efficient use of thermodynamic data in ......c2.1 schematic representation of a stage in a...

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


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.


line 19


line 7

5. page 185,

para 2 line 7


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

2 .6 . THE HYBRID APPROACH 582 .7 . CONCLUSIONS 6o

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

4 .5 . NUMERICAL EXPERIMENTS/DISCUSSIONS 1474 .6 . CONCLUSIONS 180

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

whether a model rep roduces t h e p l a n t o p e r a t in g data ©t*whethe r

the re are d i s c r e p a n c i e s i n t h e r e s u l t s . Another approach t h a t i s

commonly taken i s t h e so c a l l e d s h o r t - c u t or approximate method as

suggested by S an d le r (1980 ) , N e l s on e t a l (1983), Hernandez e t a l

( 1 9 8 4 ) , and B r i g n o l e e t a l (1985 ) . The s h o r t - c u t methods have so

f a r been a p p l i e d o n l y t o d i s t i l l a t i o n column d e s i g n s e n s i t i v i t y

s t u d i e s . The method i s based on t h e assumption t h a t the change i n

the de s ign v a r i a b l e s computed by s im p l e m ode ls p r o v i d e a good

e s t im a te of what wou ld be found from r igo rou s models . The methods

are d e r i v ed f o r l i m i t c o n d i t i o n s ( c o n s t a n t r e l a t i v e v o l a t i l i t y ,

v e r y p u r e p r o d u c t s , n o n - d i s t r i b u t i o n o f n on -key components ,

cons tant s p l i t r a t i o s , e t c ) . A base po in t p lan t de s ign i s u s u a l l y

r e q u i r e d f o r the a n a l y s i s . Because of the in h e r e n t assumpt ions i n

the s h o r t - c u t p ro cedu re , t h e i n f o r m a t i o n g e n e r a t e d wou ld be i n

e r r o r when a p p l i e d t o s y s t e m s f o r wh ich the assumptions do not

ho ld (Ne lson e t a l ( 1 9 8 3 ) , H e rn an d e z e t a l ( 1 9 8 4 ) ) . I t i s not

c l e a r , h o w e v e r , how one can e s t i m a t e t h e s e n s i t i v i t y of t h e

p r o c e s s d e s i g n t o b a s i c p a r a m e t e r s l i k e 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 s a n d c r i t i c a l c o n s t a n t s s i n c e t h e s i m p l i f i e d

c o r r e l a t i o n s are u s u a l l y independent of these parameters.

33

S t r e i ch and K i s t e n m a c h e r (1980) p resen t a more r i g o r o u s

mathemat ica l f o rm u la t i o n of the problem. Here the op t ima l p rocess

d e s i g n v a r i a b l e s a r e ob ta in ed by s o l v i n g c o n s t r a i n e d o p t im i z a t i o n

problems. The s e n s i t i v i t i e s of the de s ign v a r i a b l e s can t h e re fo re

be o b t a i n e d , 2 n_ jy 1 i n . c J j 2.Le , t h r o u g h r i g o r o u s a n a l y t i c a l

d i f f e r e n t i a t i o n a nd c h a i n- r u l i ng. The a u t h o r s a d m i t t h e i r

e xp r e s s i o n can on ly be o b t a i n e d a f t e r t e d io u s c a l c u l a t i o n s . S ince

t h e r e i s no unique o b j e c t i v e f u n c t i o n f o r a p ro ce s s , i t t h e r e f o r e

means s e n s i t i v i t i e s o b t a i n e d a t a g i v e n b a s e p o i n t w o u l d

i n v a r i a b l y depend on t h e p a r t i c u l a r c h o i c e o f o b j e c t i v e t o be

op t im iz ed .

I n t h i s t h e s i s , we show t h a t r i g o r o u s f i r s t o r d e r

s e n s i t i v i t y a n a l y s i s can be c a r r i e d out i n p ra c t i ce, q u i t e e a s i l y

and e f f i c i e n t l y f o r g e n e r a l p r o c e s s e s . We f i r s t r e v i e w the

mathemat ica l b a s i s o f the method and d e r i v e the s e n s i t i v i t i e s o f

p r o c e s s v a r i a b l e s w i t h r e s p e c t t o c o n s t a n t p a r a m e t e r s and

f u n c t i o n s o f i n d e p e n d e n t v a r i a b l e s i n C h a p t e r 4 . T hen t h e

s e n s i t i v i t i e s o f t y p i c a l p r o c e s s e s a r e s tu d i e d w i th re spec t t o

such parameters as 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 and Murphree

t r a y e f f i c i e n c y . We a l s o i n v e s t i g a t e the s e n s i t i v i t i e s o f some

e xa m p le s t o mode l e r r o r s , i n p a r t i c u l a r t o c o n s t a n t r e l a t i v e

e r r o r s i n v a p o u r e n t h a l p y and i d e a l and exce s s l i q u i d en tha lpy

models. F i n a l l y , we u t i l i z e the s e n s i t i v i t i e s t o a s c e r t a i n the

e f f e c t s o f p h y s i c a l p r o p e r t y p a r a m e t e r s u n c e r t a i n t i e s on the

l o c a t i o n of senso rs f o r c o n t r o l of d i s t i l l a t i o n columns.

34

CHAPJER_7W0

REVIEU_ 0 F_ EXISJING_ THERMODYNAMIC_PROPERTY

d m a _ in jer fa c e_s jr a ie g ie s

A c r i t i c a l e x a m i n a t i o n o f c u r r e n t l i t e r a t u r e i n

computer-a ided chemica l p ro ce s s d e s i g n i n d i c a t e s t h a t t h e r e a re

f o u r d i f f e r e n t s t r a t e g i e s a d o p te d i n the way TP c a l c u a t i o n s are

i n t e r f a c e d w i th p ro cess de s ign packages . These have been r e f e r r e d

t o i n C h a p te r One as t h e b l a c k - b o x , W e s t e r b e r g , t w o - t i e r , and

Hyb r id methods. In t h i s c h ap te r we s t a r t by s t a t i n g the 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 th e rm o d yn am ic p ro p e r t y data

i n t e r f a c e s t r a t e g y . We then p ro v id e a b r i e f d i s c u s s i o n on the TP

i n f o rm a t i o n t y p i c a l l y a v a i l a b l e from p h y s i c a l p r o p e r t y packages.

The fou r d i f f e r e n t i n t e r f a c e s t r a t e g i e s are c r i t i c a l l y r e v i e w e d .

F i n a l l y we c o n c l u d e t h a t c u r r e n t i n t e r f a c e s t r a t e g i e s a r e

inadequate s in ce TP data needs i n f l owshee t c o m p u t a t i o n s a re no t

s a t i s f ied .

2. 1. ^Qiterjj__f^or_ey aJ.jjj_ti^qn_of_Jt_h_ermodynamj_c_da ta_i nterf ace

strategy

In o r d e r t o u n d e r s t a n d and compare e x i s t i n g TP package

i n t e r f a c e s we propose the f o l l o w i n g c r i t e r i a f o r the e v a l u a t i o n of

a s t a n d a r d i s e d TP d a t a i n t e r f a c e system ( i n o rde r of dec rea s ing

p r i o r i t y ) :

35

( i ) im p r o v e m e n t i n e f f i c i e n c y . T h i s means a TP

i n t e r f a c e must r e s u l t i n smal l comput ing t ime and

p e r m i t s u b s t a n t i a l r e d u c t i o n i n t h e number of

r i g o r o u s TP c a l c u l a t i o n s . The l a t t e r can be

r e l a t e d t o the number of e q u iv a le n t a cce s se s made

t o the base TP package.

C i i ) p r o v i s i o n o f a c c u r a t e d e r i v a t i v e s necessary fo r

reasons ment ioned i n chapter one.

C i i i ) p r o v i s i o n o f t h e s e n s i t i v i t y of p rocess de s ign t o

u n c e r t a i n t i e s i n the TP models and/o r parameters .

( i v ) e f f e c t on t h e convergence of f l ow shee t numer ica l

s o l u t i o n a lg o r i t h m must be m in ima l .

(v) u s e a v a i l a b l e TP p r o c e d u r e s . A c o n s i d e r a b l e

amount of e f f o r t has gone i n t o d e v e l o p i n g such

r o u t i n e s and i t w i l l be q u i t e u n r e a s o n a b l e to

d i s c a r d them ( P e r k i n s , 1984).

( v i ) i t shou ld be p o s s i b l e t o decoup le e a s i l y the base

TP p a ckage f rom a f l o w s h e e t i n g s i m u l a t o r t h u s

mak ing i t easy to use d i f f e r e n t packages when the

need a r i ses .

36

( v i i ) s t o r a g e r e q u i r e m e n t s o f the TP i n t e r f a c e shou ld

be modest.

( v i i i ) u s e r f r i e n d l i n e s s , t h a t i s , make min imal demands

on the use r .

To e n a b l e us u n d e r s t a n d and c o m p a r e t h e d i f f e r e n t

i n t e r f a c e s (based on the c r i t e r i a s t a t e d above) , l e t us cons ide r a

h y p o t h e t i c a l f l o w s h e e t c o n s i s t i n g o f a m ix e r , s p l i t t e r , and an

i s o th e rm a l f l a s h u n i t ( F i g u r e 2 . 1 ) . Assume t h a t i t i s d e s i r e d t o

s e t up and s o l v e t h e f l o w s h e e t mode l i n v o l v i n g o n l y m a t e r i a l

f l ow s . Le t Sj-j be the f l o w r a t e of component i i n stream j , Sj the

t o t a l mo la r f low i n stream j , and T, II as f l a s h t e m p e r a t u r e and

p re s su re r e s p e c t i v e l y . The f o l l o w i n g ba lances can be w r i t t e n f o r

each u n i t :

Mi xer

Component f l o w s

1 i + ^5i ” s2 i i = 1 , ___ NC (2.1)

S p l i t t e r

Component f l o w s

^4 i - S5 -j ( 2 . 2 )

37

Figure 2.1: Simple Hypothetical Flowsheet

Splitter

Component f l o w s

s2 i = s3 i + s4 i i=1 , NC (2 .3)

Phase E q u i l i b r i u m r e l a t i o n s

methods

s4 i s3 i s4 i s3 iKi ( — , — T,ro —— = —

s3 S4 S3i / = 1 / ■ “ . .NC (2 .4 )

physi c a l p r o p e r t i e s pac kage i s a v a i l a b l e f o r c o m pu t in g

(K i ) da ta . K - v a l u e s a re u s u a l l y e v a lua ted by one of th r ee

Ki = i (2 .5 )-V

0 .1

oyX n

K i = i i ( 2 . 6 )

n

or

«i =

1 ° y 1 n.11 i

0V n(2 .7 )

. i . vwhere 0 ^, 0 a re f u g a c i t y c o e f f i c i e n t s of component i i n the l i q u i d

1 °and v a p o u r p h a se s r e s p e c t i v e l y . y i , l i g a n d II a r e 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 , vapour p ressu re and t o t a l sys tem p r e s s u r e

r e s p e c t i v e l y . L e t us however assume the K - v a lu e s a re c a l c u l a t e d

u s ing equa t i o n 2 .5 w i t h f u g a c i t y c o e f f i c i e n t s e v a l u a t e d f r om an

39

equa t i o n of s t a t e model . In o t h e r words, compos i t ion , temperature

and p ressu re d e r i v a t i v e s of the f u g a c i t y c o e f f i c i e n t s w i l l u s u a l l y

no t be a v a i l a b l e . A l t e r n a t i v e l y , suppose an i s o t h e rm a l f l a s h

p r o c e d u r e P i s a v a i l a b l e ( a g a i n f rom a p h y s i c a l p r o p e r t i e s

pa ckage ) wh ich g i v e n T , n, and S2 computes the co r re spond ing S3

and S4 . Thus e q u a t i o n s ( 2 . 3 ) and ( 2 . 4 ) can be r ep la ced by the

f o l l o w i n g :

F1i = S3 i - PCT, n, S2 ) , i = 1 , ___ NC (2.8)and

F2 i = S4l- - P(T, n , S2 ) , i = 1 , ___ NC (2.9)

where P r e f e r s t o t h e f l a s h p r o c e d u r e . Thus , we can have two

d i f f e r e n t f o r m u l a t i o n s w i t h the f o l l o w i n g groups of equa t ion s :

CASE_A

Equa t ion s ( 2 . 1 ) , ( 2 . 2 ) , (2 .3 ) and ( 2 . 4 ) .

CASE_B

Equa t ions ( 2 . 1 ) , ( 2 . 2 ) , (2 .8 ) and ( 2 . 9 ) .

Suppose a Newton-based p ro ce s s s i m u l a t o r i s a v a i l a b l e . I t w i l l

t h e r e f o r e be n e c e s s a r y t o p ro v ide the f o l l o w i n g data r e l a t i n g to

the f l a s h u n i t :

( i ) ou tpu t v a r i a b l e s , S3 and S4 ,

( i i ) p a r t i a l d e r i v a t i v e s :

40

3Fii 3F2i 3Ki_ _ _ _ _ , --- r and --3U 3U 3U

✓where u = (T, II, S2 , S3 , S4 ),

The p a r t i a l d e r i v a t i v e s o f e q u a t i o n s ( 2 . 1 , 2 . 2 , 2 .3 ) can be e a s i l y

ob ta in ed by a l g e b r a i c m a n i p u l a t i o n (Koup e t a l ( 1 9 8 1 ) , Pon ton

(1982) , P a n t e l i d e s (1986)) - The form of equa t ion s ( 2 . 4 , 2 . 8 , 2.9)

i s not known e x p l i c i t y a t the f l ow shee t l e v e l and i t i s t h e r e f o r e

3 ( F F )no t p o s s i b l e t o s e c u r e t h e p a r t i a l d e r i v a t i v e s 1 * 2 us ing

3usymbo l i c m an ip u la t i o n . B e f o r e we study the f l o w and u t i l i z a t i o n

o f i n f o r m a t i o n be tw een t h e t h r e e l e v e l s o f computat ions ( F ig u re

1 .1 ) i n t a c k l i n g t h e h y p o t h e t i c a l p ro b lem by t h e TP i n t e r f a c e

s t r a t e g i e s l e t us b r i e f l y d e s c r i b e a t y p i c a l TP package.

2 .2 . B r i e f d e s c r ip t i o n o f _ a _ t y p ic a l p h y s ic a l p r o p e r t ie s p a c kage

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 a r e d e s i g n e d t o p r o v i d e

thermodynamic and t r a n s p o r t p r o p e r t i e s o f p u re components and

t h e i r m i x t u r e s and can be used i n c o n d u c t i o n w i t h a p r o c e s s

s im u la to r or i n depen den t l y . The s t r u c t u r e of PPDS (Edmonds, 1978)

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 shown i n F igu re 2 . 2 . The

t a s k s of p h y s i c a l p r o p e r t i e s packages can be broken down i n t o fo u r

p a r t s : r e p e a t e d e s t i m a t i o n o f a d e s i r e d TP d u r i n g s i m u l a t i o n

c a l c u l a t i o n s ; p r o v i s i o n o f TP of i n t e r e s t a t c o n ve rg e d f l o w s h e e t

s o l u t i o n ; p r o v i s i o n f o r u s e r - s u p p l i ed d a t a ; and p r o v i s i o n o f a

f a c i l i t y t o e s t im a te TP where as l i t t l e as s t r u c t u r a l i n f o r m a t i o n

i s a v a i l a b l e . To be a b l e t o s a t i s f y a l l t h e s e demands t h e

p h y s i c a l p r o p e r t i e s packages r e q u i r e l a r g e compute r s t o r a g e and

41

a r e o f n e c e s s i t y d e s i g n e d w i t h c a r e . For t h i s r e a s o n , a l l

c once iv ab le p r e p r o c e s s i n g w h i c h needs t o be done t o che ck the

c o n s i s t e n c y o f t h e i n p u t d a t a i s done b e f o r e the s t a r t o f

f lowshee t s im u la t i o n . G e n e r a l l y t h e i n p u t of d a ta i s easy and

done i n f r e e f o r m a t s p e c i f i c a t i o n . Data r e q u i r e d t o d e f in e a

component o r m i x t u r e i s u s u a l l y k e p t a t a m in imum h e n c e a

s u b s t a n c e i s r e f e r e n c e d by a code number, i t s chemica l fo rmu la or

name.

P h y s i c a l p r o p e r t i e s pa ramete rs a r e p rov ided i n t h r e e ways

(see Append ix B f o r a l i s t o f pa ramete rs u s u a l l y a v a i l a b l e ) . The

f i r s t way i s f rom the permanent da tabank which p ro v id e s t he po in t

g en e ra t io n r o u t i n e s ( F i g u r e 2 .2 ) w i t h TP v a l u e s f o r e i t h e r pu re

compounds o r m i x t u r e s o f up t o say 20 components. The permanent

d a ta b a n k t y p i c a l l y has d a t a f o r 9 0 -900 s u b s t a n c e s (PPDS f o r

i n s t a n c e c o n t a i n s d a t a f o r 860 c om pound s ) . A t r u n - t im e , the

parameters f o r on ly those compounds i n v o l v e d i n the s im u l a t i o n are

c o p i e d f r o m t h e p e r m a n e n t d a t a b a n k i n t o a r u n - t im e s t o r a g e

l o c a t i o n . T h e r e a f t e r the permanent databank i s not a c c e s s e d f o r

t h e c u r r e n t s i m u l a t i o n . The second way a r i s e s f o r two reasons .

S ince the number of chemica l s p e c i e s f o r which data may be needed

i s more t h a n 900^ t h e r e i s a need t o have a f a c i l i t y whereby

u s e r - s u p p l i e d data can be u t i l i z e d . T h i s i s e q u i v a l e n t t o t he

user c r e a t i n g h i s own permanent data f i l e . T h i s user databank

can be u s e d t o c o m p l e m e n t t h e p e r m a n e n t d a t a b a n k o r

i n d e p e n d e n t l y . The o th e r reason i s t h a t a l though the TP packages

o f f e r data f o r most of the compounds which a user i s i n t e r e s t e d tn

lYl

The Physical Properties Data System

Fig. 2 *2.

43

t h e r e a r e o c c a s i o n s w h e r e one ( o r m o r e ) p r o p e r t i e s a r e

u n a v a i l a b l e . For i n s t a n c e , PPDS a l l o w s one t o i n p u t b i n a r y VLE

i n t e r a c t i o n parameters f o r use w i th the main PPDS VLE p rocedures .

I n t e r a c t i o n paramete rs may be en te red d i r e c t l y or c a l c u l a t e d f rom

expe r im en ta l data u s in g l e a s t squares f i t t i n g t e chn ique s .

The t h i r d and f i n a l means by w h ich TP p a r a m e t e r s a r e

genera ted f o r use i n t he TP p o in t g e n e ra t io n r o u t i n e s i s necessa ry

when m in im a l i n f o r m a t i o n i s s t o r e d on a c o m p o n e n t ( s ) e . g .

s t r u c t u r a l i n f o rm a t i o n , and average b o i l i n g p o i n t and d e n s i t y f o r

p e t r o l e u m f r a c t i o n s - I n t h i s c a se t h e e s t i m a t e d d a t a may be

i m p r o v e d by b l e n d i n g w i t h a n y known e x p e r i m e n t a l d a t a .

The rmodynam ic c o n s i s t e n c y t e s t s a r e somet imes c a r r i e d o u t t o

v a l i d a t e the e s t im a ted pa ramete rs .

F l o w s h e e t i n g p a c k a g e s u s u a l l y make f r equen t r e que s tS fo r

p r o p e r t i e s at s p e c i f i e d tem pe ra tu re s , p r e s s u r e s , and compos i t ions .

A c o m p re h e n s i v e l i s t o f t h e rm o d yn a m ic and t r a n s p o r t p r o p e r t i e s

t h a t may be r e q u i r e d are g i v e n i n Appendix B. V a r io u s mode ls a r e

a v a i l a b l e f o r p r e d i c t i o n o f a g iven TP p ro p e r t y . For i n s t a n c e ,

Red l i ck -Kwong Soave ( S o a v e , 1972) and P e n g -R o b in s o n (Peng and

Robinson , 1976) e q u a t i o n s o f s t a t e may be a v a i l a b l e f o r e v a l u a t i n g

f u g a c i t y c o e f f i c i e n t s . D e r i v a t i v e s of the TP models w i t h r e s p e c t

t o t e m p e r a t u r e , p r e s s u r e , and c om pos i t i o n s a re u s u a l l y not made

a v a i l a b l e . In a d d i t i o n t o the TP m ode ls phase and c h e m i c a l

e q u i l i b r i u m data ( c f . Append ix B) a re a l s o reques ted . P rocedures

f o r phase and chemica l e q u i l i b r i a do no t r e t u r n d e r i v a t i v e s o f

44

t h e i r o u t p u t v a r i a b l e s w i t h r e s p e c t t o t h e i r i n p u t s . Thus we

r e f e r t o t h i s s e c t i o n o f the p h y s i c a l p r o p e r t i e s package ( F i g u r e

2 . 2 ) as po in t g e n e r a t i o n r o u t i n e s .

R e c a l l t h a t i n chap te r one we conc luded t h a t f o r e f f i c i e n t

s o l u t i o n of f l o w s h e e t i n g problems the p a r t i a l d e r i v a t i v e s o f TP

m o d e l s , f l a s h , and d i s t i l l a t i o n p rocedure d e r i v a t i v e s e t c . , are

re q u i r e d . However, f rom our r a t h e r b r i e f d i s c u s s i o n of 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 i t i s c l e a r t h a t t h e n e c e s s a r y p a r t i a l

d e r i v a t i v e s are not c a l c u l a t e d . Thus t h e re i s the need t o p ro v id e

a TP i n t e r f a c e whi ch supplies p ro ces s s im u l a t o r w i t h TD p r o p e r t i e s

d e r i v a t i v e s not p r o v i d e d by 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 . The

o r g a n i z a t i o n o f such an i n t e r f a c e w i t h re spe c t t o the p h y s i c a l

p r o p e r t i e s and t h e f l o w s h e e t p r o p o s e d by o t h e r w o r k e r s i s

d e s c r ib e d i n t he next s e c t i o n s .

2 .3 . Jhe_Black-Box_Approach

A number o f p r o c e s s s im u l a t o r s a v a i l a b l e today have t h i s

s o r t of i n t e r f a c e w i t h TP d a t a p a ck age s ( e . g . SPEEDUP, ASPEN,

ASCEND-II, PROCESS, GENESIS, FLOWPACK).

We c o n s i d e r c a s e B o f t h e f l o w s h e e t f o r m u l a t i o n . The

p h y s i c a l p r o p e r t i e s p a ckage ( l e v e l 1 o f F i g u r e 1 . 1 ) doe s not

p ro v ide d e r i v a t i v e s of K - v a l u e s s i n c e o u r f u g a c i t y c o e f f i c i e n t

model does not have these d e r i v a t i v e s .

4b

The u n i t o p e r a t i o n L e ve l compr ises the i s o the rm a l f l a s h ,

m ixe r , and s p l i t t e r - The m ixe r and s p l i t t e r a re s o l v e d w i t h o u t

any r e f e r e n c e t o t h e p h y s i c a l p r o p e r t i e s package. Our concern

h e re i s t h e f l a s h u n i t w h i c h i n v o l v e s p h y s i c a l e q u i l i b r i a .

S e v e r a l a p p r o a c h e s h a ve been p r o p o s e d f o r t h e s o l u t i o n of the

b a s i c equa t ion s d e s c r i b i n g t he f l a s h modu le : w h a t e v e r s o l u t i o n

method t h a t i s a d o p t e d f o r t h e f l a s h , the re a re im p l i c a t i o n s f o r

the TD data a v a i l a b l e f r o m l e v e l 1 . For i n s t a n c e , u s i n g f r e e

e n e r g y m i n i m i z a t i o n o r N e w t o n ' s method w i l l n e c e s s i t a t e the

p r o v i s i o n of 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 w ith r e s p e c t t o

p r e s s u r e , t e m p e r a t u r e , and c o m p o s i t i o n f rom l e v e l 1 . These

d e r i v a t i v e s as we know are u n a v a i l a b l e . On the o th e r hand, i f the

Ra ch f o r d - R i ce t e a r i n g method i s used, then t h e re i s no d i f f i c u l t y

s in ce TD p ro pe r t y model d e r i v a t i v e s a r e no t needed. The f l a s h

p r o c e d u r e compu te s o n l y t h e v a p o u r (S3 ) and l i q u i d (S4 ) phase

m o l a r c o m p o n e n t f l o w s a t t h e s p e c i f i e d i n p u t s S2 , T, and II-

Seve ra l t e chn ique s have been adop ted f o r s e cu r in g the u n a v a i l a b l e

g r a d i e n t s o f t h e f l a s h ou tpu t v a r i a b l e s (S3 , S4 ) w ith respec t t o

i t s i n p u t s (T, II, S2 ) .

The f i r s t method o f s e c u r i n g t he p a r t i a l d e r i v a t i v e s of

the f l a s h assumes the o u t p u t v a r i a b l e s a r e w e a k l y dependen t on

t h e i r i n p u t s and hence t h e p a r t i a l d e r i v a t i v e i s s e t to ze ro .

U n f o r t u n a t e l y , n e g l e c t i n g the d e r i v a t i v e s cou ld d r a s t i c a l l y a f f e c t

the convergence o f the s o l u t i o n a lg o r i t h m (Hu tch i son e t a l (1983) ,

S t a d th e r r and H i l t o n ( 1982 ) ) . B a r r e t t and W a l sh (1984 ) r e p o r t e d

r e s u l t s where t h e s o l u t i o n a lg o r i t h m f a i l e d t o converge when the

46

p a r t i a l d e r i v a t i v e s a r e n e g l e c t e d . Our l i m i t e d e xpe r ien ce w ith

t h i s a p p r o a c h c o n f i r m s t h e s e a u t h o r s ' o b s e r v a t i o n s . The

performance of Newton 's method under thfcs> c i r cumstance i s problem

dependent and u n r e l i a b l e . In o p t im i z a t i o n the d e r i v a t i v e s c anno t

be n e g l e c t e d i f one i s t o use the power fu l s u c c e s s i v e q u a d r a t i c

programming and indeed any o t h e r d e r i v a t i v e based o p t i m i z a t i o n

a l g o r i thm.

The s e c o n d a p p r o a c h i s t h e so c a l l e d d i a g o n a l b l o c k

p e r t u r b a t i o n suggested by M a h a le c e t . a l . (1979) wh ich assumes

t h a t t h e f l o w o f component i i n t h e l i q u i d o r vapour stream i s

dependent o n l y on the f l ow of component i i n t h e f e e d . Thus the

c r o s s d e p e n d e n c e o f t h e f l o w s i s n e g l e c t e d . T h i s method o f

a p p r o x i m a t i n g t h e d e r i v a t i v e i n f o r m a t i o n haS” no t been f o u n d

s a t i s f a c t o r y ( S t a d t h e r r and Chen (1984) , F i e l d e t a l (1985 ) ) .

The t h i r d method of e s t im a t i n g the d e r i v a t i v e s and by f a r

t h e most w i d e l y used t e c h n i q u e i s f i n i t e d i f f e r e n c e s . For an

NC-component mixture, NC + 3 perturbations of the r igorous f l ash3F l

p r o c e d u r e a re needed t o s e cu re the f u l l d e r i v a t i v e m a t r i c e s _____

3 F 2 3 uand ____ . The i n t e r n a l v a r i a b l e s o f t h e f l a s h module ( e . g .

9uv ap o u r f r a c t i o n , K - v a l u e s ) can be t e m p o r a r i l y saved and u t i l i z e d

t o reduce the compu ta t iona l overhead f o r sub sequen t c a l c u l a t i o n s

o f t h e m o d u l e . The m e th o d ha s been f o u n d t o be r o b u s t and

r e l i a b l e ( S t a d t h e r r and C h e n , 1 9 8 4 ) a nd does not a f f e c t the

p e r f o r m a n c e o f N e w t o n ' s m e thod . The a u t h o r s a l s o f o u n d i t

b e n e f i c i a l i f i n t e r n a l v a r i a b l e s are saved. Another p rob lem w i t h

47

f i n i t e d i f f e r e n c i n g i s t h a t one has t o be c a r e f u l i n the cho ice of

s u i t a b l e p e r t u r b a t i o n s t e p s i z e . F i n a l l y t h e s e n s i t i v i t i e s o f

p r o c e s s d e s i g n t o p h y s i c a l p r o p e r t i e s i s ob ta in ed by expens ive

p e r t u r b a t i o n o f TP p a r a m e t e r s / m o d e l s . For t h e s e r e a s o n s , t h e

B l a c k - b o x a p p r o a c h can t h e r e f o r e be re ga rded as an i n e f f i c i e n t

t e chn ique f o r i n c o r p o r a t i n g TD data i n p rocess c a l c u l a t i o n s .

2 . 4 . The_U e s t e r b e r cj_ A p p ro a ch

T h i s a p p ro a c h was s u g g e s t e d by W e s t e r b e r g e t a l (1979)

w i t h few r e s u l t s r e po r t e d i n t h e l i t e r a t u r e ( S t a d th e r r and H i l t o n

( 1 9 8 2 ) , Locke (1981 ) ) . Aga in , l e t us t r y to app ly t h i s t e chn ique

t o the s o l u t i o n of the f l o w sh e e t problem (case A f o rm u la t i o n ) .

The p h y s i c a l p r o p e r t i e s l e v e l i s t h e same as i n both the

Westerberg and B l a c k - b o x s t r a t e g i e s .

W i t h t h e W e s t e r b e r g i n t e r f a c e s t r a t e g y , each o f the u n i t

o p e r a t i o n s module l i b r a r y c o n t a i n s o n l y the equa t ion s co r re spond ing

t o such a module. Thus, the i s o t h e rm a l f l a s h module prov ides on ly

e q u a t i o n s (2 .3) and (2 .4) i n s t e a d o f the output v a r i a b l e s (S3 and

S4 ) when in voked by the f l o w s h e e t .

The f l o w s h e e t e x e c u t i v e a s s e m b le s a l l t h e p rocess u n i t

model e qua t i o n s . The f l a s h module c a l 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

from l e v e l 1 . In o t h e r w o r d s , by t h e Wes te rbe rg approach, the

f lowshee t se t s up and so lves s im u l t an eo u s l y t h e e q u a t i o n s f o r t h e

48

p h y s i c a l p r o p e r t i e s , u n i t o p e r a t i o n s , and o th e r p rocess models.

D e r i v a t i v e s of the p l a n t model w i t h r e spe c t t o the unknown p rocess

s t r e a m v a r i a b l e s a n d e q u i p m e n t p a r a m e t e r s a r e o b t a i n e d

a n a l y t i c a l l y .

T h e W e s t e r b e r g a p p r o a c h h a s a f e w d e s i r a b l e

c h a r a c t e r i s t i c s : i t i s easy t o implement and p ro v id e s an a c c u r a t e

l i n e a r i s e d p l a n t ( f l o w sh e e t ) model. However, i t a l s o has s e ve ra l

d i sadvantages.

