixture between two liquid or between a s o l i d phase and lization, l e a c h i n g and phases (as i n l i q u i d - l i q u i d extraction), a l i q u i d phase (as i n f r a c t i o n a l crystal- a d s o r p t i o n ) , or between a l i q u i d phase and vapour phase (as i n absorbers Nor i n the d e s i g n of s e p a r a t i o n u n i t s , and a strippers). can i t s importance be g a i n s a i d i n complex chemical r e a c t i o n s where the thermodynamics of the system might be such as to l e a d to the f o r m a t i o n of more than one U n t i l about two phase. decades ago, were based on the s o - c a l l e d a l l phase-equilibrium computations 'mass-balance' method, which i n v o l v e s com- b i n i n g e q u i l i b r i u m , mass-balance and m o l e - f r a c t i o n - b a l a n c e relationships - 10 - to yield check functions whose dimensions are one less than the number of existing phases, and with phase fractions as the independent variables. The chemical-equilibrium equivalent of this is the mass-action law. In 1958, Dantzig and co-workers (see Zeleznik and Gordon, 1968), drawing on the enlightening work of J. W. Gibbs (see Shapley and Cutler, 1970) who had shown the relationship between the Gibbs free-energy function and the mass-action laws around the turn of the century, pioneered research in the application of the free-energy-minimization method to the solution of the chemical-equilibrium problem. In view of the similarities between the physical- and the chemical-equilibrium problems, the free-energy minimization approach has also been successfully applied to the former type of problem. Thus, the two methods — the mass-balance method and the free-energy minimization approach — form the core to a l l phase-equilibrium computational algorithms today. 1-2-1 The Mass-balance Approach Much of the effort that has been put into studying the applica- tion of the mass-balance method to solving the phase-equilibrium problem has been devoted to two-phase vapour-liquid equilibrium. For this type of problem, the standard form of the check function (for derivation, see Section 3-2-1) assumes the form: N N 1+(K -1)6 } " 1 i v (1-1) -11- where: is the e q u i l i b r i u m r a t i o f o r component i , N i s the number of components i n the system, x-£ denotes the l i q u i d mole f r a c t i o n of i , and zi i s the system mole f r a c t i o n of i , 6 i V s the vapour f r a c t i o n on a molar basis. R a c h f o r d and R i c e (1952) have proposed another type of check f u n c t i o n which i s r e p o r t e d to be more l i n e a r and thus to converge than the standard form. faster I t i s of the form: ( N F(6 ) = K y ^ ) v l ) Z l } l{ = i-1 V i=l 1 (1-2) V where y^ i s the m o l e - f r a c t i o n of component i i n the vapour phase. While the o r i g i n a t o r s o f the above check f u n c t i o n employed search method i n s o l v i n g f o r 0 , V its a dichotomous other a p p l i c a n t s of the method since i n t r o d u c t i o n have found i t more s u i t a b l e to apply e i t h e r the Newton method or the t h i r d - o r d e r Richmond method ( f o r a treatment of the two methods, see L a p i d u s , 1962). Sanderson and Chien (1973) have developed an a l g o r i t h m , f o r simultaneous c h e m i c a l - and p h a s e - e q u i l i b r i u m problems, that entails p e r f o r m i n g a f l a s h c a l c u l a t i o n w i t h F ( 0 ) as d e f i n e d i n E q u a t i o n V (1-2). A p o l y n o m i a l expansion of E q u a t i o n (1-2) has a l s o been s t u d i e d Rosenberg (1963, 1977). competitive. I t has turned out, however, not to be by - 12 - A multivariate Newton-Raphson formulation of the mass-balance method (Nagata and Gotoh, 1975) has also been applied to different classes of problems by various workers, including Edwards et al (1978), and Hirose et al (1978). Here the equilibrium, material-balance and mole-fraction-balance equations are individually expressed in the form of check functions, each equation contributing one dimension to the Newton-Raphson Jacobian matrix. The two main limitations of this form- ulation are: (a) The high dimensionality of the working matrix, whose inverse has to be determined. (b) The problem of a reasonable i n i t i a l guess for the independent-variable vector, without which convergence might not be achieved. Quite recently, King (1980) reported on a new formulation by Barnes and Flores which its authors claim is superior to that of Rachford and Rice. F(6 ) = y in[I y ] i=l ± It is of the form: - *n[! x ] ± i=l - ln[\ i=l ' f t ] - ln[\ i v i=l '* i v ] (1-3) In a comparative study undertaken by Rohl and Sudall (1967), the authors dismissed the standard formulation as not being capable of converging, and concluded that the best result is produced by the Rachford-Rice formulation accelerated by either the second-order Newton method or the third-order Richmond method. Their conclusion regarding the standard formulation i s , however, not right for, as Holland (1975) rightly asserts, this formulation converges to the desired solution i f the vapour fraction is initialized to 1. The conclusiveness of the study by Rohl and Sudall is further undermined by the fact that It -13- preceded and t h e r e f o r e excludes the method of Barnes and F l o r e s . Relatively l i t t l e work has been done i n the area of a p p l i c a t i o n to l i q u i d - l i q u i d s i v e and comprehensive and co-workers equilibrium calculation. computer The most e x t e n - seems to be that r e c e n t l y p u b l i s h e d by P r a u s n i t z (1980). In the a l g o r i t h m p u b l i s h e d by these workers, the check f u n c t i o n takes the f orm of E q u a t i o n (1—2) w i t h the vapour phase r e p l a c e d by a second l i q u i d phase and the phase parameters redefined. (the F o r any g i v e n set of K v a l u e s , appropriately the e q u a t i o n i s s o l v e d f o r phase f r a c t i o n f o r the second l i q u i d phase) by the Newton method. A f t e r the f i r s t (Wegstein, 1958) three i t e r a t i o n s , a Wegstein acceleration i s a p p l i e d on a l t e r n a t e i t e r a t i o n s i f some imposed d i t i o n s are s a t i s f i e d (see S e c t i o n 5-2-2 Henley and Rosen for details). (1969) proposed a method f o r h a n d l i n g phase v a p o u r - l i q u i d - l i q u i d e q u i l i b r i a , v v v =J!vvv i=l con- u t i l i z i n g the check three- functions: -*ii<v >] L N V 6 v , e L ) = In 2 i < v > V " l i < V and x 6 ^ X i=l where: ev = eL = Li/(Li + L ) , F V/F, 2 i s the t o t a l molar-content of the system, V and L denote the t o t a l vapour and l i q u i d moles and s u b s c r i p t s liquid phases '1' and respectively, '2' on x and L denote f i r s t respectively. and second - 14 - The a u t h o r s recommend a t w o - d i m e n s i o n a l Newton-Raphson method i n s o l v i n g f o r 6 it, V and 6L- acceleration The method, or a m o d i f i c a t i o n of has been implemented and a p p l i e d by a number of i n v e s t i g a t o r s f o r example, Deam and Maddox, 1969; E r b a r , Lu and c o - w o r k e r s 1973; M a u r i , 1 9 8 0 ) . (1974) have a l s o a p p l i e d a m o d i f i e d R e g u l a - f a l s i i t e r a t i o n to the v a p o u r - l i q u i d - l i q u i d p r o b l e m . authors, the method, which does not i n v o l v e the Newton-Raphson method f a i l s t o . verge, it is (see, A c c o r d i n g to derivatives, converges the where However, where the l a t t e r does c o n - faster. One g e n e r a l shortcoming of the v a r i o u s works p u b l i s h e d on the m a s s - b a l a n c e approach to p h a s e - e q u i l i b r i u m c a l c u l a t i o n s i s the l e s s t h a n - a d e q u a t e emphasis a t t a c h e d to the q u e s t i o n of the i n i t i a l i z a t i o n of variables. The f a i l u r e to f u l l y a p p r e c i a t e the e f f e c t t h a t a good i n i t i a l i z a t i o n scheme has on the s t a b i l i t y and speed of convergence of an a l g o r i t h m c o u l d l e a d to the a d o p t i o n of a l g o r i t h m s which are i n fact not the b e s t . point. The v a p o u r - l i q u i d e q u i l i b r i u m problem i s a case i n The a l g o r i t h m s f o r s o l v i n g t h i s problem are g e n e r a l l y to embody an i n n e r i t e r a t i o n loop to c o r r e c t designed f o r the dependence of the e q u i l i b r i u m r a t i o s on phase c o m p o s i t i o n s . N o n i d e a l systems f a i l to converge w i t h o u t But t h i s i n n e r i t e r a t i o n i s a great this correction. time-consumer and i t s e l i m i n a t i o n through a good and r e l i a b l e i n i t i a l i z a t i o n scheme c o u l d r e s u l t time without 1-2-2 reportedly i n a tremendous s a v i n g i n c o m p u t a t i o n j e o p a r d i s i n g the o v e r a l l s t a b i l i t y of the method. The F r e e - e n e r g y - m i n i m i z a t i o n Approach The f r e e - e n e r g y - m i n i m i z a t i o n approach i s , as the name i m p l i e s , an o p t i m i z a t i o n method i n v o l v i n g the m i n i m i z a t i o n of Gibbs f r e e e n e r g y , G, -15- d e f i n e d by M G = I p=l % I» 1=1 l p [^ «Tte(^) f p ] (1-5) where i s the moles of component i i n phase p f o r a system cont a i n i n g M phases, Np i s the number of components i n phase p, f and u denote f u g a c i t y and c h e m i c a l p o t e n t i a l respectively, T and R denote temperature and the U n i v e r s a l gas constant r e s p e c t i v e l y , and s u p e r s c r i p t 'o' denotes s t a n d a r d s t a t e . E q u a t i o n (1-5) i s o p t i m i z e d s u b j e c t to mass-balance positivity constraints and r e s t r i c t i o n s on the component molar q u a n t i t i e s , which, f o r an i s o t h e r m a l i s o b a r i c sytem, c o n s t i t u t e the Independent The d i f f e r e n t variables. a l g o r i t h m s that have been developed based on the f r e e - e n e r g y - m i n i m i z a t i o n p r i n c i p l e have as t h e i r o b j e c t i v e the s o l u t i o n of the c h e m i c a l e q u i l i b r i u m problem and have been a p p l i e d to p h y s i c a l e q u i l i b r i a merely by v i r t u e of the l a t t e r c l a s s of problems being a s p e c i a l case of the former. One of the i n i t i a l White and co-workers e f f o r t s i n t h i s area of r e s e a r c h was (1958), who that o f d e f i n e d an u n c o n s t r a i n e d o b j e c t i v e f u n c t i o n i n form of a L a g r a n g i a n based on a q u a d r a t i c approximation of T a y l o r ' s expansion of Gibbs f r e e energy. techniques: technique. They employed two optimization a s t e e p e s t - d e s c e n t search method and a linear-programming While t h e i r a l g o r i t h m s were l i m i t e d to a s i n g l e (vapour) phase and hence i n a p p l i c a b l e to the p h a s e - e q u i l i b r i u m problem, their method has s i n c e been extended to cover more phases by o t h e r workers - 16 - (see Gautam and Seider, 1979), the final form being the RAND algorithm as presented by Dluzniewski and Adler (1972). Two other algorithms, similar to those of White and co-workers in that they employ the Lagrangian of a truncated Taylor's expansion of G, have been proposed by Clasen (1965) and by Eriksson and Rosen (1973). These algorithms are applicable to multiphase systems. While the two algorithms were originally restricted to ideal systems, in a later paper Eriksson (1975) described a computer program which, while retaining the linear-programming formulation previously presented by Eriksson and Rosen (1973), allows for the consideration of nonidealities by making room for the possible inclusion of a subprogram that updates the 'constants' in the linear equations. Two other algorithms similar to that of White and co-workers, applicable to multiphase systems and allowing for nonidealities, are the 'NASA' algorithm of Gordon and McBride, and the 'quadratic-programming' algorithm of Wolfe (see Gautam and Seider, 1979). Yet another algorithm, presented by Ma and Shipman (1972), uses a modified Naphthali method — a form of the method of steepest descent — to obtain an approximate solution, which is then refined through a Newton-Raphson iteration method. Two shortcomings that undermine the robustness of a l l the freeenergy-minimization (a) methods cited above are: Their high dimensionality, which is reinforced by the introduction of Lagrange multipliers. (b) Their requirement of information on chemical potentials for every component for a l l phases in which i t is present. - 17 - George and co-workers (1976) have proposed another algorithm which is not undermined by the problem of high dimensionality. They formulate the problem in unconstrained form by employing allocation functions, then minimize the resulting objective function by the Powell optimization technique. (from - 00 The algorithm involves an open-ended search to +») for the optimal values of the unconstrained variables, and therein lies i t s weakness. Gautam and Seider (1979) have undertaken a comparative study of the major free-energy-minimization algorithms, and their conclusion was in favour of the RAND algorithm. 1-2-3 The Geometric-programming Method Geometric programming is a relatively new technique which, since its conception in 1961 and i t s subsequent development by Duffin and coworkers (1967), has enjoyed a wide application in different facets of engineering design (see, for example, Beightler and Phillips, Gupta and Radhakrishnan, 1976; 1978). A brief presentation of that aspect of the basic concept that is relevant to this work is contained in Appendix C-l. Duffin and co-workers presented a simple example on how the geometric-programming method could be applied to chemical equilibrium problems through a transformation of the Gibbs free-energy function (see Appendix C of Duffin et a l , 1967). Although the mathematical presentation implies applicability to a system containing more than one phase, i t was founded on the ideal gas law, which automatically limits i t to one phase: an ideal gas phase. Passy and Wilde (1968) have also developed a geometricprogramming algorithm for solving chemical-equilibrium problems. - 18 - T h e i r approach i n v o l v e s t r a n s f o r m i n g exp (-G/RT), which i s the o b j e c t i v e f u n c t i o n f o r the problem, i n t o i t s c o r r e s p o n d i n g an i n e q u a l i t y c o n s t r a i n t ( f o r a s i n g l e phase), s o l u t i o n by a p p l y i n g a d i r e c t the L a g r a n g i a n single, ideal function. and dual primal with s e e k i n g the d e s i r e d search to determine the saddle p o i n t of Here a g a i n , there i s the r e s t r i c t i o n to a phase. More r e c e n t e f f o r t s have been aimed at e l i m i n a t i n g the ideality r e s t r i c t i o n i n the a p p l i c a t i o n of the geometric-programming method to chemical e q u i l i b r i a (see, f o r example, L i d o r and Wilde, no attempt has h i t h e r t o been made to f o r m u l a t e problem u t i l i z i n g the geometric-programming 1978). However, the p h a s e - e q u i l i b r i u m theory. In c o n c l u d i n g t h i s s e c t i o n , i t should be s t a t e d e m p h a t i c a l l y t h a t geometric programming merely serves as a technique problem from one readily form to another to s o l u t i o n ; v i n g the r e s u l t i n g 1-3 for transforming a form which might lend i t s e l f more i t does not p r o v i d e any mathematical t o o l s f o r s o l - problem. S e n s i t i v i t y Analysis i n Phase E q u i l i b r i a Another area i n which geometric programming has patronage i n the realm of chemical e q u i l i b r i a enjoyed some i s i n the d e t e r m i n a t i o n the changes i n phase d i s t r i b u t i o n t h a t r e s u l t from changes i n the s i v e s t a t e v a r i a b l e s (temperature and theory, as proposed by D u f f i n and co-workers (1967), Appendix pressure). The inten- relevant basic i s presented in C-2. The technique presupposes t h a t a s o l u t i o n i s known at some con- d i t i o n of temperature and p r e s s u r e , and i t p r o v i d e s the mathematical t o o l s f o r e s t i m a t i n g the s o l u t i o n at a d i f f e r e n t e r a t u r e and c o n d i t i o n of temp- p r e s s u r e not f a r removed from the o r i g i n a l state. of - 19 - Dinkel and Lakshmanan (1975, 1977) have been fervent explorers in the application of the perturbation technique to the chemical equilibrium problem. However, their applications were restricted to ideal-gas systems, and the results were not very impressive where fairly large temperature changes were involved. The development of a phase-equilibrium equivalent of the sensitivity-analysis method has been undertaken in this work. No ideality assumptions are involved in the development. Various steps have been taken to improve the results that are obtained from applying the perturbation technique to different vapour-liquid systems. 1-4 Bubble- and Dew-point Calculations In process-plant operation and design, the chemical engineer quite frequently encounters the problem of determining the temperature at which a given liquid or vapour mixture experiences incipient vaporization (bubble point) or condensation (dew point) respectively. This problem could arise in the process of determining what temperature a given mixture must be made to assume in order to avoid the formation of a second phase, with the attendant complications that go with the handling — especially in a flow line — mixtures. of two-phase Or i t could rear its head when the chemical engineer i s confronted with the problem of designing some separation unit (see, for example, Lyster et a l , 1959a; Holland, 1963; Petryschuk and Johnson, 1965). Experience shows that these saturation-point calculations, being iterative, can be quite time-consuming. When, for example, they are i n corporated into an algorithm for the design of a distillation column, they could claim as much as half of the total execution time. -20- In view of the above o b s e r v a t i o n , a s i g n i f i c a n t was devoted to i n v e s t i g a t i n g most t i m e - e f f i c i e n t amount of e f f o r t the problem w i t h a view to d e t e r m i n i n g the s o l u t i o n method. For n o n i d e a l systems, a s a t u r a t i o n - p o i n t - t e m p e r a t u r e n o r m a l l y i n v o l v e s two l e v e l s of i t e r a t i o n : calculation an i n n e r i t e r a t i o n loop that c o r r e c t s f o r the dependence of the e q u i l i b r i u m r a t i o s on composition; and an outer loop that updates the temperature. The d i f f e r e n t methods f o r s o l v i n g the problems d i f f e r only i n the approach used to update the temperature. (1971) compared t h r e e methods of temperature J e l i n e k and Hlavacek update — the second-order Newton method, the t h i r d - o r d e r Richmond method, and the Chebyshev method — Richmond method was the f a s t e s t . and reached the c o n c l u s i o n t h a t the In their investigation, the f u n c t i o n a l form of the e q u i l i b r i u m r a t i o , K^, A A r i K i = they assumed to be i T^Tc + B i ( 1 " 6 ) 1 where A-^, and are c o n s t a n t s f o r any component i . r e l a t i o n s h i p presupposes that the system i s i d e a l , account f o r the dependence of K-£ on c o m p o s i t i o n . The above s i n c e i t does not I t i s , i n fact, n o t h i n g more than the combination of R a o u l t ' s law and an A n t o i n e vapourpressure equation. More r e c e n t s t u d i e s by Sobolev e t a l . (1975) and by Ketchum (1978) are a l s o g u i l t y of the i m p l i c i t assumption of i d e a l i t y . F o r a g e n e r a l s o l u t i o n method u n c o n s t r a i n e d by any assumptions as to the degree of a f f i n i t y that p r e v a i l s amongst the system's constituent components, the complexity of the problem assumes a much h i g h e r dimension. The methods s t u d i e d i n t h i s work are based on t h i s approach. rigorous - 21 - 1-5 Adlabatlc Vapour-Liquid E q u i l i b r i a The a d i a b a t i c v a p o u r - l i q u i d f l a s h problem i n v o l v e s the deter- m i n a t i o n of the temperature and the p h a s e - d i s t r i b u t i o n t h a t w i l l result i f a m i x t u r e w i t h a f i x e d heat content i s t h r o t t l e d i n t o a p e r f e c t l y i n s u l a t e d f l a s h drum. T h i s problem i s of a h i g h e r degree of difficulty than the i s o t h e r m a l v a p o u r - l i q u i d e q u i l i b r i u m c a s e , because i t has tempe r a t u r e as an a d d i t i o n a l unknown. be s o l v e d i s the e n t h a l p y The a d d i t i o n a l e q u a t i o n t h a t has to relationship. Two methods t h a t are c o n v e n t i o n a l l y employed i n s o l v i n g the a d i a b a t i c - f l a s h problem both i n v o l v e i t s f o r m u l a t i o n i n t o a problem w i t h two independent v a r i a b l e s : vapour f r a c t i o n and t e m p e r a t u r e . One of the methods ( H o l l a n d , 1975) u t i l i z e s a r e g u l a f a l s i vergence of the t e m p e r a t u r e . It con- i n v o l v e s two l e v e l s of i t e r a t i o n . the lower i t e r a t i o n l e v e l , an i s o t h e r m a l f l a s h c a l c u l a t i o n i s At performed a t the c u r r e n t v a l u e of temperature t o determine the c o r r e s p o n d i n g vapour f r a c t i o n . At the o u t e r i t e r a t i o n l e v e l , the v a l u e s of t e m p e r a - t u r e at the l a s t two i t e r a t i o n p o i n t s and the c o r r e s p o n d i n g enthalpy check f u n c t i o n s are employed i n a r e g u l a f a l s i i n t e r p o l a t i o n to mine an improved v a l u e of deter- temperature. The second method ( H o l l a n d , 1963) employs a t w o - d i m e n s i o n a l Newton-Raphson approach to s i m u l t a n e o u s l y d r i v e the v a r i a b l e s to the solution point. A r e c e n t p r o p o s i t i o n by Barnes and F l o r e s ( K i n g , 1980) employs the R a c h f o r d - R i c e f o r m u l a t i o n ( E q u a t i o n 1 - 2 ) f o r the massb a l a n c e check f u n c t i o n and adopts a d i f f e r e n t f o r m u l a t i o n of the e n t h a l p y check f u n c t i o n from the c o n v e n t i o n a l form due to H o l l a n d . The f o r e g o i n g methods of t a c k l i n g the a d i a b a t i c - f l a s h problem i n v o l v e b u b b l e - and d e w - p o i n t c a l c u l a t i o n f o r the purpose of temperature -22- initialization. These are n e c e s s a r i l y time-consuming steps. Even with t h i s i n i t i a l i z a t i o n scheme, the convergence of the Newton-Raphson method i s not assured and three rules proposed by Holland (1963) s t i l l have to be employed to avoid e r r a t i c behaviour and possible divergence of the temperature search. In t h i s work, other methods have been studied i n a d d i t i o n to the two c i t e d above. Some of the methods derive d i r e c t l y from the algo- rithms for d i s t i l l a t i o n - u n i t c a l c u l a t i o n , which d i f f e r s from the a d i a b a t i c - f l a s h c a l c u l a t i o n not i n i t s basic nature and l o g i c but only i n i t s degree of complexity. 1-6 Multicomponent, Multistage D i s t i l l a t i o n with Equilibrium Stages The d e c i s i o n to devote some e f f o r t , i n t h i s study, to the i n v e s t i g a t i o n of D i s t i l l a t i o n - u n i t computations — separation u n i t s — as opposed to other was guided by the fact that i t i s the most commonly used separation method, for reasons which have been discussed by King (1980). Computation methods for multicomponent, multistage separations generally f a l l into two classes: methods. short-cut methods and rigorous The various methods have been given ample treatment texts on separation processes (see, for example, van Winkle, i n most 1967; Smith, 1963; Holland, 1963 and 1975; King, 1980; Henley and Seader, 1981). A review of the methods i s not intended here. A discussion of any of the methods w i l l be undertaken only i n so f a r as i t r e l a t e s to t h i s study. This i n v e s t i g a t i o n pertains to the class of rigorous methods, and i t approaches the problem from an operating, rather than a design, - 23 - point of view — this being the problem-type for which most of the existing rigorous algorithms are formulated. The study is further limited to the case of a conventional, as distinct from a complex, column. The 6-converged Bubble-point (BP) method of Lyster and co-workers (1959a) was implemented, this in view of i t s proven superiority over other methods when applied to narrow-boiling mixtures (see, for example, Seppala and Luus, 1972). The modification proposed by Seppala and Luus (1972) was also implemented. The BP method involves the determination of stage temperatures through a bubble-point calculation on each tray at every iteration. This step constitutes a significant time-sink. In this study, a number of modifications of the BP method, aimed at reducing the computation time, have been introduced. The K^, method (Holland, 1963), which was deviced to avoid bubble-point calculation, was also implemented. An appreciable amount of effort has also been devoted to studying possible ways of applying a sum-rate (SR) type of approach, whereby the temperature profile is updated by means of the enthalpy balances, to distillation. The approach is known not to be applicable to narrow- boiling systems, for reasons given by Friday and Smith (1964). The study on the subject was meant to provide a complement to the BP algorithms. Because the BP and SR methods apply to opposite regions of the boiling-range axis, with an intermediate region where both of them could f a i l , the 2N Newton-Raphson method of Tomich (1970) has also been implemented. This method is applicable to narrow- as well as wide-boiling systems. It is a compromise between the methods of partitioning - 24 - (such as the BP methods) and a f u l l Newton-Raphson convergence (SC) method (Newman, 1968; N a p h t a l i and ideal, Sandholm, 1971). simultaneous- G o l d s t e i n and Unless Stanfield, the system i s extremely i t i s known to r e q u i r e l e s s computer-time than the dimensioned SC The non- more-highly- algorithms. i n v e s t i g a t i o n was extended to i n c l u d e a study of how b o i l i n g systems c o u l d p o s s i b l y be handled without J a c o b i a n m a t r i c e s and m a t r i x i n v e r s i o n s . The Tomich method was wide- getting involved i n a r e f e r e n c e f o r a s s e s s i n g the performance of the r e s u l t i n g 1-7 1970; used as algorithms. Estimation of Physical Properties The thermodynamic r e l a t i o n s i n v o l v e d i n t h i s work c o n t a i n v a r i o u s p h y s i c a l p r o p e r t i e s whose v a l u e s are r e q u i r e d as f u n c t i o n s of t u r e , p r e s s u r e and ( i n some cases) phase c o m p o s i t i o n . of these parameters h a r d l y ever e x i s t , to r e s o r t to e s t i m a t i o n methods. The and tempera- E m p i r i c a l values the c o n v e n t i o n a l approach i s problem then becomes that of making a j u d i c i o u s c h o i c e from amongst the o f t e n very l a r g e number of e s t i m a t i o n c o r r e l a t i o n s t h a t are to be found In t h i s i n v e s t i g a t i o n , guided i n the literature. the c h o i c e of e s t i m a t i o n methods was by three main f a c t o r s : (1) The q u a l i t y of the estimate. (2) The relative simplicity (3) The ease of d e t e r m i n a t i o n of the parameters i n v o l v e d i n the of the e s t i m a t i o n method. correlation. F o r any was g i v e n p r o p e r t y , the c o r r e l a t i o n r e l a t i o n s h i p u l t i m a t e l y chosen t h a t which has been w e l l t e s t e d over the years and, going by the - 25 - three factors above, is known to show an advantage over other methods. An example on this selection procedure should suffice. Let us consider the problem of estimating the liquid-phase activity coefficient. A current-state-of-the-art study reveals that the efforts of molecular thermodynamicists to predict equations of state — this quantity through such equations as the Redlich-Kwong and the rather cumbersome Benedict-Webb-Rubin have been subjected to such experimentation — have yielded very l i t t l e reward. Thus, given the present level of understanding of fluid-phase molecular behaviour, the best that can be done is employ semi-empirical correlations that relate the activity coefficients to temperature and liquid-phase composition through a number of interaction parameters determined from equilibrium data. A further review shows that the predictive correlations can be classified into three groups (Tripathi and Sri Krishna, 1976): (a) The Wohl-type equations, which include Margules, van Laar and Scatchard-Hamer equations. (b) The Redlich-Kister equation. (c) Expressions based on the concept of local composition. The last group includes the Wilson equation (Wilson, 1964) and i t s various modifications (for example: Nagata and Gotoh, 1975; Nagata et a l . , 1975a and 1975b; Tsuboka and Katayama, 1975), the Non-Random-TwoLiquid (NRTL) equation (Renon and Prausnitz, 1968) and i t s modifications (Marina and Tassios, 1973; Novak, 1974b), and the universal quasichemical (UNIQUAC) equation (Abrams and Prausnitz, 1975; Anderson and Prausnitz, 1978). Further investigation reveals that for systems that exhibit no phase splitting and no maxima in their activity coefficients, the - 26 - original Wilson equation is generally regarded as the best. It has therefore been employed for that purpose in this work. Where phase splitting is involved, the NRTL equation has usually been resorted to. However, experience has shown (Heidemann and Mandhane, 1973; Novak, 1974a) that i t has the defect of sometimes predicting multiple solutions. In view of this, i t has waned in popularity to the advantage of the UNIQUAC equation. Both the NRTL and the UNIQUAC equations have been implemented in this work. 1-8 The S t r u c t u r e o f t h i s Work While this work has been organized in such a way as to reflect a logical sequence of development, the various chapters have been so structured that they can be perused almost exclusively without much loss in comprehension. Chapter 2 presents the various correlations utilized in this work for predicting physical properties. Chapter 3 deals with the isothermal vapour-liquid flash problem. It is partitioned into four main parts, with the f i r s t three parts respectively discussing double-loop univariate methods, free-energy-minimization methods and single-loop univariate methods. The concluding part compares the best of the different classes of methods in addition to testing the effect of altering the frequency of equilibrium-ratio calculation. In Chapter 4, the effort made at applying the perturbation theory of geometric programming to vapour-liquid equilibria is documented. The investigation is taken a step further in Chapter 5 through the application of the most promising of the algorithms studied so far to -27- isothermal flash liquid-liquid algorithm implemented. one of the chapter i s i n two preceding equilibria. The liquid-liquid et a l (1980) i s a l s o main p a r t s , each p a r t d e a l i n g with problem-types. Chapter 6 takes us work i n t h i s liquid-solid r e c e n t l y proposed by P r a u s n i t z The two and chapter chapters. draws on the e x p e r i e n c e A number of a l g o r i t h m s liquid-liquid-solid i n Chapters 3 and to the problem of m u l t i p h a s e e q u i l i b r i a . f u r n i s h e d by the for a generalized The initialization schemes t e s t e d are three vapour- system are developed as l o g i c a l extensions 5. The of those similarly patterned. I n Chapter 7, are i n v e s t i g a t e d . the problems of bubble- and T h i s i s i n a n t i c i p a t i o n of t h e i r study of a d i a b a t i c v a p o u r - l i q u i d f l a s h c a l c u l a t i o n s i n succeeding and multistage The In i t s wake i s followed Chapter 10 appendices open w i t h ted the a d o p t i o n ambiguity. as Brother': the c o n c l u d i n g by short notes on s p e c i f i c that have been duly r e f e r e n c e d very serves 'Big chapter. g e n e r a l notes on the programming r e m a i n i n g appendices c o n t a i n a few The of a d i a b a t i c v a p o u r - l i q u i d logic T h i s i s the s u b j e c t matter of Appendix A. (Appendix B) the distillation-unit (Chapter 9) i s i t s w a t c h f u l distillation. employed i n t h i s work. a p p l i c a t i o n to chapters. Chapter 8 b r i n g s us to the q u e s t i o n equilibria. dew-point c a l c u l a t i o n s programs. This The t h e o r i e s and mathematical d e r i v a t i o n s i n the main body of the report. l a r g e number of symbols employed i n t h i s work n e c e s s i t a of a s p e c i a l I n Chapter 2, definition format aimed at symbols are d e f i n e d preventing immediately f o l l o w i n g the - 28 - equations in which they occur. This was considered suitable for handling correlations due to different authors, who may have used some common symbols with different meanings, without having to alter the symbols originally used by the authors. For the other chapters, the symbol definitions are presented in the form of a Nomenclature at the beginning of each chapter. This way, the same symbol could be given different identities in different chapters and a shortage of symbols does not then arise. The symbols are not redefined within the chapter, as this would make the report rather untidy. CHAPTER TWO THE DETERMINATION OF THERMODYNAMIC PARAMETERS 2-1 Introduction T h i s c h a p t e r c o n t a i n s , under s e p a r a t e s u b s e c t i o n s , the c o r r e l a t i o n s employed i n t h i s work f o r e s t i m a t i n g the f o l l o w i n g p h y s i c a l properties: saturation pressure, liquid-phase reference vapour molar volume and c o m p r e s s i b i l i t y r a t i o , vapour-phase f u g a c i t y c o e f f i c i e n t , vapour molar e n t h a l p y , l i q u i d molar volume, liquid-phase activity l i q u i d molar e n t h a l p y , fugacity, coefficient, and s o l i d r e f e r e n c e fugacity. 2-2 Saturation Pressure Where r e l e v a n t d a t a e x i s t , the pure-component saturation p r e s s u r e s a r e determined from a s i x - p a r a m e t e r e q u a t i o n of the form ( P r a u s n i t z et a l , 1967): C Un P S = C L + 2 + C T + C A 5 2 r + C *nT (2-1) 6 where P and s = pure-component saturation-pressure, T = system t e m p e r a t u r e , C^ ( i = 1, 2 , . . . , 6 ) are e m p i r i c a l c o n s t a n t s . I n the absence of the n e c e s s a r y e m p i r i c a l c o n s t a n t s , P s is e s t i m a t e d from a c o r r e s p o n d i n g - s t a t e e q u a t i o n proposed by P i t z e r and C u r l ( P r a u s n i t z et a l , 1 9 6 7 ) : -29- - 30 - l g l O o P s r - 0 . 3 4 5 6 ^ 1.454 — ^ - — 2 r r 4.318 — r + = L x .(£^81 T| + _ 2,524 Tf. + 3 . ' o 2 n 0 o 9 A + 2^008 T + Q ( 2 _ 2 r where s P r = reduced T and r reduced = a) 2-3 saturation-pressure, temperature, = P i t z e r ' s acentric factor. Liquid-Phase Reference Fugaclty F o r the purpose of e s t i m a t i n g l i q u i d r e f e r e n c e f u g a c i t y , component i s c l a s s i f i e d as condensable ( T (T r not much g r e a t e r 1). < 1), than 1) and s u p e r c r i t i c a l ( T hypothetical-liquid r much g r e a t e r than T h i s i s i n accordance w i t h the approach employed by P r a u s n i t z and co-workers 2-3-1 r each (1967). Condensable Components F o r a condensable component, the reference fugacity, d e t e r m i n e d from the fundamental thermodynamic Vi x i - relationship *±**i where Yi = activity P coefficient, = system p r e s s u r e , <t>i = f u g a c i t y coefficient, adjusted for pressure, f ^ 0 is - and x 31 - i » Y i a r e l i q u i d and vapour mole f r a c t i o n s respectively. I n the above e q u a t i o n , f ° has the P o y n t i n g f a c t o r incorporated. L When the e q u a t i o n i s a p p l i e d to the pure component a t s a t u r a t i o n c o n d i t i o n , we have L f = <^ ^ ± i i V exp( L s i i g ^ - ) x Poynting factor where Poynting factor = e x P(~gj- ), L = pure-component l i q u i d molar volume, = component p a r t i a l l i q u i d molar volume, and s u p e r s c r i p t ' s ' denotes s a t u r a t e d c o n d i t i o n . Due t o l a c k of d a t a , I s u s u a l l y r e p l a c e d w i t h v^ , the pure-component 1 l i q u i d molar volume. T h i s i s not u n r e a s o n a b l e a t low and moderate p r e s s u r e s where v a p o u r - l i q u i d e q u i l i b r i u m i s r e l a t i v e l y i n s e n s i t i v e to W i t h t h i s a l t e r a t i o n , we have ( f f P_p s ) v L - * J P J exp{ If T r ' (2-3) < 0 . 5 6 f o r the component under c o n s i d e r a t i o n , <t>| i s d e t e r m i n e d from an a p p r o p r i a t e e q u a t i o n of s t a t e ( s e e S e c t i o n 2 - 7 ) . Otherwise, the method of Lyckman and c o - w o r k e r s 1967) i s employed. (Prausnitz The l a t t e r i s a c o r r e s p o n d i n g - s t a t e et a l , r e l a t i o n which g i v e s <t>s by: M> S = 4> ( 1 ) (T ) + ox(, r ( 2 ) (T ) r (2-4) - 32 - where (1) m r 0.57335015 _ 3,076574 3 2 T T r r m ,.(2)^ \ -0.012089114 and <fr (T ) = -J2 T r J 0.024364816 -j9 r 0.10665730 ,5 T r + 0.3166137 T 2-3-2 + + m 0.14936906 ^,8 r 1.1662283 4 T r 5.6085595 _ _ T r 0.015172164 0.068603516 3 b ~ "Tl T T r r 0.18927037 _ 0.12147436 7 ^,6 r r 0.12666184 3 T r 4.3538729 _ T r r H y p o t h e t i c a l L i q u i d and S u p e r c r i t i c a l Components The r e f e r e n c e f u g a c i t y of a h y p o t h e t i c a l l i q u i d component i s e s t i m a t e d from the c o r r e l a t i o n of Lyckman, E c k e r t and P r a u s n i t z . A c c o r d i n g to the a n a l y t i c a l form of the c o r r e l a t i o n , p r e s e n t e d by P r a u s n i t z and c o - w o r k e r s f° L = P (1967), explF^+cdjF^} x Poynting factor where <o> = -1.1970522 . L3785023 T and F ± r T + 2 m ( m 8 M r - -2-7741817 , 1.5454928 = + + 1.3057555 T T r r 1 (2-5) - 33 - For s u p e r c r i t i c a l components, the r e f e r e n c e f u g a c i t y i s d e t e r - mined from the Henry's constant, H-jj, of the s u p e r c r i t i c a l i, in a r e f e r e n c e condensable component ( s o l v e n t ) , j , at the o p e r a t i n g temperature. For a mixed s o l v e n t , j i s the condensable component the h i g h e s t c r i t i c a l available. temperature f o r From the v a l u e s of H^j the r e f e r e n c e p r e s s u r e , tant as a R s u p p l i e d at two = H (l) H temperatures and and H^ ^ are 2 deter- thus: ( 2 ) T ( 2 r e f e r e n c e f u g a c i t y of component i i s then g i v e n -v" .P exp( *j ) at the r e f e r e n c e - p r e s s u r e Henry's cons- f u n c t i o n of temperature, (Pr) with which i n f o r m a t i o n on H^j i s the c o n s t a n t s H ^ ^ mined from an e x p o n e n t i a l f i t of The component, _ 6 ) by s f° = H^ L r ) x Poynting J factor (2-7) 00 v^j, is the p a r t i a l molar volume of i at i n f i n i t e d i l u t i o n i n s o l v e n t j , estimated from a l i n e a r i n t e r p o l a t i o n based on v a l u e s of v s u p p l i e d at two 2-4 temperatures. Solid-phase Reference Fugacity os The solid-phase reference fugacity, f of the l i q u i d phase by ^oL f. * ~o7> = — t f ± RT T 1 to t h a t ( P r a u s n i t z , 1969): ,,f Ah. n( , i s related - — T AC . T T-T ] - ^ ( - ^ + t l R T - -^-] T (2-8) - 34 - where Ah^ i s the l a t e n t heat of f u s i o n of T and i s the t r i p l e p o i n t of ^pi * S of the s o l i d , t * i e i, i, d i f f e r e n c e between the l i q u i d heat c a p a c i t y and t h a t both determined at T , ti. Due to the l a c k of i n f o r m a t i o n on the t r i p l e - p o i n t of most compounds, and i n view of the p r o x i m i t y of the t r i p l e p o i n t to the normal m e l t i n g p o i n t , i t i s normal p r a c t i c e to r e p l a c e T w i t h the normal m e l t i n g p o i n t , T f° *<4r f7 n i L Ahf ) - — RT mi • Equation (2-8) T [1 T , mi ] + then becomes AC . T . -^[*n(^-) R T T .-T - J5^_] T (2-9) Another impediment encountered i n e v a l u a t i n g E q u a t i o n ( 2 - 9 ) the q u a n t i t y AC . , Pi is f o r which i n f o r m a t i o n i s h a r d l y ever a v a i l a b l e . To s i d e t r a c k the p r o b l e m , the second term on the r i g h t i s n o r m a l l y ignored, the j u s t i f i c a t i o n b e i n g t h a t the f a c t o r i n b r a c k e t relatively small. f? The e q u a t i o n then reduces L Ah! S RT is to T *<4r> - — [ i ] n f° i T ( mi t L e t us take a c l o s e r l o o k at the i g n o r e d t e r m . T £n( =—) i result T about the p o i n t —~ l is - 1 , and s u b t r a c t (T I f we expand . - T)/T, mi the 2 " 1 0 ) - 35 - T ml T -T 00 = - I_ nL n=z L n [ l - T ,/T] ml J Thus, f o r the term on the l e f t - h a n d s i d e of the i g n o r e d term reasonably In i n Equation Equation (2-11), and hence ( 2 - 9 ) , to be n e g l i g i b l e , T ./T must be mi c l o s e to 1. order to extend the range of a p p l i c a b i l i t y of the relationship w i t h i n the l i m i t s of the r e s t r i c t i o n s imposed by the l a c k of i n f o r m a t i o n , an attempt was made at d e v e l o p i n g a c o r r e l a t i o n t h a t would AO f u r n i s h e s t i m a t o r s of f term. A brief it w i t h an e s t i m a t e of the n o r m a l l y - i g n o r e d U n f o r t u n a t e l y , the e f f o r t s h i e d away from the t a s t e of s u c c e s s . documentation of i t i s p r e s e n t e d i n Appendix D i n the hope t h a t might s t i m u l a t e the reader to g r e a t e r i d e a s . relationship 2-5 truncated [ E q u a t i o n (2-10)] has been employed i n t h i s work. Vapour Molar Volume and Compressibility Ratio The quantities state. vapour molar volume and I n t h i s work, two e q u a t i o n of s t a t e , and The T r u n c a t e d V i r i a l virial pressure-explicit the the W i l s o n m o d i f i c a t i o n of the state. E q u a t i o n of S t a t e e q u a t i o n of s t a t e , truncated a f t e r employed f o r systems at low and moderate p r e s s u r e s . Z = two equations of s t a t e have been employed: Redlich-Kwong e q u a t i o n of The c o m p r e s s i b i l i t y r a t i o are t h a t are normally estimated from a s u i t a b l e e q u a t i o n of truncated v i r i a l 2-5-1 The the second term, i s In the form, i t i s (2-12) - 36 - where B = the vapour-mixture second v i r i a l coefficient, v = molar volume of the vapour and Z = the c o m p r e s s i b i l i t y r a t i o of the vapour m i x t u r e . The above e q u a t i o n i s a q u a d r a t i c i n v and lends i t s e l f tion. to easy solu- However, should the r e s u l t i n g r o o t s be complex, the volume- explicit form of the v i r i a l Pv Z Equation BP Bf = e q u a t i o n i s r e s o r t e d t o , thus: " 1 RT + ( 2 (2-13) i s l i n e a r i n v. from the l e f t - h a n d - s i d e B i s determined B = I N 1 ±ii Whichever " 1 3 > form i s employed, Z f o l l o w s equality. from N y y B j£i i j i j y y where B-£i i s the second v i r i a l B ij c o e f f i c i e n t f o r pure component i , (1 * J) i s the second v i r i a l cross-coefficient between components i and j . N i s the number of components i n the m i x t u r e . B i s determined from the c o r r e l a t i o n of P i t z e r and C u r l , m o d i f i e d by P r a u s n i t z and co-workers association. P A c c o r d i n g to these (1967) t o account f o r p o l a r i t y and workers, B £|_ii = F ( 0 ) (T ) + r <o Hi F ( 1 ) (T ) + F (p , T ) + r y r r where W as Hi = t 1 i e a c e n t r : 5 - c f a c t o r of the homomorph of i , Va ^ 0 ( 2 _ U ) - 37 - y^ = the reduced d i p o l e moment of i , g i v e n by IOV P 2 y_ = i ci s ci T V>^ = the d i p o l e moment of i i n Debye u n i t s . = the a s s o c i a t i o n Subscript F ( o ) (T c o n s t a n t of i , 'c* denotes the c r i t i c a l ) - 0.1445 r F ( 1 ) (T T r ) - 0.073 Mi + r F-(T ) = exp[6.6(0.7 and _ 0,1385.. T r _ P^O T T r state, ! 0^97 T r Mm T r _ 0^0073 T r r - T )], F (y ,T ) = -5.237220 + 5.665807£ny y r' r r - 2.133816(£ny ) + 2 r + 0.2525373(£ny ) 3 r ^-[5.769770 - 6.181427Jc-ny r + 2 .28327(£ny ) r 2 r - 0.2649074(*n y ) 1 . r 3 The cross-coefficient, B c o r r e l a t i o n s , w i t h the f o l l o w i n g T , . = (T , T . ) cij c i C2 P cij = 4T cij 1 Q V n 1 / + !