1 EFFICIENT USE OF THERMODYNAMIC DATA IN PROCESS FLOUSHEETING by Godpower Iheanyi MADUABUEKE June 1987 A thesis submitted for the degree of Doctor of Philosophy of the University of London and for the Diploma of Membership of the Imperial College. Department of Chemical Engineering and Chemical Technology, Imperial College of Science and Technology, London, SW7 2AZ.
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EFFICIENT USE OF THERMODYNAMIC DATA IN ......C2.1 Schematic Representation of a Stage in a Distillation Column C2.2 Block-tridiagonal Structure of Distillation Column Model Appendix
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1
E FFIC IE N T USE OF THERMODYNAMIC DATA
IN PROCESS FLOUSHEETING
by
Godpower Iheany i MADUABUEKE
June 1987
A t h e s i s subm i t ted f o r the degree of Doc tor of Ph i l o sophy
of the U n i v e r s i t y of London and f o r the Diploma
o f Membership o f the Impe r ia l C o l l e g e .
Department o f Chemical E ng in ee r i n g and Chemical Technology,
Im p e r ia l C o l l e g e of Sc ience and Techno logy ,
London, SW7 2AZ.
2
D E D I C A T I O N
In Memory o f Mum
3
ACKNOWLEDGEMENTS
I would l i k e t o e xp r e s s my g r a t i t u d e to Drs. S. M a cch ie t t o
and R. S z c z e p a n s k i f o r t h e i r i n v a l u a b l e h e lp and encouragement
throughout the work.
I am v e r y g r a t e f u l t o t h e F e d e r a l Government of N i g e r i a
and Overseas Research Award Scheme f o r f i n a n c i a l suppo r t .
The a s s i s t a n c e r e c e i v e d f rom Ochuchu, Ch imdi, Ch inye re ,
C .C . P a n t e l i d e s , Mr. and M r s D a v i d J u a r e z , C . L . C h e n , a r e
p a r t i c u l a r l y a p p re c ia t e d .
F i n a l l y , a l a s t word o f g r a t e f u l n e s s t o Sandra Cu r ley fo r
her p a t i e n ce and c a r e f u l t y p i n g o f t h i s t h e s i s .
3 (i)
ERRATA
Abs t rac L
page 4, para 2,
line 3
Chapter 2
page 53, line 1
Chapter 3
1. page 70 line 4
2. page 70, para 2,
line 3.
3. page 90, para 3,
lint - b
4. page 123, line 16
5. page 124, para 3 ,
line 7
Chapter 4
1. page 137, para 1,
line 3
2. page 150, para 2,
line 7
Change
and/or
naphta
assume a procedure
Assuming equation (3.1)
phase splits
derivatives may
only included
environmental
absolute and
To
and perhaps
naphtha
assume that a
procedure
assuming that
equation (3.1)
the phase splits
derivatives by
perturbation may
included only
to the environmental
an absolute and a
? (ii)
Chapter 4 contd.
3. page 157, para 2,
line 19
4. page 160, para 2,
line 7
5. page 185,
para 2 line 7
6. page 187, para 1,
line 4
7. page 206 ,
3 lines below
equation Cl.6
Change
absolute and
absolute and
only secured
quickly identify
compression
To
an absolute and a
an absolute and a
secured only
quickly to identify
compressibility
References
Curtis, A.R., Powell, M.J.D., and Reid, J.K. (1974),
"On the estimation of sparse Jacobian matrices", J. Inst. Math.
Appl., 13, pi17 .
4
absjracj
The use o f t h e rm o p h y s i c a l p r o p e r t i e s (TP) data i n p ro ce s s
f l o w sh e e t i n g poses two im p o r t a n t problems: e f f i c i e n t i n c o r p o r a t i o n
o f t h e t h e r m o p h y s i c a l p r o p e r t i e s m o d e l s i n t h e o v e r a l l d e s ig n
c o m p u t a t i o n s and q u a n t i f i c a t i o n o f t h e e f f e c t o f T P m o d e l
i n a c c u r a c i e s or parameter u n c e r t a i n t i e s on p rocess de s igns .
P r o c e s s d e s i g n c o m p u t a t i o n s r e q u i r e r i g o r o u s TP p o i n t
v a l u e s a n d / o r d e r i v a t i v e s t o be p r o v i d e d . The TP p o in t v a l u e s
u s u a l l y pose no problems. A new approach i s p ropo sed and t e s t e d
f o r d e r i v i n g t h e p a r t i a l d e r i v a t i v e s w h i c h i s n o n - i t e r a t i v e , ,
a vo id s the need f o r p e r t u r b a t i o n , w i th sma l l s t o r a g e r e q u i r e m e n t
and a t t h e same t im e u s e s r i g o r o u s TP d a t a . T h e r m o p h y s i c a l
p r o p e r t i e s m o d e l s a re u s u a l l y p r o v i d e d a s p r o c e d u r e s ( f r o m a
p h y s i c a l p r o p e r t i e s p a c k a g e ) and i t i s shown how to o b t a i n t h e
ex.act p a r t i a l d e r i v a t i v e s o f t h e o u t p u t v a r i a b l e s o f a g e n e r a l
procedure w i th re spec t t o i t s i n p u t s , w i th p a r t i c u l a r r e f e r e n c e t o
VLE f l a s h and d i s t i l l a t i o n column m o d u le s . The method i n v o l v e s
t h e a n a l y t i c d i f f e r e n t i a t i o n o f c u r r e n t l y used t hermoy dnam i c
p r o p e r t i e s m o d e l s . The a p p l i c a t i o n o f t h e s e i d e a s on t y p i c a l
f l o w s h e e t i n g prob lems have r e s u l t e d i n s u b s t a n t i a l sav ings (30 % -
75 %) i n s im u l a t i o n t imes o ve r c u r r e n t methods. The r e s u l t s a l s o
i n d i c a t e t h a t p h y s i c a l p r o p e r t i e s p a c k a g e s shou ld p r o v i d e both
p o in t v a l u e s and d e r i v a t i v e s of TP models.
5
B.lS£_r_oy.§ f i r s t - o r d e r p r o c e s s d e s i g n s e n s i t i v i t y to
p h y s i c a l p r o p e r t i e s i s o b ta in ed i n an e f f i c i e n t manner by a s l i g h t
m o d i f i c a t i o n o f t h e a l g o r i t h m f o r e v a l u a t i n g the o u t p u t - i n p u t
g r a d i e n t s o f a g e n e r a l p r o c e d u r e . We a p p l i e d the te chn ique to
t y p i c a l VLE f l a s h , s u p e r f r a c t i o n a t i o n columns, and an integrated .
p r o c e s s p l a n t . The s e n s i t i v i t y i n f o r m a t i o n was used i n the
f o l l o w i n g a p p l i c a t i o n s : i d e n t i f i c a t i o n o f the c r i t i c a l p h y s i c a l
p r o p e r t i e s p a r a m e t e r s / m o d e l s on p r o c e s s de s ign ; rank ing o f the
c r i t i c a l pa ram e te r s /m ode ls i n o r d e r o f im p o r t a n c e , and t h e r e b y
i d e n t i f i c a t i o n o f which parameters o r models t o e s t im a t e / a d j u s t to
" b e s t " rep re sen t a g iv en set of e x p e r im e n t a l o r p l a n t o p e r a t i n g
d a t a ; i d e n t i f i c a t i o n o f w h e r e d e s i g n i s most s e n s i t i v e t o
u n c e r t a i n t i e s i n TP pa rameters o r mode ls ; c h o i c e o f m easu remen ts
l o c a t i o n ; cho ice o f s o l v e n t s ; and d e t e rm in a t i o n o f the e f f e c t s of
p h y s i c a l p r o p e r t i e s i n a c c u r a c i e s on t h e l o c a t i o n o f z o n e s o f
m ax im um e n r i c h m e n t f a c t o r s f o r d i s t i l l a t i o n c o l u m n s .
6
™ JrJL 9 F_ CONTENTSPa.ge
CHAPTER ONE : INTRODUCTION 14
1 .1 . ARCHITECTURES OF FLOWSHEET SIMULATORS 141 .2 . NUMERICAL SOLUTION ALGORITHMS 181 .3 . PROVISION AND USE OF TD DATA IN
PROCESS SIMULATIONS 261 .3 . 1 . E f f i c i e n t i n c o r p o r a t i o n of TD data
in p ro cess d e s ig n 271 .3 . 2 . S e n s i t i v i t y o f p ro ce s s des ign t o
u n c e r t a i n t i e s i n p h y s i c a l p r o p e r t i e s 30
CHAPTER TWO : REVIEW OF EXISTING THERMODYNAMIC PROPERTYDATA INTERFACE STRATEGIES 34
2 .1 . CRITERIA FOR EVALUATION OF TD DATAINTERFACE STRATEGY 34
2 .2 . BRIEF DESCRIPTION OF A TYPICAL PHYSICALPROPERTIES PACKAGE 40
2 .3 . THE BLACK-BOX APPROACH 442 .4 . THE WESTERBERG APPROACH 472 .5 . THE TWO-TIER APPROACH 49
2 .5 .1 . Fundamental problems a r i s i n g fromt w o - t i e r s t r a t e g y 55
CHAPTER THREE : EFFICIENT STRATEGY FOR INTERFACING THERMODYNAMIC PROPERTY DATA WITHFLOWSHEETING PACKAGES 69
3 .1 . EXACT PROCEDURE DERIVATIVES 693 .2 . DERIVATIVES OF TD PROPERTIES MODELS 743 .3 . COMPUTATION OF EXACT DERIVATIVES OF TYPICAL
VLE PROCEDURES 793 . 3 . 1 . Numer ica l e xpe r im en ts 793 . 3 . 2 . D i s c u s s i o n o f r e s u l t s and c o n c lu s io n s 87
3 .4 . APPLICATION OF NEW TD INTERFACE STRATEGY TOFLOW SHEETING EXAMPLES 943 . 4 . 1 . The SPEEDUP package 94
3 . 4 . 1 . 1 . V a r i a b l e t ypes 953 . 4 . 1 . 2 . P rocedu res 963 . 4 . 1 . 3 . Numer ica l s o l u t i o n o p t i o n s 96
3 . 4 . 2 . F low shee t in g examples 983 . 4 . 2 . 1 . S imp le d i s t i l l a t i o n column des ign 99
7
3 . 4 . 2 . 2 . Design and s im u l a t i o n o f Cavet tf l ow shee t 102
3 . 4 . 2 . 3 . Des ign o f coup led d i s t i l l a t i o ncolumns w i th energy r e c y c l e 105
3 - 4 . 2 . 4 . Design o f coup led d i s t i l l a t i o ncolumns w i t h mass and energy r e c y c l e s 108
3 . 4 . 2 - 5 - O p t im i z a t i o n of coupled f l a s h u n i t s 1103 .5 . NUMERICAL RESULTS/DISCUSSIONS 1123 .6 . CONCLUSIONS 127
CHAPTER FOUR : EFFICIENT DETERMINATION OF PROCESS SENSITIVITYTO PHYSICAL PROPERTIES DATA 130
4 .1 . SENSITIVITY TO CONSTANT PARAMETERS 1304 .2 . SENSITIVITY TO MODEL FUNCTIONS 1324 .3 . APPLICATION TO VLE EXAMPLES 137
4 .3 . 1 . Thermodynamic models 13a4 . 3 . 2 . I so the rma l f l a s h p rocedures 1414 .3 . 3 . D i s t i l l a t i o n column procedure 1414 . 3 . 4 . I n t e r g r a t e d M u l t i u n i t F lowsheet 143
4 .4 . APPLICATION TO THE LOCATION OF CONTROL MEASUREMENTSIN DISTILLATION COLUMNS- 145
CHAPTER FIVE : GENERAL CONCLUSIONS/RECOMMENDATIONS 183
REFERENCES lgg
NOMENCLATURE 2QQ
APPENDICESA : FLOW SHEETING REVIEWS AND NUMERICAL SOLUTION
METHODS. 203
B : TYPICAL TP AVAILABLE FROM A PHYSICALPROPERTIES PACKAGE. 204
C : DETAILED DESCRIPTION OF EXAMPLES.
C1 A n a l y t i c d e r i v a t i v e of f u g a c i t y c o e f f i c i e n t s( u s i n g SRK-equa t ion) with respec t t o temperature , p r e s s u r e , and compos i t ion as w e l l as p h y s i c a l p r o p e r t i e s c on s ta n t s o f components used i n t h i s s tudy . 205 '
8
C2 D i s t i l l a t i o n column procedure d e r i v a t i v e sand t e s t prob lems f o r e v a l u a t i o n o f f l a s h and d i s t i l l a t i o n column procedure d e r i v a t i v e s . 215
C3 D e t a i l e d s p e c i f i c a t i o n s o f f l o w sh e e t i n g problems. 226
D : D1 D e r i v a t i v e s o f f u g a c i t y c o e f f i c i e n t s and excesse n t h a l p i e s ( u s ing SRK-equat ion) w ith re spec t to b in a r y i n t e r a c t i o n c o e f f i c i e n t s . 230
D2 D e t a i l e d s p e c i f i c a t i o n of VLE examples. 234
D3 D e t a i l e d s p e c i f i c a t i o n of column f o r o p e r a b i l i t y and c o n t r o l s tudy . 244
E : M i c r o f i c h e programe l i s t i n g
9
List of Figures
Chapter 11.1 Levels of Flowsheet Computations1.2 Schematic Diagram of Two-Tier Strategy
Chapter 22.1 Simple Hypothetical Flowsheet2.2 The Physical Properties Data System
Chapter 33.1 Single-stage Flash Unit3.2 Breakdown of computing time for generation of flash base points and
procedure derivatives calculations3.3 Distillation Column3.4 Flowsheet of Cavett Four Flash Process3.5 Coupled Distillation Columns with Energy Recycle3.6 Coupled Distillation Columns with Mass and Energy Recycle3.7 Coupled Flash Units
Chapter 44.1 Variation of Methane Flow (Vapour Phase) with Binary Interaction
Coefficient - Flash # 14.2 Variation of Methane Flow (Vapour Phase) with Binary Interaction
Coefficient - - Flash # 24.3a Sensitivities of Ethylene Vapour Flow Profile to all the Binary Interaction
Coefficients - 8(i,j) = 0.0 (Example D2.3)4.3b Sensitivities of Ethylene Flow Profile (Vapour Phase) to all the Binary
Interaction Coefficients---- S(i,j) = 0.0 (Example D2.3)4.4 Sensitivities of Temperature Profile to all the Binary Interaction Coefficients
- S(i,j) = 0.0 (Example D2.3)4.5 Ethylene Product Purity Variation with Binary Interaction Coefficient4.6 Variation of Reboiler and Condenser Duties to Ethylene-Ethane Binary
Interaction Coefficient — 8(2,3)4.7a Normalised Sensitivities of Vapour Flow Profiles to Efficiency - Example
D2.34.7b Sensitivities of Ethane and Ethylene Vapour Flow Profiles to Efficiency4.8 Variation of Reboiler and Condenser Duties with Efficiency - Example D2.3
10
4.9 Ethylene Product Purity Variation with Efficiency4 .10a Sensitivities of Ethylene Flow Profile (Vapour Phase) to Errors in Enthalpy
Models — Example D2.34 .10b Sensitivities of Temperature Profile to Errors in Enthalpy Models.4.1 la Ethylene Product Purity Vs. Errors in Vapour Enthalpy Model — Example
D2.34.1 lb Ethylene Product Purity Vs. Errors in Ideal Liquid Enthalpy M odel-----
Example D2.34.11c Ethylene Product Purity Vs. Errors in Excess Liquid Enthalpy Model -
Example D2.34 .12a Variation of Reboiler and Condenser Duties to Errors in Vapour Enthalpy
Model - Example D2.34.12b Variation of Reboiler and Condenser Duties to Errors in Ideal Liquid
Enthalpy Model — Example D2.34.12c Variation of Reboiler and Condenser Duties to Errors in Excess Liquid
Enthalpy Model - Example D2.34.13a Sensitivities of Propylene Vapour Row Profile to all the Binary Interaction
Coefficients - S(i,j) = 0.0 (Example D2.4)4.13b Sensitivities of Propylene Vapour Row Profile to all the Binary Interaction
Coefficients - S(i,j) = 0.0 (Example D2.4)4 .14a Sensitivities of Propylene Vapour Row Profile to all the Binary Interaction
Coefficients - 8(i,j) = 0.0 (Example D2.5)4.14b Sensitivities of Propylene Vapour Row Profile to all the Binary Interaction
Coefficients - S(i,j) (Example D2.5)4.15 Variation of Propylene Product Purity with Propylene - Propane Binary
Interaction Coefficient - Example D2.44.16 Variation of Propylene Product Purity with Propylene - Propane Binary
Interaction Coefficient - Example D2.54.17 Variation of Reboiler and Condenser Duties with Propylene - Propane
Binary Interaction Coefficient — Example D2.44.18 Variation of Reboiler and Condenser Duties with Propylene - Propane
Binary Interaction Coefficient---- Example D2.54.19 Sensitivities of Vapour Row Profiles to Efficiency - Example D2.44.20 Variation of Propylene Product Purity with Efficiency — Example D2.44.21 Variation of Reboiler and Condenser Duties with Efficiency — D2.44.22 Sensitivities of Temperature Profile to Errors in Enthalpy-----Example
D2.44.23 Propylene Product Purity Vs. Errors in Enthalpy Models - Exmaple D2.4 4.24a Variation of Reboiler and Condenser Duties with Errors in Vapour Enthalpy
Model - Example D2.4
11
4.24b Variation of Reboiler and Condenser Duties with Errors in Ideal Liquid Enthalpy Model - Example D2.4
4.24c Variation of Reboiler and Condenser Duties with Errors in Excess Liquid Enthalpy Model - Example D2.4
4.25a Rigorous and Approximate values of Enrichment Factor at 5(i,j) = 0.0 4.25b Rigorous and Approximate Values of Enrichment Factor at 5(i,j) = 0.01 4.25c Rigorous and Approximate values of Enrichment Factor at 8(i,j) = 0.02
Appendix.CC2.1 Schematic Representation of a Stage in a Distillation Column C2.2 Block-tridiagonal Structure of Distillation Column Model
Appendix DD2.1 Vapour How Rate Profile — Example D2.3 D2.2 Liquid Flow Rate Profile — Example D2.3D2.3 Vapour How Rate Profile---- Example D2.4D2.4 Liquid Flow Rate Profile — Example D2.4D2.5 Vapour How Rate Profile — Example D2.5D2.6 Liquid How Rate Profile — Example D.5D3.1 Liquid How Rate Profile — Example D3.1D3.2 Vapour How Rate Profile — Example D3.1D3.3 Rigorous Enrichment Factor Profile at the Base Value of 8(i,j) = 0.0
12
LIST OF TABLES
Chapter 22.1 Effectiveness of TD data interface strategies measured against criteria stated
in section 2.1
Chapter 33.1 Relative times for fugacity/activity coefficient and its (NC+2) derivatives.