The f i r s t problem i s t h a t t h i s s t r a t e g y d i s c a r d s t h e idea

of i n c o r p o r a t i n g TP mode ls i n t o p ro ce s s s i m u l a t o r s as p r o c e d u r e s

a nd t h e r e f o r e l o s e s a l l t h e b e n e f i t s i n h e r e n t i n t h e use of

p rocedures . The second problem i s t h a t the number of equa t ion s t o

be s o l v e d i s i n c r e a s e d d r a m a t i c a l l y . L o cke (1981) g i v e s an

example of a 20 t r a y , 5-component m ix tu re d i s t i l l a t i o n w h ich can

be d e s c r i b e d by 404 e q u a t i o n s i f TD models a re not in c lu d e d . He

showed tha t the number o f e q u a t i o n s / v a r i a b l e s i n c r e a s e d by a

f a c t o r o f 1 0 when TD models a r e i n c o r p o r a t e d as equa t ion s . Thus

t h e r e i s a huge i n c r e a s e i n t h e amount o f c o m p u t e r s t o r a g e

r e q u i r e d . S in c e c om pu te r s t o r a g e i s no t l i m i t l e s s , the re i s a

l i m i t on t h e s i z e o f p r o b l e m s t h a t can be s i m u l a t e d a t a

reasonab le co s t . I t shou ld a l s o be r e a l i s e d t h a t the en la rged se t

of equa t ions poses a d i f f i c u l t problem to s o l v e f o r two r e a s o n s :

t h e com m on ly u s e d TD m o d e l s a r e h i g h l y n o n l i n e a r ; and t h e

d i f f i c u l t y of p r o v i d i n g a good initial estimate o f the new internal

v a r i a b l e s o c c u r i n g i n t h e TD m o d e l s . To ove rcome t h e s e co nd

49

problem l .o cke recommend s t a r t i n g the f l o w sh e e t c a l c u l a t i o n s w ith

s imp le TP models and u s in g the r e s u l t s t o p ro v ide an i n i t i a l guess

f o r the more complex model.

2 .5 . The_Jup^Tier_Approach

F l o w s h e e t i n g p a c k a g e s s t r i v e t o have g ene ra l numer ica l

s o l u t i o n a lg o r i t h m s w h ich a r e r o b u s t and r e l i a b l e . T h i s means

t h a t s o l u t i o n methods adop ted i n a p ro cess s im u la t o r are expected

t o o b t a i n the s o l u t i o n t o a posed problem o f t e n from poor i n i t i a l

g u e s s e s w h i c h may l i e o u t s i d e t h e r a d i u s o f conve rgence . The

consequence i s t h a t s o l u t i o n a l g o r i t h m s o f t e n tend t o t a k e r a t h e r

sma l l s teps i n t he independent v a r i a b l e s from one i t e r a t i o n t o the

next . For i n s t a n c e , w i t h f i r s t o rde r n u m e r i c a l s o l u t i o n methods

( e . g . s u c c e s s i v e s u b s t i t u t i o n , W e g s t e in ) commonly a p p l i e d i n

s eq u en t i a l -m o d u la r packages, the change i n the v a lu e of v a r i a b l e s

can be q u i t e s m a l l f o r subsequent f l ow sh ee t i t e r a t i o n s ( the same

o b s e r v a t i o n i s t r u e even f o r second o rde r Newton 's method near the

s o l u t i o n o f t h e f l o w s h e e t ) . Dynamic s i m u l a t i o n of d i s t i l l a t i o n

columns i s ano the r s i t u a t i o n where the re are s m a l l v a r i a t i o n s i n

i t e r a t e s . T h i s i s due t o t h e f a c t t h a t the d i f f e r e n t i a l models

a re u s u a l l y s t i f f and n u m e r i c a l i n t e g r a t i o n a l g o r i t h m s t a k e

r e l a t i v e l y sma l l t ime s t e p s . Thus many of the repeated c a l l s f o r

TD data t o the TP package a re made f o r p o i n t s t h a t are n e c e s s a r i l y

c l o s e i n t h e thermodynamic space . I t i s t h i s o b s e r v a t i o n t h a t we

b e l i e v e l e d t o t h e i d e a o f l o c a l a p p r o x i m a t i o n m o d e l s w h i c h

i n v o l ves r e p l a c i ng t h e r i g o r o u s " h a r d - t o - c a l c u l a t e " TD p rope r ty

50

m o d e l s w i t h s i m p l i f i e d " e a s y - t o - c a l c u l a t e " m o d e l s . The

im p l i c a t i o n s o f the t w o - t i e r methods on the t h r e e c o m p u t a t i o n

l e v e l s ( F i g u r e 1 . 1 ) i n t h e s o l u t i o n of the f l o w s h e e t i n g problem

(case A) i s d i s c u s s e d below.

At l e v e l 1 ( p h y s i c a l p r o p e r t i e s ) a l o c a l r e p r e s e n t a t i o n of

thermodynamic K - v a l u e p ro p e r t y i s proposed. The l o c a l model i s an

a c c u r a t e r e p r e s e n t a t i o n o f t h e r i g o r o u s K - v a l u e o v e r l i m i t e d

ranges of t em pe ra tu re , p r e s s u r e , and compos i t ion . The l o c a l model

can be r ep re sen ted as

i i ^3 i s4 iKi = k \ < — , — , t, n , e> <2.io)s3 S4

where i s a s e t o f a d j u s t a b l e paramete rs . In o the r words, f o r

each r i g o r o u s thermodynamic p ro p e r t y the re i s an e q u i v a l e n t l o c a l

model o f i t * A n a l y t i c d e r i v a t i v e s o f the approx imate

TP models are a v a i l a b l e s i n c e the se reduced models t y p i c a l l y have

s imp le f u n c t i o n a l fo rms. Grens (1984) and M a c ch ie t t o e t a l (1986)

have rev iewed the re cen t deve lopments i n l o c a l app rox im a te m ode ls

f o r 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 . The models s t u d i e d i n c l u d e

a c t i v i t y or f u g a c i t y c o e f f i c i e n t s ( K - v a lu e s ) , pure l i q u i d f u g a c i t y

c o e f f i c i e n t , vapour and l i q u i d e n t h a l p y . P r o p e r t i e s l i k e en t ropy ,

d e n s i t y , v i s c o s i t y among o t h e r s have no t r e c e i v e d any a t t e n t i o n

y e t . The genera l consensus by proponents of these reduced models

i s t h a t i n g e n e r a l c o m p o s i t i o n a nd t e m p e r a t u r e d e p e n d e n t

a p p r o x im a t e mode ls perform much b e t t e r than models dependent on ly

on temperature .

51

The i s o t h e r m a l f l a s h model i s w r i t t e n i n t e rm s o f the

l o c a l K - v a l u e p r o p e r t y m ode l . Thus a d i f f i c u l t and e x p e n s i v e

f l a s h model i s r e p l a c e d w i t h a s i m p l i f i e d a p p r o x im a t i o n . The

f l a s h module l i b r a r y p r o v i d e s o n l y t h e model e q u a t i o n s and does

not c o n ta in a numer ica l s o l u t i o n code t o s o lv e the f l a s h u n i t .

S i n c e an a p p r o x i m a t e f l a s h m o d e l i s a v a i l a b l e , t h e

f l o w s h e e t f o r m u l a t i o n i s t h e r e f o r e an a p p r o x i m a t i o n t o t h e

r i g o r o u s m ode l . N e w t o n ' s method can be r e a d i l y a p p l i e d s i n c e

a n a l y t i c a l d e r i v a t i v e s o f t h e s i m p l e r model f o r K - v a l u e a r e

a v a i l a b l e from the lowe r l e v e l . The i n i t i a l v a l u e s o f t h e mode l

pa rameters a r e s e t u s in g r i g o r o u s TP model data . Be fo re the s t a r t

of the f l ow shee t i t e r a t i v e c a l c u l a t i o n s , a range o f v a l i d i t y o f

the l o c a l model i s e s t a b l i s h e d . The paramete rs i n t h e l o c a l model

are updated i n an o u t e r loop u s in g r i g o r o u s K - v a lue d a ta when the

e s t a b l i s h e d range o f f i t i s v i o l a t e d . The ou te r l o op i n v o l v e s the

s o l u t i o n of the f o l l o w i n g e qua t i o n s :

1 ^3i s4 i ^3i s4 iK\ C ------ , -------- , T, n,B> = K-j ( -------- , -------- , T ,n ) C2.11)

S3 S4 S3 S4

f o r 3 by pos ing the problem e i t h e r as a l e a s t s q u a r e s e s t i m a t i o n

o r t h e s o l u t i o n o f a l g e b r a i c e q u a t i o n s . T h i s c y c l e o f

c a l c u l a t i o n s may be repeated s e v e r a l t imes be fore the s o l u t i o n o f

the f l ow shee t problem i s o b ta in ed .

To s u m m a r i s e , t h e a p p l i c a t i o n o f the two-tier method requires

the s o l u t i o n of an app rox ima te f l ow shee t model based on s i m p l i f i e d

l o c a l TP m o d e l s and t h e e v a l u a t i o n o f l o c a l model p a r a m e t e r s

52

de r i v e d from r i g o r o u s TP da ta . The range over which a g iven Loca l

model i s v a l i d i s de te rm ined and the model parameters a r e r e v i s e d

when i t e r a t i o n v a r i a b l e s f a l l o u t s i d e t h e p redete rmined range.

Fu r the r , f o r the tw o - l e v e l te chn ique t o be v i a b l e the m ode ls must

r e q u i r e r e l a t i v e l y few p a r a m e t e r s , p e rhap s t h r e e o r f o u r per

component and must a d e q u a t e l y r e p re s en t the r i g o r o u s model o v e r a

s i g n i f i c a n t range o f c o n d i t i o n s encounte red i n the course of the

c a l c u l a t i o n . M o r e o v e r , s u c c e s s i v e L o c a l TP m o d e l s m u s t n o t

e x h i b i t s t r o n g d i s c o n t i n u i t i e s . F i n a l l y , t h e use o f l o c a l

app rox im a t ion mode ls must not a d v e r s e l y a f f e c t the accu racy o f the

c a l c u l a t e d f l o w s h e e t p r o b l e m . The c o n c e p t o f t h e t w o - t i e r

s t r a te g y h a £ been i n v e s t i g a t e d by s e ve r a l workers . The consensus

i s t h a t u s i n g TD d a t a i n t h i s way im p ro v e s t h e e f f i c i e n c y of

p r o c e s s s i m u l a t o r s compa red t o t h e b l a c k - b o x a p p r o a c h w h e r e

d e r i v a t i v e s a r e o b t a i n e d n u m e r i c a l l y . The t w o - t i e r approach has

been repo r t ed t o cut r u n - t im e by a b o u t a f a c t o r o f 2 - 3 o v e r the

b la c k -box method and to i n v o l v e f a r fewer r i g o r o u s TP e v a l u a t i o n s .

H u t ch i s on and Shewchuk (1974) used the concept to s im u la t e

an i n t e g r a t e d d i s t i l l a t i o n column by u s ing l o c a l app rox im a t ion s of

r e l a t i v e v o l a t i l i t y f o r n e a r l y i d e a l m i x t u r e s . Shewchuk (1977)

e x t e n d e d t h e a p p r o a c h t o h a n d l e s y s t e m s w i t h s t r o n g

non- i d e a l i t i e s .

L e e s l e y a n d H e y e n ( 1 9 7 7 ) p r e s e n t e d a s y s t e m a t i c

a p p l i c a t i o n of t w o - t i e r s t r a t e g y u s ing the CONCEPT s im u la to r . They

a p p l i e d t h e idea t o the s o l u t i o n o f an LPG r e f r i g e r a t i o n p l a n t , a

53

gas t r e a t m e n t p l a n t , and naphta s e p a r a t i o n p lan t w ith as much as

50-90% r e d u c t i o n i n r i g o r o u s K - v a l u e c o m p u t a t i o n s and o v e r a l l

sav ings of 50 % i n t o t a l computer s im u l a t i o n t ime.

B a r r e t t and Walsh (1979) were the f i r s t workers to p resen t

a d e t a i l e d t r e a t m e n t o f r e p l a c i n g r i g o r ou s TP models w i th l o c a l

a p p r o x i m a t i o n s i n o r d e r t o h a n d l e n o n - i d e a l m i x t u r e s . T h e i r

t w o - t i e r t h e rm o d yn a m ic i n t e r f a c e package TPIF was t e s t e d u s in g

QUASILIN ( e q u a t i o n - o r i e n t e d ) and CHESS ( s e q u e n t i a l - m o d u l a r )

p r o c e s s s i m u l a t o r s . The t w o - t i e r s t r a t e g y was a p p l i e d by these

au thors t o t h e s o l u t i o n o f v a p o u r - l i q u i d e q u i l i b r i u m p ro b le m s

( b u b b l e and d e w - p o i n t t e m p e r a t u r e c a l c u l a t i o n s , and a 26 p l a t e

a ce tone/wate r d i s t i l l a t i o n column d e s i g n ) . In t he d i s t i l l a t i o n

c o l u m n c a l c u l a t i o n s , up t o 43 % r e d u c t i o n i n t h e number o f

r i g o r o u s K - v a l u e p r o p e r t y e v a l u a t i o n s was r e a l i s e d . A v e r y

impor tan t c o n c l u s i o n of B a r r e t t and Walsh i s t h a t the 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 i n f o r m a t i o n r a t h e r t h a n t h e r e p l a c e m e n t o f

r i g o r o u s by a p p r o x im a t e model c a l c u l a t i o n s i s the most impor tan t

f a c t o r f o r 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 . I t i s d i s t u r b i n g ,

however, t o note t h a t H u t ch i s o n e t a l (1983) r e p o r t u n s a t i s f a c t o r y

performance o f TPIF when i n c o r p o r a t e d i n t o t h e c a l c u l a t i o n o f an

i n t e g r a t e d f l o w s h e e t . In two examples repo r ted by Hutch inson e t

a l as much as 115 % i n c r e a s e i n the num be r o f a c c e s s e s f o r

r i g o r o u s TP d a t a i s made when t h e l o c a l mode ls o f B a r r e t t and

Walsh are used compared t o when f i n i t e d i f f e r e n c e i s used t o

g e n e r a t e c o m p o s i t i o n and tempera tu re d e r i v a t i v e s o f the r i g o rou s

K -va lue p r o p e r t y .

54

C h im o w i t z e t a l (1983) d e s c r ib e d an i n v e s t i g a t i o n of the

t w o - t i e r a l g o r i t h m i n t h e s o l u t i o n o f s m a l l - s c a l e s i m u l a t i o n ,

d e s i g n , o p t i m i s a t i o n , and d yn am ic s i m u l a t i o n o f v a p o u r - l i q u i d

e q u i l i b r i u m p ro ce s s e s . For i n s t a n c e , M a cch ie t t o e t a l (1986) have

a p p l i e d the idea of t w o - l e v e l s t r a t e g y t o s im u la te the dynamics o f

coupled f l a s h u n i t s u s in g c o m p o s i t i o n and t e m p e r a t u r e dependen t

K - v a l u e l o c a l models. I t shou ld be noted t h a t dynamic s im u l a t i o n

u s ing an i m p l i c i t based i n t e g r a t o r f a v o u r s the use o f l o c a l models

s i n c e t h e l e n g t h o f t h e i n t e g r a t i o n s t e p i s c o n t r o l l e d so t h a t

e v e r y new s o l u t i o n i s c l o s e t o a p r e v i o u s one an d t h e r e f o r e

r e q u i r e fewer o u t e r l o o p c a l c u l a t i o n s . Fo r t h e c o u p l e d f l a s h

problem as much as 50 % s a v i n g s i n s i m u l a t i o n t im e ^re o b t a i n e d .

T h e s e a u t h o r s h a v e a l s o s u c c e s s f u l l y a p p l i e d t h e method t o

l i q u i d - l i q u i d e q u i l i b r i u m c a l c u l a t i o n s . T h e i r wo rk a l s o c o n f i r m s

t h e c o n c l u s i o n o f B a r r e t t a n d W a l s h w h o s a y s t h a t t h e

a v a i l a b i l i t y o f p a r t i a l d e r i v a t i v e s o f the l i n e a r i s e d l o c a l model

o f t h e p l a n t i s t h e m a j o r s o u r c e o f b e n e f i t a r i s i n g from the

t w o - t i e r a l g o r i t h m . The l o c a l TD models proposed by Chimowitz e t .

a l , among o t h e r s y w e r e f o r low p r e s s u r e e q u i l i b r i u m s i t u a t i o n s

where v a p o u r phase n o n i d e a l i t y i s assumed t o be n e g l i g i b l e .

C h im o w i t z and L e e (1985 ) p r o p o s e d and used approximate K -va lue s

f o r h igh p re s su re compu ta t ion s .

B o s t o n and B r i t t (1978) u t i l i s e d t he t w o - t i e r approach i n

the computat ion of s i n g l e - s t a g e two-phase f l a s h e s . The a l g o r i t h m

has been m o d i f i e d t o h a n d l e m i x t u r e s a t near c r i t i c a l c o n d i t i o n

55

and t h o s e e x h i b i t i n g s im u l t aneo us phase and chemica l e q u i l i b r i u m .

B o s t o n (1980 ) ha s a l s o d e m o n s t r a t e d t h e a p p l i c a t i o n o f t h e

t e c h n i q u e t o m u l t i s t a g e s e p a r a t i o n o f h ig h l y non - idea l systems.

Bos ton and Fou rn ie r (1980) and Bos ton and Shah (1979) a p p l i e d the

t w o - t i e r s t r a t e g y t o s i n g l e - and m u l t i - s t a g e , m u l t i c o m p o n e n t

c a l c u l a t i o n s i n v o l v i n g v a p o u r - l i q u i d - l i q u i d e q u i l i b r i a . The VLE

and VLLE p rocedures deve loped by these au tho r s a re in c o r p o r a t e d i n

AS PEN-PLUS.

R e b e y r o t e ( 1 9 8 0 ) a n d S t e l l ( 1 9 8 1 ) show r e s u l t s f o r

s e p a r a t i o n p r o b l e m s . S t e l l ' s r e s u l t f o r t h e a z e o t r o p i c

d i s t i l l a t i o n o f an e t h an o l wa te r feed w i th benzene i ndi cates t ha t

i t i s p o s s i b l e t o a ch i e ve r e d u c t i o n i n computa t iona l cost of about

70 % compared t o t h e use o f r i g o r o u s TP models on l y . Bryan and

Grens (1983) extended S t e l l ' s idea t o hand le l i q u i d - l i q u i d systems

w i t h a b o u t 50 % r e d u c t i o n i n t h e use o f r i g o r o u s a c t i v i t y

c o e f f i c i e n t models.

2 . 5 . 1 . Fundamenta l_prob i em s_a r i s i n^_f rom_ tw o - 1 i e r _ st r a t e ^

Though t h e c o n c e p t o f t h e t w o - t i e r s t r a t e g y looks q u i t e

a t t r a c t i v e i t n e v e r t h e l e s s r a i s e s a few fundamental q u e s t i o n s .

I t i s g e n e r a l l y a g r e e d t h a t t h e d e t e r m i n a t i o n o f t h e

g lo ba l optimum of a problem and the p e r f o rm a n c e o f the p o w e r f u l

o p t im i z a t i o n a l g o r i t h m s a re i n f l u e n c e d s t r o n g l y by the accu racy of

the g r a d i e n t s o f t h e 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 .

56

U n f o r t u n a t e l y , t h e l o c a l m ode l s s u g g e s t e d so f a r a re concerned

p r im a r i l y w i t h t h e l o c a l r e p r o d u c t i o n o f r i g o r o u s model p o i n t

v a l u e s . Lo ca l models by t h e i r d e f i n i t i o n i n t e r p o l a t e the r i g o r o u s

p rope r t y l o c a l l y and a l l o w f o r some so r t of e x t r a p o l a t i o n o u t s i d e

t h e r e g i o n o f f i t . T h i s means the tangen ts t o the r i g o r o u s and

s i m p l i f i e d models a t a g i v en p o i n t i n the thermodynamic s pa ce a re

not n e c e s s a r i l y t h e same (Grens (1984), Chimowitz et a l ( 1983 ) ) .

G r e n s r e p o r t e d t h e e l e m e n t s o f t h e J a c o b i a n m a t r i x f r o m t h e

r i g o r o u s and a p p r o x im a t e a c t i v i t y c o e f f i c i e n t s . The J a c o b ia n

e lements have the same s i g n f o r both s i t u a t i o n s bu t show a s much

a s 27 % a b s o l u t e e r r o r i n some e lements . The p a r t i a l d e r i v a t i v e s

o f K - v a lu e re po r t e d by Ch imowi tz e t a l show b e t t e r agreement w i t h

a v e r a g e e r r o r s o f abou t 8 % i n a b s o lu t e terms# One element has a

63 % e r r o r ! . Thus the s o l u t i o n t o an o p t i m i z a t i o n p rob lem fo u n d

by the use of l o c a l models i s no t n e c e s s a r i l y the same as would be

o b t a i n e d u s i n g a r i g o r o u s mode l ( B i e g l e r e t a l , 1 9 8 5 ) . These

a u t h o r s p r e s e n t e d t h r e e m a t h e m a t i c a l e x a m p l e s w h e re t h e

a p p r o x im a t e mode l l e d e i t h e r t o t h e wrong optimum or f a i l e d to

l o c a t e the optimum even where an optimum e x i s t s u s ing the r i g o r o u s

m o d e l . They c o n c l u d e by s a y i n g t h a t t h e a p p l i c a t i o n o f t h e

t w o - t i e r s t r a t e g y l e a d s t o the optimum of the s i m p l i f i e d model a t

a p o i n t whe re p r o p e r t i e s c a l c u l a t e d by l o c a l and r i g o r o u s models

match each o th e r . I t i s a lmos t im p o s s ib l e t o d e r i v e l o c a l mode ls

w h i c h r e p r o d u c e t h e r i g o r o u s TP po in t v a l u e s and d e r i v a t i v e s at

a l l p o i n t s i n the TD space .

57

R o b u s t n e s s o f c o n t r o l sys tem s t r u c t u r e s a r e u s u a l l y

d e f in ed i n terms of the s e n s i t i v i t y o f the c o n t r o l t o m o d e l l i n g

e r r o r s . The t r a n s f e r f u n c t i o n m a t r i x used i n such a n a l y s i s w i l l

thus be a f f e c t e d by i n a c c u r a t e d e r i v a t i v e da ta and may t h e r e f o r e

lead t o the wrong c o n t r o l s t r u c t u r e be ing chosen.

L o ca l TP models used i n t w o - t i e r a l g o r i t h m s a re u s u a l l y

g iven as f u n c t i o n s o f o n l y temperatu re and compos i t i on whereas the

r i g o r o u s models depend on the se v a r i a b l e s as w e l l a s some o t h e r

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 e .g . group volumes, group su r f a c e

a r e a s , and b i n a r y i n t e r a c t i o n p a r a m e t e r s i n t h e U N I F A C

( F r e d e n s l u n d e t a l , 1977) a c t i v i t y c o e f f i c i e n t model. The re fo re

i t w i l l be r a t h e r d i f f i c u l t t o genera te the pa ram e t r i c s e n s i t i v i t y

of a des ign t o i n a c c u r a c i e s i n these s o r t o f parameters.

In o t h e r t o d e t e r m i n e w he the r a p a r t i c u l a r mode l w i l l

r e t u r n r e s u l t s o f t h e d e s i r e d a c cu ra cy , i t i s necessary t o know

the e r r o r i n the model a s a f u n c t i o n of d i s t a n c e from the p o in t i n

the TD space a t which the model was genera ted. Seve ra l approaches

have been t a k e n t o d e f i n e t h e e r r o r l e v e l o r range o v e r w h ich

l o c a l mode ls may app ly . However, the q u e s t i o n s o f when t o update

the l o c a l a p p ro x im a t io n and what r a n g e s t o t r y t o span w i t h the

approx imate models a re t o da te l a r g e l y un reso lved .

The above i s s u e s l e a d one t o t h e q u e s t i o n o f w h e t h e r

a p p r o p r i a t e s i m p l i f i e d m o d e l s can be fo u n d w h ich g i v e a good

r e p r e s e n t a t i o n of the com p l i c a te d TD model behav iou r .

58

2 .6 . Jhe_Hybrid_Approach

T h i s i s a r e c e n t i d e a due t o t h e work o f L u c i a and

M a cch ie t t o (1983) . To i l l u s t r a t e the t e c h n iq u e l e t us r e t u r n t o

the h y p o t h e t i c a l f l o w s h e e t problem (case A).

The p h y s i c a l p r o p e r t i e s l e v e l i s t h e same a s f o r t h e

b la c k -b o x approach . A n a l y t i c a l d e r i v a t i v e s as i n the se s i t u a t i o n s

are not a v a i l a b l e . A s i m p l i f i e d form f o r the r i g o r o u s K - v a l u e i s

p o s t u l a t e d t o genera te an approx imate d e r i v a t i v e of K -va lue w i t h 8

set t o ze ro ( e q u a t i o n 2 . 1 0 ) .

The f l a s h m odu le g e n e r a t e s t h e e q u a t i o n s w i th r i g o r o u s

k - va lu e p o in t da ta .

The f l o w s h e e t J a c o b i a n m a t r i x i s s p l i t i n t o two p a r t s :

computed and app rox ima ted p a r t s . That i s , the J a c o b ia n m a t r i x , J ,

can be w r i t t e n as

J = C1 + A1 (2 .12)

w h e r e and A a r e t h e c o m p u t e d and a p p r o x i m a t e d p a r t s

r e s p e c t i v e l y . The e lemen ts i n the approx imate p a r t w i l l c o n t a i n

t he p a r t i a l d e r i v a t i v e s t h a t a re u s u a l l y assumed t o be " d i f f i c u l t

and e x p e n s i v e " t o s e c u r e f r o m l e v e l 1 . T h e s e u n a v a i l a b l e

d e r i v a t i v e s a r e e s t i m a t e d u s i n g a Qua s i -N ew ton te chn ique (e .g .

59

S h u b e r t , 1970) w h ich u t i l i z e s o n l y t he r i g o r o u s K - v a l u e po in t

data . The app rox ima ted pa r t of t h e J a c o b i a n w h ich c o n t a i n s the

n o n - i d e a l c o n t r i b u t i o n s i s i n i t i a l i s e d e i t h e r t o a n u l l m a t r i x or

by numer ica l p e r t u r b a t i o n . The computed p a r t , C^, c o n t a i n s a l l the

e a s i l y c a l c u l a t e d p a r t i a l d e r i v a t i v e terms, t h a t i s , d e r i v a t i v e s

o f i d e a l K - v a l u e . The t e m p e r a t u r e d e r i v a t i v e the i d e a l K -va lues

model i s r e a d i l y c a l c u l a t e d . The e lements of the J a c o b i a n m a t r i x

c o n t r i b u t i o n s f rom t h e m i x e r , s p l i t t e r , as w e l l a s t h e mass

ba lances f o r the f l a s h a re a v a i l a b l e s t r a igh taw ay .

In s e c t i o n 2 . 5 i t was m e n t i o n e d t h a t t h e e f f i c i e n c y of

t w o - t i e r methods i s l a r g e l y due t o the p r o v i s i o n o f a p p r o x im a t e

p a r t i a l d e r i v a t i v e s and not the rep lacement of r i g o r o u s TP po in t

v a l u e s w i th app rox ima te v a l u e s . The major d i f f e r e n c e s between the

Hybr id and t w o - l e v e l approaches a re as f o l l o w s : -

( i ) w he re a s t h e s i m p l i f i e d model i s r e t a i n e d through

t h e s o l u t i o n p a th f o r t h e H y b r i d m e th o d ( i e ,

3 = 0 ) , i t i s u p d a t e d when n e c e s s a r y i n t h e

t w o - t i e r approach .

( i i ) more t h a n one pass i s made t o the TP database to

secu re the paramete rs of t h e l o c a l mode l i n the

t w o - t i e r a p p ro a c h w h i l e a s i n g l e p a s s i s made

when the H yb r i d t e chn ique i s used.

60

( i i i ) t h e " e x p e n s i v e " TP d e r i v a t i v e s a re approx imated

u s in g a Quas i-Newton update f o r the Hybr id method

u s i n g o n l y t h e r i g o r o u s p o i n t v a l u e s . On the

o t h e r h an d , s u ch i n f o r m a t i o n i s o b t a i n e d by

a n a l y t i c a l d i f f e r e n t i a t i o n o f t h e l o c a l models

f o r the t w o - t i e r a lg o r i t h m .

We ment ioned e a r l i e r t h a t L u c i a and M a c ch ie t t o f i r s t suggested the

use o f H y b r i d t e c h n i q u e i n t h e a p p r o x i m a t i o n o f q u a n t i t i e s

i n v o l v i n g p h y s i c a l prope r t 3 d e r i v a t i v e s . These au tho r s a p p l i e d

the Hyb r id method t o the s o l u t i o n of s e ve r a l dew p o i n t temperature

c a l c u l a t i o n s ^ n d to the s i r a u l a t i o n o f a f i v e e f f e c t c o u n te r - c u r r e n t

e vapo ra to r system which con cen t ra t e s a sodium h y d r o x i d e s o l u t i o n .

They r e p o r t e d t h a t the r e l i a b i l i t y of Hybr id method i s comparable

t o Newton's method. For the se examples , the a u t h o r s r e a l i s e d up

t o 50 % f e w e r r i g o r o u s K - v a l u e c a l c u l a t i o n s compared t o t h e

b l a c k - b o x t e c h n i q u e . L u c i a and M a c c h i e t t o n o t e d t h a t T a y l o r

(1982) and worke rs a t Un ion Ca rb ide C o rp o ra t i o n (USA) a p p l i e d the

Hyb r id idea t o the s o l u t i o n o f m u l t i c o m p o n e n t mass t r a n s f e r and

f l a s h c a l c u l a t i o n s r e s p e c t i v e l y w i th the same degree o f su ccess .

L u c i a and Westman (1984 ) a p p l i e d t he same i d e a t o the

s o l u t i o n o f two s e p a r a t i o n o p e r a t i o n s . One example i n v o l v e s the

v a p o u r - l i q u i d e q u i l i b r i u m s e p a r a t i o n o f m e t h a n o l , e t h a n o l ,

a c e t o n e , and w a t e r by e x t r a c t i v e d i s t i l l a t i o n w i t h water . The

second problem i n v o l v e s the d e s ig n o f l i q u i d - l i q u i d e x t r a c t i o n o f

n-heptane and cyc lohexane w i t h f u r f u r a l . In both cases the Hyb r id

61

method compared favourably with the finite difference implementation of

the b la c k -box i n t e r f a c e s t r a t e g y i n terms of number of i t e r a t i o n s .

However , t h e H y b r i d t e ch n iq u e used 60 % - 84 % fewer r i g o r o u s TP

data t o s o l v e the p r o b l e m s t o t h e same a c c u r a c y . The a u t h o r s

found i t necessa ry though to i n i t i a l i s e the approx imated p a r t (A^)

of the Ja cob ia n by f i n i t e d i f f e r e n c e s t o ensure convergence of the

method.

Westman e t a l (1984) a l s o u t i l i z e d the techn ique t o s o l v e

v a r i o u s types o f f l a s h p r o b l e m s and an e x t r a c t i v e d i s t i l l a t i o n

co lumn d e s i g n . T h e i r r e s u l t s con f i rm the o b s e r v a t i o n s re ga rd in g

robu s tnes s , r e l i a b i l i t y and com pu ta t iona l e f f i c i e n c y .

P a n t e l i d e s (1987 ) and F i e l d e t a l (1984 ) have used the

i d e a t o s o l v e s e v e r a l f l o w s h e e t i n g p r o b l e m s i n v o l v i n g t h e

i n c o r p o r a t i o n o f f l a s h p r o c e d u r e s ( t h a t i s , c ase B o f our

h y p o th e t i c a l p rob lem). T h e i r r e s u l t s a re encou rag ing . A c c o r d i n g

t o t h e s e a u th o r s t he numer o f i t e r a t i o n s f o r the Hyb r id method i s

no t s i g n i f i c a n t l y h i g h e r t h a n t h o s e o b t a i n e d u s i n g N e w to n .