£iAl](v { T ci P 1 determined from the above mixing r u l e s : 1 J V c ± Vj cij, 0.5(n also 2 - 2 ^ ' cij = s ' V g i T L i + n ), c j 3 + V / )" 1 3 cj 3 , - 38 - 0.5(1^ + 0)j) f o r both i and j nonpolar and 0) . 0.5(w H± Hi 0.5(a> 2-5-2 Hi + u)j) f o r i p o l a r and j nonpolar + co ) f o r both i and j p o l a r The W i l s o n M o d i f i c a t i o n o f the R e d l i c h - Kwong Equation of State The truncated v i r i a l s u i t a b l e f o r systems e q u a t i o n of s t a t e i s known not to be at high pressures. c a l c u l a t i o n i n v o l v i n g such systems, the v i r i a l F o r the purpose of e q u i l i b r i u m programs p a r a l l e l to those employing e q u a t i o n of s t a t e have been w r i t t e n based on the W i l s o n m o d i f i c a t i o n of the Redlich-Kwong e q u a t i o n of s t a t e (Reid et a l , 1977). T h i s e q u a t i o n takes the form Pv Z = _ v ft v-b ft b = F RT b v+b where: N F = I i=l y F ± 1 1 N 1=1 F = 1 + (1.57 + 1.62(0 ) ( T ~ * - 1 ) , ± , i b ± ft, RT b ci = - P — • ci ft = 0.4274802327 a and ft b = 0.086640350 (2-15) - 39 - E l i m i n a t i n g v from E q u a t i o n Z where - Z 3 - (A 2 (2-15) and r e a r r a n g i n g , we + A - B) Z - AB = 0 2 have (2-16) _ bP ~~ RT bPF " a " B d Equation S I T b (2-16), a c u b i c i n Z, i s s o l v e d d i r e c t l y example, La F a r a , 1973) for i t s largest real root. ( f o r method see, f o r The vapour molar volume then f o l l o w s from the l e f t - h a n d - s i d e e q u a l i t y of E q u a t i o n 2-6 (2-15). L i q u i d Molar Volume The method of e s t i m a t i o n of component l i q u i d molar volume depends, l i k e t h a t of r e f e r e n c e f u g a c i t y , on whether the component i s condensable, h y p o t h e t i c a l - l i q u i d or s u p e r c r i t i c a l . For the noncondensable components, the e s t i m a t e i s t h a t of the p a r t i a l molar volume at i n f i n i t e d i l u t i o n i n the mixed s o l v e n t . For a condensable component, the l i q u i d molar volume i s assumed to bear a q u a d r a t i c r e l a t i o n s h i p w i t h temperature, v\ where > = vj°> + v ^ T a n c * a r thus: + vfV e c o n s t ants (2-17) determined from v a l u e s of L v s u p p l i e d at three temperature supplied, points. Where only two values (2) ' i s s e t equal to z e r o , w h i l e i f only one v a l u e i s s u p p l i e d , we assume v £ ^ = = 0 are -40- The the to l i q u i d molar volume f o r a h y p o t h e t i c a l l i q u i d i s estimated by method of Lyckman and co-workers ( P r a u s n i t z et a l , 1967). According t h i s method, v where j RT . v = V.°°= " i P . ci ( 0 ) m (2-18) ' 1 L i v as determined by P r a u s n i t z e t a l (1967), i s g i v e n by vj i 0 ) = -9.5259777A + 2.9766410A., + l i 2 0.085762954 i s d e f i n e d by i _ ~ T ,P . r i ci -2 where v = ) x .v . and v<5 = £ 6 .x .v^, jeC 2 'C 2 2 being the s e t of condensable components. The s o l u b i l i t y 6^, i s g i v e n by ,2 -± P . ci where = 5<°> + u> 6 ^ i i i + u> (2) i 6< > 2 i 65 = -20.141089T .+ 57.150420T .- 60.717499T . 0) x 4 3 r i + 27.093334T 2 r i ri r i - 3.1509051, 6 ? = -52.996350T . + 151.56585T~\ - 153.64561T . X) l 4 r i + 59.821527T 2 r i . - 4.3229852, ri r i parameter, •40a- Leaf 41 missed i n numbering - 42 - and (2) i = _ 6 > 8 4 3 7 8 8 5 T 5 + 8 . 6 3 6 3 5 4 6 9 T ri ri + 32.902287T . - 89.695653T ri r i 3 2 + 73.721696T . - 18.674318 ri For a s u p e r c r i t i c a l component, the l i q u i d molar volume L — v. - v. 1 00 = — v L— from 00 I x.v>. . (2-19) where v and C are as d e f i n e d above f o r h y p o t h e t i c a l 2-7 i s estimated liquids. Vapour-phase Fugacity C o e f f i c i e n t The component vapour-phase f u g a c i t y c o e f f i c i e n t fundamental thermodynamic i s d e f i n e d by the r e l a t i o n s h i p ( P r a u s n i t z , 1969): oo * * i = RT- vM i l Ai v . nj * i " ^ ] n T h i s i n t e g r a l r e q u i r e s an equation most w i d e l y employed various modifications d V "* ^~ ^ nZ 2 of s t a t e and the two that have are the Redlich-Kwong equation ( f o r example: or any of i t s Lu et a l , 1974; Mukhopadhyay and S i n g h , 1975; R e i d et a l , 1977), and the v i r i a l equation truncated the second term ( P r a u s n i t z et a l , 1967; Dojcansky and Surovy, Nagata and Gotoh, 1975; Leach, 1977) — choice at e l e v a t e d pressures. here to play complementary the former being considered the p r e f e r r e d These two e q u a t i o n s have been adopted r o l e s , as d i s c u s s i o n i n S e c t i o n 2-5. As equation the most s u i t a b l e v e r s i o n . When the t r u n c a t e d substituted after 1975; a l r e a d y mentioned, the W i l s o n m o d i f i c a t i o n of the Redlich-Kwong was been i n t o Equation resulting relationship i s virial equation [Equation (2-12)] i s (2-20) and the i n t e g r a t i o n performed, the - 43 - * n * ± 2 = v N I 7^.. j=l J - *nZ (2-21) J With the W i l s o n m o d i f i c a t i o n of the Redlich-Kwong e q u a t i o n [see E q u a t i on ( 2 - 1 5 ) ] , we have b *n * 2-8 ± = *n(^) + ^ ft v ft Fb F *n(^) - ^ • ^ b b + ^ - *nZ ± (2-22) Liquid-Phase A c t i v i t y C o e f f i c i e n t While coefficient, vapour-phase n o n i d e a l i t i e s , as expressed by the f u g a c i t y can q u i t e a c c u r a t e l y be determined s t a t e , the same cannot coefficient. different be s a i d of i t s l i q u i d from an e q u a t i o n of counterpart: As has a l r e a d y been d i s c u s s e d i n S e c t i o n 1-7, correlations for estimating a c t i v i t y implemented i n t h i s work. the activity three c o e f f i c i e n t s have been They are the W i l s o n e q u a t i o n (Wilson, 1964), the Non-Random-Two-Liquid (NRTL) e q u a t i o n of Renon and P r a u s n i t z (1968), and the U n i v e r s a l Q u a s i c h e m i c a l Prausnitz 2-8-1 (UNIQUAC) e q u a t i o n of Abrams and (1975), as m o d i f i e d by Anderson and P r a u s n i t z (1978). The W i l s o n E q u a t i o n The W i l s o n e q u a t i o n ( W i l s o n , 1964), w i t h i t s v a r i o u s modifications (Nagata et a l , 1975a and 1975b; Tsuboka and Katayama, 1975), has been w i d e l y a p p l i e d and found to g i v e very good estimates of a c t i v i t y c o e f f i c i e n t , Y i , s u b j e c t t o two limitations: e x h i b i t s no phase s p l i t t i n g ; the a c t i v i t y t h a t the system and t h a t t h e r e be no maxima or minima i n coefficient profile. In i t s o r i g i n a l form, the e q u a t i o n - 44 - states: j j , n y = N o_i i _ ij " V..A I M i - 1- M 1 _ J - N V x, A r kki I [-*-=—] (2-23) where A V and (^^j "" This sable is a two-parameter L i e x PL S J i e m p i r i c a l l y - c o r r e l a t e d energy parameter. n form of the W i l s o n e q u a t i o n i s employed f o r conden- components. For noncondensable components, co-workers (1967) i s adopted and y modification the approach by P r a u s n i t z and determined from a one-parameter of the form X, y n = SLn[—^ ± I A where s u b s c r i p t A A x ]- -j-—* , .x, l j j I (2-24) x A . j=l * J j ' r ' denotes the chosen r e f e r e n c e solvent f o r the noncondensable component i . F o r i n t e r a c t i o n between two noncondensable components, i t i s assumed that X . - A oo A ij _ - X i j = 0, so that JJ V . - J0 V 2-8-2 = X n i The Non-Random-Two-Liquid (NRTL) Model Experience with a c t i v i t y - c o e f f i c i e n t workers has r e v e a l e d strength p r e d i c t i o n by v a r i o u s t h a t the weakness of the W i l s o n e q u a t i o n i s the of the NRTL c o r r e l a t i o n (Renon and P r a u s n i t z , 1968). While o t h e r p r e d i c t i v e c o r r e l a t i o n s such as the UNIQUAC e q u a t i o n (Abrams and -45- P r a u s n i t z , 1975) and the Orye e q u a t i o n ( B r u i n , 1970) are r e p o r t e d l y also capable of c o r r e c t l y p r e d i c t i n g phase s p l i t t i n g , — i n s p i t e of some l i m i t a t i o n s — the NRTL e q u a t i o n (Heidemann and Mandhane, 1973; Novak, 1974a) has gained by f a r the widest a p p l i c a t i o n to d a t e . A c c o r d i n g to the correlation, N N «" \ - V J + - T N (2-25) where G T 8 and j i " *P<-° i l>» e T j _ ( j i g ji ij S j " i i RT g ) j i ' x.. = o ii The energy parameters (g parameter a 2-8-3 - 8^) ^ ji g a r e determined from e m p i r i c a l ~ g j'^' t ^ 6 n o n r a n < * o m n e s s data. The U n i v e r s a l Q u a s i - C h e m i c a l (UNIQUAC) Model As a r e s u l t of the shortcomings of the NRTL model r e f e r r e d to i n S e c t i o n 2-8-2 — l i m i t a t i o n s that are r e l a t e d parameters and of s o l u t i o n s — the UNIQUAC model 1975) as m o d i f i e d by Anderson and P r a u s n i t z implemented. to m u l t i p l i c i t y of T h i s model i s g i v e n by (Abrams and P r a u s n i t z , (1978) has a l s o been - 46 - i •qi i N M l 6'T i . , •j=l J J 6'T ) + q« - q'£ 3 l j (2-26) where z I j T ^ - q j ) - ^ - ! ) . z = l a t t i c e c o o r d i n a t i o n number, and i s a s s i g n e d the v a l u e 10, v i N I r .x . j-1 J J i i N1 1 q » q X e i. = 1 I 1 j-1 q 6' = N J i 1 K i x 1 j-1 T . 1 J J = exp J [-a ./RT] k The a 's are the UNIQUAC parameters (two f o r every component kJ pair) w h i l e r , q and q' are pure-component m o l e c u l a r - s t r u c t u r e c o n s t a n t s which depend on m o l e c u l a r s i z e and e x t e r n a l s u r f a c e a r e a s . -47- 2-9 Vapour Molar Enthalpy The d e p a r t u r e of the vapour p a r t i a l molar e n t h a l p y , component i from the i d e a l - g a s v a l u e , H ° , i s g i v e n by thermodynamic r e l a t i o n s h i p (H ± the ( H o l l a n d , 1963): - H°) = 34nf 2 RT of v 9T (2-27) p,n v f S i n c e by d e f i n i t i o n , ± = t^jY^* E q u a t i o n (2-27) becomes 34n<|> 9T 2-9-1 " ( H i " i 2 H P,n The Departure (2-28) } R T Function A c c o r d i n g to R e i d et a l (1977), w h i l e the t r u n c a t e d virial e q u a t i o n i s u n r e l i a b l e i n the p r e d i c t i o n of enthalpy d e p a r t u r e s , the W i l s o n m o d i f i c a t i o n of the Redlich-Kwong e q u a t i o n i s r e p o r t e d to g i v e very good results. Combining E q u a t i o n s (2-22) and (2-28) w i t h E q u a t i o n (2-15), we have (H, - H°) i i 2 RT | -b b, i 2 (v-b) 1 v(v-b) v 8T p,n ; v ti T ; ft ft , a bF — ft v v b ft Fb, i , a i 4 + v(v+b) vfb 8T n ; 0, b 2 (v+b) p,n b ft * (v+b) <8£T> p_,.n 1 b b + v y (2-29) - 48 - where V 9v 9(TF) 9T ^ v(v+b) b 1 (v-b) (—) Q. bF(2v+b) I ft v^(v+b) 3. 3F ^ p . n ( " 3F 3T>p,n " ' (1 -1 72 = 57 + L M 1 (v-b) V T N ^7^1.57 + 1.62^)1^ N and [ ^ ] 2-9-2 The Ideal-Gas p > n = " ^ y (0.57 + ± 1.62» ) ± Enthalpy T r e a t i n g E n t h a l p y as a f u n c t i o n of temperature recognizing we the f a c t and p r e s s u r e , and that i d e a l - g a s enthalpy i s Independent of p r e s s u r e , have 9H° d H l • <-§T>p d T " C pi d T Therefore, T H? = / C°,dT (2-30) ref where T ^ i s the enthalpy r e f e r e n c e In temperature. t h i s work, C° i s r e p r e s e n t e d by a t h i r d - o r d e r p o l y n o m i a l f u n c t i o n of temperature, C ° . = a, + S T px i i thus: + C,T i + d T i 4 (2-31) - 49 - Combining E q u a t i o n s H° - a i (2-30) and (2-31), we have + ^ T T where 3 + fj. T 4 + e (2-32) ± ^ e 2-10 + !| T 2 " -t i ref i a T "T + T ref ^ + T ref + ~T T ref 1 ' Liquid Enthalpy The e s t i m a t i o n of l i q u i d t h a t of the vapour. enthalpy i s more i n v o l v e d , compared to The r e p r e s e n t a t i o n adopted here i s i n accordance w i t h the recommendation by Reid and co-workers (1977). pure-component l i q u i d \ The e n t h a l p y - d e p a r t u r e i s d e f i n e d by: ~ H ° = ( h - h j ) + ( h j - H j ) + (Hj - H°) (2-33) ± where hj^ = the l i q u i d enthalpy of pure component i at the system temperature H° and p r e s s u r e . = the i d e a l - g a s enthalpy of component i a t the system temperature h^ and a t the r e f e r e n c e p r e s s u r e , P ° . = the s a t u r a t e d - l i q u i d enthalpy a t the system tempera- ture and a p r e s s u r e c o r r e s p o n d i n g to the vapour p r e s s u r e of the component. g = the saturated-vapour enthalpy of the same c o n d i t i o n s as h^. The liquid mixture enthalpy i s g i v e n by N h - I N x , ( h , - H?) + i=l F where h i s the heat of m i x i n g . I i=l x.H° + h 1 E component a t the - 50 - The s o (H^ - H^) term i s a vapour-phase p r o p e r t y and a p p l y i n g the equations developed components. The i n S e c t i o n 2-9 I t should be noted s s presented below f o r ( h ^ - h^) c r i t i c a l components. to i n d i v i d u a l and that the relationships s ( h ^ - H^) do not apply to super- None of the a p p l i c a t i o n s i n t h i s work i n v o l v e d components. 2-10-1 Departure The ( h ^ - h^) h i " h m r e p r e s e n t s the e f f e c t of p r e s s u r e on (Reid et a l . , 1977), thus: i i" H ) C i h ci ) SL ci 'SCL' SCL denote ' s a t u r a t e d l i q u i d ' and saturated-liquid term i s d e f i n e d by *i * v-' *^ 1 oi 'sub- respectively. Six T i h C r n where the s u b s c r i p t s 'SL' and The i" H ) rp ci cooled l i q u i d ' liquid i t i s c o r r e l a t e d by the c o r r e s p o n d i n g - s t a t e s equation of and Alexander ( from S a t u r a t i o n Value g term e n t h a l p y , and Yen by other r e q u i r e d terms are estimated as d i s c u s s e d i n the following subsections. such i s obtained SL ~ 1 + d 1 i<*" ri> ' P ' w h i l e the s u b c o o l e d - l i q u i d term i s g i v e n by <V"\ T = -a (P ± ci " b ) r l ± - C i ( T - d ) r i ± - e.(T r i - f.) 2 SCL - 8 (P r l h 1 + r ) ( T rl- i 1 ) l ( l n P .)(£n T ) ± r r ± + .2 V" ri V* P + m ± + n P ri )( * > n T ri (2-35) - The constants — 'a' through i n E q u a t i o n (2-35) — compressibility vapour-pressure 2-10-2 - 'd' i n E q u a t i o n (2-34), and 'a' through are g i v e n as d i s c r e t e f u n c t i o n s of the r a t i o , Z^, R e i d et a l , 1977). 51 P 'm' critical of the component ( f o r t h e i r v a l u e s , see i n the above equations i s the reduced of component i . The E n t h a l p y of V a p o r i z a t i o n The s s e n t h a l p y of v a p o r i z a t i o n , h^ - H^, r e l a t i o n s h i p ( R e i d e t a l , 1977) i s estimated from Watson's r e f e r r e d to the normal b o i l i n g p o i n t of the component, thus: hf - H = i i S AH 1 - T , ri vbi 1 ~ T »0.38 (2-36) rbiJ where ^ T^^^ and = the reduced normal b o i l i n g temperature of the component. AH , = the component's e n t h a l p y of v a p o r i z a t i o n a t T , vbi rbi AH , . i s estimated from the c o r r e l a t i o n of V e t e r e ( R e i d et a l , 1977): vbi 0.4343£nP AH = RT ( D 1 2-10-3 - 0.68859 + 0.89584T £± 0.37691 - 0.37306T, , + 0.14878P ,T~ , bri ci bri b r l 9 ) (2-37) The E n t h a l p y of M i x i n g The molar e n t h a l p y of mixing f o r component I i n a l i q u i d i s g i v e n by the r e l a t i o n h •( - h RT o— mixture (Reid et a l , 1977): 3An Y 5 = ( 3T > P,n (2-38) - 52 - The W i l s o n e q u a t i o n al., [ E q u a t i o n (2-23)] was employed by Orye ( P r a u s n i t z e t 1967) i n the e s t i m a t i o n of excess enthalpy and was found to give reasonable r e s u l t s , e s p e c i a l l y with nonassociating s o l u t i o n s . When e q u a t i o n (2-23) i s s u b s t i t u t e d d i f f e r e n t i a t i o n i s performed, into Equation the r e s u l t i s N L (2-38) and the N X J iJ iJ 6 N A ^ V u x ^ " £ AJ X ^ . ( L X =L X j k j \ j JV 8 . where hi - hi - \ i The molar enthalpy of mixing N h = I i=l _ x (h f o r the l i q u i d mixture - h ) 1 1 When t h i s summation i s a p p l i e d to E q u a t i o n the r i g h t - h a n d s i d e reduces * N x i s g i v e n by L N T (2-39), the second term on to z e r o , and we have x 3 A j i ji j E < -*°> 2 I 3=1 x j ij A CHAPTER THREE ISOTHERMAL VAPOUR-LIQUID FLASH CALCULATION 3—1 Introduction An exhaustive done here i n an study of the v a p o u r - l i q u i d f l a s h problem has been attempt to o b t a i n c o n c l u s i v e r e s u l t s and to form the b a s i s f o r t a c k l i n g the r e l a t e d problems of l i q u i d - l i q u i d solid and l i q u i d - equilibria. The i n v e s t i g a t i o n i s i n three p a r t s . 3-2) t r e a t s double-loop u n i v a r i a t e methods. 3-3) d e a l s w i t h free-energy minimization The f i r s t part The second p a r t methods. The l a s t ( S e c t i o n 3-4) t r e a t s s i n g l e - l o o p u n i v a r i a t e methods. (Section (Section segment S e c t i o n 3-5 con- t a i n s a comparison of the d i f f e r e n t methods, a study of the e f f e c t of varying the frequency of e q u i l i b r i u m - r a t i o computation, and g e n e r a l conclusions. 3-1-1 Symbol Nomenclature Definition a M a t r i x d e f i n e d i n Eq. (3-40a). A Matrix defined i n Eq. (3-29). A Matrix defined i n Eq. (3-36). b_ S e t of column v e c t o r s d e f i n e d B_ Vectors C Parameter d e f i n e d E Deviation defined i n Eq. (3-43). i n Eq. (3-29) and i n Eq. (3-36). i n Eq. (3-40). parameter. -53- - 54 - f Partial fugacity. f Check f u n c t i o n . f° Standard s t a t e f u g a c i t y . F T o t a l molar feed r a t e . F Functions d e f i n e d i n Equations (3-28), (3-40) and (3-53). g M o d i f i e d G, d e f i n e d i n E q . (3-32). g V a r i a b l e d e f i n e d by Eq. (3-78). G N o r m a l i z e d Gibbs f r e e energy f u n c t i o n . G a G T Quadratic T a y l o r a p p r o x i m a t i o n of G. T o t a l Gibbs f r e e energy of system. I_ I d e n t i t y matrix d e f i n e d i n Eq. (3-29). J Jacobian K Equilibrium Ratio. matrix. K Q A r e f e r e n c e K, d e f i n e d i n E q . ( 3 - 1 ) . K r K ratio as d e f i n e d i n Eq. ( 3 - 1 ) . 1 Component l i q u i d molar content. L T o t a l l i q u i d molar n Number of ' s p e c i e s ' i n geometric programming formu- content. tion. N P Number of components i n system. Pressure. r V a r i a b l e s d e f i n e d i n Eq. (3-43). R U n i v e r s a l gas c o n s t a n t . Parameter d e f i n e d i n Eq. (3-78a). - 55 - t Exponent parameters d e f i n e d i n E q u a t i o n s (3-18) and (3-78a). T Temperature, u Lagrange V T o t a l ' V a p o u r molar c o n t e n t . w Geometric programming multipliers. v a r i a b l e vector, defined i n S e c t i o n 3-3-4, paragraph 1. W Geometric programming x L i q u i d - p h a s e mole f r a c t i o n , y Vapour-phase z System mole f r a c t i o n . Z Component molar content f o r system. Greek mole v a r i a b l e , r e l a t e d to w. fraction, Symbols a Mean-value a Parameter d e f i n e d Y Liquid-phase a c t i v i t y 6 Kronecker d e l t a , d e f i n e d 9 Phase 0 Temperature A Parameter v a r i o u s l y d e f i n e d and theorem parameter, i n Eq. (3-18). coefficient. i n Eq. (3-26). fraction. f u n c t i o n d e f i n e d by Eq. (3-48). i n Equations ( 3 - 3 0 ) , (3-55) (3-58). y Chemical p o t e n t i a l , v Component vapour molar c o n t e n t . <t> Vapour-phase <|> Function defined X N o n i d e a l i t y parameter, d e f i n e d Eq. ( 3 - 1 ) . fugacity coefficient. i n Eq. (3-17). Immediately f o l l o w i n g - 56 - Subscripts f Frequency. i,j,k Component. L Liquid. o Denoting h y p o t h e t i c a l primal f u n c t i o n i n geometricprogramming formulation, s s u c c e s s i v e - s u b s t i t u t i o n value, v Vapour. - (as i n x ) : denotes a v e c t o r quantity. superscripts m D e f i n e d i n Eq. (3-76). n I t e r a t i o n counter. o Standard s t a t e ; i n i t i a l ' First Second * derivative. derivative. Equilibrium; 3-1-2 Systems Employed point. optimal value. and a Measure of N o n i d e a l i t y S i x t e e n systems were used i n the a p p l i c a t i o n of the v a p o u r - l i q u i d f l a s h c a l c u l a t i o n methods. They range from b i n a r y systems to systems c o n t a i n i n g s i x components. They were chosen to span a wide spectrum of b o i l i n g ranges and to embody m i x t u r e s w i t h d i f f e r e n t degrees of nonideality. T a b l e 3-1 w e l l as a code-name by The T - v e r s u s - 6 * T h i s was c o n t a i n s v i t a l i n f o r m a t i o n on each system as which the system w i l l h e n c e f o r t h be addressed. p r o f i l e f o r each system has a l s o been g e n e r a t e d . done by choosing 12 temperatures at u n i f o r m i n t e r v a l s to cover - 57 - Table 3-1 V i t a l information on vapour-liquid systems CodeName No. Of Components Pressure (Atm.) BubblePoint (K) DewPoint (K) VA 4 1.0 336.15 342.64 VB 3 1.316 85.87 88.94 VC 5 1.0 330.83 335.34 Benzene (82.2); Acetone Acetate VD 4 1.0 351.40 365.54 Water (27.217); Ethanol (22.783); Ethyl-Acetate (27.217); Acetic-Acid (22.783). VE 4 1.0 323.24 328.85 Acetone (32.4); Chloroform (32.6); Dimethylbutane (22.2); Methanol (12.8). VF 5 1.0 352.27 360.88 Hexane (18.6); Methylcyclopentane (25.7); Cyclohexane (11.8); Benzene (15.3); Toluene (28.6). VG 4 1.0 344.21 344.69 Cyclohexane (18.6); Benzene (42); Isopropanol (35.2); Methyl-Ethyl-Ketone (4.2). VH 4 1.0 328.06 331.17 Ethanol (10); Chloroform (20); Acetone (50); N-Hexane (20). VI 4 1.0 334.05 341.08 Benzene (45.6); Chloroform (5.3); Methanol (5.4); Methyl-Acetate (43.7). VJ 6 1.0 343.08 360.05 Hexane (18.6); Methylcyclopentane (20.7); Cyclohexane (11.8); Benzene (15.3); Toluene (28.6); Ethanol (5.0). VK 3 3.7 373.97 376.77 Acetone (25.7); Methanol (64.0); Water (10.3) VL 2 1.0 383.10 542.97 Benzene (50.0); Heptadecane (50.0) VM 2 1.5 402.26 545.97 Cyclohexane (50.0); Hexadecane (50.0). VN 2 1.0 386.60 583.49 Cyclohexane (50.0); Eicosane (50.0) VO 2 1.0 379.33 529.24 Tetrachloromethane (50.0); Hexadecane (50.0). VP 4 1.0 380.50 ' 528.35 Cyclohexane (17);Hexadecane (49); Tetrachloromethane (17); Benzene (17). Composition (mole %) N-Hexane (37.1); Ethanol (3.4); Methylcyclopentane (50.1); Benzene (9.4). Nitrogen (30.36); Argon (29.48); Oxygen (40.16) (3.6); Chloroform Methanol (6.3); (4.2); Methyl(3.7) - 58 - the v a p o u r - l i q u i d two-phase r e g i o n and, at each temperature, performing * a f l a s h c a l c u l a t i o n to o b t a i n 6 . g r a p h i c a l l y i n F i g u r e s 3-1 A scheme was presented 3-2. non- I t i n v o l v e d the f o l l o w i n g s t e p s : Choose (from F i g u r e 3-1 6 r e s u l t i n g p r o f i l e s are d e v i s e d aimed at q u a n t i f y i n g the degree of i d e a l i t y of each system. (1) and The or 3-2) the temperature corresponding to = 0.5 v (2) Determine K c o r r e s p o n d i n g t o 6 = 0.0. — v (3) Vary 9^ at u n i f o r m i n t e r v a l s 8 , determine from 0.0 the c o r r e s p o n d i n g K. D e f i n e t h i s as K . —o to 1.0. For each Then d e f i n e (3-1) The cally - versus - 8^ p r o f i l e f o r each system i s presented g r a p h i - i n Figure (3-3). s i n g l e parameter x» was In view of the m o n o t o n i c i t y of the p r o f i l e s , i n t r o d u c e d as a measure of the degree of n o n i d e a l i t y of each system. X = K and r at 8 v I t i s d e f i n e d by = 1.0, i t i s a f u n c t i o n only of X has been generated different temperature. as a f u n c t i o n of temperature systems by determining i n g to d i f f e r e n t a values of 9 * V * 9^ are presented i n F i g u r e 3-4 K r from ( ^ 9 = v * V 1) at temperatures = 0 to 9 f o r a l l 16 * V =1. systems. f o r the correspond- P l o t s of x v e r s u s * - 59 - 346 • 0.0 0.1 1 0.2 1 1 0.3 0.4 1 i 1 05 0.6 0.7 1 OB 1 OS 1 1.0 »-Vapour Fraction, 8* Fig 3-1 : Two-phase region temperature profiles for narrow-boiling systems - 60 - 3 - 2 : Two-phase region temperature profiles for wide-boiling systems - 61 - 0 0.1 0.2 0 3 0.4 0.5 Q6 0.7 0.8 0.9 1.0 * Fig 3-3; K" V.C? for the different systems at Q# 0.5 r v - 62 - 0.420 0.380 4 0.340^ 0.300 0,260 4 0.220 0.180 0.140H 0.100H 0.604 0.20H 0,0 0,1 T 0.2 1 1 1 0.3 0.4 0.5 1 ' 0,6 0.7 r 0.8 0.9 1.0 a* Fig 3-4: Plot of nonidealiry parameter versus solution-point vapour fraction - 63 - 3-2 Double—Loop Univariate Methods The double-loop u n i v a r i a t e methods are based approach. As the name i m p l i e s , they i n v o l v e two on the mass-balance l e v e l s of an o u t e r i t e r a t i o n loop f o r converging the vapour f r a c t i o n ; i n n e r i t e r a t i o n loop f o r u p d a t i n g the e q u i l i b r i u m r a t i o s , mole f r a c t i o n s , at the c u r r e n t value of the vapour The efficiency following iteration: and an and hence the fraction. of the u n i v a r i a t e methods depends very much on the three f a c t o r s : (a) The way is (b) the c h e c k - f u n c t i o n f o r u p d a t i n g the vapour fraction formulated. The a c c e l e r a t i o n method employed i n u p d a t i n g the vapour fraction. (c) The way the v a r i a b l e s are initialized. The most comprehensive comparative have been undertaken authors reached study of the s u b j e c t known t o to date i s that by Rohl and S u d a l l (1967). the c o n c l u s i o n t h a t the best r e s u l t i s produced f o r m u l a t i o n of Rachford and R i c e (1952) a c c e l e r a t e d by e i t h e r second-order The Newton method or the t h i r d - o r d e r Richmond method. by the the However, the study has a number of p o i n t s a g a i n s t i t : (1) The (see authors e r r o n e o u s l y concluded t h a t the standard f o r m u l a t i o n d e f i n i t i o n below) i s i n c a p a b l e of c o n v e r g i n g . (1975) r i g h t l y a s s e r t s , t h i s f o r m u l a t i o n converges s o l u t i o n i f the vapour f r a c t i o n i s i n i t i a l i z e d (2) The study was undertaken r e p o r t e d by K i n g Rachford-Rice As H o l l a n d to the desired to 1. before a recent formulation, (1980) and said f o r m u l a t i o n , was to be s u p e r i o r to the proposed by Barnes and Flores. - 64 - (3) The study failed to a p p r e c i a t e mode of i n i t i a l i z a t i o n formulations T h i s study the The the great i n f l u e n c e that the has on the performance of the d i f f e r e n t and a c c e l e r a t i o n methods. s t a r t s by i n v e s t i g a t i n g f o u r d i f f e r e n t formulations of problem: (a) The standard formulation. (b) A l o g a r i t h m i c form of the standard (c) The Rachford-Rice formulation. (d) The B a r n e s - F l o r e s formulation. formulation. second f o r m u l a t i o n has been i n t r o d u c e d because i t i s to the standard form what the B a r n e s - F l o r e s formulation i s to the Rachford-Rice formulation. Both the Newton method and the Richmond method have been s t u d i e d as ways of a c c e l e r a t i n g convergence. problem a r e e x p l o r e d initialization 3-2-1 later Some other f o r m u l a t i o n s ( S e c t i o n 3-2-4) w h i l e schemes a r e d i s c u s s e d of the different i n S e c t i o n 3-2-5. T h e o r e t i c a l Background The centered mass-balance approach t o p h a s e - e q u i l i b r i u m computation i s on the fundamental thermodynamic e q u i l i b r i u m e q u a l i t y f where f il = i 2 f = i s the p a r t i a l = f ip= - - f i M ( 3 " I n the case of a v a p o u r - l i q u i d e q u i l i b r i u m , E q u a t i o n (3-2) f iL = iv f ) f u g a c i t y of component i , at e q u i l i b r i u m , i n phase p of an M-phase system. becomes 2 - 65 - Defining iL " Vi i f X (3-2a) and f i v f i = • Py , 1 i we have i Y If we now X i = * i P y i <"> 3 i n t r o d u c e the concept Equation of e q u i l i b r i u m r a t i o s , K^, (3-3) l e a d s to the e q u i l i b r i u m " 1 " *7~ *T The a d d i t i o n a l equations " 2 ' " ' ' 3 then relationship N ) t h a t have to be c o n s i d e r e d along w i t h ( 3 " 4 ) Equation (3-4) a r e : Component-Mass 1 + v Total-Mass L Balance: - Z = 0 (1-1, 2 N) (3-5) Balance: + V - F = 0 (3-6) Liquid Mole-Fraction Restriction: N I x = 1 (3-7) 1-1 Vapour M o l e - F r a c t i o n R e s t r i c t i o n : N I y. = i (3-8) - 66 - I n t r o d u c i n g the r e l a t i o n s v F ' = % v ± and and Z i = Fz (3-8a) i combining Equations z x. l + (3-2) through i O^-i) (3-4), we have ( i = 1, 2,..., N) e (3-9) v From E q u a t i o n ( 3 - 4 ) : y =Kx = K z. i - i 1 + (K -1)0 i v (i=l,2,...,N) (3-10) The v a r i o u s f o r m u l a t i o n s of the r e s i d u a l f u n c t i o n employ combinations of Equations The Standard Formulation It (3-9) (3-7) through different (3-10). combines the r e s i d u a l form of E q u a t i o n (3-7) w i t h E q u a t i o n to g i v e f(9 ) = N I x i=l - 1 = N z I { i } -1 i = l 1 + (K -1)6 ) i v (3-11) The Standard L o g a r i t h m i c Form: T h i s i n v o l v e s t a k i n g the l o g a r i t h m of E q u a t i o n (3-7) and combining i t w i t h E q u a t i o n ( 3 - 9 ) , thus: N f(8 ) = £n[ N z I x ] = An[ I { ± }] i=l i = l 1 + (K -1)6 i v (3-12) - 67 - The R a c h f o r d - R i c e Equation Formulation: (3-7) i s s u b t r a c t e d from E q u a t i o n combined w i t h E q u a t i o n s f(9 ) = N I {V i=l (3-9) and ~ x } = ± Barnes-Flores The N (K - l ) z I { —} i=l 1 + (K.-l)6 x I = i=l y^ of E q u a t i o n s (3-7) and v (3-8) gives y i=l the l o g a r i t h m of t h i s e q u a t i o n and from E q u a t i o n s (3-9) and -[ l { An n 1-1 v s u b s t i t u t i n g f o r x^ (3-10) r e s p e c t i v e l y , we ) =i [ I { f(6 3-2-2 (3-13) N I Taking result Formulation: combination N the (3-10) to y i e l d 1 The (3-8) and 1+(K i -1)6 i=l v 1+(K i and have i -1)6 }] vi A G e n e r a l Form of the A l g o r i t h m A g e n e r a l a l g o r i t h m f o r the double-loop u n i v a r i a t e methods c o n s i s t s of the f o l l o w i n g main s t e p s : (1) I n i t i a l i z e x, y, K and 8 . (2) Compute x c o r r e s p o n d i n g 8 v from E q u a t i o n to the c u r r e n t values of K (3-9). (3) Compute y_ from y^ = (4) Normalize (5) Using (6) I f the new K ^ ^. x x and y_ and update the new and values of K, K. update x^ x i s not w i t h i n some t o l e r a n c e of i t s o l d v a l u e , then go to Step 3. Otherwise go to Step 7. (3-U) - (7) T e s t f o r convergence. terminate (8) presented d a n go to Step the double-loop above. - I f the outcome i s p o s i t i v e the i t e r a t i o n . Update 6^ All 68 Otherwise go to Step then 8. 2. u n i v a r i a t e methods have the b a s i c s t r u c t u r e They d i f f e r only i n the way Step 8 i s implemented. Where Newton's method i s employed i n the updating of 8 , then the r e l a t i o n s h i p i s e = n+1 _ f(e )/f(e ) n 0 V V v n n v V where f(8 ) = v' d8 n d f v 8 =e n With Richmond's method, the a p p r o p r i a t e e q u a t i o n i s e^ = ej; - 2f(e^).f(e^/{2[f'(e^)] - f(e^.f e^} 1 2 ( where v e 'e =e 2 d n V f ( 8 ^ ) i s g i v e n by one V V of Equations (3-11) through (3-14) depending on which f o r m u l a t i o n i s being t e s t e d . 3-2-3 A Comparative Study of the Formulations I n order to have a b e t t e r understanding f o u r f o r m u l a t i o n s r e p r e s e n t e d by E q u a t i o n s check f u n c t i o n s were VA. 3-8. generated These have been p l o t t e d and And (3-11) through at d i f f e r e n t temperatures are p r e s e n t e d f o r the purpose of comparing the of the nature of the (3-14), f o r system i n F i g u r e s 3-5 linearity the through of the l o g a r i t h m i c f o r m u l a t i o n s r e l a t i v e to t h e i r n o n l o g a r i t h m i c c o u n t e r p a r t s , t h e i r - 69 - 0.18 -0.02 H Fig 3-5: Standard form check-function profiles for system VA - 70 - 0.16H -0.02H -Q04 J 1 Fig 3-6: Standard logarithmic check-function prof iles for system VA - 71 - 0.24 -0.20- — 1 — • « ( , Fig 3-7; Rochford-Rice check-function profiles for system VA - 72 - 0.22 -0.16H — Fig 3-8: Barnes-Flores check-function profiles for system VA -73- check-function Figures (1) 3-9 The p r o f i l e s have been juxtaposed at two and 3-10). standard by any and A quick the gradient study of the p l o t s r e v e a l s standard v — (2) The < v logarithmic method, should s o l u t i o n i f 9^ i s i n i t i a l i z e d 6* < 9 temperatures forms, that: accelerated always converge to the to 1 or any point (see right i n the range 1. — R a c h f o r d - R i c e and converge i f not Barnes-Flores properly formulations initialized. convergence i s by e n s u r i n g The way that 9° s a t i s f i e s could fail to to guarantee the their condition 0 < 9° < 9 . — v — v (3) G i v e n t h a t f o r the standard 9° = 1 and v and that f o r the other the s t a n d a r d - l o g a r i t h m i c two forms, 9° = 0, then v ' forms, one would expect the former p a i r to r e t u r n b e t t e r computation times f o r mixtures that are (4) for 'liquid-heavy' The logarithmic i n the regions 'vapour-heavy' and the l a t t e r p a i r to be better mixtures. forms do t u r n out to be where the i n i t i a l i z a t i o n l e a d to convergence. s l i g h t l y more l i n e a r of 9^ i s expected However, i t i s only through the to empirical process that i t can be a s c e r t a i n e d whether the g a i n i n linearity is the e x t r a computing e f f o r t the sufficient transformation functions In order different and to o f f s e t to a l o g a r i t h m i c f o r the check their derivatives. to t e s t out formulations initialization scale requires that the above deductions e m p i r i c a l l y , were a p p l i e d to a l l the schemes s t a t e d under D e d u c t i o n 3. were generated f o r 19 p o i n t s 16 the systems, u s i n g In each case, spread a p p r o x i m a t e l y u n i f o r m l y on the results the 0/124 •^v Fig 3-9: Comparrig the fouriormulotions for system VA at 340 K -75- 0.22 Fig 3-10 : Comporing the four formulotions for system VA at 342.5 K -76- 9v a x i s between 0.05 and 0.95. Both the Newton and Richmond methods were employed i n a c c e l e r a t i o n . The plot r e s u l t s c l e a r l y confirmed D e d u c t i o n 3 as can be seen from f o r system VC other systems). have a s l i g h t i n F i g u r e 3-11 They (A s i m i l a r t r e n d was the observed f o r a l s o r e v e a l e d that w h i l e the l o g a r i t h m i c forms edge over t h e i r n o n l o g a r i t h m i c c o u n t e r p a r t s when Newton's a c c e l e r a t i o n method i s employed, the p o s i t i o n i s r e v e r s e d when Richmond's method, which i s not a l i n e a r a c c e l e r a t i o n method and which r e q u i r e s second I t was derivatives, i s used. f u r t h e r observed requires less i t e r a t i o n s , that the Richmond method g e n e r a l l y and l e s s computation time, than the Newton method (see F i g u r e 3-12 as an example of t h i s ) . However, w h i l e the Newton method converged i n a l l cases f o r a l l the f o r m u l a t i o n s , the Richmond method was found to encounter o v e r p r o j e c t i o n i n a number of cases — systems — convergence problems due to e s p e c i a l l y with w i d e - b o i l i n g f o r a l l the f o r m u l a t i o n s except the s t a n d a r d . I n view of t h i s , no q u a n t i t a t i v e r e s u l t s are p r e s e n t e d here, t h i s having been d e f e r r e d u n t i l a f t e r an i n v e s t i g a t i o n of more e l e g a n t initilization schemes. 3-2-4 E x p l o r i n g Other Possibilities Documented i n t h i s s e c t i o n i s a summary of the attempts made a t s e e k i n g a f o r m u l a t i o n that would be s u p e r i o r to the f o u r f o r m u l a t i o n s already discussed. L e t us c o n s i d e r E q u a t i o n ( 3 - 5 ) . combining result I t can r e a d i l y be shown, by t h i s e q u a t i o n w i t h E q u a t i o n s (3-6) and over a l l components, that (3-8a) and summing the -77- TOO Legend : I II III IV 30H Standard Standard logarithmic Rachford-Rice Barnes-Flores 20 H 10H 0 r 1 1 1 1 1 1 1 " 0.1 0.2 0 , 3 0.4 0.5 0.6 0.7 0.8 0.9 IJO *- 8y Fig 3-11: Iteration units (on a scale of OtolOO) v. vapor fraction for system VC (based on Newton's method) -78- 100 90 4 80- 70 4 604 'E c o 2 504 404 Legend: Newton-acceleroted standard method Richmond-accelerated standard method Newton-occelerated Rachford-Rice method IV = Richmond-accelerated Rachford-Rice meth 30 20H 104 0 0.1 Q2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 • f t Fig 3-12: Iteration units v. vapor fraction : comparison between Newton and Richmond for system VB - 79 - N N N (3-15) Equation (3-15) can be arranged to give N , N I (x,-y.) = — ( I x i=l 6 i=l v 1 - 1) (3-16) We find that the left-hand-side term is the Rachford-Rice check function while the numerator of the right-hand-side expression is the standard check function. In view of the above relationship between the two check functions and bearing in mind the nature of the profiles in Figures 3-5 through 3-8, i t was felt that an appropriate redefinition of the denominator of the right-hand side of Equation (3-16) could produce a check function that would be more linear, and hence converge faster, than a l l the existing formulations. Let us introduce a general check function, f^, defined by 1) where <(> is some function of 0 <|> + 0 as 6 •»• 0 v Two classes of <(> were studied. satisfying the condition (3-17) - 80 C l a s s 1: In t h i s c l a s s , <f> was - g i v e n the g e n e r a l definition <f> = ^ The f^ profile for different (3-18) f o r system VA values of t . t to employ f o r any at 341.66K i s presented The i n Figure 3-13 q u e s t i o n a r i s e s as to the best v a l u e of g i v e n system. A number of t - f i x i n g methods were tried. One of these i n v o l v e d s e t t i n g t = 1 - 6°, where 6° i s v v o b t a i n e d by i n i t i a l i z a t i o n Scheme 5 d i s c u s s e d i n S e c t i o n u n d e r l y i n g r e a s o n i n g i s that that the standard f o r m u l a t i o n behaved b e t t e r than the Rachford-Rice 6° i s s u f f i c i e n t l y v standard form) near the vapour s a t u r a t i o n l i n e . testing as the one simple v s a t u r a t i o n l i n e , and would combine the best of the two On better for 0 c l o s e to 0, to 0 ( t h e T h i s way, f^ methods. the t - f i x i n g methods, i t turned out t h a t a scheme o u t l i n e d above performed, on the average, scheme s e t t i n g t = 0.5. T h i s constant only as w e l l as value of t was presence One d e f e c t of the c l a s s - 1 type of <J> proposed above i s the of the t u r n i n g p o i n t ( c h a r a c t e r i s t i c of the standard f u n c t i o n ) f o r values of t not i n the neighbourhood of 1. such the therefore employed i n subsequent i n v e s t i g a t i o n s of the f o r m u l a t i o n . C l a s s 2: form c l o s e to 0 , then d> would tend to 1 (the v form) near the l i q u i d Rachford-Rice The s i n c e the p r e v i o u s i n v e s t i g a t i o n r e v e a l e d ft ft f o r 6 c l o s e to 1 w h i l e the l a t t e r was v if 3-2-5. check This i s -81- -0.20-! 1 1 1 1 1 1 1 1 1 1 0 . 2 0 0 . 3 0 Q 4 0 0 . 5 0 0.60 0.70 0 . 8 0 0.90 1,00 0.10 *-9 y Fig 3-13 : fi \/.Q for different values of t for system VA at 341.66K v -82- a t t r i b u t a b l e t o the asymmetric n a t u r e of the <t>(9 ) p r o f i l e s v the ( 1 - 6 ) n e (see p l o t s of <> t versus v 8 f o r t = 0 . 2 and 0 . 5 i n v n Figure about 3-14). To o b t a i n a symmetric p r o f i l e , •<V = 1 (I"- + $> was d e f i n e d by 1)6 < " > 3 18 v P l o t s of <> t v e r s u s 6^ f o r a = 2 and a = 5 a r e shown i n F i g u r e 3 - 1 4 w h i l e f^ p r o f i l e s f o r the same v a l u e s of a a r e p r e s e n t e d i n F i g u r e 3 - 1 5 f o r system VA a t 3 4 1 . 6 6 K . Here, the problem i s t h a t of d e t e r m i n i n g a s u i t a b l e a f o r a given system. One of the schemes experimented w i t h (1) I n i t i a l i z i n g K w i t h x = y = z_ (2) Assuming $ = 6 and e s t i m a t i n g f ( 0 . 0 ) , A v 9 from: f (0.0) + V°- > 5 N = 1 - 1 N Kz ± ± , d-K,) , ± = 1-1 ' and N z, f^l.O) = i-1 K i involved: f,(0.5) <p and f ( 1 . 0 ) x <p -83- Fig 3-14: <p v. 0 for different forms of <£> V -84- -0.20-1 1 1 1 1 1 1 1 1 1 1 0.10 Q20 030 0.40 050 0.60 0.70 0.80 0.90 1.00 ^ Fig 3-15: f,v. £? for different forms of <£> for system VA at 341.66 K v -85- (3) C o n s t r u c t i n g a l i n e on the {9 ,f^} plane {O.O, f^(O.O)}, and JO.5, through f (0.5)} to pass through {1.0, f * ( 1 . 0 ) } . f*(1.0) (4) Setting oc = \ fjn0)\ <P When the above scheme was unreasonable lines. t e s t e d , i t was v a l u e s of a i n some cases — e s p e c i a l l y near the f o r i n the scheme. Simply s e t t i n g a equal to 2 found to perform j u s t as w e l l , on the average, schemes such as the one 0.5 as the more i n v o l v e d s h a l l r e f e r to the method u s i n g 2 as '^-normalized method 1' and the one w i t h $ = 9 v 1+9 V '^-normalized method was above. I n the a p p l i c a t i o n s to f o l l o w , we 3-2-5 saturation T h i s i s due mainly to the dependence of K on composition which i s not accounted <j) = 6 found to y i e l d as v 2'. I n v e s t i g a t i n g D i f f e r e n t I n i t i a l i z a t i o n Schemes Five i n i t i a l i z a t i o n schemes were f o r m u l a t e d and s t u d i e d . the schemes, presented below, K i s based on x = y = z. Scheme 1 - In t h i s N 9° = •XT V I *-< 1=11+ scheme, 9° i s simply g i v e n by z, i 1/K In a l l -86- Scheme 2 - T h i s scheme i s i n two steps: N Step 1: Define 6 £ i = l 1 + 1/K N z V Step 2: Set 6° = v z = £ i=l — 1-6 v i Scheme 3 - T h i s scheme i s based on a r e g u l a - f a l s i i n t e r p o l a t i o n 6 =1.0 v and 9 = 0 . 5 , u s i n g approximate v a l u e s v obtained from the standard formulation. Thus, between of f ( 6 ) v define N z > I - f ( 9 = 1.0) = i=l F 2 - f ( 9 = 0.5) = 2 ^ v and F = F /(F Then, 9° = 0.5(1 + F ) . 2 2 ^ - 1, i - 1, - F^. Scheme 4 - T h i s scheme, l i k e Scheme 3, a l s o interpolation. However, the i n t e r p o l a t i o n 9^ = 0.5 and i t i s based on involves a r e g u l a - f a l s i i s between 9° = 0.0 and approximate v a l u e s of f ( 9 ^ ) -87- o b t a i n e d from the R a c h f o r d - R i c e f o r m u l a t i o n . Thus, i f we d e f i n e N F = f(9 = 0.0) = I K z i=l 1 N F 2 = f(6 v = 0.5) (K J = 2 1=1 and F = F /(F then 9° = 0.5(1 v 2 2 - 1, 1 * - l)z 1 . i - F^), - F) When the f o u r schemes above were a p p l i e d to a good c r o s s - s e c t i o n of the systems, Schemes 1 and 2. Schemes 3 and 4 were I t was found to perform b e t t e r f u r t h e r observed than that f o r most of the systems, * Scheme 3 o u t c l a s s e s Scheme 4 i n the range 0.5 < 9 * performed was view by the l a t t e r i n the range 0.0 v plotted f o r systems VA are presented i n F i g u r e The and VP, The but i s outdisparity range of the system. data were generated In and v u s i n g the two competing schemes. These 3-16. curves of F i g u r e 3-16 reflect l e d to the c o n c e p t i o n of Scheme 5. Scheme 5: Step 1: < 1.0 — <^ 6^ < 0.5. a l s o found to depend much on the b o i l i n g of these o b s e r v a t i o n s , 9° v e r s u s 9 v Apply Scheme 4. the observed behaviour and they -88- schemes 3 and 4 -89- Step 2: I f the v a l u e of 6° from Step 1 i s g r e a t e r than 0.5, then apply Scheme 3. As expected, presented Scheme 5 was the best of them a l l , and the r e s u l t s i n the next r e s u l t s , based s e c t i o n are based on t h i s scheme (A few other on t e r m i n a l i n i t i a l i z a t i o n , are a l s o presented f o r comparison). 3-2-6 Applications In the r e s u l t s p r e s e n t e d here and i n any other s e c t i o n of t h i s c h a p t e r , the convergence c r i t e r i o n used i s ! |v" - v ? - ! < 1 0 " N i=l 1 1 4 1 Each of the s i x f o r m u l a t i o n s was a p p l i e d to each of the 16 systems at 19 p o i n t s d i s t r i b u t e d approximately u n i f o r m l y between 6 =0.05 and 9 = 0 . 