The number in parentheses are the equivalent number of fugacity/activity coefficient base points
3.2 Equivalent number of isothermal flash evaluations required to generate output sensitivities with respect to all NC+2 input variables
3.3 Computer times (CPU seconds) for distillation column procedure derivatives evaluation with respect to distillate rate and reflux ratio (analytic TD derivatives).
3.4 Summary of flowsheeting examples3.5 Solution statistics for problem C3.1 a3.6 Solution statistics for problem C3.1 b3.7 Initial Values and Solutions of problems C3.1a, b3.8 Simulation results for CAVETT problems3.9 Solution time (CPU seconds on CYBER 855)3.10 Number of iterations / function evaluations3.11 Equivalent number of thermodynamic calls3.12 Simulation results for example C3.73.13 Solution statistics for problem C3.73.14 Simulation results for example C3.83.15 Solution statistics for problem C3.83.16 Results for optimization problem3.17 Solution statistics for optimization problem (C3.9)
Chapter 44.1 Main results for ethylene/ethane splitter4.2 Main results for propy lene/propane splitter
A ppendix AA1 Reviews of process flowsheetingA2 Numerical solution methods for nonlinear algebraic equations
Appendix BTypical data available from a physical properties package
13
Appendix CC l.l Derivatives of fugacity coefficients using the SRK equation of state with
respect to temperature, pressure and composition.C1.2 Test problems for evaluation of typical TD properties derivatives C1.3 Non-zero binary interaction parameters used in SRK model C l.4 UNIQUAC binary interaction parameters (Prausnitz et. al., 1980)C l.5 UNIQUAC parameters (Prausnitz et. al., 1980)C l.6 Specific heat capacity constants used in our model (Reid et. al., 1977)C2.1 Elements of the Jacobian and right hand side matrices for distillation column
procedureC2.2 Computation of a Newton step in Naphtali-Sandholm algorithm C2.3 Test problems for evaluating the efficiency of flash and distillation
procedure derivatives computation C2.4 Specification of distillation column unit operationC2.5 Analytic distillation procedure derivatives at the base point given in Table
3.7C3.1 Distillation column design C3.2 CAVETT four flash flowsheetC3.3 Design of coupled distillation column with energy recycle - problem C3.7 C3.4 Design of coupled distillation columns with mass and energy recycles - -
problem C3.8C3.5 Optimization of coupled flash units - problem C3.9
Appendix DD 2.1 Flash and distillation specificationsD2.2 Vapour component flowrate sensitivities to binary interaction coefficients D2.3 Vapour component flow rate sensitivities to binary interaction coefficients D2.4 Sensitivities of reboiler and condenser duties to physical properties -
example D2.3D2.5 Sensitivities of reboiler and condenser duties to physical properties D2.6 Sensitivities of CAVETT four flash design (5-component example) to
binary interaction parameters D 3.1 Column specifications
14
CHAPIERONE
INJRODUCJION
P roces s f l o w s h e e t i n g i s the use o f compute r -a ids t o se t up
and s o l v e t h e hea t and mass b a l a n c e s , s p e c i f i c a t i o n s , d e s i g n
c o n s t r a i n t s , e t c . , f o r a g i v e n c h e m i c a l p ro cess . F lowshee t ing
p a c k a g e s have been d e v e l o p e d t o p e r f o rm dynam ic s i m u l a t i o n ,
o p t i m i s a t i o n , and c o s t e s t i m a t i o n s t u d i e s (W e s t e r b e r g e t a l
(1979)) . Some o f the e x t e n s i v e rev iews on the v a r i o u s a p p ro a c h e s
a d o p t e d f o r f l o w s h e e t i n g a r e l i s t e d i n A p p e n d i x A. The most
r e c e n t r e v i e w s h a v e i d e n t i f i e d t h r e e t y p e s o f a p p r o a c h :
s e q u e n t i a l -m o d u la r , e q u a t i o n - o r i e n t e d , and t w o - t i e r .
1 .1 . A rch i te c tu re s_ o f _ f lo w s h eet_ s i mul a to r s
In t h e s e q u e n t i a l - m o d u l a r approach the program s t r u c t u r e
i s m o d u la r and the c o m p u t a t i o n s a re p e r f o r m e d s e q u e n t i a l l y
d e p e n d in g on t h e c o n f i g u r a t i o n o f the f l ow shee t . The b a s i c idea
i s tha t each u n i t model c a l c u l a t i o n s are pe r fo rm ed i n p r o c e d u r e s
wh ich c a l c u l a t e o u t p u t stream (sometimes in t e rm e d ia t e ) v a r i a b l e s
f rom a u n i t g i v e n v a l u e s f o r a l l i n p u t s t r eam v a r i a b l e s and
e q u ip m e n t p a r a m e t e r s o f t h e u n i t . The re a re t h r e e l e v e l s o f
i t e r a t i o n s i n these computa t ions : t h e rm o p h y s i c a l p r o p e r t i e s (TP)
l e v e l , u n i t o p e r a t i o n (module) l e v e l , and f l ow shee t l e v e l as shown
i n F igu re 1 .1 . The t he rm ophys i ca l p r o p e r t i e s l e v e l c o m p r i s e s
c o n s t a n t t h e r m o p h y s i c a l p r o p e r t i e s ( e . g . c r i t i c a l temperature ,
15
Figure 1.1: Levels of Flowhseet Computations
Level
16
c r i t i c a l p re s su re ) and v a r i a b l e or temperature dependent p rope r ty
models (e .g . thermal c o n d u c t i v i t y , e n t rop y , a c t i v i t y c o e f f i c i e n t ,
f u g a c i t y c o e f f i c i e n t , e n t h a l p y ) . The u n i t o p e r a t i o n l e v e l as the
name i m p l i e s c o n t a i n s p h a s e and c h e m i c a l e q u i l i b r i u m m o d u le s ,
d i s t i l l a t i o n and o t h e r u n i t o p e r a t i o n modules (e .g . compressors,
e x p a n d e r s , r e a c t o r s ) . The f l o w s h e e t l e v e l c o m p r i s e s t h e
t r a n s l a t o r , d e c o m p o s i t i o n , a l g o r i t h m s , n u m e r i c a l e q u a t i o n -
s o l v e r s , e t c . A d e t a i l e d d e s c r i p t i o n o f t h e f l o w s h e e t l e v e l i s
p ro v id ed by Rosen e t a l (1977 ) , Shacham e t a l (1981) among o th e r s .
A lmost a l l the commerc ia l s i m u l a t o r s a v a i l a b l e t o d a y have t h i s
t y p e o f a r c h i t e c t u r e - PROCESS (Brannock e t . a l . (1979) , FLOWPACK
I I ( B lu c k e t . a l . ( 1 9 7 8 ) , ASPEN ( G a l l i e r e t . a l . 1980) among
o t h e r s . The a t t r i b u t e s o f a s e q u e n t i a l modular package a re : ease
of deve lopment and d e b u g g i n g ; s m a l l s t o r a g e r e q u i r e m e n t wh ich
makes i t p o s s i b l e t o h a n d l e l a r g e s i m u l a t i o n problems; and are
u s u a l l y robus t . I t i s e a s y t o i n c o r p o r a t e new m odu le s o r more
s o p h i s t i c a t e d v e r s i o n s o f e x i s t i n g u n i t o p e r a t i o n s . Fur thermore,
these types of s im u l a t o r s a re r e a d i l y a v a i l a b l e and e a s i e r t o use
by e n g i n e e r s . U n f o r t u n a t e l y , t h e s e q u e n t i a l - m o d u l a r approach
l a c k s the f l e x i b i l i t y t o hand le de s ign , o p t i m i z a t i o n and dynam ic
s i m u l a t i o n c a l c u l a t i o n s e f f i c i e n t l y ( P e r k i n s ( 1 9 8 4 ) , B i e g l e r
(1984 )) .
A c o m p l e t e l y d i f f e r e n t approach i s the e q u a t i o r r o r i e n t e d
(EO) t e ch n iq u e i n w h i c h t h e u n i t m o d u le s i n t h e f l o w s h e e t a r e
r e p l aced by s e t s o f e q u a t i o n s . Thus t he complete p lan t model i s
r e p r e s e n t e d by a l a r g e s y s tem o f e q u a t i o n s wh ich a re s o l v e d
17
s i m u l t a n e o u s l y . Howeve r , some e q u a t i o n - b a s e d s y s t em s ( e . g .
SPEEDUP ( P e r k i n s and S a r g e n t ( 1 9 8 2 ) ) have the c a p a b i l i t y f o r
h a n d l i n g mixed sys tems o f e q u a t i o n s and procedures which are a l s o
assembled and s o lv ed s im u l t a n e o u s l y . Here a l s o we can i d e n t i f y
t h r e e l e v e l s ( t h e r m o p h y s i c a l , u n i t o p e r a t i o n , and f l owshee t ) of
computa t ions as i n the s e q u e n t i a l -m o d u la r s i t u a t i o n . One o f the
a t t r a c t i v e f e a t u r e s o f the EO approach i s t h a t the user has g rea t
f l e x i b i l i t y i n s e t t i n g s p e c i f i c a t i o n s f o r h i s / h e r p r o b l e m .
I r o n i c a l l y t h i s i s a l s o one o f the weakness o f the approach s in ce
the q u e s t i o n o f c o r r e c t l y s p e c i f y i n g a p rob lem i s no t t r i v i a l
e s p e c i a l l y f o r l a r g e p ro ce s s m o d e l s . l t i s a l s o easy to fo rmu la te
o p t i m i s a t i o n , d ynam ic s i m u l a t i o n and c o n t r o l s y s t e m d e s i g n
c o m p u t a t i o n s under the same framework. A recen t d e s c r i p t i o n of
SPEEDUP which i s a t y p i c a l EO s im u l a t o r i s p rov ided by Pant e l i d e s
(1987).
The t w o - t i e r a r c h i t e c t u r e i s t h e t h i r d a p p ro a ch which
a t t e m p t s t o c o m b i n e t h e b e t t e r f e a t u r e s o f b o t h t h e
s eq u en t i a l -m o d u la r and EO systems. They are g e n e r a l l y r e f e r r e d to
as s im u l taneous -modu la r packages ( e .g . B i e g l e r ( 1 9 8 4 ) ) , Chen and
S t a d t h e r r ( 1 9 8 3 ) , T r e v i n o - L o z a n o e t a l ( 1 9 8 5 ) , and J o h n s and
Badhwana (1985 )) . They c o n t a in the same l i b r a r y o f u n i t modu les
as a s e q u e n t i a l - m o d u l a r s i m u l a t o r . They a l s o in c lu d e much l e s s
r i g o r o u s models f o r each of the u n i t s and/o r TP p r o c e d u r e s . The
s im p l e r m ode ls c o n t a i n a d j u s t a b l e pa ram e te r s and a re t y p i c a l l y
ab le t o approx imate the r i g o r o u s u n i t o p e r a t i o n performance over a
l i m i t e d r a n g e . The f l o w s h e e t i s s e t up i n terms o f the s im p le r
18
m ode ls u s i n g t h e EO a p p r o a c h . The p e r f o rm a n c e o f each o f the
s i m p l e r m o d e l s i s c h e c k e d a g a i n s t t h e p e r f o r m a n c e o f t h e
c o r r e s p o n d i ng r i g o r o u s model a f t e r each f l ow shee t i t e r a t i o n . I f
the ou tp u t s of the two models ( r i g o r o u s and a p p r o x im a t e ) do not
a g r e e t o a s p e c i f i e d t o l e r a n c e t h e n t h e p a r a m e te r s o f the
approx imate m ode ls a re u p d a ted i n an o u t e r l o o p t o remove the
d i f f e r e n c e . T h i s s o l v i n g and c h e c k i n g i s r e p e a t e d u n t i l t he
f l ow shee t i s converged and each of i t s r i g o r o u s and c o r r e s p o n d i n g
s im p le r model performance agree t o the se t t o l e r a n c e ( F ig u r e 1 . 2 ) .
The re a re a g a i n t h r e e l e v e l s o f c o m p u t a t i o n s a s i n t h e two
p re v iou s f l o w s h e e t i n g s t r a t e g i e s . The most s e r i o u s problem of the
s im u l t a n e o u s - m o d u l a r a p p r o a c h i s w i t h r e g a r d t o t h e fo rm and
a ccu racy of the s i m p l i f i e d mode ls.
The emerging consensus (Westerberg e t . a l . (1979) , Shacham
e t . a l . (1982), and P e r k i n s ( 1 9 8 4 ) ) i s t h a t the EO a p p ro a c h i s
l i k e l y t o be the f l o w s h e e t i n g method o f the f u t u r e because of i t s
f l e x i b i l i t y i n d e r i v i n g s o l u t i o n p rocedures and a p p l y i n g e f f i c i e n t
c o n ve r g e n c e a l g o r i t h m s . I t can a l s o e a s i l y h a n d le s im u l a t i o n ,
d e s ig n , o p t im i z a t i o n and dynamic s im u la t i o n p r o b le m s i n t h e same
f r a m e w o r k . Here we w i l l c o n s i d e r i n p a r t i c u l a r t h i s t y p e of
s im u la t o r s .
1.2. N um erical_Solution_Algorithm s
For s t e a d y - s t a t e s im u l a t i o n and d e s ig n , the mathemat ica l
problem the s o l u t i o n o f w h i c h i s d e s i r e d i s t h a t o f s o l v i n g a
19
Figure
r ~
Results
I__
1.2: Schematic Diagram of Two Tier Strategy
20
system o f n o n l i n e a r a l g e b r a i c e q u a t i o n s o f the form:
F(x) = 0 (1 .1)
w h e r e F ( x ) , x a r e r e a l n - v e c t o r s o f e q u a t i o n s and unknown
v a r i a b l e s r e s p e c t i v e l y .
S a rg e n t (1981 ) and Shacham (1984 ) r e v i e w e d methods f o r
s o l v i n g these e q u a t i o n s . A l i s t o f some o f the commonly used
methods i s p r e s e n t e d i n A p p e n d i x A. Newton's method seem t o be
the most w ide ly used s o l u t i o n t e chn ique i n f l o w s h e e t i n g s y s t e m s .
I t i s based on the repeated Loca l l i n e a r i z a t i o n of equa t ion (1 .1)
s t a r t i n g f rom an i n i t i a l p o i n t x ° and g e n e ra t i n g a sequence of
i t e r a t i o n s
j kAxk = - F ( x k ) (1 .2 )
3Fwhere the J a cob ia n J = __ and Axk i s the s te p c o r r e c t i o n v e c t o r
3x
used t o update the unknowns x a c c o r d in g t o the r e l a t i o n
xk+1 = x k + Axk (1 .3)
The method e x h i b i t s s e co nd o r d e r c onve rgen ce when s t a r t e d from
p o i n t s c l o s e t o t h e s o l u t i o n ( O r t e g a and R h e i n b o l d t , 1 9 7 0 ) .
U n f o r t u n a t e l y , i t has two m a jo r l i m i t a t i o n s : ( i ) t h e need f o r
p r o v i s i o n of p a r t i a l d e r i v a t i v e s a t every f l ow shee t i t e r a t i o n , and
( i i ) t h e n eed f o r h a n d l i n g s i t u a t i o n s where t h e J a c o b i a n i s
s i n g u l a r o r i l l - c o n d i t i o n e d ( i e a s o l u t i o n t o e q u a t i o n ( 1 . 2 )
cannot be found ) .
21
The second p r o b le m i s u s u a l l y d e a l t w i t h by u s i n g t h e method
proposed by Marquardt C1963) and Levenberg (1944) w h ich comb ines
t h e d e s i r a b l e c h a r a c t e r i s t i c s o f Newton's method and the s teepes t
descent m in im i z a t i o n method. As t o the f i r s t problem, we o b s e r v e
t h a t c h e m i c a l e n g i n e e r i n g p r o b l e m s i n v o l v e h i g h l y n o n l i n e a r
equa t ions ( a r i s i n g ma in ly from the use o f comp lex the rm odynam ic
(TD) mode ls ) f o r w h i c h a n a l y t i c p a r t i a l d e r i v a t i v e s a re u s u a l l y
" u n a v a i l a b l e " and a re t h e r e f o r e e i t h e r c o m p u t e d by f i n i t e
d i f f e r e n c e s or e s t im a ted by Quas i-Newton methods.
The n u m e r i c a l s o l u t i o n a l g o r i t h m w h ic h re su l t sw hen the
p a r t i a l d e r i v a t i v e s a re e s t i m a t e d by n u m e r i c a l p e r t u r b a t i o n i s
r e f e r r e d t o a s D i s c r e t e Newton method. Care must be taken with
the cho ice of f i n i t e d i f f e r e n c e i n t e r v a l i n o rder t o r e t a i n second
o r d e r p r o p e r t y o f Newton method. The D i s c r e t e Newton method i s
known t o be q u i t e e x p e n s i v e due t o t h e c o m p u t a t i o n a l c o s t
a s s o c i a t e d w i t h g e n e r a t i n g t h e J a c o b i a n m a t r i x , J / by f i n i t e
d i f f e r e n c e s .