U s u a l l y t h e number o f f u n c t i o n e v a l u a t i o n s i s much sm a l l e r than

those of N e w to n ' s m e thod . T h i s r e l a t i v e e f f i c i e n c y o f H y b r i d

compared t o Newton i n te rms o f number of e v a l u a t i o n s and e xe cu t i o n

t ime i s due t o the lower co s t per i t e r a t i o n of the fo rmer .

Our e x p e r i e n c e w i t h t h e H y b r i d t e chn ique i n the area of

dew p o i n t t e m p e r a t u r e , b u b b l e p o i n t t e m p e r a t u r e , and f l a s h

c a l c u l a t i o n s con f i rm the c o n c l u s i o n s of o the r worke rs . In a l l the

62

e x a m p l e s a t t e m p t e d s o f a r u s i n g t h e H y b r i d t e c h n i q u e , the

s i m p l i f i e d model of K - v a l u e was ba sed on i d e a l s o l u t i o n t h e o r y

g iv en as

K-j = II °/II (2 .13)

where the form of II ° i s g i v e n i n P r a u s n i t z et a l (1980).

M i l l e r and L u c i a ( 1985 ) c a r r i e d o u t d e t a i l e d n u m e r i c a l

expe r imen ts t o f i n d out why the H yb r id method per fo rms b e t t e r than

S c h u b e r t ' s method when t h e H y b r i d method i s a m o d i f i c a t i o n o f

S h u b e r t ' s method . Two d i f f e r e n t im p l e m e n t a t i o n s o f the Hyb r id

method were s tu d ie d . In t h e f i r s t case the approx imated p a r t (A^)

c o n t a i n s K - v a l u e c o m p o s i t i o n a nd t e m p e r a t u r e d e r i v a t i v e s .

H o w e v e r , f o r t h e s e c o n d i m p l e m e n t a t i o n A*! c o n t a i n s o n l y

c o m p o s i t i o n d e r i v a t i v e s o f K - v a l u e s . The computed p a r t , C^, f o r

bo th im p l e m e n t a t i o n s c o n t a i n t h e i d e a l K - v a l u e t e m p e r a t u r e

d e r i v a t i v e s . T e s t s on dew p o i n t prob lems showed t h a t t h e re are

two reasons f o r b e t t e r performance o f the H y b r i d o v e r S c h u b e r t ' s

method: a v a i l a b i l i t y of a n a l y t i c a l d e r i v a t i v e s and the s c a l i n g of

the v a r i a b l e s . F u r t h e r i n v e s t i g a t i o n s by L u c i a (1985) l e d t o the

i d e a o f im p o s i n g t h e rm o d y n a m i c c o n s t r a i n t s a s w e l l as s e c a n t

c o n d i t i o n s t o o b t a i n new update fo rmu lae . H i s new upda te f o r m u l a

makesruse o f t h e f a c t t h a t e x c e s s en tha lp y and a c t i v i t y / f u g a c i t y

c o e f f i c i e n t s a r e h o m o g e n e o u s f u n c t i o n s o f z e r o d e g r e e .

T e m p e r a t u r e d e r i v a t i v e s o f K - v a l u e s w e r e n e g l e c t e d . The

approx imated p a r t was i n i t i a l i s e d by n u m e r i c a l p e r t u r b a t i o n t o

63

a v o i d e r r a t i c b e h a v i o u r o f bo th H y b r i d v e r s i o n s . On a s e t of

s i n g l e - s t a g e VLE f l a s h p rob lems, the new Hyb r id ob ta in ed s o l u t i o n s

where t h e o r i g i n a l H y b r i d r e s u l t e d i n f a i l u r e s . The new Hyb r id

made fewer a c ce s s e s t o t h e TP package bu t used more i rions*'

than Newton’ s method.

V e n k a ta r a m a n and L u c i a (1986) i n a p p a r e n t e x t e n s i o n o f

L u c i a e t a l 's(1985) r e s u l t s deve loped new upda t ing fo rm u lae f o r the

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 . The J a c o b ia n m a t r i x i s g iv en

b y :

J = c] + + a1 (2 .14)

c l i s t h e same a s f rom e q u a t i o n ( 2 . 1 2 ) and i s a d iagona l

m a t r i x a v a i l a b l e a n a l y t i c a l l y . The u n s ym m e t r i c m a t r i x A*! i s

d i f f e r e n t f r om t h a t o f e q u a t i o n ( 2 . 1 2 ) and has t h e f o l l o w i n g

c h a r a c t e r i s t i c s :

( i ) s a t i s f i e s s e c a n t c o n d i t i o n w h i c h as we know i s

e s s e n t i a l t o a s s u r e s a t i s f a c t o r y n u m e r i c a l

p e r f o r m a n c e o f a l l Quas i-Newton updates (Denn is

and S chnabe l , 1979) .

( i i ) i t decomposes i n t o two symmetr ic p a r t i t i o n s .

64

( i i i ) t h e s y m m e t r i c p a r t i t i o n s s a t i s f y b o t h t h e

z e r o - d e g r e e h o m o g e n e i t y a n d G i b b ' s - D u h e m

equa t i o n s a t each i t e r a t i o n .

( i v ) e x c e s s e n t h a l p y p r o p e r t y d e r i v a t i v e s s a t i s f y

c e r t a i n ze r o -d eg re e homogeneity c o n d i t i o n s .

Cv) t h e method o f i t e r a t e d p r o j e c t i o n s suggested by

D enn is and S c h n a b e l i s used so t h a t t h e above

f o u r c h a r a c t e r i s t i c s a r e s a t i s f i e d s im u l t an eo u s l y

a t each i t e r a t i o n .

T h i s newes t v e r s i o n o f H y b r i d method was t e s t e d w i t h v a r i o u s

numberSof dew p o i n t , VLE ( s i n g l e - and t w o - s t a g e ) f l a s h , and LLE

f l a s h p r o b l e m s . They conc lude the new method i s more e f f i c i e n t

and r e l i a b l e than even N e w t o n ' s method . On t h e 1 , 6 5 0 p ro b lem s

a t t e m p t e d the a u th o r s found Newton 's method had a 13.82 % f a i l u r e

ra t e compared t o a modest v a l u e o f 4 .06 % f o r the H yb r i d . We a r e

not s u r p r i s e d a t t h e poo r p e r f o r m a n c e o f t h e i r Newton method.

Th i s i s because the Newton s t e p and d i r e c t i o n a re d e s t r o y e d a s a

r e s u l t o f t h e s t r a t e g y a d o p t e d f o r r e s e t t i n g t h e i t e r a t i o n

v a r i a b l e s when one o r m o re s u c h v a r i a b l e s f a l l o u t s i d e a

p h y s i c a l l y m e a n i n g f u l v a l u e . The number o f f a i l u r e s by the

o r i g i n a l H yb r id method i s reduced by a s much a s 50 % u s i n g t h i s

new H y b r i d m e t h o d . As i n p r e v i o u s H y b r i d i m p l e m e n t a t i o n s ,

N e w to n ' s method p e r f o rm e d b e t t e r i n t e r m s o f t h e num be r o f

i t e r a t i o n s . However, when compared by the number o f c a l l s t o the

65

TP package the H yb r id d i d b e t t e r .

The Hyb r id method o f i n t e r f a c i n g TD data i n p ro cess des ign

has a number of d i s advan tage s . The f i r s t d i sadvan tage i s the need

t o i n i t i a l i s e t h e a p p r o x im a t e d p a r t by f i n i t e d i f f e r e n c e s t o

a s su re r e l i a b i l i t y . I t may a l s o become necessary t o r e i n i t i a l i s e

t h e J a c o b i a n m a t r i x ( a t t h e f l o w s h e e t l e v e l ) f o r d i f f i c u l t

p r o b le m s . F i n i t e d i f f e r e n c e a s we a l r e a d y know i s u s u a l l y

e x p e n s i v e i n t e r m s o f c o m p u t i n g c o s t . T h i s p rob lem may be

e l im in a t e d by the use o f V e n k a t a r a m a n and L u c i a (1986 ) upda te

f o r m u l a . However , t h e p a r t i a l d e r i v a t i v e s genera ted when u s ing

the H y b r i d method a re o n l y a p p r o x i m a t i o n s t o t h e e x a c t o n e s

( L u c i a , 1 9 8 5 ) . Th u s , i t i s not p o s s i b l e t o use such i n f o rm a t i o n

i n power fu l o p t im i z a t i o n a l g o r i t h m s where the need f o r a c c u r a t e

d e r i v a t i v e as ment ioned e a r l i e r i s c r i t i c a l t o the performance and

e f f i c i e n c y of the o p t im i z a t i o n codes. In o t h e r words, one w i l l be

f o r c e d t o e s t i m a t e TP d e r i v a t i v e s e i t h e r by o t h e r means e . g .

n u m e r i c a l p e r t u r b a t i o n , o r use d e r i v a t i v e f r e e o p t i m i z a t i o n

a l g o r i t h m s w i t h t h e i n h e r e n t i n e f f i c i e n c i e s o f e i t h e r op t io n .

U n f o r t u n a t e l y , An to ine c o n s t a n t s ( o r c on s ta n t s i n the P r a u s n i t z e t

a l m o d e l ) a r e n o t a v a i l a b l e f o r a l l c omponents i n p h y s i c a l

p r o p e r t i e s packages and wou ld t h e r e f o r e r e q u i r e u s e r s t o p r o v i d e

them. F i n a l l y the re i s no guarantee tha t the i n i t i a l computed pa r t s o f

the J a c o b ia n m a t r i x w i l l no t pose n u m e r i c a l p r o b le m s where the

m i x t u r e i s s t r o n g l y n o n - i d e a l and p o s s i b l y i n v o l v i n g d im e r i z a t i o n

i n the vapour phase, i n h igh p r e s s u r e c o m p u t a t i o n s or even w i t h

e l e c t r o l y t e systems.

66

2 .7 . Co n c lu s io n s

T h e f o u r d i f f e r e n t w a y s c u r r e n t l y a d o p t e d f o r

i n c o r p o r a t i n g TP da ta w i t h p ro cess f l o w sh e e t i n g systems have been

e xam ined i n d e t a i l i n p r e v i o u s s e c t i o n s o f t h i s c h ap t e r . The

fundamenta l q u e s t i o n s t o wh ich each s t r a t e g y a t t e m p t e d an sw e r

can be s t a t e d as f o l l o w s : -

( i ) i s i t p o s s i b l e to o b t a i n the 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 r i g o r o u s phase and

c h e m i c a l e q u i l i b r i a p r o c e d u r e s w i t h r e spe c t t o

t h e i r i n p u t v a r i a b l e s , and

( i i ) i s i t p o s s i b l e t o d e r i v e a n a l y t i c a l p a r t i a l

d e r i v a t i v e s o f m o l e c u l a r TD m o d e l s a t a

r e a son ab le c o s t .

U n f o r t u n a t e l y , none o f t h e e x i s t i n g TP i n t e r f a c e t e c h n i q u e s

( B l a c k - b o x , Wes te rbe rg , T w o - t i e r and Hyb r id ) seem to answer e i t h e r

o f t h e above q u e s t i o n s s a t i s f a c t o r i l y . Tab le 2 . 1 . show c l e a r l y

the d e f i c i e n c i e s o f the v a r i o u s i n t e r f a c e s based on the c r i t e r i a

se t out i n S e c t i o n 2 . 1 .

R e c a l l t h a t P e r k i n s (1984) s t r e s s e d the d e s i r a b i l i t y of

i n c o r p o r a t i n g TP p r o c e d u r e s e s p e c i a l l y i n EO s i m u l a t o r s .

S t a d t h e r r and H i l t o n ( 1982 ) a l s o c o n f i r m e d t h a t p r o v i d i n g TP

TABLE 2 . 1 : Effectiveness of TD Data Interface Strategies Measured Against Criteria Stated in Section 2.1

Performance Criteria Black-Box Westerberg Two-Tier Hybrid Our Technique

(i) Efficiency X X / / /(ii) Provision of derivatives / / / / /(iii) Provision of rigorous process

design sensitivity to physical X X X X /

properties(iv) Effect on convergence of

numerical methods / X / / /(v) Use of available TD

procedures / X X X /(vi) Decoupling of base TD package / X / / /(vii) Storage requirements / X X X /(viii) User friendliness / X / / /

Legend x - unsatisfactory J - satisfactory

^Details of our technique is provided in the next chapter.

60

p r o c e d u r e s i n s t e a d o f e q u a t i o n s enhanced the performance of t h e i r

EO s im u la to r (SEQUEL). Thus p r o v i d i n g s a t i s f a c t o r y answers t o the

above q u e s t i o n s w i l l im p ro v e t h e e f f i c i e n c y and f l e x i b i l i t y of

f l o w s h e e t i n g systems.

A new TD data i n t e r f a c e s t r a t e g y with p rocess f l o w sh e e t i n g

p a ck age s i s p r o p o s e d and t e s t e d i n the nex t c h a p t e r a nd t he

i m p l i c a t i o n s o f the proposed s t r a t e g y f o r TD packages and p rocess

f l o w sh e e t i n g e x e c u t i v e s are s t u d i e d i n d e t a i l .

69

£umEB_iyREE

IffICIENJ_SJRAJEGY_FOR_INIERfACING_JHERWpYNAMIC_PBOP§Bn

I n t h i s c h a p t e r we s h a l l f i r s t show t h a t a c c u r a t e

procedure d e r i v a t i v e s can be o b t a i n e d q u i t e e a s i l y r a t h e r t h a n

a s su m in g such d e r i v a t i v e s a r e u n a v a i l a b l e . We s t a r t by d e r i v i n g

the p a r t i a l d e r i v a t i v e s o f the ou tpu t and i n t e r m e d i a t e v a r i a b l e s

o f g e n e r a l p ro cedu re s w i t h r e spe c t t o t h e i r i n pu t v a r i a b l e s . The

method i s t h e n a p p l i e d t o g e n e r a t e t he g r a d i e n t s o f f l a s h and

d i s t i l l a t i o n o u t p u t v a r i a b l e s w i t h r e s p e c t t o t h e i r i n p u t

v a r i a b l e s . The i m p l i c a t i o n s o f c o m p u t in g phase e q u i l i b r i u m

p r o c e d u r e d e r i v a t i v e s by t h i s m e th o d on e x i s t i n g p h y s i c a l

p r o p e r t i e s packages and p ro ce s s s im u la t o r s are d i s c u s sed .

The g e n e r a t e d p ro cedu re d e r i v a t i v e s a re u t i l i z e d t o so lv e

s e v e r a l s m a l l - s c a l e f l o w s h e e t i n g p ro b lem s u s i n g t h e SPEEDUP

s i m u l a t o r . C o n c l u s i o n s a r e d r a w n b a s e d on t h e n u m e r i c a l

e xpe r imen ts .

3 .1 . Exact_JProcedure_ D erivatives

L e t us d e f i n e the s e t of m + n equa t i o n s t o be so lv ed i n a

procedure as

f ( w , v , u ) = 0 (3.1)

7G

where we Rm (vector of output variables)

v e R n ( v e c t o r of in t e rm e d ia t e v a r i a b l e s )

ue RL ( v e c t o r of inpu t v a r i a b l e s )

We assume a procedure i s a v a i l a b l e t o c a l c u l a t e the s o l u t i o n w

(and v) o f e q u a t i o n s ( 3 . 1 ) c o r r e s p o n d in g t o a g iven se t of u as

ment ioned i n chap te r one. What we are i n t e r e s t e d i n i s the m a t r i x

o f p a r t i a l d e r i v a t i v e s 3 (w , v ) / a u a t the s o l u t i o n of (3 .1 ) which

r e p r e sen t the c o n s t r a in e d g r a d i e n t s ( s e n s i t i v i t i e s ) o f bo th the

o u t p u t (w) and i n t e r m e d i a t e (v) v a r i a b l e s w i th re spec t t o i n p u t s

(u ) . T h i s m a t r i x i s a f i r s t o rde r a pp rox im a t ion t o the change i n

w and v f o r a sma l l change i n u su b je c t t o e qu a t i o n ( 3 . 1 ) .

L e t w* and v* be the s o l u t i o n of the p r o c e d u r e f o r g i v e n

v a l u e s o f the in p u t v e c t o r (s tream v a r i a b l e s and u n i t p a ramete r s ) ,

u. Assuming equa t ion (3 .1) i s c o n t i n u o u s and d i f f e r e n t i a b l e , a

T a y l o r s e r i e s e x p a n s i o n a r o u n d t h e s o l u t i o n g i v e s ( n e g l e c t i n g

h ighe r o rde r te rm s) :

+ dw, v* + dv, u + du) = f(w*, V*, u)

3f af afdw + _ _ dv + — du (3 . 2 )

9w (w*, V*, u) 3v (w*, V*, u) au (w*, V*, u)

71

S i n c e t h e p e r t u r b a t i o n s a r e a round the s o l u t i o n , equa t ion (3 .2) becomes

du = 0 (3 .3)(w*, v*, u)

Le t

3f 3f 3f__ dw + — dv + L t3w (w*, v*, u) 3v (w*, v* , u) 3u

and

3fQ = —

8 w

a fR = —

3 v

3fS = —

3u

, Qe R m + n, m(w*, v* , u)

, Re R m + n, n(w*, v* , u)

, se R m + L(w*, v* , u)

w i t h a l l t h e m a t r i c e s computed a t the s o l u t i o n t o the p rocedure .

Equat ion (3 .3 ) can t h e r e f o r e be w r i t t e n as

Qdw + Rdv + Sdu = 0 (3 .3 )

Tak ing the l i m i t as du -*■ 0 , we have

3w 3 vQ — + R — = - S

3u 3u(3 .4)

o r , i n m a t r i x form

3w

l 9uQ t Ri •

3 v

iii S

M — _3 u_

(3 .5 )

T hu s , t h e d e s i r e d o u t p u t - i n p u t s e n s i t i v i t i e s 3w/3u (and 3v/3u i f

need be) can be ob ta in ed by s o l v i n g equa t ion ( 3 . 5 ) , t h a t i s , a set

o f m + 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 w i th l - r i g h t hand s i d e s . The

f o l l o w i n g t a s k s w i l l have t o be p e r f o rm e d i n o r d e r t o s e c u r e

p rocedure d e r i v a t i v e s th rough our work ing equa t ion ( 3 .5 ) :

72

Ci) s o l v e e q u a t i o n s ( 3 . 1 ) a c c o r d i n g t o t h e g i v e n

p r o c e d u r e ( t o some f i n i t e t o l e r a n c e i f t h e

s o l u t i o n i s i t e r a t i v e ) .

( i i ) g e n e r a t e m a t r i c e s Q/ R/ anc* S a t the s o l u t i o n

o b t a i n e d i n s tep ( i )

( i i i ) o b t a i n t h e LU f a c t o r s o f m a t r i x CQ : R3-

( i v ) s o l v e e q u a t i o n (3 .5) f o r 8w/3u.

For some procedures e q u a t i o n ( 3 . 1 ) can be p a r t i t i o n e d i n t o two o r

m ore g r o u p s o f e q u a t i o n s a c c o r d i n g t o t h e g r o u p i n g o f t h e

v a r i a b l e s (we s h a l l r e t u r n t o t h i s i s s u e i n s e c t i o n 3 . 3 ) .

S t e p ( i ) c o u l d be a c c o m p l i s h e d u s i n g any s u i t a b l e

numer ica l s o l u t i o n a l g o r i t h m w h ich may i n v o l v e r e f o r m u l a t i o n o r

t r a n s f o r m a t i o n o f t h e o r i g i n a l e q u a t i o n s ( 3 . 1 ) . For i n s t a n c e ,

r i g o r o u s d i s t i l l a t i o n column de s ig n can be t a c k l e d by a v a r i e t y o f

me thods : t e a r i n g , b l o c k r e l a x a t i o n , and s imu ltaneous c o r r e c t i o n s .

Temperature and sometimes c om pos i t i o n d e r i v a t i v e s of TP models may

be r e q u i r e d i n any o f t h e s o l u t i o n a lg o r i t h m s . Another example i s

the i s o th e rm a l f l a s h problem. Here r e f o rm u l a t i o n of the procedure

e q u a t i o n s may i n v o l v e d e t e r m i n a t i o n of the minimum of the G ibbs

f r e e e n e rg y s u r f a c e . The same p r o b l e m can be h a n d l e d by

t r a n s f o r m a t i o n o r t e a r i n g t h e equa t ion s as i n the Rachf o r d -R i ce

t e chn ique . On the o t h e r hand t h e f u l l s e t o f e q u a t i o n s f o r the

73

f l a s h can be s o l v e d s i m u l t a n e o u s l y u s ing Newton's method o r the

Hybr id method. D e r i v a t i v e s of TD mode ls may be r e q u i r e d depending

on t h e way t h e f l a s h mode l i s t r e a t e d . In the next s e c t i o n we

show the best way t o o b t a i n the d e r i v a t i v e s of p h y s i c a l p r o p e r t i e s

models.

M a t r i c e s Q and R w o u l d a l r e a d y be a v a i l a b l e a t t h e

converged s o l u t i o n o f the p rocedu re i f the i t e r a t i v e a l g o r i t h m i s

N e w to n ' s method ( m a t r i c e s g e n e r a t e d by Quas i-Newton updates or

t w o - t i e r te chn ique s a re i n a c c u r a t e and do not h e l p u s ) . I n t h i s

s i t u a t i o n o n l y m a t r i x S i s g e n e r a t e d a t t h e s o l u t i o n t o t h e

procedure. A l t e r n a t i v e l y , a l l t h e m a t r i c e s a r e a s s e m b le d a f t e r

convergence. The s to ra ge requ i rement f o r m a t r i c e s CQ:R3 and S are

(m + n)2 and (m + n) l r e s p e c t i v e l y . Note t ha t i n p r i n c i p l e these

m a t r i c e s are not n e c e s s a r i l y f u l l ( e .g . d i s t i l l a t i o n ) .

F i n a l l y a number o f c o d e s a r e a v a i l a b l e f o r pe r fo rm ing

ta sk s ( i i i ) and ( i v ) . The LU f a c t o r s are o b t a i n e d once and used

f o r a l l t h e r i g h t hand s i d e v e c t o r s ( i n p u t s ) . O f ten i n p rocess

s i m u l a t i o n s , some o f t h e i n p u t s t o a p r o c e d u r e a re s p e c i f i e d

v a r i a b l e s a t t h e f l o w s h e e t l e v e l . Advantage shou ld be taken of

t h i s f a c t so t ha t o n l y g r a d i e n t s o f o u t p u t s w i t h r e s p e c t t o t he

unknown i n p u t s ( o r a c t i v e ) v a r i a b l e s a r e genera ted. Th i s w i l l

ensure tha t unnecessary a r i t h m e t i c o p e r a t i o n s a r e a v o i d e d hence

e n h a n c i n g f u r t h e r t h e e f f i c i e n c y o f g e n e r a t i n g t h e p r o c e d u r e

d e r i v a t i v e s .

74

3 .2 - D e r iv a t iv e s _ o f_ J h e rm o d Y n a m ic _ P r o p e r t ie s _ M o d e ls

I n t h e l a s t s e c t i o n , we o u t l i n e d 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 procedure d e r i v a t i v e s - For p rocedures i n v o l v i n g phase

and c h e m i c a l e q u i l i b r i a one wou ld need t o generate d e r i v a t i v e s of

TP models. In t h i s s e c t i o n , we propose t h a t a n a l y t i c d e r i v a t i v e s

o f such mode ls be used. As a r e p r e s e n t a t i v e TD p rope r t y we chose

the computat ion of K - v a l u e . The the rm odynam ic p r o p e r t y m o d e l s

r e q u i r e d f o r c a l c u l a t i n g K - v a l u e s depend on whether an a c t i v i t y

c o e f f i c i e n t mode l i s u sed f o r t h e l i q u i d - p h a s e , o r w h e t h e r an

equa t ion of s t a t e i s used f o r both f l u i d phases.

When a c t i v i t y c o e f f i c i e n t models are used, n o n i d e a l i t y i n

t h e v a p o u r phase i s u s u a l l y o f minor impor tance , and vapour phase

thermodynamics i s u s u a l l y d e s c r i b e d u s ing the i d e a l gas law or the

v i r i a l model ( P r a u s n i t z e t a l , 1980) . N o n id e a l i t y from the l i q u i d

phase i s the predominant one. A c t i v i t y c o e f f i c i e n t m o d e l s t h a t

have fo u n d w idesp read use i n TP models i n c l u d e s t he NRTL-equat ion

(Renon and P r a u s n i t z , 1 9 6 8 ) , t h e U N IQ U A C -e q u a t io n (Abrams and

P r a u s n i t z , 1 9 7 5 ) , and t h e U N IFA C g ro u p c o n t r i b u t i o n model

( F r e d e n s l u n d e t a l , 1 9 7 5 , 1 9 7 7 ) . The p a r t i a l d e r i v a t i v e s o f

i n t e r e s t a r e t h e c o m p o s i t i o n and temperatu re d e r i v a t i v e s of the

a c t i v i t y c o e f f i c i e n t mode ls .

When n o n i d e a l i t y i n bo th pha se s i s o f im p o r t a n c e TD

p r o p e r t i e s f o r bo th p h a se s a re o f t e n o b t a i n e d f r o m t h e same

equa t ion o f s t a t e (£ > & ‘in e q u a t i o n 2 . 5 ) . Two parameter equa t ion s of

75

s t a t e L i k e SRK e q u a t i o n ( S o a v e , 1972) and t h e Peng R o b in s o n

e q u a t i o n (Peng and R o b i n s o n , 1976) have fo u n d w id e s p r e a d use

e s p e c i a l l y i n the hyd roca rbon p r o c e s s in g i n d u s t r y - Newer and much

more comp l i ca ted models l i k e the 6C-E0S (Skj o l d - J o r g e n s e n , 1984)

and t h e Random/Non-Random e q u a t i o n o f s t a t e (M o l l e r u p , 1985) have

r e c e n t l y been p r o p o s e d - Fo r e q u a t i o n s o f s t a t e t h e p a r t i a l

d e r i v a t i v e s o f component f u g a c i t y c o e f f i c i e n t s w i t h re spec t t o

compos i t i on , temperature and sometimes p ressu re a re re qu i re d .

We show h e re t h a t i t i s p o s s i b l e t o d e r i v e a n a l y t i c a l

d e r i v a t i v e s of TD p r o p e r t i e s o f i n t e r e s t a t the f l owshee t and u n i t

o p e r a t i o n l e v e l s a t a r e a sonab le cos t i f c a r e f u l a t t e n t i o n i s pa id

t o the u n d e r l y i n g TD s t r u c t u r e and im p l e m e n t a t i o n d e t a i l s . The

a n a l y t i c a l d e r i v a t i v e s w i l t be compared w i t h r e s u l t s ob ta in ed by

o t h e r workers and those gene ra ted by numer ica l p e r t u r b a t i o n .

L e t us chose t h e Soave m o d i f i c a t i o n o f the Redl ich -Kwong

e q u a t i o n t o compute f u g a c i t y c o e f f i c i e n t s f o r v a p o u r - l i q u i d

e q u i l i b r i a . 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 were ob ta in ed i n

tw o w a y s : by a n a l y t i c d i f f e r e n t i a t i o n (A ) a n d n u m e r i c a l

p e r t u r b a t i o n ( P ) . E x p r e s s i o n s f o r t h e 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 were d e r i v e d and a r e g i v e n i n T a b l e C1.1 (A p p e n d i x

C 1 ) . A code o f t h e r o u t i n e i s a v a i l a b l e i n m i c r o f i c h e (see the

b a c k c o v e r o f t h i s t h e s i s ) . C a re was t a k e n t o s k i p r e p e a t e d

c a l c u l a t i o n s w heneve r p o s s i b l e when n u m e r i c a l d e r i v a t i v e s are

e v a l u a t e d , f o r e x a m p l e , t h e e v a l u a t i o n o f t e m p e r a t u r e o n l y

dependen t p a r a m e t e r s . S in ce comput ing t im es depend on ly on the

76

number o f components, the r e l a t i v e t imes f o r these d e r i v a t i v e s by

methods A and P a re p r e s e n t e d i n T a b l e 3.1 f o r f o u r d i f f e r e n t

m i x t u r e s c o n t a i n i n g 5 - , 6 - , 8 - , and 16 - components r e po r t e d i n

A p p e n d i x C1 ( T a b l e C 1 - 2 ) . A l s o i n T a b l e 3 .1 we p r e s e n t t h e

e q u i v a l e n t number o f f u g a c i t y c a l l s r e qu i r e d t o eva lua te a f u l l

se t o f NC + 2 d e r i v a t i v e s n u m e r i c a l l y . For e x a m p le , f o r m i x t u r e

C1 .4 ( 1 6 - c o m p o n e n t s ) c a l c u l a t i n g t h e 18 f u g a c i t y c o e f f i c i e n t s

d e r i v a t i v e s (16 -vapour c o m p o s i t i o n s , t e m p e r a t u r e , and p r e s s u r e )

a n a l y t i c a l l y and by p e r t u r b a t i o n i s e q u i v a l e n t t o 2 .4 and 8 .3

f u g a c i t y c o e f f i c i e n t p o i n t e v a l u a t i o n s r e s p e c t i v e l y . The r e s u l t s

i n T a b l e 3 .1 i n d i c a t e t h a t t h e t im e f o r a n a l y t i c e v a l u a t i o n of

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 o n l y 2 -2 .4 f u g a c i t y c o e f f i c i e n t p o in t

v a l u e d e t e r m i n a t i o n s . In a d d i t i o n 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 by method A i s o n l y abou t one__ th i rd o f t h e t im e

r e q u i r e d f o r numer ica l e v a l u a t i o n and t h e r a t i o can be taken t o be

independent o f the number of components i n the m ix tu re . We shou ld

s t r e s s h e r e t h a t a v e r y e f f i c i e n t c o d i n g o f t h e a n a l y t i c

d e r i v a t i v e s was used t o o b t a i n the se r e s u l t s .

Tab le 3.1 a l s o shows r e s u l t s f o r e v a l u a t i o n of compos i t ion

and tempe ra tu re d e r i v a t i v e s o f a c t i v i t y c o e f f i c i e n t s f rom the

UNIQUAC model f o r 3 - and 6 - component m ix tu re s (see Appendix C1,

T a b l e C 1 .2 f o r p r o b l e m s s p e c i f i c a t i o n s ) . A g a i n t h e c o s t o f

e v a l u a t i n g a n a l y t i c d e r i v a t i v e s i s modest.

Our r e s u l t s compare f a v o u r a b l y w i t h t h o s e o b t a i n e d by

M i c h e l s e n and M o l l e r u p ( 1 9 8 5 ) ) . A c c o r d i n g t o t h e s e au tho r s ,

TABLE 3.1. 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

C1 .1 C1 .2Test problem

C1.3 C1.4 C1 .5 C1 .6

Fugacity/Activity Coefficient 1 .0* 1 .2 1 .6 3.9 1 . 0** 2.5

Fugacity/Activity Coefficientderivatives

Analytic (A) 2.0 (2.0) 2.5 (2.1) 3.4 (2.1) 9.2 (2.4) 0-7 (0.7) 3.6 (1.4)

Perturbation (P) 5.7 (5.7) 7.1 (5.9) 10.5 (6.6) 32.2 (8.3) 3.0 (0-3) 10.6 (4.2)

Ratio (A/P) 0.35 0.352 0.323 0.286 0.23 0.34

Thermodynamic Model SRK SRK SRK SRK UNIQUAC UNIQUAC

* Actual time for 100 points = 0.060 CPU seconds ** Actual time for 100 points = 0.046 CPU seconds

Note : Results for C1.6 are relative to C1.5.