9 5 . v v F o r each of the f i r s t t i o n s , both Newton's and Richmond's methods were employed i n acceleration. methods. Only four formula- the Richmond method was a p p l i e d to the ^ - n o r m a l i z e d I n a p p l y i n g i n i t i a l i z a t i o n Scheme 5, 6° was c o n s t r a i n e d between 0.05 and 0.95. The r e s u l t s are presented how much more e f f i c i e n t this i n i t i a l i z a t i o n simple t e r m i n a l i n i t i a l i z a t i o n , terminal i n i t i a l i z a t i o n been i n c l u d e d . i n T a b l e 3-2. To show scheme i s compared to the the r e s u l t s o b t a i n e d by a p p l y i n g ( 8 ^ = i.o) to the standard method have a l s o Table 3-2 Computation time (CPD seconds) for double-loop univariate algorithms System Terminal iL n l t i a l i z a tion with stand.form I'roposed init: i a l i z a t i o i I with Rj chmond Proposed i n i t i a l i z a t i o n with INewton Standard Rachford- Barnes Rachford- BarnesStandard Flores Flores Standard Logarithmic Rice Standard Logarithmic Rice ^-normal- ^-normalized 2 ized 1 Newton Richmond VA 0.9549 0.9977 1.0243 1.0231 0.9161 0.8954 0.9022 0.8980 0.9079 0.8813 1.1110 1.0169 VB 0.6432 0.6641 0.6255 0.6333 0.6257 0.6370 0.6399 0.6682 0.6487 0.6618 1.0743 0.9156 VC 1.1844 1.2021 1.0948 1.1247 1.1398 1.1639 1.1698 1.1957 1.1556 1.1684 1.9364 1.6105 VD 1.3679 1.3522 1.2152 1.2245 1.2393 1.2389 1.2106 1.2043 1.1646 1.1793 2.3183 1.6690 VE 1.1336 1.1534 1.1990 1.1550 1.1934 1.2067 1.1736 1.1590 1.1863 1.1495 2.2575 1.9408 VF 1.1246 1.1363 1.0654 1.0459 1.0932 1.0917 1.0607 1.0660 1.1169 1.0568 2.0782 1.6829 VG 0.9748 0.9891 0.9784 0.9606 0.9911 0.9897 0.9716 0.9652 0.9996 0.9842 2.1553 1.9070 VH 1.2832 1.2864 1.2850 1.2793 1.3092 1.3061 1.2869 1.2742 1.3060 1.2692 2.3846 1.9826 VI 1.0242 1.0349 1.0095 0.9994 1.0427 1.0568 1.0223 1.0087 1.0431 1.0242 1.7549 1.5226 VJ . 1.4401 1.4418 1.2869 1.2732 1.3482 1.3807 1.3209 1.3440 1.3555 1.3168 2.3292 1.8480 VK 0.9330 0.9196 0.8809 0.8884 0.9099 0.9343 0.8834 0.8898 0.8906 0.8669 1.6487 1.3901 VL 0.6187 0.5883 0.5741 0.5877 0.5474 0.5626 0.5659 0.5735 0.5592 0.5623 1.2973 0.6667 VM 0.6275 0.5853 0.5827 0.5889 0.5765 0.5891 0.5797 0.5845 0.5724 0.5822 1.1685 0.6791 VN 0.6361 0.6162 0.5954 0.6144 0.5872 0.6192 0.5941 0.5848 0.6063 0.6038 1.4750 0.6985 VO 0.6114 0.5772 0.5589 0.5828 0.5689 0.5918 0.5557 0.5519 0.5478 0.5439 1.2307 0.6574 VP 0.8405 0.7904 0.7707 0.7725 0.7802 0.7929 0.7637 0.7777 0.7658 0.7638 1.7977 0.9000 I 15.3981 15.3350 14.7567 14.7537 14.8688 15.0566 14.7010 14.7455 14.8263 14.6144 27.9976 21.0877 -91- 3-2-7 Deductions From the r e s u l t s i n T a b l e 3-2 inal initialization space and p a r t l y due i n g deductions (1) The that have not to the The been presented those — r e s u l t s f o r term- p a r t l y because of the f o l l o w - have been made: scheme seems to be q u i t e r e l i a b l e since no s i n g l e case of f a i l u r e i n a l l the p o i n t s (a t o t a l of 2,432 p o i n t s ) to which i t was (2) w e l l as f a i l u r e s previously reported — chosen i n i t i a l i z a t i o n there was as adopted I n i t i a l i z a t i o n applied. scheme r e s u l t s i n a tremendous s a v i n g i n computation time, compared to the t e r m i n a l - i n i t i a l i z a t i o n approach ( f o r example, u s i n g the standard versus formulation: 27.9976 seconds 15.3981 seconds w i t h Newton's method; 21 .0877 seconds N versus 14.8688 seconds w i t h Richmond's method). (3) While the Richmond a c c e l e r a t i o n method f a r o u t c l a s s e s the Newton method when t e r m i n a l I n i t i a l i z a t i o n i s employed (compare 27.9976 seconds versus formulation), 21.0877 seconds f o r the standard the adopted i n i t i a l i z a t i o n scheme the d i f f e r e n c e becomes q u i t e s m a l l (15.3981 seconds v e r s u s formulation (4) The above). of the l o g a r i t h m i c and (5) The 14.8688 seconds f o r the same o b s e r v a t i o n p r e v i o u s l y made r e g a r d i n g s t i l l hold the n o n l o g a r i t h m i c the r e l a t i v e performance formulations i s found to true. adopted i n i t i a l i z a t i o n eliminating with scheme a l s o has the e f f e c t of the s i t u a t i o n whereby the r e l a t i v e e f f i c i e n c y of a method depends on the l o c a t i o n of the s o l u t i o n . -92- (6) The ^ - n o r m a l i z e d methods g i v e j u s t about the same performance as the other methods, with method 2 being s l i g h t l y b e t t e r than the Rachford-Rice method. With the proposed cult scheme, i t becomes q u i t e to choose from amongst the d i f f e r e n t methods. (^-normalized 3-3 initialization diffi- On the average, method 2 has a narrow edge over the other methods. Free—Energy Minimization Methods The f r e e - e n e r g y m i n i m i z a t i o n methods i n v o l v e d e t e r m i n i n g the e q u i l i b r i u m p o i n t of an i s o t h e r m a l , i s o b a r i c system by p o i n t at which i t s Gibbs f r e e energy i s a minimum. f i n d i n g the A good number o f a l g o r i t h m s e x i s t f o r e f f e c t i n g the f r e e - e n e r g y m i n i m i z a t i o n . a l g o r i t h m s a r e g e n e r a l l y formulated f o r multiphase w i t h p o s s i b l e a p p l i c a t i o n s to n o n - r e a c t i n g systems. chemical Such equilibria Noteworthy works i n the a p p l i c a t i o n of t h i s p r i n c i p l e i n c l u d e those of White e t a l . (1958), C l a s e n (1965), Dluzniewski and A d l e r (1972), Ma and Shipman (1972), E r i k s s o n and Rosen (1973), E r i k s s o n (1975), George e t a l . (1976), and Gautam and S e i d e r (1979). A comparative study of the major a l g o r i t h m s was r e c e n t l y under- taken by Gautam and S e i d e r (1979), and the c o n c l u s i o n was t h a t the RAND method as implemented by D l u z n i e w s k i and A d l e r (1972) should be g i v e n a first-place consideration. The RAND method has been developed v a p o u r - l i q u i d phase e q u i l i b r i u m . o r i g i n a l form, was found phase e q u i l i b r i a , The here f o r the s p e c i f i c case o f But because the method, i n i t s to be r a t h e r i n e f f i c i e n t when a p p l i e d to two- a m o d i f i e d v e r s i o n of i t has been problem has a l s o been s t u d i e d from developed. the p o i n t of view o f -93- geometric programming. The report includes two methods that were developed f o r a c c e l e r a t i n g s u c c e s s i v e - s u b s t i t u t i o n i t e r a t i o n s . methods have been remarkably s u c c e s s f u l and could r e a d i l y f i n d cations A method based on vity i n other areas of p r o b l e m - s o l v i n g . analysis — is also 3-3-1 sensiti- the development of which i s presented i n Chapter 4 — G e n e r a l Theory given t o t a l Gibbs f r e e energy, of a v a p o u r - l i q u i d system i s by Vi-I>" l <V i v ml + ViL> ( 3 i s r e l a t e d to the p a r t i a l f u g a c i t y , f ^ , i n the p. = p° + 1 1 G(v,l) = 1 9 same phase by f I i=l i have { v . [ ^ + £ n ( % ] -f 1 RT f? iv 1 + RT i n ( % ] } f° iL (3-20) S u b s t i t u t i n g E q u a t i o n s (3-2a) i n t o E q u a t i o n (3-20), r e c a l l i n g = 1 (by d e f i n i t i o n ) , and i n molar terms, we I i=l Thermodynamic theory utilizing the d e f i n i t i o n of x^ that and have N G(v,l) = - > f, RT£n(—) D e f i n i n g G = G^/RT, we v appli- included. The f° The u {v o [—+ RT requires v y o 1 * (<f>.P-^-) + l . [ — + fy. RT • 3 3 n *n(Y.—)]} l l . • 3 3 that G assume a minimum v a l u e e q u i l i b r i u m , and what the free-energy minimization (3-21) at methods do is employ -94- some o p t i m i z a t i o n technique that correspond v and + 1 0 < v 3-3-2 to determine t o that minimum — = Z < the v a l u e s of v_ and 1^ subject to ( i = 1,2,....N) Z (3-22) ( i = 1,2,...,N) (3-23) The RAND Method The RAND method combines Newton's optimum-seeking technique Lagrange's p e n a l t y method of h a n d l i n g c o n s t r a i n t s (Himmelblau, First, G i s approximated with 1972). by a q u a d r a t i c T a y l o r expansion, G , about the c u r r e n t p o i n t , {v_ , l } , thus: n n G ( v , l ) = G ( v , l ) + I {|£(v - v " ) + | f ( l j=l j j a n J 1 + N §OI, J J 2 , ^ f e k=l J 2=1 J 9 G N 2 ^ + - i")} n ( 1 j ( " k V k J " V j ) ( \ " V k> ^\ - W " (3 24) J where the d e r i v a t i v e s are e v a l u a t e d at {v_ , _l } and do not take n i n t o c o n s i d e r a t i o n the dependence of D i f f e r e n t i a t i n g Equation n and T on and 1^ r e s p e c t i v e l y . (3-21), we have o If 3 2 9 G 3 A where The = # + * n 6 k * j k 3 p y i . 1 0, f o r k * j 6^ = { 1, f o r k = j d e r i v a t i v e s w i t h r e s p e c t to the l i q u i d - p h a s e composition a r e < " > 3 25 -95- o b t a i n e d from E q u a t i o n s (3-25) and (3-26) by r e p l a c i n g v by 1, V by L, and <f> .Py . by Y .x .. 3 3 3 3 A combination of Equations G (v,l) = G(v ,l ) + a n (3-24) and (3-26) y i e l d s H(^+ n t n ^ X V j - v ) n , N V°T J j J J ^, N 6 . n = 1 k = 1 n v v j J The c o n s t r a i n t E q u a t i o n s (3-22) a r e i n c o r p o r a t e d i n E q u a t i o n (3-27) by means of Lagrange m u l t i p l i e r s Lagrangian u j(J = 1»2,-»«,N) to g i v e the function N F(v,l,u) = G ( v , l ) + a I u (Z - v - 1 ) j=l J J J 3 (3-28) The d i r e c t i o n of search f o r the minimum i s g i v e n by the v e c t o r l i n k i n g {v ,.l } w i t h the p o i n t a t which VF(_v,_l,u) = 0. n n E v a l u a t i n g VF from E q u a t i o n s zero (3-27) and (3-28) and e q u a t i n g t o yields 0 A l -h K 0 0 Av -h = B —u B —V u il (3 -29) -96- where ^ i s an N x N i d e n t i t y matrix; A l i s an N-element Av i s d e f i n e d v e c t o r whose 1th element i s 1 - l ; j j ' n i n a way s i m i l a r t o A l j B i s an N-element ~u v e c t o r whose i t h element i s -Z .; 3^ i s an N-element v e c t o r whose j t h element i s 3 - y ° /RT - A n ^ y " ; B_^ i s an N-element v e c t o r whose j t h element i s o ,„„ . n n - p. /RT - AnY.x.; T J J A^ i s an N x N m a t r i x whose elements a r e g i v e n by . \i jk v." V 3 and A^ i s d e f i n e d i n a manner s i m i l a r to A , w i t h v and V v r e p l a c e d w i t h 1 and L r e s p e c t i v e l y . With { A v . A l J is obtained from E q u a t i o n (3-29), a new value of {^.lj determined from h >l } n+1 n+1 = { A l " } + X{Av ,Al } n (3-30) n A G o l d e n - s e c t i o n s e a r c h (Himmelblau, 1972), i s implemented X*, the v a l u e of X w i t h i n the l i m i t s to determine X , < X < X which min — — max minimizes G as d e f i n e d by E q u a t i o n (3-21). X and X are min max the v a l u e s of X t h a t ensure t h a t t h e mass-balance c o n s t r a i n t s and t h e positivity 3-3-3 r e s t r i c t i o n s a r e not v i o l a t e d . The M o d i f i e d RAND Method The RAND method has two l i m i t a t i o n s seek a s u i t a b l e m o d i f i c a t i o n . that called f o r the need t o One of these i s the requirement of -97- standard Gibbs f r e e e n e r g i e s , functions of temperature. dimensionality minimized. which are not n o r m a l l y a v a i l a b l e as The other l i m i t a t i o n i s the h i g h of the f i n a l o b j e c t i v e f u n c t i o n that has to be F o r a system c o n s i s t i n g of N components, there a r e 3N independent v a r i a b l e s . D l u z n i e w s k i and A d l e r , (Going s t r i c t l y by the f o r m u l a t i o n of there would have been 3N + 2 such v a r i a b l e s , s i n c e V and L would a l s o have been t r e a t e d as independent This involves The variables.) the i n v e r s i o n of a m a t r i x of the same dimension. modification discussed effects: (1) I t eliminates (2) I t reduces the number of independent v a r i a b l e s to N. (3) I t does away w i t h the m a t r i x i n v e r s i o n . Eliminating 1 the standard below has the f o l l o w i n g Gibbs f r e e energy. between E q u a t i o n s (3-21) and (3-22) and r e a r r a n g i n g , we have N G( ^> • J u°. V J u ° RT + <j>.P «T^> 1 + N Z Z + Since a constant n ( using v -«-hrr>] i l -F^r> N + 4 ^±] the f a c t o - Li U RT u° J ^ i - i f < " > 3 31 s i d e of E q u a t i o n (3-31) i s system, i t does not a f f e c t the shape of ignored. that o = - £nf i n t r o d u c i n g K , we now have 4 + term on the r i g h t - h a n d I t can t h e r e f o r e be o vi Z. - v. - v f o r any i s o t h e r m a l y and [ the l a s t the G(v) p r o f i l e . Further i * £ n v. (see S e c t i o n 3-3-4 f o r p r o o f ) -98- g ( v) = N I v [-AnK i=l 1 v. Z - v. + Arr-i- - £n(4" 1 N +J J Z where g(v) = G(v) - V Z F - v. p-V £ n ( X )+ £nY J (3-32) N £ Z.u° /RT 1=1 1 L ± From Equation (3-32): 9j_ = 9v and ag 3 k v. j 2 9v . 9 v , J - AnK. + An(-i) k (3-33) x. k Z. - v. j j V (3-34) F -V where 6^ i s as defined i n Equation (3-26). The quadratic Taylor-series approximation to Equation (3-32) then becomes N n I £n(—i)(v j=l K.x. 3 3 F(v) = G (v) = g ( v ) + a y n - v ) n J 1 N 6 N i, 6 1 i, J i + i j X[—+ !£—-i--JL_]. J - l k-1 v Z. - v V F - V J J J (v . - v . ) ( v - v ) 2 n k Setting VF(v) = 0 n n n k leads to A Av = B where Av i s the N-element vector whose j t h element i s v (3-35) - v ; J 3 n (3-36) -99- i s an N-element v e c t o r whose j t h element i s K x n n 1 1. n in 7 J and A i s an N x N m a t r i x g i v e n by ,2 ij w i t h the second 3V.9V, partial d e r i v a t i v e as v=v d e f i n e d i n E q u a t i o n (3-34). The search d i r e c t i o n i s determined by s o l v i n g E q u a t i o n (3-36) f o r Av and a new v a l u e of _v i s g i v e n by n+1 v_ n . *. = v + U v where A*, determined 1 0 A comparison identical, v , „ „_, (3-37) as p r e v i o u s l y d e s c r i b e d , s a t i s f i e s j £ Z j the c o n d i t i o n ( J = 1. 2, ...,N) between Equations (3-36) and (4-6) shows that they are with A = J(v) and B = A£nK I t w i l l be shown i n S e c t i o n 4-2-1 that the m a t r i x i n v e r s i o n step can be e l i m i n a t e d i n equations of t h i s form and E q u a t i o n (3-36) manipulated to yield v l n Av i I( n - - L i J=l Z, adopted. implementing 1 " J ) 1 N U In 13 x .y. + B. I <-A j-1 ~i the m o d i f i e d RAND method, two schemes were (3-38) -100- Scheme 1 Equation (3-37) i s s o l v e d u s i n g the o r i g i n a l approach of ft D l u z n i e w s k i and A d l e r whereby the o p t i m a l v a l u e of X i s actually solved f o r . Scheme 2 T h i s i s tagged the 'Broyden-modified RAND method' and i t 0 involves a step-limited search as proposed on quasi-Newton methods. by Broyden (1965) i n h i s work With t h i s method, the u n i d i m e n s i o n a l search the X-axis i s terminated as soon as a v a l u e of X i s encountered on such that G(v 3-3-4 + XAv) n < G(v ) n A Geometric-Programming For the convenience Formulation of the reader who might want to r e l a t e the f o l l o w i n g d e r i v a t i o n to the geometric-programming theory presented i n Appendix C . l , the f o l l o w i n g symbols have been t e m p o r a r i l y i n t r o d u c e d : w = v, w = 1, W = V, W = L and w = (w , 1, v}. —v — —L — v L — o — —' T All T other uncommon terms used 1 i n t h i s s e c t i o n are d e f i n e d i n Appendix C-l. A c c o r d i n g to E q u a t i o n N (3-20), ° ° " H'JjT* G + » i=l v Define C o = w F(w) — o =1, - (C /w ) ° o o exp(-G), W ^o C iv • 6 X P ( - L 1 [ ^ + *.(T t &]} L (3-39) -101- and C -I i L exp(^) Then Equation (3-39) becomes ' C F(w) = O N C W L i \~C. { n -it U o W i=lL Li J a and o j _iy_ W T W }w w |_ vi _ W Define wvi ~ W L (3-40) v v = -Z. (j = 1, 2, ...,N) 1, for i = j ijv, ijL (3-40a) 0, for i * j Combining these definitions with the mass-balance relationship [Equation (3-22)] leads to a N - + I (a,i j.vwvi + ai j Lw Li ) = 0 oj o w J (3-41) A close comparison of Equations (3-40) and (3-41) with the Dual-Geometric-Program formulation in Appendix C - l reveals that they are identical for a two-phase system, with in Equation (C-6) = C and n o n l "2 = q in Equation (3-40), > 1 n = N » I J = w = 1 iej(o) ° w 1 (the normality condition). The positivity condition in the Appendix i s , of course, also satisfied by virtue of the physical definition of w . Thus, the formulation is complete and the solution to the problem is given by the value of w that maximizes Equation (3-40) subject to Equation (3-41) and positivity restrictions on w. l ^ j } i s given by a By the definition above, the exponent matrix -102- N (3.42) N where Z = { z ^ and I N of dimension {a^} 2 ...,Z } N i s an N x N i d e n t i t y A first {a^j}. z , matrix. step to s o l v i n g the dual problem i s to o b t a i n N v e c t o r s 2N + 1 which are o r t h o g o n a l T h i s i s achieved through (see Duff i n et a l , 1967, complements of each column of the f o l l o w i n g o p e r a t i o n on Chapter 3): h -z h (Interchange rows '1' and p a r t i t i o n i n t o upper and segments.) ( E x t r a c t the n e g a t i v e segment) transpose of 'N+l* and lower V 7 upper segment -z Lower segment lower ^ -I N -103- Append an (N+l) d i m e n s i o n a l m a t r i x below the l a s t row) identity T Z -I N N+l 1 (Interchange rows 0 '1' and 'N+l') = {b J>} ( 0' I. where 0_ and ()' are N-element row and column n u l l v e c t o r s r e s p e c t i v e l y . The m a t r i x {b^jp} has columns which a r e o r t h o g o n a l w i t h the columns of l^j}- Furthermore, i t s f i r s t row s a t i s f i e s the n o r m a l i t y c o n d i t i o n as r e q u i r e d : n £l±o lo Hence i t f u r n i s h e s us with a n o r m a l i t y v e c t o r : b ( o ) = {i,z l f z , . . . , z , 0, 2 and N n u l l i t y v e c t o r s b ^ > } , T N 0 through b< > w i t h b<J> _b g i v e n by N -1, f o r i = j + l 1, f o r i = N + j + 1 0, f o r other i v a l u e s V J / - 1 0 4 - The g e n e r a l s o l u t i o n to the d u a l program, expressed variables, r i n terms of b a s i c ( j = 1,2,...,N), i s w = b I + ( 0 ) r.b j-1 (3-43) ( : , ) 2 By i n t r o d u c i n g the b v e c t o r s i n t o E q u a t i o n (3-43), we o b t a i n w Li = Z W vi • r i~ r i ( i = 1,2, and i (3-44) ,N) _| E q u i l i b r i u m r e q u i r e s that 8£nF(w) 3r A combination = 0 of Equations C . jv C . jL (3-45) ( j = 1,2,...,N) (3-40), (3-44) and (3-45) y i e l d s w . W vj L W * w . v Lj T = T (3-46) T From the d e f i n i t i o n of C . and C „ , we have jv JL' / o o . C. y . (u . - p. ) -JI. = - J _ ^ {-^ jv_i C. ^TP^ RT JL J x p X P 1 ; (3-47) T From the b a s i c d e f i n i t i o n of the chemical p o t e n t i a l (Smith and van Ness, 1959): p ° = 9 .(T) + RT£nf ° JL J JL (3-48) p° = 8 .(T) + RT£nf° jv 3 Jv (3-49) v and s u b t r a c t i n g E q u a t i o n (3-49) from E q u a t i o n (3-48) and u s i n g the f a c t f° = 1, we jv that have o ^-IL o ~ ^iv RT . * n f jL (3-50) -105- Combining E q u a t i o n s y -f° j J Equation (3-46), (3-47) and oi T L w . W T «. _2Ll K = _1 Ji obtains k_ (3-511 (3-51) i s , of course, the e q u i l i b r i u m r e l a t i o n The f o r two reasons. First, e q u i v a l e n c e of the mass-balance and methods. Secondly, from E q u a t i o n x i t shows c o n c l u s i v e l y the the f r e e - e n e r g y - m i n i m i z a t i o n i t r e v e a l s that the v a l u e s of r ^ [and hence (3-44)] that s a t i s f y automatically satisfy the N e q u i l i b r i u m equations the (2N + 3) equations t h a t are i n v o l v e d i n the mass-balance f o r m u l a t i o n . A s i n g l e i t e r a t i o n i s a l l that i s necessary to s o l v e the N equations r e p r e s e n t e d by E q u a t i o n the double Y^ ^ = above outcome of the geometric-programming a p p l i c a t i o n i s significant Vj, (3-50), one (3-51) — as opposed to i t e r a t i o n s r e q u i r e d f o r the c l a s s of a l g o r i t h m s s t u d i e d i n S e c t i o n 3-2. What i s needed then i s an e f f i c i e n t , convergent method f o r s o l v i n g the N n o n l i n e a r e q u a t i o n s . 3-3-5 Seeking a S o l u t i o n Method f o r the Geometric-Programming Problem (GPP). I f we rearrange eliminate 1 V and L from E q u a t i o n (3-51) and the outcome, we would have K.(Z. - v ) N N I v . - v { F I j=l j=l J v.] = 0 ( i = 1,2,...,N) (3-52) J F o r the purpose of t e s t i n g the performance of d i f f e r e n t solution techniques on the above e q u a t i o n s , a h y p o t h e t i c a l b i n a r y system w i t h composition-independent K v a l u e s was u s i n g three s o l u t i o n techniques: chosen. The problem was tackled the Newton-Raphson i t e r a t i o n method, -106- the method o f c o n t i n u i t y , and a s u c c e s s i v e - s u b s t i t u t i o n method. The Newton-Raphson Method: Check f u n c t i o n s were d e f i n e d by N - v-) J j-1 F. - K.(Z. I 1 1 1 2 N v . - v { F I j-1 1 v.}, (i=l,2,...,N) (3-53) 2 Then by the Newton-Raphson technique (Henley and Rosen, 1969); n+1 n _v = ;y_ —1. n n - J (v_ )»F(v ) x ,„ ~ i (3-54) x where J *(v_ ) denotes the i n v e r s e o f the J a c o b i a n n with respect m a t r i x of J_ to v_ a t the p o i n t v_ . n When the method was a p p l i e d t o the sample problem, i t was found to g i v e f a u l t y convergence t o the t r i v i a l solutions at e i t h e r v = 0 o r v = Z. The Method o f C o n t i n u i t y : For the purpose of a p p l y i n g the method of c o n t i n u i t y (Perry and C h i l t o n , 1973), E q u a t i o n (3-52) was r e a r r a n g e d to g i v e (K Z.-F)v + K Z I I 1 1 N N I v.+ (1-K )v I j * i j=l 2 1 v. = 0 (i=l,2,...,N) 2 Then F^ was d e f i n e d by F, = (K Z -F)v i i i i Treating v Equation N N + K Z I v . + X(l-K )v £ i lp j i i ± ( j = 1,2...,N) as f u n c t i o n s (3-55) w i t h r e s p e c t N 8F dv + J = v . = 0 (i=l,2...,N) j 1 (3-55) of X and d i f f e r e n t i a t i n g to X, we have 8F H^r -df] - r= j=l j 9 0 <> i=1 2 > N (- > 3 56 \ -107- A rearrangement of the s e t of r e l a t i o n s r e p r e s e n t e d by E q u a t i o n (3-56) leads to where J - 1 ( v ) i s as d e f i n e d i n Equation (3-54). With v_ i n i t i a l i z e d by s o l v i n g E q u a t i o n (3-55) w i t h X = 0, E q u a t i o n (3-57) i s i n t e g r a t e d numerically between the l i m i t s X = 0 and X = 1 to g i v e the d e s i r e d value of v. When t h i s method was a p p l i e d was to the sample problem, the r e s u l t s o l u t i o n ^_ - 0_> f a u l t y convergence to the t r i v i a l The Method o f S u c c e s s i v e For Substitution: convenience, l e t us d e f i n e N A I = v (3-58) Then, a rearrangement of E q u a t i o n (3-52) y i e l d s (1 " K ) v ± 2 - [F - Z K - (1 - K ) A ] v 1 ± 1 ± ± + K Z X i 1 1 = 0 The method of s u c c e s s i v e is applied to E q u a t i o n (3-59) by t r e a t i n g i t as a q u a d r a t i c and choosing the root s u b s t i t u t i o n (see, f o r example, L a Fara, 1973) Out of the three solution — t o the sample problem, albeit solution-methods d i s c u s s e d arrangements of the equations were s t u d i e d investigated. it sluggishly. above, the s u c c e s s i v e s u b s t i t u t i o n method was chosen f o r f u r t h e r i n v e s t i g a t i o n . methods were function of that l i e s between 0 and Z^. When the above method was a p p l i e d converged to the d e s i r e d (3-59) and three Five acceleration -108- 3-3-6 Various Successive-Substitution Arrangements of the GPP Arrangement 1: Represented by Equation (3-59) Arrangement 2: This i s aimed at avoiding the large number of • conditional statements involved i n the programming of Arrangement 1. i s updated from K.Z.(A°+v") vf" 1 = i Arrangement 3; term the (3-60) F + ( K . - l ) ( x £ + vj) Equation (3-60) i s rearranged to absorb i n the e x p l i c i t i n the right-hand-side numerator. »»« The result i s f i f £ _ (3-6!) F + (K - l ) ( X j + v ) - K Z n ± Arrangement 4: f(v^) and v ± Equation (3-59) i s equated to the check function updated by the Newton method, _. n. f(v ) f =? — V 1 1 1 v 1 <- > 3 62 f(vj) where 9f(v.) 9v i v i -109- Arrangement 5: T h i s i s s i m i l a r to Arrangement 4 except that the Richmond method i s employed, thus n+1 v. = v 1 n 2 f '< ft v 2[f (v )] 1 n 2 f «fr (3-63) - f(v )f"(v ) n n where . J) - £ 3-3-7 (V 9v i v i n 1 A c c e l e r a t i n g the GPP T h i s method i s due by Hyperplane Linearization to B a r r e t o and F a r i n a (1979). In i t s basic form, i t i n v o l v e s l o c a t i n g , by s u c c e s s i v e s u b s t i t u t i o n , N p o i n t s on of the N h y p e r s u r f a c e s d e f i n e d by are (1) The f o l l o w i n g steps involved: Choose a v . s e —o (2) E q u a t i o n (3-59). each t 1 = 0. J A p p l y i n g s u c c e s s i v e s u b s t i t u t i o n to v_. t generate v —j+1 . T e s t f o r convergence. I f the outcome i s p o s i t i v e (3) then t e r m i n a t e the i t e r a t i o n . Otherwise, go to Step 3. Increment j . I f j < N then go to Step 2. Otherwise go to Step (4) Determine v_ , the i n t e r s e c t i o n of the N hyperplanes that are D d e f i n e d by the p o i n t s l o c a t e d on the h y p e r s u r f a c e s , from D v f = v O . - AR . -2. — Z «A R »Av 4. -110- where A AR -2 = { A ^ , A v , Av,,,..., Q x AR and A R i s the i n v e r s e of AR - AR , -1 ^ {A^, Av , i = , 2 " " vj s n o j = 0, v_P and a l l the v e c t o r s first (5) Set v The component — p = v , \>Q = v_ 1 each of these p o i n t s involved and j = 0. exclude time). Go to Step 2. the number of successive- One i n v o l v e s employing N i n i t i a l p o i n t s , f o r a s u c c e s s i v e - s u b s t i t u t i o n step, t h i s being followed and i t then f o l l o w s using utilizing by an takes the same form as the b a s i c a l g o r i t h m a c c e l e r a t i o n step; modification after The up to the the p a t t e r n of the f i r s t that. While the b a s i c a l g o r i t h m showed s t a b i l i t y and r e l i a b i l i t y when to a number of systems, the m o d i f i c a t i o n s performed p o o r l y and were t h e r e f o r e The (1) i n i t s determination a f t e r every s u c c e s s i v e - s u b s t i t u t i o n i t e r a t i o n . second m o d i f i c a t i o n applied N-1) necessary f o r i t s update when t i n an a c c e l e r a t i o n step; a c c e l e r a t i o n step first } , that were aimed at r e d u c i n g substitution iterations. result i authors of the above a c c e l e r a t i o n method proposed two modifications the V to save on computation p N A ( j = 0,1,..., ( S i n c e a knowledge of v^ ± the Av^}, not s t u d i e d further. hyperplane l i n e a r i z a t i o n method has two major shortcomings: There are N s u c c e s s i v e - s u b s t i t u t i o n i t e r a t i o n s between -Ill- (2) accelerations. F o r l a r g e values of N, the low a c c e l e r a t i o n frequency c o u l d slow down convergence appreciably. I t i n v o l v e s a m a t r i x i n v e r s i o n a t every a c c e l e r a t i o n s t e p . interest of computation time, t h i s i s an o p e r a t i o n I n the that were b e t t e r avoided. The two a c c e l e r a t i o n methods proposed i n S e c t i o n 3-3-8 w i t h o u t these 3-3-8 Projection two v e c t o r - p r o j e c t i o n a c c e l e r a t i o n methods proposed here require only steps. defects. A c c e l e r a t i n g the GPP by V e c t o r The one s u c c e s s i v e - s u b s t i t u t i o n i t e r a t i o n between a c c e l e r a t i o n And i n a d d i t i o n to not r e q u i r i n g m a t r i x i n v e r s i o n , they are by comparison to the hyperplane method — first of the two, tagged i t e r a t i o n s before the f i r s t a c c e l e r a t i o n step, requires Euclidean vector the method, we s h a l l d e f i n e spaces. One we s h a l l c a l l such two the iterations. two N-dimensional the ' s o l u t i o n v e c t o r the other we s h a l l r e f e r t o as the ' d i s p a r i t y v e c t o r c h a r a c t e r i s t i c p o s i t i o n vector w i l l the While the the second method — three — T r i a n g u l a r - P r o j e c t i o n Method In d e v e l o p i n g and easy to implement. ' t r i a n g u l a r - p r o j e c t i o n method', r e q u i r e s ' t e t r a h e d r a l - p r o j e c t i o n method' — The below a r e r e s u l t of a p p l y i n g successive space'. A and i t w i l l be s u b s t i t u t i o n to the v e c t o r corresponding d i s p a r i t y vector w i l l d e f i n e d by be denoted by be r e p r e s e n t e d space' v_. by f_ and w i l l The be Now supposing that we have l o c a t e d v , i n the v e c t o r —sz space — two p o i n t s , v the r e s u l t s of a p p l y i n g s u b s t i t u t i o n to two p o i n t s , v_ and v_ r e s p e c t i v e l y . -si and successive I f we * denote the s o l u t i o n which we a r e seeking the by v , the p h i l o s o p h y s of t r i a n g u l a r - p r o j e c t i o n method i s to employ our knowledge of t h e points v * to v —s s JL and v _ i n l o c a t i n g a p o i n t v which i s c l o s e r S£ sn than both v and v „. — s i —sz L e t us c o n s i d e r the t r i a n g l e whose v e r t i c e s a r e the p o i n t s ft v , v and v ( s e e F i g u r e 3-17). v the d e s i r e d p o i n t , i s —s 1 —s2 ~~s —sn' ' ft such t h a t v v i s perpendicular —s - s n arises: to l o c a t e v —fin To - But then, a problem , we need to know the p o s i t i o n of v we a r e c o m p l e t e l y i g n o r a n t i n the f i r s t to v v . —si S2 — — S~' which of o r we would not be t a l k i n g of l o c a t i n g v —sn place. overcome t h i s problem, l e t us t u r n to the d i s p a r i t y v e c t o r space. There i s a one-to-one correspondence between v and f , and —s — * * c o r r e s p o n d i n g t o the p o i n t v^ i s the p o i n t f_ = (). I f we map the c o n s t r u c t i o n of F i g u r e Figure (3-18). (3-17) onto the f-space, what we have i s Since we know that f_ i s the n u l l v e c t o r , we can e a s i l y determine f . —n L e t us r e l a t e v , v and v by some parameter a a c c o r d i n g to —sn — s i —sz the equation v = av + (1 - a) v —sn — s i — s i v (3-65) ' 112a- Leaf 113 missed in numbering FIG 3-18: DISPARITY - S P A C E TRIANGLE -115- Then, by similarity, f = of + (1 - a) f —n — l —z (3-66) From F i g u r e (3-18), a = -d/c. Now, -d = aCosB and from the c o s i n e f o r m u l a , _ „ a CosB = + c - b = 2ac Hence, a a + c 2 2c 2 = 2 - b & 2 (i -ii> (i -ii> T 2(f "-l + T 2 2 £h ) 2 The above r e l a t i o n s h i p can be s i m p l i f i e d (1? « = - f u r t h e r to y i e l d " l i ) Ly T i 1 i _ (3 The implementation of the method i n v o l v e s the f o l l o w i n g main steps: (1) Assume v_ v -2 (2) and determine • Compute f± and s e t = v , . -si * Determine positive a nd t e s t f o r convergence. I f the outcome i s then stop; o t h e r w i s e compute f_ . (3) Compute a from E q u a t i o n (3-67) and (4) Set f.= f , , v , = v and v„ = v . —1—2 —si —s2 —2 —sn from E q u a t i o n (3-65). Go to Step 2. 67) -116- The above convergence method has convergence f o r an otherwise note t h a t i f we Equations the p r o p e r t y of divergent i t e r a t i o n . It i s interesting apply the approach to a u n i d i m e n s i o n a l (3-64), (3-65) and acceleration relationship (3-67) combine to y i e l d (Wegstein, 1958) — forcing to problem, the Wegstein a method which has w i d e l y a p p l i e d to u n i v a r i a t e problems and which i s known to been possess convergence-forcing properties. The T e t r a h e d r a l - P r o j e c t i o n Method; The t e t r a h e d r a l - p r o j e c t i o n a c c e l e r a t i o n method f o l l o w s the same b a s i c p h i l o s o p h y as the t r i a n g u l a r - p r o j e c t i o n t e c h n i q u e . difference lies The only i n the number of p o i n t s employed i n the p r o j e c t i o n : the t e t r a h e d r a l p r o j e c t i o n method r e q u i r e s three p o i n t s . Assume t h a t we points V. , v_ ^2 and _v a t u have l o c a t e d i n the s o l u t i o n v e c t o r - s p a c e , * — resulting respectively. The three from s u c c e s s i v e s u b s t i t u t i o n points v v and v on together with * the unknown s o l u t i o n form a t e t r a h e d r o n . S i n c e our knowledge i s * limited to the plane of v . , v that we can l o c a t e i s the image of denoted by v and —sn and To determine P^, s n o w r l v * l e t i t be r e l a t e d v = pv +Pv -sn 1-sl 2-s2 * hedron f ^ f g f g f . » 0 the n e a r e s t p o i n t to v on t h i s p l a n e . Let this be to the t h r e e known v e c t o r s by J +Pv 3-s3 a n < * ^3» (3-68) w e c o n s i d e r the f-space s c h e m a t i c a l l y i n F i g u r e (3-19). tetra- * f_ i s -117- FIG 3-19: DISPARITY - S P A C E TETRAHEDRON. -118- again the n u l l v e c t o r . From s i m i l a r i t y , i t f o l l o w s from E q u a t i o n (3-68) that f —n - P f + P f + P f 1—1 2—2 3—3 (3-69) L e t us c o n s i d e r F i g u r e 3-20, e x t r a c t e d from F i g u r e 3-19 w i t h t u rectangular coordinates 2 a = — — a + and = c^ CosB = b - * 2 2c 2 2 _ I t can be shown that (f - |_ ) e e + _ 2 2 d 2 g as d e f i n e d . t (ll - 1 ) 2 2 - h _ 2 2 2 f x (ll - 1 > T 2 ^ 2 ~ h ^ h (f - f a T 2 3 ) T (f -f ), 2 3 2 c 2ac Employing the above i n f o r m a t i o n and a p p l y i n g b a s i c mathematical principles (a) to F i g u r e 3-20, i t can be proved that: The l i n e which the v e c t o r ^12^.3 * represented on the t u c o o r d i n a t e S P 3 a r t (3-70) The l i n e which has the v e c t o r f f —15—n 0 mathematical i d e n t i t y s system by t = aa (b) * 0 f o r a segment has a on the t u plane u = aCotB + gcCosecB - tCotB g i v e n by (3-71) -119- FIG 3-20: DISPARITY - SPACE TETRAHEDRAL BASE. -120- (c) The p o i n t f_ i s d e f i n e d by 3 {t,u} = {a + cCosB, cSinB} Using the a b o v e - l i s t e d f a c t s , (d) f 1 i s d e f i n e d by {t,u} = {aa, 3 (e) i t can i n t u r n be shown t h a t : i s d e f i n e d by {t,u} = { a a , [ ( l - a ) a C o t B + BcCosecB]} 2 2ct3. l is g i v e n by i = — — (f) f \ A (g) From F i g u r e f —n It b 2 - a 2 - c 2 + b 2 - c 2 23c l 2 L 2 (3-20), we have: = «f 1 2 -13 and Y[(1-OQ6 + m i s g i v e n by m = — ^ ^—^ ^— a[4a c - 6 ] Where o - f_ v A where Y = a = *-3 + d-a) l 2 (3-72) x + -1 - mf., + (1-m) —13 ( f —12 1 0 3 _ 7 3 ) (3-74) f o l l o w s from E q u a t i o n s (3-72) through (3-74) t h a t f = [ l - a +(a-£)m] f, + a(l-m) f„ + JLmf. —n —1 —2 —J Hence, we have the u l t i m a t e results: P^ = 1 - a + (a-£)m P 2 = a(l-m) P 3 = *m (3-75) -121- where a,£ and m are as p r e v i o u s l y defined. The key steps i n the implementation of the method a r e : (1) Assume v and determine v „. —2 —s2 v . —sJ (2) Compute f . and f . . —z —3 U s i n g v, determine v Set v„ = v and determine —3 —s2 Set v — and t e s t = v . —s3 f o r convergence. I f the o u t - s come i s p o s i t i v e (3) Set = v s 2 and f_ = f . —3 —n (4) 3-3-9 then t e r m i n a t e the i t e r a t i o n ; , v - v^, v g 2 Determine o t h e r w i s e compute f . = v , f , = f_ , f_ = f. and g 3 s 2 2 3 v from E q u a t i o n (3-68). —sn Set v = v and go to Step 2. — —sn A Method Based on S e n s i t i v i t y Analysis The a l g o r i t h m p r e s e n t e d here i s based on the s e n s i t i v i t y discussed i n the next chapter (see ' P r e d i c t o r - C o r r e c t o r Method 3' i n S e c t i o n 4-6-3). The working e q u a t i o n ( d e r i v e d i n Chapter 4) i s g i v e n by m m. „ x y A£nK N i I v l i i m m 1 i1. _ n n i AJlnK . = £nK . - J l n ( - i ) y J i j z . Jj-l Z. l where analysis J »i N V mm x .y . JJ J -122- Superscript 'm' denotes a p o i n t such that v , f o r example, i s g i v e n by v m = v n + a(v - v ) n + 1 (i=l,2,...,N) n (3-77) where 0 < a < 1. The a l g o r i t h m employed i n v o l v e s the f o l l o w i n g main s t e p s : (1) v_, _1, _x and y_ are s u i t a b l y (2) K i s determined (3) With s u p e r s c r i p t v "*" 11 1 and the c o r r e s p o n d i n g AAnK computed. 'm' i s estimated. determine and y_ m replaced with 'n' i n E q u a t i o n (3-76), T h i s i s then employed i n E q u a t i o n (3-77) to v_ , u s i n g a = 0.5. m The corresponding _ l , m x" 1 are computed. (4) v_ (5) Convergence i s t e s t e d f o r . n+1 initialized. i s determined from E q u a t i o n (3-76). I f the outcome i s p o s i t i v e , the i t e r a t i o n i s terminated; otherwise c o n t r o l i s t r a n s f e r r e d to Step 2. 3-3-10 I n i t i a l i z a t i o n Schemes Four different initialization schemes, mostly based on the same l o g i c as those presented i n S e c t i o n 3-2-4, have been s t u d i e d h e r e . each of the schemes, K i s based Scheme 1: v° = 0.5Z ± (1=1,2 on x = y = z. N) T h i s scheme, simple as i t i s , performed algorithms. However, i t was q u i t e w e l l w i t h the GP u n s u i t a b l e f o r the other methods because i t Kx i m p l i e s x_ = y_ = z_ and convergence For t h i s has a great e f f e c t of these other methods s t r o n g l y on £n — , depends. on which the -123- Scheme 2: T h i s scheme d e f i n e s v° by v°1 - Scheme 3: Z i 1 + 1/K • 1,2,...,N) Three steps a r e i n v o l v e d : Z. (a) (b) Compute v = - 1 + 1/K N Compute V = £ v and L = F - V i=l i Z (c) Compute v ° = 1 + 1 Scheme 4: In this S e c t i o n 3-2-4. V o i ( i = 1,2,...,N) scheme, 9° i s determined v as i n Scheme 5 o f v ° i s then computed from ^ = ±— K.V l 1 + h o - (1=1,2,...,N) K.e° i v When the above schemes were a p p l i e d to the v a r i o u s free-energy-minimization than the other two. all the methods. a l g o r i t h m s , Schemes 3 and 4 performed b e t t e r On the average, Scheme 4 gave the best showing f o r I t has t h e r e f o r e been adopted i n a l l subsequent a p p l i c a t i o n s of the methods. 3-3-11 A p p l i c a t i o n s and Deductions The first phase of the a p p l i c a t i o n of the a l g o r i t h m s presented i n S e c t i o n s 3-3-2 through mind: 3-3-9 was c a r r i e d out with two o b j e c t i v e s i n -124- (1) To t e s t the d i f f e r e n t the most e f f i c i e n t arrangements arrangement; of the GPP so as t o determine and to compare the performance of the u n a c c e l e r a t e d s u c c e s s i v e s u b s t i t u t i o n implementation o f the GPP w i t h that of the a c c e l e r a t e d (2) version. To compare the performance of t h e 'modified RAND a l g o r i t h m ' — i s , w i t h the u n i d i m e n s i o n a l s e a r c h on X c a r r i e d a tolerance of X — that to w i t h i n t o t h a t of the 'Broyden-modified RAND algorithm'. S i x systems — test. all VA, V I , V J , VN, VO and VP — were chosen f o r t h e The chosen systems were c o n s i d e r e d to be q u i t e r e p r e s e n t a t i v e of 16 systems, i n terms of b o i l i n g range, degree of n o n i d e a l i t y , and m i x t u r e c o m p l e x i t y as measured by number of components. The e x p e r i m e n t a l d e s i g n employed methods was m a i n t a i n e d and w i l l , chapter. f o r the double-loop u n i v a r i a t e of c o u r s e , be s u s t a i n e d throughout the F o r the purpose of meeting O b j e c t i v e 1 above, the t r i a n g u l a r - p r o j e c t i o n method was used i n a c c e l e r a t i o n . The r e s u l t s o b t a i n e d a r e p r e s e n t e d i n T a b l e 3-3. lead to the f o l l o w i n g (1) The best GPP arrangement The r e s u l t s inferences: i s Arrangement 4, which i n v o l v e s applying Newton's method s u c c e s s i v e l y t o the p s e u d o - u n i v a r i a t e check functions. (2) The a c c e l e r a t i o n method a p p l i e d here t o t h e GPP does have a tremendous accelerating effect on i t s convergence. e s p e c i a l l y so f o r n a r r o w - b o i l i n g (3) This i s systems. The 'Broyden-modified RAND' method i s f a r s u p e r i o r to the 'modified RAND' method i n terms of computation time. be l e s s stable. However, i t appears to Table 3—3 Computation time (CPU seconds) for f i r s t phase of application of free-energy minimization algorithms Differ mt GPP arrangejments with tr (.angular projei:tion System Effect of acc<jleration (with arrang jment 4) Unaccelerated The mod Lfied RAND alg( jrithms Arrangement 1 Arrangement 2 Arrangement 3 Arrangement 4 Arrangement 5 VA 0.8007 0.9065 0.7658 0.7727 0.7982 VI 0.7925 0.9372 0.7583 0.7678 0.7985 2.0209 0.7882 5.3%Failure VJ 1.0463 1.0458 0.9403 0.9832 1.0001 2.3285 0.9393 VN 0.2799 0.3338 Failed 0.3210 0.3709 1.1868 0.4088 VO 0.2748 0.3333 Failed 0.3201 0.3521 1.1744 0.3957 VP 0.5175 0.4982 0.5974 0.4937 0.5269 1.5904 0.5576 I 3.7117 4.0548 3.6585 3.8467 10.1239 3.8238 Triangular 'Modified Projection RAND 2.9079 with 94.7% non convergence 0.7566 0.7727 0.4937 1.8229 'BroydenModlfied RAND' 0.7342 -126- The second phase of the a p p l i c a t i o n was designed to i n v o l v e a l l the 16 systems. the Guided by the r e s u l t s o b t a i n e d i n the f i r s t outcome of some p r e l i m i n a r y first phase and t e s t s on the methods not i n v o l v e d i n the a p p l i c a t i o n s , the f o l l o w i n g methods were s e l e c t e d f o r i n v e s t i - gation: (1) GPP (Arrangement 4) a c c e l e r a t e d by hyperplane linearization. (2) GPP (Arrangement 4) a c c e l e r a t e d by t r i a n g u l a r p r o j e c t i o n . (3) GPP (Arrangement 4) a c c e l e r a t e d by t e t r a h e d r a l (4) The s e n s i t i v i t y - b a s e d projection. algorithm. (5) < The Broyden-modified RAND algorithm. (6) A combination of the s e n s i t i v i t y method and t r i a n g u l a r p r o j e c t i o n . (7) A combination of the s e n s i t i v i t y method and t e t r a h e d r a l In the l a s t two methods, the s u c c e s s i v e - s u b s t i t u t i o n Methods 2 and 3 a r e r e p l a c e d The projection. block i n w i t h the s e n s i t i v i t y method. r e s u l t s , p r e s e n t e d i n T a b l e 3-4, l e a d to the f o l l o w i n g deduc- tions: (1) The two proposed v e c t o r - p r o j e c t i o n superior a c c e l e r a t i o n methods a r e f a r to the h y p e r p l a n e - l i n e a r i z a t i o n method. The t e t r a h e d r a l - p r o j e c t i o n method i s the b e t t e r of the two f o r n a r r o w - b o i l i n g systems, but r e t u r n s s l i g h t l y worse times f o r wide-boiling mixtures. (2) The Broyden-modified RAND a l g o r i t h m i s rather u n r e l i a b l e , recording an average f a i l u r e r a t e of 2.6%. (3) A comparison of the s e n s i t i v i t y - b a s e d a l g o r i t h m not involve with those that do t h e s e n s i t i v i t y method shows t h a t i t i s the f a s t e s t with narrow-boiling mixtures w h i l e i t i s more s l u g g i s h than the T a b l e 3-4 GPPSystem hyperplane linearization Computation times (CPU seconds) f o r f i n a l comparison o f f r e e - e n e r g y minimisation algorithms GPPtriangular projection GPPtetrahedral projection Sensitivitybased algorithm Broydenmodified RAND Sensitivity with triangular projection Sensitivity with tetrahedral projection VA 1.2712 0.7727 0.6931 0.6000 0.7342 0.5460 0.5275 VB 0.9599 0.5861 0.5323 0.4696 0.5444* (15.8% f ) 0.3744 0.3701 VC 1.7910 0.9390 0.8655 0.7961 0.9663 0.6938 0.6405 VD 1.3114 0.6825 0.6882 0.6669 0.8346 0.6187 0.5601 VE 2.2041 1.1594 0.8689 0.6629 0.8086* ( 5.3% f ) 0.5814 0.5822 VF 1.5942 0.8802 0.7959 0.6373 0.8385 (10.5% f ) 0.5648 0.5457 VG 2.4796 0.8757 0.8403 0.5382 0.7395 1.0365t 0.9139t VH 1.7547 1.0102 0.9061 0.7292 0.9021* (5.3% f ) 0.6353 0.6098 VI 1.4379 0.7678 0.6642 0.6113 0.7882* (5.3% f ) 0.5223 0.5331 VJ 1.7189 0.9832 0.9905 0.8070 0.9393 0.6627 0.6257 VK 2.1827 0.8012 0.6260 0.4772 0.6547 0.4929t 0.5097t VL 0.4307 0.3258 0.3333 0.4185 0.4158 0.3203 0.3701 VM 0.4362 0.3275 0.3352 0.4315 0.4143 0.3292 0.3545 VN 0.4204 0.3210 0.3284 0.4264 0.4088 0.3094 0.3405 VO 0.4421 0.3201 0.3250 0.4279 0.3957 0.3175 0.3464 VP 0.6800 0.4937 0.5068 0.5843 0.5576 0.4391 0.4714 21.1150 11.2461 10.2997 9.2843 10.9426 (2.6% f ) 8.4443 8.3012 I : * f = failure t O s c i l l a t o r y convergence -128- others (except the GPP a c c e l e r a t e d by hyperplane l i n e a r i z a t i o n ) f o r wide-boiling (4) systems. The a l g o r i t h m s that combine s e n s i t i v i t y w i t h v e c t o r p r o j e c t i o n r e t u r n e d the best times f o r a l l the systems except f o r and VK, the two systems w i t h the narrowest and 2.8K r e s p e c t i v e l y ) , oscillatory 3-4 f o r which they systems VG b o i l i n g ranges (0.48K are found to d i s p l a y convergence. Single—Loop Univariate Methods The reason t h a t the double-loop path i s o f t e n adopted implementation of the u n i v a r i a t e methods i s because, n o n i d e a l systems, convergence i n the for sufficiently may not be a c h i e v e d i f the dependence of K on c o m p o s i t i o n i s not i t e r a t i v e l y c o r r e c t e d f o r i n an i n n e r l o o p . s e c o n d - l e v e l i t e r a t i o n consumes much time and undermines the This advantage which the u n i v a r i a t e methods have over f r e e - e n e r g y m i n i m i z a t i o n methods — that of having to contend w i t h only one independent variable. This i n v e s t i g a t o r b e l i e v e s that the s e c o n d - l e v e l i t e r a t i o n can be s a f e l y d i s p o s e d o f , no matter initialization the system involved, given a s u i t a b l e scheme. In t h i s s e c t i o n , s i n g l e - l o o p v a r i a t i o n s of some of the u n i v a r i a t e methods are s t u d i e d . Some new methods are a l s o i n t r o d u c e d here. A l l the methods employ i n i t i a l i z a t i o n Scheme 5 of S e c t i o n 3-2-5. 