a method to extend the domain of convergence for difﬁcult

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A method to extend the domain of convergence for difficultmulticomponent, multistage separation problems

Yoshikazu Ishiia, Fred D. Otto b,*a Fraxon Technology Ltd., 2-18-6 Tanaka, Isogo, Yokohama, Japan

b Department of Chemical and Materials Engineering, Uni versity of Alberta, Edmonton, Canada T6G 2G6

Received 13 February 2002; received in revised form 21 October 2002; accepted 2 December 2002

Abstract

A new computational procedure is presented to attain con vergent solutions for multistage, multicomponent separation problems

for non-ideal fluid mixture systems when starting from very crude initial assumptions. The procedure incorporates a gradual non-

ideality enhancing method into the Ishii Á /Otto pseudo-binary-mixture (PBM) algorithm [Comput. Chem. Eng. 25 (2001) 1285].

Shifting K -values from ideal to non-ideal in the course of iterative calculations gradually approaches the correct solution to a

problem. The domain of convergence with the proposed procedure is greatly extended and very difficult problems where both non-

ideality of fluid systems and very rough assumptions for initial variables are involved can be solved with the same ease as one solves

problems with ideal physical properties. An innovative method, made possible because of the PBM concept, to simplify the Jacobian

matrix and significantly reduce the computing time for evaluation of the Jacobian is also presented. The effectiveness of the methods

is illustrated with example problems.

# 2003 Elsevier Science Ltd. All rights reserved.

Keywords: Separation; Distillation; Non-ideal systems; Process simulator; Jacobian

1. Introduction

It is well known that solutions to multistage separa-

tion calculation problems with ideal fluid mixture

systems are easily obtainable. For such problems, the

domain of convergence is usually very large. Even for

problems with moderately non-ideal systems, conver-

gence is not a problem for many of the equation-solving

methods (e.g. Naphtali & Sandholm, 1971; Ishii & Otto,

1973), provided that extremely remote assumptions forinitial variables are not made. However, for severely

non-ideal fluid systems and often for systems where the

non-ideality of both phases must be taken into con-

sideration, identification of suitable initial assumptions

is crucial for successful convergence. The domain of

convergence for such systems is limited and narrow.

The most sophisticated approaches to extending the

domain of convergence are the homotopy-continuation

methods in which the solution is gradually approached

via a step-wise integration along so-called homotopy

paths starting with an easy initial problem. Examples

are Salgovic, Hlavacek and Ilavsky (1981), Hlavacek

(1985), Byrne and Baird (1985), Chavez, Seader and

Wayburn (1986), Vickery and Taylor (1986), Kovach

and Seider (1987), Lin, Seader and Wayburn (1987).

However, the homotopy continuation methods are very

time consuming and their use does not always ensure

convergence. The thermodynamic homotopy methodproposed by Vickery and Taylor (1986) is worthwhile

mentioning in that it was successful in coping with many

problems that could not be solved with other homotopy

continuation methods. A notable feature of the method

of Vickery and Taylor (1986) is the fact that both

ideality and non-ideality for each physical property, K -

value and enthalpy are combined as a thermodynamic

homotopy and physically meaningless solutions during

the course of integration are prevented. In accordance

with their homotopy, idealities are replaced by non-

ideality gradually with the progress of integration along

the homotopy path.* Corresponding author.

E-mail address: [email protected] (F.D. Otto).

Computers and Chemical Engineering 27 (2003) 855 Á /868

www.elsevier.com/locate/compchemeng

0098-1354/02/$ - see front matter# 2003 Elsevier Science Ltd. All rights reserved.

doi:10.1016/S0098-1354(02)00271-5

mailto:[email protected]

mailto:[email protected]



Nomenclature

A Coefficient in Peng Á /Robinson equation of state

a Coefficient in Peng Á /Robinson equation of state

B Coefficient in Peng Á /Robinson equation of state

b Coefficient in Peng Á /Robinson equation of state

C Component balance function defined by Eq. (5) (kmol/h)E enthalpy balance function defined by Eq. (11) (kJ/h)

F feed rate (kmol/h) or general function in Eq. (12)

F vector of function F defined by Eq. (12)

f function defined by Eq. (14)

G Equilibrium balance function defined by Eq. (7) (mole fraction)

h enthalpy of liquid (kJ/kmol)

H enthalpy of vapor (kJ/kmol)

H F enthalpy of feed (kJ/kmol)

DH Isothermal enthalpy departure from the ideal gas state (kJ/kmol)

K Equilibrium ratio, K 0/Y /X

K * modified equilibrium ratio based on PBM concept

L flow rate of liquid (kmol/h)

M material balance function defined by Eq. (3) (kmol/h)N total number of equilibrium stages

n number of variables (Eq. (13)); parameter in Eq. (49)

NC total number of components

P pressure (kPa)

Q heat input to stage (kJ/h)

R universal gas constant (kJ/kmol per K)

S X Summation function defined by Eq. (9a) (mole fraction)

S Y Summation function defined by Eq. (9b) (mole fraction)

SL flow rate of liquid side product (kmol/h)

SV flow rate of vapor side product (kmol/h)

T Temperature (K)

t damping factorV flow rate of vapor (kmol/h)

v molar fluid volume (m3/kmol)

x Normalized liquid composition or phase composition in Eq. (48) (mole fraction), or general

variable in Eq. (13)

xc liquid composition of pseudo-binary component (mole fraction)