78

the cost of analytical derivatives is not much more than point

value evaluations even when more complicated TD models such as

UNIFAC are used. They suggest a cost factor of 1.5 - 2.5

depending on the number of components in the mixture. PPDS is

currently attempting to include analytic derivatives of some TD

models in their package based on the results of our study.

An additional benefit of having analytic derivatives of some

properties (e.g. activity or fugacity coefficients) is that some

other TD properties can easily be calculated from them. For

instance, excess molar enthalpy and entropy can be evaluated

essentially free from the temperature derivative of fugacity or

activity coefficients (see Appendix Cl, Table Cl.l).

We have established above that analytic derivatives of TD

models in process design computations can be derived at a modest

cost. The effort and time it takes to propose, test, and

efficiently incorporate local TD property approximation functions

in simulators is certainly much more than what is required for

analytic differentiation of the rigorous models. Furthermore,

there is no need to derive analytic partial derivatives of models

manually since a number of available algebraic manipuation codes

(ALTRAN (1977), REDUCE (1984), Pantelides (1987)) can be used to

automatically generate the analytic expressions

79

3 -3 . Corapu t a t i o n_ o f _ E x a c t _ D e r i v a t i v e s_ o f_ J y p i c a I V L E_ P ro c e d u re s

In t h i s s e c t i o n we want t o a s s e s s t h e e f f i c i e n c y o f

e v a l u a t i n g procedure d e r i v a t i v e s a c c o rd in g t o the method p roposed

i n s e c t i o n 3 -1 . We c o n s id e r t y p i c a l p rocedures f r e q u e n t l y used i n

v a p o u r - l i q u i d e q u i l i b r i u m c o m p u t a t i o n s and m easu re t h e t im e

r e q u i r e d t o g e n e r a t e p r o c e d u r e d e r i v a t i v e s r e l a t i v e t o the t ime

f o r a s i n g l e p rocedure e v a l u a t i o n -

3 . 3 . 1 . N j^ e r i c a l_ E x p e rim e n t s

I so therma l F lash

The f i r s t p r o c e d u r e we s t u d i e d i s t h e i s o t h e rm a l f l a s h

u n i t o p e r a t i o n . The i s o t h e r m a l f l a s h u n i t i s one o f t h e mos t

w i d e l y used m o d u le s i n p r o c e s s c a l c u l a t i o n s . S i t u a t i o n s a r i s e

q u i t e f r e q u e n t l y i n chemica l p ro ce s s des ign or o p e r a t i o n tha t c a l l

f o r t h e d e t e r m i n a t i o n o f t h e e q u i l i b r i u m d i s t r i b u t i o n o f t h e

components o f a m u l t i c o m p o n e n t m i x t u r e be tween a v a p o u r and a

l i q u i d phase a t cons tan t t em pe ra tu re and p re s su re c o n d i t i o n s .

80

A s i n g l e - s t a g e f l a s h module i s shown below:

F ig . 3 . 1 . S i n g l e - s t a g e f l a s h u n i t

81

The d e s c r i b i n g e q u a t i o n s f o r the f l a s h p rocedure a re as

f o l l o w s : -

M a t e r i a l ba lances

FX-j + FY-j = FZ-j i = 1 , 2 , ......... NC (3 .6)

Phase e q u i l i b r i u m r e l a t i o n s

Ki (

FXi FYi ---- / ------r f /■ n ) FX-j FV = FY-j FL

FL F Vi = 1 , 2 , ____NC

where FL = EFX j , FV = ZFYj andj j

K-j i s K-va lue p r o p e r t y from a p h y s i c a l p r o p e r t i e s package.

The e q u a t i o n s f o r t h e f l a s h u n i t a r e s im p l e a l t h o u g h q u i t e a

number of a l g o r i t h m s have been p r o p o s e d f o r t h e i r s o l u t i o n .

Su c ce s s i v e s u b s t i t u t i o n , G i b b ' s f r e e energy m in im i z a t i o n , s teepes t

descent , a c c e l e r a t e d s u c c e s s i v e s u b s t i t u t i o n ( u s i n g W e g s t e i n or

dom inan t e i g e n v a l u e m e t h o d s ) , N e w t o n ' s and Hyb r id methods have

b een p r o p o s e d among o t h e r s . I n t h i s w o r k we c h o o s e a

Ra ch f o r d - R i ce t y p e a l g o r i t h m w h ic h com b ine s i d e a s of Rohl and

S u d a l l (1 967) and Ohanomah an d Th om p so n ( 1 9 8 4 ) w i t h m i n o r

m o d i f i c a t i o n s . An i n i t i a l e s t im a te of vapour f r a c t i o n i s ob ta ined

u s ing r e g u l a - f a l s i i n t e r p o l a t i o n ba sed on K - v a l u e s computed by

e q u a t i n g mole f r a c t i o n s o f t h e v a p o u r and l i q u i d phases t o the

f e e d . We fo u n d t h e method t o be s a t i s f a c t o r y i n most c a s e s .

However when i t f a i l e d (K-j = 1) we used i d e a l K - va lue s ob ta ined by

82

u s i n g A n t o i n e v a p o u r p r e s s u r e c o r r e l a t i o n . The a l g o r i t h m i s

i t e r a t i v e but the i t e r a t i o n l o op i s v e r y s imp le and j u s t K - v a l u e s

are re qu i re d not t h e i r d e r i v a t i v e s .

We c l a s s i f y the v a r i a b l e s as f o l l o w s :

I n t e r n a l (v) - n i l

Output (w) - FX, FY

Input (u) - T, IT , FZ

The e x a c t 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 (m o la r

component f l o w r a t e s o f vapour and l i q u i d phases) w i t h r e s p e c t t o

t h e NC + 2 i n p u t s ( f e e d component f l o w r a t e s , t e m p e ra tu re , and

p ressu re ) can be c a l c u l a t e d by s t r a i g h t f o r w a r d a p p l i c a t i o n o f

e q u a t i o n ( 3 . 5 ) . Howeve r , an a l t e r n a t i v e and more e f f i c i e n t

f o rm u la t i o n i s used from c l o s e r e x a m i n a t i o n o f t h e fo rm o f the

f l a s h model.

Le t us s p l i t w i n t o two p a r t s w1, w11 where

w1 = FX, and w11 = FY

Equa t ion s (3 .6) and (3 .7 ) c o u ld be w r i t t e n as

(3.8)w l l = (w l , u)

f (w1, wn , u) = o (3.9)

83

U s i n g T a y l o r s e r i e s e x p a n s i o n and n e g l e c t i n g h ighe r o rde r terms

(under the usua l assumpt ions o f 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 )

equa t ion s (3 .8 ) and (3 .9 ) become

3f 3 f 3fdw * + dw11 +

3w^ 3w^ 3u

3 w1* 3W11dw** = ____ dw* + ____ du

3w 3uS u b s t i t u t i n g (3 .11) i n t o (3 .10) we have

~3 f 3f a w ^ ' 3W1 3 f gf 3 w11+ = - __+ ___ . ____

_ 3w^ 3w^ " ’ 3w^ 3 u 3 u 3w^ a u _

3w*Q . ___ = - S

3u

(3 .10)

(3.11)

(3 .12)

(3.13)

E q u a t i o n ( 3 . 1 3 ) e n a b l e us t o c a l c u l a t e the s e n s i t i v i t i e s o f the

l i q u i d phase f l o w s (FX-j) w i t h re spec t t o a l l the in pu t v a r i a b l e s .

The re fo re we need to s o l v e o n l y an set ° f l i n e a r equa t ion s w i th

NC + 2 r i g h t - h a n d s i d e s . The v a p o u r p h a s e f l o w s (FY - j )

s e n s i t i v i t i e s are ob ta in ed from equa t ion (3 .6 ) by c h a i n - r u l i n g as

f o l l o w s :

3FYi 3 FX-j____ = ” ___3T 3T~

3 FY-j 3FX-j— —

3 n 3n

3 FY-j

r Fq_ <5 .. 3 pXi - i J " ____

9 FZj

where0 i I i

« i j = 1 ’ = j

The p a r t i a l d e r i v a t i v e s i n e q u a t i o n (13.3) a re g iv en be low:-

E lements_gf_matr jx_Q

8K-Jq i j = FV( _ FXi + 6 i i K i ) - FYi

3 FXj

+ ( ____ FX-i FV + Ki FXi “ S ' i i FL) ( - S j j )8 FY j

Element s_o f_mat r i x_S

3Ki51.1 = FXi FV__

8 T

3Ki51 .2 = FXi F V ___

an8 Ki

Si 2 +j = ( FXi FV + Ki FX i - 6i i FL) ( 6 -j.j)' 8 FY j

K -va lue p ro p e r t y d e r i v a t i v e s from the lower l e v e l 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 are needed f o r the g e n e r a t i o n o f m a t r i c e s Q

and S. Note m a t r i c e s Q and S a r e s t r i c t l y no t t h e same as i n

equa t ion (3 .5 ) s in c e o n l y a s u b s e t o f the o u t p u t v a r i a b l e s a re

i n v o l v e d i n t h e l i n e a r e q u a t i o n s o l v i n g p a r t of the a lg o r i t h m .

The re qu i re d K - v a lu e d e r i v a t i v e s were ob ta in ed e i t h e r a n a l y t i c a l l y

or by p e r t u r b a t i o n o f the a c t i v i t y / f u g a c i t y c o e f f i c i e n t s models.

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 a s h t e s t p r o b le m s a re g i v e n

i n A p p e n d ix C2 ( s e e T a b l e C 2 . 3 ) . The f e e d s f o r t e s t problems

( C 2 . 1 - C 2 . 5 ) c o n s i s t s o f C j - hydrocarbon m ix tu re s c o n ta in in g

some n i t r o g e n , carbon d i o x i d e and hydrogen s u lp h i d e . M i x t u r e s o f

84

85

t h i s s o r t a r e commonly e n c o u n t e r e d i n r e s e r v o i r o i l s y s t e m s .

Aqueous-organ ic m i x t u r e s make up problems C2.6 - C 2 .7 . T y p i c a l

compu te r t im e s f o r an i s o t h e r m a l f l a s h s o l u t i o n and d e r i v a t i v e

g ene ra t ion are p re sen ted i n F i g u r e s 3 . 2 a , b and 3 . 2 c , d . The c o s t

o f g e n e r a t i n g f l a s h p rocedu re d e r i v a t i v e s f o r systems c o n t a in i n g

v a r i o u s number of components a re p resen ted i n Tab le 3 .2 .

D i s t i l l a t i o n Column

The second VLE p rocedu re we s t u d i e d i s a distillation column

m odu le . A g e n e r a l mode l i s p r e s e n t e d i n A p p e n d i x C2 . 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 i s t h e s t a n d a rd Naph ta l i -Sandho lm

techn ique as implemented i n the book by F r e d e n s l u n d e t a l , 1977 .

The a l g o r i t h m i s Newton b a sed and i s t h e r e f o r e a s imu l taneous

s o l u t i o n of the mass and energy b a la n ce s and e q u i l i b r i u m r e l a t i o n s

f o r t h e unknown v a r i a b l e s (vapour and l i q u i d component f l o w s per

s tage , temperature p r o f i l e , r e b o i l e r and conden se r h e a t d u t i e s ) .

For t h e p rob lem t o be w e l l po sed a l l f e e d component f l o w r a t e s ,

f eed temperatures and p r e s s u r e s , r e f l u x r a t e , d i s t i l l a t e r a t e , and

s i d e s t r e a m s w i t h d r a w a l r a t e s must be s p e c i f i e d (We cons ide red

these v a r i a b l e s as the i n p u t s ) . O t h e r p o s s i b l e c o m b i n a t i o n s o f

t h e i n p u t and o u t p u t v a r i a b l e s can be used f o r the problem t o be

w e l l d e f in ed . A b l o c k t r i d i a g o n a l l i n e a r sys tem i s s o l v e d per

each column i t e r a t i o n w i t h G au ss i an f a c t o r i z a t i o n performed on the

d iagona l b lo c k s o n l y . M o d i f i c a t i o n s were made t o a l l o w f o r any

s u i t a b l e K - v a l u e s and e n t h a l p y m o d e l s . The t e m p e r a t u r e and

compos i t ion d e r i v a t i v e s of the K - v a l u e s and e n t h a l p y p r o p e r t i e s

86

were e s t i m a t e d e i t h e r a n a l y t i c a l l y (A) o r by f i n i t e d i f f e r e n c e s

CP). Convergence i s assumed whenj F || < /1 .0E -6 . The v a p o u r and

l i q u i d f l o w r a t e s a r e i n i t i a l i s e d by a s sum ing c o n s t a n t m o l a l

o ve r f low per s tage . I n i t i a l guesses of the condenser and r e b o i l e r

t e m p e r a t u r e s a r e s e t t o the dew and bubble p o in t t empera tu res of

the feed m ix t u re r e s p e c t i v e l y . A l i n e a r t e m p e r a t u r e p r o f i l e i s

then assumed between the r e b o i l e r and condenser tempera tu res .

S i n c e N e w t o n ' s method i s used t o s o l v e the 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 t h e co lumn, m a t r i c e s Q and R a re r e a d i l y

a v a i l a b l e a t the s o l u t i o n o f the column. M a t r i x S i s ea sy t o s e t

up ( A p p e n d i x C2) and does not r e q u i r e TD p r o p e r t i e s p o in t v a l u e s

or t h e i r d e r i v a t i v e s . N o n - z e r o e l e m e n t s o f t h e s e m a t r i c e s a r e

shown i n T a b l e C 2 .1 ( A p p e n d i x C2) . I n o r d e r t o i n c r e a s e

e f f i c i e n c y the column p rocedure d e r i v a t i v e s were computed on l y f o r

i n p u t v a r i a b l e s s p e c i f i c a l l y i n d i c a t e d a s unknowns by t he

f lowshee t . The same b lo c k t r i d i a g o n a l code which i s used to s o l v e

f o r a c o r r e c t i o n s t e p i n e a ch Newton i t e r a t i o n o f the column i s

a l s o used t o s o l v e t h e l i n e a r s y s t em ( e q u a t i o n 3 . 5 ) f o r t h e

d e s i r e d column procedure d e r i v a t i v e s . The a lg o r i t hm f o r a Newton

s tep c o r r e c t i o n i s p ro v id ed on page 22T (Appendix C2).

The t e s t p rob lem c o n s i s t s of 7 t h e o r e t i c a l p l a t e s w i t h a%

two-phase feed stream (80 % vapour) of e thane, propane, n - b u t a n e ,

n-pentane and n-hexane in t r o d u c e d on the f o u r t h s tage . The column

i s opera ted a t a cons tan t p re s su re o f 5 ba rs .

87

C o m p u t e r t i m e s f o r a t y p i c a l s i n g l e co lumn s o l u t i o n

i n v o l v i n g t h e c o m p u t a t i o n o f t h e co lumn o u t p u t v a r i a b l e s w i t h

r e s p e c t t o o n l y two o f t h e i r i n p u t v a r i a b l e s ( r e f l u x r a t i o and

d i s t i l l a t e r a t e ) are shown in T a b le 3 . 3 . The computed a n a l y t i c

d e r i v a t i v e s o f o n l y t h e bo t tom and d i s t i l l a t e p r o d u c t s w i t h

re spec t t o t h e s e two i n p u t v a r i a b l e s a r e g i v e n i n A p p e n d i x C2

(Tab le C2 .5 ) .

3 . 3 . 2 . ^ is c u s s ig n _ o f_ R e s u lt s _ a n d _ C o n c lu s io n s

F i g u r e s 3 . 2 a , b s h o w s t h e c o m p l e t e b reakdown o f t h e

com p u t in g t im e s f o r a ba se p o i n t s o l u t i o n o f t h e f l a s h mode l

( e x a m p le C 2 .1 ) a s w e l l as e v a l u a t i o n o f i t s o u t p u t - i n p u t p a r t i a l

d e r i v a t i v e s . (Note a ,b r e f e r s t o K - v a lu e d e r i v a t i v e s genera ted by

m e t h o d s A ( a n a l y t i c ) a n d P ( n u m e r i c a l ) , r e s p e c t i v e l y ) .

C onve rgen ce o f t h e ba se p o i n t i s q u i t e f a s t r e q u i r i n g o n l y 8

i t e r a t i o n s and 9 c a l l s f o r K -va lue p o in t v a l u e s (see column 2 o f

Tab le 3 . 2 ) . From f i g u r e s 3 .2 a ,b we note t ha t about s e v e n t y - f i v e

p e r c e n t o f t h e base p o i n t c o m p u t in g t im e i s s p e n t e v a l u a t i n g

K - v a l u e s . The cos t of comput ing a n a l y t i c p rocedure d e r i v a t i v e s i s

r o u g h l y t h e same as a ba se p o i n t s o l u t i o n . U s i n g n u m e r i c a l

d e r i v a t i v e s , the cos t o f o u t p u t - i n p u t 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

1 .5 t im e s t h a t o f a base p o i n t s o l u t i o n . Roughly 47 % - 70 % of

the t o t a l t ime devoted t o the g en e ra t io n o f p rocedure d e r i v a t i v e s

i s s p e n t i n K - v a l u e d e r i v a t i v e s when t h e s e a r e o b t a i n e d

a n a l y t i c a l l y and by p e r t u r b a t i o n r e s p e c t i v e l y . The l i n e a r a lgeb ra

i n v o l v e d ( t h a t i s , s o l u t i o n o f t h e l i n e a r s y s tem o f NC = 5

k

Figure 3.2: Breakdown of computing time for generation of flashprocedure derivatives

P.D

Legend

B.P - Base Point

P.D - Procedure Derivatives

K-values + derivatives

ti

17

^ Linear Equation Solver

Arithmetic operationsA

' B.P

No. of components - 5 No. of components - 5

K-values

Generation of Q.R.S. and arithmetic operations

wrdao

(c) K-value derivatives - ATD model - UNIQUACNo. of components - 6

TD model - UNIQUACNo. of components - 6

T a b le 3 . 2 . E g u iy a le n t_ n u m b e r_ p f_ is p t h e n n a l_ f la s h _ e y a lu a t io n s _ r e g u ir e d _ t o

S § n e r a te _ p u t p u t_ s e n s it iy j t ie s _ w it h _ r e s p e c t_ t p _ a n _ N C _ + _ 2 _ in p u t_ y a r ia b L e s _

Test ProblemC2.1 C2.2 C2.3 C2.4 C2.5 C2.6 C 2.7

Exact(A) 0 .9 0 .7 0 .9 4 .8 0 . 1 0 0 . 2 0 1.4

(2 ) (2 ) ( 2 ) (2 ) (2 ) ( 1 ) ( 1 )(B) 1.5 1 . 0 1 . 1 8.5 0 . 2 0 0.25 2 . 0

(7) ( 8 ) (1 0 ) (8 ) (7) (3) (4)

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

o p e r a t i o n models i n g e n e ra l .

3 .4 . APPIC AJJO_N _OF_NEW_ _TD I NJERJACE JS'TR ATEGY FLOW SHEETING

EXAMPLES

In t h i s s e c t i o n we u t i l i z e p rocedure d e r i v a t i v e s developed

i n the p rev iou s s e c t i o n s f o r the s o l u t i o n o f s e v e r a l s i m u l a t i o n ,

d e s i g n , and o p t i m i z a t i o n prob lems. Four f l o w sh e e t s a re mode l led

i n v o l v i n g t h e i s o t h e r m a l f l a s h and d i s t i l l a t i o n c o lu m n u n i t

m odu le s . The f l o w s h e e t i n g p a ckage SPEEDUP i s used i n a l l the

e x p e r im e n t s ( e x c e p t the o p t i m i z a t i o n t e s t p r o b l e m ) a n d i s

t h e r e f o r e b r i e f l y d e s c r i b e d i n t he f o l l o w i n g .

3-4 .1 . Th e_ S PE EDU P_ Package

S P E E D U P a l l o w s p r o b l e m s t o be e n t e r e d i n an

e n g in e e r i n g - o r i e n t e d language. A t r a n s l a t o r p rog ram ( w r i t t e n i n

PASCAL) c o n v e r t s t h e i n p u t language i n t o FORTRAN programs which

a re t h e n e x e c u t e d t o o b t a i n a s o l u t i o n . The EO package a l s o

95

f e a t u r e s an i n t e r a c t i v e i n t e r f a c e making i t easy f o r the user to

r u n , change , and r e s t a r t a p r o b l e m . The sys tem can r e t r i e v e

i n f o r m a t i o n f rom a d a t a b a s e w h i c h i s t h e c o r e o f t h e SPEEDUP

package where a l l t h e i n f o r m a t i o n p e r t a i n i n g t o a p rob lem are

s t o r e d . The d a t a s t o r e d i n t h e database i s updated a f t e r every

change i s made and, t hu s , the p rob lem can be r e s t a r t e d from any

p o in t w i th a minimum amount of r e p ro c e s s in g .

The i n p u t language c o m p r i s e s seven s e c t i o n s , namely

FLOWSHEET, MODEL, UNIT, OPERATION, TITLE, DECLARE and OPTIONS. A

d e t a i l e d d e s c r i p t i o n o f each o f these s e c t i o n s i s p r o v id e d i n the

SPEEDUP U se r Manual (1986 ) . Three f e a t u r e s o f the in pu t language

and t h e i r i n f l u e n c e on t h e s i m u l a t i o n phase r e q u i r e f u r t h e r

c l a r i f i c a t i o n : v a r i a b l e type d e c l a r a t i o n i n the DECLARE s e c t i o n ,

p r o c e d u r e s p e c i f i c a t i o n i n t h e MODEL s e c t i o n , a n d f i n a l l y

numer ica l s o l u t i o n a l g o r i t h m s i n t h e OPTIONS s e c t i o n .

3 .4 .1 -1 - V a r ia b le jy p e s

W i t h i n t h e DECLARE s e c t i o n g l o b a l v a r i a b l e t y p e s a re

d e c l a r e d . V a r i a b l e s t h a t a p p e a r i n t h e MODEL s e c t i o n s a r e

a s s i g n e d a s one t y p e o f v a r i a b l e or another . The u s e fu ln e s s of

t h i s f a c i l i t y i s t h a t i t g i v e s t o e a ch v a r i a b l e t y p e a d e f a u l t

i n i t i a l guess , a lower bound, and an upper bound. The form of the

type d e c l a r a t i o n i s :

TYPE

PRESSURE = 10:1 :100 UNIT = "BAR"

96

where 1 0 i s t h e d e f a u l t i n i t i a l guess , and 1 , 100 a re lower and

upper bounds f o r v a r i a b l e t y p e PRESSURE, r e s p e c t i v e l y - (These

d e f a u l t s can be o v e r w r i t t e n , i f s p e c i f i e d d i f f e r e n t l y i n t h e

OPERATION s e c t i o n ) .

3 .4 .1 -2 . P rocedures

When a p ro ced u re i s used i n the s im u l a t i o n a r e f e r en ce to

a Fo r t r an r o u t i n e i s i n c l u d e d i n t h e MODEL s e c t i o n - P r o c e d u r e s

a re d e c l a r e d i n the MODEL s e c t i o n i n the f o l l o w i n g manner:

(ou tpu t v a r i a b l e s l i s t ) NAME OF PROCEDURE ( i n p u t v a r i a b l e s l i s t ) .

I t i s assumed t h a t each o u t p u t v a r i a b l e i s dependent on a l l the

in p u t v a r i a b l e s .

3 . 4 . 1 . 3 . Numer ica l_ S o lu t i o n _ O p t i o n s

SPEEDUP i s c apab le o f s o l v i n g s e t s o f e q u a t i o n s , equa t ion s

and p r o ce d u r e s , o r p r o c e d u r e s s i m u l t a n e o u s l y . Four d i f f e r e n t

s o l u t i o n a lg o r i t h m s (Newton, H y b r i d , Schube r t , and QN methods) are

a v a i l a b l e i n the s im u l a t o r . P a n t e l i d e s (1987) p ro v id e s a d e t a i l e d

d e s c r i p t i o n of the s o l u t i o n methods and s c a l i n g o p t i o n s a v a i l a b l e

i n SPEEDUP. QN i s P a l o s c h i ' s (1982) im p le m e n t a t i o n o f B r o y d e n ' s

method . Here we d e s c r i b e on l y Newton's and the Hyb r id methods which

are used i n our s tudy . We choo se t h e s e two methods b e cau se of

c o n c l u s i o n s by s e v e r a l w o r k e r s , e . g . P a n t e l i d e s , t h a t QN and

97

S c h u b e r t ' s me thods a r e Less r e l i a b l e and e f f i c i e n t i n compar ison

t o the fo rmer methods.

S p a r s e m a t r i x t e c h n i q u e s a r e used t o s t o re the J a cob ia n

m a t r i x and perform the l i n e a r a lg eb r a . E q u a t i o n s p r e s e n t a t the

f l ow shee t l e v e l a re d i f f e r e n t i a t e d s y m b o l i c a l l y and exac t a n a ly t i c

d e r i v a t i v e s o b t a in e d . As t o the e q u a t i o n s i n v o l v i n g p r o c e d u r e s

t h e 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 w i t h r e s p e c t t o t h e i r

unknown i n p u t s are o b t a i n e d i n two ways depending on the s o l u t i o n

method. Newton and H y b r i d methods d i f f e r o n l y i n the way they

o b t a i n and update the J a c o b i a n app rox im a t ion du r ing the c o u r s e o f

the i t e r a t i v e c a l c u l a t i o n s .

N e w to n ' s me thod e m p lo y s a n a l y t i c d e r i v a t i v e s genera ted

a u t o m a t i c a l l y complemented by f i n i t e d i f f e r e n c e s , i f necessary* .

When f i n i t e d i f f e r e n c e s a r e u s e d , t h e C u r t i s e t a l . (1974)

a lg o r i t hm i s used t o m in im ize the number of f u n c t i o n e v a l u a t i o n s .

No te t h e C u r t i s e t a l a l g o r i t h m i s s u i t a b l y m od i f i e d t o take i n t o

a c c o u n t t h e a v a i l a b i l i t y o f some d e r i v a t i v e i n f o r m a t i o n

a n a l y t i c a l l y . Thus t h e p r o c e d u r e d e r i v a t i v e i n f o r m a t i o n i s

ob ta in ed from o u t s i d e the p rocedure . I f procedure d e r i v a t i v e s are

ob ta in ed w i t h i n by the t e chn ique suggested e a r l i e r i n s e c t i o n 3 . 1 ,

they are passed back t o t he s o l v e r and t r e a t e d j u s t l i k e o t h e r

a n a l y t i c a l l y a v a i l a b l e d e r i v a t i v e s . SPEEDUP has r e c e n t l y been

p rov ided w i t h a f a c i l i t y i n the in p u t language which identifies and

s u p p o r t s such p r o c e d u r e s . N e w to n ' s method r e c a l c u l a t e s a l l the

e lements o f the J a c o b ia n a t subsequent i t e r a t i o n s u s i n g the same

98

procedure employed f o r the f i r s t i t e r a t i o n .

The Hybr id method a l s o u se s the s y m b o l i c a l l y g e n e r a t e d

d e r i v a t i v e i n f o r m a t i o n . The p r o c e d u r e d e r i v a t i v e s a r e o n l y

c a l c u l a t e d by f i n i t e d i f f e r e n c e s a t the f i r s t f l owshee t i t e r a t i o n

and when a r e s t a r t i s n e c e s s a r y . A g a in t h e i d e a suggested by

C u r t i s e t . a l . i s u t i l i z e d t o m i n im i z e t h e number o f f u n c t i o n

e v a l u a t i o n s . Fo r s u b s e q u e n t i t e r a t i o n s a l e a s t change secant

fo rm u la i s used t o update the " u n a v a i l a b l e " procedure d e r i v a t i v e s .

3 . 4 . 2 . F lo w s h e e t! ng_Exa5p j.e s

F o u r d i f f e r e n t f l o w s h e e t s have been chosen f rom the

l i t e r a t u r e . The examples were chosen t o i n c l u d e :

- mass ba la n ce s

- energy ba la n ce s

- complex m o le c u l a r TD mode ls,

r i g o r o u s VLE p ro cedu re s .

The e x a m p le s a r e d e s c r i b e d b r i e f l y i n t h e f o l l o w i n g s e c t i o n .

D e t a i l e d s p e c i f i c a t i o n s may be found i n Appendix C3. The problems

a r e by no means l a r g e bu t a r e s u f f i c i e n t l y l a r g e t o t e s t t h e

p r i n c i p l e s of e f f i c i e n t i n c o r p o r a t i o n o f TD p r o p e r t i e s and VLE

p r o c e d u r e s . The SRK ( S o a v e , 1972) e q u a t i o n o f s t a t e i s used to

compute K - v a l u e s i n a l l p r o b l e m s a t t e m p t e d . The f l o w s h e e t

s im u l a t i o n s i s assumed t o have converged when

•99

|| F 1^ < 1 .0E -5

The f o l l o w i n g n u m e r i c a l s o l u t i o n c o m b i n a t i o n s were

cons ide red : S o l u t i o n method a t t h e f l o w s h e e t l e v e l - Newton (N)

and H y b r i d CH); c a l c u l a t i o n o f p rocedure d e r i v a t i v e s a t the un i t

o p e r a t i o n l e v e l - a n a l y t i c (A) and p e r t u r b a t i o n ( P ) ; c a l c u l a t i o n

o f p r o c e d u r e d e r i v a t i v e s a t t h e the rmodynam ic p rope r t y l e v e l -

a n a l y t i c (A) and p e r t u r b a t i o n (P) . Thus, w i t h NAP we i n d i c a t e a

s o l u t i o n u s i n g N e w t o n ' s method a t t h e f l o w s h e e t l e v e l w i t h

a n a l y t i c u n i t o p e r a t i o n procedure d e r i v a t i v e s and t h e r m o p h y s i c a l

p r o p e r t y d e r i v a t i v e s by p e r t u r b a t i o n , e t c . The c r i t e r i a f o r

a s s e s s in g the pe r fo rmance o f t h e d i f f e r e n t s t r a t e g i e s a r e : ( i )

s i m u l a t i o n t im e , ( i i ) number o f c a l l s t o t h e TD r o u t i n e s , and

( i i i ) number o f f l o w sh e e t i t e r a t i o n s and f u n c t i o n e v a l u a t i o n s .

3 . 4 . 2 . 1 . S imp le D i s t i l l a t i o n Column Des ign

T h i s i s t h e s im p l e d e s i g n o f a r i g o r o u s d i s t i l l a t i o n

column and can be argued not t o be a f l o w s h e e t . The " f l o w s h e e t "

i s g i v e n i n F i g u r e 3 . 3 . We p r e f e r t o re ga rd i t as a f l ow shee t

however, s in ce i t i n v o l v e s covergence of d e s i g n v a r i a b l e s a t the

f l owshee t l e v e l .