3-4-1 Richmond-Accelerated Methods The methods c l a s s i f i e d under t h i s heading are drawn d i r e c t l y S e c t i o n 3-2. A f t e r a c a r e f u l assessment that s e c t i o n , the a l g o r i t h m s i n v o l v i n g of the r e s u l t s o b t a i n e d i n the Newton method and those from -129- i n v o l v i n g l o g a r i t h m i c f o r m u l a t i o n s have been l e f t out. Thus we have the f o l l o w i n g f o u r methods ( a l l employing Richmond's a c c e l e r a t i o n ) : The (1) The (2) The R a c h f o r d - R l c e (3) <t>-normalized method 1. (4) ^-normalized 2. key standard formulation. method steps of a g e n e r a l a l g o r i t h m are: (1) I n i t i a l i z e x, y, K and (2) Compute _x c o r r e s p o n d i n g 9 ^ from E q u a t i o n (3) Normalize x_ and i n Step (4) to the c u r r e n t v a l u e s of K y, and Compute y_ from y^ = update K. Then recompute x and y_ as Update 9 ^ and go otherwise to Step go to Step 3-4-1 5. the Mean-Value Theorem methods d i s c u s s e d i n t h i s s e c t i o n employ an a c c e l e r a t i o n ( f o r MVT, The then 2. method that i s based on the mean-value theorem (MVT) calculus and K^x^ I f the outcome i s p o s i t i v e the i t e r a t i o n ; Methods U t i l i z i n g The . (3-9). T e s t f o r convergence. (5) 9 2. terminate 3-4-2 formulation. see H o l l a n d , of differential 1975). a l g o r i t h m i n v o l v e d i s the same as that presented above. The 9 ^ update i n Step 5 now iteration): (1) Computing f ( 9 * ) and (2) Computing 9 n f'(9^). af(6 ) — . f»(9 ) v n m V = 9 V n i n Section i n v o l v e s ( a f t e r the nth -130- where a, the MVT 0 < a < 1. (3) Calculating (4) Updating 8 In order s e c t i o n search parameter, s a t i s f i e s the c o n d i t i o n from to determine the best value of a to employ, a technique (Himmelblau, 1972) and VP w i t h a as the search v a r i a b l e and tions (over 19 p o i n t s f o r each result of ct = of the search was golden- a p p l i e d to systems the sum VA of the number of i t e r a - system) as the o b j e c t i v e f u n c t i o n . i s p l o t t e d i n F i g u r e 3-21 and i t leads to a The choice 0.6. Two v e r s i o n s of the MVT on the standard a l g o r i t h m were implemented: f o r m u l a t i o n of f ( 8 ) v the other on the one i s based Rachford-Rice formulation. 3-4-3 Wegstein-projected Methods D i f f e r e n t ways of a p p l y i n g the Wegstein a c c e l e r a t i o n method (Wegstein, 1958) and studied The to the v a p o u r - l i q u i d f l a s h problem have been developed here. general a l g o r i t h m i n v o l v e s the f o l l o w i n g s t e p s : s (1) I n i t i a l i z e x, y_, K and direct-iteration 8 . Determine 6^ methods d i s c u s s e d through one i n subsequent of the paragraphs. -131- 100 • 99 H 98 | "a 97- CD 96 H * • 95 H 94 1 0.0 0.1 1 1 1 1 1 r- 0,2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 MVT parameter.a Fig 3-21: Golden-section search for optimal MVT parameter -132- Compute g from g = 6 - 8 v v (3-78) s Set (2) g_ = g, 8 = 8 and 8 =' 8 2 °' v2 v v v s s U s i n g the c u r r e n t v a l u e of 8 determine 8 v T e s t f o r convergence. (4) the i t e r a t i o n ; o t h e r w i s e go to Step 4. Compute g from E q u a t i o n (3-78) and s e t (5) = g , g = g, e 2 2 vl as In Step 1. v (3) g l s I f the outcome i s p o s i t i v e =e v2 and e v2 v then t e r m i n a t e = ej Define S 2 " S l and compute 8 from v 8 = 8 + (1 - p)8 v vl v2 P (6) Go to Step 2. Different direct-iteration updated v a l u e of 8 were s t u d i e d . v f o l l o w i n g steps i n common: schemes f o r o b t a i n i n g 6 , ar A l l the schemes have the (a) Compute x_ from E q u a t i o n (3-9) and y_ from y^= K^x^ (b) Normalize x_ and y_ and update (c) U s i n g the new K, recompute x and y_. (d) O b t a i n 8 from one of the schemes d i s c u s s e d below. v s K. -133- Scheme 1: In this scheme, we employ (3-78a) 0 = 6 S v v y S t where N S y y = i i=l 1y 1 and t i s some p o s i t i v e exponent. exemplifying F i g u r e 3-22 shows the i n f l u e n c e of t on 0^ v* Q^. profiles F i g u r e s 3-23 and 3-24 i n t u r n compare 6 ( t = 1) and 8 ( t = 20) f o r s m a l l and l a r g e © y for v systems VA and VP. The curves r e v e a l that the n a r r o w - b o i l i n g system (VA) w i l l be f a v o u r e d by a l a r g e t w h i l e the w i d e - b o i l i n g system (VP) w i l l do b e t t e r w i t h a s m a l l t . A method of computing t was t r i e d . ensuring that f o r any g i v e n system, t s a t i s f i e s conditions I t was aimed a t the f o l l o w i n g two simultaneously: C o n d i t i o n 1: C o n d i t i o n 2: 0.04 f o r S < 1 y 0 - 0 v v s 0 s 0 =0.05 v - 0 0.8 f o r S > 1 y < 0.4 0 =0.5 v Thus, i f we d e f i n e 1, f o r S y 0.05 2.8332/^nS =1 y -1.6094/^nS , for S > 1 y y , for S < 1 y -134- Fiq 3-22: 8 w.Q for different values of t for system VA at T corresponding to a * 0 3 5 v -135- 0 0.2 0.4 0.6 »• # 0.8 v Fig 3-23: 0* v.Q fort =1 and t = 20 for system VA y 1.0 -136- .0 -137- 1, for S = 1 y and t 0.5 0.587787/£nS for S > 1 y -1.609438/^nS , for S < 1 y y with S for t and t, based on 0 =0.05 and 0.05 0.5 v y 0 =0.5 respectively, then 1 The determination of t is done just before the f i r s t calculation of v The t-profiles presented in Figure 3-25 resulted from applying the above method to systems VA and VP. The scheme was applied to systems VA, VI, VJ, VN, V0 and VP using: (1) The above t-fixing method (2) A constant value of t = 1. The second method was found to be better for the wide-boiling systems, and only slightly worse than the f i r s t for the narrow-boiling systems. The scheme was therefore adopted in the form 0 = 0 S v v y S It was also found that in computing x and y.for the purpose of updating K at each iteration, i t is better to keep x and y_ in mass, rather than equilibrium, balance. follows: y i 1 + (K, - 1)0 » i v They were therefore determined as -138- -139- i Scheme 2 N T h i s scheme i n v o l v e s d e t e r m i n i n g s 6^ from 9^ by a p p l y i n g Richmond's method to the standard f o r m u l a t i o n . Some p l o t s of 8 v v versus 9^ based systems VA on t h i s scheme are p r e s e n t e d and VP. An attempt i n F i g u r e 3-26 for to r e p l a c e the Richmond method w i t h Newton method i n t h i s scheme was d i s c o u r a g e d by the l a t t e r ' s the poor performance w i t h w i d e - b o i l i n g systems (see p r o f i l e s i n F i g u r e 3-27). Scheme 3: T h i s scheme i s s i m i l a r to the p r e c e d i n g one Rachford-Rice f o r m u l a t i o n r e p l a c e s the standard form. Richmond's a c c e l e r a t i o n methods were t r i e d and the former was Scheme 4: found to be except t h a t the Both Newton's and (see F i g u r e s 3-28 and 3-29) unreliable. T h i s scheme combines the R a c h f o r d - R i c e mean-value theorem a c c e l e r a t i o n method. formulation with As the p r o f i l e s the i n Figure 3-30 show, poor i n i t i a l i z a t i o n c o u l d l e a d to d i v e r g e n c e . Scheme 5: Here, the Richmond a c c e l e r a t i o n technique i s a p p l i e d to '^-normalized method 2'. 3-4-4 A Q u a d r a t i c Form of Wegstein's P r o j e c t i o n Because the e f f i c i e n c y of an a c c e l e r a t i o n method g e n e r a l l y i n c r e a s e s w i t h i t s order (compare: S e c t i o n 3-2; 3-3), s and t r i a n g u l a r p r o j e c t i o n methods i n S e c t i o n a q u a d r a t i c form of the Wegstein p r o j e c t i o n method has developed 9 t e t r a h e d r a l and Richmond's and Newton's methods i n here. The been method employs the three most c u r r e n t values of g [defined i n Equation (3-78)] i n a c c e l e r a t i o n . -140- Fig 3-26: # v.# for the Richmond-accelerated standard formulation v v -141- Legend: System VA System VP *6L Fig 3-27: $ V S v. O forthe Newton-accelerated standard formulation v -142- 1.4- 1.2- 1.0 0.8H 0.6i to > CD A 0.4H 0.2H —i Q1 1 1 0.2 0.3 ;—i 0.4 1 1 0.5 0,6 1—: 1 07 '\0,8 1— OS -0.24 Legend: System VA System VP -0.41 •0.64 - a Fig 3-28; 6 v. f9 for the Richmond-accelerated Rachford-Rice form y v - 1 4 3 - Fig3-29; (9%. 6 for the Newton-accelerated Rachford-Rice form -144- -145- L e t us r e p r e s e n t the three most c u r r e n t v a l u e s of 0 by s v 8^, 9 and 8^ and the c o r r e s p o n d i n g v a l u e s of g by g^, 2 g^ and g^. the p o i n t 8 ^ c o r r e s p o n d i n g to Then to determine g = 0, an a p p l i c a t i o n of Lagrange's i n t e r p o l a t i o n formula J e f f r e y s , 1 9 6 3 ) leads to 6 v So So " r=r S i So i " -ATir 1 4> 4> 9 x (Jenson and S-i 8o 9+ r / o 2 * * 3 (3-79) e 9 2 3 v 3 ' where and •l = S 2 " 81, $ = g 3 - g 3 2 The v a l u e of 8 ^ o b t a i n e d from E q u a t i o n direct (3-79) i s used i n the next s a new 8 . i t e r a t i o n to determine v There i s only one d i r e c t i t e r a t i o n between a c c e l e r a t i o n s . i n Section 3 - 4 - 3 , a l g o r i t h m r e q u i r e s the same main steps as are o u t l i n e d except off. t h a t three d i r e c t The i t e r a t i o n s are r e q u i r e d f o r the scheme to take At every stage, a f t e r a new 8 ^ i s computed, the f o l l o w i n g assignments are made: 8, = 8 3 v s Q 1 = Q ; 2 9 2 = V 6 3 = 6 v' 8 1 = 8 2 ; 8 2 = 8 3 ; 8 . v In a p p l y i n g the method, d i r e c t - i t e r a t i o n Scheme 4 was used. c h o i c e was based on the outcome of t e s t s of the d i f f e r e n t the ( l i n e a r ) Wegstein a c c e l e r a t i o n . This schemes u s i n g -146- 3-4-5 Applications and Deductions The various methods discussed in Sections 3-4-1 through 3-4-4 were applied to the 16 systems, using the same experimental design hitherto employed. The computation times obtained are presented in Table 3-5. It is significant to note that there was no single failure or oscillatory convergence. The results lead to the following inferences: (1) Of the four Richmond-accelerated methods, ^-normalized method 2 i s the best. It has a thin edge over the Rachford-Rice formulation — i t being better about 60% of the time. (2) The MVT acceleration scheme gives better results than the Richmond acceleration method. A comparison of the two methods for corresponding problem formulations shows that the former produces better computation times for every one of the systems. (3) The Wegstein-projected methods are generally superior to the other methods. (4) The proposed quadratic form of the Wegstein method is better than the linear form only 50% of the time; and i t yields an overall performance that is slightly worse, compared to the latter form. (5) Everything considered, the best performance is obtained with the Wegstein-projected form of the MVT-accelerated Rachford-Rice formulation. By way of comment, i t should be noted that the relative performances of the quadratic and linear forms of the Wegstein projection method depend very much on the nature of the function being considered. Although the linear form has turned out to be slightly more Table 3-5 Computation times (CPU seconds) for single-loop univariate algorithms MVT-acce lerated Metr ods Richmond-accelerated Methods System Standard RachfordRice •-normalized 1 •-normalized 2 Standard VA 0.5553 0.5549 0.5490 0.5452 0.5433 VB 0.4364 0.4071 0.3987 0.3973 VC 0.7619 0.7421 0.7376 VD 0.6809 0.6801 VE 0.6895 VF RachfordRice Wegstein-•projected Methods Quadratic Wegstein (+ MVT + RachfordRice) Scheme 5 Scheme 1 Scheme 2 Scheme 3 Scheme 4 0.5416 0.4885 0.4883 0.4763 0.4906 0.4725 0.4878 0.4111 0.3804 0.3881 0.3824 0.3536 0.3530 0.3598 0.3434 0.7348 0.7480 0.7259 0.7601 0.6757 0.6943 0.6864 0.6943 0.6718 0.6828 0.6855 0.6536 0.6536 0.5928 0.6529 0.6396 0.6176 0.6587 0.6438 0.6782 0.6869 0.6836 0.6652 0.6489 0.5582 0.6037 0.5855 0.5886 0.5999 0.6125 0.6524 0.6552 0.6575 0.6482 0.6411 0.6433 0.6301 0.5598 0.5703 0.5674 0.5610 0.5946 VG 0.5728 0.5598 0.5614 0.5578 0.5451 0.5396 0.5494 0.5119 0.4850 0.4842 0.5007 0.4939 VH 0.7758 0.7881 0.7930 0.7817 0.7504 0.7569 0.6540 0.7393 0.7135 0.7039 0.7359 0.7507 VI 0.6441 0.6042 0.6043 0.6096 0.6265 0.5794 0.5335 0.5663 0.5443 0.5484 0.5554 0.5201 VJ 0.7785 0.7307 0.7428 0.7337 0.7438 0.7107 0.7268 0.6813 0.6557 0.6494 0.6681 0.6504 VK 0.5496 0.5556 0.5620 0.5492 0.5238 0.5346 0.5284 0.4683 0.4573 0.4685 0.5144 0.4483 VL 0.3878 0.3646 0.3657 0.3590 0.3576 0.3442 0.2963 0.3228 0.3256 0.3123 0.3248 0.3069 VM 0.3754 0.3655 0.3669 0.3740 0.3704 0.3489 0.2931 0.3300 0.3143 0.3072 0.3180 0.2989 VN 0.3858 0.3749 0.3727 0.3800 0.3670 0.3533 0.3040 0.3287 0.3223 0.3221 0.3358 0.3228 VO 0.3763 0.3636 0.3625 0.3590 0.3603 0.3361 0.3039 0.3237 0.3175 0.3127 0.3241 0.3132 VP 0.5067 0.4977 0.5059 0.4961 0.4897 0.4726 0.4322 0.4629 0.4468 0.4480 0.4563 0.4426 I 9.1292 8.9223 8.9497 8.8947 8.7969 8.5700 8.0394 8.0980 7.9019 7.8603 8.0797 7.9017 -148- time-efficient here, the quadratic form could prove superior with some other class of problems. 3-5 How the Different Methods Compare In this section, the best in the three broad classes of methods are compared. Attention is next turned on investigating the effect of varying the frequency of K-computation. The section is terminated with general conclusions on the contents of the chapter as a whole. 3-5-1 Applications and Deductions Based on the outcome of the studies reported in Sections 3-2 through 3-4, the final comparison was designed to involve. (1) The double-loop univariate approach based on '^-normalized method 2' and employing Richmond's acceleration technique. (2) The free-energy-minimization algorithm based on the sensitivity method accelerated by tetrahedral projection. (3) The single-loop univariate method based on the MVT-accelerated Rachford-Rice formulation projected by Wegstein's technique. The results obtained are presented in the three columns under the heading "'Normal' K-computation frequency" in Table 3-7. The following inferences are drawn: (1) The double-loop univariate approach is far less efficient than the other approaches. (2) The free-energy-minimization algorithm is slightly better than the single-loop univariate algorithm for most of the narrow-boiling mixtures, notable exceptions being systems VG and VK for which i t -149- e x h i b i t s o s c i l l a t o r y convergence. For wide-boiling mixtures, the former algorithm i s s l i g h t l y out-performed by the l a t t e r . 3-5-2 Varying the Frequency of K-Computation The free-energy minimization algorithms and the single-loop u n i v a r i a t e methods have been developed on a structure whereby the e q u i l i b r i u m r a t i o s are computed once i n an i t e r a t i o n . There i s nothing sacrosanct i n t h i s l o g i c and there i s no reason not to expect good performance from a scheme that updates K less frequently. But how much less frequently should t h i s be without s a c r i f i c i n g s t a b i l i t y and r e s u l t - q u a l i t y f o r convergence-speed? The answer to t h i s question w i l l depend on how composition-dependent the e q u i l i b r i u m r a t i o s are. To answer i t i n q u a n t i t a t i v e terms, the f o l l o w i n g algorithms modified were so that the frequency of K-computation could be varied: (1) The Wegstein-projected MVT-accelerated Rachford-Rice method. (2) The tetrahedral-projection-accelerated (3) The t e t r a h e d r a l - p r o j e c t i o n - accelerated GP method. (4) The s e n s i t i v i t y method. s e n s i t i v i t y method• The algorithms were applied to systems VA, VD, VH, V J , VN, VO and VP. I n each case, the frequency of K-computation was varied from 'once every i t e r a t i o n ' to 'once every four i t e r a t i o n s ' . The tetrahedral-projection-accelerated s e n s i t i v i t y method was observed to encounter problems near the s a t u r a t i o n l i n e s f o r frequencies other than 'once every i t e r a t i o n ' . three methods have been subjected loss i n q u a l i t y — results. The r e s u l t s obtained f o r the other to some analysis to determine how much r e l a t i v e to a frequency of 'once every i t e r a t i o n ' A deviation parameter, E , was introduced and defined by f — -150- where the s u b s c r i p t and ' f denotes i s the vapour flow f o r f=1. The average E^ v a l u e s — f o r each system — a frequency of 'once every f i t e r a t i o n s ' averaged over 19 temperature are presented i n T a b l e 3-6. points E^ f o r the u n i v a r i a t e method and E^ and E^ f o r the s e n s i t i v i t y method are not i n c l u d e d i n the T a b l e due to l a c k of convergence at these f r e q u e n c i e s . I f we assume a v a l u e of E < 0.001 the r e s u l t s used as a t o l e r a b l e d e v i a t i o n , then i n d i c a t e that a frequency of '1 every 2 i t e r a t i o n s ' can be f o r the u n i v a r i a t e and the s e n s i t i v i t y methods i n t h e i r to any of the systems. application They a l s o i n d i c a t e that f o r the t e t r a h e d r a l - p r o j e c t i o n - a c c e l e r a t e d GP method, a frequency of '1 every 3 i t e r a t i o n s ' would be s a t i s f a c t o r y . Having thus gained some e x p e r i e n c e as to the s a f e f r e q u e n c i e s t o use f o r each of the methods, the t h r e e methods were executed, based on their limiting f r e q u e n c i e s , f o r a l l 16 systems. r e s u l t s o b t a i n e d were, i n every case, checked K-computation a g a i n s t those f o r 'normal' frequency, and were found to y i e l d w i t h i n the assumed t o l e r a b l e l i m i t . obtained i n S e c t i o n d e v i a t i o n s that are The computation from the a p p l i c a t i o n are presented i n T a b l e 3-7. same T a b l e f o r the purpose The p h a s e - d i s t r i b u t i o n of comparison times resulting A l s o i n c l u d e d i n the are the e x e c u t i o n times 3-5-1. The main d e d u c t i o n to be made here i s that r e d u c i n g the Table 3-6 Absolute deviations (E) f o r various K-computatlon frequencies W e g s t e i n - p r o j e c : t e d , MVT-acce- T e t r a h e d r a l - p r c > j e c t i o n - a c c e l e i rated GP Method l e r a t e d Rachfoir d - R i c e Method Sensitivity Method System E2 E3 E2 E3 Ek VA 0.000069 0.005057 0.000110 0.000282 0.002418 0.000017 VD 0.000017 0.007604 0.000077 0.000053 0.001537 0.000013 VH 0.000206 0.023116 0.000205 0.000525 0.001846 0.000030 VJ 0.000147 0.001845 0.000574 0.000172 0.001600 0.000035 VN 0.000016 0.023967 0.000023 0.000032 f 0.000004 VO 0.000009 0.012801 0.000007 0.000051 f 0.000003 VP 0.000011 0.012414 0.000023 0.000022 0.000538 0.000003 f = failure E2 Table 3-7 Final comparison of vapour—liquid algorithms: computation times (CPU seconds) Limiting K-computation frequency (tolerable E < 0.001) 'Norma 1' K-computation fre quency System Double-loop Richmond-accelerated •-normalized method 2 Sensitivity method with tetrahedral projection Single-loop Wegstein-projected MVT-accelerated Rachford-Rice Meth. GP with tetrahedral projection Single-loop Wegstein-projected MVT-accelerated Rachford-Rice Method Sensitivity Method VA 0.8556 0.5227 0.5027 0.5651 0.4902 0.4311 VB 0.6612 0.3700 0.3574 0.4188 0.3769 0.3390 VC 1.1560 0.6420 0.6763 0.7941 0.6777 0.5812 VD 1.1455 0.5682 0.6248 0.7278 0.6477 0.5398 VE 1.1294 0.5755 0.5951 0.8404 0.6195 0.5391 VF 1.0525 0.5357 0.5448 0.6619 0.5926 0.4954 VG 0.9657 0.9207* 0.4826 0.7244 0.5018 0.4129 VH 1.2392 0.6145 0.7232 0.9288 0.7305 0.5900 VI 1.0066 0.5207 0.5565 0.6796 0.5506 0.5361 VJ 1.2903 0.6260 0.6504 0.7433 0.6524 0.5629 VK 0.8761 0.5100* 0.4830 0.6560 0.5272 0.4208 VL 0.5679 0.3432 0.3270 0.3195 0.3387 0.3060 VM 0.5677 0.3437 0.3179 0.2961 0.3328 0.2954 VN 0.5816 0.3506 0.3246 0.3135 0.3409 0.3091 VO 0.5508 0.3454 0.3178 0.3059 0.3337 0.2982 VP 0.7586 0.4676 0.4609 0.4350 0.4761 0.4112 I 14.4046 8.2565 7.9450 9.4102 8.1893 7.0682 * Oscillatory convergence Note: Equivalent total time for double -loop Newton-accelerated Rachford-Rice with terml nal i n i t i a l i z a t i o n (6° = 0.0) i s 19.5157 seconds. 1 — — — . _* 1 -153- frequency of K-computation for the best single-loop univariate method by half results i n an average time-saving of about 12% and makes i t d i s t i n c t l y better than the best of the free-energy minimization methods. 3-5-3 (1) Conclusions Solving the vapour-liquid f l a s h problem by a double-iteration algorithm i s comparatively (2) inefficient. The speed, s t a b i l i t y and r e l i a b i l i t y of the computational algorithms are a strong function of the i n i t i a l i z a t i o n scheme employed. With proper i n i t i a l i z a t i o n , even highly nonideal mixtures can be handled by a s i n g l e - i t e r a t i o n computational algorithm. (3) It i s possible to reduce the frequency of K-computation from the 'normal' value of 'once every i t e r a t i o n ' , i n most cases by half, for even highly nonideal systems without incurring intolerable loss in the quality of r e s u l t s . (4) The geometric-programming formulation of the free-energy minimization method compares favourably with other methods, given a suitable acceleration method. (5) The two vector-projection methods proposed i n this work have put up a quite good showing and seem to have a future. (6) The quadratic form of the Wegstein accelerat ion method proposed here has not been a disappointment and could find useful application i n other classes of problems. CHAPTER FOUR SENSITIVITY ANALYSIS IN VAPOUR-LIQUID EQUILIBRIA 4-1 Introduction This chapter presents the attempts made to apply sensitivity analysis to vapour-liquid equilibria, the object being to be able to accurately project from an equilibrium solution at some temperature and pressure condition to the equilibrium solution at a different level of temperature and pressure. The study was undertaken in the belief that the technique, i f successful, could constitute a powerful tool in the design and automatic control of process separation units. The geometric-programming-based perturbation theory of Duffin and co-workers (1967), the gist of which is presented in Appendix C-2, is put to the test and i t s weaknesses unveiled. The subject is* then viewed from other angles. The system code employed here refers to Table 3-2. 4-1-1 Nomenclature Note: Any symbol not defined below and not clearly defined where i t occurs within this chapter retains the definition of Section 3-1-1. Symbol Definition A,B Parameters in K versus T model. s Partition parameter. Subscripts i»j.k.n Components. Superscripts j Interval. -154- -155- 4-2 The Vapour-liquid Formulation A p p l i c a t i o n of the theory presented i n Appendix C-2 t o the v a p o u r - l i q u i d problem leads to the r e l a t i o n s h i p j=l + b< > k k=l n=l nL nv -gS.}] (4-1) o where J i s g i v e n by , b «b< ( N k ) ^ ( w * ) - - ^w r 2 - b<J>b< > k + iI ^ = l[ ^ V1. i k o b<«b< + - ^ h v.r ^ ] i N i i + k ) b - t- U j 1=1 L * 1 = " 1 i N t i - i ] ( 4 . 2 ) V In the above e q u a t i o n s , the s t a r s u p e r s c r i p t denotes the e q u i l i b r i u m s o l u t i o n from which the p r o j e c t i o n i s being made, w h i l e w v e c t o r tw , _1 , _v }. S e c t i o n 3-3-4. denotes the A l l other parameters a r e as d e f i n e d i n S u b s t i t u t i n g the values of the n o r m a l i t y and n u l l i t y v e c t o r s Equation ( 4 - 1 ) , we have d^= where - dv = J ( w * ) • [dJlnf^ - d£nC_ ] s v = [dlnC^, ,d£nC i4 J " 1 i X ] . 'v' s u b s t i t u t e d f o r 'L'. s u b s t i t u t i o n into Equation 6. J dlnC^, s i m i l a r l y defined, with A similar (4-3) - 1 dinC^ dAnC^ i into (4-2) y i e l d s 6. + - J " ) " (— + — ) v L V i ( i = 1,2,...,N) (4-4) -156- 1 for j = i , where 6.= { 0 for j J M From Equation (3-46), C. •"• • K. . • = Therefore, |"d£nC - dJlnC 1 = -dJtnK L v — (4-5) T where d£nK ,... .dJlnl^] dlvK = [dZnK^ 2 Combining Equations (4-3) and (4-5), we have dv = - dl = J (w*) • dfcnK (4-6) -1 Equations (4-4) and (4-6) constitute the working equations. 4-2-1 Eliminating Matrix Inversion Equation (4-6) shows that an application of the sensitivity analysis method would involve the inversion of an N x N matrix — a time-consuming step, especially for large N values. However, Equation (4-4) reveals that a l l the off-diagonal elements of the matrix are identical. Advantage was taken of this special form of the Jacobian matrix by doing some matrix manipulation that led to the elimination of any matrix inversion. Using the facts that L* + V* = F and 1, + v i ± L = Z ( i = 1,2 N), Equation (4-4) was rearranged to give T J ij - 1 J * * V i F *~* L V (4-7) -157- Let us i n t r o d u c e the r e l a t i o n s h i p s -,* 1 = *L *x , T x ± ± V* = V*v* i and Z v i ' y = Fz . i i Then the m a t r i x J can be manipulated to g i v e J -F = (4-8) (1 - D) * * L V where 1_ i s an N x N m a t r i x whose elements are a l l i d e n t i c a l l y and D i s the N x N d i a g o n a l m a t r i x d e f i n e d by ii X (4-9) * * i i y Combining E q u a t i o n s (4-6), (4-8) and ( 4 - 9 ) , we -J-^ L V Q unity, have - D)dv = dJtnK * ic -L V - Ddv = — - — ldv or d£nK (4-10) The product l d v i s a column v e c t o r whose elements are dV, w h i l e the product Ddv i s a column v e c t o r whose 1th element i s z.dv./x*y*. J J J J Hence, E q u a t i o n (4-10) leads to * * X d V i = i y i d z ic ic V i X i y L*V* ' + d ^ n K i ( 1 Summing E q u a t i o n (4-11) from 1 to N, we N I dv. j=l or dV - N x.y. = dV = dV L V F I -J-J- + J=l N V J=l 3 L V —-— x.y.d^nK. 3 3 3 z. J N I v j=l = 1 , 2 N (4-11) ) have x.y.dfcnK. -J-J z. 3 J * * . , Z. J-1 J (4-12) -158- S u b s t i t u t i n g Equation (4-12) i n t o E q u a t i o n (4-11) and r e a r r a n g i n g , we obtain * * N x*y*d*nK. J ] J V 3=1 J * it x .y. N 1 " + d^nK. -JJ3 I 3=1 T h i s i s the f i n a l working e q u a t i o n , w i t h finite r a t h e r than an i n f i n i t e s i m a l 4-2-2 A p p l i c a t i o n s and Outcome In 'A' r e p l a c i n g 'd' to s i g n i f y a change. the a p p l i c a t i o n of the method, w h i l e the programs employed were w r i t t e n to be a b l e to handle p e r t u r b a t i o n s i n both temperature p r e s s u r e , o n l y v a r i a t i o n s i n temperature justified (4-13) by the f a c t were a c t u a l l y t e s t e d . and This i s t h a t the thermodynamic parameters y, $ and f° are most s e n s i t i v e to temperature changes, and much l e s s so to p r e s s u r e variations. The of (1) f o l l o w i n g are the main steps i n v o l v e d i n the implementation the method: An i s o t h e r m a l v a p o u r - l i q u i d f l a s h c a l c u l a t i o n i s c a r r i e d out a t the base temperature, (2) T , to o b t a i n the 'known s o l u t i o n ' . A step change i n T, AT, i s i n t r o d u c e d and the c o r r e s p o n d i n g A&nK (based on the composition at T*) i s determined. (3) The molar compositions at the new T a r e computed v = v* + A^v, and 1_ = Z_ - v_, w i t h A\> c a l c u l a t e d from E q u a t i o n (4-13). from -159- The method was t e s t e d by a p p l y i n g i t to systems VA and VB. The r e s u l t s f o r system VA are summarized i n T a b l e 4-1 i n the form of an e r r o r term, E, d e f i n e d by where V r e s u l t s from s e n s i t i v i t y and V i s the expected v a l u e , problem a t T. I t should s e n s i t i v i t y method — analysis obtained be noted that i n the a p p l i c a t i o n s of the here and i n subsequent s e c t i o n s — s o l u t i o n i s determined only a t the i n i t i a l relevant Tables). by r e - s o l v i n g the The estimated T = T* + nAT, base p o i n t an exact (the T* i n the r e s u l t f o r any p o i n t , n = 2,3,4... r e s u l t s from u s i n g the estimated 'known s o l u t i o n ' . The e r r o r s f o r system VB are of the same order of magnitude. r e s u l t f o r T = T* + (n-1)AT as the The e r r o r s are d e f i n i t e l y s i g n i f i c a n t , and t h i s c a l l s f o r a c l o s e r look a t the s e n s i t i v i t y method. 4-3 A Quadratic Taylor Approximation A c l o s e i n s p e c t i o n of the s e n s i t i v i t y r e l a t i o n s developed i n the preceding section reveals that they are i d e n t i c a l to what would r e s u l t from a two-term t r u n c a t i o n of the T a y l o r A n K ^ i = 1,2,...,N) w i t h r e s p e c t AnK ± = AnK ± + I ^ j=l j In view of t h i s o b s e r v a t i o n , series-expansion, next studied. of to v_, that i s N 9£nK A series-expansion ^ 1 (v v. A J - v ) ( i = 1,2,...N) (4-14) J a three-term t r u n c a t i o n of the T a y l o r manipulated to e l i m i n a t e second-order d e r i v a t i v e s , was -160- Table 4-1 AT(K) 0.1 0.2 0.8 2.0 Errors f o r S e n s i t i v i t y Analysis ( o r i g i n a l GP version) f o r system VA, with T* - 336.5K T(K) E 340.5 -0.0946 341.3 -0.1267 340.5 -0.1612 342.5 -0.2433 340.5 -0.3437 342.1 -0.5503 340.5 -0.4483 342.5 -0.5880 -161- The T a y l o r e x p a n s i o n i s g i v e n by A AnKj^v) T & 1 & = JtnK (v ) + V [ j l n K ^ v ) ] ( v + 2<v - v ) V A [£nK (v )](v - ± + [ 4 t h term + . . . . . . ] - v ) v ) (1-1,2, ,N) (4-15) T a k i n g the g r a d i e n t of E q u a t i o n ( 4 - 1 5 ) and p o s t m u l t i p l y i n g each term by 0.5(^v - v*), we have yV [AnK.(v)](v T 1 & T + v*) 0 v*) $c )V[£nK (v i +. [j x 4 t h Term + If = yV^JtnK^v*)]^ - )](v - v ) ] (4-16) we assume t h a t [4th Term + . . . ] i n E q n . ( 4 - 1 5 ) = [ -^x 4 t h Term + . . . ] then s u b t r a c t i n g E q u a t i o n ( 4 - 1 6 ) from ( 4 - 1 5 ) AnK ± - T 1 (4-16), gives *nK* = | { V [ £ n K ( v ) ] - V [ £ n K ( v * ) ] } ( v T i n Eqn. ± v*) (4-17) A p p l y i n g to E q u a t i o n ( 4 - 1 7 ) a m a t r i x m a n i p u l a t i o n s i m i l a r t o performed i n S e c t i o n 4 - 2 - 1 r e s u l t s N Av. = 2v 1 11 I v.l. 11 in A£nK. J j=l + A£nK. v F "A that i i i (1=1,2,....N) (4-18) -162- * * where LV = and V L .VL * * V L + VL The above r e l a t i o n s h i p i s u n f o r t u n a t e l y recursive i n v thus n e c e s s i t a t i n g some form of i t e r a t i o n i n i t s e v a l u a t i o n . In a p p l y i n g (ranging the r e l a t i o n s h i p s , a f i x e d number of i t e r a t i o n s from 1 to 10) was the method was employed i n updating a p p l i e d t o systems VA and VB, be q u i t e d i s a p p o i n t i n g , JL, V, L and K. the r e s u l t s turned not showing any s i g n i f i c a n t When out to improvement i n r e s u l t q u a l i t y over the two-term T a y l o r - s e r i e s t r u n c a t i o n to j u s t i f y e x t r a computation i t i n v o l v e s . investigated 4-4 I n view of t h i s , the the approach was not further. A Mean-value—theorem Approach The next c o n s i d e r a t i o n g i v e n a p p l i c a t i o n of the mean-value to the s e n s i t i v i t y technique was theorem of d i f f e r e n t i a l c a l c u l u s the (Holland, 1975). Still t r e a t i n g InY.^ as a f u n c t i o n of the vapour molar v e c t o r , we have, a c c o r d i n g Zn K = in K* + V [ j l T ± where 0 < a < 1 n to the mean-value K (v*+ aAv)](v ± theorem, - v*) ( i = l ,2,... ,N) (4-19) -163- S o l v i n g the above N equations f o r v, we have v_ = v* + J A £ n K (4-20) - 1 where J i s the J a c o b i a n of AnK w i t h r e s p e c t to v™, where v l m m * * = v + a(v - v ) } = Z - v (4-21) m and jc*, y* are o b t a i n e d by summation and n o r m a l i z a t i o n . With a = 0, the above r e l a t i o n s h i p reduces the geometric-programming t h e o r y . except t h a t the s t a r s u p e r s c r i p t Thus, E q u a t i o n i s replaced with to t h a t o b t a i n e d from (4-13) s t i l l a p p l i e s 'm'. To e v a l u a t e v™, one needs r e a s o n a b l e e s t i m a t e s of the unknown q u a n t i t i e s a and v^. F o r the l a t t e r q u a n t i t y , t h i s i s o b t a i n e d by doing an Initial c a l c u l a t i o n based on a = 0. 4-4-1 O b t a i n i n g a Best a Attempts were made to determine the v a l u e of a, as a f u n c t i o n of 8 , t h a t would be the most s u i t a b l e i n the a p p l i c a t i o n of the mean-value theorem. S t a r t i n g w i t h the r e l a t i o n s h i p JlnK = Znv^ + £nL -£nV -Znl i and we (4-22) s u b s t i t u t i n g f o r l . i n terms of v and f o r L and V i n terms of 8 , i i v have 4nK Treating = Anv - £n(Z ± - v ) + A n ( l - 6 ) - InQ^ as a f u n c t i o n of theorem to E q u a t i o n and 8^ and a p p l y i n g the mean v a l u e (4-23) r e s u l t s i n (4-23) -164- A*nK = [-j + v^^ + aAv^^ Z ]Av. - ( v ^ + aAv^) s ± - [—3j—^ 6 + aA9 + ^ ]A6 1 - (6 + aA6 ) (4-24) v V V V E q u a t i o n (4-24) can be reduced t o the form f.(a) = C i > 4 a + C 4 i > 3 a + C ^ a 3 2 + C.^a + C (4-25) = 0 1 > 0 where C. . = A£nK • A 9 « A v , i,4 i v i ' 2 C i,3 " = - A 9 v A v i 2 - ^ n K [ A v ( l - 2 6 * ) + A B ^ Z ^ v * ) ], i i C. = [A6 -Av.(Z - 2 v * ) ( l - 2 9 * ) - 8 * ( l - 6 * ) - A v i,2 v i i i v v v i 2 0 L o & & - A6 - v ( Z - v ) ] A £ n K v c i , i 1 [ v e = i , A v 1 ( -v r i i i ( z 2 v + Av .A9 [(Z -2v*) 1 and C v + 1 i Z i ) + Av^A9 (Z A9 -Av ), v A e # v v J( 1 z v - i v 1 i ) ( i " 2 e C ] ) M n K i (l-29*)], - = v*^9*(Z -v*)(l-9*)A£nK. + v*»A9 (Z.-v*) 1,0 i v i i v i 1 v I 1 4 * * - Z «e «Av.(l-9 ) . i v i v Summing E q u a t i o n (4-25) f o r a l l components N 4 f(«) = I f , ( a ) = I C a i-1 j-0 1 where C. = 3 N I i=l C J = 0 J 1»3 (j=0,l,2,3,4) i , ( i = l , 2 ,...,N), we have -165- U s i n g one of the v a p o u r - l i q u i d r e s u l t s were generated f o r system VA the two-phase r e g i o n . case. The at d i f f e r e n t temperatures p o i n t s were chosen such that A0 E q u a t i o n (4-26) was method, to o b t a i n The then s o l v e d the v a l u e essentially same as that shown i n F i g u r e T h i s agreement j u s t i f i e s and realized A £ n K. i n t o that of 0 . i v serve tested: (1) An shown i n s was were 4-1. However, i t must ' and be the p l o t of F i g u r e 4-1 can only as a rough guide. Guided by F i g u r e 4-1 were ± the i n c o r p o r a t i o n of the dependence of a t h a t a a l s o depends on A0^ therefore the Newton of a c o r r e s p o n d i n g to each 6 . i n d i v i d u a l components, the p r o f i l e s obtained the i n each C^ When, i n s t e a d of E q u a t i o n (4-26), E q u a t i o n (4-25) a p p l i e d to the 3, spanning « 0.1 f o r each p o i n t , u s i n g r e s u l t i n g p r o f i l e of a as a f u n c t i o n of 6 F i g u r e 4-1. i of Chapter These r e s u l t s were employed i n the e v a l u a t i o n of the terms. on v f l a s h algorithms exponential the f o l l o w i n g a - v e r s u s - 0 ^ r e l a t i o n s h i p s f i t of the p r o f i l e i n F i g u r e 4-1, resulting in the relationship 0.46 - 0.31 exp(-15.5273 0.46 + 5.5968(10~ ) e x p ( 1 5 . 5 2 7 3 ) , a = v ), 8 0 < — e v (2) A l i n e a r r e l a t i o n s h i p of the a = m0 6 v + 8 v < 0.5 — 0.5 y < 1 form were t e s t e d , i n c l u d i n g {o.1,0.45}, {0.1,0.4}, {0.2,0.35}, {0.2,0.4}, {0,0.5}, {0,0.45}, {0,0.48} and A p a r a b o l i c r e l a t i o n s h i p of the a = c + 4m6 e c. 12 d i f f e r e n t s e t s of {m,c} (3) < v (1-0 •) v y form {0,0.52}. -166- 1.0 I a E o i_ 0.8 D CL | o OJ JZ rI CD I > i 0.6 0.4- sz D CU 0.2H — i 0 0.1 1 1 1 1 0.2 0.3 0.4 0.5 •Bi 1 1 0.6 0.7 1 1 0 . 8 0.9 1.0 Fig 4-1: Fit of mean-value -theorem parameter, a, for system VA -167- 13 s e t s of {m,c} were t e s t e d . These i n c l u d e d p o s i t i v e and n e g a t i v e values of m. The d i f f e r e n t a-versus-6^ r e l a t i o n s h i p s were compared on the b a s i s of the maximum a b s o l u t e e r r o r s i n the mole f r a c t i o n s and the percentage e r r o r i n the t o t a l flow r a t e s , that r e s u l t e d when they were a p p l i e d to system VA a t the same temperatures same temperature changes. v better i s remarkable and w i t h the The best r e s u l t s were o b t a i n e d w i t h a = 0.53 - 0 . 0 8 9 ( l It (x_ and y ) , - 6 ) (4-27) that the constant r e l a t i o n a = 0.5 performed than most of the 8-dependent r e l a t i o n s h i p s and was i n f a c t as good as t h a t r e p r e s e n t e d by E q u a t i o n (4-27). With an a - r e l a t i o n s h i p thus determined, determining almost the a l g o r i t h m t h a t minimizes the next the s e n s i t i v i t y c o s t i n g too much i n terms of e x e c u t i o n time. step i s that o f errors without Three a l g o r i t h m s were studied. 4-4-2 Algorithm A T h i s a l g o r i t h m i n v o l v e s the f o l l o w i n g s t e p s : (1) O b t a i n the s o l u t i o n a t T* through an i s o t h e r m a l f l a s h calculation. (2) Set T = T* + AT and compute K based determine (3) Hence A&nK. C a l c u l a t e v_ m Equation on T, x* and y*. from E q u a t i o n (4-21) where Av_ i s e s t i m a t e d (4-13) and a i s computed from E q u a t i o n the c o r r e s p o n d i n g v a l u e s of _ l , m x m and y . m (4-27). from Calculate -168- (4) Employing the q u a n t i t i e s computed i n Step 3 and r e p l a c i n g the '*' s u p e r s c r i p t w i t h 'm' i n E q u a t i o n (4-13), determine the f i n a l v a l u e of v_. The r e s u l t s o b t a i n e d from a p p l y i n g t h i s a l g o r i t h m to system VA are p r e s e n t e d at two T a b l e 4-2. temperatures and three AT v a l u e s i n Column 3 of A comparison of these r e s u l t s w i t h those of T a b l e 4-1 shows t h a t the former i s an improvement over the l a t t e r . 4-4-3 Algorithm B In A l g o r i t h m A, K at T was computed u s i n g x* and y*. tue of i t s d e f i n i t i o n , K ought to be based on x and y at T. these v a l u e s are unknown. 2£ » ^m^ m steps i n v o l v e d are e x a c t l y Column 4 of T a b l e 4-2 t h i s a l g o r i t h m to system VA. using the same as f o r A l g o r i m and y™. shows the r e s u l t s o b t a i n e d from a p p l y i n g Compared to the r e s u l t s i n Column 3 of the T a b l e , these r e s u l t s are not s i g n i f i c a n t l y d i f f e r e n t . felt vir- However, In t h i s algorithm, K i s c a l c u l a t e d thm A except that i n Step 3, K i s recomputed u s i n g x i t was By I n view of t h i s , that K could be t r e a t e d as being a p p r o x i m a t e l y independent of c o m p o s i t i o n over a temperature range as l a r g e as 2K f o r system A l g o r i t h m C i s based on t h i s 4-4-4 VA. assumption. Algorithm C By the v e r y nature of s e n s i t i v i t y perturbation, analysis, the more a c c u r a t e the r e s u l t the s m a l l e r the of the a n a l y s i s . If K is assumed to be independent of x and y_ over a temperature range T* < T < (T* + AT), then the p r o j e c t i o n from T* to T = (T* + can be e f f e c t e d i n a s e r i e s of steps by p a r t i t i o n i n g the temperature range i n t o a number of i n t e r v a l s and computing AJtnK f o r each from a s u i t a b l e K-versus-T fit. AT) interval Table 4-2 Errors f o r s e n s i t i v i t y - a n a l y s i s algorithms A and B f o r system VA, with T* = 336.5K E \/alues AT(K) T(K) Algorithm A Algorithm B 340.5 -0.0110 -0.0185 342.5 -0.0134 -0.0281 340.5 -0.1170 -0.1214 342.1 -0.3492 -0.3497 340.5 -0.3281 -0.2907 342.5 -0.3613 -0.3725 0.2 0.8 2.0 -170- Two types of K-versus-T f i t were c o n s i d e r e d : (1) A l i n e a r AnK v e r s u s ^- r e l a t i o n s h i p . (2) A l i n e a r AnK v e r s u s T r e l a t i o n s h i p . L i n e a r AnK v e r s u s T * For t h i s we have AnK i = A ± B. + -y (4-28) I f we d i v i d e AT i n t o s equal i n t e r v a l s Equation (see F i g u r e 4-2), then a p p l y i n g (4-28) t o the p o i n t s T* and T and t a k i n g the d i f f e r e n c e , we have -T TAAnK i B = AT ( 4 By s i m i l a r l y a p p l y i n g E q u a t i o n points, T i+1 J (4-28) to any two a d j a c e n t " 2 9 ) interval i and T , we have -T^T^AAnK^ B. = (4-30) : AT 1 J where AJlnKJ = £ n K j and AT j = T - T J + 1 Combining E q u a t i o n s + 1 - £nK^ ( j = 0,1,...,s-l). j (4-29) and (4-30) and r e a r r a n g i n g , we have T*.T.AT^.AAnK M n K i = — n + i — - Combining E q u a t i o n (4-31) w i t h the r e l a t i o n s h i p s AT J = AT/s ( 4 _ 3 1 ) -171- s-2 Temperature Fig 4 - 2 : I n K - v e r s u s - T partitioning scheme -172- T j = T and T we have MnK 3 + [ l _ C - T * [ l 1 = 2 i - ( 3 3)A sT + + ^ sT A T ] T ] t A£nK Is r r — (j+DAT i r _ (s - j)ATi s(T - A T ) sT r L J 1 v (4-32) J L i n e a r £nK v e r s u s T A l i n e a r £nK v e r s u s T r e l a t i o n s h i p £nK If i = A ± implies + B/T (4-33) we apply the same p a r t i t i o n i n g approach as i n the f i r s t f i t and f o l l o w a s i m i l a r mathematical procedure, the f i n a l r e s u l t i s A£nK;j = A £ n K / s (4-34) i B e a r i n g i n mind that to s, a comparison ( j + 1) and (s - j ) are always of Equations (4-32) and (4-34) r e v e a l s that f o r r e a s o n a b l y s m a l l AT and f o r noncryogenic systems, equivalent. The l e s s than or equal the two f i t s are The l a t t e r was employed i n the t e s t s that follow. f o l l o w i n g steps are i n v o l v e d i n the ensuing a l g o r i t h m : (1) O b t a i n the s o l u t i o n a t T* through an i s o t h e r m a l f l a s h a l g o r i t h m . (2) Set T = T* + AT. Compute K based on T, x* and y*. Calculate A£nK and hence determine AAnK-', from E q u a t i o n (4-34). (3) Calculate v m f o r the f i r s t i n t e r v a l as i n Step 3 o f A l g o r i t h m A. (4) C a l c u l a t e v_ f o r the f i r s t Step 4 of A l g o r i t h m A. i n t e r v a l by the same procedure as -173- (5) Repeat Steps 3 and 4 f o r i n t e r v a l s 2 to s, u s i n g the r e s u l t i n t e r v a l as the succeeding The (with '*' s t a t u s ) f o r the interval. a l g o r i t h m was w i t h AT = 2K. different 'known s o l u t i o n ' The a p p l i e d to system VA f o r s = 2,3,....,6 and r e s u l t s o b t a i n e d are p r e s e n t e d temperature levels. i n T a b l e 4-3 They are a d e f i n i t e those f o r a l g o r i t h m s A and B w i t h the same i n v o l v i n g systems VA, VD, systems, the temperature VH, V J , VN, exact s o l u t i o n was and VP. to 9* = 0.05 on the estimated s o l u t i o n at T c o r r e s p o n d i n g The e r r o r i n each estimate was where s u p e r s c r i p t s 's' and were 0.65 o n l y a t the base Thus, f o r example, the e s t i m a t e at T c o r r e s p o n d i n g to 0* based computed a to 0* « then and 0.65 was 0.35 from subscript 'n' s i g n i f i e s the p o i n t (1,2 and 3 f o r the p o i n t s c o r r e s p o n d i n g approximately to 8* v a l u e s of 0.35, The e r r o r v a l u e s are p r e s e n t e d and 0.95 i n T a b l e 4-4. respectively). The symbol denotes cases where an i n d e t e r m i n a t e l o g a r i t h m i c term, due composition quantity, r e s u l t e d was temperature. ' r ' are as p r e v i o u s l y d e f i n e d , and 0.65 out, F o r each of the to 0* v a l u e s of 0.35, determined over next c a r r i e d Three other temperatures chosen to correspond approximately two AT. c o r r e s p o n d i n g approximately chosen as the base temperature. The VO, at improvement A more comprehensive t e s t of the method was 0.95. f o r any ,00 ' to a n e g a t i v e i n premature t e r m i n a t i o n of e x e c u t i o n . -174- Table 4-3 E r r o r s f o r S e n s i t i v i t y - a n a l y s i s Algorithm C f o r System VA, with T* = 336.5K E Va Lues s T = 340.5K T = 342.5K 2 -0.1919 -0.2625 3 -0.0937 -0.1066 4 -0.0381 0.0009 5 -0.0040 0.0627 6 0.0182 0.1199 Table 4-4 Errors for a more comprehensive application of sensitivity-analysis Algorithm C. E Values System 8 VA VD VH VJ VN vo VP Note: 00 s =1 8 =2 8 s = 4 =3 8 = 5 8 = 6 1 - 0.4080 - 0.1966 - 0.0927 - 0.0323 0.0054 0.0302 2 -0.4459 -0.2246 -0.0869 0.0024 0.0618 0.1024 3 -0.4982 -0.2763 -0.1137 0.0003 0.0630 0.1262 1 -0.1928 -0.0427 -0.0091 0.0048 0.0122 0.0166 2 -0.2029 -0.0070 0.0283 0.0425 0.0499 0.0543 3 -0.2929 -0.0086 0.0287 0.0444 0.0528 0.0577 1 -0.2355 -0.1235 -0.0926 -0.0790 -0.0717 -0.0673 2 -0.1994 -0.0612 -0.0248 -0.0088 -0.00004 0.0045 3 -0.1783 -0.0381 -0.0019 0.0143 0.0230 0.0280 1 -0.3213 -0.0714 0.0089 0.0473 0.0689 0.0821 2 -0.2268 -0.0442 0.0214 0.0555 0.0754 0.0878 3 -0.2170 -0.0456 0.0181 0.0510 0.0586 0.1156 1 -1.1449 -0.0513 0.3646 0.9343 2.2471 3.5916 2 10.6047 6.0xl0 3 2.5xl0 1 -1.0043 2 8.8900 3 -9.4xl0 1 -0.9409 2 -1.3343 3 -6.1xl0 5 8 OO 00 OO OO 00 CO oo oo oo 0.0062 0.2744 5.9x10"* 3.76xl0 OO 0.0006 265.8 5 5 0.4182 OO 0.2123 4.8xl0 oo premature termination due to indeterminate logarithm OO 32 0.6773 OO oo oo oo CO oo 0.2998 27 0.5480 0.3623 0.4141 OO oo OO oo oo oo -176- The e r r o r s , as can be seen from the T a b l e , are e s p e c i a l l y so f o r the w i d e - b o i l i n g The next step 4-5 levels. They systems. taken was to attempt to t r a c k down the e r r o r w i t h a view to knowing whether reasonable negligible are q u i t e s i g n i f i c a n t . steps can be taken to reduce i t t o T h i s i s the s u b j e c t matter of the next s e c t i o n . Error-tracking I n attempting applying Algorithm to a p p o r t i o n C discussed the e r r o r i n the r e s u l t s o b t a i n e d by above, two major sources were (a) The e r r o r due to the composition-dependence of K. (b) The e r r o r due t o the i n e x a c t i t u d e of a. considered: Three computer programs were w r i t t e n f o r the e r r o r - t r a c k i n g . first (henceforth error. designated 'Program A') e l i m i n a t e s the f i r s t The type of The second e r r o r - t y p e i s e l i m i n a t e d by the second program ('Program B ' ) . The t h i r d program ('Program C ) eliminates the e f f e c t of both sources of e r r o r . Program A T h i s program i n v o l v e s the f o l l o w i n g key s t e p s : (1) Algorithm given (2) C i s employed i n p e r f o r m i n g the s e n s i t i v i t y a n a l y s i s on a system. With T r e p l a c i n g T*, AT s e t equal to 0, and the c u r r e n t v a l u e s of the molar q u a n t i t i e s r e p l a c i n g t h e i r values of A l g o r i t h m (3) A ( S e c t i o n 4-4-2) are executed. Step 2 i s repeated until at T*, steps 2 to 4 -177- Program B T h i s i n v o l v e s the f o l l o w i n g s t e p s : (1) Algorithm C i s employed as i n Step 1 of Program A (2) With K f i x e d at i t s c u r r e n t value ( c a l c u l a t e d under A l g o r i t h m C u s i n g x* and y * ) , the problem i s r e - s o l v e d u s i n g a m o d i f i e d v e r s i o n of one of the f l a s h a l g o r i t h m s modified and of Chapter 3. This v e r s i o n uses the r e s u l t s of Step 1 as i n i t i a l values does not r e c a l c u l a t e K throughout the i t e r a t i o n . Program C (1) Same as Step 1 o f Program A. (2) Same as Step 2 of Program B. (3) Steps 2 and 3 of Program A are implemented. The three programs were executed f o r system VA based on T* = 336.5K and AT = 2K. The r e s u l t s a r e presented a t two temperatures i n T a b l e 4-5. The negligibly non-consideration s m a l l e r r o r s recorded f o r Program C j u s t i f i e s the of any other p o s s i b l e e r r o r s o u r c e s . The s l i g h t e r r o r s that i t y i e l d s can a c t u a l l y be a t t r i b u t e d t o the s u c c e s s i v e , rather that simultaneous, c o r r e c t i o n of the two e r r o r types. While the e r r o r s f o r Program A are due to the i n e x a c t i t u d e of a, those f o r program B r e s u l t from the composition-dependence of K. the r e s u l t s , the two e r r o r - t y p e s are both s i g n i f i c a n t . the dependence of K on composition incurring The e r r o r due t o could be c o r r e c t e d , p o s s i b l y two much i n c r e a s e i n computation time. From without However, the same cannot be s a i d of the e r r o r due to a, e s p e c i a l l y as the s i t u a t i o n i s f u r t h e r complicated by the dependence of t h i s e r r o r on the v a l u e of the Table 4-5: Error-tracking results (system VA with T* = 336.5 K and AT = 2TC) T = 342.5 K T = 340.5 K Program E E X Maximum Absolute Error In y Maximum Absolute Error i n Maximum Absolute Error In X Maximum Absolute Error i n Z A 0.1418 0.0004 0.0003 0.0097 0.0014 0.0004 B -0.6258 0.0020 0.0008 0.0205 0.0007 0.0004 C 0.0281 0.0001 0.0001 0.0032 0.0000 0.0000 -179- p a r t i t i o n parameter, s, employed. towards a 4-6 A t t e n t i o n was at t h i s p o i n t d i r e c t e d ' p r e d i c t o r - c o r r e c t o r ' approach. Predictor-corrector Approach T h i s approach employs the s e n s i t i v i t y - a n a l y s i s method as ' p r e d i c t o r ' step which p r o v i d e s The estimate i s then used as an i n i t i a l method to s o l v e f o r the exact exact results, i t s worth. discussed 4-6-1 p o i n t by a s u i t a b l e ' c o r r e c t o r ' Since computation time becomes the t h i s approach y i e l d s critical factor i n assessing Three methods have been i n v e s t i g a t e d here, and i n the three first (1) Algorithmic (2) Using subsequent they are subsections. method i n v o l v e s the f o l l o w i n g C of S e c t i o n 4-4-4 steps: i s implemented. the r e s u l t s from Step 1 as the i n i t i a l of the i s o t h e r m a l obtaining f l a s h algorithms the f i n a l p o i n t , the best of Chapter 3 i s employed i n results. Method 2 It best solution. of the d e s i r e d s o l u t i o n . Method 1 The 4-6-2 an estimate a was felt that i n s t e a d of a of the f l a s h a l g o r i t h m s c o r r e c t o r step might be of Chapter 3, T i that employs a s h o r t e r and the faster possible. L e t us express E q u a t i o n A£nK* = V [ £ n K ( v ' c o r r e c t o r ' step n (4-19) i n the i t e r a t i v e + aAv )](v n n + 1 - v ) 11 ( i = 1,2,...,N) where n i s an i t e r a t i o n counter ( n = 0 , form 1, 2, . . . ) , (4-35) -180- n and AAnK? - AnK? 1 i An(—) n x K n i s computed at T, I f we replace Equation '*' by (4-35). x 11 i and y . n 'n' i n E q u a t i o n Thus, the key (4-13) then i t becomes i d e n t i c a l steps f o r t h i s method are: (1) Algorithm (2) D e n o t i n g the r e s u l t s from Step 1 by Equation C of S e c t i o n 4-4-4 (4-35) i s s o l v e d to i s executed. the s u p e r s c r i p t 'o', i t e r a t i v e l y u n t i l convergence i s achieved. 