X liquid composition (mole fraction)

x vector of variable x

y Normalized vapor composition (mole fraction)

Y c vapor composition of pseudo-binary component (mole fraction)

Y vapor composition (mole fraction)

Z feed composition (mole fraction)

Z Compressibility factor

Greek letters

a Coefficient of Eq. (22)

b coefficient of Eq. (22)

g constant of Eq. (22) or activity coefficient

d numerical constant defined by Eq. (26)

o M convergence criterion for material balance

o T overall convergence criterion

u non-ideality enhancing parameter

8 fugacity coefficient in mixture

n pure component fugacity coefficient

L Wilson parameter

F vector of zeros

Y. Ishii, F.D. Otto / Computers and Chemical Engineering 27 (2003) 855 Á / 868856



Ishii and Otto (2001) recently improved the original

formulation of their simultaneous correction procedure

(Ishii & Otto, 1973) by incorporating the pseudo-binary-

mixture (PBM) concept to account for the composition

dependence of the equilibrium K -values. The method

assumes that the pseudo-component derivative of the K -

value is the composition weighted average of the

composition derivatives for all the components repre-

sented by the pseudo-component. The PBM concept

makes it possible to account for compositional effects

and yet retain the advantages and simplicity of the

original formulation of the Ishii and Otto algorithm, i.e.

the Jacobian is simplified by replacing a large number of

cross-component elements by a single element. This

greatly reduces computational time because of very

efficient matrix manipulations. The Ishii Á /Otto PBM

procedure also enjoys a large domain of convergence

even for difficult problems with non-ideal mixture

systems (Ishii & Otto, 2001). However, difficulties in

obtaining convergent solutions are encountered when

very poor guesses are made for the initial variables.

For multicomponent separation calculation problems

with non-ideal fluid mixture systems, the bulk of the

computing time, typically more than 80%, is consumed

for the evaluation of the thermodynamic properties

when Newton’s method or any of its variants is used

(Westman, Lucia & Miller, 1984; Byrne & Baird, 1985).

This means that most of the time for the calculation is

spent in the evaluation of the Jacobian and the majority

of this time is used to obtain all the compositional

partial derivatives of each K -value.

Many studies directed to reducing the computing time

for the evaluation of the Jacobian have been published.

Quasi-Newton methods such as represented by the

method of Tomich (1970) a hybrid approach presented

by Lucia and Macchietto (1983), and Westman et al.

(1984), the dogleg method of Chen and Stadtherr (1981)

are typical examples. The major purpose of model

simplification approaches proposed by such as Perre-

gaard (1993), Ferraris (1983) is also for the reduction of

the computational burden associated with the evalua-

tion of the Jacobian.

These methods have contributed to improvement in

handling of the Jacobian. However, it should be noted

that this improvement is generally attained at theexpense of the quality of calculations, i.e. the stability

of convergence or the domain of convergence.

The present paper presents enhancements to the

Ishii Á /Otto PBM algorithm that extend the domain of

convergence for separation problems involving non-

ideal fluid mixture systems and reduce the time required

to evaluate the Jacobian without degradation of the

quality of calculations. The enhanced PBM method can

be employed to successfully solve difficult problems

starting from very crude initial assumptions and offers a

significant advantage over other methods with respect to

stability of convergence and computational time.

2. Linearization of model equations

The generalized multistage model is shown in our

previous paper, Ishii and Otto (2001). The model

represents a complex column configuration in which

multiple feeds, side streams and/or side heat exchangers

are considered. For linearization, all the model equa-

tions are Taylor series expanded with respect to the

unknown variables and truncated after the linear terms.

The linearized material balance, equilibrium relation-ship, summation and enthalpy balance (MESH) equa-

tions for any component i and any stage j are expressed

as follows:

Overall material balance

DL j (1'DV j '1(DL j (DV j 0(M j (1)

from which it follows that

DL j 0X j

k 01

M k 'DV j '1(DV 1 (2)

where

Subscripts

i component number

j stage number, component number

k component number

Superscripts

A actual

I idealk iteration number

L liquid

V vapor

T transpose of vector

Y. Ishii, F.D. Otto / Computers and Chemical Engineering 27 (2003) 855 Á / 868 857



M j F j 'L j (1'V j '1((L j 'SL j )((V j 'SV j ) (3)

Component material balance

L j (1DX i ; j (1'V j '1DY i ; j '1((L j 'SL j )DX i ; j

((V j 'SV j )DY i ; j 'X i ; j (1DL j (1'Y i ; j '1DV j '1

(X i ; j DL j (Y i ; j DV j

0(C i ; j (4)

where

C i ; j F j Z i ; j 'L j (1X i ; j (1'V j '1Y i ; j '1((L j 'SL j )X i ; j

((V j 'SV j )Y i ; j (5)

Equilibrium relationship

DY i ; j (K i ; j DX i ; j (X i ; j

XNC

k 01

@ K i ; j

@ X k ; j

DX k ; j (X i ; j

ÂXNC

k 01

@ K i ; j

@ Y k ; j

DY k ; j (X i ; j

@ K i ; j

@ T j

DT j

0(G i ; j (6)

where

G i ; j Y i ; j (K i ; j X i ; j (7)

Summation equation

XNC

i 01

DX i ; j 0(S X j (8a)

or

XNC

i 01

DY i ; j 0(S Y j (8b)

where

S X j

XNC

i 01

X i ; j (1:0 (9a)

or

S Y j

XNC

i 01

Y i ; j (1:0 (9b)

