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Computational algorithms for multicomponent phase equilibria and distillation Ohanomah, Matthew Ochukoh 1981

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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