4-6-3 Method 3 T h i s method cannot s t r i c t l y one. be d e s c r i b e d as a p r e d i c t o r - c o r r e c t o r I t i s a m o d i f i c a t i o n of Method 2 i n which the p r e d i c t o r step been e l i m i n a t e d . Thus, i n essence, i t i n v o l v e s the f o l l o w i n g (1) The s o l u t i o n at T* (2) Denoting the r e s u l t s from Step 1 by Equation 4-6-4 (4-35) i s s o l v e d A p p l i c a t i o n s and To i s obtained test VO the s u p e r s c r i p t 'o', i t e r a t i v e l y as i n Step 2 of Method the performance of the p r e d i c t o r - c o r r e c t o r a l g o r i t h m s were a p p l i e d to systems VA, VD, and pursuing VH, VJ, VN, VP. For each system f o u r temperatures, one temperature, were chosen along 4-4-4. 2. Deductions f u r t h e r , the a l g o r i t h m s and steps: through a f l a s h c a l c u l a t i o n . to a s c e r t a i n whether the s e n s i t i v i t y a n a l y s i s study i s worth any has The s o l u t i o n f o r any the next p o i n t . the l i n e s of which i s the of what i s presented given p o i n t serves For Methods 1 and as the i n i t i a l 2 which i n v o l v e the base i n Section point f o r partitioning -181- parameter, s, the v a l u e of 2 was comparison of the r e s u l t s i n T a b l e 4-4 In a d d i t i o n versions of used. to the of the Chapter 3 were a l s o implemented. as to i n i t i a l i z e the f o r the p r e d i c t o r - c o r r e c t o r initialization c h o i c e was isothermal One a version algorithms); The algorithms, uses the the of s o l u t i o n at any succeeding point (just other v e r s i o n uses r e s u l t s are 1. two f l a s h algorithms computation f o r the scheme of Chapter 3. based on f o r d i f f e r e n t s values at n = three p r e d i c t o r - c o r r e c t o r the most e f f i c i e n t given point This the contained i n Table 4-6. The from the r e s u l t s show t h a t where the known s o l u t i o n i s f a r removed s o l u t i o n being sought, as i s the Methods 2 and 3 quite S i n c e the v e r s i o n of the predictor-corrector a i l m e n t , and since those f o r the as good as to converge to the isothermal initialization the f l a s h algorithm desired the than the that uses computation times f o r t h i s program as w e l l methods are as generally the initialization scheme of Chapter 3, latter initialization i t can scheme y i e l d s a p o i n t not be that is base s o l u t i o n . s e n s i t i v i t y - a n a l y s i s approach to the h a n d l i n g of vapour-liquid f l a s h problem y i e l d s r a t h e r the ideal. As has been r e v e a l e d s e n s i t i v i t y based a l g o r i t h m by the the poor e s t i m a t e s of the s o l u t i o n , except of course i n a world n e i g h b o u r i n g on the the Conclusion The or solution. scheme a l s o s u f f e r s from t h i s sensitivity predictor-corrector using deduced that 4-7 fail t e s t s here, those o b t a i n e d from a d i r e c t r e - s o l u t i o n of the problem at every p o i n t better often case i n the desired the i n f i n i t e s i m a l study i n t h i s chapter and of Chapter 3, the method i s r e a l l y no by Table 4-6: Predictor-Corrector Algorithm 1 System Computation times (CPU seconds) for predictor-corrector algorithms Predictor-Corrector Algorithm 2 Predictor-Corrector Algorithm 3 Flash Algorithm with Base-point Initialization Flash Algorithm without Base-point Initialization VA 0.0825 0.0900 0.0896 0.0660* 0.0848 VD 0.0981 0.0711* 0.0730* 0.1150 0.1027 VH 0.1206 0.1155 0.0803* 0.1172 0.1175 VJ 0.1166 0.1228 0.0817* 0.1431 0.1171 VN 0.0671 0.0*** 0.0305** 0.0542* 0.0644 VO 0.0644 0.0244** 0.0671 0.0478* 0.0649 VP 0.0900 0.0263** 0.0375** 0.0678* 0.0872 I 0.6393 {0.6111} 0.6386 (0.4463) (0.4481) [0.4597] [ 0 . 4 5 9 7 ] { 0 . 6 I I I ) In the i failure *i failure ** k failure Notes: summation row, the different quantities in last column correspond to similarly-represented points i n other columns. at 3rd point at 2nd and 3rd points at a l l three points. -183- more than one so a p p l i e d , i t e r a t i o n of a m u l t i - i t e r a t i o n s o l u t i o n path. i t has to be s u i t a b l y i n i t i a l i z e d then, i t i s not n e c e s s a r i l y When i t i s to be r e l i a b l e — the 'king' of t i m e - s a v e r s . and even CHAPTER FIVE ISOTHERMAL LIQUID-LIQUID AND 5-1 LIQUID-SOLID FLASH CALCULATIONS Introduction This chapter i s devoted to the liquid-solid for the study of l i q u i d - l i q u i d equilibrium calculations. liquid-solid study i n Chapter 3, While the a l g o r i t h m s problem f o l l o w as a l o g i c a l the liquid-liquid study has, The deals the i n a d d i t i o n to those l i q u i d - l i q u i d problem i s t r e a t e d i n S e c t i o n w i t h the and liquid-solid case, w h i l e S e c t i o n 5-4 5-2. Section contains 5-3 conclusions studies. Nomenclature Note: it by P r a u s n i t z (1980). drawn on both 5-1-1 presented consequence of d e r i v i n g from Chapter 3, a method r e c e n t l y p u b l i s h e d co-workers and Any occurs w i t h i n Symbol symbol not this defined below and not clearly chapter r e t a i n s the d e f i n i t i o n Definition f Partial fugacity f Check f° Standard s t a t e F T o t a l feed rate I,J Components identifiers K Equilibrium L L i q u i d - p h a s e flow x Mole function fugacity ratio rate fraction -184- defined of S e c t i o n where 3-1-1. -185- Greek Symbols Y Activity coefficient 6 Phase (j) F u n c t i o n d e f i n e d i n Eq. (3-17) fraction Subscripts i,j Components L L i q u i d or l i q u i d - l i q u i d Ll L i q u i d phase 1 L2 L i q u i d phase 2 s Solid 1 L i q u i d phase 1 2 L i q u i d phase 2 phase Superscripts 5-2 L Liquid phase s Solid s S u c c e s s i v e - s u b s t i t u t i o n value T Transpose phase Liquid-liquid Equilibria The l i q u i d - l i q u i d e q u i l i b r i u m study was h e a v i l y c o n s t r a i n e d by the s c a r c i t y of e m p i r i c a l or s e m i - e m p i r i c a l data needed f o r the a p p l i c a t i o n of the necessary property-predicting correlations. Table 5-1 c o n t a i n s v i t a l i n f o r m a t i o n on the systems t h a t were f i n a l l y put together f o r the purpose of t h i s i n v e s t i g a t i o n . Throughout t h i s work, they w i l l be r e f e r r e d to by the codes g i v e n i n Column 1 of the T a b l e , any such r e f e r e n c e i m p l y i n g the mixture otherwise s t a t e d ) . at the g i v e n p r e s s u r e (unless Table 5-1: Identification Code No. of Components V i t a l Information on l i q u i d - l i q u i d systems System Pressure (Atm.) Temperature range (K) Components and molar composition (%). LA 4 1.0 298-298 Water (9.4); Methanol (6.0); Benzene A n i l i n e (47.0). LB 3 1.0 248-348 Water (40.0); Methanol (20.0); Benzene (40.0) LC 3 1.0 253-353 B u t y l - a l c o h o l (20.2); Water P r o p y l - a l c o h o l (5.62). LD 3 1.0 278-378 Water (23.5); Benzene (57.5); E t h a n o l LE 3 0.980 314-414 Methanol (3.38); Water (77.19); B u t a n o l LF 3 1.009 315-415 E t h a n o l (3.17); Water (76.72); B u t a n o l (20.11). LG 4 1.0 248-348 F u r f u r a l (40.0); 2,2,4-trimethylpentane Benzene (10.0); Cyclohexane (20.0). (30.0); LH 3 1.0 283-383 Water (60.0); A c r y l o n i t r i l e A c e t o n i t r i l e (5.0). (37.6); (74.18); (35.0); (19.0). (19.43) -187- 5-2-1 T h e o r e t i c a l Background The mathematical relationships for l i q u i d - l i q u i d equilibria q u i t e s i m i l a r to those f o r v a p o u r - l i q u i d e q u i l i b r i a . equilibria The two are types of d i f f e r only to the extent that vapour-phase p r o p e r t y r e l a t i o n s h i p s are d i f f e r e n t from v a r i o u s r e l a t i o n s h i p s presented those f o r the l i q u i d phase. i n Chapter The 3 apply, with the f o l l o w i n g modifications: (1) x i s r e p l a c e d w i t h JC^ (2) JL i s r e p l a c e d w i t h JL (3) L i s replaced with L (4) The and y w i t h x ^ • a n d v w i t h JLj. and V with (6) 2 parameters f o r l i q u i d phase 1 are s u b s c r i p t e d w i t h those f o r l i q u i d phase 2 with (5) L . K and 9^ are r e p l a c e d w i t h ' L l ' and 'L2 . 1 and 8^ respectively. i s g i v e n by !L2i L F o r condensable i W U I = l Y L21*1.21 components, f ^ ^ = f^2i a n d Equation (5-1) reduces to = 1 1 L 5-2-2 ± ^L2i _ iLli X Y *Lli Y ( 5 - 2 L2i O u t l i n e of A l g o r i t h m s Based on the performances of the d i f f e r e n t methods of Chapter as measured by both e x e c u t i o n time and iteration count, w i t h special 3 ) -188- s i g n l f i c a n c e attached to t h e i r s t r e n g t h s with highly nonideal systems, the and weaknesses when d e a l i n g f o l l o w i n g methods were s e l e c t e d : (1) The GP method w i t h t r i a n g u l a r p r o j e c t i o n . (2) The GP (3) The s e n s i t i v i t y - b a s e d method. (4) The s e n s i t i v i t y method w i t h t e t r a h e d r a l (5) The W e g s t e i n - p r o j e c t e d MVT-accelerated R a c h f o r d - R i c e method. (6) The Wegstein-projected d i r e c t i t e r a t i o n method w i t h t e t r a h e d r a l projection. projection. employing j=l (7) The quadratic-Wegstein-projected MVT-accelerated R a c h f o r d - R i c e method. In a preliminary the study, the 5th method was double-iteration logic. disappointing. conclusion, Prausnitz regarding liquid-liquid and implemented based I t s performance was, to say on the l e a s t , co-workers (1980) have a l s o reached a the a p p l i c a t i o n of d o u b l e - i t e r a t i o n methods to e q u i l i b r i a , which i s i n consonance w i t h the above observation. In a d d i t i o n to the above seven methods, the method r e c e n t l y proposed by P r a u s n i t z and co-workers (1980). method employs the R a c h f o r d - R i c e f o r m u l a t i o n c o m p o s i t i o n by a p p l y i n g using the study i n c l u d e d and converges the the This phase Wegstein's a c c e l e r a t i o n on a l t e r n a t e i t e r a t i o n s f u n c t i o n <|> d e f i n e d by N • | L11 L11 " L 2 i L 2 i | Y X Y x < 5 _ 2 a > -189- The main steps of the a l g o r i t h m a r e : (1) Initialize x^, (2) Using ' a n d 9 L and X J ^ J s o l v e the check f u n c - the c u r r e n t v a l u e s of K^, x ^ tion iteratively f o r an updated v a l u e of 8 , u s i n g Newton's method. A f t e r every step of t h i s i t e r a t i o n , t e s t the v a l u e of 8 . JLi I f E l 9 s L > K^IR) e l f 9 - 1 t h e n S e L < K^IE) - 1 ' t 6 t h L = L 6 e n S e t + 9 0 L ' < L 5 = 9 6 L + + K J I R ) - 1> °' < L 5 6 + KjIE) - 1> 8° denotes the p r e v i o u s v a l u e of 8^ w h i l e IR and IE denote r a f f i n ate ( l i q u i d phase 1) and e x t r a c t ( l i q u i d phase 2) ' s o l v e n t s ' respectively. (3) Compute x ^ i (4) Apply Wegstein a c c e l e r a t i o n to X Q a n a < jS^2 a n d n o m ^ z e them. and x ^ i f the f o l l o w i n g c o n d i t i o n s are s a t i s f i e d : (a) The i t e r a t i o n number, n, i s g r e a t e r than 3 and there was no a c c e l e r a t i o n f o r (n-1). < l . o w h i l e n < 5. (b) 0 < 8 (c) The c u r r e n t $ i s l e s s than 0.2 and Is l e s s than i t s p r e v i o u s L value. Set n e g a t i v e mole f r a c t i o n s to zero and normalize x ^ (5) Compute (6) S t o r e the c u r r e n t value of <|> and compute a new v a l u e . and and Y. Ifi t i s an a c c e l e r a t e d i t e r a t i o n then go to Step 8; otherwise Step 7. x^* go t o -190- (7) T e s t f o r convergence. I f the outcome i s p o s i t i v e terminate the i t e r a t i o n ; (8) then otherwise go to the next s t e p . S t o r e c u r r e n t v a l u e s of the phase compositions f o r the of a c c e l e r a t i o n and go to Step 2. The a l g o r i t h m p r e s e n t e d above was adopted convergence convergence c r i t e r i o n , which criterion, purpose structured to f i t i t s a u t h o r s ' i s based on (j). In t h i s work, the f o r a l l a l g o r i t h m s , i s based on 1 and d e f i n e d i n a manner s i m i l a r to that f o r v a p o u r - l i q u i d a l g o r i t h m s . view of t h i s , convergence it the a l g o r i t h m was properties): restructured In (without undermining i t s Step 7 i s p l a c e d immediately a f t e r Step 3 and i s executed f o r a l l i t e r a t i o n s ; Step 6 then l e a d s to Step 8 i n a l l cases. 5-2-3 The NRTL Model and the Problem of M u l t i p l e Solutions The NRTL and UNIQUAC models are among the few excess-Gibbs-energy models that are capable of p r e d i c t i n g l i m i t e d m i s c i b i l i t y . i n t e n t i o n was to use the NRTL model i n t h i s work. three d i f f e r e n t original However, when i t was i n c o r p o r a t e d i n t o the a l g o r i t h m s p r e s e n t e d i n S e c t i o n 5-2-2 to system LA, The and applied s o l u t i o n s were o b t a i n e d , depending on the a l g o r i t h m employed and the i n i t i a l i z a t i o n scheme used. The f ( 9 ) IJ v e r s u s 6^ p r o f i l e - w i t h f ( l a t i o n - was we 8 L ) based on the R a c h f o r d - R i c e consequently generated and p l o t t e d formu- (see F i g u r e 5-1). What have i s a c l e a r case of m u l t i p l e s o l u t i o n s , which a few other workers ( G u f f e y and Wehe, 1972; Heidemann and Mandhane, 1973) w i t h the NRTL model and which model. have a l s o observed c o n s t i t u t e s a s e r i o u s shortcoming of the -191- 0.16 0.124 0.081 0.04 A •0.04H - 0.08H •0.12H Q10 0.20 0^30 0.40 0.50 0.60 0.70 0.80 Q90 1.00 Q!O ~0L Fig.5- 1 f (8 ) v. 6 L L for the system LA at 298 K (based on NRTL) -192- I n l i g h t of t h i s o b s e r v a t i o n , the d e c i s i o n pendulum swung i n the d i r e c t i o n of the UNIQUAC model. Prausnitz (1978) was (Rachford-Rice presented chosen. The v e r s i o n proposed by Anderson Some check f u n c t i o n p r o f i l e s type) r e s u l t i n g from a p p l i c a t i o n s of the model are i n F i g u r e s 5-2 and 5-3. While F i g u r e 5-2 was v a l u e s c o r r e c t e d f o r composition-dependence, F i g u r e 5-3 uncorrected —Li v a l u e s and presented based on involved i s thus more r e p r e s e n t a t i v e of the s e a r c h p a t h of the a l g o r i t h m s under The and investigation. c o r r e s p o n d i n g p l o t s r e s u l t i n g from u s i n g the NRTL model are i n F i g u r e s 5-4 and 5-5. Here a g a i n , we a l s o p r e d i c t s m u l t i p l e s o l u t i o n s f o r system L E . f i n d t h a t the model E x i t NRTL; e n t e r UNIQUAC. 5-2-4 I n i t i a l i z a t i o n Schemes The initialization schemes of Chapter u n s u i t a b l e here, which i s not s u r p r i s i n g , 3 were found to be c o n s i d e r i n g the h i g h l y c o n v o l u t e d nature of the p r o f i l e s d i s p l a y e d i n F i g u r e 5-3. f o c u s e s on the steps taken to overcome the problem. scheme employed by P r a u s n i t z and The This initialization co-workers, which served as c o r n e r s t o n e f o r the schemes developed here, w i l l section the r e c e i v e prime discussion. The Scheme of P r a u s n i t z and The scheme i s i n two l e a s t mutual s o l u b i l i t y having z > 0.1; (2) Co~workers parts: (1) i s determined The components, w i t h each 98% The p a i r of components w i t h from amongst the set of components p a i r i s assumed to be the only rich i n one of the two existing phases, and K —Tj computed based on t h i s assumed the composition. i s -193- Fig 5-2: f(# ) v.fusing the UNIQUAC model, with K's corrected for composition dependence L -194- -0.4-1 1 1 0.0 0.1 , 1 0.2 0.3 0.4 1 0.5 1 1 I i 0.6 0.7 0.8 0 3 1 1.0 Fig 5 - 3 : f{Q ) v. 6 , using the UNIQUAC model,wifh K's uncorrected for composition dependence L L -195- -196- -197- An outline of the scheme is as follows: (1) For every pair of components, I and J, with both z(I) and z(J) greater than 0.1: (a) Set x ( I ) = x ( J ) = 0.02 and x ^ J ) = x ( I ) = 0.98. Set u L2 L2 a l l other mole fractions to zero and compute K^. (b) Define KS = max { K ( I ) , ^—^-jy} and IS as the I or J corresL ponding to KS. (2) Set X ^ C I S ) = 0.02 and x (IS) = 0.98. Then for every component J L2 such that J * IS and z(J) > 0.1, set * ( J ) =0.98 and L l x ( J ) = 0.02. Set a l l other mole fractions to zero and compute L2 K . Define KP as the smallest of a l l the K (J)'s and define IP as —TJ LI the corresponding value of J . (3) Set IE = IS and IR = IP. IE and IR denote the extract and raffinate solvents respectively and they constitute the least soluble pair. (4) Initialize x ^ and x ^ by setting x ^ I R ) = x (IE) = 0.98, L2 x^^ClE) = x (IR) = 0.02 and a l l other mole fractions to zero. L2 (5) Compute an i n i t i a l value for K (6) Define z(IE) a = z(IE) + z(IR) based on the composition in Step 4. -198- and initialize 8 Q from Ju L Constrain The , -<* = fl ^(IR) 8° w i t h i n the - 1 (a - l ) K^IE) l i m i t s 0 < 8° £ above scheme i s q u i t e ingenious reliable convergence f o r systems LB included due the importance of e x e c u t i o n for i n i t i a l i z a t i o n time-scale, and i t yielded a stable through LH :> J; and (system LA was not that is. For speed. enjoying could seem to have been d e v i s e d with a For a large-component system c o m p o s i t i o n prominence, the be a s t r o n o m i c a l - by a time computer example, an eight-component system w i t h a c o m p o s i t i o n d i s t r i b u t i o n so c l o s e that every component has require, ' " 1.0. However, i t does not w i t h many of the c o n s t i t u e n t s required ( to u n a v a i l a b i l i t y of UNIQUAC parameters f o r some of i t s component-pairs). mind on - 1 i n the process of i n i t i a l i z a t i o n , 72 z > 0.1 will activity-coefficient evaluations! The eliminate initial initialization the above shortcoming w h i l e seeking The Set x ^ f o l l o w i n g main steps (2) and compute T Set X j ^ = JLlj,!.* Normalize x ^ (3) Compute (4) Define the not to to o b t a i n a worse point. Scheme 1: (1) schemes p r e s e n t e d below were designed are involved: = £ = T IE and L l i IR L 2 i ( i = 1,2, such that K smallest K ( I ) L /Y and IJ compute Y j ^ * , N). (IE) i s the largest K f o r a l l I such that z ( I ) > ( I ) and Li 0.1. K (IR) T L -199- (5) Initialize x and x by s e t t i n g x f I R ) = x „(IE) = 0.98, —Lz Ll L2 T Ll x ( I E ) = x ( I R ) = 0.02, and a l l other mole f r a c t i o n s to zero L 1 L 2 (6) Initialize K (7) Initialize 9 based on the compositions i n Step 5. from, Li Z(IE)-(1 0° L * ) + K (IR)-Z(IR) E) L Ji = Z(IE) + Z(IR) ^ (5-A\ * ; subject to 0 < 8° < 1.0 Equation (5-4) derives from assuming that only components IE and IR e x i s t , each c o n t r i b u t i n g Z(IE) and Z(IR) moles to the system; and that Z ( I E ) / K ( I E ) moles of IE dissolve i n the I R - r i c h phase 1 while L K^(IR) Z(IR) moles of IR dissolve i n the I E - r i c h phase 2. A p p l i c a t i o n of the above scheme to the hypothetical eightcomponent system would require only 4 a c t i v i t y - c o e f f i c i e n t evaluations the same f o r any other number of components. Scheme 2: For t h i s scheme, Steps 1 through 6 of Scheme 1 are r e t a i n e d , but the computation of 9° i n Step 7 i s based on the method of Prausnitz Li and co-workers [Equation (5-3)] Scheme 3: (4) — This scheme r e t a i n s Steps 1 through 3 of Scheme 1. Define x = N I . .« Z K T _^ ^ 1 set x Then f o r every component, I , with K ( I ) < x» Li LiX ( I ) = z ( I ) and x LiL set X L 1 Then: ( I ) = 0,0; f o r every other component, J , Lii~ ( J ) = 0 and x ( J ) = z ( J ) . L 2 -200- (5) Initialize 6° from Li 5 (6) N o N o r m a l i z e x ^ and x ^ Scheme 4 a n d compute K^. T h i s scheme i s s i m i l a r t o Scheme 3 except f o r t h e f o l l o w i n g modifications: I n Step ( 4 ) : F o r a l l I w i t h 1^(1) < x, s e t ^ ( 1 ) = z ( I ) and x ^ C l ) = K^(I)*z(I). x L 2 (J) = (J) F o r every o t h e r component, J , s e t and x ( J ) Z = z(J)/K (J). u L I n Step 5: O b t a i n 6 ° from Li N 9° = 1 L A = ^ 1 N i N ^ ^ l i 1=1 L i l + ^i L 2 i X 1 = L 2 i Scheme 5: The f i r s t 5 steps of t h i s scheme a r e i d e n t i c a l t o those o f Scheme 1 . (6) Assume L 2 = 1 ( I E ) and F = Z ( I E ) + Z ( I R ) . 2 Then s i n c e -201- it f o l l o w s that aK (IE)6: L 6° = a = By e x p r e s s i n g L [ ^ ( I E ) - l]9° + 1 L where L Z(IE) Z ( I E ) + Z(IR) the above e q u a t i o n e x p l i c i t l y f o r 0°, we have L aK ( I E ) - 1 ,o _ L L K, (IE) - 1 V > = IE Scheme 6: I t f o l l o w s an approach s i m i l a r to Scheme 5, but i t more p r e c i s e l y d e f i n e s 0° by the r e l a t i o n s h i p Li QO L _ 1 ( I E ) + 1 (IR) _2 2 Z ( I E ) + Z(IR) where Z(IE) 1 (IE) = 2 1-0° 2 1 + 0°K (IE) L and 1 (IR) = 9 - 2 1 + Z(IR) 1-0° L 0° K J I R ) A combination of the above three equations leads to the q u a d r a t i c equation [ ^ ( I E ) - l J t K j I R ) - l](0°) 2 + {aK (IE)[2 - ^ ( I R ) ] - a - 1 + a[l - ^(IE)] = 0 L -202- Writing t h i s as a(6 °) Li + b6 ° + c = 0, Li 2 T then the q u a d r a t i c both r o o t s T i s solved f o r i t s root that l i e s between 0 and 1. l i e i n s i d e t h i s range or both are o u t s i d e v a l u e i s taken and c o n s t r a i n e d w i t h i n 0 < 9°< 1. If i t , the average Also, i f by any Ju 2 chance, the d i s c r i m i n a n t to b - 4ac i s n e g a t i v e , i t i s a r b i t r a r i l y set zero. Scheme 7: This scheme implements Steps 1 through 6 of Scheme 1 and complements t h i s w i t h the ef'-determining procedure of Scheme 4. Li When any of the above requires initialization algorithms, 1 ± schemes i s a p p l i e d to an a l g o r i t h m of 1 that as i s the case w i t h the GP computed from s Z — 1-M 2 1 1 + If the i n i t i a l i z a t i o n (i = 1,2 ,N) 1 - 6° 9 Xi of 1^, and i s a l s o r e q u i r e d (the s e n s i t i v i t y method i s a case i n p o i n t ) , then: N L and 2 L = 1 1=1 hi = F - L 2 The seven schemes o u t l i n e d above, as w e l l as the method of Prausnitz and co-workers, were t e s t e d u s i n g a c c e l e r a t e d GP method. The t e s t of systems LB through LH. the t e t r a h e d r a l - p r o j e c t i o n - i n v o l v e d one temperature p o i n t I n each case, 9° and the number of Li f o r each -203- i t e r a t i o n s r e q u i r e d to reach a s o l u t i o n were r e c o r d e d . iterations, the i n i t i a l i z a t i o n step was taken as one i t e r a t i o n , i r r e s p e c t i v e of the number of each c a l c u l a t i o n i s counted presented It i n Table I n c o u n t i n g the computations i n v o l v e d . as one i t e r a t i o n . Thereafter, The r e s u l t s are 5-2. i s s i g n i f i c a n t to note gave p o s i t i v e r e s u l t s t h a t a l l the schemes except Scheme 4 i n every case. A c l o s e i n s p e c t i o n of 6° * v a l u e s f o r Scheme 4 shows that they are i n f a c t than f o r most of the other schemes. to c l o s e r to 6^ I t s f a i l u r e must t h e r e f o r e be due the method i t employs i n d e t e r m i n i n g K . This deduction i s Li supported is by the f a c t i n the It that the o n l y d i f f e r e n c e between Schemes 4 and 7 determination. i s apparent o v e r a l l best r e s u l t . from the i t e r a t i o n counts t h a t Scheme 1 g i v e s the I t was t h e r e f o r e i n c o r p o r a t e d i n t o the v a r i o u s algorithms. 5-2-5 Applications I n comparing the a l g o r i t h m s , f i v e for each of the systems LB through LH. over temperature p o i n t s were used The p o i n t s were spread the temperature ranges g i v e n i n T a b l e 5-1. P r a u s n i t z et a l was implemented i n two forms: original initialization 1'. It of one employing i n terms of computation their 'Scheme time. s h o u l d be mentioned, i n p a s s i n g , t h a t r e d u c i n g the frequency K^-computation — very The a l g o r i t h m of scheme; the other u s i n g the proposed T a b l e 5-3 c o n t a i n s the r e s u l t s , uniformly as was done f o r v a p o u r - l i q u i d systems — s l u g g i s h convergence. l e d to -205a- Leaf 204 missed in numbering Table 5-2: 8? and iteration count (I.C.) for different i n i t i a l i z a t i o n schemes System T (K) e* L 9° L I.C. 6 L I.C. 9 L I.C. 6" L I.C. 9° L I.C. 9° L Prausrtitz et £L l . Schenie 7 Schenie 6 Schenie 5 Schenie 4 Scheme 3 Scheme 2 Scheme 1 I.C. 9° L I.C. 9° L I.C. LB 298 0.4144 0.5884 8 0.6573 8 0.4000 9 0.4982 f 0.6066 8 0.6551 8 0.3869 9 0.5013 9 LC 303 0.5034 0.4136 8 0.2681 9 0.2582 8 0.4888 f 0.1925 11 0.2362 10 0.3756 9 0.2681 9 LD 328 0.2715 0.2911 8 0.2864 8 0.2350 11 0.4494 f 0.2851 9 0.2882 8 0.3589 10 0.7136 8 LE 364 0.4522 0.4114 10 0.2477 10 0.2281 10 0.4794 f 0.1726 11 0.2178 10 0.3707 10 0.2477 10 LF 365 0.4981 0.4161 9 0.2569 12 0.2328 11 0.4829 f 0.1793 11 0.2256 12 0.4015 10 0.2569 12 LG 298 0.3939 0.5621 8 0.5738 8 0.4000 7 0.4917 f 0.5514 8 0.5717 8 0.4212 7 0.4262 8 LH 333 0.4431 0.4088 7 0.3814 7 0.4000 7 0.4949 f 0.3487 8 0.3735 7 0.5392 7 0.3814 7 f = failure T a b l e 5-3: System GP with triangular projection GP with tetrahedral projection Computation times (CPD seconds) f o r l i q u i d - l i q u i d systems Sensitivity method Sensitivity with tetrahedral projection Wegstein-projection Rachford-Rice with with MVT 9 L " L 9 \\2t Prausnitz et a l Quadratic Wegstein + Rachford-Rice Original I n i - Proposed Initializatialization + MVT tion LB 0.3359 0.2696 0.5831 0.2906 0.6848 0.5511 0.7878 0.5001 0.5048 LC 0.4729 0.2819 0.8466 0.2872 0.9424 0.7250* 0.8524 0.4157 0.3952 LD 0.3753 0.3168 1.1938 0.3640 1.4237 1.1881 1.7872 0.7755 0.6906 LE 0.3486 0.3302 1.0776 0.3073 0.7985 0.8635 1.1918 0.4170 0.3969 LF 0.5129 0.3444 1.1445 0.3132 0.8465 0.9139 1.3242 0.4280 0.4010 LG 0.3435 0.2946 0.4840 0.3011 0.5309 0.5399 0.6980 0:3862 0.3015 LH 0.3197 0.2444 0.4824 0.2531 0.5225 0.5495 0.5764 0.3025 0.2845 I 2.7088 2.0819 5.8120 2.1165 5.7493 5.3310 (2.9%fallure) 7.2178 3.2250 2.9745 * faulty convergence to e = 0 at the 5th tempera ture-point. L -207- 5-2-6 (1) Deductions The v e c t o r - p r o j e c t i o n - a c c e l e r a t e d a l g o r i t h m s o u t c l a s s the o t h e r methods, the t e t r a h e d r a l - p r o j e c t i o n v e r s i o n being by f a r the b e t t e r of the (2) On the average, two. the t e t r a h e d r a l p r o j e c t i o n does slightly b e t t e r w i t h the GP method than w i t h the s e n s i t i v i t y (3) As expected, the a l g o r i t h m of P r a u s n i t z and more e f f i c i e n t w i t h the proposed the a u t h o r s ' o r i g i n a l between the two scheme. method. co-workers i s i n i t i a l i z a t i o n scheme than w i t h One should expect the disparity v e r s i o n s to e s c a l a t e as the number of components increases. 5-3 Liquid-solid The Equilibria problem of the d e a r t h of data t h a t was study of l i q u i d - l i q u i d equilibria encountered i n the assumed a much h i g h e r dimension the s e a r c h l i g h t was turned on l i q u i d - s o l i d excruciating effort that s u f f i c i e n t equilibria. data was garnered I t was ( I t should be noted that e l e c t r o l y t i c with to make p o s s i b l e the a p p l i c a t i o n of the a l g o r i t h m s i n v o l v e d i n the study to two systems. when binary systems are not c o n s i d e r e d here.) V i t a l data on the two systems are p r e s e n t e d i n T a b l e 5-4. names, by which the systems w i l l h e n c e f o r t h be addressed, i n the T a b l e . The temperature-solid-fraction profiles have a l s o been prepared 5-3-1 and are to be found i n Figure Code are i n c l u d e d f o r the systems 5-6. T h e o r e t i c a l Background The v a p o u r - l i q u i d e q u i l i b r i u m r e l a t i o n s h i p s presented i n Chapter Table 5-4: V i t a l Information on l i q u i d - s o l i d systems No. of components System pressure (Atm.) Melting point (K) SA 2 0.221 77.74 79.45 N i t r o g e n (43.17); Argon (56.83). SB 2 0.2105 89.82 93.83 N i t r o g e n (50.0); Methane (50.0). Identificat i o n code Freezing point (K) Components and % Molar Composition -209- 95.0-1 85.0H 80.OH SA 75.0-1 0,0 1 1 1 1 1 " r 0.1 0.2 Q3 Q4 0,5 0,6 0.7 0.8 0.9 1.0 Fig 5-6: Temperature-solid-fraction profile for the Solid Systems -210- 3 transform directly to g i v e the l i q u i d - s o l i d equilibrium equivalents, w i t h the f o l l o w i n g r e d e f i n i t i o n s : (1) y i s r e p l a c e d w i t h x^, (2) Subscript (3) K i s replaced is (4) given w i t h £ and V with S. ' s ' takes the p l a c e with and the l i q u i d - p h a s e mole f r a c t i o n by x^. The l i q u i d activity coefficient d i s t i n g u i s h i t from the s o l i d The of s u b s c r i p t 'v'. i s subscripted with 'L' to coefficient, Y . s p a r t i a l f u g a c i t y i s n o r m a l l y d e f i n e d by solid-phase activity OS f si = Y si i f x s i <* - L —-.*) 2 When t h i s i s combined w i t h the c o r r e s p o n d i n g l i q u i d - p h a s e r e l a t i o n s h i p , we have ,oL X K si o-l Y evaluation relevant data. f = _ s i = J±LJ_ ( ..os *Li The T-I -I of Y Y 5_ 5 ) si i f poses a s e r i o u s problem due to l a c k of As a r e s u l t , a number of a l g o r i t h m s i n v o l v i n g the s o l i d phase have been based on the assumption that a l l s o l i d - p h a s e c o n s t i t u e n t s e x i s t as s e p a r a t e pure-component e n t i t i e s example, Gautam and S e i d e r , = 1 f o r a l l i i n the s o l i d In i s o t h e r m a l 1979). This implies (see, f o r that Y „ = x . si si phase. liquid-solid equilibria i n v o l v i n g the p r e c i p i t a t i o n of more than one component, the p u r e - s o l i d assumption would be h i g h l y unrealistic, unlikely f o r i t would presuppose that e i t h e r there event of an o c c l u s i o n - f r e e c o n s e c u t i v e some m y s t e r i o u s f o r c e - f i e l d ensures that while 'unlike absolutely repels unlike'. i s the h i g h l y p r e c i p i t a t i o n , or that 'like perfectly attracts like' -211- I n t h i s work, i t has been c o n s i d e r e d more l o g i c a l to assume, i n the manner of E r b a r (1973), that the s o l i d phase i s a mixture w i t h no i n t e r a c t i o n e f f e c t s between i t s components. £ K si E q u a t i o n (5-5) then becomes oL . - 5 ! . _1LA_ x_ . *L1 I n t h i s model, ( 5 OS f_os i _ 6 ) v those components that do not 1 undergo s o l i d i f i c a t i o n assume a ' n u l l p r e s e n c e ' i n the s o l i d phase a t e q u i l i b r i u m - i n s i m i l i t u d e t o h i g h l y n o n - v o l a t i l e components i n a vapour-liquid system. The l i q u i d activity coefficients i n E q u a t i o n (5-6) are e v a l u a t e d from the W i l s o n model. 5-3-2 Choice of A l g o r i t h m s The seven a l g o r i t h m s drawn from Chapter 3 and a p p l i e d to liquid-liquid e q u i l i b r i a were c o n s i d e r e d to show the g r e a t e s t and have been implemented 5-3-3 here. I n i t i a l i z a t i o n Schemes In o r d e r to f o r m u l a t e s u i t a b l e i n i t i a l i z a t i o n liquid-solid a l g o r i t h m s , i t was schemes f o r the n e c e s s a r y to g a i n a v i s u a l knowledge of the nature of the f u n c t i o n s i n v o l v e d i n the problem s o l u t i o n . R a c h f o r d - R i c e check f u n c t i o n was have been p l o t t e d at d i f f e r e n t 5-7 promise and 5-8. entirely used f o r t h i s purpose. The Some p r o f i l e s temperatures f o r each system i n F i g u r e s While the p l o t s f o r system SA ( F i g u r e 5-7) a r e based on K v a l u e s c o r r e c t e d f o r composition-dependence, —s profiles based on u n c o r r e c t e d K v a l u e s (aimed at s i m u l a t i n g the a c t u a l —s s o l u t i o n path) are a l s o p r e s e n t e d f o r system SB ( F i g u r e 5-8). -212- 77.80 K 0.032 0.024^ 0.016 0.008A CD -0.008 H -0.016H -0.024 -0.032- i 0.1 0.2 r 0.3 0.4 0.5 0.6 0.7 +~ 6s Fig 5-7: f (# ) s v e r s u s $s f o r t n e s y s t e m S A ~\ 0.8 r Q9 1D -213- 0.030 0.024 4 •• T. = 91.59 K 0.016 0.008 4 0 if) eg •4- 4 •0.008 4 -0.016 4 -0.0244 •0.032 Legend: •f(t9) employing composition-corrected K's -0.0404 s f(6 ) employing composition-uncorrected K s S -0.048 4 0.0 0.1 0.2 Fig 5-8- f ($ ) v. 6 s 5 Q3 0.4 0.5 •eSL for system SB Q6 0.7 0-8 0.9 1.0 -214- The shapes of the curves suggest that the i n i t i a l i z a t i o n developed f o r v a p o u r - l i q u i d e q u i l i b r i a w i l l be s u i t a b l e . schemes These schemes were t h e r e f o r e a p p l i e d to the two systems w i t h a view to d e t e r m i n i n g the best scheme. I t was found that Scheme 5 (see S e c t i o n 3-2-5), which, i t w i l l be recalled, gives best h e r e . liquid one the best results for vapour-liquid However, w i t h 0° c o n s t r a i n e d i s a l s o the the way 6° was f o r vapour- systems, the v e c t o r - p r o j e c t i o n methods were each found to y i e l d f a u l t y convergence, due to o v e r - p r o j e c t i o n , liquid flash, s a t u r a t i o n l i n e of system SB. c o n s t r a i n t s on 6° were r e d e f i n e d s This f o r a p o i n t c l o s e to the f a u l t was e l i m i n a t e d when the thus: 0.2 f o r 6° < 0.02 6° = { S 5-3-4 Applications The is, 0.80 f o r 9° > 0.98 s using e x p e r i m e n t a l d e s i g n was s i m i l a r to that of Chapter 3 — temperatures a t 19 p o i n t s u n i f o r m i n t e r v a l s of 8 obtained 5-3-5 from 0.05 to 0.95. The computation times are presented i n T a b l e 5-5. O b s e r v a t i o n s and d e d u c t i o n s The results g c o r r e s p o n d i n g a p p r o x i m a t e l y to following observations and i n f e r e n c e s were made from the i n T a b l e 5-5 as w e l l as the raw computer r e s u l t s iteration information: containing that Table 5-5: System GP with triangular projection Computation times (CPU seconds) f o r l i q u i d - s o l i d systems GP w i t h tetrahedral projection Sensitivity method Sensitivity with tetrahedral projection Wegstein ]j r o j e c t i o n Rachford-Rice w i t h MVT with 9 s s = 6 £x . s £si Quadratic Wegstein + RachfordR i c e .+ MVT SA 0.3813 0.3125 0.4616 0.7071* 0.2621 0.4185** 0.2696 SB 0.5944 0.3861 0.6021 0.4613* 0.4484 0.4684** 0.4683 I 0.9757 0.6986 1.0637 1.1684 0.7105 * O s c i l l a t o r y convergence a t some p o i n t s , with g r e a t e r o s c i l l a t i o n ** I n each case, f a i l s f o r 1 point near the s o l i d saturation line. f o r SA. 0.8869 (5.3%failure) 0.7379 -216- (1) The tetrahedral-projection-accelerated s e n s i t i v i t y method exhibits o s c i l l a t o r y convergence at some points. It manifested this same behaviour for the two vapour-liquid systems with the narrowest boiling-ranges. It would seem that the problem here i s also related to the narrowness of the melting-ranges of the systems. (2) On a computation-time scale, the quadratic form of the Wegstein acceleration technique i s again s l i g h t l y worse than the l i n e a r form. (3) The tetrahedral-projection-accelerated GP method and the Wegstein- projected MVT-accelerated Rachford-Rice method give the best results. 