Enthalpy balance

L j (1

XNC

k 01

@ h j (1

@ X k ; j (1

DX k ; j (1((L j 'SL j )

Â XNC

k 01 @ h j

@ X k ; j DX

k ; j ((V

j

'SV j )

ÂXNC

k 01

@ H j

@ Y k ; j

DY k ; j 'V j '1

XNC

k 01

@ H j '1

@ Y k ; j '1

DY k ; j '1

'L j (1

@ h j (1

@ T j (1

DT j (1

(

(L j 'SL j )

@ h j

@ T j '(V j 'SV j )

@ H j

@ T j

DT j

'V j '1

@ H j '1

@ T j '1

DT j '1'h j (1DL j (1(H j DV j

(h j DL j 'H j '1DV j '1

0(E j (10)

where

E j F j H F j 'Q j 'L j (1h j (1'V j '1H j '1((L j 'SL j )h j

((V j 'SV j )H j (11)

The Ishii and Otto PBM algorithm is an efficient

simultaneous correction procedure for solving these

linearized equations for severely non-ideal systems.

The following sections describe (i) a gradual enhance-

ment of non-ideality procedure which can be employed

to achieve a greater domain of convergence when using

the PBM concept; and (ii) a procedure made possible by

the PBM concept to significantly reduce the computing

time required to evaluate the Jacobian.

3. Gradual enhancement of non-ideality

This procedure uses the concept of combining ideality

and non-ideality but in a way that is different from that

which was employed by Vickery and Taylor (1986) for

their thermodynamic homotopy.Let

F (F 1; F 2; . . . . . . ; F n; f )T0F (12)

and

x(x1; x2; Á Á Á Á Á Á ; xn; u)T (13)

where F is a vector of zeros, and F is a vector of

functions whose variables are represented by a column

vector x . In contrast to typical and customary expres-

sions for the Newton Á /Raphson equations, function f

and variable u are introduced into Eqs. (12) and (13),

respectively.




Provided that f is only a function of variable u as,

f 0 f (u) (14)

then the Newton Á /Raphson equations are obtained by

expanding Eq. (12) through the linear terms only in a

Taylor series as follows,

@ F 1=@ x1 @ F 1=@ x2 × ×@ F 1=@ xn

@ F 2=@ x1 × × × ×

× × × × ×

× × × × ×

@ F n=@ x1 × × × @ F n=@ xn

266664

377775

Dx1

Dx2

×

×

Dxn

266664

377775

0(

F 1'(@ F 1=@ u)Du

F 2'(@ F 2=@ u)Du

×

×

F n'(@ F n=@ u)Du

266664

377775 (15)

In place of Eq. (14), introduce the following equation,

f 01:0(u (16)

then Du in Eq. (15) is determined by Eq. (16) indepen-

dently of the rest of the equations and is expressed as,

Du01:0(u (17)

Although u is actually a variable to be sought in a

similar manner to the rest of the variables, x ?s, we have

chosen to use u as a ‘parameter’ which controls the

degree of non-ideality of the model equation system at

each iteration level. As will be illustrated later, u is given

particular values as the calculation progresses.

Suppose the following equation for a vapor Á /liquid

equilibrium relationship as a function of u ,

K i 0(1:0(u)K Ii 'uK Ai (18)

where K i I stands for the ideal K -value of a pure

component i and K i A is the actual K -value in a fluid

mixture system for which non-ideality is prevailing. By

means of Eq. (18), a hypothetical K -value can be

evaluated as a combination of ideality and non-ideality.

The value of u , which varies from zero through unity,

represents the degree of non-ideality for the hypothetical

K -value. K -values at both ends of the spectrum are

identical to an ideal K -value and a non-ideal K -value,

respectively, as follows,K i 0K Ii when u00:0 (19)

K i 0K Ai when u01:0 (20)

By the introduction of Eq. (18), the linearized model

equations expressed by Eqs. (1) Á /(7), (8a), (8b), (9a),

(9b), (10) and (11) may be transformed into a general-

ized form of Eq. (15). However, only Eqs. (6) and (7) are

subject to corrections by the introduction of Eq. (18).

The remaining equations are unchanged, since terms

containing a K -value are not present, and thus most of

@ F 1/@ u , @ F 2/@ u , . . . in Eq. (15) are set to zero. The

modified form of Eq. (6) follows as,

DY i ; j (K i ; j DX i ; j (X i ; j

XNC

k 01

@ K i ; j

@ X k ; j

DX k ; j (X i ; j

ÂXNC

k 01

@ K i ; j

@ Y k ; j

DY k ; j (X i ; j

@ K i ; j

@ T j

DT j

0(

G i ; j '

@ G i ; j

@ u

Du

(21)

Eq. (21) can be converted to Eq. (22) by applying a

linearization based on the PBM concept. Details of the

PBM concept were reported in our previous paper, Ishii

and Otto (2001).