T h e e x a m p l e i n v o l v e s t w o h y p o t h e t i c a l d e s i g n

s p e c i f i c a t i o n s f o r the column. The d i s t i l l a t i o n column p ro c e d u r e

d e s c r i b e d i n s e c t i o n 3 .3 was used. For problem C3.1a, the p u r i t y

of the l i g h t key component (ethane) i n the d i s t i l l a t e i s s p e c i f i e d

1 G'U

Figure 3.3: Distillation Column

DEST

101

and t h e r e f l u x r a t i o r e q u i r e d t o meet t h e s p e c i f i c a t i o n i n the

d i s t i l l a t e i s t o be c a l c u l a t e d . In the s e cond p ro b lem ( C 3 . 1 b ) ,

t h e mo le f r a c t i o n o f n -Hexane i n t h e bot toms i s a l s o s p e c i f i e d

w i th both r e f l u x r a t i o and d i s t i l l a t e r a t e regarded as unknowns to

be determ ined. The d i s t i l l a t i o n procedure i s t r e a t e d l i k e a b l a c k

box module w i t h d i s t i l l a t e and b o t t o m s r e g a r d e d a s the o u t p u t

v a r i a b l e s a n d t h e f e e d m o l a r c o m p o n e n t f l o w r a t e s , f e e d

tempera tu re , r e f l u x r a t i o , and d i s t i l l a t e r a t e a s i n p u t s . The

d i s t i l l a t i o n r o u t i n e i s i n v o k e d by i n c l u d i n g t h e f o l l o w i n g

re f e r e n c e i n the MODEL d e s c r i p t i o n :

( D i s t i l l a t e , B o t tom ) 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 ed - tem pe ra tu re , f e e d - f l o w r a t e s )

I n i t i a l guesses f o r the f o l l o w i n g v a r i a b l e t ypes were used:

F lowra te = 5 : 0 : 1000 u n i t = ' k m o l e s / h r 1

R e f l u x - r a t i o = 0 .5 : 1 .E -15 : 100 u n i t = ' c o n s t a n t '

D i s t i l l a t e - r a t e = 100 : 10 : 500 u n i t = ' k m o le s /h r '

Temperature = - 0 . 4 0 : -100 : 100 u n i t = , 0 C'

We t e s t e d the f o l l o w i n g comb ina t ions of s o l u t i o n m e thods :

No d e r i v a t i v e s a v a i l a b l e a t any l e v e l s - NPP and HPP; a n a l y t i c a l

d e r i v a t i v e s a v a i l a b l e o n l y from the p h y s i c a l p r o p e r t i e s l e v e l -

NPA and HPA; a n a l y t i c a l d e r i v a t i v e s o f d i s t i l l a t i o n procedure

a v a i l a b l e w i t h n u m e r i c a l d e r i v a t i v e s f rom p h y s i c a l p r o p e r t i e s

102

L e ve l - NAP; and f i n a l l y a n a l y t i c a l d e r i v a t i v e s from a l l l e v e l s -

NAA.

3 . 4 . 2 . 2 . Des j3 n_and_S imu la t2 on_o f_Cave t t_F lowshee t

The se cond f l o w s h e e t i s t h e s t a n d a r d f o u r f l a s h CAVETT

process ( C a ve t t , 1963; F i g u r e 3 . 4 ) . T h i s p r o c e s s has been used

e x t e n s i v e l y i n t h e f l o w s h e e t i n g l i t e r a t u r e to study a number of

i s s u e s such as t e a r i n g , d e c o m p o s i t i o n , e t c - F i v e d i f f e r e n t

v a r i a t i o n s o f t h e p rob lem were a t tempted w i th feed c o n s i s t i n g of

5 - , 6 - , 8 - and 1 6 - c o m p o n e n t m i x t u r e s and c o m p r i s i n g 4

s i m u l a t i o n s a n d a d e s i g n s p e c i f i c a t i o n ( see T a b l e C3 .2 of

A p p e n d i x C3) . The f e e d s tream c o n s i s t s o f hydrocarbons C«j - C-|2 /

carbon d i o x i d e , n i t r o g e n and hydrogen s u l p h i d e . The i n t e g r a t e d

f l a s h e s o p e r a t e d a t w i d e l y d i f f e r e n t c o n d i t i o n s p rov ide a good

t e s t of the performance o f procedure d e r i v a t i v e s gene ra ted by ou r

method i n s t e a d y - s t a t e f l o w s h e e t i n g .

The i s o t h e r m a l f l a s h u n i t s a r e m o d e l l e d w i th the f l a s h

r o u t i n e de s c r ib ed i n s e c t i o n 3 . 3 . The r o u t i n e i s i n c l u d e d i n the

MODEL d e s c r i p t i o n a s :

(Vapour f l o w r a t e s , L i q u i d f l o w r a t e s ) FLASH (Temperature, P re ssu re ,

Feed f l o w r a t e s ) .

The m i x e r s a r e m o d e l l e d by c o n s i d e r i n g on ly the f l ow of

m a t e r i a l (That i s , ene rgy ba la n ce s are not c on s ide red ) . A n a l y t i c

d e r i v a t i v e s o f e q u a t i o n s co r r e spond ing t o the m ixe rs a re r e a d i l y

103

FEED

Figure 3.4: Flowsheet of Cavett Four Flash Process

VAPOUR PRODUCT11

Flash 1

T i - n i

211 MixerO

Flash 3

T3’ n3

6

f

0

7

w

Flash 2

T2* n2

Mixer

Flash 4

T4> "4

LIQUIDPRODUCT

10

104

a v a i l a b l e by s y m b o l i c m a n i p u l a t i o n . A p a r t f rom t h e i n i t i a l

g u e s s e s l i s t e d be low a l l f l o w r a t e s w e r e i n i t i a l i s e d t o t h e

f o l l o w i n g d e f a u l t v a l u e :

F lowra te = 1 : 1 .E -15 : 5000 u n i t = ' km o le s /h r '

Flows i n t o FLASH 1 or FLASH 2 were i n i t i a l i s e d a c c o r d i n g t o the

f o l l o w i n g s t r a t e g y :

Problem FLASH 1 FLASH 2

C3.2 Same as feed d e f a u l t

C3.3 same as feed d e f a u l t



C3.6 d e f a u l t same as feed

The f o l l o w i n g com b ina t io n s o f s o l u t i o n methods were t e s t e d : NP, HP

(no a n a l y t i c a l d e r i v a t i v e s u sed ) ; NAA ( a n a l y t i c a l d e r i v a t i v e s used

a t a l l l e v e l s ) ; NAP ( a n a l y t i c u n i t o p e r a t i o n l e v e l d e r i v a t i v e s and

numer ica l d e r i v a t i v e s of p h y s i c a l p r o p e r t i e s ) .

We a l s o u sed an i s o t h e r m a l f l a s h r o u t i n e a v a i l a b l e f r o m ' f e

PPDS . The same p h y s i c a l p r o p e r t i e s m o d e l ( S R K ) i s u s e d

t h r o u g h o u t . Howeve r , 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 s w i t h i n

PPDS databank are o v e r w r i t t e n by v a lu e s r e p o r t e d i n Ap p en d ix C1.

I t i s n o t p o s s i b l e t o i n p u t n o n - z e r o 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 s f o r a l l p a i r s i n t h e 16 -componen t m i x t u r e as the

d a t a b a n k i s capable of handling only 120 b in a r y i n t e r a c t i o n p a i r s

105

f o r a m i x t u r e . The i n i t i a l e s t im a t e s are the same f o r our f l a s h

code and tha t of PPDS.

3 . 4 . 2 . 3 . De_si_gn._of _ CojjjDljed_Di s_tjj._l_atign_ C^J._umns_wj th__Ejne r gy

Recyc le

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.

BottomsCkmol ./hr)Ethane 5 .0 8 .58 5 .0 9.31Propane 5 .0 127.99 5 .0 128.19Butane 5 .0 96.56 5.0 96.55Pentane 5 .0 67.75 5 .0 67.75Hexane 5 .0 49.13 5.0 49.13

Re f lu x R a t i o 0 .50 0.726 0.50 0.704

D i s t i l l a t e Rate 150* 150.0* 1 0 0 . 0 149.07

* Set Va lue , (kmol /h r )

116

f i r s t c a l l . For s u b s e q u e n t c a l l s t h e i n t e r n a l v a r i a b l e s saved

from the p re v iou s p ro cedu re c a l l are used as the i n i t i a l guess f o r

t h e co lumn c a l c u l a t i o n s - Thus numer i ca l d e r i v a t i v e s o f the u n i t

o p e r a t i o n module are o b t a i n e d i n an e f f i c i e n t manner . The same

d e s i g n i s f o u n d by a l l t h e s o l u t i o n methods and a re as shown i n

Tab le 3 -7 - The r e s u l t s show t h a t the i n i t i a l g u e s s e s a r e f a i r l y

bad . In f a c t , we found t h a t the same performance c h a r a c t e r i s t i c s

i s o b ta in ed even when a l l the unknown f l o w r a t e v a r i a b l e s a r e s e t

t o d e f a u l t v a l u e s o f 1 . E - 1 5 , 1 - 0 , or 5 - 0 .

A n a l y t i c and n u m e r i c a l d e r i v a t i v e s computed a t a g i v e n

p o i n t have e x a c t l y t h e same n um e r i c a l v a l u e s , so s o l u t i o n s w i th

t h e same f l o w s h e e t l e v e l o p t i o n f o l l o w e x a c t l y t h e same

convergence path and t a k e the same number of i t e r a t i o n s . The on ly

f a c t o r s a f f e c t i n g the computer t imes are t h e r e f o r e t h e number o f

e x e c u t i o n s o f the column and p h y s i c a l p rope r t y procedures and the

way procedure d e r i v a t i v e s are computed- I f we compare methods NPP

and HPP we o b s e r v e t h a t t h e s e co nd i s f a s t e r , a l t h o u g h more

f l ow shee t i t e r a t i o n s a r e r e q u i r e d / b e c a u s e t h i s d i s a d v a n t a g e i s

more t h a n c o u n t e r b a l a n c e d by f e w e r e v a l u a t i o n s o f t h e co lumn

p rocedure . These r e s u l t s a r e i n a g reem en t w i t h t h o s e o f o t h e r

w o r k e r s ( P a n t e l i d e s (1987 ) , M a c ch ie t t o (1985) , Westman and L u c i a

(1984 ) , e t c ) . U s in g a n a l y t i c d e r i v a t i v e s of p h y s i c a l p r o p e r t i e s

( K - v a l u e s a n d e n t h a l p i e s ) r a t h e r t h a n n u m e r i c a l d e r i v a t i v e s

(methods NPA and HPA) r e s u l t s i n s a v i n g s o f 25 % - 28 % i n both

p r o b l e m s . The use o f a n a l y t i c d e r i v a t i v e s o f j u s t the column

p rocedure and p e r t u r b a t i o n d e r i v a t i v e s o f p h y s i c a l p r o p e r t i e s

117

(method NAP) a l s o r e s u l t s i n t ime s a v i n g s , even when p e r t u r b a t i o n

o f o n l y one i n p u t v a r i a b l e t o t h e co lumn i s p e r f o r m e d (15 %

s a v i n g s f o r p r o b le m C 3 . 1 a ) . When two column p e r t u r b a t i o n s a re

performed a t each f l o w s h e e t i t e r a t i o n ( p ro b le m C 3 .1 b ) the t im e

s a v i n g s are 38 % compared t o NPP. When a n a l y t i c d e r i v a t i v e s a re

used a t the p h y s i c a l p r o p e r t i e s , u n i t o p e r a t i o n s , and f l o w s h e e t

l e v e l s (method NAA) t he two r e d u c t i o n s a re compounded, produc ing

the best o v e r a l l r e s u l t - T h i s method p e r f o r m s even b e t t e r t h a n

t h e d e r i v a t i v e f r e e H y b r i d method- The o v e r a l l r e d u c t i o n i n

computer t ime f o r NAA i n problems C3-1a ,b i s 39 % and 55 % w i t h

re spec t t o NPP and 33 % and 38 % w i th re spec t t o HPP.

When the s o l u t i o n c o m b in a t i o n s a r e compared i n t e rm s o f

e q u i v a l e n t number o f K - v a lu e and en tha lp y p r o p e r t i e s e v a l u a t i o n s ,

NAA o u t p e r f o r m s t h e o t h e r m e thod s - The o r d e r o f d e c r e a s i n g

e f f i c i e n c y i s a s f o l l o w s : NAA, HPA, NPA, NAP, HPP, NPP- Method

NPP needed a b o u t a f a c t o r o f 3 . 2 9 and 4 -85 more TD p r o p e r t i e s

e v a l u a t i o n s f o r p r o b l e m s C 3 . 1 a , b r e s p e c t i v e l y compared t o NAA.

The d e r i v a t i v e f r e e H yb r id method a l s o re q u i r e d more e v a l u a t i o n s

o f t h e p h y s i c a l p r o p e r t i e s mode ls than NAA- When on ly a n a l y t i c

d e r i v a t i v e s of TD p r o p e r t i e s are u s e d as i n methods NPA and HPA,

s u b s t a n t i a l s a v i n g s a r e made w i t h r e s p e c t t o u s i n g n u m e r i c a l

d e r i v a t i v e s of such models (NPP, HPP).

118

Cave t t problems (Prob lems C3.2 - C3.6)

The r e s u l t s f o r problems C3 .2 , C3 .3 , C3.4 , C3 .5 , and C3 .6

a r e p r e s e n t e d i n T a b l e s 3 .8 - 3 . 1 1 . From the s im u la t i o n r e s u l t s

shown i n Tab le 3 .8 we note t h a t v e r y poo r i n i t i a l e s t i m a t e s f o r

a lm o s t a l l t h e p r o c e s s s t r e a m v a r i a b l e s were assumed. Thus the

f l a s h module was execu ted f o r a w ide range of in p u t c o n d i t i o n s *

When 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 a r e c o m p u t e d by

p e r t u r b a t i o n t he H y b r i d method ( H P ) , as one wou ld e x p e c t , i s

a lw a y s f a s t e r than Newton 's method o p t i o n NP. I t a l s o uses fewer

a c c e s s e s t o the TP p a c k a g e a l t h o u g h i t u s u a l l y t a k e s more

i t e r a t i o n s t o converge. When e xa c t p rocedure d e r i v a t i v e s a re used

at a l l l e v e l s (method NAA) computer t imes a re up t o 50 % s m a l l e r

t h a n t h o s e w i t h NP. Even when d e r i v a t i v e s o f the rm odynam ic

q u a n t i t i e s a re no t a v a i l a b l e and a r e o b t a i n e d by p e r t u r b a t i o n

(method NAP) the comput ing t im e s a re a lmost as good as method NAA.

For a l l the Newton method o p t i o n s (NPP , NAA, NAP) t h e f l o w s h e e t

c o n ve r g e d a f t e r t h e same number of i t e r a t i o n s (Tab le 3 .10) which

con f i rms t h a t a c c u ra t e p ro cedu re d e r i v a t i v e s a r e computed by our

method. That p rocedure d e r i v a t i v e s genera ted by the Hybr id method >

a re o n l y a p p r o x i m a t i o n s g o o d i n t h e Q u a s i - N e w t o n s e n s e i s

r e f l e c t e d i n the l a r g e r number of i t e r a t i o n s t o converge. Of a l l

the methods t e s t e d NAA and NAP c o n ve rg e d i n s i g n i f i c a n t l y f ew e r

f u n c t i o n e v a l u a t i o n s than t h e o the r methods s i n c e p e r t u r b a t i o n of

the f l a s h procedure i s a vo id ed .

TABLE 3.8. Simulation Results for Cavett Problems

ProblemTop Product, ( H o w rates - k m o 1 /hr) Bottom Product (Flowrates - k m o1 /hr)

C3.2 C3.3 C3.4 C3.5 C3.6 C3.2 C3.3 C3.4 C3.5 C3.6

ComponentsNitrogen 451.976 358.185 24.233E-3 14.986E-3Carbon Dioxide 511 .916 511 .916 2253.521 6638.451 4930.073 0.175 0.175 3.479 59.549 35.527Hydrogen Sulphide ” 295.679 43.721Methane 1361.712 3776.971 2992.297 0.288 5.029 3.203Ethane 361.475 361.475 2776.021 2226.392 1 .666 1 .666 248.979 169.108Propane 781,626 1517.150 1230.115 259.374 1375.850 1060.885Iso-butane 146.669 457.431Butane 90.580 90.580 189.788 471.394 262.389 30.685 30.685 510.212 2235.606 1277.511Iso-pentane 50.270 740.130Pentane 53.164 1076.736Hexane 21.497 21.497 12.654 114.839 21.671 71.912 71 .912 789.346 7829.161 1734.029Heptane 7.682 2599.018Octane 1 .284 1843.216Nonane 0.122 0.253 758.878 1668.747Decane 0.123 0.123 0.277 26.844E-3 26.931 26.931 7019.723 831.673Undecane 8.364E-3 1214.492

1Z0

Table 3.9

Solution Time (CPU seconds on CDC CYBER 855)

Source of Flash Routine

SolutionAlgorithm C3.2 C3.3

ProblemC3.4 C3.5 C3.6

This Work NP 0.998 0.973 2.548 3.447 14.958

HP 0.778 0.818 1.791 2.768 10.896NAP 0.580 0.539 i.q3i* 1 .626 6.780

NAA 0.502 0.469 1 .33^ 1 .490 6.234

P.P.D.S. NP 2.915 2.835 6.578 11.089 NA

HP 1.906 2.014 3.763 6.345 NA

*NA - Not availableTable 3.10

Number of Iterations/Function Evaluations



Problem

C3.4 C3.5 C3.6

This Work NP 5/31 4/29 8/57 7/64 7/120

HP 11/18 11/19 20/28 24/34 27/45

NAP 5/6 4/5 8/9 7/8 7/8

NAA 5/6 4/5 8/9 7/8 7/8

P.P.D.S. NP 5/31 4/29 8/57 8/73 NA

HP 11/18 11/19 20/28 26/36 NA

NA - Not applicableTable 3.11

Equivalent Number of Thermodynamic Calls



ProblemC3.4 C3.5 C3.6

This Work NP 713 667 1968 1990 3452

HP 413 430 782 974 1283

NAP 306 255 561 497 817

NAA 234 195 310 315 319

In terms o f c a l l s t o the TP package (Tab le 3 .11 ) NAA i s by

f a r the bes t method f o l l o w e d by NAP, HP, and NP i n t h a t o r d e r .

F a c t o r s f rom T a b l e 3 . 1 were used t o c a l c u l a t e t h e e q u i v a l e n t

number of c a l l s f o r a n a l y t i c TD p rope r t y d e r i v a t i v e s .

When a s t a n d a r d i s o t h e r m a l f l a s h modu le f rom t h e PPDS

package i s used f o r compar ison , s im u l a t i o n t im e s a r e f o u n d t o be

2 - 3 t im e s l a r g e r t h a n t h o s e o b t a i n e d w i t h our f l a s h a lg o r i t h m

im p lem en ta t io n . F u n c t i o n e v a l u a t i o n s and i t e r a t i o n s f o r bo th

r o u t i n e s a r e t h e same i n d i c a t i n g t h a t i n e f f i c i e n c i e s r e s u l t

t o t a l l y from the f l a s h p ro cedu re .

Coupled columns w i t h energy r e c y c l e (Problem C3.7)

In example C3.7 we found t h a t reasonab le i n i t i a l e s t im a te s

o f t h e i n p u t v a r i a b l e s i n t o t h e second column were necessa ry to

a s s u r e c o n v e r g e n c e (u se o f d e f a u l t v a l u e s g i v e n i n s e c t i o n

3 . 4 . 2 .3 . r e s u l t e d i n f a i l u r e o f the column procedure t o conve rge) .

B e c a u s e o f t h e way o u r p r o c e d u r e i s f o r m u l a t e d , we c h o s e t o

s p e c i f y the d i s t i l l a t e ra t e as opposed t o b o i l - u p ra te as g iv en by

H o l l a n d . For b r e v i t y , we r e p o r t t h e s i m u l a t i o n r e s u l t s o f o n l y

t h e o u t p u t and in p u t v a r i a b l e s f o r each column (Tab le 3 . 1 2 ) . Our

s o l u t i o n s compare v e r y w e l l w i t h t h o s e r e p o r t e d i n t h e book by

H o l la nd t a k i n g account t h a t we used d i f f e r e n t K -va lue and en tha lpy

mode ls.

122

Table 3.12 Simulation Results for Example C3.7

Variable Column I Column II

This work Holland (1981) This work Holland (1981)

Qc (GJ/hr) 4.456 5.559Qr (GJ/hr) 5 . 5 5 9 6.458Top temperature (°C) 33:81 83.20Bottom temp. (°C) 65.73 112.76Feed temp. (°C) 33.09* 65.73Distillate (Kmol /hrEthane 14.998 15.0 2.417E-3 1.076E-4Propylene 27.280 28.- 276 7.679 6.722Propane 14 .081 13 .083 15.765 16.90Iso-butene ' 1 .'241 E-3 1.637E-4 11.064 10.88Bottom (Kmol /hr)Ethane 2.471E-3 1.076E-4 0.0 2.74E-10Propylene 7.720 6.725 40.334E-3 3.0E-3Propane 15.919 16.92 0:154 1.85E-2Iso-butene 19.999 20.0 8.936 9.12Reflux Ratio 4.388 4.0 8.00* 8.00Distillate Rate(Kmole/hr) 56.36* 56.36 34.51* 34.51Feed (Kmol /hr)Ethane 15.0* 15.0 2.471E-3 1.076E-4Propylene 35.0* 35.0 7.720 6.725Propane 30.0* 30.0 15.919 16.92Iso-butene 20.0* 20.0 19.999 20.0

* Set values'

Table 3.13 Solution Statistics for Problem C3.7

Solution Algorithm Simulation Time (CPU seconds)

Flowsheet Iteration/ Function Evaluations

Equivalent no. TD properties Calls

NPP 36.042 • 4/25 45,682HPP 17.805 2/9 21,798NPA 23.657 4/25 15,554HPA 12.429 2/9 6,834NAP 15.478 3/4 17,668NAA 10.648 3/4 4,744

The s i m u l a t i o n s t a t i s t i c s f o r t h i s example (Tab le 3-13)

show th a t NAA i s the bes t s o l u t i o n o p t i o n i n terms of s i m u l a t i o n

t im e , number o f f u n c t i o n e v a l u a t i o n s and number of c a l l s t o the

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 . When n u m e r i c a l d e r i v a t i v e s o f

p h y s i c a l p r o p e r t i e s o n l y a r e used, method NAP t ake s a lmost 50 %

e x t r a computng t ime and over t h r e e t imes more TP c a l l s than method

NAA. For bo th s o l u t i o n o p t i o n s (NAA, NAP) t h e same number of

f l ow shee t i t e r a t i o n s / f u n c t i o n s e v a l u a t i o n s was r e q u i r e d . W i t h

N e w to n ' s method a t t h e f l o w s h e e t l e v e l and numer ica l d e r i v a t i v e s

o f lower l e v e l r o u t i n e s (NPP) we see t h a t an enormous number o f

c a l l s i s made t o t h e 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 . S ince the cost

i s r e l a t e d t o t he number o f u n i t o p e r a t i o n s and TP p r o c e d u r e s

e v a l u a t i o n s the computer t ime i s r e l a t i v e l y h igh . Our r e s u l t s a re

i n agreement w i t h t h o s e o f G r e n s (1984 ) and Westman and L u c i a

(1984) who c o n c l u d e t h a t f o r computa t iona l e f f i c i e n c y TP model

d e r i v a t i v e s may no t be u sed f o r d i s t i l l a t i o n co lumn i t e r a t i v e

c a l c u l a t i o n s . I f o n l y a n a l y t i c a l 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 e n t h a l p y m o d e l s a r e u t i l i z e d (method NAP) a

t h i r t y t h r e e p e r c e n t (33 %) r e d u c t i o n i n comput ing cos t and one

t h i r d f e w e r TP c a l l s a r e r e a l i s e d w i t h r e s p e c t t o NPP. W i t h

method NAA, 70 % and 8 9 . 6 % r e d u c t i o n s i n s i m u l a t i o n t ime and

number of a cce s se s t o p h y s i c a l p r o p e r t i e s i n compar ison t o method

NPP was r e a l i s e d . The i m p l i c a t i o n o f t h i s i s c l e a r . I f f o r

i n s t a n c e a d e r i v a t i v e f r e e num e r i c a l s o l u t i o n method, e . g . H y b r i d

me thod , i s a p p l i e d t o t h e ba se p o i n t s o l u t i o n o f t h e co lumn,

a n a l y t i c p r o c e d u r e d e r i v a t i v e s s h o u ld be u sed , e v e n i f t h i s

i n v o l v e s comput ing num er i c a l TP d e r i v a t i v e s . For Newton's method,

n u m e r i c a l p r o c e d u r e d e r i v a t i v e s r e q u i r e d an e x t r a i t e r a t i o n to

s a t i s f y the c o n v e r g e n c e c r i t e r i a c o m p a r e d t o t h e a n a l y t i c

p rocedure d e r i v a t i v e s cases .

From T a b l e 3 . 1 3 we o b s e r v e t h a t H y b r i d method a l s o

r e q u i r e d fewer i t e r a t i o n s (2) t o converge the f l owshee t a g a i n s t 3

o r 4 i t e r a t i o n s t a k e n by the Newton methods. Methods HPP and HPA

each used more f u n c t i o n e v a l u a t i o n s i n c o m p a r i s o n t o N ew to n

methods w i t h e x a c t p r o c e d u r e d e r i v a t i v e s (NAA, NAP). In f a c t ,

u s ing a n a l y t i c a l d e r i v a t i v e s o f p h y s i c a l p r o p e r t i e s w i t h H y b r i d

m e th o d (HPA) p r o d u c e s r o u g h l y 30 % s a v i n g s i n c o m p u t a t i o n a l

overhead compared t o HPP (The r e d u c t i o n i n TP r o u t i n e c a l l s i s by

a f a c t o r of 3 ) .

Coupled columns w i t h mass and energy r e c y c l e (Problem C3.8)

The r e s u l t s f o r t h e s e co n d e xam p le t a k e n f rom H o l l a n d

( e x a m p l e C 3 . 8 ) i n v o l v i n g b o th mass and e n e rg y r e c y c l e s a r e

d e t a i l e d i n T a b l e s 3 . 1 4 and 3 . 1 5 . He re a g a i n we f o u n d t h a t a

r e a s o n a b l y good i n i t i a l guess o f the bottom product of the f i r s t

column or the in p u t t o the second co lumn was r e q u i r e d t o o b t a i n

c o n v e r g e n c e . R e c a l l t h a t t h e s e co nd i n p u t t o co lumn I I i s

f i c t i t i o u s ( i . e . s e t t o z e r o ) and i s o n l y i n c l u d e d so as t o enab le

use t h e same d i s t i l l a t i o n co lum n p r o c e d u r e inpu t s p e c i f i c a t i o n

d e s c r ib e d i n s e c t i o n 3 . 4 . 2 . 3 . The s i m u l a t i o n r e s u l t s a re a g a i n

s i m i l a r t o v a l u e s r e p o r t e d by H o l la nd (Tab le 3 . 1 4 ) .

Table 3.14 Simulation Results for Example C3.8

Cc)lumn I Column II

This work Holland (1981) This work Holland (1981)

Qc (GJ/hr) 5.082 6.253QR (GJ/hr) 6.253 7.134Distillate (Kmol ./hr)Ethane 14.998 15.0 1.661E-3 6.99E-5Propylene 29.790 30.695 5.210 4.306Propane 17 .440 16.540 12.560 13.46Iso-butene • 1.566E-3 2.067E-4 19.998 20.00Bottom (Kmol /hr)Ethane 1.662E-3 6.99E-5 0.0 1.675E-10Propylene 5.249 4.307 39.294E-3 1.659E-3Propane 12.730 13.48 0.170 1.224E-2Iso-butene 35.757 29.97 15.759 9.975Reflux Ratio 4.526 4.0 8.0* 8.0Distillate Rate(Kmol /hr) 62.230* 62.23 37.77* 37.77Feed 1 (Kmol /hr)Ethane 15.0 * 15.0 1.989E-3 6.99E-5Propylene 35.0 * 35.0 5.468 4.307Propane 30.0 * 30 .0 12.584 13.48Iso-butene 20.0 * 20.0 40.293 29.97Top temperature (°C) 35.08 93.95Bottom temp. (°C) 76.33 113.26Feed temp. - 1 (°C) 33.09* 76.33Feed temp. - 2 (°C) 113.26 0.0

* set valuesTable 3.15 Solution Statistics for Problem C3.8

Solution Algorithm Simulation Time (CPU seconds)

Flowsheet Iteration/ Function Evaluations

Equivalent no. TD properties Calls

NPP 46.567 4/29 59,767HPP 27.137 4/12 33,276NPA 28.276 4/29 18,199HPA 14.985 4/12 8,580NAP 23.960 4/5 23,623NAA 11.268 4/5 5,087

When n u m e r i c a l TD and d i s t i l l a t i o n procedure d e r i v a t i v e s

were used t h e H y b r i d method (HPP) as u s u a l p e r f o rm s b e t t e r i n

t e rm s o f number o f TD c a l l s , , f u n c t i o n e v a l u a t i o n s , and computer

t ime. Method HPP uses j u s t o v e r h a l f t h e e f f o r t t a k e n by NPP.

W i th o n l y a n a l y t i c d e r i v a t i v e s o f p h y s i c a l p r o p e r t i e s l e v e l used

(NPA, HPA) we a g a i n have a n o t h e r l e v e l o f r e d u c t i o n i n t h e

computa t iona l overhead. The r e d u c t i o n i n number of TD a ccesses by

NPA(HPA) over NPP(HPP) a r e 3 . 2 8 ( 3 . 8 8 ) r e s p e c t i v e l y . T h i s a g a i n

e m p h a s i s e s t h e d e s i r a b i l i t y o f hav ing a n a l y t i c d e r i v a t i v e s o f TD

p r o p e r t i e s m o d e l s . I f on t h e o t h e r h a n d we h a v e n u m e r i c a l

K - v a l u e s 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 i n f o r m a t i o n but

p ro v id e exac t ( a n a l y t i c ) p ro cedu re d e r i v a t i v e s t o t h e f l o w s h e e t

(method NAP) we make e ven f u r t h e r r e d u c t i o n s i n e f f i c i e n c y than

NPA and thus much b e t t e r than NPP (49 % sa v in g s i n computer t im e ) .

C l e a r l y t h e a v a i l a b i l i t y o f e x a c t d i s t i l l a t i o n p rocedure and TP

d e r i v a t i v e s o f f e r s the most s i g n i f i c a n t r e d u c t i o n o v e r t he o t h e r

s o l u t i o n methods. S e v e n t y - s i x pe rcen t l e s s computer t ime i s taken

by NAA over NPP w i th a c o r r e s p o n d i n g m a s s i v e 91 % r e d u c t i o n i n

e q u i v a l e n t number of r i g o r o u s p h y s i c a l p rope r t y e v a l u a t i o n s .

Qf i t im isa t i gn_J^Prgbl em_C3_.9)_

The numer ica l r e s u l t s f o r the o p t im i z a t i o n a re repo r t ed in

Tab le s 3 .16 and 3 .17 t o g e t h e r w i t h the i n i t i a l e s t im a t e s shown i n

b r a c k e t s . The o p t im a l s o l u t i o n found (Tab le 3 .16) compares very

w e l l w i th those ob ta in ed by Ch imowitz e t a l (1983).

