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A

Robust Iterative Method

for

Flash Calculations

Using

the

Soave Redlich Kwong or

the

Peng Robinson

quation o

State

L.X. Nghiem,

SPE, Computer Modelling Group

K Aziz,*

SPE, Computer Modelling Group

Y.K. Li,

SPE, Computer Modelling Group

Abstract

A robust algorithm for flash calculations that uses an

equation of state (EOS) is presented. t first uses a

special version

of

the successive substitution (SS)

method and switches

to

Powell s method if poor con

vergence is observed. Criteria are established for an effi

cient switch from one method to the other. Experience

shows that this method converges near the critical point

and also detects the single-phase region without com

puting the saturation pressure. The Soave-Redlich

Kwong

1

(SRK) and the Peng-Robinson

2

(PR) EOS s are

used in this work, but the method is general and applies

to any EOS.

Introduction

The calculation

of

vaporlliquid equilibrium using an

EOS

in

multi component systems yields a system

of

nonlinear equations that must be solved iteratively. The

SS method is commonly used, but it exhibits poor rate of

convergence near the critical point. To overcome con

vergence problems, Newton s method has been used by

Fussell and Yanosik

3

to solve the equations. The

drawback of Newton s method is the necessity

of

com

puting a complicated Jacobian matrix and its inverse at

every iteration. Hence, for systems removed from their

critical point it involves more work to arrive at the solu

tion than the

SS

method. Furthermore, the radius

of

con

vergence of Newton s method is relatively small when

compared to that of the SS method; hence, a good initial

Now with Stanford

u

01977520/83/00068285 00.25

Copyright 1983 Society of Petroleum Engineers of AIME

JUNE 1983

guess is required before convergence can be achieved.

The single-phase region usually

is

determined

by

com

puting the saturation pressure and comparing

it

with the

pressure

of

the system. This requires additional work,

and it is sometimes difficult to decide whether a dew

point or bubblepoint pressure, which involve different

equations, should be computed.

This paper presents a robust iterative method for flash

calculations using either the SRK or the PR EOS, both

of

which have received much interest in recent years. The

proposed method combines SS with Powell s iteration,

which

is

a hybrid algorithm consisting

of

a quasi-Newton

method and a steepest-descent method.

4

The SS method

is

used initially and

is

replaced

by

Powell s method if

it

demonstrates poor convergence, thus taking advantage

of the simplicity of the former method and the robustness

of

the latter. The

SS

method has been modified so that

the single-phase region can be detected without having to

compute the saturation pressure.

The nonlinear equations to be solved by

an

iteration

scheme could behave differently, depending on their

form and the variables for which they are solved. In this

paper three different approaches are considered with

Powell s method. One

of

the three approaches

is

based

on the minimization of the Gibbs free energy.

The convergence properties

of

the proposed schemes

are demonstrated by three example problems.

Equations in Flash Calculations

At a given pressure and temperature, flash calculations

yield the number

of

moles

of

component

in

the liquid

phase, NiL and

in

the vapor phase,

N

w

given that

N

i

=1 nc represents the number of moles of each com-

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ponent

in

the feed. The thermodynamic criterion

of

vaporlliquid equilibrium is that the Gibbs free energy,

G be minimized, which gives the following necessary

conditions.

aG

aG

--=--=0 i=1...ne ................

1)

aN

iL

aN

iV

Since NiL

+

NiV = N

i

, it easily can be shown

5

that

aG aG

--=---=G

iL

iV

,

i=l.

..

ne

aN

iL

aN

iV

............................. 2)

where iL is the partial molar Gibbs free energy of com

ponent i in the liquid phase. When an EOS is used to

compute_the pa.Itial molar Gibbs free energy, the dif

ference GiL - G V is related to the fugacities fiL and fiV

as follows.

- - fiL.

GiL -GiV=RT In , 1=

1.. .ne ..........

(3)

fiV

The expressions for the fugacities using the SRK and

PR EOS s are given in Appendix A.

The equilibrium conditions of Eq. I yield

InfiL

=0

i=1...nc

.......................

4)

fiV

or

Fi - I =0 i= 1...ne> ....................... (5)

where Fi is the fugacity ratio,fiLlfiV Eq. 5 represents a

system of ne nonlinear equations that must be solved

iteratively for NiL or N

iV

.