DY i ; j 0ai ; j DX i ; j 'bi ; j DT j 'gi ; j (22)

where

ai ; j fK i ; j ( xi ; j (@ K i ; j 1=@ xc

i ; j )gd

i ; j

(23)

bi ; j X i ; j (@ K i ; j =@ T j )

di ; j

(24)

and

gi ; j (fG i ; j ' (@ G i ; j =@ u)Dug

di ; j

(fS X

j x2i ; j (@ K i ; j 1=@ x

ci ; j ) ' S Y

j X i ; j yi ; j (@ K i ; j 1=@ yci ; j )=(1:0' S Y

j )gdi ; j

(25)

further

di ; j 1:0'X i ; j (@ K i ; j 1=@ yc

i ; j )

(1:0 ' S Y j )

(26)

and

G i ; j '

@ G i ; j

@ u

Du0Y i ; j (K Ai ; j X i ; j (27)

The modification to be made by the introduction of

Eq. (18) is limited to the first term in Eq. (25). As a

result, K i , j in Eq. (7) is modified to K i , j A as expressed by

Eq. (27) where Du is eliminated at the end result. The

equation solving procedure described in our previouspaper (Ishii & Otto, 2001) can be employed to solve the

newly derived equations. Actual rather than hypothe-

tical K -values must be used for Eq. (27).

The basic concept used to extend the domain of

convergence for difficult problems is to approach

solutions through the hypothetical K -values by shifting

equilibrium relations gradually from ideal to actual K -

values as the calculation progresses. This gradual shift is

made via a change of u in Eq. (18) from zero through

unity. When this parameter reaches unity, then K -values

are identical to non-ideal K -values of the actual fluid

system and strict calculations follow thereafter.




Similar functional relations that combine ideal and

non-ideal enthalpies can be developed. However, our

experience (Ishii & Otto, 2001) is that the non-linearity

caused by enthalpy relations is insignificant compared

with that of equilibrium relations so that enthalpy

relations are all evaluated based on actual fluid condi-

tions throughout the computational procedure.Intermediate solutions are always sought so as to

satisfy not a transient and hypothetical condition but

the final non-ideal condition even before u reaches

unity. This is understood by the fact that u behaves as

one of the variables and always advances toward unity,

whatever the value of u at the previous iteration level is,

u'Du0u'(1:0(u)01:0 (28)

Thus, corrections to all other variables for the model

equations are sought simultaneously in order to meet a

condition given by Eq. (28), which is the final non-ideal

condition.Corrections to the variables of Eq. (12), i.e. Dx ?s to be

obtained from Eq. (15) and Du from Eq. (17) at the k th

iteration should be used to improve all the variables for

the succeeding iteration as,

xk '1 j 0xk

j 't × Dx j 15 j 5n (29)

and

uk '1

0uk 't × Du (30)

where a damping factor, t , is introduced in order to

prevent the variables from correcting excessively. The

variables in Eq. (29), i.e. x ?s, correspond to the phase

compositions, flow rates and temperatures associatedwith the problem. The recommended criterion to

determine the damping factor for non-ideal systems is

jDT j j5/20 8C. If the factor becomes unity, and this is

most likely to be the case, Eq. (30) is identical to Eq. (28)

and uk '1 also becomes unity. When improving the

variables for the succeeding iteration, however, we

intentionally do not use Eq. (30) and set the value of

u so as to delay its approach to unity at the early stages

of the iteration process. The state of u is predetermined

by Eq. (28) and it always advances towards unity.

Furthermore, the other variables are entirely governed

by Eq. (15) wherein they are directly influenced by thevalue of u . Thus, u is used as a parameter to control the

whole equation system via Eq. (15). Negative values for

the variables are avoided by confining changes to flow

rates to a region one-half to two times the value at the

previous iteration level and by setting any negative

phase compositions to zero.

Difficulties in solving multistage equilibrium separa-

tion problems involving non-ideal mixture systems and/

or remote initial assumptions are obviously incurred

because of a non-linearity effect that is attributable to

such problems. And this degrades the quality of

compositional and often temperature derivatives for

K -values in the Jacobian. A significant feature with

the proposed method is the fact that the extent of non-

linearity accompanying the equilibrium relationship can

be alleviated in the intermediate calculation stage (0.0B/

uB/1.0), where the solution is searched to meet the final

non-ideal condition even though the value of u is yet less

than unity. For example, the partial derivatives of all K -values with respect to composition which appear in the

Jacobian are moderated by multiplying by u (0.0B/uB/

1.0) as follows,

@ K

@ x0(1:0(u)

@ K I

@ x'u

@ K A

@ x0u

@ K A

@ x(31)

In other words, even for calculations at the inter-

mediate stage, a solution is always sought to satisfy the

final condition (u'/Du0/1.0), by using the moderated

compositional derivatives, i.e. u @ K A/@ x . This is also the

case when @ K /@ y is concerned. By the same token, K -

value itself and @ K /@ T , which are evaluated in ahypothetical form at u (0.0B/uB/1.0) are also moder-

ated and influence the Jacobian through a in Eq. (23)

and b in Eq. (24). However, g in Eq. (25) does not

influence the evaluation of the Jacobian, since it is not a

coefficient of any unknown variable.

Prior to entering into the calculation based on

hypothetical K -values combined with ideal and non-

ideal, it is recommended that several calculations using

ideal K -values be initially made. The domain of

convergence for problems with ideal fluid mixture

systems is large and convergence is sufficiently insensi-

tiv

e to initialv

alues so that the solution can be easily

obtained for a wide range of initial values. If the

calculation starts with ideal K -values, it will advance

with ease in the direction of convergence because the

model equations behave more linearly and are thus

easier to solve. But it will not be meaningful to achieve

convergence in this manner, since this solution is

imaginary. Even if the solution based on ideal K -values

is used for the initial guess, it does not guarantee

convergence for actual problems. Thus it will be

necessary to change the direction of convergence by

shifting the quality of K -values from ideal to non-ideal

gradually prior to arriving at a solution based on ideal

K -values.Thus the computational procedure consists of the

following three calculation stages.