127

T a b l e 3 . 1 7 shows t h a t abou t 75 % r e d u c t i o n i n computer

t ime and TD c a l l s are a ch i e ved when exac t d e r i v a t i v e s a re used a t

a l l l e v e l s o f c o m p u t a t i o n s (method OAA) compared t o r e p e a t e d

p e r t u r b a t i o n o f t h e f l a s h modu le (method OP) . The r e s u l t i s

s i g n i f i c a n t s i n c e t h e exam p le i s s m a l l w i t h j u s t two d e c i s i o n

v a r i a b l e s . For r e a sonab ly l a r g e - s c a l e o p t im i z a t i o n problems which

may i n v o l v e t en s o f d e c i s i o n v a r i a b l e s s u b s t a n t i a l improvements i n

e f f i c i e n c y s h o u ld be p o s s i b l e . Method OAP i s t h r e e p e r c e n t

m a r g i n a l l y w o r s e t h a n OAA i n c o m p u t a t i o n a l o v e rh e a d and i s

p robab ly w i t h i n the a c cu ra c y o f t im in g .

3-6- CONCLUSIONS

A s m a l l s e t o f f l o w s h e e t i n g p r o b l e m s have been s o l v e d

u s ing SPEEDUP w i t h Newton 's and Hyb r id s o l u t i o n methods and the TP

d a ta i n t e r f a c e s u g g e s t e d i n t h i s t h e s i s . The t e s t problems a re

sma l l by f l o w s h e e t i n g s tanda rd s but r e p r e s e n t a t i v e of the t ype s of

problems encounte red .

For t y p i c a l TD p ro cedu re s a t the p h y s i c a l p r o p e r t i e s l e v e l

( a c t i v i t y / f u g a c i t y c o e f f i c i e n t s ) 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

c o m p u t e d a t a r e a s o n a b l e c o s t i n c o m p a r i s o n t o p o i n t v a l u e s

e v a l u a t i o n s . We found the co s t of a n a l y t i c d e r i v a t i v e s ( f o r SRK

and UNIQUAC models) i s a f a c t o r of about 2.1 - 2.5 those of po in t

v a l u e s and e s s e n t i a l l y independent o f the number o f componen ts .

Fur thermore , g e n e r a t i o n o f a n a l y t i c d e r i v a t i v e s o f TP models t a k e s

on the average on l y one t h i r d o f the t ime needed f o r n u m e r i c a l

p e r t u r b a t i o n .

TABLE 3.16. Results for optimization problem

Variable Initial Guess Solution

Decision

T1 (K) 360.0 342.81t2 (K) 340.0 324.66Tear (Recycle)Pentane flow rate (kraol /hr) 20.0 21 .08Hexane flow rate (kmol /hr) 10.0 19.12Octane flow rate (kmol /hr) 10.0 56.40

TABLE 3.17. Solution statistics for optimization problem (C3.9)

Optimisation Method Simulation Time (CPU seconds)

Total no. of function evaluations

No. of gradient evaluations

Equivalent no. of TD property calls

0 P 5.329 186 31 1482

0 A P 1 .401 28 27 503

0 A A 1 .365 31 31 371

E x a c t d e r i v a t i v e s o f VLE procedures s tu d ie d ( s i n g l e - s t a g e

i s o th e rm a l f l a s h and m u l t i c o m p o n e n t , m u l t i - s t a g e d i s t i l l a t i o n

co lumn) a r e ob ta in ed e f f i c i e n t l y by s t r a i g h t f o r w a r d m o d i f i c a t i o n s

of e x i s t i n g codes. S o l v i n g the l i n e a r a l g e b r a i c system ( e q u a t i o n

3 . 5 ) and g e n e r a t i n g t he p a r t i a l d e r i v a t i v e s (m a t r i c e s Q,R and S)

can be a c c o m p l i s h e d e a s i l y . Even i n comp lex 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 t h e o v e r h e a d s f o r d e v e l o p i n g t h e p r o c e d u r e

d e r i v a t i v e s a re v i r t u a l l y z e r o ( o f t h e o r d e r o f a s i n g l e co lumn

i t e r a t i o n ) . The r i g h t - h a n d s i d e m a t r i x f o r t h e d i s t i l l a t i o n

modu le (S) i s q u i t e s i m p l e . Fo r most o f t h e f l a s h e x a m p l e s ,

g e n e r a t i o n o f a c o m p l e t e s e t o f p r o c e d u r e 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 o n l y about one f l a s h base p o i n t e v a l u a t i o n . Even

when a n a l y t i c TD p r o p e r t i e s d e r i v a t i v e s a r e not a v a i l a b l e , i t i s

s t i l l e f f i c i e n t t o use n u m e r i c a l TP d e r i v a t i v e s when c o m p u t in g

o u t p u t - i n p u t d e r i v a t i v e s o f VLE p rocedu res .

A p p l i c a t i o n s of the e x a c t VLE and TP procedure d e r i v a t i v e s

i n t h e f l o w s h e e t i n g e x a m p l e s r e s u l t i n s i g n i f i c a n t sav ing s i n

comput ing cos t (up to 75 %) f o r t h e t e s t p r o b le m s . S u b s t a n t i a l

r e d u c t i o n s i n the number of TD p r o p e r t i e s c a l l s a re r e a l i s e d when

a n a l y t i c d e r i v a t i v e s are used a t a l l l e v e l s . Moreover, the number

o f f l o w s h e e t i t e r a t i o n s / f u n c t i o n e v a l u a t i o n s i s much l e s s than

w i t h o t h e r m e t h o d s . A l t h o u g h we a t t e m p t e d o n l y a s i n g l e

o p t im i z a t i o n problem we b e l i e v e t h a t o p t im i z a t i o n c a l c u l a t i o n s can

be done more e f f i c i e n t l y i f e x a c t procedure g ra d i e n t s genera ted by

our t e chn ique a re used.

130

CHAPTER FOUR

E F F IC IE N T DETERMINATION OF PROCESS

SE N S IT IV IT Y TO PHYSICAL PROPERTIES DATA

In t h i s c h ap t e r we show t h a t p rocess des ign s e n s i t i v i t y t o

p h y s i c a l p r o p e r t i e s can be e a s i l y and e f f i c i e n t l y genera ted f o r

genera l p ro ce s se s .

In t h e f i r s t s e c t i o n we s t a t e the mathemat ica l b a s i s o f

the method and show how s e n s i t i v i t ie s t o cons tan t pa ramete rs i n TP

m ode ls and t o e r r o r s i n TD model f u n c t i o n s a re ob ta ined from the

b a s i c work ing e qua t i o n .

R e s u l t s on the a p p l i c a t i o n of the method a re d e s c r i b e d i n

t h e f o l l o w i n g s e c t i o n s i n v o l v i n g v a p o u r - l i q u i d e q u i l i b r i u m

p r o c e s s e s ( f l a s h and 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 s ) and an

i n t e g r a t e d f l o w sh e e t . C o n c lu s i o n s on t e s t problems a re drawn.

4 . 1 . SEN S I J IV I J Y_ IP _ CON STANX_ PARAMETERS

L e t us c o n s i d e r a g a i n t h e s t e a d y - s t a t e m ode l o f a

procedure g iven in c h ap t e r 3 .

f ( w , v , u ) = 0 (3 .1)

where w, v , u a re t h e o u t p u t , i n t e r n a l and i n p u t v a r i a b l e s .

131

However, f o r g e n e r a l i t y , the model would be w r i t t e n as

f (w ,v ,u ,p ) = 0 (4-1)

where p i s a v e c t o r (o f d im ens ion t ) o f p ro cess paramete rs which

are constant and c on t inuous - F o l l o w in g s im i l a r a n a l y s i s as i n the

l a s t c h a p t e r , t h e f i r s t o rd e r s e n s i t i v i t y of the s o l u t i o n v e c t o r

w* t o t h e p a r a m e t e r s p a t t h e i r base p o i n t v a l u e s (p°) may be

w r i t t e n as

~3w"3 f 3 f ___ a f___ —— 8 p = -9 w 9v 3 v 3p

9 P_

\I 3 w

Ii

Q I R l

■3p 3 V = - S1

iL ( J 3 P

(4 .2)

(4-2)

where Q and R a r e d e f i n e d a s i n chap te r 3- M a t r i x (d imens ion

mxt) i s ob ta in ed by d i f f e r e n t i a t i n g f u n c t i o n f w i t h r e s p e c t t o

constant parameters o f i n t e r e s t i n t h e model.

The s e n s i t i v i t i e s o f the ou tpu t and i n t e r n a l v a r i a b l e s t o

p a r a m e t r i c u n c e r t a i n t i e s i n t h e model a r e e va lu a ted by s o l v i n g

equa t ion (4-2) which i s a l i n e a r sys tem o f s i z e m w i t h t r i g h t

hand s i d e s - The a l g o r i t h m f o r comput ing 3w*/3p and 3v*/3p i s the

same a s t h a t f o r e v a l u a t i n g e x a c t d e r i v a t i v e s o f g e n e r a l

132

p r o c e d u r e s a s d e t a i l e d i n s e c t i o n 3.1 of l a s t chap te r . In f a c t ,

s teps ( i ) , ( i i i ) and C iv ) are the same f o r bo th a l g o r i t h m s w i t h

o n l y a m in o r d i f f e r e n c e i n s t e p ( i i ) r e l a t i n g t o the generated

m a t r i c e s . In both a lg o r i t h m s , the m a t r i c e s a r e computed a t the

s o l u t i o n t o p rocedure ( 3 . 1 ) . Aga in the s o l u t i o n t o the procedure

can be accomp l i shed u s in g any s u i t a b l e numer ica l s o l u t i o n method.

M a t r i c e s Q and R wou ld be a v a i l a b l e a t the s o l u t i o n o f procedure

C3.1) i f Newton's method i s used i n s tep ( i ) i n which case on ly

i s c a l c u l a t e d a t t h e s o l u t i o n t o the p r o c e d u r e c a l c u l a t i o n s .

O t h e r w i s e t h e t h r e e m a t r i c e s (Q ,R , and ) a r e g ene ra ted from

s c r a t c h i n s tep ( i i ) .

4-2. SENSITIVITY TO MODEL FUNCTIONS

We h a v e j u s t shown how t o compute s e n s i t i v i t i e s o f a

p r o c e s s d e s i g n ( u n i t o p e r a t i o n s , e n t i r e f l o w s h e e t s ) t o

u n c e r t a i n t i e s i n c o n s t a n t p a r a m e t e r s o f TD m o d e l s , such as

c r i t i c a l t e m p e r a t u r e s , c r i t i c a l p r e s s u r e s , 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 s , e t c . However p r o p e r t i e s such as d e n s i t y , en tha lpy

or r e l a t i v e v o l a t i l i t y are s i n g l e - v a l u e d f u n c t i o n s dependen t on

o t h e r p r o c e s s v a r i a b l e s ( c o m p o s i t i o n s , t e m p e r a t u r e s , and

p re s su r e s ) . To con s id e r each o c c u r r e n c e o f , s a y , e n t h a l p y i n a

p r o c e s s model as an i n d i v i d u a l parameter sub je c t t o independent

v a r i a t i o n s ( S t r e i c h and K i s tenm ache r , 1980) i s n e i t h e r p r a c t i c a l

nor, i n our o p i n i o n , c o r r e c t . The problem we a re i n t e r e s t e d i n i s

t o e s t a b l i s h the e f f e c t t h a t the same change i n a l l t h e f u n c t i o n

model o c c u r e n c e s have on the s o l u t i o n , w*, i . e . t o determine the

133

s e n s i t i v i t y o f t h e s o l u t i o n t o the f u n c t i o n . M a th e m a t i c a l l y , we

have the procedure

f ( w , v , u , p, h(w)) = 0 (4 .3)

where h(w) i s a f u n c t i o n of dependent v a r i a b l e s such as d e n s i t y ,

e n t h a l p y , e n t r o p y , r e l a t i v e v o l a t i l i t y , e t c . To c a r r y out the

a n a l y s i s we p ro p o se t h a t a s u i t a b l e i n f i n i t e s i m a l d e v i a t i o n

f u n c t i o n Sh(w) be c h o s e n t o r e p r e s e n t e i t h e r e x p e r im e n t a l or

assumed d e v i a t i o n s f r om h(w) o v e r the domain o f the s o l u t i o n

v e c t o r , w*. Va r iou s c h o i c e s can be made about the form of 6 h(w).

Thermodynamic p r o p e r t i e s m o d e l l e r s o f t e n r e p o r t the e r r o r s

be tween e x p e r im e n t a l da ta and v a l u e s c a l c u l a t e d by t h e i r models.

Zudkev i t c h (1975) noted t h a t L y d e r s e n ' s method p r e d i c t e d a v e r a g e

+ + •r e l a t i v e e r r o r s o f _ 1 .5 % and _ 5 % i n c r i t i c a l t emperatu res and

p re s su r e s , r e s p e c t i v e l y . A l s o , a c c o r d in g t o Z u d k e v i t c h , M e i s s n e r

and R ead in g ' s method gene ra ted v a l u e s o f c r i t i c a l tempera tu res and

p r e s s u r e s w i t h r e l a t i v e e r r o r s o f t 5 % and 1 10 %. As noted by

Z u d k e v i t c h t h e s e s o r t o f e r r o r s i n c r i t i c a l t e m p e r a t u r e a n d

p r e s s u r e may w e l l r e s u l t i n 25 % e r r o r i n the s p e c i f i c hea t , cp,

f o r c o n d i t i o n s near the c r i t i c a l p o in t of the system. Z u d k e v i t c h

r e p o r t e d a n o t h e r examp le r e l a t i n g t o the compress ion of e thy lene

from 8.95 - 24.12 ba rs t o 3 4 4 . 5 6 - 4 8 2 . 3 8 b a r s . Here a b s o l u t e

e r r o r s be tween e x p e r i m e n t a l d a ta and p r o p e r t i e s genera ted u s ing

t h e R e d l i c h - K w o n g e q u a t i o n o f s t a t e w e re c o m p u t e d . A t a

t e m p e r a t u r e o f 3 7 . 8 ° C and p r e s s u r e r a n g i n g from 24.81 - 447.93

134

b a r s , a v e r ag e d e v i a t i o n s i n s p e c i f i c heat and compress ion f a c t o r

(Z ) o f between ( - 6 .6 1% ) and (1.21%) and between ( -0 .33% ) and

( 8 .6 4% ) , r e s p e c t i v e l y , were re p o r t e d . At a h ig h e r temperature of

1 1 4 . 4°C e r r o r s i n Cp and compress ion f a c t o r s range from ( - 6 .5 9 ) -

(11 .59) and (-1 .07) - (2 .03) r e s p e c t i v e l y . G ib b o n ' s e t a l (1 978)

r e p o r t average e r r o r s based on compar isons o f data f o r 2300 p o i n t s

and 54 d i f f e r e n t m ix t u re s composed e s s e n t i a l l y of n i t r o g e n , carbon

d i o x i d e and p a r a f f i n s r ang ing from methane t o hexane. Us ing SRK

model these au tho r s produced e r r o r s i n gas and l i q u i d d e n s i t i e s of

1 % and l e s s t h a n 10 % r e s p e c t i v e l y . For K - v a lu e s and dew po in t

tempera tu res , e r r o r s o f l e s s than 20 % and 5 % r e s p e c t i v e l y were

g i v e n . These f o rm s o f e r r o r r e p r e s e n t a t i o n i n TP models can be

w r i t t e n as

Ah(w) = h(w) + <$h(w) (4 .4)

Awhere h(w) i s the new m od e l . The e r r o r s , <$h(w), we have been

d i s c u s s i n g so f a r can be lo oked a t as be ing p r o p o r t i o n a l t o h(w)

t h a t i s ,

<$h(w) = con s tan t . h(w) (4 .5)

AT h i s means the new model h(w) has a c o n s t a n t r e l a t i v e d e v i a t i o n

from the o ld one. Hence,

Ah(w) = h(w) + constant? h(w) (4.6)

135

A n o th e r c h o i c e f o r the e r r o r i s what seve ra l worke rs have done i n

the past (Ne lson e t . a l (1983), Shah and Bi s hno i (1 9 7 8 ) , E l l i o tA

e t a l ( 1 9 8 0 ) , e t c . ) . He re a new model f u n c t i o n h(w) i s used i nA

equa t ion (4 .1 ) and a new s o l u t i o n w* ob ta ined . I f the new and o ld

model f u n c t i o n s a r e n e a r l y e q u a l , then one would expec t the two

As o l u t i o n s (w*, w*) t o be i d e n t i c a l . Thus the two model f u n c t i o n s

d i f f e r f rom each o t h e r by 6h(w) b e i n g t h e d i s t a n c e between the

models. Thus,

h(w) = h(w) + cons tan t (4 .7)

I t i s a l s o p o s s i b l e t o assume d i s t r i b u t i o n o f e r r o r s i n t h e

f u n c t i o n model over the range of the independent v a r i a b l e s such as

f o r

0 < w < w-j, no e r r o r , ft(w) = h(w);

w-j < w < W2 , c o n s t a n t r e l a t i v e e r r o r , h (w) = h(w) +

c o n s ta n t . h (w ) ;A

and W2 < w , cons tan t e r r o r , h(w) = h(w) + c on s tan t .

These may be p o s s i b l e f o r example i n the c r i t i c a l r e g i o n s o r i n

the d i l u t e s o l u t i o n r e g ion .

To e v a l u a t e the s e n s i t i v i t y of s o l u t i o n w* w i th respec t to

the f u n c t i o n ( t h a t i s , t o a l l occu rences of h(w) i n eve ry pant of

the sys tem), we in t r o d u ce a c o n t r o l parameter q (which i s a s i n g l e

va lued v a r i a b l e ) i n t o the s t e a d y - s t a t e model o f t h e p r o c e d u r e .

136

Thus, equa t i o n s (4 .6) and (4 .7) can be w r i t t e n as

Ah(w) = (1 + q) h(w) (4.8)

and

Ah(w) = h(w) + q (4 .9)

The above e r r o r model c o n s t r u c t i o n i s a p a r a m e t r i z a t i o n i n terms

of a new parameter , q. Thus equa t i o n (4 .1 ) can be w r i t t t e n as

f o r cons tan t e r r o r and con s tan t r e l a t i v e e r r o r s r e s p e c t i v e l y .

T hus , a n o m in a l v a l u e o f q° = 0 of the i n t r o d u c e d parameter w i l l

no t a f f e c t t h e base p o i n t s o l u t i o n i n a n y w ay . H o w e v e r a

s e n s i t i v i t y a n a l y s i s can now be p e r f o rm e d w i th respec t t o q by

add ing i t t o the l i s t of c on s tan t pa ramete rs ,p , and p r o c e e d i n g as

i n s e c t i o n 4 . 1 . The r i g h t hand s i z e m a t r i x §1 does not r e q u i r e

d i f f e r e n t i a t i o n of the t h e rm o d yn a m ic m o d e l s . The a l g o r i t h m i s

t h e r e f o r e i d e n t i c a l t o t h a t w i t h r e s p e c t t o t h e c o n s t a n t

f (w , u, v, p, h(w) + q) = 0 (4.10)

and

f (w , u, v , p, h(w) ( 1 + q)) = 0 (4.11)

parameters .

137

4 - 3 - A P PL IC A J 10 N_ T0_ V L E_ E X A M PL E S

Nowadays i n d u s t r i a l p r a c t i c e has moved towards extreme

p ro ce s s in g temperatures and p re s su re s due t o t h e need f o r e n e rg y

c o n s e r v a t i o n and e n v i r o n m e n t a l p o l l u t i o n abatement p o l i c i e s of

s e v e r a l g o v e rn m en ts - Fo r i n s t a n c e , p o l l u t i o n c o n t r o l p o l i c y

g u i d e l i n e s l e d t o t h e development of r e l i a b l e phase and chemica l

e q u i l i b r i a c a l c u l a t i o n methods f o r s ou r -w a te r s t r i p p e r s . The goal

was t o p rocess f o u l water c o n t a i n i n g ammonia, hydrogen s u lp h id e ,

and carbon d i o x i d e . F lash c a l c u l a t i o n s are of pr imary im p o r t a n c e

i n t h e d e s i g n o f m i s c i b l e gas f l o o d s i n enhanced o i l re cove ry .

Due t o the h ig h t e m p e r a t u r e s and p r e s s u r e s such p r o c e s s e s a r e

u s u a l l y o p e r a t e d i n t h e v i c i n i t y o f t h e c r i t i c a l r e g i o n and

r e g io n s where r e t r o g rade phenomena i s l i k e l y t o occur . S epa ra t ion

p r o c e s s e s such as d i s t i l l a t i o n and l i q u i d - l i q u i d e x t r a c t i o n are

a l s o h i g h l y im por tan t i n the p e t r o l e u m and c h e m i c a l i n d u s t r i e s .

Phase e q u i l i b r i u m c a l c u l a t i o n s a re a l s o r e q u i r e d i n heat exchanger

and two-phase p i p e l i n e s de s ig n . Thus, v a p o u r - l i q u i d e q u i l i b r i u m

o p e r a t i o n s a re of v i t a l impor tance i n chemica l eng inee r ing .

A c c u r a t e phase e q u i l i b r i u m ( o r c h e m i c a l , c h e m i c a l and

phase e q u i l i b r i u m ) p r e d i c t i o n s a re n e c e s s a r y f o r a c c e p t a b l e

d e s i g n / s i mul a t i on o f t h e s e VLE u n i t o p e r a t i o n s . The problem i s

compounded f o r i n d u s t r i a l p ro ce s se s s in c e a c cu ra te p r e d i c t i o n o f

phase e q u i l i b r i a i s made more d i f f i c u l t because of the inadequacy

of a v a i l a b l e TP m o d e l s . F u r t h e r m o r e , some o f t h e r e q u i r e d TP

138

p a r a m e t e r s may no t be a v a i l a b l e or a re not v a l i d a t the process

o p e r a t i n g c o n d i t i o n s .

We have t h e r e f o r e r e s t r i c t e d o u r s t u d i e s t o p r o c e s s e s

i n v o l v i n g v a p o u r - l i q u i d e q u i l i b r i u m c a l c u l a t i o n s . S i x e xam p le s

taken from the literature were chosen f o r study:

- VLE procedures

- u n i t o p e r a t i o n s of i n d u s t r i a l impor tance

- r i g o r o u s TD models o f w idesp read use

- d i f f i c u l t phase e q u i l i b r i u m s e p a r a t i o n s .

The examples a r e d e s c r i b e d i n the f o l l o w i n g s e c t i o n s . A l l

computat ions were performed u s in g an IBM 4341 computer.

4.3.1. Jhermo^nami c_ Propert i es__ Model s

F u g a c i t y c o e f f i c i e n t s ( o r K - v a lu e s ) and e x ce s s e n t h a l p i e s

were c a l c u l a t e d f rom t h e S o a v e m o d i f i e d R e d l i c h -Kw ong two

parameter equa t ion o f s t a t e (SRK) exp ressed as :

Rg T an = ------ - ---------- (4.12)

v -b v(v+b)

where v i s s p e c i f i c volume and Rg i s u n i v e r s a l gas cons tan t .

The SRK model i s used e x t e n s i v e l y i n t h e h y d r o c a r b o n i n d u s t r y .

P a r a m e te r a i s d ependen t on pu re component c r i t i c a l p r o p e r t i e s

(tempera ture and p re s su re ) and P i t z e r ' s a c e n t r i c f a c t o r w h i l e b

depends on the c r i t i c a l pa rameters o n l y .

139

For m u l t i c om ponen t systems a and b i n equa t ion (4.12) a re

r e p l a c e d by t h e i r m i x t u r e e q u i v a l e n t s am xt and bmi x-t- The TD

p r o p e r t i e s p r e d i c t e d u s in g equa t i o n (4 .12) depend s t r o n g l y on the

way t h e m ix t u r e parameters am-jxt and b ^ x t are eva lua ted from the

pure component parameters ( a ^ b - p . For the purpose of t h i s work,

the f o l l o w i n g u b i q u i t o u s m ix in g r u l e s have been used:

ami xt = Z Z x-j xj a-jj (4 .13)i j

kmi xt = Z x i ^i (4.14)i

a i j = (a i a j ) 5 (4 .15)

A 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 ( ^ j ) i s u s u a l l y i n t r o du ced as

follows:

a-j j = (1 - 6-j j ) (a-j aj )2 (4 .16)

The i n t r o d u c t i o n o f a 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 t o the

m ix ing r u l e g r e a t l y im p ro v e s the a p p l i c a b i l i t y o f t h e SRK and

i n d e e d o f a l l c u b i c e q u a t i o n s o f s t a t e . The i n t e r a c t i o n

c o e f f i c i e n t i s s om e t im es c a l l e d the c r o s s - t e r m i n t e r a c t i o n

p a r a m e te r w h i c h c o r r e c t s t h e depa r tu re s from the geometr ic mean

a p p r o x i m a t i o n f o r t h e c r o s s t e rm s a j wh ich was known t o be

i n c o r r e c t . I t has been suggested (Chueh and P r a u z n i t z , 1967) tha t

t o a good a p p r o x i m a t i o n <$i j i s a t r u e m o l e c u l a r c o n s t a n t

c h a r a c t e r i s t i c of the i - j i n t e r a c t i o n independent of t e m p e r a t u r e ,

d e n s i t y , and c o m p o s i t i o n . D i f f e r e n t v a lu e s f o r ^i j are u s u a l l y

p u b l i s h e d f o r d i f f e r e n t e q u a t i o n s o f s t a t e . Even f o r the same

140

equa t ion o f s t a t e , w id e l y d i f f e r e n t 6 i j v a lu e s have been pub l i s hed

( R e i d e t a l ( 1 9 7 7 ) , G m e h l i n g e t a l ( 1 9 7 7 , 1 98 2 ) . G e n e r a l l y ,

b i n a r y i n t e r a c t i o n p a r a m e t e r s a r e t he c o n s ta n t s sub jec t t o the

most u n c e r t a i n t y compared t o the o the r p a ram e te r s i n e q u a t i o n o f

s t a t e m o d e l c o m p u t a t i o n s . The number o f 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 s i n a mu l t i componen t m ix t u re may be ve ry h igh . For an

NC c o m p o n e n t m i x t u r e t h e t o t a l number o f p a r a m e te r s e q u a l s

NC(NC-1)/2. I t i s im po r tan t t o de te rm ine wh ich , i f any, o f t h e se

6 i j c an be a s s u m e d t o be z e r o a nd w h i c h must be e s t im a t e d

a c c u r a t e l y . There are s e v e r a l s i t u a t i o n s where s e t t i n g S i j t o

ze ro i s reasonab le :

( i ) i n p r e l i m i n a r y d e s i g n c a l c u l a t i o n s where g ro ss e r r o r s i n

TD models a re a c c e p t a b l e .

( i i ) when m o d e l l i n g m i x t u r e s c o n t a i n i n g a la r g e number of not

w e l l d e f in ed components, i . e . i n the c h a r a c t e r i s a t i o n of

heavy o i l s .

( i i i ) when t h e n um be r o f c o m p o n e n t s i s h i g h , s i g n i f i c a n t

s i m p l i f i c a t i o n s i n the c a l c u l a t i o n of phase e q u i l i b r i a being

p o s s ib l e (M i ch e l s en , 1986) .

( i v ) when d r a s t i c r e d u c t i o n i n compute r t im e i s o f h i g h e r

importance than the d i s a d v a n t a g e o f a s l i g h t d e c r e a s e i n

accuracy e. g. r e s e r v o i r s im u l a t i o n s .

141

In a l l the examples a t tempted we eva lua ted the s e n s i t i v i t y

of the p ro cesses t o 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 . Fo r t h i s ,

a n a l y t i c 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 exce s s en tha lpy

w i th re spec t t o b in a r y i n t e r a c t i o n paramete rs are r e q u i r e d . These

a n a l y t i c d e r i v a t i v e s a re p re sen ted i n Append ix D1.

4 . 3 . 2 . I s o t herm a l _ FIa sh_ P ro c e d u re s

Two i s o t h e r m a l f l a s h t e s t p r o b l e m s were t a k e n f rom

M i c h e l s e n ( 1 9 8 0 ) . The f i r s t p r o b l e m i s a 7 - c o m p o n e n t

n i t r o g e n - h y d r o c a r b o n m ix t u re . The second t e s t problem i s a f i v e

component h y d r o c a r b o n m i x t u r e r i c h i n h yd rogen s u l p h i d e under

r e t r o g r a d e c o n d i t i o n ^ The complete s p e c i f i c a t i o n i s d e t a i l e d i n

Appendix D2. The f l a s h p rocedure c a l c u l a t i o n a lg o r i t hm d e s c r i b e d

i n t h e p r e v i o u s c h a p t e r i s u t i l i z e d t o compute t h e o u t p u t

v a r i a b l e s ( v a p o u r and l i q u i d phase f l o w r a t e s ) . The n o n - z e r o

6 i j ' s used i n t h e c o m p u t a t i o n s a r e g i v e n i n A p p e n d i x C1. The

s e n s i t i v i t i e s o f t h e v a p o u r and l i q u i d p r o d u c t s t o a l l t h e

i n d i v i d u a l paramete rs were c a l c u l a t e d as p resen ted i n s e c t i o n 4 . 1 .

A n a l y t i c compos i t i on 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 were

used i n the computa t ion o f the J a c o b ia n m a t r i x w i th the r i g h t hand

s id e v e c t o r s g iv en i n Append ix D1.

4 . 3 . 3 . Di s t i l l a t i on_ C o l umn_ P ro c e d u re

E t h y l e n e and p r o p y l e n e s e p a r a t o r s i n o l e f i n p l a n t s a re

used t o p ro d u ce p o l y m e r i z a t i o n g rade e t h y l e n e and p r o p y l e n e .

142

These d i s t i l l a t i o n u n i t s have t o separa te e th y lene and propy lene

from chemical s p e c i e s w i t h v e r y s im i l a r b o i l i n g p o i n t s (ethane and

p ropane r e s p e c t i v e l y ) through the inpu t of c o n s id e ra b le amountsof

e n e r g y . These t y p e s o f VLE u n i t o p e r a t i o n s h a ve been u s e d

e x t e n s i v e l y t o s t u d y t h e e f f e c t s o f 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 co lumn d e s i g n (A n g e l e t a l ( 1 9 8 6 ) , Z u d k e v i t c h

(1975) , and Hernandez e t a l ( 1984 ) ) .

T h ree t e s t prob lems taken from Hernandez e t a l (1984) are

d e t a i l e d i n A p p e n d i x D2. The d i s t i l l a t i o n c o lu m n code o f

F r e d e n s l u n d e t a l ( 1977 ) m e n t i o n e d i n c h a p t e r 3 was used t o

compute the o u t p u t v a r i a b l e s ( v a p o u r f l o w , l i q u i d f l o w , and

t e m p e r a t u r e p r o f i l e s , r e b o i l e r and condenser d u t i e s ) . S ince the

code i s based on Newton 's method, both Ja cob ia n and l i n e a r a lgeb ra

codes were a v a i l a b l e . At the base p o in t s o l u t i o n o f the columns,

on ly minor changes were r e q u i r e d t o generate t h e r i g h t hand s i d e

m a t r i x . The e l e m e n t s o f the r i g h t hand s id e m a t r i x a re g iv en i n

Appendix D1. S e n s i t i v i t i e s w i t h r e s p e c t t o b i n a r y i n t e r a c t i o n

p a r a m e t e r s , Murphree t r a y e f f i c i e n c y , cons tan t r e l a t i v e e r r o r s i n

vapour e n tha lp y (combined i d e a l and excess c o n t r i b u t i o n s ) , i d e a l

l i q u i d en tha lp y and e x ce s s l i q u i d e n tha lp y were s tud ied .