Q u a s i - N e w t o n m e t h o d s on t h e o t h e r hand do not need
a n a l y t i c a l d e r i v a t i v e s t o be p r o v i d e d . They s t a r t by t a k i n g an
a p p r o x im a t i o n t o the J a c o b i a n which i s updated a t every i t e r a t i o n
t h e r e a f t e r u s i n g o n l y f u n c t i o n v a l u e s . B r o y d e n ' s ( 1965 ) and
S h u b e r t ' s (1970) methods a r e t h e most w i d e l y used i n chemical
e n g i n e e r i n g . This class ■ exhibits the s U p e r L i n e a r
c o n v e r g e n c e p r o p e r t y ( O r t e g a a n d R h e i n b o l d t , ( 1 9 7 0 ) ) . The
performance of the methods depend t o a l a r g e ex ten t on the way the
22
i n i t i a l a p p r o x i m a t i o n t o t h e J a c o b i a n m a t r i x i s e s t i m a t e d .
S eve ra l approaches have been t e s t e d , n am e ly , f u l l p e r t u r b a t i o n ,
d iagona l p e r t u r b a t i o n and the i d e n t i t y m a t r i x .
D u r i n g t h e c o u r s e o f an i t e r a t i v e c a l c u l a t i o n , the
J a c o b i a n m a t r i x may be r e i n i t i a l i s e d u s i n g any o f t h e a b o v e
m e thods , e s p e c i a l l y f o r d i f f i c u l t p ro b le m s i n o rde r to a ch ie ve
c o n v e r g e n c e . F u l l p e r t u r b a t i o n , a l t h o u g h t h e m o s t a c c u r a t e
J a cob ian app ro x im a t io n t e c h n iq u e , i s u s u a l l y not recommended s in c e
i t imposes a la r g e compu ta t iona l c o s t .
A r e c e n t num e r i c a l s o l u t i o n a lg o r i t h m i n chemica l p ro cess
d e s i g n i s t h a t s u g g e s t e d by L u c i a a n d M a c c h i e t t o ( 1 9 8 3 ) i n
p a r t i c u l a r f o r a pp rox im a t in g p h y s i c a l p r o p e r t i e s d e r i v a t i v e s . The
method i s Newton-based w i t h t h e J a c o b i a n c o n s t r u c t e d by m ak ing
c o m b i n e d u se o f a l l r e a d i l y a v a i l a b l e a n a l y t i c a l d e r i v a t i v e
i n f o r m a t i o n ( computed p a r t ) and u s i n g a m o d i f i c a t i o n o f t h e
Q u a s i -N e w to n upda te f o r m u l a o f S c h u b e r t (1970) t o e s t im a te the
u n a v a i l a b l e d e r i v a t i v e s ( a p p r o x im a t e d p a r t ) . The a p p r o x im a t e d
p a r t i s d e r i v e d so t h a t i t s a t i s f i e s t h e secant c o n d i t i o n which i s
necessary f o r s a t i s f a c t o r y p e r f o rm a n c e o f Q u a s i - N e w t o n methods
(Denn is and Schnabe l , 1979) . P a n t e l i d e s (1987) a p p l i e d the method
t o the s o l u t i o n o f f l o w s h e e t i n g p r o b le m s w i t h t h e a p p r o x im a t e d
p a r t o f t h e J a c o b ia n i n i t i a l i s e d by f i n i t e d i f f e r e n c e s . Note some
of P a n t e l i d e s t e s t p r o b le m s do no t i n v o l v e t he rm odnam ic (TD)
p r o c e d u r e s . L u c i a ( 1 9 8 5 ) , L u c i a e t a l . (1985) and Venkataraman
and L u c i a (1986) extended the o r i g i n a l Hyb r id method of L u c i a and
23
M a c c h i e t t o . Howeve r , t h e e x a m p le s r e p o r t e d so f a r by t h e s e
a u t h o r s h a ve b e e n r e s t r i c t e d t o p h y s i c a l p r o p e r t i e s a n d
v a p o u r - l i q u i d e q u i l i b r i u m p rocedu res .
A number o f EO s i m u l a t o r s such as SPEEDUP ( P a n t e l i d e s ,
1987), QUASILIN ( F i e l d s e t a l (1984), and ASCEND (Ben jam in e t a l ,
1983) a l l o w t h e u s e r t o p r o v i d e some of the system equa t ion s i n
the form of p rocedu res .
A p rocedure i s a s u b - s e t of equa t ion s which g iven a s e t of
i n p u t v a r i a b l e s and p a r a m e t e r s u, c a l c u l a t e s a s e t o f o u t p u t
v a r i a b l e s w, and i n t e r n a l v a r i a b l e s , v. I t i s e q u i v a l e n t t o
w = P (u) (1 .4 )
and v = P (u) (1 .5)
where P rep re sen t the p rocedu re . T h i s i s the same as w r i t i n g the
se t of equa t ion s
f (w ,u ) = w - P (u) (1 .6)
i n the f l o w s h e e t mode l ( N o t e : t h e i n t e r n a l v a r i a b l e s , v , a r e
u s u a l l y om i t ted a t the f l o w s h e e t l e v e l ) . Thus when a procedure i s
used t o g e th e r w i t h the o th e r equa t ion s t o s im u l a t e or o p t i m i z e a
f lowshee t w i th a Newton-based s im u l a t o r , a l i n e a r i z e d model of the
procedure i s needed a t each i t e r a t i o n f o r which
24
3f af aP— = i , — = - __aw au au
must be p rov ided . The major problem i s the e f f i c i e n t p r o v i s i o n of
9P /8u as TP and u n i t o p e r a t i o n p rocedures do no t u s u a l l y p r o v i d e
t h i s m a t r i x . T y p i c a l examples of p rocedures a re those f o r phase
e q u i l i b r i u m c o m p u t a t i o n s ( e . g . VLE f l a s h , d i s t i l l a t i o n ) and
the rm odynam ic d a ta (eg K - v a l u e s , en th a lp y , e t c ) . P e r k in s (1984)
s t r o n g l y recommends t h a t e q u a t i o n - b a s e d f l o w s h e e t i n g s y s t em s be
a b l e t o s o l v e s i m u l t a n e o u s l y a m i x e d s e t o f e q u a t i o n s and
p r o c e d u r e s . The use o f p r o c e d u r e s h a s s e v e r a l a d v a n t a g e s .
P rocedures can be used t o re p r e s e n t e qua t i o n s which are d e f i n e d by
d i f f e r e n t a l g e b r a i c fo rms i n d i f f e r e n t doma ins ( P e r k i n s , 1 9 8 4 ) .
P r o c e d u r e s can a l s o be u s e d t o im p le m e n t s p e c i a l i s e d s o l u t i o n
a l g o r i t h m s f o r p a r t i c u l a r p r o c e s s u n i t o p e r a t i o n s , i f s u c h
a l g o r i t h m s o f f e r a d v a n t a g e s o v e r g e n e r a l pu rpo se s o l u t i o n
a lg o r i t h m s . P rocedu res are a l s o u s e f u l when the c a l c u l a t i o n o f
t h e i r o u t p u t v a r i a b l e s i n v o l v e s s e ve ra l in te rm ed ia te v a r i a b l e s ,
the v a l u e s o f which are no t needed o u t s i d e the p r o c e d u r e ( e . g .
d i s t i l l a t i o n ) . In such s i t u a t i o n s t h e s i z e o f t h e a l g e b r a i c
s y s tem s s o l v e d a t t h e f l o w s h e e t l e v e l i s r e d u c e d t h r o u g h the
e l i m i n a t i o n o f t h e s e i n t e r m e d i a t e v a r i a b l e s . The s a v i n g s i n
s to rage space and t ime r e q u i r e m e n t s a r e o f t e n s i g n i f i c a n t . The
o t h e r a d v a n t a g e s o f u s i n g p r o c e d u r e s a r e t h e ease of t r a c k i n g
p a t h o l o g i c a l s i t u a t i o n s , l o c a l i z a t i o n of d i a g n o s t i c i n f o r m a t i o n ,
and ease of i n i t i a l i s a t i o n s ( P e r k i n s , 1984).
25
The im p o r t a n t c o n c l u s i o n from f l o w s h e e t i n g l i t e r a t u r e i s
t h a t Newton's method i s i n genera l the most e f f i c i e n t and r e l i a b l e
numer ica l s o l u t i o n a l g o r i t h m p r o v i d i n g cheap a n a l y t i c a l d e r i v a t i v e
i n f o r m a t i o n i s a v a i l a b l e ( P a n t e l i d e s ( 1 9 8 7 ) , P e r k i n s ( 1 9 8 4 ) ,
Shacham et a l (1982 ) ) .
R e v i e w s o f a l g o r i t h m s f o r p e r f o r m i n g f l o w s h e e t
o p t i m i z a t i on p r o b l e m s have been made by B i e g l e r (1985), Lasdon
( 1 9 8 1 ) , and S a r g e n t ( 1 9 8 0 ) . E f f i c i e n t o p t i m i z a t i o n a lg o r i t h m s
based on the Han-Powe l l method (Han ( 1 9 7 5 ) , P o w e l l ( 1 9 7 8 ) ) have
r e c e n t l y been deve loped f o r p ro ce s s f l o w s h e e t i n g (Hu tch i son e t a l
(1983 ) , B i e g l e r e t a l (1982) , S t a d t h e r r and Chen (1984), and Locke
e t a l ( 1 9 8 3 ) ) . T h e s e a u t h o r s h a v e r e p o r t e d s a t i s f a c t o r y
p e r f o rm a n c e o f t h e a l g o r i t h m i n the s o l u t i o n o f o p t i m i z a t i o n
problems. However, one of the prob lems a s s o c i a t e d w i th s u c c e s s i v e
q u a d r a t i c programming a lg o r i t h m s i s t h a t t h e i r performance depends
c r i t i c a l l y on the p r o v i s i o n o f a ccu ra te p a r t i a l d e r i v a t i v e s o f the
o b j e c t i v e f u n c t i o n and c o n s t r a i n t s . F i n i t e d i f f e r e n c e s and
c h a i n - r u l i n g ( S h i v a r a m and B i e g l e r , 1 983) have been suggested
a l though these methods are bound t o be e x p e n s i v e f o r l a r g e s c a l e
p r o b le m s o r even f o r prob lems c o n t a in i n g complex u n i t o p e r a t i o n s
( e .g . f l a s h , d i s t i l l a t i o n ) .
The dy_namic s i m u l a t i o n and des ign of a chemical p lan t i s
o b t a i n e d by s o l v i n g mixed s e t s o f p a r t i a l d i f f e r e n t i a l equa t ions
(PDE), o r d in a r y d i f f e r e n t i a l e qua t ion s (ODE), and coup led o rd in a r y
d i f f e r e n t i a l and a l g e b r a i c e q u a t i o n s ( D A E ) . Some o f the
26
t e chn ique s f o r s o l v i n g PDEs (e .g . f i n i t e d i f f e r e n c e s ) a re based on
t r a n s fo rm in g the PDEs t o ODEs.
Two c l a s s e s o f methods a r e a v a i l a b l e f o r the numer ica l
s o l u t i o n of ODE 's : e x p l i c i t and i m p l i c i t . The l a t t e r c l a s s o f
methods a r e g e n e r a l l y r e g a r d e d as more e f f e c t i v e and can e a s i l y
hand le the a d d i t i o n of a l g e b r a i c e q u a t i o n s under t h e f r a m e w o r k .
Most o f t h e a v a i l a b l e code s f o r n u m e r i c a l s o l u t i o n o f dynamic
models t y p i c a l l y use Newton 's method o r i t s v a r i a n t s t o s o l v e the
i m p l i c i t e q u a t i o n s . Thus, as w i th steady s ta t e f l o w s h e e t i n g and
o p t im i z a t i o n , t h e r e i s t h e need t o p r o v i d e p a r t i a l d e r i v a t i v e s
a n a l y t i c a l l y , n u m e r i c a l l y , or by any o ther methods i f the i m p l i c i t
i n t e g r a t i o n schemes are t o be used.
I t can t h e r e f o r e be c o n c l u d e d t h a t i t i s necessa ry and
d e s i r a b l e t o p ro v ide p a r t i a l d e r i v a t i v e s c h e a p l y f o r e f f i c i e n t
s o l u t i o n o f d e s i g n , s i m u l a t i o n , o p t i m i z a t i o n , and dynam ic
s im u la t i o n prob lems which a r i s e i n p ro cess f l o w sh e e t i n g .
1 . 3 . P r o y i s i on _ an d _ u se_o f_ JP _ da ta _ in _ p r o c e s s _ s im u la t io n s
I r r e s p e c t i v e o f t h e f l o w s h e e t i n g s t r a t e g y adopted , the re
i s a lways the need f o r i n t e r f a c i n g TP d a t a . Tha t i s , p r o c e s s
c a l c u l a t i o n s u s u a l l y i n v o l v e the repeated use of thermodynamic and
p h y s i c a l p r o p e r t i e s d a t a . The use o f TP m o d e l s i n p r o c e s s
s i m u l a t i o n s g i v e s r i s e t o two impor tan t prob lems. The f i r s t i s
the d i f f i c u l t y i n v o l v e d i n e f f i c i e n t l y i n c o r p o r a t i n g the TD model
27
i n t h e d e s i g n c o m p u t a t i o n s . The second problem i s the e f f e c t of
TP model i n a c c u r a c i e s or parameter u n c e r t a i n t i e s on the s im u la t i o n
r e s u l t s .
1 . 3 . 1 . E f f j c i e n t _ i n co rpo ra t i on_g f_TP_da ta_ in_p ro ce s s_de s ign
Seve ra l packages have been deve loped to p rov ide TD data i n
f l o w sh e e t i n g packages. D e t a i l e d d e s c r i p t i o n o f the s t r u c t u r e of
p h y s i c a l p r o p e r t i e s data systems have been made by Westerberg e t .
a t . ( 1979 ) and E van s e t . a l . ( 1977 ) and a b r i e f o u t l i n e o f a
t y p i c a l p h y s i c a l p r o p e r t i e s package i s made i n the next chapte r .
In f a c t , i t can be argued t h a t the n o n l i n e a r i t y o f t h e e q u a t i o n s
wh ich a r i s e i n compute i— a i d e d p r o c e s s d e s ign problems can t o a
g rea t e x t en t be a s s o c i a t e d w i th the comp lex i t y of TP models. These
n o n l i n e a r TP m o d e l s pose a s e r i o u s p rob lem i f one i s t o use
N e w to n ' s method o r i n d e e d any o f the o t h e r methods t h a t need
d e r i v a t i v e i n f o r m a t i o n a t t h e f l o w s h e e t and u n i t o p e r a t i o n s
l e v e l s . The h i g h l y n o n l i n e a r equa t ion s of TP models are t y p i c a l l y
s o l v e d t o g e t h e r i n a p r o c e d u r e ( o r s u b - r o u t i n e ) i n a p h y s i c a l
p r o p e r t i e s package.
I t has been r e c o g n i s e d by s e v e r a l a u t h o r s ( B a r r e t t and
Walsh (1979), L e e s l e y and Heyen (1977), Shacham e t . a l . ( 1 9 8 2 ) ) ,
t h a t the e f f i c i e n c y and r e l i a b i l i t y of f l o w sh e e t i n g systems depend
s t r o n g l y on how TP d a t a a re t r e a t e d i n the o v e r a l l s o l u t i o n
scheme. In f a c t , i t has been r epo r t ed by Westerberg e t a l (1979) ,
Shacham e t a l (1982) , Rosen e t a l (1980), and Gibbons e t a l (1978)
28
t h a t up t o 95 % of the bu lk s im u l a t i o n t ime i s o f t e n spent i n the
gene ra t ion of TD and p h y s i c a l p r o p e r t i e s da ta . S eve ra l approaches
have been p r o p o s e d f o r i n c o r p o r a t i n g TP d a t a i n t h e f l ow shee t
s o l u t i o n schemes.
The f i r s t m e t h o d we t e r m e d t h e B l a c k - b o x a p p ro a c h .
R igo rous TP p o in t v a l u e s are p ro v ided from a p h y s i c a l p r o p e r t i e s
p a ckage . M a t r i x 3 p / 9 u i s e i t h e r n e g l e c t e d o r app rox ima ted by
f i n i t e d i f f e r e n c e s . Most s i m u l a t o r s a v a i l a b l e t o d a y have t h i s
s o r t o f i n t e r f a c e w i t h p h y s i c a l p r o p e r t i e s packages (e .g . PROCESS,
ASPEN, e t c . ) . The s e co nd a p p r o a c h i s what we have c a l l e d t he
W e s t e r b e r g t e c h n i q u e ( W e s t e r b e r g e t a l ( 1 9 7 9 ) ) . Here t he
equa t ion s of the TP model a re w r i t t e n and s o l v e d s i m u l t a n e o u s l y
w i t h t h e o t h e r p r o c e s s mode l e q u a t i o n s a t the f l ow shee t l e v e l .
P a r t i a l d e r i v a t i v e s of the TP mode ls are g e n e r a t e d a n a l y t i c a l l y .
The t h i r d a p p r o a c h i s r e f e r r e d t o a s t h e t w o - t i e r t e c h n i q u e
(Hu tch i son and Shewchuk (1974 ) , B a r r e t t and Walsh (1979 ) , L e e s l e y
a n d H e y e n ( 1 9 7 7 ) , B o s t o n and B r i t t ( 1 9 7 9 ) , C h im o w i t z e t a t
(1983)) . The te chn ique i n v o l v e s replacement of complex TP m ode ls
by a p p r o x i m a t e o n e s f o r most o f t h e i t e r a t i v e c a l c u l a t i o n s
p a r t i c u l a r l y i n the e s t i m a t i o n o f d e r i v a t i v e i n f o r m a t i o n . The
f l o w s h e e t c a l c u l a t i o n s a r e done u s ing approximate TD data i n an
i n n e r lo op , w h i l e r i g o r o u s TD c a l c u l a t i o n s occur on l y i n the ou te r
t i e r .