1) Initial stage (u0/0.0).

Initiate the calculation by using ideal K -values and

proceed with several iterations until the convergence

criterion reaches a certain value.In this stage, the

calculation advances toward an imaginary solution

based on ideal K -values.

2) Intermediate stage (0B/uB/1.0).

Switch the equilibrium relationship from ideal K -

values to combined (hypothetical) K -values. Calcu-




lations for a few additional iterations are made

using the hypothetical K -values. The degree of non-

ideality increases gradually based on the increase in

u for which particular values are given as the

calculation progresses.

3) Final stage (u0/1.0).

Continue calculations based on actual K -values

setting u equal to unity until the convergence

criterion meets the final specification.

A recommended and more detailed procedure is as

follows,

1) Initial stage (u0/0.0).

a) Assume initial temperature and overall flow

rate profiles (e.g. see Table 1) for the entire

column and estimate the initial phase compo-

sitions using ideal K -values.

b) Continue the calculations using ideal K -values

until the following criterion is satisfied,

o M 50:01)N (32)

where o M is defined as

o M

XN

j 01

f(S X j )2

'(S Y j )2g (33)

This represents a degree of convergence for the

material balance calculation.

2) Intermediate stage (0B/uB/1.0).

a) Conduct the first run by setting u equal to

0.25.

b) Continue with two more runs changing u to

0.5 and 0.75, respectively.3) Final stage (u0/1.0).

a) Continue the calculations by using actual K -

values until the overall convergence criterion,

Eq. (34), is satisfied,

o T510(6)N (34)

where o T is defined as,

o To M

Table 1

Problem statements and solutions

Problem number 1 Problem number 2

Type of column Extractive distillation Reboiled absorption

Number of components 4 12

Number of stages 42 30Pressure (kPa) 101.3 3275.4

Top product (kmol/h) 260.0 1520.0

Reflux ratio 3.0

Feed(s) (kmol/h) 500.0 1820.0

Stage/temperature (8C) (6/48.9) (1/(/20.0)

1000.0 2890.0

(21/37.8) (15/(/20.0)

2960.0

(22/(/20.0)

Final temperature (8C )

Condenser 56.2

Column top 56.3 (/13.5

Reboiler 71.8 95.5

Final v

apor (kmol/h)Condenser 0.0

Column top 1040.0 1520.0

Reboiler 966.8 5959.0

Case A B A B

Initial assumptions

Temperature (8C)

Condenser 40.0 20.0

Column top -35.0 80.0

Reboiler 90.0 110.0 110.0 80.0

Vapor rate (kmol /h )

Condenser 0.0 0.0

Column top 1040.0 1040.0 1520.0 1520.0

Reboiler 1040.0 1040.0 3000.0 1520.0




'XN

j 01

E j

(F j H F j ' Q j ' L j (1h j (1 'V j '1H j '1)

2

(35)

4. A simplified PBM Jacobian matrix

The PBM concept introduced by Ishii and Otto (2001)

uses an overall compositional partial derivative of K -

values in the Jacobian where only a single element

collectively represents all the component derivatives

with respect to every composition as follows,

@ K i 1

@ xci

0(@ K i

@ xi

'1:0

(1:0 ( xi )

XNC

k 01;k "i

xk

@ K i

@ x k

(36)

Eq. (36) can be further transformed to the following,

@ K i 1

@ xci

0(1:0

(1:0( xi )

@ K i

@ xi

'1:0

(1:0 ( xi )

XNC

k 01

xk

@ K i

@ xk

(37)

On examination, it was found that some of the

correlation equations for predicting vapor Á /liquid equi-

librium, where activity coefficients are used, conform to

Euler’s homogeneous function. For example, the Wilson

equation is a homogeneous function for which the

degree is (/1. Accordingly, the second term in Eq. (37)

can be expressed as,

XNC

k 01

xk

@ K i

@ xk

0(K i (38)

The derivation of Eq. (38) is detailed in the Appendix

A. Consequently, when the Wilson equation is em-

ployed, Eq. (36) is equivalent to the following simple

relation which excludes all cross-component derivatives,

@ K i 1

@ xci

0(1:0

(1:0( xi )

@ K i

@ xi

'K i

(39)

where

@ K i @ xi

0(K i

2:0XNC

j 01

(xi Li ; j )

(XNC

k 01

xk L

2

k ;i XNC

j 01

x j Lk ; j

2

(40)

Thus, Eq. (36) can be calculated with great ease

through the significantly simplified form of Eq. (39).

This equation eliminates the need for the time-consum-

ing calculation of all the cross-component partial

derivatives. By using the new functional form of Eq.