4 . 3 . 4 . Jn te c jr a t e d M u l t i u n i t_ F lo w s h e e t

So f a r we have c o n ce rn e d o u r s e l v e s w i th the study of 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 a s i n g l e p i e c e o f

e q u ip m e n t . Because of the c o u p l i n g o f d i f f e r e n t p ie c e s o f u n i t s

143

t h r o u g h t h e r e c y c l e o f mass and e n e r g y i n a f l o w s h e e t i t i s

e q u a l l y impor tant t o s tudy the s e n s i t i v i t y o f a f l o w s h e e t d e s i g n

u s ing a p rocess s im u l a t o r .

The C a v e t t f o u r f l a s h f l o w s h e e t i s a s tandard i n t e g r a t e d

p ro ce s s o f t e n used a s benchmark i n s i m u l a t i o n s t u d i e s ( F i g u r e

3 . 3 ) . We c o n s i d e r e d a f e e d o f f i v e components i n v o l v i n g fo u r

i s o t h e rm a l f l a s h e s . The f l o w s h e e t was s i m u l a t e d u s i n g SPEEDUP.

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 o f t h e f l a s h p r o c e d u r e o u t p u t

v a r i a b l e s t o t h e i r i n p u t v a r i a b l e s ( t e m p e r a t u r e , p r e s s u r e , and

f e e d f l o w r a t e s ) were computed a s d e t a i l e d i n c h a p t e r 3 . The

s e n s i t i v i t i e s o f t h e unknown p r o c e s s s t r e a m v a r i a b l e s t o t h e

n o n - z e r o 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 s between carbon d io x id e

and the hydrocarbons a re r e q u i r e d .

The cu r r e n t v e r s i o n o f SPEEDUP (SPEEDUP User Manual , 1986)

does not g e n e r a t e f l o w s h e e t s e n s i t i v i t y i n f o r m a t i o n d i r e c t l y .

Howeve r , t h e e q u a t i o n - o r i e n t e d s im u l a t o r has t h e c a p a b i l i t y fo r

h an d l in g o p t im i z a t i o n p rob lems. Some or a l l of the SET v a r i a b l e s

i n o p e r a t i o n s e c t i o n o f SPEEDUP in p u t data f i l e can be t r e a t e d as

d e c i s i o n v a r i a b l e s f o r t h e f e a s i b l e pa th based o p t i m i z a t i o n

a l g o r i t h m im p lem en ted i n t h e s im u l a t o r . Reduced g ra d i e n t s with

r e s p e c t t o t h e d e c i s i o n v a r i a b l e s a r e c a l c u l a t e d f o r t h e

o p t i m i z a t i o n p r o c e d u r e a t each i t e r a t i o n . The d e s i r e d p rocess

des ign s e n s i t i v i t i e s t o p h y s i c a l p r o p e r t i e s ( 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 s ) a r e i n f a c t t h e reduced g r a d i e n t s , where t h e S i j ' s

are t r e a te d as d e c i s i o n v a r i a b l e s . A s p e c i a l i n t e r f a c e program

144

was w r i t t e n by Dr . C .C . P a n t e l i d e s t o enab le SPEEDUP to generate

o n l y the s e n s i t i v i t i e s and a vo id p e r f o r m i n g a f u l l o p t i m i z a t i o n

c a l c u l a t i o n - Exac t p a r t i a l d e r i v a t i v e s of f l a s h output v a r i a b l e s

t o b in a r y i n t e r a c t i o n paramete rs are p rov ided by our f l a s h r o u t i n e

f o r the s e n s i t i v i t y a n a l y s i s .

4 . 4 . APPLIC£T 10N _J9_ THE. _LOC A J10 N__0 F _C0 NJROL__ME ASU R EME_NTS _ IN

DISJILLAJigN_COLUPWS

B r i g n o l e e t a l (1985) argued t h a t i n a d i s t i l l a t i o n column

t h e r e a r e r e g i o n s i n t h e c o l u m n f o r w h i c h t h e p r o f i l e s

( c o m p o s i t i o n s and t e m p e r a t u r e ) change q u i t e s i g n i f i c a n t l y fo r

r e l a t i v e l y sma l l changes i n the t o p or bottom product q u a l i t i e s or

t e m p e r a t u r e s . These zones , they reasoned, p ro v ide i d e a l l o c a t i o n

o f s e n s o r s f o r t h e c o n t r o l o f t h e co lumn . T h i s i n f e r e n t i a l

approach l e d the a u th o r s t o propose o b j e c t i v e f u n c t i o n s c a l l e d the

en r ichment f a c t o r s pe 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 i n o r d e r t o d e t e r m in e t h e z o n e s o f maximum enr ichment

f a c t o r and t h u s t h e i d e a l s e n s o r L o c a t i o n s . The o b j e c t i v e

f u n c t i o n s a re d e f i n e d as f o l l o w s :

r e c t i f y i n g s e c t i o n :

NC-1ui = £ I y*,i - yi,i+ii =1 1 1

(4.17)

s t r i p p i n g s e c t i o n :

(4.18)

where y* . , i s t h e mo le f r a c t i o n s o f the vapour e x i t i n g from an 7 i, l'/

145

i d e a l s t age l ( t r a y e f f i c i e n c y equa l t o u n i t y ) and cor responds t o

t h e e q u i l i b r i u m c o m p o s i t i o n w i t h t h e l i q u i d x - j^ . Furthermore

y i^ L+1 and y-\r i+'\ a r e the average com pos i t i on of vapour e n t e r in g

p l a t e l i n the r e c t i f y i n g or s t r i p p i n g s e c t i o n s , r e s p e c t i v e l y .

Thus, on s o l u t i o n o f a r i g o r o u s model the exact enr ichment f a c t o r s

per s t a g e a re o b t a i n e d d i r e c t l y u s i n g e q u a t i o n s ( 4 . 1 7 , 4 . 1 8 )

a l l o w i n g ea s y i d e n t i f i c a t i o n o f the r e g io n of maximum enr ichment

f a c t o r . By making s i m p l i f y i n g assumpt ions e. g. non d i s t r i b u t i o n

o f n o n - k e y c o m p o n e n t s , p r e s e n c e o f o n l y t h e l i g h t and key

components i n the d i s t i l l a t e , p r e s e n c e o f t h e h e a v y and key

components i n t h e bo t tom p r o d u c t , and the n e g l i g i b l e amounts of

the heavy components a few p l a t e s above the f e e d , e t c . , B r i g n o l e

e t a l d e r i v e d s im p le e x p r e s s i o n s f o r comput ing the dominant zones

i n a co lumn. They showed t h a t t h e l o c a t i o n o f the s e n s i t i v e

p l a t e s n ea r or on the feed p l a t e i s u n d e s i r a b l e and e q u i v a l e n t t o

the c o n d i t i o n o f " p i n c h " i n h e a t e x c h a n g e r n e t w o r k d e s i g n .

A c c o r d i n g t o the a u th o r s , the optimum v a lu e s of enr ichment f a c t o r

shou ld be l o c a t e d i n both s e c t i o n s of the column and away from the

f e e d l o c a t i o n . A l s o the a u th o r s showed how d e v i a t i o n s ( abso lu te

ba s i s ) i n the de s ign and o p e r a t i n g v a r i a b l e s from the base v a l u e s

a f f e c t e d t h e l o c a t i o n o f t h e dominant zones i n both s e c t i o n s of

the column. In o rde r words, they were a b l e t o s tu d y the co lumn

o p e r a b i l i t y and c o n t r o l s t r u c t u r e s by s tud y ing the v a r i a t i o n of

t h e l o c a t i o n o f z o n e s o f maximum e n r i c h m e n t w i t h c h a n g e s i n

o p e r a t i n g v a r i a b l e s . They p e r f o r m e d t h e s e s t u d i e s u s i n g t h e

s i m p l i f i e d e x p r e s s i o n s . H e re we s tu d y t h e v a r i a t i o n s o f t h e

d o m in a n t z o n e s d i r e c t l y , u s i n g a s e n s i t i v i t y a n a l y s i s , i n

146

pa r t i c u l a r f o r u n c e r t a i n t i e s i n 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 .

D e r i v a t i v e s of e q u a t i o n s (4 .17) and (4 .18) w ith re spec t to

t h e 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 s were ob ta in ed by cha in r u l i n g

of the column p r o f i l e s s e n s i t i v i t i e s , t h a t i s ,

and

3UL NC-1y

3 y ? ,i 3 y i,l+ 1

80 i =1 8 0 8 0

3u{ NC-1z

a 13 y i,l+ 1

~80 i =1 8 0 8 0

(4.19)

(4.20)

f o r the 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 , r e s p e c t i v e l y . Note we

have i n t r o d u c e d 0 f o r 6 i j f o r c o n v e n i e n c e . U s i n g t h e s e

s e n s i t i v i t i e s , a l i n e a r a p p ro x im a t io n of the enr i chment f a c t o r s a t

p e r t u r b e d v a l u e s o f 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 s can be

ob ta in ed thus

U^(new) = U^lbase) + — . A® (4.21)

and

1 1Ui (new) = u l ( b a s e ) + -----

^0A0 (4.22)

These e q u a t i o n s enab le one t o de term ine whether or not e r r o r s i n

the p h y s i c a l p r o p e r t i e s would a f f e c t the l o c a t i o n of the dom inant

z o n e s u s i n g o n l y t h e base p o i n t s o l u t i o n of the column and the

s e n s i t i v i t i e s . Note t h a t t h i s i s o n l y a f i r s t o rder approx imat ion .

147

A d e t a i l e d d e s c r i p t i o n o f a h y p o t h e t i c a l d i s t i l l a t i o n

column i s g iv en i n Append ix D3. S e n s i t i v i t i e s o f the e n r i c h m e n t

f a c t o r s t o o were ob ta in ed and a re r epo r t ed i n F igu re D3.3.

4 -5 - J^jfiBJC^-EXPERIjJCNJS/DISCySSIONS

Some comments a b o u t t h e g e n e r a t e d s e n s i t i v i t e s a r e i n

o r d e r b e f o r e d i s c u s s i o n o f t h e r e s u l t s o f o u r n u m e r i c a l

e x p e r i m e n t s . I n s e c t i o n s 4 .1 and 4 . 2 we d e r i v e d n u m e r i c a l

a l g o r i t h m s f o r the e v a l u a t i o n o f s e n s i t i v i t i e s o f the s o l u t i o n

v e c t o r , w*, t o c o n s t a n t p a r a m e t e r s (p) and c o n s t a n t r e l a t i v e

e r r o r s i n model f u n c t i o n s ( 6 h(w)) , t h a t i s , Sw*/Sp and 6 w*/ 5h(w).

These s e n s i t i v i t i e s o r c o n s t r a i n e d g r a d i e n t s shows the change i n

ou tpu t v a r i a b l e s which co r re spond t o a change i n the p a ra m e te r o r

m o d e l . D e s i g n v a r i a b l e s u s u a l l y have d i f f e r e n t o r d e r s o f

m a g n i t u d e , c o n s e q u e n t l y l a r g e s e n s i t i v i t i e s a re computed f o r

v a r i a b l e s o f l a r g e magni tude ( c o n v e r s e l y , sm a l l e r s e n s i t i v i t i e s

are ob ta in ed f o r v a r i a b l e s p re sen t i n sma l l amounts).

From t h e above s e n s i t i v i t i e s we a re a b le t o c a l c u l a t e the

n o r m a l i s e d s e n s i t i v i t y i n f o r m a t i o n , t h a t i s , <$w*/w*<5p ( o r

6w*/w* s h ( w ) ) . T h i s i s e q u i v a l e n t t o s c a l i n g the s e n s i t i v i t i e s .

The s c a l e d s e n s i t i v i t y i n f o r m a t i o n shows the r e l a t i v e change i n

t h e d e s i g n v a r i a b l e s (w*) due t o an a b s o lu t e change i n paramete rs

o r model f u n c t i o n s . T h i s i n f o r m a t i o n e n a b l e s one t o r a n k the

o r d e r o f i m p o r t a n c e o f e v e r y p a r a m e t e r and m o d e l u n d e r

i n v e s t i g a t i o n f o r a g i v e n ou tpu t v a r i a b l e . However, s ince t h i s i s

a r e l a t i v e m e a su re , i t t e n d s t o o v e r e s t im a t e the impor tance of

w* 's p resen t i n sma l l amounts e .g . i m p u r i t i e s . In t h a t c a se , we

s im p l y i g n o r e t h e s e n s i t i v i t i e s p e r t a i n i n g t o such v a r i a b l e s . On

the o the r hand, i f we a re i n t e r e s t e d i n the i m p u r i t i e s , t h e n i t

w ou ld be wrong t o n eg le c t t h e i r c o r r e s p o n d i n g s e n s i t i v i t i e s * These

are not the o n l y way s e n s i t i v i t i e s can be e x p r e s s e d . O t h e r ways

o f e x p r e s s i n g s e n s i t i v i t y i n f o r m a t i o n a r e : r a t i o o f p e r c e n t

change of o u tp u t / p e r c e n t change o f u n c e r t a i n p a r a m e te r s ( o r model

f u n c t i o n ) ; o r the r a t i o o f p e rcen t change o f output t o a r e l a t i v e

or ab so lu t e change i n paramete rs (mode l f u n c t i o n s ) . T h e r e f o r e ,

once t h e s e n s i t i v i t y m a t r i x 9w*/3p i s a v a i l a b l e i t i s up t o the

i n v e s t i g a t o r t o p re s en t h i s / h e r r e s u l t s i n a fo rm t h a t s u i t s a

p a r t i c u l a r p r o b le m . We w i l l however r e s t r i c t our d i s c u s s i o n s t o

the f i r s t r e p r e s e n t a t i o n and sometimes use s c a l e d s e n s i t i v i t i e s .

For our prob lems, these two v e r s i o n s l e a d t o the same con c lu s ion s .

The o t h e r r e a s o n i s t h e f a c t t h a t s i n c e we a r e i n t e r e s t e d i n

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 s whose base v a l u e s a re o f t e n ze ro ,

i t i s not p o s s ib l e t o d e f i n e a pe rcen t change i n such p a r a m e te r s

from ze ro base p o i n t .

Flash 1 (D2.1)

The base p o i n t d e s i g n u s ing v a l u e s o f 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 g iv en i n Append ix C1 a re r e po r t e d i n Tab le D2 .2 . We

a l s o p r o v i d e t h e s e n s i t i v i t i e s o f t h e v a p o u r and L i q u i d molar

component f l o w r a t e s t o t h e 2 1 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 s

148

149

( A p p e n d i x D2) - From t h e s e s e n s i t i v i t i e s i t i s e a s i l y seen t h a t

t h e b i n a r y i n t e r a c t i o n between n i t r o g e n and methane ( & | 2 = 0 . 0 2 )

h a v e t h e g r e a t e s t i n f l u e n c e o n t h e b a s e p o i n t d e s i g n .

U n c e r t a i n t i e s on h y d r o c a r b o n - h y d r o c a r b o n i n t e r a c t i o n

c o e f f i c i e n t s have sma l l e f f e c t s on the s im u l a t i o n r e s u l t s . T h i s

c o n f i r m s t h e t r a d i t i o n a l p r a c t i c e o f s e t t i n g

hydroca rbon-hydroca rbon i n t e r a c t i o n s t o ze ro (G rabosk i and Daubert

( 1 9 7 8 ) , R e i d e t a l ( 1 9 7 7 ) , e t c . ) . Gmeh l ing e t a l (1977, 1982)

r e p o r t e d a v a l u e o f 2 = 0 . 0 2 7 8 wh ich c o r r e s p o n d t o 39 % and

+0.0078 r e l a t i v e and a b s o lu t e e r r o r s r e s p e c t i v e l y compared t o the

v a l u e <5^2 = 0 . 0 2 u sed i n t h e s e n s i t i v i t y c a l c u l a t i o n s . I t i s

d i f f i c u l t t o o b t a i n data r e g a r d in g the range of u n c e r t a i n t i e s , i n

b i n a r y i n t e r a c t i o n s u sed i n c u b i c e q u a t i o n s o f s t a t e . However,

f r o m a s u r v e y o f t h e l i t e r a t u r e we f o u n d t h a t f o r t h e

S R K - e q u a t i on , <5 — v a l u e s g e n e r a l l y l i e between 0 - 0 .15 (R e id e t

a l (197) , G m e h l i n g e t a l ( 1 9 7 2 , 1 9 8 2 ) ) . On t h e o t h e r hand , a

number o f w o r k e r s p r e s e n t r e s u l t s f o r wh ich they conclude tha t

i n a c c u r a c i e s o f b e t w e e n 2% - 3 0 % i n p h y s i c a l p r o p e r t i e s

p r e d i c t i o n m o d e l s a r e t o l e r a b l e (G ibbons e t a l (1978) , S t r e i ch &

K is tenmacher (1980) , Shah and B i shno i (1978), e t c . ) . However , we

s i m u l a t e d t h e f l a s h u n i t w i t h v a l u e s o f 6^2 = 0 . 0 , 0 . 0 2 , 0 .04 ,

0 . 0 6 , 0 . 0 8 . T h i s i s t o t r y and a s c e r t a i n how f a r t h e l i n e a r

a p p r o x i m a t i o n o f t h e o u t p u t s can be c a r r i e d w i t h c o n f i d e n c e

w i thou t g ross e r r o r s i n the p r e d i c t i o n of t h e r i g o r o u s p r o c e s s

des ign v a lu e s . In f a c t , we performed t h i s k ind o f a n a l y s i s i n a l l

the t e s t p r o b l e m s . T h e r e f o r e t h e r i g o r o u s c a l c u l a t i o n s w e re

p e r f o rm e d u s in g v a l u e s o f 6^2 which we know are g r o s s l y i n e r r o r .

150

F i g u r e 4 .1 shows t h e v a r i a t i o n o f the amount of methane i n both

f l u i d phases from r i gorous d e s ig n and by l i n e a r e x t r a p o l a t i o n of

the exact s e n s i t i v i t i e s - C l e a r l y t h i s e x t r a p o l a t i o n i s v a l i d even

i n t h e r e g i o n f o r wh ich we know v a lu e s a re f a r too h igh . The

above r e s u l t s show tha t u n c e r t a i n t i e s i n the p h y s i c a l p r o p e r t i e s

parameters have n e g l i g i b l e e f f e c t on the f l a s h u n i t des ign .

F la sh 2 (D2.2)

The base s o l u t i o n and s e n s i t i v i t y i n f o rm a t i o n a re g iv en i n

A p p e n d i x D2, T a b l e D2 .3 (H e r e t h e r e a r e 10 d i f f e r e n t 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 s ) . An e xam ina t io n o f the s e n s i t i v i t i e s

shows t h a t the i n t e r a c t i o n paramete r between hydrogen s u lph id e and

methane (<$23 = 0-08) i s t h e parameter w i t h the g re a t e s t i n f l u e n c e

on the amount of m a t e r i a l i n t h e vapour and l i q u i d pha se s . Note

t h i s o b s e r v a t i o n i s t r u e b o t h on a b s o l u t e and r e l a t i v e b a s i s .

R i g o r o u s s i m u l a t i o n o f t h e f l a s h module w i th 6 2 3 v a lu e s between

0 . 0 - 0 . 1 6 a t i n t e r v a l s o f +0 .02 w e r e made . F i g u r e 4 . 2 i s

ob ta ined as i n t h e p re v io u s example. From t h i s f i g u r e we see tha t

methane f l o w r a t e i n the vapour phase v a r i e s n o n l i n e a r l y with <$23-

U n l i k e i n the p r e v io u s example , t h e amount of methane p r e d i c t e d

f rom l i n e a r e x t r a p o l a t i o n i s ' v a l i d only around the base va lue of

6 2 3 - I f we assume e r r o r s o f up t o - 30 % i n 6 2 3 are t o l e r a b l e ,

then we see t h a t a l i n e a r a p p ro x im a t i o n of the t rue process des ign

t o be s a t i s f a c t o r y . E x t r a p o l a t i o n s o u t s i d e t h i s range are i n

e r r o r .

Meth

ane

flow rate Ckmol

/hr)

Methane

flow rate (Kmol./hr)

151

FIGURE 4.1: VARIATION OF METHANE FLOWCVAPOUR PHASE) WITH BINARY INTERACTION COEFFICIENT — FLASH # 1

FIGURE 4.2 : VARIATION OF METHANE FLOW (VAPOUR PHASE) WITH BINARY INTERACTION COEFFICIENT — FLASH # 2

152

E th y le n e / E th a n e S u p e r f r a c t io n a t o r (D2.3)

The e t h y l e n e - e t h a n e s p l i t t e r was s im u l a t e d a t the base

p o i n t w i t h t h e s i x 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 s se t to ze ro .

Tab le 4 -1 . p r o v i d e s a summary o f the main s im u l a t i o n r e s u l t s . We

have a l s o shown the r e s u l t s o f Hernandez e t a l f o r compar ison.

We c a r r i e d ou t a s e n s i t i v i t y a n a l y s i s of the column de s ign

t o t h e b i n a r y i n t e r a c t i o n p a r a m e t e r s i n o r d e r t o e n a b l e us

a s c e r t a i n which of the i n t e r a c t i o n terms have s i g n i f i c a n t e f f e c t

on t h e co lumn p r o f i l e s , e t h y l e n e product p u r i t i e s , and r e b o i l e r

and condenser d u t i e s . From a s tudy of the s e n s i t i v i t i e s , we found

t h a t t h e e t h a n e - e t h y l e n e p a r a m e te r ( 6 2 3 ) i s t h e p r e d o m in an t

c o e f f i c i e n t ( T h i s i s i n agreement w i th the r e s u l t s o f Hernandez e t

a l ) . F ig u re s 4 .3 a and 4 .4 shows the s e n s i t i v i t i e s o f the e thy lene

vapour f low and t e m p e r a t u r e p r o f i l e s t o a l l t h e 6 i n t e r a c t i o n

parameters under i n v e s t i g a t i o n . F i g u re 4 .3b i s ob ta in ed from 4 .3a

w i t h 6 2 3 c u r v e o m i t t e d . From t h e l a t t e r f i g u r e we see t h a t

p a r a m e te r 6 2 4 ( e t h y l e n e - p r o p y l e n e ) and 634 ( e thane -p ropy lene ) are

the n e x t most im p o r t a n t s e t o f p a r a m e t e r s . S e n s i t i v i t i e s t o

p a r a m e t e r s 6 2 4 and 6 3 4 a re about two o rde r s o f magnitude sm a l l e r

t h an t h a t t o p a r a m e t e r <5 2 3 . The u n c e r t a i n t i e s of the rema in ing

p a r a m e t e r s 6^2 ( m e th a n e - e t h y l e n e ) , 613 (methane -ethane), and S14

( m e t h a n e - p r o p y l e n e ) can be i g n o r e d . T h u s we c an r a n k t h e

paramete rs i n t h e f o l l o w i n g o r d e r of impor tance on column des ign :

6 23 > <6 24/ 6 34> > ((S12 6 13/ 6 14) -

TABLE 4.1. Main results for ethylene/ethane splitter

Components Distillate Composition Bottom Composition

This work 6ij = 0

Hernandez et al 6ij = 0.0123

This work<5ij = 0

Hernandez et al 6ij = 0.0123

Methane 0.0002 0.0002 0.0 0.0Ethylene 0.9988 0.9960 0.0060 0.0096Ethane 0.0010 0.0037 0.9865 0.9832Propylene 0.0 0.0 0.0070 0.0072

Reboiler Duty(GJ/hr) 46.19

Condenser Duty(GJ/hr) 38.18

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)

47-,

4b- * * *

_ 45-JZ

*—> 44 —

; 43-

taQ)-*= 40-

QC(EXTRAPOLATION)

QC(ACTUAL)

-h- QR(EXTRAPOLATION)

* - QR(ACTUAL)

39-

.0000 .0050 .0100 .0150 .0200 .0250 .0300 .0350ethylene-ethane binary interaction coefficient

. 0400

157

A s e n s i t i v i t y a n a l y s i s o f the column a t a new base po in t

w i t h 623 = 0.0123 ( a l l o th e r <$ i j ' s s e t t o ze ro ) as recommended by

Hernandez e t a l gave e s s e n t i a l l y the same c o n c l u s i o n s r e a c h e d

above.

(<$2 3) “ Figure 4.6.

The no rm a l i s ed s e n s i t i v i t i e s o f the column p r o f i l e s t o the

Murphree t r a y e f f i c i e n c y a t the base v a l u e o f u n i t y f o r a l l t r a y s

were computed and p l o t t e d i n F i g u re 4 .7 a . The f i g u r e shows tha t

u n c e r t a i n t i e s i n p l a t e e f f i c i e n c y a f f e c t s t h e l i g h t k e y

( e t h y l e n e ) , heavy key (e thane) components, and methane s l i g h t l y .

The s e n s i t i v i t y o f one o f t h e i m p u r i t i e s ( p r o p y l e n e ) i s more

p ro noun ced t h a n t h a t o f t h e o t h e r components i n the r e c t i f y i n g

s e c t i o n . The e f f e c t o f t r a y e f f i c i e n c y as one would expect covers

e s s e n t i a l l y a l l o f the c o i unm<t The e f f e c t on

e th y l e n e f low i s s l i g h t l y more than t h a t on ethane ( F i g u r e 4 . 7 b ) .

The r e v e r s e i s t h e c ase a t t h e r e b o i l e r end o f the column. The

s e n s i t i v i t i e s i n both s e c t i o n s of the column i s about the same (on

both ab so lu t e and r e l a t i v e b a s i s ) . The e f f e c t of u n c e r t a i n t i e s i n

e f f i c i e n c y on r e b o i l e r and c o n d e n s e r d u t i e s i s shown i n F i g u r e

4 . 8 . E f f i c i e n c y a f f e c t s t h e r e b o i l e r d u t i e s much more than the

c o n d en se r d u t y . In f a c t , t h e e f f e c t on the c o n d e n s e r du ty i s

v i r t u a l l y n e g l i g i b l e . P u r i t i e s o f t h e po lym e r grade e thy lene

computed a t f i v e d i f f e r e n t e f f i c i e n c i e s r ang ing from 0 .5 - 1 .0 are

p l o t t e d i n f i g u r e 4 . 9 . P r e d i c t i o n s from l i n e a r a n a l y s i s a re a l s o

shown. The p u r i t y o f e t h y l e n e v a r i e s q u i t e d r a m a t i c a l l y w i t h

158

NORMALISED SENSITIVITIES OF VAPOUR FLOW PROFILES TO EFFICIENCY-- EXAMPLE D2.3

Figure 4,7a;

SENSITIVITIES OF VAPOUR FLOW PROFILES TO EFFICIENCY — EXAMPLE B2. 3

159

VARIATION OF RE BOILER AND CONDENSER DUTIES WITH EFFICIENCY---- EXAMPLE D 2 . 3

Figure 4.8

ETHYLENE PRODUCT PURITY VARIATION WITH EFFICIENCY

160

e f f i c i e n c y and a t t a i n s a maximum v a l u e a t e f f i c i e n c y of u n i t y .

T h i s i s so f r om the d e f i n i t i o n o f e f f i c i e n c y as the deg ree t o

wh ich e q u i l i b r i u m be tw een t h e vapour and l i q u i d f l u i d phases i s

a c h i e v e d ( t h e c o n d e n s e r i s t r e a t e d a s an e q u i l i b r i u m s t a g e ) .

P r e d i c t a b l y , p roduct p u r i t y i n c r e a s e s w i th i n c r e a s i n g e f f i c i e n c y .

Computat ion of column p r o f i l e s s e n s i t i v i t i e s t o c o n s t a n t

r e l a t i v e e r r o r s i n the e n tha lp y models as d e t a i l e d i n s e c t i o n 4 .3

were c a r r i e d ou t . We obse rved t h a t the order of impor tance of the

e r r o r s i n t h e model f u n c t i o n s a re as f o l l o w s ( F ig u r e 4 .1 0 a ,b ) :

i d e a l l i q u i d e n t h a l p y , v a p o u r e n t h a l p y ( i d e a l and e x c e s s

c o n t r i b u t i o n s ) , and e x ce s s l i q u i d e n th a lp y . The s e n s i t i v i t i e s a re

q u i t e sma l l (on r e l a t i v e and a b s o l u t e b a s i s ) compared t o t h o s e

w i t h r e s p e c t t o b i n a r y i n t e r a c t i o n p a r a m e te r 6 3 3 and t r a y

e f f i c i e n c y T h i s r e s u l t i s not s u r p r i s i n g s i n c e t h e m i x t u r e

c o n s i s t s o f hydrocarbons a t moderate p re s su re . T h i s con f i rms the Ange l

e t a l /S (1986) o b s e r v a t i o n t h a t t h e use o f v a r i o u s m ode ls f o r

e x c e s s e n t h a l p y p r e d i c t i o n had v i r t u a l l y no e f f e c t on t h e i r

s im u l a t i o n r e s u l t s . The same o b s e r v a t i o n was made by F r e d e n s l u n d

e t a l , 1 977 . However , t h i s c o n c l u s i o n may not ho ld f o r m ix t u re s

e x h i b i t i n g d i m e r i z a t i o n i n t h e v a p o u r p h a s e o r u n d e r h i g h

p r e s s u r e s . Once a g a i n t h e dom inan t zone i s e v i d e n t f rom t h e

f i g u r e s . F igu re 4.11 a , b , c shows the p u r i t y o f e t h y l e n e i n t h e

d i s t i l l a t e to be a f f e c t e d m a r g i n a l l y by e r r o r s o f as much as t

20 % i n the l i q u i d and vapour e n tha lp y models . 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 re v i r t u a l l y u n a f f e c t e d by e r r o r s i n the vapour

o r indeed e x c e s s l i q u i d e n t h a l p y f u n c t i o n s ( F i g u r e 4 . 1 2 a , c ) .

•161

SENSITIVITIES OF ETHYLENE FLOW PROFILE (VAPOUR PHASE) TO ERRORS IN ENTHALPY MODELS --- EXAMPLE D2.3

Figure 4.10a:

Figure 4.10b: SENSITIVITIES OF TEMPERATURE PROFILE TO ERRORSIN ENTHALPY MODELS

stage5

mole

fracti

on

Figure 4.11a: e t h y l e n e p r o d u c t p u r i t y v s . errors in v a p o u r e n t h a l p y m o d e l--- EXAMPLE D2.3

Figure 4.11c: . e t h y l e n e p r o d u c t p u r i t y v s . e r r o r s in e x c e s s l i q u i d e n t h a l p yMODEL --- EXAMPLE DT. 3

1. 0 b 0 —i

1. 040 -

C 1-020- o

u.I: 1- 000~:------------ «------------*------------«-------<u

e 0. 980-

0.9b0 -

0. 94 0-1-------- .-------- ,-------- -------. 200 -.150 -.100 -. 05C

c o n s t a n t re', a

— t- EXTRAPOLATION

actual

152 n ne p r r c

mole

fracti

on

Figure 4.11b: e t h y l e n e p r o d u c t p u r i t y v s . e r r o r s in i d e a l l i q u i d e n '-a l p yMODEL --- EXAMPLE D2.3

cnro

200

heat

duty

— (

GJ/hr

) He

at Du

ty CG

J/hr)

163

FIGURE 4.12a : VARIATION OF REBOILER AND CONDENSER DUTIES TO ERRORS IN VAPOUR ENTHALPY MODEL — Example D2. 3

VARIATION OF REBOILER AND CONDENSER DUTIES TO ERRORS IN IDEAL LIQUID ENTHALPY MODEL --- EXAMPLE D2. 3

Heat Duty

CGJ/hr)

FIGURE 4.12c : VARIATION OF REBOILER AND CONDENSER DUTIESTO ERRORS IN EXCESS LIQUID ENTHALPY MODEL — Example D2.3

1/91

Effects of u n certa in t ies in the ideal l iqu id enthalpy are much more

pronounced as shown in Figure 4-12b.