A f o u r t h a p p ro a ch t o the i n t e r f a c e of TP data i n p rocess
des ign computat ions i s the H yb r id method s u g g e s t e d by L u c i a and
29
M a c c h i e t t o , ( 1 9 8 3 ) . An a pp ro x im a te model of the r i g o r o u s TP i s
a l s o p o s t u l a t e d a s i n the t w o - t i e r t e c h n i q u e . H o w e v e r , t h e
s i m p l i f i e d m o d e l s d o no t c o n t a i n a d j u s t a b l e p a r a m e te r s but are
based on the l i m i t i n g model b e h a v i o u r ( e . g . i d e a l K - v a l u e u s i n g
A n t o i n e vapou r p re s su r e c o r r e l a t i o n ) . The d e r i v a t i v e i n f o rm a t i o n
i s c on s t ru c t e d i n two p a r t s : a "computed p a r t " g i v e n by a l l the
a v a i l a b l e a n a l y t i c a l d e r i v a t i v e s , and an "approx imated p a r t " which
i s e s t i m a t e d u s i n g a Q u a s i - N e w t o n t e c h n i q u e ( e . g . S h u b e r t ' s
m e th o d ) . L u c i a e t a l ( 1985 ) e x t e n d e d t h e o r i g i n a l Hyb r id idea
b e c a u s e i t ( o r i g i n a l H y b r i d ) w a s f o u n d t o p e r f o r m
u n s a t i s f a c t o r i l y . We p o s t p o n e f u r t h e r d i s c u s s i o n s o f t h e s e
te chn ique s u n t i l l a t e r i n Chap te r 2.
The f u n d a m e n t a l p r o b l e m i s how t o d e r i v e a c c u r a t e
d e r i v a t i v e s o f a jge_neral p r o c e d u r e w i th p a r t i c u l a r r e f e r e n c e to
t h e r m o p h y s i c a l p r o p e r t i e s and phase and c h e m i c a l e q u i l i b r i u m
procedures . In a l l the e x i s t i n g f o u r TP data i n t e r f a c e s t r a t e g i e s
i t i s assumed t h a t p r o c e d u r e d e r i v a t i v e s a re not a v a i l a b l e and
t h e r e f o r e must be app rox ima ted . S i n c e a c c u r a t e d e r i v a t i v e s a re
n e c e s s a r y i n p ro ce s s f l o w s h e e t i n g , t h e re i s t h e r e f o r e the need t o
dev i s e a method f o r g e n e r a t i n g the d e s i r e d d e r i v a t i v e i n f o r m a t i o n
e f f i c i e n t l y .
T h e d e s i r a b l e c h a r a c t e r i s t i c s o f a s t a n d a r d i s e d
thermodynamic data i n t e r f a c e are s u g g e s t e d i n the nex t c h a p t e r .
These a re compared w i t h the s a l i e n t f e a t u r e s o f a t y p i c a l p h y s i c a l
p r o p e r t i e s package a v a i l a b l e today . A c r i t i c a l l i t e r a t u r e r e v i e w
30
o f t h e e x i s t i n g TP d a t a i n t e r f a c e s t r a t e g i e s i s p r e s e n t e d .
F i n a l l y a new TD data i n t e r f a c e s t r a t e g y i s p r o p o s e d and t e s t e d
e x t e n s i v e l y i n Chapter 3 .
1.3 .2 . Sensi t i v ity_of_proce ss_design_to_uncertai nti es_i n_physi caj.
properties
As p a r t o f t h e i n p u t t o a p ro cess s im u l a t i o n program the
use r must s p e c i f y the TD model o p t i o n s used i n t h e c a l c u l a t i o n s
( e . g . e q u a t i o n o f s t a t e or a c t i v i t y c o e f f i c i e n t s ) . Sometimes the
u se r i s a l l o w e d t o s u p p l y v a l u e s f o r some p h y s i c a l p r o p e r t i e s
c o n s t a n t s o v e r r i d i n g those a v a i l a b l e ' i n the da tabanks. These TD
p r o p e r t y c o r r e l a t i o n s a n d / o r d a ta a r e o f t e n i n a c c u r a t e and
t h e r e f o r e p r o c e s s d e s i g n s a r e c a r r i e d o u t b a s e d on t h e s e
i n a c c u r a t e data . E x i s t i n g p ro ce s s s im u l a t o r s do no t p r o v i d e the
s e n s i t i v i t y o f t h e d e s i g n t o u n c e r t a i n t i e s i n the TD model or
paramete rs i n a r o u t i n e way w i t h the r e s u l t t h a t s i m u l a t o r u s e r s
a r e g e n e r a l l y unaware o f how s e n s i t i v e t h e i r d e s i g n i s t o TD
i n f o rm a t i o n . In f a c t , the de s ig n of chemica l p rocesses can depend
s t r o n g l y on t h e TD m o d e l u t i l i z e d i n c a l c u l a t i n g d ep en d en t
q u a n t i t i e s s u c h a s K - v a l u e , e n t h a l p y / e n t r o p y , a nd p h a s e
e q u i l i b r i a . Z u d k e v i t c h ( 1980 ) made a q u a l i t a t i v e s tudy of the
e f f e c t o f d i f f e r e n t TD m o d e l s and p a r a m e te r s ( e . g . c r i t i c a l
c o n s ta n t s , e q u i l i b r i u m r a t i o , en th a lp y , and en t ropy) on the d e s ig n and
economics o f v a r i o u s chemica l p rocesses ( r e a c t o r s , e x t r a c t o r s , and
d i s t i l l a t i o n , e t c ) . A d l e r and S p e n c e r (1980) and S t r e i c h and
K is tenmacher (1979) s t u d i e d the e f f e c t s of model i n a c c u r a c i e s on
31
d i s t i l l a t i o n u n i t o p e r a t i o n d e s ig n s - These a u tho r s s i n g l e d out
d i s t i l l a t i o n o p e r a t i o n as the u n i t most s e n s i t i v e t o p r o p e r t y
i n a c c u r a c i e s - In a l a t e r paper, Zudkev i t ch (1980) p re sen ted cases
of p l a n t s rendered in o p e r a b le due t o i n a c c u r a c i e s i n TD p r o p e r t y
d a t a . For i n s t a n c e , t h e a u t h o r c i t e d t h e case o f a l i q u i f i e d
n a tu r a l gas p r o c e s s in g p l a n t wh ich was shut-down immed ia te ly a f t e r
s t a r t - u p . On the o the r hand, t h e r e a re many s i t u a t i o n s where even
g ro ss assumpt ions and rough a pp ro x im a t io n s t o TD d a t a may r e s u l t
i n l i t t l e or no e f f e c t on the de s ign (Mah, 1977) .
The p ro b lem t h e r e f o r e a r i s e s o f e s t a b l i s h i n g w h i c h
p r o p e r t i e s and p a r a m e t e r s i f any a r e c r i t i c a l l y impor tant i n a
g iven p rocess p la n t (o r u n i t o p e r a t i o n ) and o f q u a n t i f y i n g t h e i r
e f f e c t on des ign and/o r p r e d i c t e d performance. One way around the
problem i s t o perform repea ted s i m u l a t i o n s o f t h e w ho le p r o c e s s
p l a n t a d j u s t i n g t h e in p u t TD p ro pe r t y parameters i n d i v i d u a l l y (o r
i n c o m b i n a t i o n s ) and u s i n g a v a r i e t y o f m o d e l s f o r t h e same
p ro pe r t y . T h i s i s r e f e r r e d t o as the case s t u d i e s method. E l l i o t
e t a l ( 1980 ) s t u d i e d t h e e f f e c t s o f u s i n g d i f f e r e n t K - v a l u e ,
e n t h a l p y , and e n t r o p y models on the economics and des ign of high
p ressu re d i s t i l l a t i o n u n i t s and tu rbo -expande r p l a n t s . Shah and
B i s h n o i (1978) s im u la t e d ab so rb e r s and d i s t i l l a t i o n columns u s ing
d i f f e r e n t TD m o d e l s f o r f u g a c i t y c o e f f i c i e n t s a n d e n t h a l p y
p r e d i c t i o n s . A n g e l e t . a l . ( 1 9 8 6 ) a l s o s i m u l a t e d f o u r
d i s t i l l a t i o n c o l u m n s ( d e e t h a n i z e r , d e b u t a n i z e r , e t h y l e n e
d i c h l o r i d e s t a b i l i z e r , and an e x t r a c t i v e d i s t i l l a t i o n ) with a
v a r i e t y of TP models . The case s t u d i e s method i s no t e f f i c i e n t
32
s i n c e i t i n v o l v e s many s i m u l a t i o n s o f complex p l a n t o p e r a t i o n s
i n v o l v i n g the r e c y c l e of mass and energy . For i n s t a n c e , f o r a 10
component m i x t u r e , 46 r i g o r o u s d i s t i l l a t i o n column c a l c u l a t i o n s
must be c a r r i e d ou t i n o rde r t o e va lu a te the des ign s e n s i t i v i t y t o
b in a ry i n t e r a c t i o n c o e f f i c i e n t s i f a cub ic equa t i o n of s ta te model
i s used. U s ing d i f f e r e n t TD models enab le s one to a s c e r t a i n on ly
P e r t u r b a t i on 2 .7 2.3 2 . 6 6.3 3 .0 2 .9 4 .7C223 C403 C533 ■ C373 C67I C303 1293
No. of i t e r a t i o n s / b a s epoi ntNo. of TD c a l l s / b a s e
8 17 20 5 38 17 7
poi nt 9 18 21 6 39 18 8Thermodynamic Model SRK SRK SRK SRK SRK UNIQUAC UNIQUAC
C 3 E q u iv a l e n t number of thermodynamic c a l l s f o r d e r i v a t i v e s(A) k - v a lu e d e r i v a t i v e s ob ta in ed a n a l y t i c a l l y(B) k - v a lu e d e r i v a t i v e s ob ta in ed by p e r t u r b a t i o n
90
e q u a t i o n s f o r NC + 2 = 7 r i g h t hand s i d e s f o r l i q u i d phase
g r a d i e n t s f o l l o w e d by e v a l u a t i o n o f v a p o u r p h a se p a r t i a l
d e r i v a t i v e s ) t a k e r e l a t i v e l y sma l l comput ing t ime as d e p i c t e d i n
F igu res 3-2 a ,b .
W i th K - v a l u e s e v a l u a t e d u s i n g e qua t ion (2-6) and l i q u i d
phase a c t i v i t y c o e f f i c i e n t s m o d e l l e d by t h e more c o m p l e x
UNIQUAC-equa t i on , we a g a i n o b s e r v e t h a t the base po in t s o l u t i o n
s t i l l converges q u i c k l y , e t c - ( F i g u r e s 3 - 2 c , d ) . Here the base
p o i n t c o m p u t i n g t i m e i s due e s s e n t i a l l y t o TD p r o p e r t i e s
e v a l u a t i o n . F lash o u t p u t - i n p u t d e r i v a t i v e s u s i n g a n a l y t i c and
n u m e r i c a l K - v a l u e d e r i v a t i v e s a r e bo th more expens ive than the
re fe rence p o in t f l a s h c a l c u l a t i o n s - Use o f numer ica l K - va lue data
i s s l i g h t l y worse t h a n when t h e VLE f l a s h procedure d e r i v a t i v e s
c a l c u l a t i o n s u t i l i z e a n a l y t i c i n f o r m a t i o n . S e v e n t y - s e v e n and
f i f t y - s i x p e r c e n t o f t h e t o t a l d e r i v a t i v e s ( p r o c e d u r e )
c a l c u l a t i o n s a re d e v o t e d t o l i n e a r a l g e b r a u s i n g a n a l y t i c and
numer ica l TD p ro pe r t y d e r i v a t i v e s , r e s p e c t i v e l y .
The e f f o r t r e q u i r e d t o genera te i s o th e rm a l f l a s h procedure
d e r i v a t i v e s w i t h s e v e r a l m i x t u r e s and d i f f e r e n t TD m od e l s i s
p r e s e n t e d i n T a b l e 3 . 2 a s e q u i v a l e n t num be r o f b a s e p o i n t
c a l c u l a t i o n s / t h a t i s , t a k i n g t h e t ime f o r a base po in t s o l u t i o n as
r e f e r e n c e . In e v e r y c a s e we d i d n o t p e r f o r m an y t e s t s t o
a s c e r t a i n w h e t h e r phase s p l i t s p r e d i c t e d by our f l a s h r o u t i n e
a c t u a l l y s t a b l e or no t . G e n e r a l l y the t im e f o r a n a l y t i c f l a s h
p r o c e d u r e d e r i v a t i v e s v a r i e s from 0 -3u - 4-8 and 0 -2L - 8-5 base
91
p o in t e v a l u a t i o n s when K -va lue d e r i v a t i v e s a re computed by methods
A and P r e s p e c t i v e l y .
For m ix t u re s C2 .1 , C2 .2 , C2.3 and C2.7 gene ra t io n of f l a s h
d e r i v a t i v e s i s rough ly equa l t o a s i n g l e base p o i n t c a l c u l a t i o n .
For t h e s e m ix t u re s , the number of i t e r a t i o n s taken t o converge to
a s p e c i f i e d t o l e r a n c e a r e 8 , 1 7 , 20 , and 7 r e s p e c t i v e l y . When
n u m e r i c a l K - v a l u e d e r i v a t i v e s a r e u s e d a s l i g h t l y w o r s e
performance i n te rms o f the comput ing co s t i s r e a l i s e d compared to
use o f a n a l y t i c TD p r o p e r t y d a ta . In a l l cases , a n a l y t i c f l a s h
d e r i v a t i v e s p e r f o rm b e t t e r t h a n when f l a s h d e r i v a t i v e s a r e
c a l c u l a t e d by p e r t u r b a t i o n o f the NC + 2 i n p u t s t o the f l a s h which
r e q u i r e a b o u t 2 . 3 - 6 . 3 ba se p o i n t e v a l u a t i o n o f t h e f l a s h
p r o c e d u r e . Over n i n e t y pe rcen t sav ings are r e a l i s e d i n the number
of K -va lue c a l l s when a n a l y t i c d e r i v a t i v e s o f the p r o p e r t y a re
u s e d c o m p a r e d t o t h e r e p e a t e d p e r t u r b a t i o n o f a l l t h e i n p u t
v a r i a b l e s . Numer ica l K - v a lu e d e r i v a t i v e s on the o t h e r hand make
be tween 51 % - 90 % fewer TP c a l c u l a t i o n s . S p e c i f i c a l l y , f o r the
16-component m ix tu re (C2 .4) g e n e ra t io n of procedure d e r i v a t i v e s i s
e q u i v a l e n t t o 4 . 8 and 8 . 5 f l a s h base p o i n t e v a l u a t i o n s when
p h y s i c a l p r o p e r t i e s d e r i v a t i v e s a re c a l c u l a t e d a n a l y t i c a l l y and
n u m e r i c a l l y , r e s p e c t i v e l y . S i n c e t h e measure of computa t iona l
co s t i s r e l a t i v e , i t t ends t o o v e r e s t im a t e the c o s t o f a n a l y t i c
d e r i v a t i v e s when t h e base po in t s o l u t i o n converge e a s i l y ( o n l y 5
i t e r a t i o n s taken t o a ch ie ve convergence - Tab le 3 . 2 ) . Even unde r
the c i r cum s tance , a n a l y t i c p ro cedu re d e r i v a t i v e s a re s t i l l cheaper
by r o u g h l y 24 % i n c o m p a r i s o n t o r e p e a t e d f l a s h p r o c e d u r e
92
p e r t u r b a t i o n ( P ) - M i x t u r e s C 2 . 5 and C2 .6 r e p r e s e n t more
d i f f i c u l t p r o b l e m s t h a n the o t h e r s y s t e m s . The c o n d i t i o n o f
m i x t u r e C 2 . 5 , f o r i n s t a n c e , i s such t h a t r e t r o g r a d e phenomena
occur and convergence i s hard t o a c h i e v e (38 i t e r a t i o n s o f the
base p o i n t ) . In t h i s c a s e , t h e t im e f o r procedure d e r i v a t i v e s
when a n a l y t i c K - v a lu e d e r i v a t i v e s a r e used i n o u r method i s 30
t im e s f a s t e r t h a n by p e r t u r b a t i o n ! (Even when numer ica l K -va lue
d e r i v a t i v e s are used ou r method t a k e s o n l y 1/15 o f the t im e f o r
f i n i t e d i f f e r e n c e ) .
For t h e d i s t i l l a t i o n column procedure , on ly r e s u l t s us ing
a n a l y t i c a l K - v a l u e and e n t h a l p y p r o p e r t i e s d e r i v a t i v e s a r e
r e p o r t e d . The b u l k o f the comput ing e f f o r t (Tab le 3 .3 ) i s spent
i n the base p o in t c a l c u l a t i o n s w i th o n l y 20 % of t h a t t ime devoted
t o t h e e v a l u a t i o n o f t h e r e l e v a n t TD p r o p e r t i e s . The dominant
p a r t o f the c a l c u l a t i o n s i s i n t h e m a in i t e r a t i o n l o o p . The
e f f o r t r e q u i r e d t o g e n e r a t e t h e co lumn o u t p u t - i n p u t g ra d ie n t s
amounts t o o n l y 18 % of the t o t a l t ime taken f o r e v a l u a t i o n of the
d i s t i l l a t i o n column, and i s e q u i v a l e n t t o a s i n g l e e x t r a i t e r a t i o n
o f t h e c o n v e r g e n c e l o o p . S i n c e e v e n t h e m o s t e f f i c i e n t
p e r t u r b a t i o n a lg o r i t h m would r e q u i r e a t l e a s t one pass through the
loop to determ ine the d e r i v a t i v e s o f the o u t p u t v a r i a b l e s , ou r
m e t h o d i s b o u n d t o be m o re e f f i c i e n t . T w e n t y - f i v e and
s e v e n t y - f i v e pe rcen t of the procedure d e r i v a t i v e s t ime i s spent in
TD p rope r ty c a l c u l a t i o n s and l i n e a r system s o l u t i o n r e s p e c t i v e l y .