(39) in place of Eq. (36), the relative computational

frequencies required to obtain the compositional deri-

vatives of K -value is reduced to 1/NC where NC

represents the number of components in the mixture

system. When a large number of components is in-

volved, the effect of Eq. (39) is decisively significant

since a huge volume of calculations corresponding to the

square of the number of components is necessary for the

evaluation of the compositional derivatives in the

existing Newton’s methods including the Ishii-Otto

(2001) method.The functional form of the UNIQUAC equation and

the UNIFAC equation is also homogeneous of degree

(/1 and thus Eq. (39) can be used similarly to the case of

Wilson’s equation. However, the NRTL equation is of

degree zero so that the following relation can be derived,

@ K i 1

@ xci

0(1:0

(1:0 ( xi )

@ K i

@ xi

(41)

However, for equations of state such as the Peng Á /

Robinson equation, all the functions for correlating

vapor Á /liquid equilibrium are not homogeneous so that

simplified equations based on Euler’s theorem such as

Eqs. (39) and (41) can not be used. When equations of

state are used, @ K i 1/@ xi c and @ K i 1/@ yi

c are expressed as

follows (Ishii & Otto, 2001),

@ K i 1

@ xci

0(K i

(1:0 ( xi )

@ ln 8 Li

@ xi

'K i

(1:0 ( xi )

Â XNC

k 01

xk

@ ln 8 Li

@ xk

(42)

@ K i 1

@ yci

0K i

(1:0 ( yi )

@ ln 8 Vi

@ yi

(K i

(1:0 ( yi )

ÂXNC

k 01

yk

@ ln 8 Vi

@ y k

(43)

The following relation is obtained for the second term

in Eq. (42) when the Peng Á /Robinson Equation of Stateis used.

Table 2

Feed streams for problem number 1

Problem number 1

Feed stage 6 21

Component

Acetone 0.0 250.0

Methanol 0.0 650.0

Ethanol 0.0 50.0

Water 500.0 50.0




@ ln 8 Li

@ xk

0bk

b

bi

b(1

'

1

z ( B (

bi

b

bk

bz(

@ z

@ xk

(A

(z ' 2:414)(z ( 0:414)

2XNC

j 01

(x j a j ;i )

a

(bi

b

Â

bk

bz(

@ z

@ xk

(

A ffiffiffi2

p B

ln

z' 2:414B

z( 0:414B

Â

ak ;i

a'

bk bi

b2(

bi

b

XNC

j 01

(x j a j ;k )

a(

bk

b

Â

XNC

j 01

(x j a j ;i )

a

(44)

where

The value of @ K i 1/@ xi c can be obtained by substituting

Eq. (44) into Eq. (42). However, this calculation,especially when many components are involved, is very

time-consuming. By taking advantage of mutual com-

pensation and/or complementarity among the composi-

tional partial derivatives of all components in the second

term of Eq. (42), this term can be simplified as follows,

XNC

k 01

xk

@ ln 8 Li

@ xk

0bi

b(1' 1

z( B (

bi

bz(X

NC

k 01

xk

@ z

@ xk

(A

(z ' 2:414B )(z ( 0:414B )

2XNC

j 01

(x j a j ;i )

a(

bi

b

Â

z(XNC

k 01

xk

@ z

@ xk

(46)

where

XNC

k 01

xk

@ z

@ xk

0B '2(B ( z)(A ' 2Bz) ( 4Bz

(A ( 3B 2 ( 2B ) ' z(3z ' 2B ( 2)(47)

Eq. (46) together with Eq. (47) clearly indicates that

the same degree of simplification can be obtained when

using the Peng Á /Robinson Equation of State even

though Euler’s theorem cannot be applied. To be more

specific, the evaluation of all the individual cross-

component derivatives is not needed and they can be

represented by a single value obtained from Eq. (46). A

similar relation can be obtained for @ K i 1/@ yi c by repla-

cing liquid composition with vapor composition in Eqs.

(44) Á /(47).

Other equations for the correlation of vapor Á /liquid

equilibrium, where Euler’s theorem is not applicable,

including Chao Á /Seader, Soave Á /Redlich Á /Kwong,Scatchard Á /Hildebrand, Margules, van Laar and

ASOG were examined in the same way. As a result,

the compositional derivatives for all the equationsexamined to date, which cover essentially all of the

existing equations for vapor Á /liquid equilibrium correla-

tion, can be expressed in simplified forms so that the

evaluation of all individual derivatives are not required

whenever the PBM concept is employed. Several equa-

tions such as ASOG contain Euler’s homogeneous

function therein partly and such cases can be more

easily simplified. Whichever correlation equation is

used, the reduction in the computing time required to

obtain the derivatives is decisively significant in that the

greater part of the computing time is spent for the

evaluation of the Jacobian in the existing methods. Thisis a significant outcome of the introduction of the PBM

concept, since the greatest drawback with any of the

Newton-type methods is this huge time requirement for

the evaluation of the Jacobian.

It is our experience that even though the composi-

tional derivatives of K -value are neglected completely,

i.e. neglecting Eqs. (42) and (43), it is possible to solve

the problems with moderately non-ideal mixtures for

which the Peng Á /Robinson Equation of State is used.

Any detrimental effect on convergence has not been

experienced, at least for cases when the gradual non-

ideality enhancing method is applied. Further studies to

@ z

@ xk

0

2A(B ( z)XNC

j 01

(x j a j ;k )=a ' B f(A ( 3B 2 ( 2B ) ( z(z ( 6B ( 2)gbk =b

(A( 3B 2 ( 2B ) ' z(3z ' 2B ( 2)(45)




establish the criteria to determine cases where the

compositional derivatives of K -value can have negligible

effect are warranted. Though it is conservative, our

present recommendation for moderately non-ideal mix-

ture systems is to evaluate the overall derivatives by

using a simplified equation such as Eq. (46) since their

evaluation can be made very quickly.