Instead of solving directly for NiL or N

iV

,

it

is often

more convenient to solve for mole fractions, which are

combinations

of

N

i

, NiL, and N

iV

.

The definitions of mole fractions are given in Appen

dix B. We also define moles of component

i

in liquid

phase per mole

of

feed as

niL =/LXi

=NiLIN,

........................

6)

and moles of component

i in

vapor phase per mole of

feed as

niV=fvYi

= N/ N

. .........................

7)

Eq. 5 can be solved for one of the following sets of

variables.

/L,x i i= 1...n c - I (LX iteration).

/v,Yi,

i=

1...ne

- I

(VY iteration).

niL, i=1...nc

(NLiteration).

niV,

i=

1...ne

(NV iteration).

It easily can be shown that the variables in each of

these sets are independent and completely define the

system. Later in this paper, we will show how Powell s

method is used to solve Eq. 5 for these sets of variables.

LX or NL iterations will be used for predominantly liq

uid systems.

522

The equilibrium-phase compositions also can be ob

tained by minimizing the Gibbs free energy. It is shown

below how Powell s method readily can be used to ob

tain the solution from this approach.

Since niL =NiLIN and niV=NiVIN, we obtain from

Eqs. 2 and 3:

I aG I aG fiL

- In ......... 8)

NRT an iL NRT an iV fiV

We see from Eq. 8 that In(jiLlfiV) is equal to the gra

dient

of

GINRT with respect to niL and the gradient of

-GINRT with respect to niV. Hence, by applying

Powell s method to solve Eq. 4 instead of Eq. 5, we are

minimizing

GINRT

by using the quasi-Newton method.

However, in doing so, we do not take advantage of the

fact that

In(IiLlfiV), i= 1...n

e

,

form the gradient of

GINRT(-GINRT) with respect to niL(niV), i=1...ne

since we merely were trying to reduce the values of

In(fiLlfiV) to O. Hence, the refinements that could

be

ob

tained from this additional information

6

were not in

troduced. We define the iterations on

niL

and

niV,

i = 1...n e as NLM iteration and NVM iteration, respec

tively. NLM iteration will be used for predominantly

vapor systems and NVM iteration for predominantly liq

uid systems.

In practice, the equilibrium ratios K = Y/x

i= 1...ne> are introduced. The following equations can

be derived.

xi=Z/[I+fv(Ki- l )] , i=1...ne . . . . . . . . . 9)

Yi=Kiz/[I+fv(Ki-l)], i=1...ne

. . . . . .

10)

ne

zi(K

i

-I

g(fv)=

=0. . . . . . . . . . . . . . (11)

i=i

I +fv(K

i

- I

Eq. 5 also can be solved iteratively for K

i

, i=1...ne

Once a set

of

values has been determined,

fv

is

ob

tained by solving Eq. II and x i and Y are obtained

from Eqs. 9 and

10.

The SS method usually

is

used in

this case and is described

in

the next section.

ethod o Successive Substitution

It can be shown that values are related to the fugacity

coefficients iL and iV as follows.

= iL = fiLI(XiP) F . ~

iV fiV1(YiP)

Xi

. . . . . . . . . .

12)

In this solution, estimated

K

values are used to flash

the mixture at the specified pressure and temperature.

The liquid and vapor fugacities then are determined from

the phase compositions, and new estimates for values

are obtained from Eq. 12. The procedure is assumed

converged when

(Fi

_1 2

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Solution of

g(fv

= 0

Every time

K

values are determined, Eq.

11

has to be

solved for

Iv(O:5lv:5

1).

g O

is a monotonically

decreasing function of

d g l d ~

L and liquid

otherwise.

Criteria for Switching From

SS

to Powell s Method

The important step of the proposed algorithm is the

switch from

SS

to Powell s method. This should lead to

a decrease in execution time; otherwise the switch is not

necessary. The criteria for an efficient switch from one

method to another were determined using a ternary

system and were then tested on 10- and IS-component

mixtures.

The ternary system used was C0

2

/nC

5

/nC

I6

Fig. 2

shows the calculated phase envelope at 1,500 psia (10.3

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85

a Calculafed

15

mol nC

1

6

Fig 2 Phase envelope of C0

2

/nC

s

/nC

16

system at 1,500 psia

and 122F. Flash calculations

in

Table 1 are for points

on

Line

A.