Tab le 3 . 3 . Computer^ t im e s _ ( C P U _ s e c o n d s ) _ f o r _ d i s t i l l a t i on_ co l j^n_p rocedu re d e r i v a t i v e s _ e v a lu a t i on_ui th__ re spe c t_ to_d i s t i l l a t e _ r a te _ a n d r e f l u x ~
r a t i o_ (.a n a l y t i c _ JD _ d e r i y a t i y e s )
I n i t i a l i sa t i o nBase p o i n t 0.001c a l c u l a t i o n I t e r a t i v e c a l c u l a t i o n
(6 column i t e r a t i o n sT o t a l t ime 0 . 708 0 . 562f o r D i s t i l l a t i on Thermodynamic c a l c u l a t i o n
( k - v a l u e and e n t h a l p y )0 . 145
P r o cedu r e
0 . 865 P rocedu re Thermodynamic c a l c u l a t i o nD e r i v a t i v e 0 .04c a l c u l a t i o n
S o l u t i o n of l i n e a r sys t ems0. 157 f o r p r o c edu r e d e r i v a t i v e s
0 . 117
Note : The base p o i n t s o l u t i o n of t he d i s t i l l a t i o n column conve rges i n 6 i t e r a t i o n s .
94
To s u m m a r i s e , we c o n c l u d e t h a t e x a c t d e r i v a t i v e s o f
r i g o r o u s VLE p rocedures can be genera ted e f f i c i e n t l y as s u g g e s t e d
i n t h i s t h e s i s . The m e th o d a v o i d s t h e need f o r e x p e n s i v e
p e r t u r b a t i o n o f p h y s i c a l p r o p e r t i e s r o u t i n e s o r t h e u se o f
a p p r o x im a t e p r o p e r t y m o d e l s . The method i s a l s o n o n - i t e r a t i v ^ ,
u t i l i z e s l i t t l e s t o r a g e , and above a l l , p r o v i d e s e x a c t a n a l y t i c
d e r i v a t i v e s . We b e l i e v e t h e c o n c l u s i o n s h o l d s no t o n l y w i t h
p h y s i c a l p r o p e r t i e s and VLE p r o c e d u r e s b u t a l s o i n o t h e r u n i t
T h i s e x a m p le i s t a k e n f rom t h e book by Ho l l a nd (1981).
Two i n t e r c o n n e c t e d d i s t i l l a t i o n co lumns s e p a r a t e a s a t u r a t e d
l i q u i d f e e d s t r e a m o f h y d r o c a r b o n s (e thane , p ropy lene , propane,
and is obu tene ) as shown i n F i g u r e 3 .5 . Feed t o co lumn I (number
o f s t a g e s , NST = 20) i s i n t r o d u c e d a t s tage 11 w ith d i s t i l l a t e
w i thd rawa l r a t e se t a t 62 .23 kmo le s /h r . The se cond co lumn ha s a
t o t a l o f 1 2 t h e o r e t i c a l s t age s w i t h feed ( i . e . bottom product of
column I) f e d a t s tage 6 . Both c o lum ns have p a r t i a l c o n d e n s e r s
and a re assumed t o ope ra te a t 100 % thermodynamic e f f i c i e n c y . The
d e t a i l e d s p e c i f i c a t i o n s a r e g i v e n i n T a b le C 3 .3 ( A p p e n d i x C3 ) .
The o b j e c t i v e h e r e i s t o use t h e ene rgy r e c o v e r e d f rom the
condenser of the second column t o meet the h ea t d u t y o f co lumn I
r e b o i l e r . Thus, the e qu a t i o n
QC , TT = Qr . _ (3.14)column I I column I
i s added a t the f l o w s h e e t l e v e l .
The d i s t i l l a t i o n p ro cedu re de s c r ib ed e a r l i e r i s used w i th
the same i n i t i a l i s a t i o n s t r a t e g y . The r e p r e s e n t a t i o n i s s i m i l a r
w i t h t h e o u t p u t v a r i a b l e s l i s t extended by i n c l u s i o n o f r e b o i l e r
Figure 3.5: Coupled Distillation Columns with Energy Recycle
DEST2
oXT)
5
107
and c o n d e n s e r d u t i e s and tempe ra tu re s . The inpu t v a r i a b l e s l i s t
remain the same, t hu s , the procedure re f e ren ce used i s :
( R e b o i l e r - d u t y , C o n d e n s e r - d u t y , T o p - t e m p e r a t u r e ,
D i s t i l l a t e , B o t t o m - t e m p e r a t u r e , B o t t o m ) COLUMN ( R e f l u x - r a t i o ,
D i s t i l l a t e - r a t e , Feed - tempe ra tu re , F e e d - f l o w r a t e ) .
T h i s examp le e n a b l e s us t o use and e v a l u a t e the performance of
u s ing column procedure d e r i v a t i v e s w i t h r e s p e t ^ t o a l l t h e i n p u t
v a r i a b l e s i n a more r e a l i s t i c problem.
We f o u n d t h a t a r e a s o n a b l e i n i t i a l e s t ima te of the in p u t
t o the second column ( s t r e a m 3) i s n e c e s s a r y f o r t h e i t e r a t i v e
co lumn c a l c u l a t i o n s t o c o n v e r g e . In o r d e r t o p r o v i d e such a
guess , we s im u la te d each column i n d i v i d u a l l y by t e a r i n g s t r e a m 3 .
From t h e s e s im u l a t i o n r e s u l t s , the f o l l o w i n g i n i t i a l guess o f the
v a r i a b l e types were made:
F lowra te = 1 . 0 : 1E-15 : 100 u n i t = ' km o le s /h r '
D i s t i l l a t e - r a t e = 35 : 1 . 0 : 100 u n i t = ' k m o le s /h r '
R e f l u x - r a t i o = 30 : 0 .20 : 30 u n i t =
Temperature = 38 : -100 : 600 u n i t =, 0 C'
Condenser-duty = 30 : -100 : 100 u n i t = ' G J / h r 1
R e b o i l e r - d u t y = 30 : -100 : 100 u n i t = 'G J / h r '
We a l s o i n i t i a l i s e d t h e f e e d i n t o t h e s e cond co lumn w i t h the
va lu e s below:
108
F low ra te = C 1 .E -02 , 4 . 0 , 2 0 . 0 , 20.0.11 u n i t = ' k m o le s /h r '
Temperature = 64°C.
NAA, NAP, NPA, HPA, NPP, and HPP comb ina t ions of s o l u t i o n methods
were te s ted .
3 . 4 . 2 . 4 . De sj^n_ _of __cojjjp_l_ed__di stj_l_l_ajti_on__col u r n ns_wi th_mass_and
e n e r^ _ r e c y c l e s
T h i s e x a m p le i s a l s o t a k e n f rom H o l l a n d , 1981 ( F i g u r e
3 . 6 ) . I t i s s i m i l a r t o t he p r e v i o u s e xam p le w i t h the bo t tom
p r o d u c t f r om t h e s e co n d c o lum n a s t h e second feed t o the f i r s t
column. The energy e x t r a c t e d f rom t h e c o n d en se r o f t h e se cond
co lumn , as i n t h e p r e v i o u s example , i s used as t h e r e b o i l e r duty
of column I. The o u tpu t v a r i a b l e s of the d i s t i l l a t i o n p r o c e d u r e
and t h e s o l u t i o n me thod c o m b i n a t i o n s a r e t h e same a s b e f o r e .
However, the i n p u t s l i s t i s e x t e n d e d t o h a n d l e m u l t i p l e f e e d s
s i n c e t h e f i r s t c o l u m n ha s two f e e d s . Tha t i s , t h e co lumn
procedure r e p r e s e n t a t i o n i n t h e MODEL s e c t i o n o f SPEEDUP i s a s
f o l l o w s :
( R e b o i l e r - d u t y , C o n d e n s e r - d u t y , T o p - t e m p e r a t u r e , D i s t i l l a t e ,
Bo t tom- tempera tu re , Bottom) COLUMN
( R e f l u x - r a t i o , D i s t i l l a t e - r a t e , F e e d - t e m p e r a t u r e - 1 ,
F e e d - f l o w ra t e -1 , F eed - tem pe ra tu re -2 , Feed - f low ra te -2 )
D e f a u l t i n i t i a l e s t im a t e s o f the f o l l o w i n g g lo ba l v a r i a b l e types
were used:
Figure 3.6: Coupled Distillation Columns with Mass and Energy Recycle
o<03
11U
F low ra te = 5 : 1E -15 : 100 u n i t = ' km o le s /h r '
D i s t i l l a t e - r a t e = 35 : 1 -0 : 100 u n i t = ' k m o le s /h r '
R e f l u x - r a t i o = 8 : 0 .2 : 30 u n i t = ' c o n s t a n t '
Temperature = 38 : - 1 0 0 : 600 uni t = i o c «
Condenser-duty = 30 : - 1 0 0 : 100 ' uni t = G J /h r '
R e b o i l e r - d u t y = 30 : - 1 0 0 : 100 ' u n i t = GJ/ h r '
The feed t o the second column were i n i t i a l i s e d t hu s :
F l o w r a t e = C1 .E -10 , 6 . 5 , 1 6 . 0 , 2 9 .9 3 , Temperature = 64°C.
We in t r o d u ced a dummy s p l i t t e r on t h e d i s t i l l a t e p r o d u c t s t r e am
from t h e f i r s t column so a s t o use the same column s p e c i f i c a t i o n s
i n the MODEL s e c t i o n of SPEEDUP. The second i n p u t t o the s e cond
column i s f i c t i t i o u s and i s s e t t o z e ro .
3 . 4 . 2 . 5 . Opt i_m j z a t l on_ o f _ Co ug]. ed_ £1 a sh_JJ ni t s
We con s id e red t h e coup led f l a s h u n i t s example of Chimowitz
e t . a l . (1983) as d e p i c t e d i n F i g u r e 3 .7 . The components i n the
m ix tu re and c o n d i t i o n s o f the f l a s h e s a re d e t a i l e d i n Appendix C3.
We do not c o n s i d e r e n e r g y b a l a n c e s i n t h e m o d e l . The d e s i g n
o b j e c t i v e i s t o produce a vapour stream from u n i t 2 c o n s i s t i n g of
a 60 % recove ry of the mos t v o l a t i t e component ( n - p e n t a n e ) and
w i t h a t l e a s t a 78 % p u r i t y . The problem i s t h e r e f o r e t o o b t a i n
t h e t e m p e r a t u r e s i n bo th u n i t s (T-j, T2 ) and r e c y c l e stream tha t
meet these o b j e c t i v e s .
Figure 3.7: Coupled Flash Units
r \
FeedMixero
Vapour Product
Tr ni
Flash 1
T2 * n2
Flash 2
/\Torn Stream
Liquid Product
uz
The p rob lem f o r m u l a t i o n i s s i m i l a r to t h a t of Chimowitz
e t . a l . e x c e p t t h a t t h e e q u a l i t y c o n s t r a i n t s a r e r e p l a c e d by
equa t ion s deno t ing f l a s h p rocedures i n the form of equa t ion ( 1 . 6 ) .
The problem compr ises o f 15 e q u a l i t y c o n s t r a i n t s , 24 i n e q u a l i t y
c o n s t r a i n t s , 2 d e c i s i o n v a r i a b l e s , and 3 t e a r v a r i a b l e s . U n l i k e
i n p rev ious e x a m p le s SPEEDUP i s no t used f o r the o p t i m i z a t i o n
s i n c e a t the t ime t h i s exper iment was conducted, the EO s im u la to r
had only a f e a s i b l e path based o p t im i z a t i o n a lg o r i t h m implemented.
The su c c e s s i v e q u a d r a t i c p ro g ram m ing a l g o r i t h m code o f Powe l l
(1982 ) was used i n t h i s s t u d y . We t e s t e d t h r e e d i f f e r e n t
c o m b i n a t i o n s o f t h e o p t im i z a t i o n method and procedure d e r i v a t i v e
e v a l u a t i o n s t r a t e g i e s . L e t us d e n o t e o p t i m i z a t i o n a l g o r i t h m as
method 0 a t t h e f l o w s h e e t l e v e l . The d e f i n i t i o n o f A and P f o r
the u n i t o p e r a t i o n s and p h y s i c a l p r o p e r t i e s l e v e l s s t i l l a p p l i e s
here . Thus, the n um e r i c a l o p t im i z a t i o n methods a re denoted by OP,
OAP, and OAA.
A summary o f a l l t h e f l o w s h e e t i n g examples attempted i s
presented i n Tab le 3 . 4 .
3 .5 - J^]?ERIC^_RESyLJS/DISCySSIONS
D i s t i l l a t i o n Column Des ign (Problem C3.1 a ,b )
The numer i ca l r e s u l t s f o r example C3.1 a ,b a re d e t a i l e d in
T a b l e s 3 . 5 and 3 . 6 . The d i s t i l l a t i o n co lumn p r o c e d u r e i s
i n i t i a l i s e d a c c o rd in g t o F r e d e n s l u n d e t a l (1977) f o r t h e v e r y
TABLE 3.A : Summary of flowsheeting problems
Problem Title Type of Problem Number of variables/ equations.
C3.1 a Distillation column design Design 11C3i1b Distillation column design Design 12C3.2 Cavett four flash flowsheet (5-component mixture) simulation 55C3.3 Cavett four flash flowsheet (5-component mixture) design 55C3.4 Cavett four flash flowsheet (6-component mixture) simulation 66C3.5 Cavett four flash flowsheet (8-component mixture) simulations 88C3.6 Cavett four flash flowsheet (16-component mixture) simulation 184C3.7 Design of coupled distillation columns with energy
recycledesign 24
C3.8 Design of coupled distillation columns with mass and energy recycle
design 29
C3.9 Optimization of coupled flash units optimization 2 decision variables
114
TABLE 3.5 Solution Statistics for Problem C3.1a
MethodCPU Time (seconds) Flowsheet Iterations
FunctionEvaluations
Equivalent no. of thermodynamic calls
NPP 4.710 4 9 3705
HPP 4.337 4 7 3325
NPA 3.466 4 9 1549
HPA 3.197 4 7 1323
NAP 3.992 4 5 3050
NAA 2.899 4 5 1125
TABLE 3 6 Solution Statistics for Problem C3.1b
MethodCPU Time (seconds) Flowsheet Iterations
FunctionEvaluations
Equivalent no. of thermodynamic calls
NPP 8.741 5 16 6820
HPP 6.312 6 10 4840
NPA 6.244 5 16 2788
HPA 4.709 6 10 1886
NAP 5.402 5 6 3975
NAA 3.912 5 6 1406
T a b le 3 - 7 - I n i t i a L - y a lu e s _ a n d _ S o lu t io n s _ o f_ p ro b le m s _ C 3 -1 _ a Jlb .
Problem C3.1 a I n i t i a l P o in t S o l u t i o n
Problem C3.1 b I n i t i a l P o in t S o l u t i o n
D i s t i l l a t e C k m o l ./hr)Ethane 127.08 118.50 127.08 117.77Propane 159.37 31.38 159.37 31.19Butane 96.67 0 . 1 2 96.67 0 . 1 2Pentane 67.75 199 .06E-6 67.75 207.10 E- 6Hexane 49.13 0. 49.13 0.
Figure 4.3a: s e n s i t i v i t i e s o f e t h y l e n e v a p o u r f l o w p r o f i l e t o a l l t h e b i n a r yINTERACTION COEFFICIENTS--- SCi.j) * 0.0 (EXAMPLE D2.3)
Figure 4 .= 4 : sensitivities of temperature profile to all the binary interactionCOEFFICIENTS--- SC i. j) = 0.0 EXAMPLE D2.3
100-
■t*v
■ 100 -
-200-
£ -300 —| \-400- <--1---1---r ■
5 10 15 20
-hh- S (3. o
SC2. o
-a- SCI. 4)
SC2.
* SCI. -
tn j
t 1
___1
r:
40 45 50 55r j
25 30 35*.a j e number
sensitivity--
-dFY/dS(i.j J
Figure 4.3b: SENSITIVITIES OF ETHYLENE FLOW PROFILE ( VAPOUR PHASE 1 TO AU_ THE BINARY INTERACTION COEFFICIENTS --- BCi.j) - 0 . 0 EXAMPLE D2. 3
155
These r e s u l t s c o n f i r m s The c o n c l u s i o n s o f G rabosk i and Daubert
C1978) t h a t s e t t i n g t h e i n t e r a c t i o n be tween h y d r o c a r b o n s o f
s i m i l a r m o l e c u l a r w e i g h t and d i s s i m i l a r mo le cu la r s t r u c t u r e to
ze r o i s not a good p r a c t i c e . The p r e s e n c e o f a v e r y s e n s i t i v e
s e c t i o n a ro u n d t r a y s 8 - 2 0 ( i n the s t r i p p i n g s e c t i o n ) i s e v id en t
from f i g u r e s 4 .3 and 4 - 4 . H e rnande z e t a l a l s o i d e n t i f i e d the
same r e g i o n i n t h e column from computat ion of the column p r o f i l e
de s ign s e n s i t i v i t i e s t o u n c e r t a i n t i e s i n r e l a t i v e v o l a t i l i t y . The
f i g u r e s a l s o shows a much l e s s s e n s i t i v e r e g i o n i n the r e c t i f y i n g
s e c t i o n .
From the o p t im i z a t i o n o f p l a n t o p e r a t i n g data Hernandez et
a l recommend a v a l u e o f <$23 = 0 .0123 . S t r e i c h and K is tenmacher
(1979) note t h a t O l l e r i c h recommends i n t e r a c t i o n c o e f f i c i e n t s
be tween 0 . 0 2 - 0 . 0 4 f o r h y d r o c a r b o n - hydrocarbon i n t e r a c t i o n s .
F i g u r e 4 . 5 show s t h e v a r i a t i o n o f e t h y l e n e p u r i t y i n t h e
d i s t i l l a t e product o b ta in ed by repeated r i g o r o u s s im u l a t i o n s w ith
v a l u e s o f ^23 be tw een 0 .0 - 0 .04 . A l s o shown on the sample p lo t
are the p u r i t i e s p r e d i c t e d from l i n e a r e x t r a p o l a t i o n s based on the
s e n s i t i v i t i e s d e t e r m in e d a t ^23 = 0 . I t i s e v i d e n t t h a t the
l i n e a r a n a l y s i s i s o n l y v a l i d a round the base p o s i t i o n . The
p u r i t y o f t h e d i s t i l l a t e p roduct ob ta in ed by r i g o r ou s s im u l a t i o n
d e t e r i o r a t e s w i th ^ 23 v a l u e s g r e a t e r than ze ro .