5. Illustrative examples

Two test problems are presented in this paper to

demonstrate the effectiv

eness of the proposed method.

Problem number 1 is identical to the example 1 of our

previous paper (Ishii & Otto, 2001). Problem number 2

is a three-feed demethanizer used to separate hydrogen

rich feeds in a reboiled absorber. Shah and Bishnoi

(1978) reported the details of this problem. The compo-

nents for Problem number 2 cover a wide range of

boiling points from light ones including hydrogen,

nitrogen and methane through heavy lean oils. It is

interesting to test this problem for the following reasons:

1) The column is operated in the vicinity of the critical

pressure and its fluid system is non-ideal both in

liquid and vapor phases.2) A wide range of components in terms of boiling

point is involved in the system

3) Molar enthalpy values for individual components

are distributed very widely.

4) A broad temperature profile in the column is

expected.

5) A broad flow rate profile is expected.

6) Sudden changes in temperature and flow rate

profiles will be caused by a large quantity of sub-

cooled feed.

7) Convergence is very sensitive to the assumed initial

variables.

Problem statements are presented in Table 1, and

initial temperatures and flow rates are compared with

those of converged solutions for all problems in the

same table. Two cases (Case A and Case B where Case B

is more difficult) are presented for each problem. Initial

temperature and flow rate profiles were assumed to be

linear by interpolating values given at the top and

Table 3

Feed streams for problem number 2

Problem number 2

Feed stage 1 15 22

Component

Hydrogen 0.0 0.0 562.40Nitrogen 0.0 0.0 207.20

Methane 0.0 28.90 651.20

Ethylene 0.0 57.80 414.40

Ethane 0.0 144.50 414.40

Propylene 0.0 491.30 384.80

Propane 23.66 317.90 118.40

1-Butene 196.56 520.20 88.80

Cis -2-Butene 444.08 173.40 14.80

n -Butane 777.14 982.60 59.20

n -Pentane 331.24 158.95 35.52

Heptane 47.32 14.45 8.88

Table 4

Convergence characteristics for problem number 1

Iteration number Case A Case B

u eT u eT

0 0.0 0.529)/103 0.0 0. 876)/106

1 0.0 0.546)/103 0.0 0. 922)/106

2 0.0 0.482)/103 0.0 0. 906)/106

3 0.0 0.466)/103 0.0 0. 856)/106

4 0.0 0.353)/103 0.0 0. 892)/106

5 0.0 0.331)/103 0.0 0. 780)/106

6 0.0 0.142 0.0 0.773)/106

7 0.25 0.808 0.0 0.666)/106

8 0.5 0.120)/102 0.0 0. 393)/106

9 0.75 0.434)/10(1 0.0 0. 270)/106

10 1.0 0.426)/10(3 0.0 0. 110)/106

11 1.0 0.116)/10(3 0.0 0. 294)/104

12 1.0 0.330)/10(4 0.0 0. 308)/103

13 1.0 0.474)/10(6 0.0 0. 113)/103

14 0.0 0.147)/10

15 0.0 0.250)/

10

(1

16 0.25 0.462

17 0.5 0.664)/10

18 0.75 0.371)/10(2

19 1.0 0.114)/10(1

20 1.0 0.220)/10(2

21 1.0 0.620)/10(4

22 1.0 0.135)/10(5

Table 5

Convergence characteristics for problem number 2

Iteration number Case A Case B

u eT u eT

0 0.0 0.394)/10 0.0 0.153)/102

1 0.0 0.302)/10 0.0 0.109)/102

2 0.0 0.274)/10 0.0 0.552)/10

3 0.0 0.101)/10 0.0 0.489)/10

4 0.25 0.184)/10(1 0.0 0. 491)/10

5 0.50 0.150)/10(2 0.0 0. 431)/10

6 0.75 0.124)/10(3 0.0 0. 261)/10

7 1.0 0.525)/10(3 0.0 0. 295)/102

8 1.0 0.766)/10(3 0.25 0.481)/10

9 1.0 0.702)/10(4 0.50 0.313

10 1.0 0.810)/10(5 0.75 0.224)/10(1

11 1.0 0.308)/10(2

12 1.0 0.105)/10(1

13 1.0 0.513)/10(2

14 1.0 0.331)/10(3

15 1.0 0.310)/10(4

16 1.0 0.191)/10(5




the values for u calculations in the intermediate stage

are as follows,

u u'/Du

0.0 0.25

0.25 0.5

0.5 0.750.75 1.0

Although the method presented in this paper is

seemingly similar to homotopy-continuation methods

it is actually a modified version of the Newton Á /

Raphson method in which the Newton Á /Raphson calcu-

lations are made by using a modified Jacobian matrix.

The method has several distinguishing features superior

to homotopy-continuation methods. In the homotopy-

continuation methods the solution is approached by

successive integration of the homotopy differential

equations. At a given parameter level many iterations

are required until convergence is attained. Intermediatesolutions are, in many cases, physically meaningless, and

the Jacobian often becomes singular so that a counter-

measure to cope with the difficulties is required. The

new computational procedure does not require the

integration of differential equations and no iteration-

wise convergence is required at each parameter level.