5—4 On the average, the former is s l i g h t l y better. Conclusion The studies undertaken i n this chapter lead to the following conclusions: (1) Unless one i s dealing with a system for which the NRTL activity-model has been proven s a t i s f a c t o r y , employing the model entails the r i s k of one encountering multiple solutions, which i s just as bad as no solution at a l l . It i s safer to use the UNIQUAC model. (2) The i n i t i a l i z a t i o n scheme presented i n Section 5-2-4 and tagged 'Scheme 1' seems to be very e f f i c i e n t for handling the highly nonlinear l i q u i d - l i q u i d equilibrium problem. (3) The tetrahedral-projection-accelerated GP method has given a quite good account of i t s e l f i n the l i q u i d - l i q u i d and l i q u i d - s o l i d e q u i l i b r i a applications here. l i q u i d - l i q u i d case. This i s especially true of the CHAPTER SIX MULTIPHASE EQUILIBRIA 6-1 Introduction In t h i s chapter, e q u i l i b r i u m problem. able to handle any liquid-liquid, light i s turned A l l the a l g o r i t h m s on the m u l t i p h a s e s t u d i e d here are designed of the f o l l o w i n g phase e q u i l i b r i a : liquid-solid, s o l i d - l i q u i d - v a p o u r and The the search algorithms liquid-liquid-vapour, solid-liquid-liquid, solid-liquid-liquid-vapour. are s t r u c t u r e d i n such a way on i n f o r m a t i o n s u p p l i e d by any the user f o r a s o l u t i o n , any eliminated. And that they start (see Appendix B). by based In the process of phase whose f l o w - r a t e reduces to zero i s i t i s an important phase e l i m i n a t i o n , the search u s i n g the composition be liquid-vapour, assuming the maximum number of phases that c o u l d p o s s i b l y e x i s t , searching to f e a t u r e of the a l g o r i t h m s i s not r e s t a r t e d but simply that at continues, of the other phases at the e l i m i n a t i o n p o i n t . Methods based on the p h a s e - f r a c t i o n approach are t r e a t e d i n S e c t i o n 6-2. S e c t i o n 6-3 programming (GP) d i s c u s s e s methods d e r i v i n g from the formulation. An extension of the s e n s i t i v i t y m u l t i p h a s e systems i s the s u b j e c t matter of S e c t i o n 6-4. t r e a t s some a l t e r n a t i v e f o r m u l a t i o n s based on the dimension of the GP dimension form. S e c t i o n 6-6. while The question S e c t i o n 6-7 S e c t i o n 6-8 presents geometricmethod to Section 6-5 which employ v e c t o r p r o j e c t i o n f o r m u l a t i o n as w e l l as i n a reduced- of i n i t i a l i z a t i o n i s b r i e f l y considered c o n t a i n s v a r i o u s a p p l i c a t i o n s of the methods the c o n c l u s i o n s -217- reached. in -218- For reasons d i s c u s s e d i n S e c t i o n 1-1-2, the r e s u l t s in this chapter are presented only i n terms of i t e r a t i o n requirements. mixtures employed are drawn from Chapters names are used 3 and 5, and The the same code here. 6-1-1 Nomenclature Note: Any symbol not d e f i n e d below and not c l e a r l y d e f i n e d where i t occurs w i t h i n t h i s chapter r e t a i n s the d e f i n i t i o n of S e c t i o n 3-1-1. Symbol Definition f Partial fugacity f Check f u n c t i o n o f Fugacity coefficient ^l'^2'^3 Check f u n c t i o n s f o r p h a s e - f r a c t i o n f o r m u l a t i o n H Negative J Jacobian matrix K Equilibrium ratio 1 Liquid-phase component L Liquid-phase total s S o l i d - p h a s e component S S o l i d - p h a s e t o t a l flow x Mole f r a c t i o n of i n v e r s e of J a c o b i a n m a t r i x Phase fraction Subscripts i,j Component flow flow f o r a phase other than the vapour phase Greek Symbol 9 flow -219- L L i q u i d or l i q u i d - l i q u i d Ll L i q u i d phase 1 L2 L i q u i d phase 2 s Solid v Vapour phase 1 L i q u i d phase 1 2 Liquid - (As i n K ) : phase phase 2 Vector quantity Superscripts m Mean-value p o i n t d e f i n e d f o l l o w i n g Eq. (6-42) o Standard s t a t e ; 6-2 initial point The Phase-fraction Approach The p h a s e - f r a c t i o n f o r m u l a t i o n has been a p p l i e d by a number of workers to v a p o u r - l i q u i d - l i q u i d systems ( f o r example: Osborne, 1964; Henley and Rosen, 1969; Deam and Maddox, 1969; E r b a r , 1973; Peng and Robinson, 1976; Mauri, 1980). Most of these a l g o r i t h m s use a two-dimensional Newton-Raphson method of convergence. employ {i^, ^ } , {i^/F, I^/F}, the check v a r i a b l e s . (1969), ensures that The l a s t {V/I^ , and {v/F, They v a r i o u s l y ^ / ( L ^ ) } as form, which i s due to Henley and Rosen the check v a r i a b l e s have c l e a r l y - d e f i n e d l i m i t s of 0 and 1. I n the more g e n e r a l development f o r s o l i d - l i q u i d - l i q u i d - v a p o u r e q u i l i b r i a presented below, the approach of Henley and Rosen has been employed, w i t h one important the e q u i l i b r i u m r a t i o s by departure: the above-named authors defined -220- l i K and In K = y = 2 i i / y ; L x l i /x 2 i t h i s work, the second ; r a t i o has been r e p l a c e d by T h i s m o d i f i c a t i o n i n t r o d u c e s two 2±^ l±' K x advantages: (1) The working (2) I t makes f o r easy a p p l i c a t i o n to a two-phase l i q u i d - l i q u i d 6-2-1 equations assume a s i m p l e r form. system. The Problem F o r m u l a t i o n The total-mass balance e q u a t i o n i s g i v e n by F = V + L 1 + L 2 + S (6-1) F o r the component-mass balance we have Z i • V h± + hi + i s ( 6 E x p r e s s i n g E q u a t i o n (6-2) In terms of t o t a l ~ 2 ) flows and mole f r a c t i o n s , we have Fz = V ± + L x 7 l u + L x 2 2 1 + Sx s l (6-3) Define e = -J- (6-4) e =^ (6-5) v L and 8 = s . p S V . ^ Combining E q u a t i o n s V = F6 L and 2 = F 8 (6-6) (6-4) through (6-7) v L (6-6), we o b t a i n ( 1 " V ( 6 _ 8 ) S = F6 (1 - 9 ) ( 1 - 8 ) s v L t (6-9) -221- S u b s t i t u t i n g Equations (6-7) through (6-9) i n t o E q u a t i o n (6-1) l e a d s to L. - F ( l - 6 ) ( 1 - 9 )(1 - 9 ) 1 v L s Let y X and i (6-10) the e q u i l i b r i u m r a t i o s be d e f i n e d by = vi li K 2i < - > x 6 " ^1*11 ( 6 x . = K . si si l i When E q u a t i o n ~ U 1 2 > (6-13) (6-1) and E q u a t i o n s (6-7) through (6-13) a r e combined, we have - z,/[e x. li i K. + e (i - e v vi L v )K . L i + e (i - e )(i - e )K s L v si + (1 - 8 ) ( 1 - 9 ) ( 1 - 6 ) ] S (6-14) V Li . Now, d e f i n e the check f u n c t i o n s N f,(8 ,6,8 ) = 1 v L s I (x - y ) l i i (6-15) (x - x ) l i 2i (6-16) (x - x ) l i si (6-17) N f,(9 ,6 ,9 ) = 2 v L s and f,(9 , 8 6 ) j v L s Equations N £ (6-11) through (6-17) c o n s t i t u t e the working The main steps (1) I Information to e x i s t of a g e n e r a l equations. algorithm are: i s s u p p l i e d as to which of the f o u r phases a r e expected (where there i s doubt, the maximum p o s s i b l e number of phases i s assumed). (2) 9 , 9 and 9 a r e a p p r o p r i a t e l y i n i t i a l i z e d , v L s T f r a c t i o n s are i n i t i a l i z e d exist. and mole f o r phases that have been assumed t o -222(3) Appropriate Equations (4) The K values (6-7) are computed and through (6-14) are the r e l e v a n t ones amongst solved. r e l e v a n t check f u n c t i o n s are computed and new values r e l e v a n t phase f r a c t i o n s are determined through an acceleration Any (6) Convergence i s t e s t e d f o r . phase whose phase f r a c t i o n i s n o n p o s i t i v e The should Step 3 w i t h that a double-loop i n v o l v i n g an i t e r a t i v e implementation of the o b j e c t i v e of c o r r e c t i n g f o r the a l s o s t u d i e d and was composition-dependence abandoned because i t performed poorly. S o l v i n g by a Newton-Raphson Approach In applying the Newton-Raphson a c c e l e r a t i o n method to converge the p h a s e - f r a c t i o n f o r m u l a t i o n of the m u l t i p h a s e problem, the d e r i v a t i v e s of the check f u n c t i o n s w i t h f r a c t i o n s are e v a l u a t e d For any and i n the c o n t r o l i s t r a n s f e r r e d to be mentioned, f o r the r e c o r d s , of the K v a l u e s , was 6-2-2 i s eliminated. 3. v e r s i o n of the above a l g o r i t h m , very appropriate i t e r a t i o n i s terminated event of a p o s i t i v e outcome; otherwise, It the technique. (5) Step of what we respect partial to the r e l e v a n t phase analytically. two-phase system, only one p a r t i a l derivative i s involved have i s the Newton method. For a three-phase system, four d e r i v a t i v e s are r e q u i r e d and i n v e r s e of the r e s u l t i n g 2 x 2 the f a c t square m a t r i x i s obtained that '11 '21 l 12 22 22 -a 21 a -a 12 '11 a — a a 11 22 12 21 directly, the using -223- A four-phase system r e q u i r e s a 3 x 3 J a c o b i a n m a t r i x . The m a t r i x i n v e r s i o n i s performed through a Gauss-Jordan e l i m i n a t i o n method (La Fara, 1973). With the i n v e r s e of the J a c o b i a n m a t r i x thus determined, the p h a s e - f r a c t i o n v e c t o r , 9_, i s updated from 6 = 6_ - J ( f / ^ ) . f _ ( ^ ) ... n+1 _ 1 n n (6-18) n where 6 i s a v e c t o r w i t h the dimension of 1,2 the system has 2,3 6-2-3 or 3 depending on whether or A phases. Employing a Quasi-Newton Approach The quasi-Newton method was a p p l i e d to the m u l t i p h a s e problem i n order to c a p i t a l i z e on the two advantages that i t has over the Newton method: i t s avoidance, a f t e r the f i r s t i t e r a t i o n , of the e v a l u a t i o n of p a r t i a l d e r i v a t i v e s ; and i t s e l i m i n a t i o n of the m a t r i x - i n v e r s i o n step of the Newton method. As a f u r t h e r improvement, the s t e p - l i m i t e d u n i d i m e n s i o n a l s e a r c h on t , as proposed by Broyden (1965), i s employed e — t being d e f i n e d by = e + t H f(e ) n + 1 n N n (6-19) n In Equation (6-19), H° = - J ( f / e ° ) _ 1 For a two-phase system, H 'quasi-Newton' N (n > 0) i s determined from the method of S e c t i o n 7-8, the a p p r o p r i a t e r e c u r r e n c e formula being H n = H (n-D / ( 1 _ f n / f (n-l) } F o r a system c o n t a i n i n g more than two phases, the r e c u r r e n c e formula of -224- Broyden (1965) I s employed, thus r (n-l) n_ (n-l) R H = H N f - N _ 1 ~ ( H + (n-l) (n-l) (n-1)i H f ~ ~ (n-l) (n-l) T (n-l) f ) H (n-l) (n-l)^(n-l) f ~ n _ ( J ( f n - 1 f ~ )) ~ T I n p e r f o r m i n g the u n i d i m e n s i o n a l s e a r c h on t , f_ i_ i s used as the objective (discussed and f u n c t i o n to be minimized. i n some d e t a i l i n S e c t i o n A q u a d r a t i c - f i t method 9-7-1) i s employed i n the s e a r c h , the s e a r c h i s t e r m i n a t e d as soon as a t i s o b t a i n e d t h a t g i v e s an improved value of the o b j e c t i v e 6-2-4 function. P a r t i t i o n i n g Method w i t h MVT and T e t r a h e d r a l This Is an attempt at d i r e c t l y Projection e x t e n d i n g the W e g s t e i n - p r o j e c t e d M V T - a c c e l e r a t e d u n i v a r i a t e method f o r two-phase systems to the m u l t i p h a s e problem, the only p h i l o s o p h i c a l d i f f e r e n c e being that the t e t r a h e d r a l p r o j e c t i o n method has r e p l a c e d the t r i a n g u l a r p r o j e c t i o n method, which i s the m u l t i v a r i a t e e q u i v a l e n t The method i n v o l v e s t h a t matches 6 ^ 6L a n through a s u c c e s s i v e d 9 g with Equations (6-17) r e s p e c t i v e l y . relevant updating substitution (6-15), (6-16) and F o r each s u c c e s s i v e - s u b s t i t u t i o n s t e p , the 9 i s updated by means of the mean-value-theorem t e c h n i q u e . new v a l u e o f the 6_ v e c t o r projection. The Geometric Programming (GP) Formulation For a geometric-programming f o r m u l a t i o n problem, the f o l l o w i n g g. s p e c i a l symbols w i l l be of the m u l t i p h a s e introduced: = a thermodynamic a t t r i b u t e ( s e e d e f i n i t i o n below) of component i i n phase p. A h a v i n g been o b t a i n e d , a f u r t h e r improvement of the _9_ v a l u e s i s sought through t e t r a h e d r a l 6-3 of the Wegstein method. -225- M = t o t a l number of phases i n the system. = number of components i n phase p w ^p = number of moles of component i i n phase p. = t o t a l number of moles i n phase p p = s u b s c r i p t denoting All other strange definitions For symbols 'phase'. that f e a t u r e i n t h i s s e c t i o n r e t a i n the of S e c t i o n 3-3-4. the M-phase system, G (as d e f i n e d i n S e c t i o n 3-3-4) i s g i v e n by g = N N ° i X * p ~ ^ * ~!oV w p=l [ + n c i=l (6 f " 2o) Define C and g = - — exp{-y° /RT} ip (6-21) 1 f W = -iE-2. . ip (6-22) j.O ip i p ^ip n a s D e e defined n such that i t c o n t a i n s we s u b s t i t u t e E q u a t i o n s no molar q u a n t i t i e s . If (6-21) and (6-22) i n t o Equation(6-20) and we define C o = w o = 1 w and the v(w) = ( C / W ) °exp{-G/RT}, O q result i s N w M p v(w) = (C /w ) ° H { n (C. /w p=l i = l p p w ) l p W W } p p (6-23) -226- I f we f u r t h e r d e f i n e a . = -Z ., oj J 1 for i = j a. . = { i-JP 0 for i * j and then a mass balance f o r component M a w ° 2 ° p N I a i=l I p=l + i .w j p i (6-24) = 0 ( j = 1,2,...,N) p A comparison of Equations and j yields (6-23) and (6-24) w i t h E q u a t i o n s (C-6) (C-12) (Appendix C - l ) shows that the above equations c o n s t i t u t e the d u a l program f o r a problem w i t h M c o n s t r a i n t s and w i t h n = 1, so o that £ w iejfo] The p o s i t i v e l y w ^p» = w = 1 ° c o n d i t i o n s are, of course, a u t o m a t i c a l l y b i n d i n g s i n c e being a molar q u a n t i t y , i s always p o s i t i v e . The a p p l i c a t i o n of E q u a t i o n s (6-23) and (6-24) t o l i q u i d - l i q u i d - v a p o u r e q u i l i b r i a i s presented i n S e c t i o n 6-3-1 below w h i l e i t s e x t e n s i o n to the s o l i d - l i q u i d - l i q u i d - v a p o u r case i s o u t l i n e d i n S e c t i o n 6-3-2. 6-3-1 Liquid-liquid-vapour Equilibria For an N-component l i q u i d - l i q u i d - v a p o u r components system where a l l are s o l u b l e i n a l l t h r e e phases, E q u a t i o n (6-23) becomes: _ *>_»±i >_2 - n Ii L l O li C i _ o_ w ]L l w n i=l L il2, 2i L^iJ V "JrT C L iv v i _ I V l 2 L L 1L 1 L 2 V (6-25) V S i m i l a r l y , E q u a t i o n (6-24) becomes " j Z + + X 2j + V j = 0 ( j = 1.2.-...N) (6-26) -227- Following the procedure employed i n S e c t i o n 3-3-4, we have -z N and to o b t a i n the o r t h o g o n a l transformation, we f o l l o w the s t e p s : -Z -Z Interchange rows 1 and ( N + l ) and p a r t i t i o n as shown Extract the n e g a t i v e transpose of lower segment Append a (2N + 1) i d e n t i t y below the l a s t matrix row Interchange rows 1 and (N+l) 1 ~h ~h ^(2N+1) 1> U) 1 -22.8- { t / j ) } i s the d e s i r e d t r a n s f o r m a t i o n . I t i s a (3N + 1) by (2N + 1) m a t r i x whose j t h column i s g i v e n by _ b ^ \ b The v e c t o r i s d e f i n e d by ( o ) ^(o) = [ 1 while b ^ , Z , • •. ,Zjj,0,0,... ,0 ] , 2 ( j = 1,2 2N) i s the column v e c t o r { b ^ } such that f o r ( j = l , 2 , . . . , N ) , ^ -1 f o r i = j+1 = 1 f o r i = N+j+1 _ and 0 f o r a l l other i ' s f o r ( j = N+l, N+2,...,2N), f _ -1 f o r i = j-N+1 1 f o r i = N+j+1 L_ 0 f o r a l l other i ' s The v e c t o r w i s d e f i n e d by ( w.= b ^ i 1 where, > 2 0 ; ( -\ N I r.b) ,, ]i 1=1 + i n this case, W = [W 1- , 1,1 — o L 1, _ V . . . . V 1,N 1 N T Substituting for b ^ 1 l i V 1 i Z = r ,2N) 1_ . .... 1_ • 2,1 2,N (6-27) T J ( j = 0,1,...,2N) i n E q u a t i o n J (6-27) we have i " N+i r r ( i = 1,2,...N) i (6-28) = r N+i 2i Equilibrium H^ i " (1=0,1, 3) 1 requires (Il^»I 2 ) = 0 ( J " 1.2.....2N) (6-29) -229- Combining E q u a t i o n s (6-25), (6-28) and (6-29) and s i m p l i f y i n g the r e s u l t , we have: for ( j = 1,2 N), C. - v. L v ITT 1 jL <«-*» 1j and f o r ( j = N+l, N+2,...,2N), jLl 1j 2 From the d e f i n i t i o n s of the C s , i t can r e a d i l y be shown that C C a n d C jv „ JLl jLl Y f 3L1 * j _JL2 j Y m L jLl jLl 1 Y = V f K .= Lj Equations (6-30) and (6-31) become L J T 1 ( J = 1 1j 21 l L l » 2 » ' - - » N ) < 2 L " 3 2 > (6-33) j (6-32) and (6-33) a r e the e q u i l i b r i u m relationships. they are combined w i t h E q u a t i o n (6-28) and expressed variables 6 L X and m jL2 jL2 v. v i ^ f Hence, E q u a t i o n s K = P v_ and 1^ i a n When i n terms of the manner s i m i l a r to the v a p o u r - l i q u i d case, the r e s u l t i n g equations are ( f o r i = 1,2,...,N): ? N (1 - K , ) v , - [ F vi i ., L I 1- . - K , ( Z , - 1„.) - (1 - K ,)A . ]v. 2i vi i 2i vi v i i J + K v i ( Z i - V ^ i = ° ( 6 ~ 3 4 ) -230- and (1 - K L i )l 2 2 v. - K . - [F - L ± (Z - (1 - ± V Li X *Li< ± ~ + - v ) 1 Z = K L i )X L i ]l ° 2 1 ( " 6 3 5 ) where N I X . = ^ X L i = Equations v Ij * i h 2J (6-34) and (6-35) c o n s t i t u t e solved simultaneously f o r 6-3-2 Solid-liquid-liquid-vapour the 2N equations that have to be and 1^• Equilibria The development i s s i m i l a r to that equilibria. The d e t a i l s have t h e r e f o r e of the f i n a l r e s u l t s i s presented. The 3N equations that (for ( 1 been omitted and only r e s u l t from a p p l y i n g a summary the t r a n s f o r m a t i o n a r e i = 1,2,... ,N): _ K ) v 2 . [ F _ j ( 1 J + "" L i ^ l ' f " \ F j _ + J=l (1 of l i q u i d - l i q u i d - v a p o u r ( V K ( _ ^ z _ } _ ( 1 . K v ± )X ]v J v i K j + S ( i " hi Z j ) - i S "^ i ) X vi = °> " i- i V S ) < ' ) 6 " " 3 6 hl^hl 1=1 + *L1< 1 " Z V i" i S ) X Li = ° ( 6 " 3 7 ) -231- and (1 - K 2 ) s^ N -[F - ^ v s i i (v. 1 .) + 2 - K 8 l (Z ± s i V i - hiKi + K ( V - v - l ± 2 i ) - (1 - " ° K s l )X s i ] S i <" > 6 38 where N * - I s. J*i J I n s o l v i n g the above equations, on the the f o l l o w i n g bounds are variables 0 < v 0 < l < Z i < (Z 2 °< 1< Finally, 1 ± ± - (6-39) v ) ± < l- i-l2i> 8 Z v _J i s computed from (6-40) l ^ Z - v - ^ - ^ 6-3-3 S o l u t i o n Methods A general basic (1) imposed algorithm f o r handling the GP problem has the following structure: A number of phases i s assumed, based on i n f o r m a t i o n user, and (2) The (3) K values storage relevant i s appropriately component flows are are computed and s u p p l i e d by allocated. initialized. the r e l e v a n t component flows are updated through a s u c c e s s i v e - s u b s t i t u t i o n approach a p p l i e d to E q u a t i o n (6-36) through (6-38). within I n the update, the the bounds d e f i n e d the flows are by E q u a t i o n (6-39). constrained -232- (4) Convergence i s t e s t e d f o r . is (5) execution terminated; otherwise Step 5 i s implemented. I f any phase composition reduces to zero ( w i t h i n some t o l e r a n c e ) , that phase i s (6) I f the outcome i s p o s i t i v e , Acceleration eliminated. i s performed on v^, through an a p p r o p r i a t e 1 and method and _1 s_> as applicable, i s updated from E q a u t i o n (6-40). (7) Control The i s t r a n s f e r r e d to Step two v e c t o r - p r o j e c t i o n methods developed i n S e c t i o n 3-3-8 employed i n Step 6. defined 3. For the by w = {y_, 1^ > s}» than f o u r phases. The an & general four-phase system, the v e c t o r i t i s appropriately a p p l i c a t i o n of the to E q u a t i o n s (6-36) through (6-38) c o u l d (1) Phase-by-phase Arrangement. in the This reduced f o r less successive-substitution take one involves of two is method forms: u p d a t i n g the variables order [v ( j = 1,2,...N), 1 (2) are ( j = 1,2 Component-by-component Arrangement N), s ( j = 1,2,...,N)] I n t h i s case, the updating order i s [v , 1 The two To applied 2 , s ( j = 1,2....,N)]. arrangements are test the i d e n t i c a l f o r any r e l a t i v e performance of the two-phase system. two structures, they were to four l i q u i d - l i q u i d - v a p o u r systems, w i t h i n i t i a l i z a t i o n on Scheme 1 of S e c t i o n employed. The 6-6. The t e t r a h e d r a l p r o j e c t i o n method r e s u l t s , presented i n T a b l e 6-1, performance depends on based on Arrangement 1. the system being handled. r e v e a l t h a t the based was relative Subsequent s t u d i e s are Table 6-1 I t e r a t i o n c o u n t s o b t a i n e d from a p p l y i n g t h e two GP to l i q u i d - l i q u i d - v a p o u r systems arrangements Component-bycomponent arrangement System Code Temp (K) Phase-by-phase arrangement LB 335 .50 16 13 LC 364.60 66 52 LE 364 .0 21 28 LF 365 .0 22 37 -234- The Equation next step taken was to t e s t the e f f e c t of the c o n s t r a i n t s (6-39) on the convergence p r o p e r t i e s of the GP a l g o r i t h m . f o l l o w i n g f o u r c o n s t r a i n i n g methods were Method 1: No Method 2: The c o n s t r a i n t s are values values obtained The studied: applied of the v a r i a b l e s r e s u l t i n g from a c c e l e r a t i o n step are c o n s t r a i n e d of according the to E q u a t i o n (6-39). from the s u c c e s s i v e - s u b s t i t u t i o n step are The not constrained. Method 3: The values of the v a r i a b l e s obtained s u c c e s s i v e - s u b s t i t u t i o n step as w e l l as a c c e l e r a t i o n step are c o n s t r a i n e d Method 4: v a r i a b l e that The l i m i t i n g value scheme and defined 6-4 systems. and to E q u a t i o n (6-39). the e x a c t l y the any value constraints. systems: 5 vapour-liquid, 4 liquid-liquid-vapour. As i t turned out, the same the same number of i t e r a t i o n s f o r each of not applying 5 The the a c c e l e r a t i o n method employed are T h i s outcome favours four the the c o n s t r a i n t s at a l l . The S e n s i t i v i t y Approach While the vapour-liquid liquid-solid — by structure test e a r l i e r reported. methods r e q u i r e d 16 according the of i t s bounds i s set to i t s p r e v i o u s liquid-liquid, 2 liquid-solid, as f o r the those r e s u l t i n g from f o u r methods were a p p l i e d to 16 initialization the T h i s method i s s i m i l a r to Method 2 except that s t r a y s out i n s t e a d of the from s e n s i t i v i t y method performed q u i t e w e l l w i t h systems, i t was systems. As f a r less s u c c e s s f u l with l i q u i d - l i q u i d to how i t will perform w i t h multiphase w e l l , i t i s only by e a t i n g the pudding that one taste. To be able to r e s t assured that n o t h i n g has and systems gets to know i t s t r u e been l e f t untested -235- which ought to have been t e s t e d , a m u l t i p h a s e v e r s i o n of the sensitivity method has been developed here, i n d e f i a n c e of the cumbersome a l g e b r a involved. The f o l l o w i n g three s u b s e c t i o n s p r e s e n t , i n a n u t s h e l l , development the of the working r e l a t i o n s h i p s f o r a g e n e r a l two-phase system, a g e n e r a l three-phase system, and a four-phase system r e s p e c t i v e l y . 6-4-1 A G e n e r a l Two-phase Problem The development combinations: i n t h i s s e c t i o n covers the f o l l o w i n g vapour-liquid, l i q u i d - l i q u i d two-phase and l i q u i d - s o l i d . The method has a l r e a d y been a p p l i e d to these phase-combinations i n Chapters 3 and 5, based on the c o n t e n t s of Chapter 4. here i n a form that v i v i d l y I t i s merely summarized emphasizes i t s l o g i c a l e x t e n s i o n to m u l t i p h a s e systems. For second the convenience, l e t us use vapour-phase symbols to denote the which c o u l d be a vapour, l i q u i d or s o l i d phase — phase normal l i q u i d - p h a s e symbols to denote the base l i q u i d F o l l o w i n g from Chapter 4, we and phase. have ,m. n . . ,,n J Av = A£nK —v (6-41) J (6-42) where with 1 for j = i 6 { * i ; 0 for j Av n v n+l n - v -236- and n y Jtn(—). n i A £ n K , = JlnK^ vi vi n X m i s g i v e n by v m = v n , , n+1 n. + a(v - v ) m m 1^ , V w i t h 0 < a < 1. in and L are s i m i l a r l y I f we s u b s t i t u t e E q u a t i o n manipulate the r e s u l t i n a way defined. (6-42) i n t o E q u a t i o n similar (6-41) and to what was done i n S e c t i o n 4-2-1, the outcome i s Av - n j , .AV vlj (j + e _. v2j n e V J = 1,2,...,N) ' ' ' (6-43) where e . . = —j- ( vlj d. m + v 2 and 1 J d. J A a. =— v ), m^' v L ^nK ,, r n VJ ^ x m j By summing E q u a t i o n m j (6-43) over a l l j and r e a r r a n g i n g the r e s u l t , we have AV n = (6-44) where N F v - ~ I . , J=l e . . v2j N and D = I • i J-1 e - 1 vlj J -237- A s u b s t i t u t i o n of E q u a t i o n (6-44) i n t o E q u a t i o n (6-43) leads to the v a l u e of Av 6-4-2 and hence of 11 n+ The Three-phase Problem The r e l a t i o n s h i p s the v ^, developed here are capable of h a n d l i n g any of f o l l o w i n g three-phase combinations: v a p o u r - l i q u i d - l i q u i d , liquid-solid and l i q u i d - l i q u i d - s o l i d . here by the normal l i q u i d The base l i q u i d vapour- phase i s denoted symbols w h i l e the other two phases a r e , f o r convenience, denoted by the symbols f o r vapour and s o l i d phases respectively. F o l l o w i n g the same approach as f o r two-phase systems, we JAr = AJtnK 11 have (6-45) n where A_r and = {Av^As } n 11 A£nK = {A£nK ,A£nK } —v n n n E x p r e s s i n g £nK as a f u n c t i o n of _r, we have fcnK . = Jim* - £nV + £n(F - V - S) - An(Z, - v vi i 1 i j S j ) _ i i ^ (6-46) and £ n K = s £ i n s i " £ n S + £ n ( F ~ V - S) - Zn(Z ± - v ± - s ) ± . The J a c o b i a n , J , i s determined by d i f f e r e n t i a t i n g E q u a t i o n with respect the to _r. i n t o E q u a t i o n (6-45) and r e s u l t s i m p l i f i e d , the outcome i s + JL)Av v When t h i s i s s u b s t i t u t e d (6-46) m 3 x m 3 n j + -Lm As" - ( ^ - + J - ) ± i 2 v * AV — n ^ AS n = AAnK . n * (6-47) vj L and I l Av m J n + (— s m + —) l As m 2 n - — L AV m n - (^ S m + -1 L m ) AS n = AJlnK . n S J (6-48) -238- I f Equations Av n 3 (6-47) and (6-48) are s o l v e d s i m u l t a n e o u s l y f o r and As*!, the r e s u l t i s 3 i v % j and e , .AV vlj + e n + e „ ., v3j' n 0 (6-49) + e , .AS* + e - , s2j s3j A s = e , .AV j slj n .AS v2j n (6-50) 1 where e e .. « vlj v2 j d. j = ( i- m \.m V 1. L i- + 3 _L.], l_r_i m m L s. J T 3 3 L.1 d. m m j L s. „m, m ' S 1 . L J T 3 J - J T [ ( — + ^ - ) A £ n K . - ^ - AJlnK .], e n 3 e ) + m s. IT 1 slj d. m m szj d . _,m ,m j S 1. L L n v v 1 3 m i m + 3 m v. m m' L v. J T 3 3 , = ! _ [ ( ! - + !-)AiLnK , - — A £ n K . l , d S s3j j 1 J l m VJ n n m v and d. = — — j mm + — — + ,m m m, m V.s . 1.s . v . l . j 3 3 3 3 3 J By summing E q u a t i o n s 1 (6-49) and (6-50) r e s p e c t i v e l y for a l l j ( j = 1,2,..,N) and s o l v i n g the r e s u l t s s i m u l t a n e o u s l y AS , we have n j=l J j-1 J j=l J j=l J f o r AV n and -239- M» - and where N »l< U H l v3i N D = ( £ e j = 1 N N N U ) - lU^ll^-n] (6-52) sli N N N - 1)( £ e - 1) - ( £ e )( £ e vlj s2j slj v2j j = 1 j = 1 j = ) 1 A combination of E q u a t i o n s (6-49) through (6-52) g i v e s the adjustments i n the phase molar q u a n t i t i e s f o r the n t h i t e r a t i o n . 6-4-3 The Four-phase Problem For the g e n e r a l v a p o u r - l i q u i d - l i q u i d - s o l i d system, the phase symbols are e x a c t l y the same as those employed i n S e c t i o n s 6-2 and 6-3. The development f o l l o w s the same path as f o r the three-phase system, except that the process i s f a r more i n v o l v e d here. the final results Only a summary of i s presented. The changes i n the phase component flows are g i v e n by Av = e . .AV + e „ . A L " + e - .AS + e , . vlj v2j 2 v3j v4j n n j n Al" = e .AV + e .AL? + e , . A s " + e . . 2j Llj L2j 2 L3j L4j T (6-54) As^ = e s 4 j (6-55) n T 1 and (6-53) T 0 T .AV + e ^ .AL* + e n s l g 3 .A n S + e where e vl • " T~[ — J e v2j L j + —(C " M 1 L _m T •3 ^ m V . + C, . + C .) 1, 0 J = kj - l l - J I d. t J ' 3 1 2 ..j-[.si-fa. ], v3j d. m m J ' J 2 J -240- 6j L V 1_ d. r _ 5 j _ _3_J L J 1 d [ ( C T m m L 5j d. J 2_ d. J 1 TT l + L T 3j T, 2j sj J ' 5j 3j 2 3j C m L V L + r T vj l J 1 m S- C j m \ L i_ ly 3j C lj> T 1 V i_ [ _ H j L m m J L C l j M n K s j " C 3j *<j]» A ' J _ _ll m l!j " 1 ,.m V x M n K 1 ' 2 1 + i_ S (C m + . + C l J J c_,)], J — [(C + C + C )AJlnK - C A£nK - C A £ n K ], dj *j l j 2j sj l j Lj 2j vj n -^-(C. . + C. . + C m l j *J j g m,m ' v .1. . J l j m m ' l j 2j 0 .) + - ^ J - , i m l j n n -241- ,m m' 1, .s . '31 41 m,m '51 ' mm' v .s . J J and 6i J J l m . s . 2j J ,1 0 The simultaneous s o l u t i o n of E q u a t i o n s (6-53) through (6-55) f o r ,„n .,n , ,„n _ AV , A L and AS leads to 2 F AV AL n n = (6-56) =^, (6-57) F and AS - jp-, n (6-58) where F v " t E L3 E s2 " + F L ( E t v3 L2 " E (E " tELl(Es3 " + L2 - [ ( E v l" X 1 ) E ) 1 ) ( E s3 " + t v2 E ( E 8 3 " ^ " ^ 2 ^ 4 " v2 L3^ s4' 1 } E E E - L3 sJ v4 E L3 - E E E + K v3EJ s4> E E 3 sl " ( E vl " 1 ) ( E S 3 " - 2 4 2 - F s " [ L2 " (E t + E 1 ) 2 E E E ^ v l " + 2 " < E v s l" L l s ] v 4 E L 1 " E vi D(\2 " 1 ) ] E 1 ) E s2 " E V 2 E S 1 ^4 s4> and ° " t L2 " ( E 1 ) ( E [ Ll s2 " + E E s3 " ( E L2 " L3 s2^ vl " X ) E 1 ) E - E E X ) + [ L3 sl " L l E E E ( E Kl s3 " l) sJ v3 E In the above d e f i n i t i o n s , the E's are g i v e n by N E = pn j I = e 1 pnj where p = v, L or s, and n = 1,2,3, or Equations 6-4-4 (6-53) through The General 4. (6-58) c o n s t i t u t e the working Algorithm In the implementation r e l a t i o n s h i p s of S e c t i o n 6-4-1 algorithm. The equations. of the s e n s i t i v i t y method, the through 6-4-3 are b u i l t c l a s s of r e l a t i o n s h i p s a p p l i e d to any into a general system at any stage of the i t e r a t i v e s o l u t i o n then depends on how presumably e x i s t . a l g o r i t h m i s , as i n p r e v i o u s a p p l i c a t i o n s of the The many phases s e n s i t i v i t y method, based on the mean-value-theorem (MVT) the MVT parameter, a, i s taken to be The (1) and 0.5. g e n e r a l a l g o r i t h m c o n s i s t s of the f o l l o w i n g main steps Based on i n f o r m a t i o n s u p p l i e d as to the phases that c o u l d p o s s i b l y exist, initialize a l l r e l e v a n t e q u i l i b r i u m r a t i o s , mole f r a c t i o n s , phase component flows and phase t o t a l (2) technique, Apply the equations flows. of S e c t i o n 6-4-1, 6-4-2 or 6-4-3, -243- depending on whether there a r e 2, 3 or 4 phases. the q u a n t i t i e s s u p e r s c r i p t e d with the e q u a t i o n s w i t h I n every case, o b t a i n 'm' i n the equations by f i r s t solving 'm' r e p l a c e d by 'n' and then i n c r e m e n t i n g the c u r r e n t v a l u e s of the unknown molar flows by h a l f the r e s u l t i n g A v a l u e s . (3) E l i m i n a t e any phase whose t o t a l flow i s w i t h i n a t o l e r a n c e of z e r o . (4) T e s t f o r convergence. the i t e r a t i o n ; In the event component — I f the outcome i s p o s i t i v e , then terminate o t h e r w i s e go to Step 2. of an e x i s t i n g phase not c o n t a i n i n g a p a r t i c u l a r a very r e a l p o s s i b i l i t y w i t h the s o l i d phase — , flow i n q u e s t i o n i s s e t to a v a l u e w i t h i n a t o l e r a n c e of z e r o . the phase The method does not admit of zero flow v a l u e s as t h i s would make some of the terms i n the working equations 6-5 Some A l t e r n a t i v e indeterminate. Formulations Two a l t e r n a t i v e f o r m u l a t i o n s are i n t r o d u c e d here. designed and such They a r e that they c o u l d employ 1^ as the independent thus peg the dimension variable of the problem a t N, i r r e s p e c t i v e of the number of phases i n v o l v e d . Alternative 1 The GP a l g o r i t h m of S e c t i o n 6-3-3 a p p l i e s , except t h a t the s u c c e s s i v e - s u b s t i t u t i o n s t e p takes the f o l l o w i n g form ( a p p l i e d t o i from 1 to N): Step 1 Compute 1 (6-59) si v i -244- hi 3nd \ihi = i Si !!' S = 1 where A and For any Step 2 vi = vi V K L 2 Li \i * A = SK si / L K si l' / L l /L, 1 phase that does not e x i s t , A i s set to Update the t o t a l phase-flows based on zero. the updated flows f o r component i . Alternative It 2 a l s o d i f f e r s from the GP successive-substitution introducing By of S e c t i o n rearranging 6-3-3 E q u a t i o n (6-59) in the and the d e f i n i t i o n s . = X X X and we step. algorithm X s h i ' = V - v L l " L = L 2 v " ± > h i ' = S - s , i would have hi\ X + hi + K vi ( X v - Z (X ± Substituting the + V + + 1) 1± K Li ( X L + 1 2i> + K si ( X s + ±» 8 = 0 relationships for A l t e r n a t i v e 1 i n t o E q u a t i o n (6-60) and (6-60) ^ a n d u rearranging, n d e we r have -245- t 1 + K vi vi A ^ A i + + K 1^ K si s X Z si si^li A ± \ ± - X l X + Z i + \ i \ + \ i \ + • ° ( 6 ' 6 1 ) i s updated by a p p l y i n g a Newton i t e r a t i o n step to E q u a t i o n (6-61). The r e s u l t i n g 1 when employed i n updating does not y i e l d v ± 2 ± nd s r e l a t i o n s h i p s under A l t e r n a t i v e 1. employ according ± a component-mass balance to the To ensure component-mass balance, we the r e l a t i o n s h i p s V i X 2i = \iV = V i ' and s. = A . 8 ., I si I where 3 ± = (Z ± - + \ ± + A^) E a c h of the two a l t e r n a t i v e methods Is implemented In two forms: Form 1: The t e t r a h e d r a l - p r o j e c t i o n method i s employed, u s i n g the same v a r i a b l e v e c t o r as i n S e c t i o n 6-3-3. Form 2: Here, the reduced-dimension form e a r l i e r mentioned i s implemented. T h i s i n v o l v e s a p p l y i n g t e t r a h e d r a l p r o j e c t i o n t o the v e c t o r JL . For e i t h e r a l t e r n a t i v e , the r e s u l t i n g 1± i n updating the other component-mass flows r e l a t i o n s h i p s under 6-6 from the r e l e v a n t ' A l t e r n a t i v e 2' above. Initialization Schemes Three i n i t i a l i z a t i o n Scheme 1: (1) i s employed schemes were s t u d i e d . T h i s scheme i n v o l v e s the f o l l o w i n g s t e p s : Set the phase f r a c t i o n s f o r phases that cannot e x i s t Initialize the other p h a s e - f r a c t i o n s according to z e r o . to the adopted -246- initialization schemes f o r the a p p r o p r i a t e two-phase systems each case assuming (2) the e x i s t e n c e of only two phases. U t i l i z i n g E q u a t i o n s (6-7) through (6-10), compute appropriate): 3 =_! = _e ) i-e )(i-e ), (1 a s ( V + L = — — v F L a = L L v = 6 + 3 , v + L e(i-e) + g, -2-_-l = L v S + L and o — - = 9 (1-9 ) ( l - 9 ) + B F S L V S (3) Compute (where V i appropriate): " V i ^ 1+ ( 1 - v Q ) / e v vi^ K i° = ^ / { i + d - e ) / 9 K } , ± L L Li s° = a Z , / { l + (1-9 )/9 K } i s i s s s i 1 and Scheme 2: 1° -h - (vj l| + ± + J .J) The f o l l o w i n g steps a r e i n v o l v e d : (1) Same as Step 1 o f Scheme 1. (2) Compute 3 as i n Scheme 1 and ^I V* 1 = + A v i + Si Si> + where A -9 K vi ./e, vvi Si • - V L L i ( 1 9 K from / 0 (where -247- and (3) A = (1-9 si ) ( 1 - 9 )6 K /8 L s si t Compute (where a p p r o p r i a t e ) : 0 _ V Scheme 3: (1) v i o " o _ v i h i ' 2± o o _ \± li L ' L T h i s method e n t a i l s the f o l l o w i n g s t e p s : Then compute v ° = . / { i + (i-e°)/e°K } z 1 i v 1 v vi Determine 9° as i n l i q u i d - s o l i d s (Z - v°) as the system m i x t u r e . •j • ' i z (3) J equilibria ^y^j + Determine 9° as i n l i q u i d - l i q u i d Li Compute 1^ by component-mass equilibria using Then compute balance. Where t o t a l phase flows and initialized, using Then compute (Z^ - v ° - s°) as the system m i x t u r e . any o sihi Determine 9° as i n v a p o u r - l i q u i d e q u i l i b r i a u s i n g Z_ as the system mixture. (2) i~ s phase mole f r a c t i o n s need to be they are computed from the component flows o b t a i n e d from of the schemes above. The three i n i t i a l i z a t i o n two-phase systems. schemes g i v e i d e n t i c a l r e s u l t s f o r To compare t h e i r performances w i t h multiphase systems, they were a p p l i e d to the f o u r l i q u i d - l i q u i d - v a p o u r systems f o r which i n f o r m a t i o n i s a v a i l a b l e . limited application. The Table 6-2 r e s u l t s favour contains the r e s u l t s of the c h o i c e of Scheme 1. this Table 6-2 I t e r a t i o n counts f o r d i f f e r e n t i n i t i a l i z a t i o n schemes, applied to liquid-liquid-vapour systems System Code Temp (K) Scheme 1 Scheme 2 Scheme 3 LB 335.50 16 21 21 LC 364.60 66 f 71 LE 364.0 21 30 22 LF 365.0 22 30 44 f = failure -249- 6-7 Applications For the purpose of a p p l y i n g the v a r i o u s methods d i s c u s s e d i n the p r e c e d i n g p a r t s of t h i s chapter, the f o l l o w i n g a l g o r i t h m s have been implemented: (1) The p h a s e - f r a c t i o n method with Newton-Raphson a c c e l e r a t i o n . (2) The p h a s e - f r a c t i o n method w i t h quasi-Newton a c c e l e r a t i o n . (3) The p a r t i t i o n e d p h a s e - f r a c t i o n method w i t h MVT tetrahedral projection. (4) The GP method w i t h t r i a n g u l a r (5) The GP method w i t h t e t r a h e d r a l p r o j e c t i o n . (6) The s e n s i t i v i t y method. (7) A l t e r n a t i v e F o r m u l a t i o n 1 w i t h reduced (8) A l t e r n a t i v e F o r m u l a t i o n 1 with GP (9) A l t e r n a t i v e F o r m u l a t i o n 2 with reduced (10) r e d u c i n g to zero w h i l e is the base l i q u i d dimension. dimension. each of the i s g r e a t e r than z e r o . phase. liquid of I n such an event, phase 2 i s r e d e f i n e d as Should reduce liquid to zero w h i l e a l s o equal to zero, e x e c u t i o n i s a u t o m a t i c a l l y t e r m i n a t e d . The restricted result dimension. i s made f o r the p o s s i b i l i t y phase 1 i s e l i m i n a t e d and phase 1, dimension. should be mentioned that i n implementing a l g o r i t h m s above, allowance liquid projection. A l t e r n a t i v e F o r m u l a t i o n 2 with GP It and scope of the a p p l i c a t i o n of the a l g o r i t h m s has been h i g h l y by the l a c k of i n f o r m a t i o n on multiphase t h a t only the l i q u i d - l i q u i d - v a p o u r multiphase systems, with combination the has been s o l v e d f o r f o u r systems. The a p p l i c a t i o n s are i n two forms. One form i n v o l v e s assuming the r i g h t number of phases known to e x i s t at e q u i l i b r i u m . The other -250- form i s based on assuming more phases to be present Assuming the R i g h t Phase T h i s form was Combination a p p l i e d to 16 systems: liquid-liquid, 2 liquid-solid 5 vapour-liquid, 5 and 4 l i q u i d - l i q u i d - v a p o u r . i n f o r m a t i o n on the systems and the r e s u l t s obtained of the 10 a l g o r i t h m s 6-3. (1) vapour-liquid (2) Appropriate from a p p l y i n g to them, at one p o i n t each, are presented The r e s u l t s l e a d to the f o l l o w i n g The quasi-Newton than a c t u a l l y e x i s t . i n Table observations: p h a s e - f r a c t i o n a l g o r i t h m i s u n r e l i a b l e except f o r systems. F o r two-phase systems, the ' A l t e r n a t i v e f o r m u l a t i o n s ' performances each that are comparable show to those of the 'GP algorithms' that are based on the same convergence method. (3) The reduced-dimension forms of the ' A l t e r n a t i v e f o r m u l a t i o n s ' very (4) give s l u g g i s h convergence f o r m u l t i p h a s e systems. F o r m u l t i p h a s e systems, the ' P a r t i t i o n e d p h a s e - f r a c t i o n and the ' S e n s i t i v i t y a l g o r i t h m ' T h i s must be the e f f e c t algorithms run i n t o convergence algorithm' problems. of n o n i d e a l i t i e s , f o r when these two as w e l l as the 'Newton-Raphson p h a s e - f r a c t i o n and the 'GP a l g o r i t h m s ' were t e s t e d w i t h l i q u i d - l i q u i d - v a p o u r problem which was the sample algorithm' quarternary s o l v e d by Osborne (1964) and by Deam and Maddox (1969) — the problem i s based on constant values — converged s p e e d i l y to the r i g h t a l l the a l g o r i t h m s K solution. (5) I n the m u l t i p h a s e a p p l i c a t i o n s , none of the a l g o r i t h m s all Assuming excels f o r systems. Redundant Phases Ten systems were i n v o l v e d i n t h i s a p p l i c a t i o n . Table 6-4 Table 6—3 Iteration counts for the various multiphase algorithms, based on assumption of the right number of phases GP algoi-ithms Phase--fraction algor thms System Code ro Temp (K) Sensitivity algorithm Phases NewtonRaphson Quasi-Newton Partitioned Triangular Projection Alternt itives with re•duced dimeiis ion Form 1 Tetrahedral Projection Form 2 Alternt itives witt l GP dimeiision Form 1 Form 2 VD 356.0 LV 7 5 7 6 6 7 7 7 7 7 VH 329.5 LV 9 6 9 9 10 9 8 7 8 7 VJ 353.0 LV 5 5 5 6 7 5 7 7 7 7 VN 515.5 LV 5 5 4 2 3 4 4 4 4 4 VP 470.0 LV 4 4 3 3 5 3 4 5 4 5 LB 298.0 LL 21 fc 22 9 7 22 7 8 7 8 LC 303.0 LL 33 13 32 13 7 32 10 8 10 8 LD 328.0 LL 54 f 54 8 10 53 9 11 9 11 LE 330.0 LL 36 fc 35 14 8 35 9 8 9 8 LF 330.0 LL 38 fc 38 19 9 38 11 9 11 9 SA 78.7 SL 5 6 6 7 5 6 5 6 5 6 SB 92.66 SL 14 fc 14 13 :6 14 8 7 8 7 LB 335.50 LLV 16 f f 17 16 f NC NC 13 21 LC 364.60 LLV 42 f f 58 66 f NC NC 44 35 LE 364.0 LLV 36 f 42 38 21 f NC NC 29 33 LF 365.0 LLV 41 fc f 38 22 f NC NC 36 32 f = failure; fc faulty convergence; NC = convergence not attained i n the set iteration limit of 100 : -252- c o n t a i n s i n f o r m a t i o n on: (1) The system code-name and (2) The expected and assumed phases 'solid , 1 ' l i q u i d ' and temperature. i n the system 'vapour' phases The (4) The average i n c r e a s e i n number of i t e r a t i o n s — denote outcome of the i t e r a t i o n . applications liquid-liquid r e l a t i v e to based on the r i g h t phase combination — systems w i t h l i q u i d - l i q u i d - v a p o u r assuming assumed, and phases and l i q u i d - l i q u i d - v a p o u r assumed. systems s o l i d - l i q u i d - l i q u i d - v a p o u r have not been s i m i l a r l y a n a l y z e d due The for phases l i q u i d - v a p o u r systems w i t h s o l i d - l i q u i d - v a p o u r (The r e s u l t s f o r l i q u i d - l i q u i d (1) 'L', 'V respectively). (3) for ('S', to the h i g h f a i l u r e following rate). o b s e r v a t i o n s d e r i v e from the r e s u l t s : The quasi-Newton and s e n s i t i v i t y methods almost always f a i l when the wrong phase combination i s assumed. (2) A l l the a l g o r i t h m s except those mentioned i n O b s e r v a t i o n 1 perform q u i t e w e l l when a l i q u i d - l i q u i d - v a p o u r phase combination i s assumed for l i q u i d - l i q u i d systems. Some of the a l g o r i t h m s — triangular-projection-accelerated GP algorithm — n o t a b l y , the actually require l e s s i t e r a t i o n s w i t h t h i s wrong phase combination. (3) The a l g o r i t h m s commended under O b s e r v a t i o n 2 a l s o show good convergence behaviour when a s o l i d - l i q u i d - v a p o u r assumed f o r l i q u i d - v a p o u r systems. as good as when l i q u i d - l i q u i d - v a p o u r combination i s However, the performance i s assumed f o r l i q u i d - l i q u i d systems. (4) i s not For cases where s o l i d - l i q u i d - l i q u i d - v a p o u r i s assumed f o r Table 6—4 Iteration counts for the various multiphase algorithms, based on assumption of redundant phases Phase--fraction algor Lthms System Code Temp (K) Expected phases GP algoirithms Sensitivity algorithm As sumed phases Newt onRaphson Quasi-Newton Partitioned Triangular Projection Tetrahedral Projection Alternj itives with r«•duced dimeris ion Form 1 Form 2 Alternj itives W i t t l GP dimeiision Form 1 Form 2 LB 298.0 LL LLV 28 fc 23 9 8 fc 8 9 8 9 LC 303.0 LL LLV 33 fc 32 10 9 fc 9 8 9 8 LF 330.0 LL LLV 38 fc 38 14 10 fc 10 10 10 10 VD 356.0 LV SLV 9 8 8 9 7 fc 13 10 8 9 VJ 353.0 LV SLV 8 fc 6 9 9 11 7 8 8 8 VN 515.5 LV SLV 10 fc 5 6 7 fc 6 6 5 5 LB 298.0 LL SLLV 31 f fc fc fc fc f f 11 fc LF 330.0 LL SLLV 42 fc 39 16 11 fc 18 f 11 11 LB 335.5 LLV SLLV fc f fc fc fc fc fc fc 26 28 LF 365.0 LLV SLLV fc f fc 47 41 fc NC NC 36 35 Average iteration increase for assuming LLV for LL systems 2.3 - 0.3 -2.7 1.3 - -0.3 0.7 -0.3 0.7 Average iteration increase for assuming SLV for LV systems 3.3 - 1.0 3.3 2.3 - 2.7 2.0 1.0 1.3 f - failure; fc = faulty convergence NC = convergence not attained i n the set iteration limit of 100 ' -254- either liquid-liquid 'Alternative 6—8 Conclusions (1) For formulation liquid-liquid that solid place, or l i q u i d - l i q u i d - v a p o u r 1 w i t h GP systems where i t i s known with a b s o l u t e approach whereby the a f t e r a phase e l i m i n a t i o n (2) Where the and f o r m a t i o n of two a solid formulation one. As 1 w i t h GP where v a p o r i z a t i o n could take restarted alright. l i q u i d phases as w e l l as a vapour phase the algorithm based on dimension' appears to be generally certainty s o l u t i o n procedure i s not seems to be phase i s p o s s i b l e , f o r the only dimension' emerges unscathed. f o r m a t i o n cannot occur but the systems, the only 'Alternative reliable poor performance of the a l g o r i t h m s when an o v e r s p e c i f i c a t i o n of a s o l i d - l i q u i d - l i q u i d - v a p o u r combination i s used, i t i s p o s s i b l e that the poor s t a t e of the r e l a t i o n s h i p s employed c o n s t i t u t e s , (3) Everything be that solid-phase as i t were, an A c h i l l e s ' c o n s i d e r e d , the best m u l t i p h a s e f l a s h a l g o r i t h m based on 'Alternative formulation 1 w i t h GP heel. seems to dimension'. CHAPTER SEVEN BUBBLE- AND DEW-POINT CALCULATION. 