MPa) and 122F (SOC). To determine the convergence

behavior

of

the

SS

method for mixtures with different

compositions, a series of flash calculations was per

formed for mixtures containing 88 CO

2

and various

mole percents of nC 5 and nC

16.

These mixtures lie

along Line A in Fig. 2 and for notational convenience

will be referred

to

using only their mole percent of

nC

5.

This line contains mixtures in the fully developed two

phase region (e.g.,

0

nC 5),

mixtures near the critical

point (e.g., 8 nC

5

),

and mixtures in single-phase

region (e.g., 12

nC

s

).

Plots

of

log

~ F i

-1)2 vs. the number of SS iterations

(SSI) for diffelrent mixtures are shown in Fig.

3.

The

following can be observed.

1. The convergence of the SS method deteriorates as

the critical point

is

approached.

2. After a few iterations the slope

of

each curve re

mains almost constant for a l a r ~ e number of iterations.

This implies that the ratio

~ [ F l

)

I F / ~ [ F l k - 1 ) I F

1 1

is also almost constant during that period and can be

used as an indication of the rate

of

convergence

of

the SS

method for a particular mixture.

3. For the

8

nC

s

mixture, ~ F i _1)2 increases after

44 iterations and then d e c r e s e ~ again after S6 iterations

(not shown). This type of behavior of the

SS

method oc

curs frequently in the critical region.

4. For the 11

nC

5 mixture, the single-phase region

is

detected after 2S iterations.

Fig. 4 shows the change of ~ F i _1)2 and

v

with

the number of SSI for a mixturb in the fully developed

524

2

\ 11 nC5

\

0

1

-2

N

..-

I

-3

u..

1- 1 -

5

4

01

.2

-5

-6

-7

-8

_ 9 L L_ _

L L

_ _

L L

_ _L L__L L__

o

5

10

15

20

25 30

35 40 45 50

No. of SST

Fig.

3 Convergence

rate of 55 for different mixtures of

C0

2

/nC

s

/nC

16

at 1,500 psia and 122F.

two-phase region (2 nC

5),

a mixture near the critical

point (8

nC s),

and a single-phase mixture (11

nC 5)

For a two-phase mixture, ~ F i _1)2 decreases, n l

v

1

reaches its ultimate value rapidly. For a mixture near the

critical point, the change

of

~ F i _1)2 and

v is

slow.

1

For a single-phase mixture, the change of v is very

rapid and a value for

v

outside [0,1]

is

obtained after a

few iterations. In this case, we set

v

to either zero or

unity and proceed with the SS method as discussed

earlier.

Based on extensive testing

of

the flash program, the

following criteria are proposed to switch from the SS

method

to

Powell's method.

~ [ F l k ) - I F / ~ [ F l k - 1 ) - I F > E R (lSa)

1 1

Ilv k)

-lv k-1)

i =

fugacity coefficient of component i in liquid

phase

Wi

=

acentric factor

of

component i

JUNE 1983

T BLE 5 FLASH C LCUL TIONS FOR THE RESERVOIR OIL

Pressure

SS Plus Powell

Newton

(psia)

_ _v_

SSI Plus

FE

SSI Plus FE Iterations

4,950 0.0116 22 +

111

2+55

2

4,925 0.1144

21

+ 56

2+53

4

4,900 0.1770 21 + 49

2+30

4

4,875 0.2206

20+

33

2+28 3

4,850 0.2536

20+

30

2+27

3

4,800 0.3016 19+ 28

2+27

4

4,750 0.3358

18+

26

2+25

4

4,700 0.3623

18+

26

2+26

3

4,650 0.3838

17+

25

2+25

3

4,600 0.4019

16+

25

2+24

3

4,550 0.4175

16+

25

2+24 3

4,500 0.4313

15+

25

2+24 3

-2

,{O 4875 psio

1.2

8{ 4875 psio

fIR

-1 t:J

4750

psio

...

4750

PSiO

1.0

-4

C\I

0.8

.,....