The r e b o i l e r and c o n d e n s e r d u t i e s a r e o n l y s l i g h t l y
a f f e c t e d by u n c e r t a i n t i e s i n the b in a r y i n t e r a c t i o n c o e f f i c i e n t
156Figure 4.5: ETHYLENE PRODUCT PURITY VARIATION WITH BINARY
INTERACTION COEFFICIENT
Figure 4.6:
VARIATION OF REBOILER AND CONDENSER DUTIES TO ETHYLENE-ETHANE BINARY INTERACTION COEFFICIENT S(2, 3)
FIGURE 4.24a : VARIATION OF REBOILER AND CONDENSER DUTIES WITH ERRORS IN VAPOUR ENTHALPY MODEL — - Example D2. 4
.150 . 200
Heat Duty CGJ/hr)
176
FIGURE 4.24b ; VARIATION OF REBOILER AND CONDENSER DUTIES WITH ERRORSIN IDEAL. LIQUID ENTHALPY NOBEL ... Example D2. 4
FIGURE 4.24c : VARIATION OF REBO IL ER AND CONDENSER DUTIES WITH ERRORS IN hXCESS LIQUID tNiHALP'l MUDtL hxample D2. 4
2b. 350 —
2b. 300-
2b. 250-
£ 2b. 2'
;x 2 b. 150
^ 2b.100 »•<
"" 2b. 050
2b.000
25.950- . 200 -.150 -.100 -. 050 .000 .050 .10
Constant relative error in model.150 . 200
17 7
Cavett problem (D2.6)
The s e n s i t i v i t e s of a l l the process stream var iab les to
the non-zero in teract ion parameters between carbon dioxide and the
s a tu ra te d hydrocarbons (<$'12/’ 13/- ^14 $15) are evaluated using
SPEEDUP. The e f fec t s of the remaining hydrocarbon-hydrocarbon
p a r a m e t e r s a re ignored i n view of our e a r l i e r r e s u l t s . The
s e n s i t i v i t i e s are g iven in Appendix D2, Table D2.6. For the
f lo w s h e e t , the product rates (streams 10 and 11) were found to be
remarkably in se n s i t ive to the binary in t e ra c t io n parameters with
chan g es of even 100 % in the most important parameter (S-^)
r e s u l t i n g i n only a 6 . 6 x 1 0“ 3 percent change in the top carbon
dioxide product rate.
1 78
Location of Dominant Zones (D3.1)
The s teady-s ta te design (vapour and Liquid flow p r o f i l e ) ,
enrichment factor per stage, and s e n s i t i v i t i e s of the enrichment
f a c t o r to the 3 b inary in te ra c t io n c o e f f i c ie n ts (Table D3.1) are
presented in Appendix D3. Figures 4-25 a ,b ,c shows the v a r i a t i o n
of enrichment fac to rs with deviat ions in the base value of binary
co ef f ic ie n ts of zero. The r ig o ro u s enrichment f a c t o r s a t t a i n s
maximum v a lu e s of 0-4370 at the top p la te (condenser) in the
r e c t i fy in g section and 0.1961 located on stage 7 ( i e feed s t a g e ) .
L o c a t io n of the maximum enrichment factor near the condenser i s
good for contro l purposes s in c e time d e la y s would be small in
magnitude where temperature sensors placed at the top plate are
used to control the purity of the d i s t i l l a t e product. However, a
maximum in the rec t i fy ing s e c t i o n lo c a t e d on the feed t r a y i s
undesirable based on the c r i t e r i a d ef in ed e a r l i e r . The f i g u r e s
also show that the locat ion of the dominant zones are not affected
by u n c e r t a i n t i e s of 0 . 0 0 5 , 0 . 0 1 , and 0 . 0 2 i n the b i n a r y
i n t e r a c t i o n c o e f f i c i e n t b e t w e e n i s o p e n t a n e - p e n t a n e ,
i so pentane-hexane and pentane-hexane. The l i n e a r approximations
of U from the base point se nsi t i v i t i es are v a l id for er rors of
0.005 in a l l the parameters. These r e s u l t s mean that the chosen
co n tro l s t r u c t u r e s (or l o c a t i o n of sensors) using the physical
c r i t e r i o n of B r ig n o le e t a l i s not a f f e c t e d by e r r o r s in the
binary interact ion constants.
enri
chme
nt fa
ctor
en
rich
ment
factor
Figure 4.25a Rinorous and approximate values of enrichment factor at 6(i,j) = 0.005
Figure ,4.25b Rigorous and approximate values of enrichment factor at 6(i,j) = 0.01
i
enri
chme
nt f
acto
r
Figure 4.25c Rigorous and approximate values'of enrichment factor at S(i,j) = 0.02
1CU4.6. CONCLUSIONS
Exact s e n s i t i v i t i e s of f u g a c i t y coef f ic ie nts and excess
enthalpies to binary in te ra c t io n parameters were computed. The
a n a l y t i c e x p r e s s io n s g iven i n Appendix D1 are f a i r l y easy to
derive.
S e n s i t i v i t i e s of r igorous f la sh and d i s t i l l a t i o n columns
to binary in teract ion c o e f f i c ie n t s have also been generated q u i te
e a s i ly and e f f i c i e n t l y . In f a c t , in some cases, the generation of
th e r i g h t hand s i d e s i n e q u a t i o n 4 . 2 d o e s n o t r e q u i r e
d i f f e r e n t i a t i o n of the physica l propert ies models (Murphree tray
e f f i c i e n c y and co n stan t r e l a t i v e e r r o r s i n e n t h a l p y model
funct ions) .
For the f la sh and Cavett processes, the interact ion between
the nonhydrocarbon and most v o l a t i l e hydrocarbon parameter was
a lw a y s t h e most c r i t i c a l parameter . With the d i s t i l l a t i o n
examples , the most important parameters were the c r o s s - t e r m
i n t e r a c t i o n between the l i g h t and heavy key components. The
s e n s i t i v i t i e s to the other parameters are 1-2 order of magnitude
s m a l le r than the l i g h t - h e a v y key i n t e r a c t i o n coef f ic ie nt . The
s e n s i t iv i t y of product p u r i t i e s to Murphree t ra y e f f i c i e n c y was
found to be a p p r e c i a b l e . In t h e examples considered here
condenser duty i s q u i t e i n s e n s i t i v e to u n c e r t a i n t i e s i n the
p h y s ic a l propert ies parameters and enthalpy models. The reboi ler
duty on the o t h e r hand i s s l i g h t l y more s e n s i t i v e to the
181
parameters and model functions.
The i m p o r t a n c e of s e n s i t i v i t y of process des ign to
uncerta int ies in physical propert ies cannot be over-emphasized.
S e n s i t i v i t y d a ta have been used here to i d e n t i f y the most
important param eter (s ) and or model f u n c t i o n s . I t was a l s o
p o s s i b l e to rank the order of importance of the i n t e r a c t i o n
parameters. I d e n t i f i c a t i o n of the c r i t i c a l parameters have an
important use in the l ig h t of recent research in TD model building
( U r l i c et a l , 1985). Accurate phase equi l ibrium p r e d i c t i o n s are
n ecessa ry for any a c c e p t a b le de si gn/s i mul a t i ons of these unit
o p e r a t io n s . For i n d u s t r i a l p r o c e s s e s , ho w ever , a c c u r a t e
p r e d ic t i o n of phase e q u i l i b r i a becomes d i f f i c u l t , e ither because
the ava i lab le models or some of the req u ire d parameters are not
a v a i l a b l e or a re not v a l i d a t the conditions of operation of the
process. The est imat ion and/or adjustment of model parameters
t h e r e f o r e becomes necessary. Therefore, the a b i l i t y to quantify
and rank the e f fe c t s of u n c e r t a in t i e s in p h y s i c a l p r o p e r t ie s on
process design i s cruc ia l in such studies.
The s e n s i t i v i t y data was used to evaluate the va r ia t io n of
the l o c a t io n of the maximum enrichment factor to uncerta int ies in
the binary in te ract io n parameters. Consequently, the d e s ig n er i s
a b le to a s c e r t a i n whether or not the designed contro l sensor
locat ions are affected by e r ro rs in physical properties and hence
the o p e r a b i l i t y of the column. B a s i l i (1986) a l s o used the
s e n s i t i v i t y of l iq u id - l iq u id e q u i l ib r ia simulations to the number
of groups in a molecule for the purpose of choosing solvents.
The l i m i t a t i o n s of the l i n e a r a n a l y s i s should not be
o v e r l o o k e d . In a l l the e x a m p le s , we d e m o n st ra te d t h a t
extrapolat ion of s e n s i t i v i t y information far away from the point
at which i t was generated i s not recommended since i t can give
inaccurate r e s u l t s . However, some of the e x t r a p o l a t i o n s were
quite extreme.
XUAfTIB-flVE
G .§ ne r s i _ Co n c L u s i o n s_ a nd_ R e c oom e nda t i o n s
In t h i s t h e s i s , we proposed and tested a new thermodynamic
property data i n t e r f a c e s t r a t e g y . We co n c e rn e d o u r s e l v e s
p r im a r i l y with the e f f i c i e n t provision of rigorous thermophysical
properties and phase e q u i l ib r iu m procedure d e r i v a t i v e s and the
g e n e ra t io n of process design s e n s i t i v i t i e s to uncerta int ies in TP
models or parameters. Several conclusions and recommendations for
future work can be made based on our re s u l t s ,
Prov is ip n _p f_e xact_p rp ced u re_d erivatiyes
Our r e s u l t i n d i c a t e s t h a t a n a l y t i c d e r i v a t i v e s of TD
properties can be e a s i l y ob ta ined a t a cost of about 1 -2 - 2 .5
base point evaluat ions.
The new technique f o r computing isothermal f lash (VLE)
procedure d e r i v a t i v e s i n v o l v e s r e l a t i v e l y small computational
overhead compared to a base point evaluat ion. We bel ieve further
improvements could be r e a l i s e d i n the computation of procedure
d e r i v a t i v e s f o r our d i s t i l l a t i o n module. One possible method i s
to use a technique s i m i l a r to that adopted for the iso therm al
f l a s h procedure. That i s , we p a rt i t io n the 2NC + 1 equations per
stage into two parts- F i r s t , the equilibrium re lat ions (equat ion
C3-2) and energy balance (equation C3.3) are grouped together in
184
th e form of equat ion 3-6 (Chapter 3 ) . The second p a r t i t i o n
comprises of the component mass ba lances (equat ion C 3 .1 ) . The
l i q u i d flow and temperature p r o f i l e s design s e n s i t i v i t i e s to the
inputs can therefore be obtained by solving a l inear set of NC + 1
eq uat io ns with the vapour flow p ro f i le s s e n s i t i v i t i e s obtained by
chain-ru ling using equation 3-7.
At the moment, d i s t i l l a t i o n procedure d e r i v a t i v e s are
eva lu a ted in about 1/4 of the time fo r a r igo rous base point
determination. Our method of generating VLE procedure der ivat ives
i s by no means r e s t r i c t e d to phase and chemical e q u i l ib r iu m unit
modules but can be a p p l ie d t o any given procedure- In f a c t , we
used the method to der ive exact o u tp u t- in p u t g ra d ie n t s of heat
exchanger models.
Applications of our TD property data interface strategy to
flowsheeting examples produced encouraging re s u l t s in terms of the
c r i t e r i a s t a t e d i n C hap ter two- The r e s u l t s i n d i c a t e that
numerical der ivat ives of TP models should be used in procedure
der ivat ives ca lcu la t io n s where a n a ly t ic der ivat ives of such models
are unavailable- The use of numerical TP der ivat ives between the
th ree l e v e l s of computation (Figure 1.1) should be avoided. In
expensive VLE ca lcu la t ions (e .g . d i s t i l l a t i o n ) we suggest the use
of any s u i tab le "der ivat ive f ree" numerical solut ion method (e.g.
Hybrid) to obtain base point so lut ions . In these cases, numerical
d e r i v a t i v e s should only be used to secure accu ra te procedure
gradients. Thus, even though i t i s b e n e f ic ia l to use d e r i v a t i v e
185
based methods for the solut ion of procedures, our method gives one
the f l e x i b i l i t y of us ing any s u i t a b l e method in the s o l u t i o n
algori thm.
The r e s u l t s of our work have c e r t a i n i m p l i c a t i o n s on
current flowsheet executives- Procedure re p re s e n t a t io n s need to
be extended to in c lu d e the matrix of output-input der ivat ives in
the output v a r i a b l e l i s t with the in p u ts not l i m i t e d to o n ly
temperature , p r e s s u r e , and composit ion- Unknown inputs at the
flowsheet level should be flagged so that procedure d e r i v a t i v e s
a r e o n ly s e c u r e d f o r such v a r i a b l e s t h e r e b y e l i m i n a t i n g
unnecessary ca lcu la t io n s . In other words, the number of columns
in the r ig h t hand s id e m a t r i x , S, i s modified according to the
number of " a c t i v e " in p u t v a r i a b l e s . F o r t u n a t e l y , such
in fo rm at io n i s r e a d i l y a v a i l a b l e when the C u r t i s e t a l (1974)
algorithm i s used to minimize the number of f u n c t i o n e v a lu a t io n s
in the generation of numerical flowsheet Jacobian matrix.
At each flowsheet i t e r a t i o n , i t was i m p l i c i t l y assumed
th a t input c o n d i t io n s i n t o the f l a s h and d i s t i l l a t i o n routines
would r e s u l t i n v a p o u r and l i q u i d p h a s e s w i t h p r o c e d u r e
d e r i v a t i v e s e v a l u a t e d a c c o r d i n g l y . U n fo r t u n a te ly , most TP
packages do not have routines which determine the number of phases
present in any g iven mixture at a prescribed condition- Thus, a
problem a r i s e s when the assumed number of phases i s incorrect- We
do not know how to cope with the discontinuity a r i s in g as a resu l t
of the disappearance of a phase or indeed the appearance of more
186
t h a n th e number of p h a s e s assumed a p r i o r i . Under such
c i r c u m s t a n c e s , p r o c e d u r e d e r i v a t i v e s g e n e r a t e d would be
m e a n i n g l e s s s i n c e t h e a s s u m p t i o n of c o n t i n u i t y and
d i f f e r e n t i a b i l i t y i s not t r u e . In o ther words, the f lowsheet
model chan g es when t h e r e i s a change i n the number of f l u i d
phases. We recommend a d e t a i l e d study of t h i s problem i n the
future.
The f l o w s h e e t s i m u l a t i o n r e s u l t s i n d i c a t e t h a t
thermophysica l p r o p e r t i e s packages s h o u ld p r o v i d e a n a l y t i c
d e r i v a t i v e s i n a d d i t i o n to point v a l u e s of p r o p e r t ie s a s a
standard feature. Such exact an a ly t ic d e r iv a t i v e in fo rm a t io n i s
a l s o needed i n other a r e a s , such a s , phase s t a b i l i t y ana ly s is
(Michelsen, 1982a) , and computation of other p r o p e r t ie s ( e .g .
e x c e ss entha lpy from temperature der ivat ive of fu g a c i ty /a c t iv i ty
c o e f f i c i e n t s ) . F u r th e rm o re , the a v a i l a b i l i t y of a n a l y t i c
der ivat ives removes the need to develop numerical solution methods
which attempt to s a t i s f y the re le v a n t TD c o n s t r a i n t s a t each
i te ra t io n CVenkataraman and Luc ia , 1986).
S®JD§iiiyity_to_physi ca l_prop erties
E x a c t s e n s i t i v i t i e s of f u g a c i t y c o e f f i c i e n t s , excess
enthalpy, f l a s h procedure , d i s t i l l a t i o n module and in t e g ra t e d
process units were obtained. Suitable l in ear algebra and gradient
chain-ruling y ie ld the desired s e n s i t i v i t i e s of a process to basic
physical property q u an t i t ie s . All the s e n s i t i v i t i e s were obtained
•187
e f f i c i e n t l y by solving a s ingle l inear system. The method avoids
repeated perturbations of the rigorous process model or the need
to make d r a s t i c assumptions. The s e n s i t i v i t y information enables
one quickly ident i fy important parameters (or models) in a process
d e s i g n a s w e l l as t h e l o c a t i o n of p o s i t i o n ( s ) w ith high
s e n s i t i v i t i e s . We also used the s e n s i t i v i t y data to ascerta in the
e f f e c t s of u n c e r t a i n t i e s i n b in ary in t e ra c t io n c o e f f i c ie n ts on
column control s t ructures . I t would be i n t e r e s t i n g to f in d out
the p o s s i b i l i t y of determ in ing v a r i a t i o n s of zones of maximum
enrichment factor to changes i n the operating v a r i a b l e s by using
s e n s i t i v i t i e s of column p r o f i l e s to such input or operat ing
v a r ia b le s .
A s e r io u s problem with our method i s the requirement for
p r o v i s io n of TP model d e r i v a t i v e s to p h y s i c a l p r o p e r t i e s
c o n s t a n t s . We b e l i e v e i t would be u n r e a l i s t i c to demand that
physical properties packages provide temperature , p r e s s u r e , and
composition d e r i v a t i v e s a s w e l l as d e r iv a t i v e s of TD models to
constant parameters e . g . T c, nc, S i j - We suggest the use of
numerical d er ivat ives when a n a ly t i c information i s not a v a i l a b l e .
The a d d i t i o n a l computational overhead a r i s i n g from the use of
numerica l TP d e r i v a t i v e s may not be p r o h i b i t i v e s i n c e t h e s e
der ivat ives are only needed at the solution of the model.
The current s t a t e of a f f a i r s whereby u se rs of process
simulators are large ly unaware of the impact of errors in physical
properties assumptions on t h e i r des ign i s u n s a t i s f a c t o r y . We
188
t h e r e f o r e recommend t h a t p ro ce ss s e n s i t i v i t e s be c a r r i e d out
r o u t in e ly by p ro ce ss s im u la t o r s - The adoption of our method
im p l i e s tha t the input v a r i a b l e l i s t of procedures (or argument
l i s t of s u b ro u t in e s ) f o r un it o p e r a t i o n s and thermodynamic
p r o p e r t ie s be extended. We recommend a general representation of
the form:
■Coutputs, o u t p u t - i nput g r a d i e n t s } P { i n p u t s to i n c l u d e
temperature, pressure , composition, c r i t i c a l parameters, e t c . }
T h i s w i l l make i t e a s y f o r u s e r s t o d i r e c t l y s p e c i f y the
parameter(s) or model f o r which s e n s i t i v i t i e s are desired.