7-1 Introduction The study of bubble- and dew-point treated c a l c u l a t i o n undertaken and i n t h i s chapter embraces s i x d i f f e r e n t methods: (1) A regula-falsi i n t e r p o l a t i o n method. (2) A q u a d r a t i c i n t e r p o l a t i o n method (3) A dynamic Lagrange (4) Newton's method (5) The (6) A quasi-Newton i n t e r p o l a t i o n technique. t h i r d - o r d e r Richmond approach. approach. Of the s i x methods only two — the Newton and Richmond methods a r e known to have h i t h e r t o been s u b j e c t e d ( J e l i n e k and Hlavacek, 1971; Sobolev et a l , 1975; c o n c l u s i o n s from which are undermined on the i m p l i c i t to any comparative studies Ketchum, 1978), the by the f a c t t h a t they were founded assumption of i d e a l i t y . While the r e g u l a - f a l s i method has p r e v i o u s l y been d i s c u s s e d by H o l l a n d (1963) as one way the problem, — of s o l v i n g i t has never been compared a g a i n s t other methods. The q u a d r a t i c i n t e r p o l a t i o n method, though a f a i r l y well-known mathematical technique f o r s o l v i n g u n i v a r i a t e problems, Is here making i t s debut i n the arena of s a t u r a t i o n - p o i n t c a l c u l a t i o n . The o t h e r two methods are completely new The developments introduced here. study a l s o i n c l u d e s an i n v e s t i g a t i o n of the p o s s i b i l i t y of h a n d l i n g the s a t u r a t i o n - p o i n t problem through a s i n g l e - l o o p i t e r a t i o n opposed to the c o n v e n t i o n a l double-loop approach. -255- as -256- The mixtures However, the system p r e s s u r e s p r e s e n t e d p r e s s u r e i s now 7-1-1 i n Table 3-1. there do not apply here s i n c e a variable. Nomenclature Note: it employed are the same as those l i s t e d Any symbol not d e f i n e d below and not c l e a r l y d e f i n e d where o c c u r s w i t h i n t h i s chapter r e t a i n s the d e f i n i t i o n of S e c t i o n 3-1-1. Symbols Definition f Fugacity f Check f u n c t i o n f° Standard H Enthalpy n Iteration r,t Parameters d e f i n e d i n Eq. (7-34) x L i q u i d mole x,y General v a r i a b l e s used y Vapour mole state fugacity count fraction i n Lagrange i n t e r p o l a t i o n fraction Greek Symbols 6 Symbol denoting a s m a l l change T V a r i a b l e d e f i n e d i n Eq. (7-24) <|> Fugacity coefficient Subscripts b Boiling-point bp Bubble-point dp Dew-point i Component value formula -257- i,j,l Summation and n Normal v Vaporization v Vapour product parameters Superscripts a Average 7-2 Theoretical Background The given b a s i c check f u n c t i o n f o r the bubble p o i n t of a mixture i s by N I f(T) = i=l The corresponding K z 1 - 1 (7-1) 1 equation f o r the dew-point i s N I f(T) = z /K - 1 (7-2) i=l For the b u b b l e - p o i n t c a l c u l a t i o n , x i s set equal i y Vi' = = 1,2 (i N) ' dew-point problem i n v o l v e s s e t t i n g y_ = z_ and X The loop y^ computed j from The to z^ and i = Z i updating / K i ( ± = 1 (7-1) of K at any or ( 7 - 2 ) . from . »--'» )2 N temperature i n v o l v e s an i n n e r to e l i m i n a t e the e f f e c t of c o m p o s i t i o n . from E q u a t i o n obtaining x The The outer iteration loop updates T only d i f f e r e n c e between the various methods i s i n the temperature update. As c o r r e c t l y pointed out by Sobolev and co-workers (1975) and Ketchum (1978), a more l i n e a r r e l a t i o n s h i p i s o b t a i n e d check f u n c t i o n s as: by d e f i n i n g by the -258- 1 N f(^) for = *n[ K z ] (7-3) x bubble-point determination, and l f(^) for I 1-1 N = M I z./K.] i=l (7-4) 1 the dew-point case. The preference of ^ over T as the check variable is born out by Figures 7-1 and 7-2 where the logarithmic check functions have been plotted with respect to both T and }~ for system VA at standard atmospheric pressure. Equations (7-3) and (7-4) have aptly been used in this work with ^ the a s check variable. The main steps of a general double-iteration algorithm are: (1) Initialize T and set x = y_ = z_ (2) Compute K. y For = K i i Z ( i = 1 » »-". )2 N dew-point calculation, update x from X At i For bubble-point calculation, update y_ from i = Z i / K i ( ± = 1 » »"-» ) 2 N the current T value, iteratively update the appropriate components of K for bubble point; only for noncondensable components — and f° — the latter for dew point) until i t is constant within some tolerance. (3) Update the temperature and test i t for convergence. If the outcome is positive, terminate the iteration; otherwise go to Step 2. -259- -260- -261- A m o d i f i c a t i o n of t h e above a l g o r i t h m which was aimed a t e l i m i n a t i n g the i n n e r i t e r a t i o n i s d i s c u s s e d i n S e c t i o n Brief subsequent 7-3 7-10-2. t r e a t m e n t s o f the s i x methods a r e p r e s e n t e d i n t h e sections. The Regula-falsi Interpolation Method A p p l i c a t i o n of t h e r e g u l a - f a l s i o r f a l s e - p o s i t i o n method t o e i t h e r of E q u a t i o n s ( 7 - 3 ) and ( 7 - 4 ) y i e l d s an updated v a l u e of temperature g i v e n by n T = f T n n-1 - f (7-5) n n ~ n-1 T I n t h e event of a n e g a t i v e convergence (T n + 1 max ,f n + 1 {(f 1 1 ) test, f n + ^ i s determined and i s s u b s t i t u t e d f o r the point corresponding to - !,!^}. 1 I n a p p l y i n g t h e method, two i n i t i a l p o i n t s a r e r e q u i r e d . methods f o r o b t a i n i n g these were i n v e s t i g a t e d . T ° and T^" a r e r e s p e c t i v e l y I n the f i r s t Two method, s e t e q u a l t o T, . and T, , bmin bmax t h e computed b o i l i n g p o i n t s f o r t h e most and t h e l e a s t volatile components, determined by the method t o be d i s c u s s e d i n S e c t i o n 7 - 9 . The second method chooses T ° as w i l l be d i s c u s s e d i n S e c t i o n 7 - 9 , and o b t a i n s through a Newton's a c c e l e r a t i o n step as s u b s e q u e n t l y d i s c u s s e d under "Newton's m e t h o d " . The second method was found t o be the b e t t e r of the t w o . 7-4 The Quadratic Interpolation Method The q u a d r a t i c i n t e r p o l a t i o n method, which r e q u i r e s 3 p o i n t s f o r its i m p l e m e n t a t i o n , i s of a h i g h e r o r d e r than t h e r e g u l a - f a l s i method. A p p l i e d to Equation ( 7 - 3 ) or ( 7 - 4 ) , given three points (T^ } f ^ -262(T , f ) and (T , f „ ) » a new temperature value results from 1 T (7-6) + where and After a new f is evaluated based on the updated T, and i n the event of a negative convergence-test, the new point ( T , f ) is substituted for the point corresponding to max The {|f 1, 1 |f |> 2 f f3| } - f i r s t three temperature points are obtained as for the r e g u l a - f a l s i method. 7—5 The Dynamic Lagrange Interpolation Method This method has been devised as an extension of the polynomial i n t e r p o l a t i o n technique, of which r e g u l a - f a l s i interpolation and quadratic interpolation are f i r s t - and second-order respectively. iterations method. versions It was conceived i n the hope that i t w i l l require fewer to reach a solution than does the quadratic-interpolation It i s dynamic i n the sense that the order of the polynomial depends on the i t e r a t i o n number, no points being rejected as the i t e r a t i o n progresses. Thus, i t i s equivalent to a r e g u l a - f a l s i method after the f i r s t i t e r a t i o n and to a quadratic interpolation after the second. The well-known Lagrange interpolation method for f i t t i n g experimental data (Jenson and J e f f r e y s , compact form 1963) can be expressed i n the -263- n n ... n 2, r y. i (x. - x ) 1 n+1 ^ V - where x i s the v a r i a b l e whose v a l u e i s known a t the p o i n t 'n+1', y i s the v a r i a b l e whose v a l u e i s unknown at the p o i n t 'n+1' and s u b s c r i p t s 1 through n denote the n p o i n t s f o r which full information i s available. Now, setting x i f we a r e seeking the v a l u e of y a t which x = 0, then n + 1 = 0 i n Equation s e t t i n g x = f and y = n The we have n —rr = n f T k-1 j j I J=l — n ( iteration i s initiated temperature Let (7-7), doing some rearrangement and f i _ (7-8) f 3 ) as i n the r e g u l a - f a l s i method and the f i r s t u p d a t i n g i s c a r r i e d out s i m i l a r l y . us d e f i n e n k k-1 n (7-9) and 4 n = n (f - f ) i * j 1 n J A f t e r the n t h i t e r a t i o n , q u a n t i t i e s are merely J only ( | > ( j = 1,2,...n) i s calculated f u l l y . The other updated a c c o r d i n g to the e q u a t i o n s : -264- • n 1 • .(f - f ) n and * n = n 1 n—1 f (j = J 1,2,...,n-l) (7-10) n Then T i s updated a c c o r d i n g t o the e q u a t i o n 1 n+l T n n m 7-6 (7-11) . . 1=1 T f d> . nj L J J J Newton's Method I n a p p l y i n g the p o p u l a r Newton's method to bubble- and calculations, dew-point two p o s s i b l e computation paths p r e s e n t themselves: one i n v o l v i n g an a n a l y t i c a l d e t e r m i n a t i o n of the d e r i v a t i v e s of K w i t h r e s p e c t to temperature; the other u t i l i z i n g i n d e t e r m i n i n g the d e r i v a t i v e s . a finite-difference approach L e t us l o o k i n t o the former case w i t h a view to a s s e s s i n g i t s f e a s i b i l i t y . From the b a s i c d e f i n i t i o n of the e q u i l i b r i u m _oL| 9*nf 9T P,x,y Differentiating ° K +• |P,x,y i ~3f L Bin*, "5T" P,x,y ~5T~ ratio, P,x,y ] (7-12) the W i l s o n a c t i v i t y model w i t h r e s p e c t to temperature leads to N 9 Any, 9T -1_ P,x,y RT N 2 I \~ J-1 + x A . J i j . \ ~N { k = 1 J-1 J k j N *k \ i \ A A J X 3-1 N 1 (7-13) -265- where 3,. ij = A A similar d i f f e r e n t i a t i o n of the W i l s o n m o d i f i c a t i o n of the R e d l i c h - . - X i j i i Kwong e q u a t i o n combining 3*n<|> —~=-1 ) yields o l |r,x,y ( t o determine Equations (2-28) and 4^ [Equations (2-23) and the equations programming e f f o r t , computer s t o r a g e requirement The + ST) - ± " time — K (T) ± &T where ST i s a p e r t u r b a t i o n i n T. 0.001K (Employing computation method uses K (T 8f-|p,x,y and i n terms of approach. finite-difference 9K defining (2-22)] shows that e v a l u a t i n g the d e r i v a t i v e s n u m e r i c a l l y i s f a r more advantageous — than u s i n g an a n a l y t i c a l as (2-29). A comparison of the d e r i v a t i v e s w i t h and the same r e s u l t ST = 0.01K was ( In t h i s work, ST was found not to a f f e c t 7 _ 1 4 ) set equal to the iteration requirement). Applying bubble-point T n + 1 the Newton method to E q u a t i o n = (7-15) 1 + I n 1 = Z± K? 1 N Z i=l i 3K. i 8T T the purpose of programming, i t was Equation yields, for determination, f For (7-3) (7-15) i n the form n c o n s i d e r e d neater to employ -266- N N I z K (T + 6T) - Zn I i=l i=l ^ £n 3f where 1 = 1 z K.(T) 1 , 1 , (7-17) 7 1 7 For the dew-point p r o b l e m , E q u a t i o n (7-16) i s a l s o a p p l i c a b l e , w i t h the derivative term d e f i n e d by N 9 F * N L ± =1 z^ VT I N «T) - 3T 7-7 J =1 z^^ yr) ST . ( 7 " 1 8 ) A Third-order Richmond Approach The main h i n d e r a n c e to the a p p l i c a t i o n of the t h i r d - o r d e r Richmond method i n the r i g o r o u s s o l u t i o n of the b u b b l e - and dew-point problems l i e s i n i t s r e q u i r i n g the e v a l u a t i o n of the second of K w i t h r e s p e c t to T, which cannot be r e a d i l y done by the finite-difference approach. ( I t would i n v o l v e t h r e e K derivative evaluations.) I n the i m p l e m e n t a t i o n of the method, t h i s problem was overcome assuming, o n l y f o r the purpose of e v a l u a t i n g the second d e r i v a t i v e s , by a model of the form AnK. = a , + b . / T I i x where a i and ^ i a r e (7-19) f u n c t i o n s of c o m p o s i t i o n but not of F i g u r e 7-3 shows a comparison of the K v a l u e s temperature. obtained from E q u a t i o n (7-19) w i t h those o b t a i n e d from a r i g o r o u s c a l c u l a t i o n system VD. The r e s u l t s are based on a c o n s t a n t parameters, a^ and b^, were e v a l u a t e d c o m p o s i t i o n and the at a base temperature The p e r c e n t a g e a b s o l u t e d e v i a t i o n i s d e f i n e d by v < % Absolute deviation = IOOIK. «. - K. I i,exact x,approx. • —— i,exact 1 for L of 370 K. -268- The p l o t shows t h a t the % a b s o l u t e d e v i a t i o n tends to i n c r e a s e p a r a - b o l i c a l l y w i t h the a b s o l u t e d i f f e r e n c e between the base temperature ( t h e temperature f o r the c u r r e n t i t e r a t i o n ) and the p r o j e c t e d (the temperature could r e s u l t is less f o r the next iteration). temperature Thus, w h i l e an e r r o r of 40% f o r a AT of +100K, the c o r r e s p o n d i n g e r r o r f o r a AT of +50K than 7.5%, and f o r a AT of +25K, i t i s only about 1.5%. I t f o l l o w s t h a t p r o v i d e d the temperature t i o n i s not very l a r g e , b a s i n g the second w i l l not r e s u l t i n very s i g n i f i c a n t change from i t e r a t i o n t o i t e r a d e r i v a t i v e on E q u a t i o n errors that w i l l (7-19) tend t o r e t a r d convergence. I f we d i f f e r e n t i a t e E q u a t i o n (7—19) twice w i t h r e s p e c t t o T and c a r r y out some a l g e b r a i c m a n i p u l a t i o n , we come out w i t h (7-20) 3T The 3T K. 3T l temperature-updating relationship i s ( f o r both bubble and dew points) ,n T 1 - (7-21) n D where D -f ff 2f' and For b u b b l e - p o i n t d e t e r m i n a t i o n , f " i s g i v e n by -269- For the dew-point case, , NN x x, d3 K, K _ 2 TI ^ {^[a^/ar] L_L_J: L_ 4 f I --|} li f . ( f . + 2 T ) (7-23) x 1-1 7-8 2 1 A Quasi-Newton Approach A p r e l i m i n a r y comparison of the f i v e methods r e v e a l e d that the Newton and Richmond methods iterations discussed above r e q u i r e the same number of i n most cases, and t h a t , i n a good number of cases, each of these two methods, w h i l e consuming more time than the three non-gradient methods, r e q u i r e d one i t e r a t i o n l e s s than t h e s e . I n view of t h i s o b s e r v a t i o n , and c o n s i d e r i n g t h a t a s u b s t a n t i a l amount of time i s d r a i n e d by the d e r i v a t i v e e v a l u a t i o n , i t was might r e s u l t derivative At 1 = two 2 + bx + c (7-24) T" f(x ) = f f(x ) = f n n + 1 f(T ) n and of the check f u n c t i o n p r o f i l e between any the end of the ( n + l ) t h i t e r a t i o n , we have the f o l l o w i n g I f we l e t us i s a q u a d r a t i c of the form f ( T ) = ax where method evaluation. the purpose of d e v e l o p i n g a quasi-Newton method, assume that the segment iterations that a 'best' from a quasi-Newton approach that would do away w i t h the finite-difference For felt information: (7-25) n n + = (f') 1 (7-26) n (7-27) s u b s t i t u t e E q u a t i o n s (7-25) through (7-27) i n t o E q u a t i o n s o l v e f o r a and b, we have (7-24) -270- a = ( . n - n+1. , „ .. n „ n+1 ><f> ^ , n+1 n 2 (T - T ) T + n v (7-28) f N and - f(T b = ^ n + 1 ) - 2 (T ) 1(f') +2T (f f > 1^ > ^ ^ i ( T n+1 - T n. )2 n V 2 n N + 1 From E q u a t i o n n n ) L n + 1 L N : (7-29) ' v (7-24), we have (f ) n + Combining E q u a t i o n s n T - T = 2ax 1 + b n + 1 (7-30) (7-28) through n+1 we have as a f i n a l f = (7-30) and u s i n g the r e l a t i o n n (f ^ , result n+1 (f') n + 1 = (1 - — )(f') f Thus, w i t h f n e v a l u a t e d f o r n = 1, s u c c e s s i v e v a l u e s of f e s t i m a t e d from E q u a t i o n Equation can be (7-31). (7-31) i s exact f o r q u a d r a t i c p o l y n o m i a l s . f u n c t i o n s , one would expect f For other a m u l t i p l i c a t i v e f a c t o r d i f f e r e n t from 2 n+l a t t a c h e d to the r a t i o f relationship (7-31) n n . I f f o r , example, one were to assume a of the form f(T) = ax + j + c a similar derivation (7-32) to the one above would lead to a g r a d i e n t r e c u r r e n c e formula of the form +1 (f') n + i n+l = [ l - (1 + - ! — ) i _ - ] ( f - ) x f Thus we have a v a r i a b l e n f factor n that i s g r e a t e r than or l e s s than 2 (7-33) -271- depending on whether the solution i s being approached from a higher value of T ( T > x ^") or from a lower one (x n+ N 11 < x ^"). n+ I t must be observed that as the solution i s approached, the factor tends to 2. For example, applying the method to the problem £(t)-20T !220T 1 0 o f with T i n i t i a l i z e d to 100, yields the solution x = 10 i n 5 Iterations with (1 + x ^ x * ) values of 17.750, 1.660, 1.908 1 44 and 1.997 in that order. Following from the above observations, l e t us, for general purposes, express the recurrence formula i n the form (f') = (1 - t r ) ( f ) n + 1 n n (7-34) where r n = .n+l f f n and t i s a parameter whose magnitude would depend on the nature of the function, f. To test the general a p p l i c a b i l i t y of Equation (7-34), t was determined as a function of r from n+l t = - (U (f) ] ( n for polynomials of degree varying from 2 to 6. i n i t i a l point was 7_35) In each case, the chosen very far from the real solutions. versus r i s shown i n Figure 7-4 A plot of t r for the different polynomials. While the second degree polynomial yields a straight l i n e of slope 2, i n agreement with Equation (7-31), the other polynomials y i e l d curves -272- Fig 7-4:tr versus r for polynomials of different degrees -273- ( d i v e r g e n t p o i n t s are not p l o t t e d ) w i t h s l o p e s ( v a l u e s of t , that i s ) that v a r y from 1.875 <_ t _< 2.0 f o r the t h i r d - d e g r e e p o l y n o m i a l to 1.552 _< t _< 2.0 f o r the s i x t h degree p o l y n o m i a l . every case that as the s o l u t i o n i s approached approaches I t was observed i n ( r •*• 0 ) , the value of t 2.0. Now, we t u r n a t t e n t i o n to the bubble- and dew-point problems. V a l u e s of r and t were generated f o r systems VA through VE and systems VL through VP by a p p l y i n g Newton's method i n the d e t e r m i n a t i o n of t h e i r bubble and dew p o i n t s . every i t e r a t i o n a f t e r For each system, the f i r s t r and t were generated f o r a t 20 d i f f e r e n t system c o n d i t i o n s ( p r e s s u r e i s the v a r i a b l e ) . A n a l y s i s of the data r e s u l t i n g than 95% of the p o i n t s y i e l d e d from the above r e v e a l e d that more t v a l u e s l y i n g between 1.5 and 2.4, w i t h more than 70% l y i n g i n the narrower range 1.9 _< t _< 2.1. The p o i n t s , c o n s t i t u t i n g l e s s than 5%, that l a y o u t s i d e the range 1»5 <_ t <_ 2.4 were observed to r e s u l t the Newton method. from an i n i t i a l An attempt o s c i l l a t o r y or d i v e r g e n t tendency i n to c o r r e l a t e f as a g e n e r a l f u n c t i o n of r for these anomalous p o i n t s d i d not meet w i t h much success, f o r , w h i l e for n a r r o w i n g - b o i l i n g systems these p o i n t s - and they only - corresponded demarcation In to e i t h e r r < - 0.1 o r r > 0.02, no such f i n e l i n e of c o u l d be drawn f o r w i d e - b o i l i n g systems. the process of t r y i n g to o b t a i n a best t - p r o f i l e f o r the poorly-behaved p o i n t s , i t was observed gives e s s e n t i a l l y that u s i n g a constant t of 0.5 the same number of i t e r a t i o n s f o r the well-behaved p o i n t s as u s i n g the v a l u e of 1.8, which i s the approximate average -274- e m p i r i c a l v a l u e f o r these p o i n t s . that T h i s can be e x p l a i n e d by the fact the r e c u r r e n c e g r a d i e n t - f o r m u l a has a s e l f - a d j u s t i n g e f f e c t , as the f o l l o w i n g argument shows: Assume, f o r example, t h a t ( f ' ) i s exact and that n a t o o - s m a l l value of t i s used. (7-34), would r e s u l t (f') n + 1 /(f') . i n a too-large value f o r T h i s would i n t u r n l e a d to n a too-small A t n + 1 which, f o r a w e l l - c o n d i t i o n e d f u n c t i o n , would y i e l d result tend a too-large r i n a too-small ( f ) n + which would 1 /(f) to compensate f o r the t o o - l a r g e F o r the poorly-behaved 1.25 T h i s , from E q u a t i o n t h a t would (f ) 1 n + 1 p o i n t s , i t was /(f') . n observed that t = g i v e s a performance that i s i n t e r m e d i a t e between t = 0.5 1.8. T h i s suggests an o p t i m a l t w i t h i n the range 0.5 was t = < t < 1.8. determine the o p t i m a l t, a g o l d e n - s e c t i o n s e a r c h technique 1972) and To (Himmelblau, employed, u s i n g the number of i t e r a t i o n s f o r a l l the poorly-behaved p o i n t s f o r f i v e of the systems as the o b j e c t f u n c t i o n . The performed w i t h i n the l i m i t s 0.5 s e a r c h was plotted i n F i g u r e 7-5, +1 f = (1 - f Let i s 1.0. results, Hence, f o r the purpose of the r e c u r r e n c e g r a d i e n t - f o r m u l a was (f') The l e d to the c o n c l u s i o n t h a t the value of t t h a t g i v e s the most s t a b l e behaviour study, <^ t <_ 2.0. n + d e f i n e d simply as 1 )(f') n (7-36) n us attempt to c o n s t r u c t a t h e o r e t i c a l f o u n d a t i o n f o r the above empirical relationship. According to Newton, this -275- Fig7-5: Iteration-units versus-t prof ile resulting from the Golden section optimization scheme -275a- Leaf 276 missed i n numbering -277- T I f we (7-37) assume t h a t ( f ) linking the n+1 ( x , f ) and n n (T relationship n+1. f + (f) A combination Now, Equation n (7-38) (7-37) and (7-38) l e a d s to E q u a t i o n the argument l e a d i n g to E q u a t i o n method. (7-38) i m p l i e s that iteration, the secant method i s r e p l a c e d w i t h the Newton S i n c e t h i s i s the scheme adopted i n the r e g u l a - f a l s i 'quasi-Newton' method r e p r e s e n t e d by E q u a t i o n identical to the r e g u l a f a l s i e l i m i n a t i o n of p o i n t s : the two larger (7-36). (7-36) i s a secant convergence r e l a t i o n s h i p p r o v i d e d t h a t i n the f i r s t the of Equations AT iteration, (7-36) would have been method but f o r the d i f f e r e n c e i n the the former method always e l i m i n a t e s the o l d e r of s t o r e d p o i n t s ; the l a t t e r method e l i m i n a t e s the one w i t h | f | . For t h i s s o l e d i f f e r e n c e , the the 'quasi-Newton' method has been r e t a i n e d as a s e p a r a t e method. 7-9 I n i t i a l i z a t i o n Schemes For the purpose of the f i r s t K c a l c u l a t i o n , y_ ( f o r bubble or x ( f o r dew point) i s i n i t i a l i z e d to z. Three schemes f o r initialization were s t u d i e d . Scheme scheme i n v o l v e s the f o l l o w i n g s t e p s : (1) 1 The Estimate temperature the b o i l i n g p o i n t s of the pure components at the system p r e s s u r e from the C l a u s i u s - C l a p e y r o n e q u a t i o n , i n t e g r a t e d — the b a s i c assumptions that l e a d to the vapour-pressure Clapeyron point) (Reid et a l , 1977) — with r e l a t i o n of between normal atmospheric pressure -278- and the system p r e s s u r e to y i e l d T T = bi l RT .Hn P nbi AH . vi (7-41) W ; u i where T , . = Normal b o i l i n g p o i n t of component i nbi T, . = the estimated b o i l i n g bi p o i n t of component i at p r e s s u r e P, and AH (2) = the normal enthalpy of v a p o r i z a t i o n of component i I n i t i a l i z e T from T° = 0.5(T + T . ) bmax bmin V (7-42) u where T, and T, . a r e the computed b o i l i n g p o i n t s f o r bmax bmin the l e a s t Scheme 2: Equation and the most v o l a t i l e components I n t h i s scheme, T respectively. v a l u e s are computed a c c o r d i n g to (7-41) but T i s i n i t i a l i z e d t o T° = i Scheme 3: N I l Z i 1 T 0 b (7-43) i 1 = Here T ° i s o b t a i n e d d i r e c t l y using mole-fraction-weighted from E q u a t i o n average v a l u e s of T ^ a n c (7-41) by * ^H^, thus: T ° = ( 7 RT , £n P ^ _ nb a AH a where N nb ^ i nbi _ 4 4 ) -279- N and AH = v Y a The systems — Z j AH i vi t three i n i t i a l i z a t i o n schemes were a p p l i e d to 11 of the s i x n a r r o w - b o i l i n g (VA through VK) and 5 w i d e - b o i l i n g (VL through VP) — over a p r e s s u r e range of 1 atm. to 14.5 atm. done f o r both bubble- and dew-point d e t e r m i n a t i o n . 7-8 r e p r e s e n t the three d i f f e r e n t This F i g u r e s 7-6 trends that are found was through to be assumed by the dependence, on p r e s s u r e , of the a b s o l u t e d i f f e r e n c e between T ° and T bp or T, . dp - The t r e n d r e p r e s e n t e d by F i g u r e 7-6, whereby Scheme 3 i s c o n s i s t e n t l y b e t t e r than the o t h e r two schemes f o r both bubble- and dew-point c a l c u l a t i o n , was found to predominate. systems, t h i s l e d to fewer i t e r a t i o n s f o r Scheme 1. The Schemes 2 and 3 than f o r t r e n d of F i g u r e 7-7, where t h e r e i s a c r o s s - o v e r between Scheme 1 and Schemes 2 and 3, was l e s s common. the t r e n d observed The For some of the F i g u r e 7-8 represents f o r w i d e - b o i l i n g systems. r e s u l t s show a net c r e d i t i n f a v o u r of Scheme 3. T h i s scheme a l s o has the f o l l o w i n g p o i n t s i n i t s f a v o u r : (1) I t i s computationally l e s s demanding than the other two schemes. (2) I t i s , l i k e Scheme 2, a b e t t e r c h a r a c t e r i z a t i o n of the system i s Scheme 1 which i s b l i n d than to the r e l a t i v e d i s t r i b u t i o n of the c o n s t i t u e n t s of the system. All 7-10 the succeeding e m p i r i c a l work i s based on Scheme 3. Applications The s i x methods were implemented f o r both bubble- and dew-point calculations. I n each case, two a l g o r i t h m s were s t u d i e d : the -280- 18.0 16.0- 14.0- 12.0- 10.0- 6.04 Legend: Bubble point Dew point 4.0 Initialization sheme 1 • Initialization scheme 2 o Initialization scheme 3 x 2.0- &•' -o I-::::?-' 1.0 2.0 X o- o x 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 1E0 13.0 14.0 15.0 •Pressure ( Atm.) Fig7-6: Variation of lTj n j — T o I with system Pressure for system VC -281- 30.01 Legend: Bubble point Dew point 26.0 J Initialization scheme 1 • Initialization scheme 2 ° Initialization scheme 3 x 22.CH 18.0H 8 .JO 6.0 ox- o • ; x. . . • X 0 « " 2.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 • Pressure (Atm.) Fig 7-7: Variation of I T. , - T f I with system Pressure for system VA 15.0 -282- 75.0! 52.0 -I 1.0 1 1 1 2.0 3D 4.0 1 1 5.0 6.0 1 1 7.0 8.0 1 1 1 1 1 Pressure (Atm.) Fig 7-8; Variation of l T i n i t -T S 0 L 1 1 9.0 10.0 11.0 12.0 13.0 14.0 15.0 l with system Pressure for system VP -283- d o u b l e - i t e r a t i o n a l g o r i t h m presented i n S e c t i o n 7-2, and a new ( d i s c u s s e d i n S e c t i o n 7-10-2) that aims at e l i m i n a t i n g the algorithm inner iteration. 3-1). The methods were a p p l i e d to systems VA The s a t u r a t i o n p o i n t s f o r each system were determined at pressure atm., VL (see c o n d i t i o n s u n i f o r m l y d i s t r i b u t e d over a p r e s s u r e with the p r e s s u r e v a l u e s through VP i n T a b l e 3-1 (the w i d e - b o i l i n g systems), dew-point c a l c u l a t i o n was 10 range of as the minima. the p r e s s u r e Table For 27 systems range used f o r narrowed s l i g h t l y because some convergence problem (common to a l l the methods) was 28 through VP observed at p r e s s u r e s close to atm. The s e a r c h was terminated after the nth i t e r a t i o n i f I n n-11 T - T | _< 0.001K The above t o l e r a n c e i s c o n s i d e r e d to be i n tune w i t h r e a l i t y , b e a r i n g i n mind the degree of p r e c i s i o n of the best i n common use. considered temperature-measuring In the i n n e r loop, the composition-dependence of K i s to have been f u l l y c o r r e c t e d f o r when the average change i n the a p p r o p r i a t e p h a s e - m o l e - f r a c t i o n s 7-10-1 devices Double-Loop absolute i s l e s s than 10 ^. Algorithms In the a p p l i c a t i o n of the methods u s i n g the d o u b l e - i t e r a t i o n a l g o r i t h m of S e c t i o n 7-2, (1) The the f o l l o w i n g o b s e r v a t i o n s were made: Newton and Richmond methods r e q u i r e the same number of i t e r a t i o n s i n almost a l l cases. method concedes favour I n view of t h i s , the Richmond to the Newton method to the extent that former r e q u i r e s more c a l c u l a t i o n i n the temperature update. the -284- (2) The Lagrange to (3) method i s to the q u a d r a t i c method what Richmond's i s Newton's a l a O b s e r v a t i o n ( 1 ) . I n a good number of cases, the r e g u l a - f a l s i method r e q u i r e s the same number of i t e r a t i o n s as the q u a d r a t i c i n t e r p o l a t i o n method. (4) The 'quasi-Newton' in method g i v e s a performance terms of i t e r a t i o n count and s t a b i l i t y — r e g u l a - f a l s i method. computation t h a t i s comparable — t o that of the The former thus r e t u r n s s l i g h t l y better times than the l a t t e r due to i t s s i m p l e r p o i n t - e l i m i n a t i o n scheme. The above o b s e r v a t i o n s on i t e r a t i o n requirements are r e f l e c t e d i n the computation-time v a l u e s p r e s e n t e d i n T a b l e s 7-1 and 7-2 f o r bubble and dew p o i n t s r e s p e c t i v e l y . falsi, q u a d r a t i c , Lagrange The r e s u l t s and 'quasi-Newton' performances, w i t h the 'quasi-Newton' terms i n d i c a t e that the r e g u l a - of the t o t a l computation methods have method h a v i n g a s l i g h t comparable edge — i n times f o r both bubble- and dew-point c a l c u l a t i o n s as w e l l as of the ease o f programming. 7-10-2 Fixed-Inner-Loop A l g o r i t h m s The first attempt made at improving the double-loop a l g o r i t h m of S e c t i o n 7-2 i n v o l v e d j e t t i s o n i n g the i n n e r i t e r a t i o n . Step 3 then i n v o l v e d u s i n g K v a l u e s ( o b t a i n e d i n Step 2) that have not been corrected f o r their composition-dependence. When t h i s s i n g l e - l o o p a l g o r i t h m was t e s t e d on the v a r i o u s systems, i t was found to be q u i t e u n r e l i a b l e , gradient than the g r a d i e n t methods. and more so f o r the non- F i x e d - i n n e r - i t e r a t i o n algorithms were next c o n s i d e r e d . In the f i r s t v e r s i o n of the f i x e d - i n n e r - i t e r a t i o n a l g o r i t h m , the number of i n n e r i t e r a t i o n s was f i x e d at 2. The r e s u l t s o b t a i n e d w i t h Table 7-1 Computation times (CPU seconds) f o r bubble-point c a l c u l a t i o n based on the double-iteration algorithms Lagrange Newton Richmond 'Quasi-Newton' 0.3180 0.3225 0.3836 0.4084 0.3111 0.2030 0.2060 0.2073 0.2564 0.2638 0.2003 VC 0.4190 0.4156 0.4191 0.5416 0.5385 0.4200 VD 0.4041 0.4109 0.4117 0.5066 0.5061 0.4033 VE 0.3720 0.3810 0.3841 0.4410 0.4410 0.3707 VF 0.3103 0.3047 0.3155 0.4041 0.4045 0.3075 VG 0.3302 0.3376 0.3403 0.4044 0.4214 0.3301 VH 0.3500 0.3573 0.3602 0.4200 0.4333 0.3508 VI 0.3556 0.3619 0.3686 0.4178 0.4303 0.3544 VJ 0.4627 0.4626 0.4666 0.5900 0.5881 0.4596 VK 0.2520 0.2553 0.2658 0.3195 0.3391 0.2478 VL 0.2670 0.2649 0.2722 0.3139 0.3159 0.2652 VM 0.2241 0.2334 0.2383 0.2932 0.2982 0.2259 VN 0.2855 0.2885 0.2950 0.3257 0.3357 0.2843 VO 0.2582 0.2596 0.2610 0.3023 0.3002 0.2666 VP 0.3982 0.3775 0.3744 0.4593 0.4762 0.3848 I 5.2026 5.2348 5.3026 6.3794 6.5007 5.1824 System Regula-falsi VA 0.3107 VB Quadratic Table 7-2 Computation times (CPU seconds) f o r dew-point c a l c u l a t i o n based on the double-iteration algorithms System Regula-falsi Quadratic Lagrange Newton Richmond 'Quasi-Newton' VA 0.3531 0.3544 0.3621 0.4203 0.4334 0.3439 VB 0.2929 0.2903 0.3009 0.3235 0.3351 0.2820 VC 0.5368 0.5190 0.5225 0.6260 0.6231 0.5271 VD 0.54523 0.5474 0.5574 0.6273 0.6438 0.5302 VE 0.7167 • 0.6918 0.7238 0.7041 0.7247 0.6943 VF 0.5669 0.5645 0.5758 0.6072 0.5914 0.5566 VG 1.0078 0.9369 0.9648 0.8908 0.9094 0.9423 VH 0.7214 0.6980 0.8214 0.7280 0.7740 0.6945 VI 0.5045 0.5189 0.5207 0.5627 0.5891 0.4956 VJ 0.6765 0.6841 0.7135 0.7346 0.7495 0.6623 VK 0.4000 0.4007 0.4190 0.4733 0.4906 0.4048 VL 0.3213 0.3193 0.3224 0.3703 0.3675 0.3219 VM 0.3480 0.3426 0.3553 0.3918 0.3836 0.3534 VN 0.3797 0.3749 0.3811 0.4152 0.3985 0.3847 VO 0.3483 0.3391 0.3422 0.3881 0.3838 0.3568 VP 0.5728 0.5699 0.5728 0.6341 0.6499 0.5661 I 8.2890 8.1518 8.4557 8.8973 9.0474 8.1165 -287- t h i s a l g o r i t h m are presented i n Tables 7-3 point calculations (1) respectively. The Where i t does converge, time-efficient (3) 7-4 f o r bubble- and The the dew- results reveal that: T h i s a l g o r i t h m , y i e l d i n g a great improvement over the i t e r a t i o n algorithm, s t i l l (2) and single- r e c o r d s a number of f a i l u r e s . the a l g o r i t h m i s , almost than the double-loop always, more algorithm. a l g o r i t h m behaved w e l l f o r the b u b b l e - p o i n t c a l c u l a t i o n of a l l n a r r o w - b o i l i n g systems (VA methods except the through VK) Richmond method (which, and w i t h a l l the a t any r a t e , i s the most s l u g g i s h of them a l l ) . O b s e r v a t i o n 3 i s q u i t e i n t e r e s t i n g because i f the r e s u l t s f o r the 11 n a r r o w - b o i l i n g systems used b o i l i n g mixtures, here can be g e n e r a l i z e d to a l l narrow- then the v e r s i o n of the fixed-inner-iteration a l g o r i t h m as a p p l i e d above can be i n c o r p o r a t e d i n t o a BP a l g o r i t h m f o r distilation-unit calculation (see treatment i n Chapter 9), s i n c e the BP method i s n o r m a l l y a p p l i c a b l e to o n l y n a r r o w - b o i l i n g systems. To ensure the s t a b i l i t y of the f i x e d - i n n e r - i t e r a t i o n a l g o r i t h m , h i g h e r numbers of i n n e r i t e r a t i o n were t r i e d . a l s o y i e l d some f a i l u r e , but a v a l u e of 4 was performance. A v a l u e of 3 was found In s e t t i n g the i n n e r i t e r a t i o n a t 4, bourhood of the s o l u t i o n . — The v a l u e of 4 was one was m i n d f u l of especially r e q u i r e d to i n the n e i g h - t h e r e f o r e used as an upper l i m i t , w i t h convergence t e s t s being c a r r i e d out a f t e r every i t e r a t i o n . In a p p l y i n g t h i s new v e r s i o n of the a l g o r i t h m , r e s u l t s were generated to to g i v e r e l i a b l e the p o s s i b i l i t y of l e s s than f o u r i t e r a t i o n s being a c t u a l l y do the K-composition-dependence c o r r e c t i o n found fixed-inner-iteration only f o r those systems which, when : Computation tines (CPD seconds) for bubble-point calculation based algorithms with inner iteration fixed at 2. System Regula-falsi Quadratic Lagrange Newton Richmond 'Quasi-Newton' VA 0.2739 0.2746 0.2862 0.4050 f 0.2814 VB 0.1788 0.1823 0.1884 0.2410 0.2460 0.1801 VC 0.3826 0.3853 0.3892 0.5271 0.6353 0.3834 VD 0.3569 0.3617 0.3645 0.5484 0.6863 0.3624 VE 0.3224 0.3243 0.3302 0.4611 0.5861 0.3309 VF 0.2772 0.2813 0.2860 0.3937 0.3956 0.2794 VG 0.2804 0.2891 0.2995 0.3829 f 0.2918 VH 0.3158 0.3235 0.3310 0.4221 0.6483 0.3284 VI 0.3313 0.3466 0.3432 0.4227 0.5107 0.3294 VJ 0.4124 0.4124 0.4232 0.5858 0.7521 0.4142 VK 0.2259 0.2272 0.2375 0.3077 0.3174 0.2289 VL f f f 0.3965 f f VM 0.2182 f f 0.3540 f 0.2192 VN f f f 0.4372 f f VO 0.2316 f f 0.3995 f 0.2364 VP 0.3538 f f 0.5208 f 0.3691 for VA' through VK 3.3576 3.4083 3.4789 4.6975 ? 3.4103 Corresponding 1 for Doubleiteration 3.7696 3.8109 3.8617 4.6794 I f => failure i n at least one point. 3.7556 Table 7-4 Computation times (CPU seconds) for dew-point calculation based on algorithms with inner iteration fixed at 2 System Regula-falsi Quadratic Lagrange Newton Richmond VA f f f f f 'Quasi-Newton' f VB* 0.2130 0.2181 0.2193 0.2704 0.3259 0.2173 VC* 0.5241 0.5153 0.5151 0.6188 0.7846 0.5168 VD* 0.4116 0.4362 0.4486 0.5970 f 0.4116 VE f 0.4060 0.4272 f f 0.4133 VF f 0.4377 0.4551 f f 0.4437 VG 0.5151 f f f f 0.4918 VH f f f f f f VI* 0.3970 0.4740 0.4898 0.5526 0.5829 0.3943 VJ* 0.5167 0.5203 0.5348 0.6965 0.7342 0.5096 VK 0.3067 0.3166 0.3202 f f 0.3100 VL* 0.2459 0.2347 0.2366 0.3127 f 0.2387 VM* 0.2563 0.2392 0.2448 0.3374 f 0.2656 VN f f f f 0.4078 f VO* 0.2584 0.2424 0.2426 0.3299 0.4086 0.2590 VP* 0.4270 0.4133 0.4063 0.5696 f 0.4347 £ for starred systems 3.2500 3.2935 3.3379 4.2849 corresponding i for Doubleiteration 4.1434 4.1306 4.2077 4.6584 f => f a ilure i n at leas t one point 3.2476 - 4.0954 -290s o l v e d w i t h the f i r s t least one of the methods, e x c l u d i n g Richmond's. T a b l e s 7-5 The v e r s i o n , r e c o r d e d at l e a s t a f a i l u r e w i t h at total ( b u b b l e - p o i n t c a l c u l a t i o n ) and times 7-6 The r e s u l t s are i n (dew-point calculation). summed over the same systems are a l s o i n c l u d e d f o r the double-iteration algorithms. The o b s e r v a t i o n s made on the r e s u l t s i n T a b l e s 7-1 apply to those i n T a b l e s 7-5 and 7-6. And comparing the loop method w i t h the double-loop method, one the former — and excluded). to 9.0 f i n d s t h a t the t i m e - g a i n iterations. The calculation ordinarily c o r r e c t i o n s f o r K t o take much more than f o u r time s a v i n g f o r the m o r e - c o m p o s i t i o n - s e n s i t i v e dew- p o i n t c a l c u l a t i o n i s more s i g n i f i c a n t , assuming magnitudes of 18.4 21.1 percent ( t h i s excludes to the Newton and Richmond methods which r e g i s t e r e d some f a i l u r e f o r system 7-11 by percent f o r the v a r i o u s methods (Richmond's T h i s i s not s u r p r i s i n g because one would not expect vapour-composition also fixed-inner- over the l a t t e r i s r a t h e r s m a l l f o r b u b b l e - p o i n t r a n g i n g from 6.7 7-2 VG). Conclusions J (1) The nongradient methods are on the same s c a l e of performance, and they o u t c l a s s the Newton and Richmond methods. (2) Imposing a c e i l i n g of 4 on the i n n e r i t e r a t i o n that c o r r e c t s K v a l u e s f o r composition-dependence h e l p s to cut down on time. T h i s g a i n i n computation dew-point (3) The time i s more s i g n i f i c a n t f o r calculation. r e g u l a - f a l s i method, based method adopted here, i n computation on the f i r s t - i t e r a t i o n acceleration i s b e t t e r implemented i n the form of the 'quasi-Newton' method. results computation T h i s not o n l y l e a d s to a s l i g h t r e d u c t i o n time but i t a l s o seems to improve s t a b i l i t y , i n T a b l e 7-4 suggest. as the Table 7-5: Computation times (CPU seconds) f o r bubble-point c a l c u l a t i o n based on algorithms with Inner i t e r a t i o n bounded at 4. System Regula-falsi Lagrange Newton Richmond 'Quasi-Newton' VA 0.3081 0.3192 0.3150 0.3755 0.3929 0.3084 VG 0.3384 0.3412 0.3383 0.4028 0.4176 0.3367 VL 0.2410 0.2348 0.2451 0.2829 0.3167 0.2366 VM 0.2089 0.2051 0.2118 0.2663 0.3096 0.2037 VN 0.2549 0.2468 0.2497 0.3022 0.3927 0.2529 VO 0.2404 0.2254 0.2332 0.2617 0.3501 0.2322 VP 0.3442 0.3247 0.3217 0.4149 0.5333 0.3455 I 1.9359 1.8972 1.9148 2.3063 2.7579 1.9160 Corresponding I f o r Doubleiteration 2.0739 2.0795 2.1037 2.4824 2.5560 2.0680 % time g a i n 6.7 8.8 9.0 7.1 Quadratic -7.9 7.4 Table 7-6: Computation times (CPU seconds) f o r dew-point c a l c u l a t i o n based on algorithms with inner i t e r a t i o n bounded at 4. Lagrange Newton Richmond *Quasi-Newton 0.3283 0.3347 0.3980 0.4083 0.3230 0.5237 0.4927 0.5209 0.5448 0.7829 0.4947 VF 0.4906 0.4804 0.5001 0.5345 0.5905 0.4808 VG 0.6918 0.6424 0.6567 VH 0.6393 0.6056 0.6461 0.6518 0.7518 0.5723 VK 0.3343 0.3307 0.3370 0.4040 0.4790 0.3254 VN 0.3718 0.3539 0.3569 0.3879 0.3802 0.3569 I 3.3826 3.2340 3.3524 ? ? 3.1817 4.1456 4.0212 4.2480 4.2389 4.3220 4.0211 1 1 System Regula-falsi VA 0.3311 VE Corresponding I f o r Doubleiteration %Time gain 18.4 f => f a i l u r e i n a t l e a s t Quadratic 19.6 21.1 one p o i n t ; ? => unknown f f 0.6286 20.9 1 CHAPTER EIGHT ADIABATIC VAPOUR-LIQUID FLASH CALCULATION 8-1 Introduction The c l o s e a t t e n t i o n g i v e n to the a d i a b a t i c v a p o u r - l i q u i d problem, as documented i n t h i s chapter, i s due i n the d e s i g n to the f a c t of f l a s h drums and partly t i v e form of the d i s t i l l a t i o n - u n i t Holland (1963) presents employed i n h a n d l i n g literature that i t i s a diminu- two methods which are conventionally a two-dimensional a double-iteration regula-falsi i s without any reported the best method f o r s o l v i n g the In a d d i t i o n to the two problem. methods mentioned above, a number of from m u l t i s t a g e algorithms, study. been developed i n t h i s the Newton-Raphson approach are presented regula-falsi approach. comparative study to determine methods, some of which d e r i v e d i r e c t l y has to i t s importance problem. the a d i a b a t i c f l a s h problem: Newton-Raphson method, and The partly flash distillation Three a l g o r i t h m s i n S e c t i o n 8-3 method i s t r e a t e d i n S e c t i o n 8-4. while S e c t i o n 8-5 based on the investigates the p o s s i b i l i t y of a p p l y i n g a p a r t i t i o n method of the sum-rates type ( K i n g , 1980) to the problem. bubble-point type are v a r i o u s methods w h i l e Vital out 8-1-1 S e c t i o n 8-7 S e c t i o n 8-8 p a r t i t i o n methods of contains concludes the a p p l i c a t i o n s of i s contained the chapter. on the systems which were employed i n algorithms the i n Table testing 8-1. Nomenclature Note: it studied. information the v a r i o u s I n S e c t i o n 8-6, Any symbol not occurs w i t h i n t h i s chapter d e f i n e d below and not c l e a r l y d e f i n e d where r e t a i n s the d e f i n i t i o n of S e c t i o n 3-1-1. -293- Table 8-1: V i t a l Information on a d i a b a t i c - f l a s h systems System Code No. of Components HA 4 Same as f o r system VA 370 - 410 8-8 HB 3 Same as f o r system VB 108 - 144 20 - 40 1.316 HC 5 Same as f o r system VC 378 - 478 21 - 37 1.0 HD 4 Same as f o r system VD 360 - 508 10 - 42 1.0 HE 4 Same as f o r system VE 350 - 486 10 - 38 1.0 HF 5 Same as f o r system VF 370 - 490 10 - 38 1.0 HG 4 Same as f o r system VG 350 - 486 10 - 38 1.0 HH 4 Same as f o r system VH 340 - 488 10 - 46 1.0 HI 4 Same as f o r system VI 350 - 498 10 - 46 1.0 HJ 6 Same as f o r system VJ 370 - 490 10 - 38 1.0 HK 3 Same as f o r system VK 380 - 480 10 - 46 3.7 HL 2 Same as f o r system VL 400 - 580 10 - 22 1.0 HM 2 Same as f o r system VM 410 - 598 10 - 46 1.5 HN 2 Same as f o r system VN 400 - 592 10 - 38 1.0 HO 2 Same as f o r system VO 390 - 578 10 - 46 1.0 HP 4 Same as f o r system VP 385 - 585 10 - 46 1.0 Feed Components and % Composition Temperature Range (K) Feed Pressure Range (Atm.) Operating Pressure (Atm.) 1.0 -295- Symbol Definition f Check f u n c t i o n F T o t a l feed h Liquid-phase H Vapour-phase e n t h a l p y rate enthalpy Greek Symbols a v > Damping parameters f o r T and a T 0 Phase V fraction Subscripts bp Bubble-point dp Dew-point F Feed H Enthalpy 1 Component; e q u a t i o n number M Mass balance 1,2,3 I t e r a t i o n points Superscripts E Excess o I d e a l gas * Equilibrium point - (as i n H): denotes p a r t i a l 8-2 value; initial point value Theoretical Background A p h y s i c a l d e f i n i t i o n of the a d i a b a t i c v a p o u r - l i q u i d lem i s represented diagrammatically c o n s i d e r a t i o n makes i t s way i n Figure towards the 8-1. As flash prob- the mixture under f l a s h drum from some other part -296- FIG 8-1: AN ADIABATIC FLASH DRUM -297- of the p r o c e s s p l a n t loaded w i t h some amount of heat at a f i x e d c o n d i t i o n of temperature, T and p r e s s u r e , P , p i t encounters a t h r o t t l e v a l v e and undergoes i s e n t h a l p i c expansion to a p r e s s u r e P. mixture e x p e r i e n c e s a f l a s h a l r e a d y e x i s t ) at t h i s ( o r a phase r e d i s t r i b u t i o n , reduced p r e s s u r e . added nor removed, the system And i f two The phases because heat i s n e i t h e r e q u i l i b r i a t e s at some lower temperature, T. U n l i k e the case of i s o t h e r m a l f l a s h , the a d i a b a t i c f l a s h problem r e q u i r e s the s o l u t i o n of the e n t h a l p y - b a l a n c e e q u a t i o n as w e l l as the mass-balance and e q u i l i b r i u m r e l a t i o n s — s i n c e temperature i s an addi- t i o n a l unknown. The e n t h a l p y balance assumes the N f H _ v H I = 1=1 H 1 1 N + _ l h I i=l 1 1 - H form = 0 (8-1) F A c c o r d i n g to the r e l a t i o n s h i p s i n Chapter 2, the l i q u i d - p h a s e heat of mixing i s not computed f o r the i n d i v i d u a l components. it has been found convenient to d e f i n e h^ by h i - h i + \ T h i s d e f i n i t i o n i n t r o d u c e s no d i f f e r e n c e long as N I 1 -L ± i =l X N or However, I x -1 (8-2) to the enthalpy balances as -298- Where the feed to the t h r o t t l e v a l v e i s subcooled l i q u i d , we N is t _ Z h , i Fi I ^ (8-3) the g e n e r a l case where the feed i s a v a p o u r - l i q u i d mixture, given N supplied T = F It v ^ _ H + Fi Fi N y £ Ty not Hp. v H te is determined by performing appropriately-computed To logically r and P r , and as a known q u a n t i t y the a d i a b a t i c f l a s h an i s o t h e r m a l e q u i l i b r i u m and flash employing the r e s u l t s i n Equation determine the s t a t e (temperature and i n Chapter calcula- sound s i n c e the known q u a n t i t y i s phase m o l a r - e n t h a l p i e s the a d i a b a t i c system, E q u a t i o n balance, (8-4) ' A more r e a l i s t i c approach i s adopted i n t h i s t i o n on the feed at T (8-1) and (8-3) or(8-4). phase d i s t r i b u t i o n ) of i s s o l v e d along w i t h mole-fraction-balance calcula- the mass- r e l a t i o n s h i p s presented 3. Newton-Raphson Methods The Newton-Raphson f o r m u l a t i o n of the a d i a b a t i c - f l a s h problem s t u d i e d here employs two temperature. and _ h Fi Fi to a computer program f o r h a n d l i n g T h i s i s , however, not study: i i s common p r a c t i c e to assume Hp tion. and H F by H 8—3 saturated have H„ = F For or a new outlined The form 'conventional and vapour f r a c t i o n formulation' according r e c e n t l y proposed by Barnes and i n S e c t i o n s 8-3-1 at r e d u c i n g independent v a r i a b l e s : 8-3-2 to H o l l a n d F l o r e s (King, 1980) respectively. computation time i s d i s c u s s e d and (1963) are A m o d i f i c a t i o n aimed i n S e c t i o n 8-3-3. -299- The c o n v e n t i o n a l method and the method of Barnes and r e q u i r e that the i n i t i a l Flores temperature be i n the two-phase r e g i o n , that is: T, < T° < T, . bp dp In view of t h i s , both bubble- and dew-point performed the to be on the system b e f o r e the i t e r a t i o n can take o f f . Even w i t h temperature so i n i t i a l i z e d , Holland c a l c u l a t i o n s need t h e r e i s no guarantee of convergence. (1963) recommends some simple r u l e s to reduce the chances of divergence: Rule 1: For a n e g a t i v e Rule 2: For T n + 1 R u l e 3: For T n + 1 The < T 9 n + 1 v , use 9 v , use T n + 1 = 0.5(T > 1\ , s e t dp T n + 1 =0.5 bp = 0.5 n + 1 9 ) v n + T. ) bp n (T (6° + v n + T\ ) dp f o l l o w i n g steps are common to the c o n v e n t i o n a l and the Barnes-Flores algorithms. (1) Compute H„ and determine T, and T, at the o p e r a t i n g F bp dp pressure. (2) Initialize 9 , T and v K. — (3) Compute the check (4) (5) Compute the d e r i v a t i v e terms. Update 9^ and T. Apply Rules 1 through 3 (6) T e s t f o r convergence. the iteration. functions. above. I f the outcome i s p o s i t i v e then t e r m i n a t e Otherwise go to Step 3. -300- I n the implementation obtained of the two a l g o r i t h m s , i n i t i a l v a l u e s are from: • °-(b V ' T 5 T P+ 6° = 0.5 o r V ° = 0.5F, v ' K 8-3-1 o i s based on T o The C o n v e n t i o n a l and x = y = z. Formulation I n the r e l a t i o n s h i p s i 1 + (K, - 1) i that f o l l o w , x. i s g i v e n by Z x, i ~ e v and y_ by y The i • i K X i mass-balance check f u n c t i o n i s g i v e n by M V f T ) = ( I " i=l x ± C") 1 8 1 5 By s u b s t i t u t i n g the r e l a t i o n s h i p s v and = Z i i 1 - 1 i x = F(l - 6 ) into Equation (8-1) and r e a r r a n g i n g , we have the enthalpy check function f H ( e v T ) = X z i * i " ( 1 " v 9 ) X x i ( V V " By a p p l y i n g the Newton-Raphson method to E q u a t i o n s have r- ^ (8-5) and (8-6), we -301- n "9 ~ n+1 "B V 3f. H - 9f. M ~3f V = T 9f. H T 9 — — f M (8-7) f, M —H_ where 3f 3f. M D = 80 The 9f H 3T ~ 9f. H 30 M 9T d e r i v a t i v e terms are g i v e n by 9f, M 30 X i + " i ( K - l ) «vj y ± 3K, 3f. M 3T 9f. H 30 9f 9T H N v i 3T (K -l) 0 i=l ± ' v (8-8) ? y A i ( H i ' i h > i + Oc - i ) e > ± N 9H U* 3^" ± y 3(H. - (1 " V*J i v i - i> -W i + K - i) ev 9 9 ( H h.) 3T K h ( ± ]} I n the above r e l a t i o n s h i p s , the d e r i v a t i v e s of K and r e s p e c t to T are determined through a f i n i t e - d i f f e r e n c e approach, p e r t u r b a t i o n of 0.001K on 8-3-2 enthalpy w i t h T. The B a r n e s - F l o r e s Form The B a r n e s - F l o r e s method i s based on the Rachford-Rice with a -302- mass-balance check f u n c t i o n and an enthalpy check f u n c t i o n which assumes a d i f f e r e n t form from E q u a t i o n (8-6). I n the f o l l o w i n g r e l a t i o n s h i p s , x and y a r e a g a i n d e f i n e d as i n S e c t i o n 8-3-1. Mass Balance: Employing the R a c h f o r d - R i c e ye ,T) = ^ ( v E n t h a l p y Balance: Equation y ± -x r e l a t i o n s h i p , we have (8-9) ± ) The enthalpy check f u n c t i o n can be o b t a i n e d from (8-6) by combining the terms i n 6 K x f o r z. - (1 - 0 ) x . v i i l v i a n d substituting The r e s u l t i s N W T H_ = 1=1 \ J v i i ) x Equation e K H + ( 1 " V h J - -i (8-10) (8-7) a p p l i e s , w i t h the d e r i v a t i v e s g i v e n by N 3f. M 36 (K -l)(y - x ) 1 1 ± H 3K, xi N 3f, M 3T l 3T + (VD ± x 1 6 i=l 1 + (K - 1)6 i v v 3f. 36 H 3f_ N H _ y 3T , . i=l L + 6 (1-6 )(H -h,)3K, rv v i i i 1 + (K -1) 0 3T I v x X i t (8-11) 3H / x 3h t V i — < " J — ]} 3T + X Q 3T -303- 8-3-3 A M o d i f i e d Approach The modified time-saving The approach proposed below embodies a number o f devices. vapour f r a c t i o n , variables. 