6

LL

.-

0.6 t)

c>

-8

0

0.4

-10

0.2

~ 2 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

16 20 24 28 32 36

No

of

FE

Fig. 5-Convergence characteristics of Powell s method.

Subscripts

c = critical

i,j,k = component number

= subscript

of

largest equilibrium ratio

L =

liquid

r

=

reduced

s = subscript of smallest equilibrium ratio

V

= vapor

Superscripts

k) = iteration level

k

T

=

transpose

- =

results from SS method

cknowledgments

This research is supported

by

the general members of

Computer Modelling Group and the Alberta/Canada

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Energy Resources Research Fund administered by the

Dept.

of

Energy and Natural Resources of the Province

of Alberta. We thank R.A. Heidemann and P.K.W. Vin

some for their helpful comments on this work. The ver

sions of the SRK and PR equations used are from the

Hyprotech fluid properties program,

HYPROP.

References

1. Soave, G.: Equil ibrium Constants from a Modified Redlich

Kwong Equation of State, Chern Eng Sci. (1972) 27,

1197-1203.

2. Peng, D.Y. and Robinson, D.B.: A New Two-Constant Equa

tion of State, Ind Eng. Chern Fund (1976) 59-64.

3. Fussell, D.D. and Yanosik, J.L.: An Iterative Sequence for

Phase-Equilibria Calculations Incorporating the Redlich-Kwong

Equation of State, Soc Pet. Eng. J (June 1978) 173-82.

4. Powell, M.J.D.: A Hybrid Method for Nonlinear Equations,

Numerical Methods for Nonlinear Algebraic Equations, P.

Rabinowitz (ed.), Gordon and Breach, London (1970).

5. Sandler, S.I.: Chemical and Engineering Thermodynamics, John

Wiley

Sons Inc., New York City (1977) 300-04.

6. Broyden, C.G.: Quasi-Newton Methods and their Application to

Function Minimization,

Math

Compo (1967) 368-81.

7. Wilson, G.: A Modified Redlich-Kwong Equation

of

State, Ap

plication to General Physical Data Calculations, paper 15C

presented at the 1968 AIChE National Meeting, Cleveland, OH,

May

4-7.

8. Jarrat, P.:

A

Review of Methods for Solving Non-linear

Algebraic Equations in One Variable, Numerical Methods for

Nonlinear Algebraic Equations, P. Rabinowitz (ed.), Gordon and

Breach, London (1970).

9. Nghiem, L.X. and Heidemann, R.A.: General Acceleration Pro

cedure for Multiphase Flash Calculation with Application to Oil

Gas-Water System s, paper presented at the 1982 European Sym

posium on Enhanced Oil Recovery, Paris, Nov. 8-10.

10. Michelsen, M.L.:

The

Isothermal Flash Problem, Parts I and

II,

Fluid Phase Equilibria (1982) 1-40.

11. Baker, L.E., Pierce, A.C., and Luks, K.D.: Gibbs Energy

Analysis

of

Phase Equilibria,

Soc Pet. Eng

J (Oct. 1982)

731-42.

12. Broyden, C.G.: A Class

of

Methods for Solving Nonlinear

Simultaneous Equations,

Math Camp.

(1965) 577-93.

13. Peng, D.Y. and Robinson, D.B.: A Rigorous Method for Pre

dicting the Critical Properties of Multicomponent Systems from an

Equation

of State,

AIChE

J

(1977) 23, 137-44.

14. Bergman, D.F.: Predicting the Phase Behavior of Natural Gas in

Pipelines, PhD dissertation, U.

of

Michigan, Ann Arbor (1976).

15. Katz, D.L. and Firoozabadi, A.: Predicting Phase Behavior

of

Condensate/Crude-Oil Systems Using Methane Interaction Coeffi

cients,

J

Pet.

Tech

(Nov. 1978) 1649-55.

16. Young, L.C. and Stephenson, R.E.: A Generalized Composi

tional Approach for Reservoir Simulation, paper SPE 10516

presented at the 1982 SPE Symposium on Reservoir Simulation,

New Orleans, Feb. 1-3.

PPENDIX A

The SRK and PR EOS s

The details of the development of the SRK and PR

EOS's are given in Refs. 1 and 2. The final results are

the following.