F i n a l l y , we b e l i e v e t h e i m p l e m e n t a t i o n of our
thermodynamic i n t e r f a c e s t r a t e g y in p r o c e s s f l o w s h e e t i n g
( i r r e s p e c t i v e of the f lo w s h e e t a r c h i t e c t u r e ) w i l l r e s u l t in
s i g n i f i c a n t improvement i n the e f f i c i e n c y of c o m p u t e r - a id e d
process ca lcu la t ions .
1U9
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ZOO
NOMENCLATURE
f , H/ f/ g f u n c t io n v e c t o r s
p gene ra l p rocedure r e p r e s e n ta t io n
U, U1 e n r i c h m e n t f a c t o r p e r s t a g e f o r t h e r e c t i f y i n g and s t r i p p i n g s e c t i o n s o f a column.
J Ja cob i an m a tr i x
C1, c], cl " c o m p u te d " p a r t of t he J a c o b i a n m a t r i x i n H yb r id methods
A1 " a p p r o x i m a t e d " p a r t o f t he ja c o b ia n m a t r i x i n h y b r id methods
Q,R, S, S1 m a t r i c e s d e f i n e d as the p a r t i a l d e r i v a t i v e s o f a gene ra l procedure model wi th re spe c t t o o u t p u t , i n t e r n a l , i n p u t v a r i a b l e s , and cons tan t parameters.
Rg u n iv e r s a l gas constant
w, v , u, p, h v e c t o r s o f o u t p u t v a r i a b l e s , i n t e r n a l v a r i a b l e s , i n p u t v a r i a b l e s , c o n s t a n t pa ram e te rs , and model fu n c t io n
X v e c t o r o f unknown v a r i a b l e s ; l i q u i d phase mole f r a c t i o n
y v e c t o r o f unknown v a r i a b l e s ; v ap o u r phase mole f r a c t i o n
FX, FY, FZ v e c t o r s o f l i q u i d , v a p o u r , and f e e d component f low ra te
FL, FV t o t a l l i q u i d , vapour f low ra te
HL, HV, HF t o t a l l i q u i d vapour, and feed en tha lpy
He exce s s e n th a lp y per mole
SL, SV d i m e n s i o n l e s s l i q u i d , v ap o u r s id e - s t r e a m f low r a t e
mass b a lan ce of component i on stage l
Ql , i e q u i l i b r i u m r e l a t i o n o f component i on stage l
ID I d e a l
EX Exce ss
ElQc
201
El
Qc
en tha lp y ba lance on stage l .
condenser duty
Q r r e b o i l e r duty
Murphree p la t e e f f i c i e n c y
DEST d i s t i l l a t e ra te
RFLX r e f lu x r a t i o
K v a p o u r - l i q u id e q u i l ib r iu m r a t i o
NST number o f e q u i l i b r iu m s tages
NC number o f components
Qi f u g a c i t y c o e f f i c i e n t o f component
T tem pera tu re
TF feed tem pe ra tu re
n p re s su re
PF feed p re s su re
no s a tu ra te d p re s su re o f a pure s p e c ie s
O) P i t z e r a c e n t r i c f a c t o r
RU r e l a t i v e number o f segments per m o lecu le as used i n th e UNIQUAC equa t ion
QU, QP r e l a t i v e s u r f a c e area o f a m o lecu le as used i n the UNIQUAC equa t ion
Z c o m p r e s s i b i l i t y f a c to r
q p a ra m e t r iz e d v a r i a b le s f o r e r r o r in en tha lpy model
3 v e c to r o f l o c a l model param eters
A energy o f i n t e r a t i o n i n UNIQUAC equ a t io n
6 b i n a r y i n t e r a c t i o n c o e f f i c i e n t i n S o a v e - R e d l i c h - K w o n g e q u a t i o n ; k r o n e c k e r d e l t a
II II i n f i n i t y norm
m, n, l , d im e n s io n o f o u t p u t , i n t e r n a l , i n p u t , and param eter v e c t o r s ;
Z 0 2
A s u b - d i a g o n a l b l o c k m a t r i x o f d i s t i l l a t i o n column J a c o b ia n m a t r ix ; m ix tu re parameter i n SRK-equa t i on.
B d ia g n a l b l o c k m a t r ix o f d i s t i l l a t i o n column J a c o b i a n m a t r i x ; m i x t u r e p a r a m e t e r i n SRK-equa t i on.
C s u p e r - d i a g o n a l b lo c k m a t r ix o f d i s t i l l a t i o n column J a c o b ia n m a t r ix .
s i i m o la r f lo w ra te of component i i n stream j .
d , d , d ,d s p e c i f i c hea t c a p a c i t y c on s tan ts .
a, b, m, pa ram ete rs d e f in e d i n
km ix t/ ’ am ix t SRK-equa t i on.
S u b s c r i p t
i n t e r a c t i o n between component i and j
component i i n a stream le a v in g stage l
L stage in d e x
S u p e r s c r ip t
k i t e r a t i o n i ndex
0 base p o in t ; pu re component p rope rty
e exce ss p ro p e r t y
c c r i t i c a l c o n d i t io n
APPENDIX A
T a b le A1 : R e v ie w s _ p f_ P r o c e s s _ F lo w s h e e t in g
Author Cs) Approaches D iscussed
Motard , Shacham, and Rosen (1975)
(S e q u e n t ia l ) - modular; Equat i on -O ri ented.
H lavacek (1977) M odu la r , G loba l
Rosen (1980) Sequenti a l - (M o d u la r ) ; S im u ltaneous ,S im u ltaneous-m odu la r (Tw o-T ie r)
Evans (1981) S e q u e n t ia l -M o d u la r ; E q u a t io n -O r ie n te d ; Two-T ier
Evans (1982) Sequenti a l-m o d u la r ;
Shacham, M a c ch ie t to , S tutzman, Babcock (1982)
S e q u e n t ia l -m o d u la r ; Des ign - O r ie n te d ; E q u a t io n -O r ie n te d ;
P e rk in s (1984) Equa t i on -O r i ented
B ie g l e r (1985) S im u ltaneous-m odu la r
T^ble_A2_j_ .fh jmeri Ceil S o l u t i o n Methods f o r N o n l in e a r A lg e b r a ic
Method
Equat i ons
Re fe rence
NewtonD is c r e t e NewtonBroydenShubertBrownB ren tHybr idM o d if ie d Hybr id
Broyden (1965)Shubert (1970)Brown (1969)B ren t (1973)L u c ia and M a cch ie t to (1983) L u c ia e t a l (1985)
APPENDIX B
T y p i c a l Data A v a i l a b l e from a P h y s i c a l P r o p e r t i e s Package
C o n s tan t_ p rp p e r t ie s
C r i t i c a l temperature C r i t i c a l p re ssu re C r i t i c a l volume M e l t in g p o in t B o i l i n g p o in t M o le cu la r we igh t ParachorVapour heat o f v a p o u r i s a t io n L iq u id heat o f v a p o u r i s a t io n F lash p o in t F lam m a b i l i t y l i m i t A u t o ig n i t io n temperature S o l u b i l i t y parameter A c e n t r i c f a c t o r D ip o le moment
^3£ijble_prppertiesFu ga c ity c o e f f i c i e n t s Vapour heat c a p a c it y L iq u id heat Vapour v i s c o s i t y Su r fa ce te n s io n L iq u id v i s c o s i t y L iq u id d e n s i t y Vapour d e n s it y Vapour en tha lpy L iq u id en tha lp y G ibbs f r e e energy En tha lp y o f v a p o u r is a t io n L iq u id the rm a l c o n d u c t iv i t y Vapour thermal c o n d u c t iv i t y Vapour en tropy L iq u id en tropyC o e f f i c i e n t o f c u b i c a l expan s ionS a tu ra te d vapour p re s su re Heat o f fo rm a t io n A c t i v i t y c o e f f i c i e n t s
TP P rocedu res
Iso the rm a l f l a s h (VLE)L i q u i d - l i q u i d f l a s h Bubb le p o in t Dew p o in t I s o c h o r i c f l a s h I s e n t h a lp i c f l a s h I s e n t r o p ic f l a s h D i s t i l l a t i o n column VLLE f l a s h
Optigna l_packacjes
Steam packagePetro leum f r a c t i o n package R e f r ig e r a n t packageS p e c ia l i s e d equa t ion of s t a t e package
205APPENDIX Cl
Analytic derivatives of fugacity coefficients (using SRK-equation) with respect to temperature, pressure, and composition as well as physical properties constants of components used in this study.
Table Cl.1: Derivatives of fuaacitv coefficients using theSRK equation of--s.t_a.t.e--with respect totemperature, pressure and composition.
The fugacity coefficient of component i in a mixture can be calculated from the SRK equation of state (equation 4.12, Soave, 1971):
1 biIn 0 . = -- -— (Z-l) - In (Z-B)1 b ,mixt
A r _ kB ’ L
a-k ika .mixt
- b . JmixtBIn (1 + — ) Z
k / i 11 2,where
, NC
T .cb. = 0.08 664Rg — —1 7C .C
(R T.C)^a. = 0.42747. — -— --- . a.i 7C. c i
2 c —a. = 1 + m. (1 - (T (T. ))2i i l
m-ii = 0.480 + 1.574 0) - 0.176 CO2b . = T x bmixt
amixt
k k k
“ ? ? xi xk aiki k
Cl.l
Cl.2
Cl .3
Cl. 4
Cl. 5
4.144.13
206
a i k = C1 - 6 i k 5 <ai ak ) 1 / 2 4 .16
I f we d e f in e co m p re s s io n f a c t o r as Z=-------- th e n th e S R K -e q u a t io nHgT'( e q u a t i o n 4 .12) can be t ran s fo rm ed i n t o the f o l lo w in g c u b ic equ a t io n in
Z:
Z3 - Z2 + (A - B - B2 ) Z = 0 C1.6
Equa t ion C1 . 6 y i e l d one o r th re e ro o ts depending on the number of phases
p re sen t i n th e system. In th e twcrphase r e g io n , the la r g e s t ro o t i s the
com p re s s io n f a c t o r o f t h e v a p o u r , w h i l e t h e s m a l l e s t p o s i t i v e r o o t
c o r r e s p o n d s t o t h a t o f th e l i q u i d . We o b t a in e d t h e c o m p r e s s ib i l i t y
f a c t o r o f systems used i n t h i s s tudy u s in g the a lg o r i t h m o f G u n d e rs e n ,
1982.
D e r j w a t i v e s o f p a r a m e t e r s i n S R K -equa t i o n w i t h r e s p e c t to_tem perature^.
p r e s s u r e _ a n d _ c o m p g s i t io n -
Le t G-j-j = 2 x a k k k i
C1.7
and
G2 i =
2G-| i _ bi_ ;am ix t ^mixt
C1.8
The f o l lo w in g d e r i v a t i v e s w i l l be needed la t e r ;
3 a-j 3t
1/2 mn* a-;1/2
2(T T i c ) 2
C1.9
207
am ixt 9a i k 86i -j______= 1 J * i *k = Z x-j c i . i o
8T i k 8 T i a t
where
3ai k(1 - 6jk) 1/ 2
3a ^/2 8a/ / 2
8T 8Tak C1.11
8 am ix t 8 ^mixt 8 ^mixt
an " " ’ “ a / " an” "
3am ixt
3xk= 2 G1 k C1.13
3^mi xt _ = bk
9 xk /
C1.14
3A A ^mi xt A= -------- ---------- — 2 —
3T ami xt 3T TCl .15
3b B
3T TCl .16
9 A am ix t
8n <RgT)2C1.17
9b ^mixt
an Rg TC1.18
208
9 A n ami xt
9*k (RgT)2 3 x k
SB
-Q
II
3xk RgT
£ 3A9Z 63 3T " (Z--B) 3T
3 T G4
where
G3 = A + Z (1 + 2B)
G4 = Z (3Z - 2) + A - B - B2
C1.19
C1.20
C1.21
C1.22
C1.23
P re ssu re and com pos it ion d e r i v a t i v e s o f Z a re o b t a in e d by r e p l a c i n g
te m p e ra tu re d e r i v a t i v e s w ith the a p p ro p r ia t e d e r i v a t i v e e x p re s s io n i n
equa t ion C1.21.
J e m p e ra tu r e _ a n d _ g r e s s u r e _ d e r iv a t iv e _ p f_ fu g a c i t y _ c g e f f ic ie n t
From equa t ion C1.1 we have the f o l l o w in g tem peratu re d e r i v a t i v e s :
31 n 0 -j L_ bi 3 Z 1 3Z 3b
T t ^mi xt ^ (Z-B) 9T 9t
where
C1.24
GA = 65 g2T + g5T g2 C1.25
A BG5 = i1+c C1.26
B Z
Z09
g5T =B
g6 T * G7 T In (1
1 “ 3 B B
g6 T =Z+B
— •
9 T Z
1 3 A A 3 B
g7T =B 9 T B^ 9 T
g2T = 2
1 9GlI-
B
Z
3z
ami xt 9T
9am ix t
9T
C1.27
C1.28
C1.29
C1.30
N o t e : P r e s s u r e d e r i v a t i v e s o f f u g a c i t y c o e f f i c i e n t s a re ob ta in ed by
r e p la c in g the tem peratu re d e r i v a t i v e s by t h e i r p r e s s u r e d e r i v a t i v e s
e q u iv a le n t .
Cgmposi t ig n ^ d e r i v a t i v e s_g f __f u g a c i t y _ c g e f f i c i ent
The c o m p o s i t i o n d e r i v a t i v e s a r e e v a l u a t e d from th e f o l l o w i n g
e xp re s s io n s :
(Z-1)
where
31 n 0 -j*- ___ = b-j
1 3 Z bk
3 x k bmi xt 3 x k bmixt
1 ~3 Z 3b ”
Z-B
1,
X
l 1
ixk- 6B
GB = G5 G2x + Gsx G2
C1.31
C1.32
g5X = - g8 X + g9X + C1.33
210
g8X "
69X “
1 3B B 8 Z
Z+B _3xk Z 8 xk_
1 8 A A 8 B
B xk B 8 xk
C1.34
C1-35
G2x - 2a i k 8a,
- G*i -j bi bk+D rmxt
C1.36Jmi x t
am ix t
Vapour phase d e r i v a t i v e s are o b ta in ed by r e p la c in g x ' s by y ' s i n a l l
the above e x p re s s io n s .
The f o l l o w in g eq u a t io n r e l a t e s d e r i v a t i v e s w ith re spec t t o molar f low
ra te s to d e r i v a t i v e s w i t h re sp e c t t o mole f r a c t i o n :
8 q 8 q
FI______ = ___ _ Z x n‘ ----- C1.373FXk 3 x k i 3 Xi
where Q i s In 0 -j or A He
P a r t i a l m o la r exce ss e n t h a l p i e s (AHa ) a r e o b t a in e d from fu g a c i tyic o e f f i c i e n t s u s in g the fundam enta l r e l a t i o n s h i p :
8 l n 0 -j
3 TAHie= - RgT2 C1.38
Tab le C1.2 : T e s t Prob lems f o r E v a lu a t io n o f T y p ic a l TD P r o p e r t i e s D e r i v a t i v e s
ProblemComponents (Kmol /h r)
C1.1 C1.2 C1.3 C1.4 C1.5
N it ro g e n 451 .97 20.04Carbon D io x id e 511.83 1361.32 6637.16 1356.21Hydrogen S u lp h id e 206.72Methane 2253.67 3776.69 456.12Ethane 361.33 2772.74 1273.33Propane 782.16 1510.93 1341 .34Iso -bu tane 203.05Butane 90 .32 189.44 474.94 387.85I so-pen tane 86.46Pe nta ne 53.73He xa ne 12.83 113.04 49.76Hepta ne 24.52Octane 6 .05Nonane 0 . 1 2 1.87Decane 6 9 .5 E-3 0 .28 0 .32Unde caneMethyl cyc lopen tane Be nze ne Cy c lohexa ne To luene
0 .16
27.00
E th an o l 23.00Water 50.00
Temperature (K) 322.0 311.0 311.0 309.0 345.15
P re ssu re (bars) 19.0 56.2 56.2 4 .39 1.013
Phase Vapour Vapour Vapour Vapour L iq u i d
Thermodynamic Model SRK SRK SRK SRK UNIQUAC
C1.6
12.60
20.7015.3011.8022.60
5.00
360.15
1.013
L iq u i d
UNIQUAC
*12
T a b le C1 .3 : N o n -Z e r o B i n a r y I n t e r a c t i o n Parameters usedModel (Re id e t a l . , 1977)
N it ro g e n Hydrogen CarbonSul phi de Di ox i de
Carbon D io x id e -0 .0315* 0 . 1 2
Methane 0 . 0 2 0 .08 0 . 1 2
Ethane 0.06 0.07 0 .15Propane 0 .08 0 .07 0.15Iso -bu tane 0 .08 0.06 0.15Butane 0 .08 0.06 0 .15Iso -pen tane 0 .08 0 .06 0.15Pe nta ne 0 .08 0.06 0.15He xane 0 .08 0 .05 0 .15Heptane 0 .08 0 .04 0 .15Octane 0 .08 0 .04 0 .15No na ne 0 .0 8 0 .03 0 .15De ca ne 0 .08 0 .03 0.15Unde cane 0 .0 8 0 .03 0.15
★ Taken from Gmehling Onken and A r l t (1982)
T a b le C1.4 : UNIQUAC B in a r y I n t e r a c t io n Parameters( P r a u s n i t z et.. a l . „ 1980)
D i_s_t i L l_at_l2 j. u jm _Pjrjj_ce. du_r_e _D e r i vjjtj_v.es_a nd_ J_e_st _Prpb_l_em_s _f o r
E v a lu a t io n o f F L a sh_and_D is t iL La t ign_C gLum n_P ro cedu re_D e r iva t iv e s
The e q u a t i o n s w h i c h d e s c r i b e c o n t i n u o u s , m u l t i c o m p o n e n t
d i s t i l l a t i o n a r e w e l l known (H o l la n d , 1981; N a p h ta l i & Sandholm, 1971,
e t c ) . For c om p le ten e ss and d i s c u s s i o n of g e n e r a t i o n of m a t r i c e s i n
e q u a t i o n 3 . 5 ( c h a p t e r 3 ) we c o n s i d e r t h e case o f a co lumn w i t h NST
plates separating NC-components where p la te 1 i s a r e b o i le r and p la te
NST i s a p a r t i a l c o n d e n s e r . F u r t h e rm o r e , l e t s ide s tream s (SL,SV) be
s p e c i f i e d as the r a t i o o f th e s i d e s t r e a m t o t he s t r ea m wh i ch r e m a i n s
a f t e r they are withdrawn. Figure C2.1 shows a schematic representation
o f a t y p i c a l p la t e . F X ^ , F Y ^ , T[ a re the unknown v a r i a b le s w ith FLL,
FVl re p re s e n t in g the t o t a l phase f lo w s .