8 , i s r e p l a c e d by V as one of the check The check f u n c t i o n s are d e f i n e d by f (V,T) M N I (K i=l = - l)x 1 and f (V,T) H In Equation Equation N - V ^ K . x ^ (8-13), (8-12), relationship (8-12) X N - h ± ) + ^ - H, x^ i s NOT t r e a t e d as a f u n c t i o n of V or T. i t i s t r e a t e d as a f u n c t i o n of (8-13) In V o n l y , the being Z. x. *i F + (K This relationship - 1) V * i s used only f o r the purpose of d i f f e r e n t i a t i n g x^ w i t h r e s p e c t to V. The d e r i v a t i v e s presented i n Equation (8-15) below are based on the f o l l o w i n g d e f i n i t i o n s of x and y: Z, (1) v ± = 1 + K (2) (3) 1 ± = Z x, = i N ± Il (4) y.i N 1=1 V - v ± (8-14) li i=l i i -304- T h i s approach, which ensures that x and stage of the i t e r a t i o n , was y are i n mass balance found to be more e f f i c i e n t than at every the a l t e r n a t i v e approach that i s based on an e q u i l i b r i u m agreement between them. The d e r i v a t i v e s are determined from 3f 9V 3f. M 3T g N L i-1 M - i r ± 7 N 3K = I i=l 3f„ H 3V . = N I 3f. H 3T - V ± x , 1 K i 3T , x , ( H . - h ), i-1 (8-15) x N I i=l 3 x [(H -h 1 1 ) 1 K i 3T a + K i ( H l -h.) 3T N ] 3h In a p p l y i n g the above r e l a t i o n s h i p s , three time-saving were i n t r o d u c e d : enthalpy with (1) Holding r e s p e c t to T, constant the d e r i v a t i v e s of K a f t e r a number of i t e r a t i o n s ; the number of K e v a l u a t i o n s per i t e r a t i o n ; e v a l u a t i o n of the K-T discussed below. derivatives. The (3) measures and (2) E l i m i n a t i n g the three m o d i f i c a t i o n s are Reducing direct -305- Holding K-T, H-T and h-T D e r i v a t i v e s constant F i g u r e 8-2 shows some p l o t s of p h a s e - e n t h a l p i e s , system m o l e - f r a c t i o n s . N All the p l o t t e d p o i n t s are based on the constant x = y_ = z. Similar i s g i v e n by _ z H I H = For example, H i n the p l o t weighted by p l o t s of average K v a l u e s , compositions, d e f i n e d by N I K = z.K 1-1 are presented i n Figure 8-3. From the c u r v a t u r e the first of the p r o f i l e s i t was s p e c u l a t e d few i t e r a t i o n s have brought T w i t h i n a reasonable * — _ that when d i s t a n c e of the d e s i r e d s o l u t i o n , T , the H-T, h-T and K-T d e r i v a t i v e s c o u l d be h e l d constant without i n c u r r i n g an i n c r e a s e i n the number of i t e r a t i o n s . 9K To ted test t h i s out e m p i r i c a l l y , v a l u e s as f u n c t i o n s of i t e r a t i o n number narrow- and w i d e - b o i l i n g It constant therefore modified three i t e r a t i o n s , succeeding to and —g^- were genera- f o r some systems, i n c l u d i n g both case, a f t e r the t h i r d compute these the values iteration. of the d e r i v a t i v e s The a l g o r i t h m was d e r i v a t i v e s only f o r the f i r s t and to use the v a l u e s iterations. , 8^ mixtures. was observed that i n every were r e l a t i v e l y of S-H^ i f o r the t h i r d iteration for a l l Fig8-2: Variation of Entholpies with temperature at constant composition -307- o > < °T 0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 • Temperature( Kelvin units) Fig 8 — 3 : Variation of system average K with temperature at constant composition -308- R e s u l t s were generated for system HL based at 10 p o i n t s f o r system HA on t h i s m o d i f i c a t i o n . compared w i t h those o b t a i n e d without When the r e s u l t s were the m o d i f i c a t i o n , i t was the m o d i f i e d form d i d not i n c u r e x t r a i t e r a t i o n s and average t i m e - s a v i n g of about Reducing and at 6 p o i n t s found d i d l e a d to an 23%. the Number of K E v a l u a t i o n s per Iteration T h i s m o d i f i c a t i o n takes advantage of the f a c t t h a t the K employed i n temperature u p d a t i n g at every i t e r a t i o n are based same composition as the K v a l u e s used The i n the next l a t t e r K v a l u e s are t h e r e f o r e estimated a s u i t a b l e K-T model. S e c t i o n 8-6-3, and The that approach adopted composition from the former values on the updating. by means of i s discussed l a t e r i n the a p p l i c a b l e r e l a t i o n s h i p i s p r e s e n t e d as E q u a t i o n (8-35). E l i m i n a t i n g D i r e c t E v a l u a t i o n of K-T D e r i v a t i v e s Let us c o n s i d e r the thermodynamic r e l a t i o n s h i p SAnf, iL 3T The " (h p,x - ± RT ( P r a u s n i t z , 1969): H°) 2 (8-16) c o r r e s p o n d i n g vapour-phase e q u a t i o n i s ( H o l l a n d , 1963): 3£nf, iv 3T I f we -<H - ± P,7 RT H°) 2 (8-17) employ the b a s i c d e f i n i t i o n s of f * n f iL " * n f iv = £ n K i + £ n x i " £ n y iL, i f . and K , we would have iv i (8-18) -309- From E q u a t i o n 9 * f (8-17) and K have 9^nf, iv iL 3T n Eliminating 9 (8-18), we HnK p,y p,x ~3T i (8-19) P,x,y the f u g a c i t y - d e r i v a t i v e terms between Equations (8-19) leads i 9T P,x,y K (8-16), to i i 2" —V (H R (8-20) T I n the implementation of the d i f f e r e n t v e r s i o n s of the 'modified Newton-Raphson' method (see a l g o r i t h m s s e r i a l l y numbered 5 through 7 i n Table initialized 8-3), the temperature was saturation-point simply c a l c u l a t i o n and temperature-constraining the three flow- r u l e s of S e c t i o n 8-3 as i n the and were not applied. The method performed w e l l i n t h i s form and this successfully eliminates determination points. of system bubble and i n i t i a l i z i n g L and V and these dew The method r e q u i r e s are each set to h a l f the feed v a l u e . the purpose of computing i n i t i a l K v a l u e s , x and y_ are set equal 8-4 the For to z_ A Double-iteration Regula-falsi Method T h i s method i n v o l v e s the s o l u t i o n of the mass-balance, e q u i l i b r i u m and isothermal mole-fraction-balance r e l a t i o n s h i p s through f l a s h c a l c u l a t i o n at the c u r r e n t an temperature, f o l l o w e d by a temperature update through the a p p l i c a t i o n of the r e g u l a - f a l s i method to the two most c u r r e n t temperature p o i n t s , u s i n g the enthalpy f u n c t i o n as the temperature-dependent v a r i a b l e . check -310- The following are the key steps involved: (1) Determine the state of the feed through an isothermal f l a s h . Then compute Hp. (2) Choose two temperatures, T and T , such that 2 T, < T. < T„ < T, . bp — 1 — 2 — dp At each of these temperatures, perform an isothermal f l a s h c a l c u l a t i o n and determine the corresponding enthalpy check functions, f (3) and HI f Hz from Equation (8-1). Carry out an i n t e r p o l a t i o n regula f a l s i between points T^ and T 2 to obtain T from T f T - f T = J L i Hl__2 H2 HI ( g _ Perform an isothermal f l a s h at T. (4) Test f o r convergence. Terminate i t e r a t i o n i f the outcome i s p o s i t i v e ; otherwise go to Step 5. (5) (6) Determine f at T. H Replace the larger of f , f w i t h f and the corresponding HI H2 H temperature with T. Go to Step 3. Step 2 requires that both the dew and bubble points be determined. To save on computation time, T^ and T^ i n t h i s step are set equal to T, and T, r e s p e c t i v e l y . bp dp This way, the isothermal f l a s h at the two i n i t i a l temperatures are avoided and fewer enthalpy terms are also computed. 2 1 ) -311- Sum-Rates P a r t i t i o n i n g Methods 8-5 The methods d e s c r i b e d i n t h i s s e c t i o n have been l o o s e l y branded 'sum-rates' methods because l i k e the d i s t i l l a t i o n - c a l c u l a t i o n methods, so i d e n t i f i e d by F r i d a y and Smith (1964), they i n v o l v e matching of temperature w i t h e n t h a l p y . when i t comes to d e t e r m i n i n g phase The The sum-rates i d e n t i t y i s l o s t to develop a method that the d o u b l e - i t e r a t i o n s t r u c t u r e of the method of S e c t i o n 8-4 t a t i o n of K and i n the e v a l u a t i o n of d e r i v a t i v e s and enthalpies that they e n t a i l that have been s t u d i e d eliminates without the double compu- (as i n the Newton-Raphson Methods) or i n time-consuming s a t u r a t i o n - p o i n t algorithms the distribution. main o b j e c t i v e here was g e t t i n g Involved sum-rates calculations. The three i n t h i s s e c t i o n t r e a t _v and T as the independent v a r i a b l e s . 8-5-1 Successive-Substitution Method w i t h V e c t o r L e t us c o n s i d e r E q u a t i o n ( 8 - 1 ) . A simple m a n i p u l a t i o n leads N _ N N _ I v,(H-H°) + I v H° + I 1 (h.-H?) + i-1 1-1 1=1 i-1 1 1 Combining the h we + V 1 1 ideal-gas i = Z 1 1 terms and 1 1 using Projection the N I IH? 1 fact 1 - H to - 0 F that i> have f H The = X±1 Z i=l E + J/i^i- i) H i=l + Xi =hl ^ P scheme adopted here i n v o l v e s temperature s u c c e s s i v e l y w i t h the highly non-linear, ~ *F " ° <- > 8 22 s o l v i n g E q u a t i o n (8-22) f o r r e l a t i o n s f o r v_. non-polynomial nature of the Because of the three middle terms of -312- Equation (8-22), some assumption was made i n order to a v o i d a f i n i t e - d i f f e r e n c e d e r i v a t i v e e v a l u a t i o n i n u p d a t i n g T. t h a t a l l the enthalpy terms, except the c u r r e n t temperature. temperature, I t was assumed the i d e a l - g a s term, are constant at S i n c e H° i s a f o u r t h - d e g r e e p o l y n o m i a l i n t h i s assumption converts Equation (8-22) i n t o an e a s i l y - d i f f e r e n t i a b l e p o l y n o m i a l i n T. The main steps of the a l g o r i t h m a r e : (1) Perform Pp. an i s o t h e r m a l f l a s h c a l c u l a t i o n on the feed a t T F and Hence compute Hp. (2) I n i t i a l i z e v_ and T. (3) Compute the enthalpy departures at the c u r r e n t T, x_ and y_. Employing these i n E q u a t i o n (8-22) and t r e a t i n g the i d e a l - g a s enthalpy term as a f u n c t i o n of temperature, perform a Newton step on the e q u a t i o n to o b t a i n a new estimate of T. (4) Using the c u r r e n t values of T, x and y_, c a l c u l a t e K and o b t a i n new e s t i m a t e s of v_ and 3. by performing a single successive- s u b s t i t u t i o n step of the GP a l g o r i t h m of S e c t i o n 3-3. (5) Perform an a p p r o p r i a t e v e c t o r - p r o j e c t i o n on algorithm. as i n the GP ( I n t e s t i n g the method, the t r i a n g u l a r - p r o j e c t i o n method was employed.) (6) T e s t f o r convergence. I f the outcome i s n e g a t i v e then go to Step 3; otherwise execution. terminate For temperature saturation-point found initialization, calculation the three schemes developed f o r ( S e c t i o n 7-9) were t e s t e d and Scheme 3 was to be s l i g h t l y b e t t e r than the other two. Also, i n i t i a l i z i n g v -313- f rom V l = w i t h K based 1 (i=l,2,...,N) on T° and x = y_ = z_ gave s l i g h t l y better results than w i t h K c a r r i e d over from the i s o t h e r m a l f l a s h on the feed at {T ,P F and a l s o than from simply s e t t i n g v° = When the a l g o r i t h m was o s c i l l a t o r y behaviour was a l g o r i t h m , the temperature reduce F 0.5Z^ a p p l i e d i n the above form to system observed. }, HA, By the very nature of the i s always f o r c e d i n such a d i r e c t i o n as to the e r r o r i n the flow q u a n t i t i e s , as the f o l l o w i n g argument shows. Assume that _v i s l e s s than i t s e q u i l i b r i u m v a l u e at some stage i n the i t e r a t i o n and that T i s not too f a r from i t s e q u i l i b r i u m v a l u e . The t o t a l e n t h a l p y d e p a r t u r e would be l e s s than i t s e q u i l i b r i u m v a l u e . For E q u a t i o n (8-22) to be s a t i s f i e d , the i d e a l - g a s term would have to be h i g h e r than i t s e q u i l i b r i u m v a l u e . S i n c e H° i s an i n c r e a s i n g f u n c t i o n of T, a h i g h e r T would r e s u l t . T h i s would i n t u r n d r i v e up the v a l u e of _v — i n the d i r e c t i o n of e q u i l i b r i u m . A s i m i l a r argument o b t a i n s f o r a s i t u a t i o n where c u r r e n t l y l a r g e r than . The is source of the o s c i l l a t i o n then i s the d i s p a r i t y between the convergence speed of T and that of _v. To e l i m i n a t e the o s c i l l a t o r y a c c o r d i n g to the T = a behaviour, _v and T were damped relationships n+l , . n T + (1 - a ) T , m T T and v — n+l = a v v— n+l ... . n+l + ( l - a ) v v — -314- where 0 < a < 1 T — — and 0 < a < 1 — v — Different combinations of a T a n d a.^ r e w e experimented upon and the best r e s u l t s were o b t a i n e d w i t h the combination To see what e f f e c t the embodiment of the t o t a l e n t h a l p y r e s i d u a l i n the i d e a l - g a s term has on the convergence of the method, an approach that takes account of the dependence of a l l the e n t h a l p y terms temperature was tried. The d e r i v a t i v e was determined on from of - g | = 1 0 0 0 [ f ( T + 0.001) - f ( T ) ] R The approach was H found not to e l i m i n a t e the o s c i l l a t o r y b e h a v i o u r . t h e r e f o r e a l s o r e q u i r e d temperature-damping, and s i n c e i t i n v o l v e s e n t h a l p y computations f o r every temperature update, It two i t i s more time- consuming . 8-5-2 Absorption-Factor Formulation with Vector P r o j e c t i o n T h i s i s an attempt to e l i m i n a t e the damping of v^, as was i n the p r e c e d i n g s e c t i o n , by u s i n g a more s l u g g i s h formulation. Arrangement very s l o w l y , was 2 of S e c t i o n 3-3-6, which tried. T h i s arrangement employed flow-convergence i s known to converge leads to Z. v. = i F - V + 1 K V 1 ± Z ° ' r V i = A i + 1 ( where A i s a b s o r p t i o n 1 = 1 > »--->N) 2 factor. (8-23) -315- The steps i n v o l v e d i n the r e s u l t i n g a l g o r i t h m are the same as those o u t l i n e d i n S e c t i o n 8-5-1, w i t h the f o l l o w i n g e x c e p t i o n : 4, v_ i s computed from E q u a t i o n ted; 8-5-3 i n Step (8-23) and Step 5 i s a p p r o p r i a t e l y execu- no damping of _v i s done. A b s o r p t i o n - F a c t o r F o r m u l a t i o n Without A c c e l e r a t i o n Still temperature i n search of a mass convergence r a t e t h a t w i l l match the convergence speed s i o n of the a l g o r i t h m was and eliminate o s c i l l a t i o n s , studied. T h i s v e r s i o n excludes another ver- the v e c t o r - p r o j e c t i o n a c c e l e r a t i o n as r e p r e s e n t e d by Step 5 of the a l g o r i t h m of S e c t i o n 8-5-1. v° = as t h i s was Also, i s simply i n i t i a l i z e d from 0.5Z, found here to be S e c t i o n 8-5-1. j u s t as good as the scheme adopted in In a l l other r e s p e c t s , t h i s a l g o r i t h m i s the same as t h a t of S e c t i o n 8-5-2. 8-6 Bubble-point (BP) P a r t i t i o n i n g Methods Bubble-point distillation mixtures (BP) p a r t i t i o n i n g methods are w i d e l y a p p l i e d to calculations involving the s e p a r a t i o n of n a r r o w - b o i l i n g (see, f o r example, K i n g , 1980; Henley and Seader, 1981). p a r t i t i o n i n g matches enthalpy w i t h e i t h e r V or L and e n t a i l s temperature In through a b u b b l e - p o i n t this been developed critical updating calculation. s e c t i o n , an a l g o r i t h m based ( S e c t i o n 8-6-1). The The on the b a s i c BP concept has a l g o r i t h m has been s u b j e c t e d to a study w i t h a view to e l i m i n a t i n g p o i n t c a l c u l a t i o n step ( S e c t i o n 8-6-2). the time-consuming Steps are next bubble- taken to reduce -316- the number of K e v a l u a t i o n s per i t e r a t i o n ( S e c t i o n 8-6-3). Finally, in S e c t i o n 8-6-4, the a l g o r i t h m i s m o d i f i e d t o embody the method i n S e c t i o n 8-3-3 f o r e l i m i n a t i n g the d i r e c t c a l c u l a t i o n of the proposed K-T derivatives. The study r e p o r t e d i n t h i s s e c t i o n was undertaken ledge of the K^-method ( H o l l a n d , 1975), a technique employed i n BP i n f u l l know- currently a l g o r i t h m s to e l i m i n a t e b u b b l e - p o i n t c a l c u l a t i o n . a l g o r i t h m based An on the technique has, however, not been implemented because, as l a t e r shown i n S e c t i o n 9-4-2, i t i s — of a base component — simply one with suitable here choice of the f o r m u l a t i o n s t h a t have been s t u d i e d i n S e c t i o n 8-6-2. 8-6-1 The C o n v e n t i o n a l Form I n the f o r m u l a t i o n , l e t us choose V as the o t h e r check v a r i a b l e i n a d d i t i o n to T. dated V i s updated from the e n t h a l p y balance w h i l e T i s up- from a b u b b l e - p o i n t c a l c u l a t i o n , based composition, x. For the purpose of computing V, rearranged i To achieve t h i s , constant-composition s u b s t i t u t e the form equation V i K = into Equation for V the enthalpy balance i s i n t o the r e p o r t e d l y more e f f i c i e n t ( H o l l a n d , 1975). x on the c u r r e n t l i q u i d - p h a s e (8-13), and set the r e s u l t equal to z e r o . Then solving yields H V = F " N _ 1=1 i - i \ — - h ) 1 I y (H ± 1=1 1 z J 1 (8-24) -317- The main steps of the c a l c u l a t i o n a l procedure (1) Determine (2) I n i t i a l i z e L, V, T and K. the (3) (4) as i n p r e v i o u s a l g o r i t h m s . (The i n i t i a l i z a t i o n 'modified Newton-Raphson' method was Compute v_ from E q u a t i o n balance. are: (8-23) and scheme a p p l i e d to adopted.) determine 1 from component mass Hence compute x_ and y_. C a l c u l a t e T as the b u b b l e - p o i n t based on x and the o p e r a t i n g pressure. (5) Compute the e n t h a l p i e s at the c u r r e n t T, x_ and y_. (6) Determine V from E q u a t i o n (7) T e s t f o r convergence. the i t e r a t i o n ; 8-6-2 (8-24) and L from total-mass I f the outcome i s p o s i t i v e , otherwise go to Step E l i m i n a t i n g Bubble-point I t w i l l be r e c a l l e d terminate 3. Calculation that the b u b b l e - p o i n t of a l i q u i d of composi- t i o n x i s normally determined that reduces then balance. by i t e r a t i v e l y s o l v i n g f o r the temperature the check f u n c t i o n N f(T) = I K x i=l 1 - 1 (8-25) 1 or an i n v e r s e - l o g a r i t h m i c form of i t to z e r o . When E q u a t i o n (8-25) i s employed i n u p d a t i n g T, the r e s u l t would be i n * error liquid to the extent that x i s removed from x , the composition. There i s t h e r e f o r e no equilibrium j u s t i f i c a t i o n i n performing t i m e - d r a i n i n g b u b b l e - p o i n t c a l c u l a t i o n f o r the purpose of o b t a i n i n g an i n e x a c t v a l u e of T. a -318- In the b u b b l e - p o i n t - e l i m i n a t i n g methods experimented upon here, T i s updated by a p p l y i n g a s i n g l e N e w t o n - i t e r a t i o n temperature-point T h i s not step at the current to check f u n c t i o n s of the form of E q u a t i o n (8-25). only r e s u l t s i n tremendous t i m e - s a v i n g f o r each i t e r a t i o n , c o u l d a c t u a l l y l e a d to a r e d u c t i o n i n the number of i t e r a t i o n s , as but the f o l l o w i n g argument shows. By c o n s i d e r i n g the r e l a t i o n s h i p : Z x.i = 1 + (K i i - 1) 6 v ie it will Now be seen that T suppose that {6 1 v ie ie = T, f o r 6 =0.0 bp v while T ie = T,, f o r 9 dp v =1. ,T} i s c u r r e n t l y such that 0 < 6 < 9 . ' v v Carrying out a bubble-point i n a new T such that T, c a l c u l a t i o n at the c u r r e n t x_ would ie < T < T • result ie I f T was greater than T bp before the updating, t h i s would amount to an o v e r c o r r e c t i o n . single-step Newton-iteration The approach proposed here would at l e a s t damp ie the o v e r c o r r e c t i o n . I f T was l e s s than T , the one-step method would a l s o y i e l d a b e t t e r updated v a l u e i f the displacement of 9 ie from 9^ was v ie such as to s h i f t T f a r t h e r away from T * argument a p p l i e s to the case where 0 < 9 v Seven d i f f e r e n t < . A similar 1. v check f u n c t i o n s were s t u d i e d i n the temperature-up-date a l g o r i t h m . I n the p r e s e n t a t i o n of the seven formu- l a t i o n s which f o l l o w s , the f o l l o w i n g r e l a t i o n s h i p s w i l l necessary: one-step- be invoked where -319- i x. i = 1 + (K - 1)6 ' i v Z V i- i 1 + (K 1)9 » 9 i 9 x 8 T _ 1 3T V " [1 + ( i K - l)6 ] K l ' 2 v 3K 9 ^ i 8 T and ^ " V i I f Z [1 + ( K - ± l)9 ] 2 v N Form 1: f(T) = I i-1 K x - 1 (8-26) E q u a t i o n (8-26) i s a p p l i e d w i t h x^ t r e a t e d as independent of T, so that N f (T) - ^ Form 2: Here, 3K x - t ? f(T) - £ i-1 y K i -±. - l i (8-27) i s t r e a t e d as temperature-independent, w i t h the r e s u l t N f(T) = - y i 4 i i-1 K 2 3K 2_i 3T that -320- Form 3: f(T) = N I (K x 1=1 1 1 K y - -±) i (8-28) where the dependences of x^ and y^ on T are ignored. The resulting derivative is N I (x f'(T) = i=l y 8K -gi + KT i 1 v N Form 4: f(T) = I x i=l 3 _1 - 1, with x treated as a funtion of T. J- Thus, we have N z ± i + (K - i ) e " i=l i v f ( T ) z and 1 - I f(T)= < 9 V (8-29) 1 ^ 8T- i=i [ l + (K - i ) e j ' ± Form 5: N £ f(T) = i=l with y y - 1, 1 treated as a function of T. In this case, the working equations become N f ( T ) " * i=l K z 1 + (K - 1) 6 i v " 1 i) e ] i 8K and f(T)= IN (i - e ) i=i [l + v z ± (K ± -gi y <- > 8 30 -321- Form 6 : f(T) This leads N £ i=l = (y - x ), 1 and x 1 are f u n c t i o n s of T. 1 to :(T) - N (K I ^ 1-1 - and where both y 1 f'(T) i " K J i-i [l 1) z ± (8-31) ( = - ± l ) 6 v i 3T + (K - i) e ]' ± y I n a p p l y i n g each of the above f o r m u l a t i o n s , the temperature i s updated a c c o r d i n g to the Newton i t e r a t i o n f o r m u l a T n+1 = T n _ f(T") (8-32) f'(T ) n Form 7 : An i n v e r s e - l o g a r i t h m r e l a t i o n s h i p of the form 1 f(—) N r I = in I K x 1=1 with x t r e a t e d as independent of T. A p p l y i n g the Newton i t e r a t i o n f o r m u l a to t h i s form y i e l d s N r—- ( I T n+1 T n 1 + N K x )(*n i=l I K x 1=1 m N n r I i =L l x ) 1 (8-33) i i^T Only two K computations are r e q u i r e d f o r each temperature u p d a t e , the second b e i n g n e c e s s i t a t e d by the f a c t t h a t f i n i t e - d i f f e r e n c e method. 8K i s determined by a -322- The above r e l a t i o n s h i p s were a p p l i e d to 8 of the systems i n T a b l e 8-1. S e c t i o n 8-6-1 The a l g o r i t h m s a r e , of course, d i f f e r e n t o n l y i n Step 4 where T i s updated relationships. For each system, the f e e d - p r e s s u r e and used. The listed from that of through one of the above f i v e points, uniformly d i s t r i b u t e d temperature ranges i n d i c a t e d i n T a b l e 8-1, r e s u l t s o b t a i n e d are presented i n T a b l e 8-2 over were i n the form of the number of i t e r a t i o n s r e q u i r e d , summed over the f i v e p o i n t s involved. As can be seen from the T a b l e , the d i f f e r e n t q u i t e c l o s e i n t h e i r performance, f o r a l l systems. they are found to g i v e i d e n t i c a l perform- S i n c e F o r m u l a t i o n 4 r e q u i r e s l e s s computational e f f o r t than the other two, 8-6-3 w i t h no s i n g l e f o r m u l a t i o n e x c e l l i n g On the average, F o r m u l a t i o n s 4 through 6 show a s l i g h t edge over the other f o u r , and ances. f o r m u l a t i o n s are i t was Reducing d e c l a r e d the winner. K Computation I n f o l l o w i n g the computational path of S e c t i o n 8-6-2, K i s computed t h r e e times i n every i t e r a t i o n : from E q u a t i o n (8-23), and once f o r the purpose twice f o r the temperature of e v a l u a t i n g update. Now, a s c r u t i n y of the path r e v e a l s that t h e r e i s no u p d a t i n g of x_ and y_ between any was temperature update and the next v_ c a l c u l a t i o n . Advantage taken of t h i s o b s e r v a t i o n to e l i m i n a t e the d i r e c t e v a l u a t i o n of the K r e q u i r e d f o r c a l c u l a t i n g v_ from E q u a t i o n (8-23), by e s t i m a t i n g 9K K from a simple K-T model. In so doing, the f a c t i n the process of u p d a t i n g T i s b e n e f i c i a l . that i s evaluated Table 8-2 I t e r a t i o n c o u n t s f o r d i f f e r e n t check f u n c t i o n f o r m u l a t i o n s f o r o n e - s t e p N e w t o n - I t e r a t i o n update of t e m p e r a t u r e i n BP a l g o r i t h m System Form 1 Form 2 Form 3 Form 4 Form 5 Form 6 F orm 7 HA 30 26 27 27 27 27 30 HD 34 36 35 34 34 34 33 HE 37 37 37 37 37 37 37 HG 43 37 42 40 40 40 42 HH 39 39 39 39 39 39 39 HI 28 31 30 30 30 30 29 HJ 28 29 28 27 27 27 29 HK 24 26 24 24 24 24 24 I 263 261 262 258 258 258 263 -324- L e t us c o n s i d e r expansion of £nK^ = InK *nK 1 I f we with a two-term t r u n c a t i o n of the T a y l o r s e r i e s respect (T ) + n 1 apply 1 r e s u l t i n g equation, n 1 9 J l n K (A - ±-) T T Equation to ^- about p o i n t T . < -± 3(1) n (8-34) n T=T (8-34) to the p o i n t T we This r e s u l t s i n n + * and we s i m p l i f y the would have 3K K i (T n + 1 where the the p o i n t ) = K xp{T (l - T /T n i e n q u a n t i t i e s to the r i g h t n + 1 ) ^ / K j (8-35) of the e q u a l i t y s i g n r e f e r to T . n I n the new form of the a l g o r i t h m , (see S e c t i n 8-6-1) i s f o l l o w e d , the convergence t e s t of Step 9 i f n e c e s s a r y , by the updating of 3K from E q u a t i o n process The (8-35), of a p p l y i n g ^-1 having been p r e v i o u s l y e v a l u a t e d the temperature-updating scheme of S e c t i o n 8-6-2. K computation i n Step 3 i s thus 8-6-4 E l i m i n a t i n g K-T eliminated. Derivative Calculation Seeking a f u r t h e r improvement on s t e p s were taken to e l i m i n a t e the d i r e c t doing, every the a l g o r i t h m of S e c t i o n 8-6-1, i e v a l u a t i o n of and, by so f u r t h e r reduce the number of K computations by one ( t o one) in iteration. To 8-3-3 i n the and achieve t h i s o b j e c t i v e , the r e l a t i o n s h i p developed i n S e c t i o n presented as E q u a t i o n (8-20) was employed. r e l a t i o n s h i p r e q u i r e s a knowledge of the enthalpy according to the a l g o r i t h m However, s i n c e terms and since, as p r e v i o u s l y o u t l i n e d , the e n t h a l p i e s are the -325- 3 not e v a l u a t e d u n t i l a f t e r the temperature needed), some r e t h i n k i n g was i 7 ^ —directly 9 determine Equation necessary. The approach adopted i is was to K f o r the f i r s t i t e r a t i o n and (8-20) f o r a l l subsequent i t e r a t i o n s , r e s u l t i n g from the p r e v i o u s to estimate i t from u s i n g the enthalpy values iteration. A f t e r a l l the bludgeoning, following K update ( f o r which the a l g o r i t h m now assumes the shape: (1) Determine H (2) I n i t i a l i z e L, V, T, x and y. (3) Compute F as b e f o r e . from E q u a t i o n mass-balance. Compute K. (8-23) and determine 1^ from component Hence compute x and y_. 3K (4) Compute K. by f i n i t e I f i t i s the f i r s t difference; iteration, otherwise, then compute 3K compute — ^ from E q u a t i o n (5) Update T by the method recommended i n S e c t i o n 8-6-2. (6) Compute the e n t h a l p i e s at the c u r r e n t T, x and y_. (7) Update V from E q u a t i o n (8-24) and L (8) T e s t f o r convergence. I f the outcome i s p o s i t i v e t e r m i n a t e the i t e r a t i o n ; otherwise (9) Update K from E q u a t i o n (8-35) and 8-7 Applications The t h e o r i e s presented from total-mass go to Step go to Step balance. then 9. 3. i n the f o r e g o i n g s e c t i o n s of t h i s have been o r g a n i z e d i n t o 15 d i f f e r e n t algorithms. (8-20). The chapter r e f e r e n c e names -326- for, Table as w e l l as s h o r t d e s c r i p t i o n s o f , these algorithms are presented in 8-3. Both the c o n v e n t i o n a l and the B a r n e s - F l o r e s formulations Newton-Raphson method have each been implemented i n two forms: s i o n that employs K v a l u e s t h e i r composition dependence; and a v e r s i o n that does a c o r r e c t i o n on the K v a l u e s — references that have not been i t e r a t i v e l y at the c u r r e n t v a l u e s two v e r s i o n s , i n each case, 8-7-1 Convergence terminating i s the recommended different the a d i a b a t i c f l a s h i t e r a t i o n : and T. The choice. a temperature c r i t e r i o n , flash calculations, f o r terminating that i s s i m i l a r l y chosen to match that employed i n saturation-point calculations, v i z : | n T - T n _ 1 I t e r a t i o n i s terminated and criterion. The temperature c r i t e r i o n was satisfied composition c r i t e r i a have been employed i n t h i s work f o r The v_ c r i t e r i o n i s the same as that employed terminating corrected f o r Criteria a vapour component-flow isothermal v a ver- on the two methods c o n t a i n no i n f o r m a t i o n as to which of these Two of 0 of the < 0.001 | a f t e r the nth i f the above two c r i t e r i a are simultaneously. T a b l e 8-3 Reference names and d e s c r i p t i o n s o f a d i a b a t i c - f l a s h Reference Name Serial Number algorithms Description 1 Conventional Newton-Raphson 1 Based on the conventional Newton-Raphson method discussed in Section 8-3-1, and employs K values that are corrected for composition dependence. 2 Conventional Newton-Raphson 2 Same as 1 but uses K values that are NOT corrected for composition effects. 3 Barnes-Flores 1 Based on the Barnes-Flores version of. the Newton-Raphson method discussed i n Section 8-3-2, with K values that are corrected for composition-dependence. 4 Barnes-Flores 2 Same as 3 but with K values that are NOT corrected for composition effects. 5 Modified Newton-Raphson 1 Based on proposed formulation of Section 8-3-3, with evaluation of K-T, H-T and h-T derivatives only for f i r s t 3 iterations, and without application of Equations (8-20) and (8-35). 6 Modified Newton-Raphson 2 Same as 5 but with application of Eq.(8-35) to estimate K values employed i n composition update. 7 Modified Newton-Raphson 3 3K Same as 6 but with ^ estimated from Eq.(8-20). 8 Double-iteration Regula-falsi Based on the method described i n Section 8-4. 9 SR 1 Based on the method discussed i n Section 8-5-1. 10 SR 2 Based on the method described i n Section 8-5-2. 11 SR 3 Based on the method discussed i n Section 8-5-3. 12 BP with Bubble point Based on the algorithm of Section 8-6-1. 13 Proposed BP 1 Same as 12 but with T update as described i n Section 8-6-2 14 Proposed BP 2 Same as 13 but with reduction i n K computation application of Equation (8-35). 15 Proposed BP 3 Same as 14 but with K-T derivatives estimated from Equation (8-20). through -328- 8-7-2 R e s u l t s and O b s e r v a t i o n s I n a l l the a p p l i c a t i o n s f o r which r e s u l t s have been p r e s e n t e d here, f i v e p o i n t s were u t i l i z e d f o r each system. p o i n t s were spread u n i f o r m l y on the l i n e feed-temperature and l i n k i n g the p o i n t of minimum and - p r e s s u r e to the p o i n t of maximum - p r e s s u r e i n the ranges As a f i r s t I n each case, the f i v e given i n Table step i n the comparison, to 10 of the systems. feed-temperature 8-1. a l l 15 a l g o r i t h m s were a p p l i e d The r e s u l t s are p r e s e n t e d i n T a b l e 8-4 and they l e a d to the f o l l o w i n g o b s e r v a t i o n s : (1) A check of the e q u i l i b r i u m temperature, generated by the d i f f e r e n t flows and compositions a l g o r i t h m s showed t h a t a l l the converged s o l u t i o n s are e x a c t l y the same i n a l l c a s e s . (2) The BP methods, as expected, f a i l when a p p l i e d to w i d e - b o i l i n g systems. (3) None of the three SR a l g o r i t h m s i s t o t a l l y r e l i a b l e f o r handling n a r r o w - b o i l i n g systems, and o n l y 'SR a l g o r i t h m 3' performs wide-boiling (4) well for systems. The c o n v e n t i o n a l Newton-Raphson and the B a r n e s - F l o r e s algorithms perform b e t t e r when the K v a l u e s are c o r r e c t e d f o r c o m p o s i t i o n dependence i n an i n n e r i t e r a t i o n - l o o p . The ' M o d i f i e d Newton-Raphson' a l g o r i t h m s , implemented as they are without c o m p o s i t i o n - c o r r e c t i o n of the K v a l u e s , are q u i t e r e l i a b l e and fast. (5) A comparison of the i t e r a t i o n requirements f o r the c o n v e n t i o n a l Newton-Raphson and the B a r n e s - F l o r e s a l g o r i t h m s ( v e r s i o n 1 i n e i t h e r case) r e v e a l e d that t h e i r i t e r a t i o n requirements are q u i t e c l o s e , the l a t t e r being b e t t e r f o r o n l y about 50% of the systems. This i s r e f l e c t e d i n the e x e c u t i o n times. Table 8-4 Computation times (CPU seconds) for adiabatic-flash algorithms Newton-Raphson Algorithms System Conventional 1 2 Modified Barnes-Flores 1 2 1 2 BP Algorithms SR Algorith ms Doubleiteration Regula-falsi 1 f 1.6541 2 3 3 with bubble point Proposed 1 2 3 1.4905 0.5807 0.5241 0.4600 f 1.9511 0.8400 0.8076 0.6356 f 1.0503 1.7668 0.8386 0.7543 0.5667 2.3349 f 0.9961 2.1930 0.8708 0.7885 0.6070 1.4005 2.2859 f 1.0265 1.7926 0.6992 0.6568 0.5504 1.4982 2.2945 2.5324 f 1.1266 2.2215 0.8322 0.7573 0.7122 0.9677 0.8372 1.7679 f f 2.3173 0.9870 0.4729 0.4434 0.3856 >3.6381 1.5334 1.4006 1.6895 1.3565 f f 0.5327 f f f f >3.2032 1.5082 >3.2419 1.2715 1.1450 1.2742 1.3006 f f 0.5636 f f f f >4.7989 1.9005 >4.9861 l l .3975 1.0689 1.2453 1.3712 1.8211 f 1.0826 f f f f 0.8956 HA 1.3333 2.8010 1.2250 2.4143 1.0709 0.7990 0.7777 HD 2.1308 >5.9456 2.2373 >5.9202 1.4234 1.0837 0.8378 1.7538 f f HG 1.9908 1.9358 2.2158 2.1779 1.3669 1.0317 1.0476 1.9536 1.9365 HH 1.9040 2.3249 2.2000 2.5166 1.3142 1.0047 0.9683 1.7696 HI 1.7066 2.4211 1.7304 2.5637 1.2079 0.9694 0.9278 HJ 2.5608 >4.5484 2.4282 >6.3566 1.8122 1.6929 HK 1.3765 1.4789 1.5082 1.5440 1.0959 HN 1.6435 >3.5117 1.6134 HO 1.6060 HP 2.0733 0.5696 ast one joint. f => failure in at le int, within the set upper bound on number of iterations > => convergenee was not reac ied, for at leas t one po -330- (6) The i t e r a t i o n requirements for the different BP algorithms are presented i n Table 8-5. A comparison of the iterations for 'BP with Bubble point' and 'Proposed BP 1' vindicates the argument presented i n Section 8-6-2 regarding possible 'overcorrection' with the bubble-point algorithm. The results i n the Table further show that the time-saving steps r e s u l t i n g i n Proposed BP algorithms 2 and 3 respectively do not have any s i g n i f i c a n t adverse effects on their i t e r a t i o n requirements. (7) An i t e r a t i o n comparison (not presented) for the 'Modified Newton-Raphson' algorithms showed that Algorithm 2 required less iterations, on the average, than Algorithm 1. It also showed that Algorithm 3 required about the same number of iterations Algorithm 2 for narrow-boiling systems, but required much more — sometimes more than 20% more — for wide-boiling (8) as systems. The proposed BP methods showed greater s t a b i l i t y with wide-boiling systems than does the 'BP with bubble point' method; for while the l a t t e r method f a i l e d i n every case of wide-boiling mixture application, the former algorithms were found to converge for one out of every five points when applied to systems HO and HN. This suggests that they could show good behaviour with simple damping techniques. However, this option was not pursued, i n view of the experience with the SR methods where damping does not lead to good performance with ALL narrow-boiling systems. Guided by the results from the f i r s t stage of the applications, the f i n a l comparison was designed to exclude some of the 15 algorithms, and to involve separate comparisons for narrow-boiling and wide-boiling systems. Table 8-5 Comparison of I t e r a t i o n count f o r BP algorithms System With Bubble Point Proposed 1 Proposed 2 Proposed 3 HA 31 27 27 29 HD 34 34 36 36 HG 38 40 40 37 HH 40 39 39 38 HI 31 30 31 33 HJ 32 27 27 33 HK 23 24 24 26 -332- Narrow-boiling Systems F o r the n a r r o w - b o i l i n g systems (systems HA through HK), the f o l l o w i n g a l g o r i t h m s were i n v o l v e d : (a) Conventional Newton-Raphson 1. (b) M o d i f i e d Newton-Raphson 2. (c) M o d i f i e d Newton-Raphson 3. (d) Double-iteration regula (e) BP w i t h (f) Proposed BP 3. The computation times falsi. bubble-point. are presented b e s t r e s u l t s are o b t a i n e d w i t h i n T a b l e 8-6. The o v e r a l l the 'Proposed BP a l g o r i t h m 3', w h i l e , i n the c l a s s of methods that apply to both narrow- and w i d e - b o i l i n g systems, the 'Modified Newton-Raphson a l g o r i t h m 3' i s the p r e f e r r e d choice. W i d e - b o i l i n g Systems For the w i d e - b o i l i n g systems (systems HL through HP), the comparison was amongst the f o l l o w i n g a l g o r i t h m s : (a) Conventional (b) M o d i f i e d Newton-Raphson 2 (c) M o d i f i e d Newton-Raphson 3 (d) Double-iteration regula (e) SR a l g o r i t h m 3 Before Newton-Raphson 1 falsi the comparison was made, a f u r t h e r look was taken a t 'SR a l g o r i t h m 3' w i t h a view to determining how i t would f a r e with w i d e - b o i l i n g systems without any te
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Computational algorithms for multicomponent phase equilibria and distillation Ohanomah, Matthew Ochukoh 1981
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Title | Computational algorithms for multicomponent phase equilibria and distillation |
Creator |
Ohanomah, Matthew Ochukoh |
Date Issued | 1981 |
Description | This work has two major objectives: (1) To develop reliable, stable, easily-programmable and fast computational algorithms that would apply to process operation and design. (2) To determine the best algorithms for specific operations through comparative studies. The study covers the following areas: (1) Isothermal phase equilibria in up to four phases — one vapour, two liquid and one solid — with emphasis on vapour-liquid, liquid-liquid, liquid-solid and vapour-liquid-liquid systems. (2) Sensitivity analysis in vapour-liquid equilibria. (3) Saturation-point calculation methods. (4) Adiabatic vapour-liquid equilibria. (5) Conventional equilibrium-stage distillation-unit calculations. The study deals with multicomponent systems and, except where constrained by lack of information, all nonidealities are rigorously accounted for. The phase-equilibria studies embody both mass-balance and free-energy-minimization methods and some of the algorithms are, like the sensitivity-analysis study, based on geometric programming. Vector projection methods, for accelerating multivariate successive-substitution iteration, and a quadratic form of Wegstein's projection method have been developed and successfully applied. The saturation-points study includes three interpolation methods — regula falsi and quadratic interpolations and a dynamic form of Lagrange interpolation — a quasi-Newton formulation and the generally applied Newton and Richmond methods. The results favour the interpolation methods. For adiabatic-flash and distillation-unit calculations, algorithms employing two-dimensional Newton-Raphson, BP-partitioning and SR-partitioning methods have been developed. Methods for reducing equilibrium-ratio calculation to once per iteration per stage in the first two classes and for reducing enthalpy evaluation In the third class, have also been developed. For every problem-type, relevant algorithms have been applied to a wide class of systems and compared. However, the lack of data excluded any applications on solid-liquid-liquid-vapour equilibria and on wide-boiling distillation. |
Subject |
Phase rule and equilibrium Algorithms |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-04-15 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
IsShownAt | 10.14288/1.0058997 |
URI | http://hdl.handle.net/2429/23665 |
Degree |
Doctor of Philosophy - PhD |
Program |
Chemical and Biological Engineering |
Affiliation |
Applied Science, Faculty of Chemical and Biological Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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