Basing initial assumptions on ideal K -values and

gradually introducing non-linearity as the solution

procedure progresses is effective in avoiding the problem

of a singular or near singular Jacobian matrix which

may occur when dealing with non-linear functions and

the initial variables are remote from the true solution. If

such a problem does occur the recommendation is tostep back the calculation to a point halfway between the

present and previous parameter levels representing the

degree of non-ideality.

Flow sheet simulation is very instrumental in design-

ing and analyzing complex chemical processes where

many separation and/or reactor units are involved. It is

especially effective when many interconnected streams

are involved and loops exist by recycle streams. Most

commercial process simulators employ computational

procedures in a sequential modular fashion where each

unit accepts the information necessary to initiate

calculation from outside and its output streams aretransferred sequentially to other connected units. For

these simulators reasonably good initial estimates for all

input parameters in a chemical process such as a

complex distillation system have to be provided to

insure convergence, especially for highly non-ideal

mixtures (Wasylkiewicz & Alva-Argaez, 2002). For

downstream units in a complex distillation separation

system good initial assumptions are often not known

and/or are subjected to the changes of other units so that

it is often difficult and very time consuming to achieve

convergence. The improved Ishii Á /Otto algorithm with

the non-ideality enhancement method, for which the

convergence is less-dependant on initial assumptions,

can be an effective tool for flow-sheet simulation where

difficulties are anticipated. It is also noted that the

algorithm does not require a huge amount of storage

memory and it is able to achieve convergence very

quickly.

Both the PBM concept and the gradual non-idealityenhancing method are applicable to other types of

separation problems such as three-phase distillation,

liquid Á /liquid extraction and even to reactive distillation.

In the papers authored by us to date, the model

equations have been exclusively grouped by equation

type rather than stage location. This is to avoid the

toilsome manipulation of a large number of block

matrices where elements in each constituent matrix are

scattered widely. However, it is worthwhile to apply the

PBM concept alone or together with the gradual non-

ideal enhancing method to cases where the equations are

grouped by stage location. The following situations willbias the choice in favor of grouping by individual stages:

Á / A large number of theoretical stages, such as a

propylene-propane splitter where more than 200

stages are specified and the number of components

involved is quite limited. In such cases, relative

memory storage required becomes smaller and thus

the computing time can be lessened accordingly.

Á / Reactive distillation where relations among compo-

nents governing the chemical reactions can be speci-

fied with ease. All chemical species can be present in

the same stage-wise matrix when the equations are

formulated in accordance with stage location.

7. Conclusions

Incorporating the gradual non-ideality enhancing

method into the Ishii Á /Otto PBM algorithm provides a

method that works well for difficult problems even when

starting with very crude initial assumptions. This

significant expansion of the domain of convergence is

meaningful in that very difficult problems can be solved

with the same ease as if they are identical to problems

where ideal mixture systems are involved. It was alsoobserved that the proposed method is effective for

problems where the influence of critical pressures may

become significant. The method is, in effect, a modified

version of the Newton Á /Raphson method and does not

require the integration of differential equations and

iteration-wise convergence at each parameter level such

as is required when using homotopy-continuation meth-

ods.

Provided that the linear approximation based on the

PBM concept is applied, the number of the composi-

tional derivatives of K -value required in the Jacobian

matrix can be greatly reduced for problems for essen-




tially all the correlating equations for non-ideal vapor Á /

liquid equilibrium. The significant reduction in the

computing time due to the above fact is meaningful in

that Jacobian evaluation is a major consumer of

computational effort for multicomponent, multistage

separation problems.

When equations of state are involved for the predic-tion of K -values, no compositional derivatives may be

required in the Jacobian, provided that the gradual non-

ideality enhancing method is employed. More specifi-

cally, the following can be assumed when the method is

employed for problems with moderately non-ideal

mixture systems.

@ K i 1

@ xCi

00:0 and@ K i 1

@ yCi

00:0

Appendix A

Derivation of Eq. (38)

The activity coefficient for any component i in the

Wilson equation is expressed by,

gi 0exp

(ln

XNC

j 01

(xi Li ; j )'1:0(XNC

k 01

xk Lk ;i XNC

j 01

x j Lk ; j

(A1)

Introducing the following variables,

x?10lx1; x?20lx2; . . . ; x?NC0lxNC

and substituting them into Eq. (A1), we can obtain a

new relation as,

g?i 0exp

(ln l(ln

XNC

j 01

(xi Li ; j )'1:0

(XNC

k 01xk Lk ;i

XNC

j 01 x j Lk ; j

0gi l(1 (A2)

Thus Eq. (A1) conforms to Euler’s homogeneous

function in which its degree is (/1. Differentiating Eq.

(A2) with respect to l gives

@ g?i

@ l0(

gi

l20XNC

j 01

@ x? j

@ l

@ g?i

@ x? j

0XNC

j 01

x j

@ g?i

@ x? j

(A3)

If l is set equal to 1.0, then x j ?0/x j and Eq. (A4)

follows,

XNC

j 01

x j

@ gi

@ x j

0(gi (A4)

K i is expressed by the following fundamental relation,

K i gi n0

i

8 Vi (A5)

where, ni 0 is the fugacity coefficient of pure liquid

component i and 8 i V is the fugacity coefficient of

component i in the vapor mixture. Thus, ni 0/8 i

V is

independent of liquid composition so that the following

relation can be derived,

XNC

j 01

x j

@ K i

@ x j

0(K i (A6)

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a method to extend the domain of convergence for difﬁcult

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