Propylene/Propane Superfractionator (D2.4, D2.5)

The two propylene/propane s u p e r f r a c t i o n a t i o n u n i t s were

s imulated assuming once again the in teract ion coeffc ients have a

value of zero. As in the ethylene/ethane test problem, our aim i s

to be a b le to i d e n t i f y which of the 6 p a r a m e t e r s a f f e c t

substant ia l ly the computed column p r o f i l e s , product p u r i t i e s , and

r e b o i l e r and condenser d u t i e s . The main simulation r e s u l t s as

well as the values reported by Hernandez e t a l are presented in

Table 4 .2 .

The generated s e n s i t i v i t i e s of the two columns to S i j ' s

are plotted in Figures 4 .13 (a ,b ) and 4 . 1 4 ( a , b ) . Both p lo t s are

s i m i l a r and show that the c r i t i c a l parameter i s between propylene

and propane - $1 2 - The complete order of importance of the s ix

parameters i s as fo l low s :

612 > <614/ 523/ $13/ <s24) > 534

This i s again in agreement with the co n c lu s io n of Graboski and

Daubert that only in te ra c t io n s between hydrocarbons of ident ica l

molecular weight but d i f f e r e n t m olecu lar s t r u c t u r e are of any

importance. The presence of a s e n s i t ive region within the columns

i s e as i ly id en t i f ie d ( trays 15-40 f o r example D2.4 and 10-34 in

TABLE 4.2. Main results for propylene/propane splitters

Distillate Composition Bottom Composition

This work Hernandez et al This work Hernandez et alSij = 0 Sij = 0.0075 Sij = 0 6ij = 0.0075

No. of stages = 117ComponentPropylene 0.9998 0.9998 0.024 0.024Propane 0.0002 0.0002 0.888 0.888Propadiene 0. 0.0 0.044 0.044Propyne 0. 0.0 0.044 0.044Reboiler duty (GJ/hr)

26.25 — —

Condenser duty (GJ/hr) 25.97 - —

No. of stages = 96ComponentPropylene 0.9998 0.9999 0.0377 0.0376Propane 0.0002 0.0001 0.8457 0.8458Propadiene 0.0 0.0 0.0615 0.0613Propyne 0.0 0.0 0.0552 0.0552Reboiler duty (GJ/hr) 26.25 — _

Condenser duty(GJ/hr) 25.97 *—

167SENSITIVITIES OF PROPYLENE VAPOUR FLOW PROFILE TO ALL THE

BINARY INTERACTION COEFFICIENTS 6(i,j) = 0.0

Figure 4.13a

Example D2.4

Figure 4.13b

SENSITIVITIES OF PROPYLENE VAPOUR FLOW PROFILE TO ALL THE BINARY INTERACTION COEFFICIENTS — 6(i,j) = 0.0 EXAMPLE D 2 .4

sensitivity

— dFY/d8(i,j)

Figure 4.14a 168SENSITIVITIES OF PROPYLENE VAPOUR FLOW PROFILE TO ALL THE BINARY

INTERACTION COEFFICIENTS — 6 ti, j) = 0.0 C EXAMPLE D2.5 )

35, 000-,

30, 000

25, 000

2 0, 000

15, 000

1 0 , 0 0 0

5, 000

0

10 20 30 40 50 b0 70 80 90stage number

Figure 4.14b

SENSITIVITIES OF PROPYLENE VAPOUR FLOW PROFILE TO ALL THE BINARY INTERACTION COEFFICIENTS — S C i, j) = 0.0 C EXAMPLE D2. 5 )

169

the case of example D2.5). The s e n s i t i v i t i e s of column p ro f i le s

t o 6 -j 2 a t t a in ' a maximum va lue at s tag e s 30 and 25 (from the

reboi ler end) for problems D2.4 and D2.5 respect ive ly . Since the

i n t e r a c t i o n between the l i g h t and heavy key components ( ^ 2) i s

the most c r i t i c a l of a l l the f iv e cross-term co ef f ic ie n ts we

s imulated the column with various values of 612 ranging from 0 -

0.0375 at in te rv a l s of 0.0075. The p u r i t i e s of propylene in the

d i s t i l l a t e were then computed from resu l t s of repeated rigorous

s im u la t io n s and p lo t t e d in F igure 4 . 1 5 and 4 . 1 6 . P u r i t i e s

p re d ic te d by l i n e a r e x t r a p o l a t i o n a re a l s o shown on the same

graph. The shortcomings of the l inear a n a l y s i s i s ev ident s in c e

i t p r e d i c t s in a c c u r a t e p u r i t i e s at points f a r away from the base

va lue of = 0 . From the actual simulation re s u l t s we note an

a p p r e c i a b l e e f f e c t of u n c e r t a i n t i e s i n &12 on p u r i t y . The

p re d ic te d p u r i t i e s as i n the e th y le n e /e th a n e case d e c r e a s e

sh a rp ly with i n c r e a s in g va lues of 612- Hernandez et al found an

optimum value of 2 = 0.0075 which i s within the l inear region.

A s e n s i t i v i t y ana ly s is of both columns at a new base value of

= 0.0075 gave the same c o n c lu s io n s ( F i g u r e s 4 .15 and 4 .16 bear

t h i s o u t ) . The e f f e c t s of u n c e r ta in t ie s in 6^2 on reboi ler and

condenser duties are roughly the same - Figures 4.17 and 4.18.

In f a c t , the r e s u l t s for both propylene/propane examples

a re s i m i l a r . From T a b l e 4 . 2 and f i g u r e s 4 . 1 3 , 4 . 1 4 one

immediately n o tes th at the same propylene product p u r i t i e s are

(GJ

/hr)

m

o 1e

fr

ac

t i

on

Figure A.15:

VARIATION OF PROPYLENE PRODUCT PURITY WITH PROPYLENE-PROPANF BINARY INTERACTION COEFFICIENT --- EXAMPLE 2.4

Figure 4.16:VARIATION OF REBOILER AND CONDENSER DUTIES WITH PROPYLENE-PROPANE

BINARY INTERACTION COEFFICIENT --- EXAMPLE D2. 4

heat d

uty

(GJ/

hr)

mole

fr

acti

on

171VARIATION Of- PROPYLENE PRODUCT PURITY WITH PROPYLENE -PRUPANh.

BINARY INTERACTION COEFFICIENT --- EXAMPLE D2. 5

Figure A.17:

Figure 4.18:VARIATION OF REBOILER AND CONDENSER DUTIES WITH PROPYLENE-PROPANE

BINARY INTERACTION COEFFICIENT EXAMPLE D2. 5

heat

du

ty ---

(GJ/

hr)

figure 4.19: sensitivities of vapour flow profiles to efficiency EXAMPLF 02 . 4

FIG4.21 VARIATION OF REBOILER AND CONDENSER DUTIES WITH EFFICIENCY --- EXAMPLE D2.4

2b. 300n

2b. 250--

2b.200-

2b.150-

2b. 100-

2b. 050-

-e- QC-EXTRAPOLATION

— QC-AC7UAL

QR-EXTRAPOLATION

q r -actlai

2b. 00C------------- «------ •------ »-____ii------------- o- ■ — ........o— ... a a---- .2 5 . 950-4------ 1--------------------------------------- 1---------1------------------ ;------------------

. 5BC . 550 . bCC . b5C . 7CC . 750 . 800 . 850 . 90C . 950 1 . 000p r * . c i e n c y

mole

fr

acti

on

Figure 4.20 VARIATION 0" pROPYLENE PRODUCT PURITY WITH EFFICIENC --- EXAMPLE D2.4

174

obtained when S-jj = 0.0 or 0.0075. In other words, the purity of

d i s t i l l a t e product i s in se n s i t iv e to errors of as much as + 0.0075

in 6.,-j v a lu e s . S e n s i t i v i t i e s of propylene to Sjj are far from

negl igib le near the reboi ler and would therefore, have s ig n if ican t

e f f e c t on t ra y des ign or control design. We confine our re su l t s

henceforth to only the 117 stage example problem.

S e n s i t i v i t i e s of the vapour flow p ro f i le to e f f ic ie n c y i s

shown in Figure 4.19. Actual propylene p u r i t i e s were determined

at f i v e d i f fe re n t values of e f f i c ie n c y and plotted in Figure 4.20

(The data obtained by l i n e a r e x t r a p o l a t i o n are a l s o d e p ic t e d ) .

Here a g a i n the p u r i t y i n c r e a s e s with i n c r e a s i n g e f f i c i e n c y .

Linear predict ions of pur ity are accurate for errors of about 15 %

from the base va lu e of e f f i c i e n c y . The reboi ler and condenser

d u t ie s are a f f e c t e d on ly s l i g h t l y by e f f i c i e n c y a c c o rd in g to

Figure 4 .21 .

F igure 4 .2 2 shows the s e n s i t i v i t y of temperature prof i le

to constant re la t i v e errors in the enthalpy models. (Temperature

p r o f i l e s e n s i t i v i t y i s shown because temperature measurements are

usually taken for composit ion c o n t r o l ) . Once more we observe that

e r r o r s in these models have n e g l i g i b l e e f f e c t on column flow

p r o f i l e s , and product p u r i t i e s (F igures 4 .23) . Also the r e b o i l e r

d u t ie s a re a f f e c t e d more by u n c e r t a i n t i e s in vapour and l iquid

entha lpy models in comparison to condenser heat load (F igu re

4 .2 4 ) .

Mole fraction

175FIGURE 4.23 : PROPYLENE PRODUCT PURITY Vs. ERRORS IN ENTHALPY MODELS

-- Example D2. 4

1. 0b0-i

1. 040-

1.020- *

vap-enth C i d+ex) l i q-enth C i d) l iq-enth(ex)

1. 000

0. 980-

0. 960-

0. 940------------- 1—-. 200 “.150

— ,------ 1------- 1------ 1------ 1- . 100 - . 050 . 000 . 050 . 100Constant relative error in models

— I— .150 .200

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

REFERENCESABRAMS, D .S. and PRAUSNITZ, j 7 m7 ~ ( 1 9 7 5 ) ,

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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)

Com oone nt s j j h i

1 2 -138 .84 162.131 3 132.43 - 77.131 4 -145 .56 172.731 5 218.17 -147.811 6 1441.57 -108 .93

N-hexane (1) 2 3 147.22 - 76.06M e th y lc y c lo pen tane (2) 2 4 -118 .82 144.37Benzene (3) 2 5 89.77 - 48.05Cyclohexane (4) 2 6 1385.93 -118 .27To luene (5) 3 4 - 85.00 192.72E th an o l ( 6 ) 3 5 -220 .57 330.67Water (7) 3 6 947.20 -138 .90

3 7 2057.42 115.134 5 83.67 - 44 .044 6 1269.49 -113 .705 6 1009.48 -141 .166 7 - 71.06 387.38

I§i?Le_C1 -5_ i_ yN IQ yAC_Pa ra rae te rs_ (P rausn itz_e t-_a l-^ _1980 )

Com pone nt RU QU QP

N-Hexa ne 4 .50 3 .86 3.86

Me they c lo pentane 3.97 3.01 3.01

Benzene 3 .19 2 .40 2.40

Cyclohexane 3.97 3.01 3.01

To luene 3 .92 2.97 2.97

E th an o l 2 . 1 1 1.97 0.92

Water 0 .92 1.40 1 . 0 0

T ab le C1.6 : S p e c i f i c h ea t c a p a c i t y c o n s ta n ts used i n ou r model

(R e id e t a l . , 1977)

Cp = d0 + d-j T + d2 + d3 T3

wher'e Cp i s i n J /m o le and

T i s in K e lv in

Com pone nt d 0 d 1 d 2 d3

Methane 4 .598 1 .245 E -2 2 .860E-6 -2 .7 03E -9Ethane 1.292 4 .254E-2 -1 .657E-5 2.081 E-9E th y le n e 0 .909 3 .740 E-2 -1 .994E-5 4 .192E -9Propane -1 .0 0 9 7 .315E -2 -3 .7 8 9 E-5 7 .678E -9P ropy lene 0 . 8 8 6 5 .6 0 2 E-2 -2 .771 E-5 5 .2 66E-9Pro pa d iene 2.366 4.723 E-2 -2 .8 22E -5 6.645 E-9Propy ne 3.513 4.453 E-2 - 2 .8 0 3 E-5 7.701 E-9Butane 2.266 7 .913 E-2 -2 .6 4 7 E-5 -0 .6 74E -9Is c rB u tan e -0 .3 3 2 9.1 89 E-2 -4 .4 09E -5 6 .915E -9Pe nta ne - 0 . 8 6 6 1 .1 64E-1 -6 .1 6 3 E-5 1 .267E-8Iso -Pen tane -2 .2 7 5 1 .2 1 0E-1 - 7 .4 4 9 E-5 1 .5 5 1 E-8

APPENDIX C2

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) .

SNST.N2.1 = - DESTSNST.N2.2 = - RFLX S1.N2.2 = - 1

S1.N2.3dHF1

dTF1

221

L co r re sp on d in g to a feed L o ca t io n

s L . i . 3 + k = " 1

9HFL

s l.N2.3+K = “ - r --------3FZL.k

L co r re sp o n d in g t o a feed Lo ca t io n

S1.N2.3+K = 1

The bLock t r i d i a g o n a L s ys t e m f o r the coLumn procedure i s soLved u s ing

Thomas e L im in a t io n aLgo r ithm a t each i t e r a t i o n to o b t a i n a Newton s t e p

c o r r e c t io n x-

T a b L e C 2 . 2 : C_o.mg ut_a_ti_o_n__o f _a__Newton_ s t e p _i_n_ NjjpJvt aL i _ - __S_ajldholm

aLggri.thm

f 2 £W3£d_eL im in a t i on__step

L = 1C-j = C ^ ) " 1 C-j

F-i = F-j

JL_=_2i:__i _i _i __NSJ-1

(C^) - (B — A | _ C ^ (F “ A(_F^^

( F L) = <BL - AL CL- - , ) “ 1 CF l - A iF i - ' i

i_=_NSJ

-1f NST = (bNST - aNST cNST-1} ( f NST “ aNST Fn ST-^

§ac]<ward_ S u b s t i t u t io n

At the end o f the fo rw a rd s tep s

XNST = " f NST

and s u c c e s s iv e backward s u b s t i t u t i o n

X L = - C F - CL F L+1)

222

The derivatives of reboiler and condenser duties with respect to the reflux ratio (RFLX) is given as:

5q cd (RFLX) = I

dHVNST-1 dFY.

dFYNST-1, i

NST-1, i d (RFLX)

+dHVNST-1dTNST

dT „ „NST-1d (RFLX)

-XdHL.NST

i dFX.NST, i

dFX . dHL NST,i , NST dT.'NSTd (RFLX) ^TNst ^ (RFLX)

- XdHVNST

dFYNST, i

dFYNST, i dHVNSTd(RFLX) dTNST

dT'NSTd(RFLX)

(C2.8)

d2;d (RFLX) - X-

dHV. dFY If ii dFY.If i d (RFLX)

dHV

dT.

dT

d (RFLX)

dHL. dFXIf i dHL

i dFX. . d(RFLX) dT1,1 1

dT

d (RFLX)

- X -dHL.

i dFX,2 f i

dFX 2,id(RFLX)

dHL.

dT.

dT,

d (RFLX)(C2.9 )

The derivatives of Q^, Qr with respect to other input variables are obtained in a similar way.

Tab le C2.3Test Prob lems f o r E v a lu a t in g th e E f f i c i e n c y o f

F la sh and D i s t i l l a t i o n P rocedure D e r i v a t iv e s Computat ion

For F lash Procedure :

Problem C2.1 C2.2 C2.3 C2.4 C2.5 C2.6 C2.7Components

N it ro g e n 477.00 20.50Carbon D io x id e 511.92 1482.40 6878.00 1546.90 6 . 0 0

Hydrogen S u lp h id e 304 .10 24.00Methane 2581 .80 3861.00 488.30 6 6 . 0 0Ethane 361 .48 2944.00 3 .00Propane 1164.50 1710.00 2891.40 1 . 0 0

Iso -bu tane 745 .60Butane 90.58 380.40 607.00 1831.80Iso -pen tane 864.20Pentane 1180.10 'Hexane 63.50 245.00 1813.50 18.60Heptane 2633.30Octane 1851 .60Nonane 3 .70 1671.30Decane 0 . 1 2 5.00 832.10Undecane 1214.70Methyl cyc lopen tane 20.70Benzene 27 .0 15.30Cyclohexane 11.80Toluene 28 .60E thano l 23 .0 5 .0Water 50.0

Temperature (K) 332.0 311.0 3 .110 309.0 225.0 345.15 360.15

P re ssu re (bars) 19.0 56.2 56.2 4 .39 60 .78 1 .013 1.013Thermodynamic Model SRK SRK SRK SRK SRK UNIQUAC UNIQUAC

£27

Problem :

224

Tab le C2.4S p e c i f i c a t i o n o f D i s t i l l a t i o n Column U n i t O pe ra t ion

C2.8

ComponentFeed Com pos it ion (mole p e rcen t)

EthanePropaneButanePentaneHexane

25.4231.8719.3313.559 .83

Feed temperature (K) Feed P ressu re (bars) Feed ra te (Kmol /h r) Vapour f r a c t i o n in feed

Column C o n f ig u ra t io n

= 272.75 = 5 .00 = 500.0 = 0 .2042

Number o f s tages Feed s tageCondenser p ressu re (ba rs )

= 7 = 3 = 5 .0

P re ssu re d rop /s tage (ba rs ) = 0-0D i s t i l l a t e Rate (kmol / h r ) = 150.0R e f lu x Ra t i o Type o f Condenser

P h y s ic a l P r o p e r t ie s

= 0 .80 P a r t i a l

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 t h a lp ie s - 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 Appendix C1)

B in a ry in t e r a c t i o n c o e f f i c i e n t s a re s e t t o z e r o .

Table C2.5 : Analytic distillation procedure derivatives at the base point given in Table 3.7

&(•)*aCKFLX). v (kmol /hr)a(.)*

Distillate

Ethane 14.7515 0.3161Propane -14.6402 0.6825Butane - 0.1113 0.1378E-2Pentane - 0.4816E-4 0.6487E-6Hexane - 0.2535E-7 , 0.3881E-9

Bottom

Ethane -14.7515 -0.3161Propane -14.6402 -0.6825Butane 0.1113 -0.1378E-2Pentane 0.4816E-4 -0.6487E-6Hexane 0.2535E-9 -0.3881E-9

*(.) Flowrate of component i in the distillate or bottom product stream

226

A P P E N D IX C 3

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 lo w s h e e t in g P ro b le m s

Tab le C3.1 : Di s t i 11at ion_Colurnn_Desi cjn

These prob lems d e f i n i t i o n s a re the same as g iv en i n problem C2.8

w i t h the f o l lo w in g d i f f e r e n c e s - (The p h y s i c a l p r o p e r t i e s a s s u m p t io n s

are a ls o the same).

Problem C3.1a C3.1b

D i s t i l l a t e Rate (K mol / h r ) 150-0

Mole Pe rcen t o f l i g h t - k e y 7 9 .0 79 .0component (Ethane) i n the D i s t i l l a t e p roduct

Mole Pe rcen t o f heavy-key 14.0component (Hexane) i n the Bottom Product

Tab le C3.2: CAVETT fo u r f l a s h f lo w sh e e t

Problems components

C3.2 C3.3 C3.4Feed

F low ra te s

C3.5

(Km ol/h r)

C3.6

N it ro g en 452 .0 358.2Carbon D io x id e 512.0 512.0 2257.0 6698.0 4965.6Hydrogen s u lp h id e 339 .4Methane 1362.0 3782.0 2995.5Ethane 363 .0 363 .0 3025.0 2395.5Propane 1041.0 2893.0 2291.0Iso -bu tane 604.1Butane 1 2 1 . 0 1 2 1 . 0 700 .0 2707.0 1539.9Iso -Pen tane 790.4Pentane 1129.9Hexane 93 .0 93 .0 802.0 7944.0 1764.7Heptane 2606.7Octane 1884.5Nonane 759 .0 1669.0De cane 27.0 27.0 7020.0 831.7Undecane 1214.5

227

S p e c i f i c a t i o n s

T1 (K) 322.0 342 .0* 311.0 311.0 311.0T2 (K) 311.0 311-0 322.0 322.0 322.0T3 (K) 301.0 301.0 309.0 309.0 309.0T4 (K) 293.0 293.0 303.0 303.0 303.0

1 (bar) 19.0 19.0 56.2 56.2 56.22 (bar) 1 0 . 0 1 0 . 0 19.6 19.6 19.63 (bar) 3 .5 3 .5 4 .39 4 .39 4 .394 (bar) 1 .013 1.013 1.91 1.91 1.91

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 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

Feed Flow ra te s (Kmol /h r )

Components

Ethane 15.0P ropy lene 35.0Propane 30 .0Iso -bu tene 2 0 . 0

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 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

i j3FY i 3 FY 2

3 6 . .ij3 FY 336. .ij

3 FYu3 6 . .ij

M i36. . ij3 FY C36. .ij

3 FY736. .ij

1 2 2.01162 59.3183 0.256742 0.114858E-1 0.117398E-2 0.366691E-4 0.157901E-41 3 0.122951 2.77254 • 0.133581E-1 0.537393E-3 0.548959E-4 0.171347E-5 0.737257E-42 3 0.555337 36.0998 0.689035 0.725937E-2 0.793895E-5 0.263154E-4 0.119639E-41 A 0.471923E-1 1.02992 0.446253E-2 0.285913E-3 0.203377E-4 0.634165E-6 0.272562E-62 4 0.3A3028E-2 2.79692 0.824224E-2 0.353726E-1 0.598811E-4 0.215319E-5 0.102751E-53 A -0.162229 -8.16655 -0.186043E-1 0.119838E-2 -0.166805E-3 -0.529174E-5 -0.232038E-51 5 0.399823E-1 0.867763 0.375895E-2 0.167812E-3 0.290716E-4 0.533766E-6 0.229339E-62 5 -0.32755E-1 0.827451 -0.145689E-3 0.658634E-4 0.482744E-2 0.629025E-6 0.324708E-63 5 -0.140055 -7.05441 -0.163261E-1 "0.139543E-2 0.243155E-3 ' -0.458779E-5 -0.201361E-5A 5 -0.6018A6E-1 -3.03769 -0.133177E-1 0.336805E-3 0.898359E-4 -0 .199894E-5 -0.879874E-61 6 0.999665E-2 0.216771 0.938948E-3 0.419156E-4 0.427663E-5 0.617118E-6 0.572727E-72 6 -0.934152E-2 0.146328 -0.322835E-3 0.275275E-5 0.214041E-5 0.194981E-3 0.577105E-73 6 -0.351187E-1 -1.76910 -0.410456E-2 -0.350088E-3 -0.362552E-4 0.145441E-4 -0.505493E-6A 6 -0.150880E-1 -0.761617 -0.333970E-2 0.83880E-4 -0.157470E-4 0.567766E-5 -0.220803E-65 6 -0.1300A3E-1 -0.656758 -0.288239E-2 ~0.130960E-3 0.188709E-4 0.482285E-5 -0.191128E-61 7 0.319393E-1 0.692487 0.299949E-2 0.133899E-3 0.136615E-4 0.425835E-6 0.444018E-62 7 -0.304120E-1 0.437504 -0.117680E-2 0.171290E-5 0.604882E-5 0.313914E-6 0.105327E-33 7 -0.112254 -5.65492 -0.131255E-1 -0.111915E-2 -0.115904E-3 -0.368126E-5 0.685330E-5A 7 -G.482259E-1 -2.43442 -0.106754E-1 0.267855E-3 -0.503391E-4 -0.160334E-5 0.262844E-55 7 -0.415657E-1 -2.09924 -0.921354E-2 -0.418626E-3 0.602800E-4 -0.138677E-5 0.222550E-56 7 -0.104200E-1 -0.526306 -0.231040E-2 -0.104992E-3 -0.109133E-4 0.385444E-5 0.556343E-6

Note : Liquid component flow rate sensitivities are obtained by multiplying the corresponding vapour component flow rate sensitivities by -1 .

236

TABLE D2.3 : 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

i J 9FY i96. .ij

9FY296. .ij

9FY396. .ij

9FYh9 6. . ij

9FY?9 6. .13

1 2 5.83904 1.85494 -18.9670 -1.09143 -0.2371321 3 3.37494 2.52377 36.5907 1.77467 0.3510642 3 17.6099 18.5406 232.1640 11 .4272 2.358171 A 0.880910 0.424329 2.6886 0.678904 0.400452E-12 4 -.243508 2.00563 -4.17247 3.87528 -0.703264E-23 4 1.87344 1.73401 24.3334 1 .51832 0.2428341 5 .445607 0.835546E-1 -0.166930E-1 0.77638E-2 0.1483292 5 -1.54889 -0.300917 -17.5094 -0.989856 0.9595263 5 1.11157 1.08462 14.5382 0.726027 ■ 0.2418374 5 .512464E-1 0.101413 0.489164 0.278313 0.107529

Note : Liquid component flow rate sensitivities are obtained by multiplying the corresponding vapour component flow rate sensitivities by -1»

237

238

Table D2 .4 Sensitivities of Reboiler and Condenser Dutiesto Physical Properties - Example D2.3

— —— (GJ/hr)a9

dQr--- (GJ/hr)a 9

ta -i n Pt r V .Interaction Coefficients.,

512 0 .241413E-02 0.307803E-02

513 0.9722 91E-5 0.12 6538E-4

523 1.10691 1.54601

«14 -.198072E-9 -.321180E-0

524 -.14 9632E-2 -.514 690E-3

§34 .223327E-2 -.279358

Mnrnhree Trav Efficiency. 13.8651 13.5503

Model FunctionVapour enthalpy (ideal + excess) .230659E-2 -.2.58899

Liquid enthalpy (ideal) 0.199235E-1 -20.8274

Liquid enthalpy (excess) -0.33037 4E-4 0.3391123E-1

239

Table D2.5: Sensitivities of Reboiler and Condenser Dutiesto Physical Properties

D 2 .4Test Problem

D 2 .5

——— (GJ/hr)a9

— (GJ/hr)' d6

(GJ/hr)a9

(GJ/hr)a9

Binary Interaction Coefficient

512 .51007 IE-2 -.553162 . 310043E-1 -.843521

S13 -.157972E-4 -.198724E-1 -. 165097E-3 -.457325-1

523 .149269E-4 -.663503 .156050 -.882229

814 -.141919E-4 -.271509E-1 -.133661E-1 — .56203 6E-1

524 .13417 6E-4 -.806568 .12 66412E-3 -.965578

534 -.240142E-7 -.251405E-1 -.324 638E-6 -.44114E-1

Mnrphree Tray Efficiency Modal Function

1051.271 1045.28

Vapour enthalpy (ideal + excess) .2 9432 9E-7 -1.04808

Liquid enthaply (ideal) .108830E-7 .744571

Liquid enthalpy (excess) -.36004E-10 .414028E-3

CO CO

240

Table D2.6 Sensitivities of Cavett four flash design (5-component example) to binary interaction parameters

S *ji

s l l , l S1 1 ,2 sll,3 sll,4511.5

510.1510.2S10,3s i 0 ;a510.5

S9 ,1s9 2 q * b9 3 c * s9,459.5

5*,2

* Sji~

asjid612(kmol /hr)

56i3(kmol /hr)

asjid614(kmol /hr)

asjiasp

(kmol /hr)

2.6610E-2 3.1704E-1 9.8453E-1 6.0427E-1-4.6113E-2 -5.0992E-1 -1.3913 -8.7111E-18.7264E-1 1.1852 -1.2534 -1.12178.5369E-1 9.4420E-1 2.0119 3.09681E-11.8500E-2 1.5776E-2 2.1507E-2 1.0250E-2

-2.2881E-2 -3.1265E-1 -9.8164E-1 -6.0432E-15.0846E-2 5.1832E-1 1.3982 8.7128E-1

-3.5238E-1 -1.1520 1.2782 1.1219-1.0026 -9.4476E-1 -2.1701 -3.8587E-1-6.4155E-3 -1.1373E-2 2.7519E-2 -2.9371E-2

-2.6175E-1 -2.2525 -6.3716 -3.88003.3316E-2 1.7066 5.5269 3.4959

-1.2450 -2.2396 6.4558E-1 8.7847E-1-1.0805 -1.1551 -2.3992 -4.9262E-1-1.7765E-2 -2.3273E-2 2.8659E-2 -2.7813E-2

-5.7069E-2 1.4839E-1 6.7078E-1 5.2965E-1-1.5307E-1 -6.3835E-1 -1.5697 -9.1874E-16.9446E-1 9.3952E-1 -1,5667 -1.20877.0567E-1 7.3314E-1 1.6453 2.3690E-18.6226E-3 2.3565E-3 -2.2445E-3 3.5649E-3

fl«Dwrate of compt)nent i in streai i j of flowsheet shown in Figure 3.4

flow

rat

e Ckmol

/hr)

flow

ra

te

(kmol

/hr)

241

Figure U2.1: VAPOUR FLOW RATE PROFILE — EXAMPLE D2.3

Figure D2.2: LIQUID FLOW RATE PROFILE — EXAMPLE D2.3

flow

rat

e (kmol

/hr)

flow

rat

e (kmol

/hr)

242

Figure D2.3: VAPOUR FLOW RATE PROFILE---- EXAMPLE D2. 4

Figure D2.4: LIQUID FLOW RATE PROFILE ------- EXAMPLE D2.4

flow

rat

e Ckmol

/hr)

flow

rate

Ckmol

/hr)

243

Figure D2.5: VAPOUR FLOW RATE PROFILE -- EXAMPLE D2.5

Figure D2.6: LIQUID FLOW RATE PROFILE — EXAMPLE D2.5

80 90

244

APPENDIX D3

D e t a i l e d S p e c i f i c a t i o n o f Colinnn f o r O p e r a b i l i t y and C o n t r o l S tu d y

Tab le D3.1:: Column S p e c i f i c a t i o n s

Feed Test Problem

D3.1

Isopentane (kmol / h r ) 0 .30

Pentane " 0 .30

Hexane " 0 .30

Feed f lo w ra te (kmol /h r ) 1 . 0 0

R e f lu x r a t i o 5 .33

D i s t i l l a t e r a t e 5.33

Feed s tage 6

Number o f s tages 6

Vapour i n feed 13

Top p ressu re (bar) 0 .

P re ssu re d rop /s tage (bar) 1.013

Condenser type P a r t i a l

Feed Bubb le p o in t

flow

rate

( kmol /hr

245Figure D3.1: LIQUID FLOW RATE PROFILE --- EXAMPLE D3.1

Figure D3.2: VAPOUR FLOW RATE PROFILE — - EXAMPLE D3. 1

4. 0

3. 0

2. 0

1. 0

0 . 0

b 7 8

stage number10 11 12 13

50

40

30

20

10

00

JRE D3.3 RIGOROUS ENRICHMENT FACTOR PROFILE AT THE BASE VALUE OF S(i,j) = 0 . 0

f2

f3 b

s ta g e7 8

number10 11 12 135 9

246

efficient use of thermodynamic data in ......c2.1 schematic representation of a stage in a...

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