FY • F x .ri£,i 1+1, l
SL
F i g u r e C2.1 : S c h e m a t i c r e p r e s e n t a t i o n o f a s t a g e i n a d i s t i l l a t i o n
column
215
Three t y p e s o f e q u a t i o n s w h i ch d e s c r ib e p h y s ic a l p ro cesses on
p la t e l (assuming the p re s su re i s f i x e d and the p la t e i s a d i a b a t i c ) a re
as f o l l o w s :
Component m a te r ia l b a la n ce s
MLi = (1 + SVL) FYL / i + C1 + SLL> FXL / i - FYl+ 1 / i
" ™ l - 1 , i - FZM
i = 1 , .........NC, l = 1 , . . . NST
E q u i l ib r iu m r e l a t i o n s
fxl,!QL i = nL FVl KL. i -------- - FYl/ r i
Fk
+ (1 - nL) FVtFYL+1,Fvl+1
(C2.1)
i = 1 , ......... .. NC, 1 = 1 , ____NST (C2 .2)
En tha lpy ba lances
El = (1 + SVL)HVL + (1 + SLL)H L L - HVL+1 - HLL- 1 - HFl
1 = 1 , ___ NST (C2.3)
These equa t ion s app ly to a l l i n t e r i o r p la t e s o f the column as w e l l as t o
a p a r t i a l r e b o i l e r ( o r c o n d e n s e r ) . T h e r e a r e 2 N C + 1
e q u a t i o n s / v a r i a b l e s p e r s t a g e , t h a t i s , a t o t a l o f NST (2NC + 1)
e q u a t io n s / v a r ia b le s . S in ce the hea t lo ads on r e b o i l e r and condenser are
unknown, o t h e r s p e c i f i c a t i o n s a re w r i t t e n in s t e a d o f en tha lp y b a lan ce s
f o r p la t e s 1 and NST. We chose the f o l lo w in g s p e c i f i c a t i o n s w h i c h were
cons ide red in the F redens lund e t a l (1977) code.
216
Condenser ( P a r t i a l )
eNST ” f l NST " RFLX • DEST (C2.4)R e b o i le r
E-j = - FL-i - DEST + E E FZ j_ ^Li
(C2.5)
where RFLX and DEST a re r e f l u x - r a t i o and d i s t i l l a t e ra te r e s p e c t i v e ly .
Note RFLX = FLNST/ DEST
The r e b o i l e r and condense r d u t ie s a re c a l c u la t e d a f t e r the s o lu t i o n t o
the above se t o f e q u a t io n s from the f o l l o w in g r e l a t i o n s :
The f u l l se t of unknown v a r i a b l e s o r a subse t t h e re o f can be t r e a t e d as
t h e d i s t i l l a t i o n p r o c e d u r e o u t p u t v a r i a b l e s . We have cho sen th e
f o l lo w in g groups o f v a r i a b l e types s im p ly f o r conven ience
I n te rn a l (v) - HV i, HL , K -jl = 1 ,2 , . , NSTi = 1 ,2 . . , NC
FXL i , FYLi l = 2 ,3 . . , NST-1i = 1 ,2 - . . NC
Output (w) - FX-j .j, FY^jsTi' TNST' Q° ' qP
Input (u) “ RFLX, DEST, TF^, FZ^
Qc - HVn s t _ i -H L NSt “ HVn s T
Qr = HV-j + HL*i “ HL2
(C2.6)
(C2.7)
1 = 1 , - . . NST
The s t e a d y - s t a t e de s ign o f the column i s ob ta in ed by f i n d in g the s e t of
i n d e p e n d e n t v a r i a b l e s x ( F X ^ , FY^O wh i ch s a t i s f i e s t h e model
rep re sen ted by e q u a t io n s C 2 .1 -C 2 .3 . In o th e r w o rds , we a re i n t e r e s t e d
i n s o lv in g a n o n l in e a r a lg e b r a i c system of the form
F(x) = 0 (2 .1 )
N a p h t a l i - Sandho lm (1971) a lg o r i t h m im p lem en ta t ion i s u t i l i s e d by the
code o f F redens lund e t a l (1971) to s o lv e the eq u a t io n s .
The s t r u c t u r e o f t h e J a c o b i a n i s o f t h e b l o c k t r i d i a g o n a l form (see
f i g u r e C 2 .2 ) .
B 1 C 1—
1> X h-* ___
I rF1
a 2 b 2 C2 a x 2 f 2
a3 B3 C3 a x 3 f 36 » »
A 9 0
* » « _
• 9 <■
» « <
o 6 o
« » *
An ST-1 b n s t - i CNST-1 Ax n s t - i f n s t - i
a n s t b n s t A x n s tf n s t
Figure C2.2 : Blocktridiagonal structure of distillation
column model
218
The block tridiagonal structure arises because conditions on stage 1 are only influenced directly by the conditions on stages 1+1 and 1-1. The diagonal elements of the Jacobian, B, contain derivatives for stage 1 with respect to the variables on stage 1. The elements below the diagonal, A, contain the derivatives for stage 1 with respect to the variables on stage 1-1. The elements above the diagonal, C, contain derivatives for stage'1 with respect to the variables on stage 1+1. The non-zero elements of matrices A, b, C (ie. Q + R) are given below:
Let N1 = NC + 1, N2 = 2NC + 1
Table C 2 .1: Elements of the Jacobian and right hand sidematrices for distillation column procedure.
Elements of Matrix A
A1 .i .kwhere
= - 8k. i
ki = 1 k = i0 k * i
1 2. ..... NST - 1
II 1, •..... NCdHLi-i
a i .n i . Nl* 1-1
1 .NST- 1
1.N2.kdHL
dFX1-1 1 = 2 , . . . NST—1
1-1.k
219
Elements of Matrix C
Cl.i.Nl+k = - 8 k .idHV
C 1+11.N2.Nl+k dFY 1+1.k
C1.NC+i.Nl+k =d-^)FV1+1
FY8k. i - 1+1. i
FV 1 + 10HV.1 + 1
1 .N2. N1 dT1+1Elements of Matrix B
B1 .i. kFXi.i(EL1) 2 . SL + 5, . (1 + ------ )k. i f l
B. F Y l.i1 .i. Nl+k SV, +(EV1> 2 1 W 1 svi
+ E r >
FXB = T] FV. l.il.NC+i.k ‘1 1 FL
'3K. . K .l.i l.idFX FL.
l.kK
+ 8k . i 11 FV^ l.iFL.
B = T1 FV.FX. . .l.i l.i
l.NC+i.Nl ‘1 1 FL 1 dT.
SL-B HL.1. N2 . k (FLX)2
1 +SL.FL.
dHL.
dFX l.k
220
1 = 2, NST-1
B1 .N2.N1 1 +SL.FL.
dHL.
dT.
+ 1 +SV.FV.
dHV.
0T.
1 = 2, NST-1
B1 .N 2 . N1 +Ksvi----- o HV-,( FV1) 2 1
1 +
1 = 2,
SV.FV.
0HV
3FYi.k.NST-1
B1.N2.k = “ 1
BNST.N2.k = 1
Elements of Matrix SNon-zero elements of matrix S, the right hand side are presented below. Note the order of the columns (ie. differentiation) is the same as the input vectoru 1 = (RFLX, DEST, TF]_, FZ1>k) .
Amount o f Decane i n the vapour p roduct(kmol /h r ) 0-070
F u g a c i ty c o e f f i c i e n t s model - SRK
* I n i t i a l Guess
T ab le C3.3 : D e s i3 n_o f_ coup led_d js t iL la t ion_ co ] .um ns_w jth_ene r_gy_ re cyc le- Problem C3.7
Feed F lo w ra te s (Kmol . /h r )
Components
Ethane 15-0 P ropy lene 35-0 Propane 30-0 Iso -b u ten e 20-0
Vapour f r a c t i o n L iq u id a t bubb le p o in t
Col. umn_ conf ig u ra tnonColumn I Column I I
Number o f s tages 20 12Feed s tage 9 6Condenser P re ssu re (ba rs ) 17-23 24.12P re ssu re d rop /s tage (bars) 0 .0 0 .0D i s t i l l a t e Rate ( K m o l / h r ) 56 .36 34-51R e f l u x R a t i o 8-0Type o f Condenser P a r t i a l P a r t i a l
P |]Ys lca l_ P r o p e r t ie s
V a p o u r /L iq u id f u g a c i t y c o e f f i c i e n t s - SRK
V a p o u r / l iq u id e n th a lp y - i d e a l ( P o ly n o m ia l fo rm f o r s p e c i f i c heatc a p a c i t y , See T a b le C1.6 Appendix C1)
228
Binary interact ion co e f f ic ie n ts set to zero.
★ Initial value
Tab le C 3 .4 : D e s i3 n- o f_ c o u p le d _ d is t i l la t io n _ c o lu m n s_ w i th _ m a s s3 nd_Energy_Recy les_-_P rgb lem _C3 .8
Vapour f r a c t i o n L iq u id a t bubb le p o in t
T ab le C3. 4 : Cont inued
Column c o n f ig u r a t io nColumn I Column I I
Number o f s tages 2 0 1 2Feed stage 9.5 6Condenser P re ssu re (bars) 17.23 24.12P ressu re d rop /s tage (ba rs ) 0.0 0.0D i s t i l l a t i o n Rate (KmoL / h r ) 62 .23 37.77R e f lu x R a t io 4 .0 8 . 0Type o f Condenser P a r t i a l P a r t i a l
fh^slcal_Properties
V a p o u r /L iq u id f u g a c i t y c o e f f i c i e n t s - SRK
V a p o u r / l iq u id e n th a lp y - I d e a l (P o ly n om ia l form f o r s p e c i f i c heatc a p a c i t y , See Tab le C1.6 Append ix C1).
Binary interact ion coe f f ic ien ts set to zero.
★ Initial value
Tab le C3.5: O p t jm is a t io n _ o f_ c o u p le d _ f la sh _ u m ts_ -_ p ro b le m _ C 3 -.9
Feed F lo w ra te s (Kmol /h r)
Component
Pentane 40 .0Hexane 30 .0Octane 30.0
P re s su re in both f l a s h u n i t s = 1 .0 bar
Z29
Obj_e_ctj_ve : 60 % r e c o v e r y and a t L e a s t 7 8 % p u r i t y o f p en tane from second f l a s h .
fhysicaL_Properties
V a p o u r / L iq u id f u g a c i t y c o e f f i c i e n t s - SRK
B in a r y i n t e r a c t i o n c o e f f i c i e n t se t t o ze ro .
APPENDIX DlDerivatives of fugacity coefficients and excess enthalpies (using SRK-equation) with respect to binary interaction coefficients
Derivatives of SRK-equation mixture parameters with respect to 8ij
Let us denote 8ij by 0.
da.. da ..1 3 = ___D i
de de <ai a j>
1/2
Where 6 = 1,2,.....NC(NC-l)/2
mixtde S I
i jx . x
da. .13
d0
db. db .1 m ixt------ = ------------- = 0de de
dz B-Z dAde Z( 3Z-2 ) + A-B-B2 de
Using the above derivatives, the derivatives of fugacity
coefficients to 0 are obtained thus:
231
3ln0-j b-j _3Z _ 1 3 Z
3e (Z-B) 36
where
(DA.DB' + DA'.DB)
DA
DA'
BLn(1+ -)
B z
rr B -|In (1 + -----) p_
Z 3 A A
B■
“ “36 Z(Z+B)
3Z
36
DB
DB'
2
jX j a j j
bi
ami xt bm ix t
am ix t £ xjj
3a i j" ami x t
2 ! I cb
I! i
___
1
a m ixt
D1.1
D1.2
D1.3
D1.4
D1.5
Excess_en thaLp^ _de r i\ /a t ive s
The model f o r excess en th a lp y i s g iv e n as f o l lo w s :
1 BA He = RgT (Z-1) - _ In (1 + - ) .
B Z
f Z Z x i
L 1 j
rXj a-jj D1.6
232
twhere a-jj
aij 7T(Rg T)2
and r=1 2 C( ,
m-i (Tp-j)
T72
1/2
The o th e r v a r i a b le s i n the
TTr = -
Tc ,
mj ( T r j ) 1
model a re as p r e v io u s ly d e f in e d
3 A 3 7-____ = R g T ------ - (DC. DD + DC'DD)
3 g 3-6
where DC = In (1+B/Z)/B
Z(Z+B) 3 6
D1.7
D1.8
D1-9
DD Z Z
i jx i xj a i j D1.10
DD'D1.11
i j 36
Note: Vapour phase p r o p e r t ie s a re ob ta in e d by r e p la c in g x ' s by y ' s .
S e n s i t i v i t y o f f l a s h p ro cedu re to
The model o f an i s o t h e rm a l VLE f l a s h p ro cedu re was g iv e n i n Chapter 3 .
M a t r i c e s Q and R a re unchanged. The r i g h t hand s id e m a t r ix S 1 c o n ta in s
the d e r i v a t i v e s o f the f l a s h model (equa t io n s 3 .6 and 3 .7 ) w it h r e s p e c t
to 6 ^j. The non -ze ro e lem ents o f m a t r ix S* a re as f o l l o w s :
3"f i 3 ^i____ = FX-j F V _____ D1.1236 T 6_The s e n s i t i v i t i e s o f the vapour phase f lo w s (FY-j) t 0 ^ a re ob ta in ed by
c h a in - r u l i n g o f l i q u i d phase f lo w s (FX-j) - 5^ s e n s i t i v i t i e s .
233
S en j j jM j^ it^ _ Lation_procedure_tg__^^_t j i_and_e rro rs_ jn _ en th a lp y
models.
A g e n e r a l i s e d model o f a d i s t i l l a t i o n column was g iv en i n appendix C2
and m a t r ic e s Q and R are g iv en t h e r e in .
The non -ze ro e lem ents o f m a t r ix S a re g iven below f o r parameters 6 i j "
3 <k,i 3Kl . i
L 3$ FL-l9 0
l = 1 , 2 - . . . , NST; i = 1 , 2 ,
3EL 3HV i 3HLL------ = (1 + S V i ) --------+ ( 1 + SL i) --------
30 36 30
3HVL+1 3H L L_ 1
30 30
D1.13
D1.14
where l = 2 ,3 . . . . , NST-1
The s e n s i t i v i t e s o f the r e b o i l e r and condenser d u t ie s t o 5 ^ j(0 r -0) are
o b t a in e d by s u b s t i t u t i n g 6 j f o r r e f l u x - r a t i o (RFLX) in equa t ion C2.8
and C2 .9 .
D i f f e r e n t i a t i o n o f t h e d i s t i l l a t i o n column model w ith re spec t to t ra y
e f f i c i e n c y and con s tan t r e l a t i v e e r r o r i n e n t h a lp y m ode ls a re e a s i l y
ob ta in ed i n a s im i l a r manner.
234
APPENDIX D2
D e ta i le d S p e c i f i c a t i o n o f VLE Examples
Tab le D2.1 : F la sh and d i s t i l l a t i o n s p e c i f i c a t i o n s
F lash Test Problem
D2.1 D2.2
Ni t rogen (kmol /h r ) 1 .40 (1)
Carbon D io x id e II 6 .0 CD
Hydrogen Su lph id e It 24.0 (2)
Methane II 94.30 (2) 66.0 (3)
Ethane II 2 .70 (3) 3 .0 (4)
Propane II 0 .74 (4) 1 .0 (5)
Butane II 0 .49 (5)
Pentane II 0 . 1 0 (6 )
Hexane II 0 .27 (7)
F la sh C o n d it io n s
Temperature (K) 175.0 225.0
P re s su re (bar) 27.01 60.78
Note: Component number in p a re n th e s is
V
235
D is t i L L a t io n
Feed Compos it ion*
(Kmol /h r )
( 1)
( 2)
(3)
(4)
Feed Rate (kmol /h r )
R e f lu x R a t io
D i s t i l l a t e Rate
Feed Stage
Vapour f r a c t i o n
No. o f P la t e s
Top P re ssu re (bar)
P re ssu re d rop /s tage (bar)
Condenser Type
Feed Temperature (k)
Feed Temperature (bar)
Components (1) Methane(2) E th y lene(3) Ethane(4) P ropy lene
Test Problem
D2.3 D2.4 D2.5
0 .16 9.43 9.88
894.54 1.82 1 .38
717 .58 0 .0 9 0 . 1 0
5 .34 0 .09 0 .09
1617.62 11.43 11.45
3 .27 157.05 150.02
891.14 9.383 9.82
31 54 54
0 .27 - -
55 117 96
9 .79 15.22 15.22
0 . 0 . 0 .
P a r t i a l P a r t i a l P a r t i a l
247.5 251.8 250.8
9 .79 15.22 15.22
P ropy lene Propane P ropad iene Propyne
P ropy lene Propane P ropad iene Propyne
Cave tt Problem
T h is problem s p e c i f i c a t i o n s i s the same as example C2.2 (Appendix C3 ) .
TABLE D2.2 : Vapour component flow ra te s e n s it iv i t ie s to b ina ry in te ra c t io n c o e ff ic ie n ts