SRK

528

RT a T)

P

v-b v v+b)

RTc

b=0.08664-

Pc

and

PR

and

R2T2

a Tc)

=0.42747-_

c

-

Pc

m

=0.480

+1.574w

-0.176w

2

.

RT a T)

p

v-b

v(v+b)+b(v-b)

RTc

b=0.07780-

Pc

R2Tl

a(T

c

=0 . 45724 - -

Pc

m=0.37464+ 1.54226w-0.26992w

2

.

f the compressibility factor, Z =pvIR1), is intro

duced, the following cubic equations are obtained. For

SRK,

Z3 -Z2

+ A -B -B

2

)z-AB=0.

. (A-1a)

For PR,

Z3 _(1-B)Z2

+ A

-3B

2

-2B)z-(AB-B

2

-B3)=0

,

. (A-1b)

where A=apIR

2

T

2

and B=bpIRT.

Eqs. A 1a and A 1b yield one

or

three real roots

depending on the number

of

phases in the system. In the

two-phase region, the largest root

is

the compressibility

factor of the vapor, while the smallest positive root cor

responds to that of the liquid.

For mixtures, the following mixing rules are

recommended.

2

b=

2. >ibi, (A-2)

a= ~ i j a i j

, A-3)

i

and

where 0 j are binary interaction coefficients obtained by

fitting the predicted binary bubblepoint pressures to ex

perimental data.

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The fugacity coefficient of component

k

in a mixture

can be calculated from the following equations.

. . . . . . . . . . . . . . . . . . . . . . (A-5a)

For PR,

b

k

In k=- z - I ) - ln z -B)

b

__

[

2: . Xiaik

~ B a - :kJ

. In Z+2.414B) , (A-5b)

z-0.414B

APPENDIX B

Mole Fractions Definitions

Let

The mole fractions are defined as follows.

Global mole fraction of component i,

i

=N;lN. .

(B-1)

Liquid mole fraction of component

i,

xi=NiLIN

L

.

(B-2)

Vapor mole fraction of component i,

Yi =NiVIN

v

. B-3)

Liquid mole fraction

of

the system,

fL

=N

LIN.

. B-4)

Vapor mole fraction

of

the system,

fv=NvIN. . (B-5)

From these definitions we obtain

i

=fLXi +fVYi, B-6)

fL = f v =

1 , B-7)

JUNE 1983

and

2 :

Z

i=

2:Xi=

2:Yi=1 .

i i

B-8)

APPENDIX C

Powell s Hybrid Method

Consider the problems of finding the root x* of a system

of

n

nonlinear equations in

n

unknowns .

f x)=O

C-l)

given a starting iterate x 0) .

Various methods have been studied and shown to be

quadratically or superlinearly convergent if x 0) is close

enough to x*. For many problems, however, a good in

itial estimate

is

not available and therefore a more robust

algorithm is needed.

Two well-known methods for solving Eq. C-l are

Newton s method and the steepest-descent method .

Newton s method is based on a linear approximation

of f around the kth iterate x k) .

f[X k) +a k)] =f[x k)] +J k)a k) ,

(C-2)

where

(

af ) k)

J k)=

-

ax

is the Jacobian matrix

of

f at

x k).

The Newton s step, a k) =x k +

I)

-x k), is obtained

by setting the left side of Eq. C-2 to 0 and solving for

a k) .

a k)

= - J -I k)f k) ,

(C-3)

where f k) denotes f x

k)]

for convenience. Newton s

method is quadratically convergent if x k) is close to x*.

I f

x k)

is

not close to x*, there

is no

guarantee that

x k+ I)

will be any better than

x

k) and Newton s method

need not converge. A method that guarantees that

is the steepest-descent method where

11 11

denotes the

Euclidian norm.

Consider the functional equation

F x) = z

T

x ) f x). . (C-4)

The steepest-descent direction g is

g= -

\ l F x)

=

-JTf x) .

.

(C-5)

Given the kth iterate x k) , the steepest -descent step

p

k)

=

x k+I)

- x k) is defined by

p k)

=J1. k)g k)

,

C-6)

with J1 k) chosen so that the functional C-4 with f x)

replaced by f k) + J k)p k) is minimized.

5 9

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It can be shown that the value of p, k) that minimizes

F[X(k)] =

1 2

[f(k)

+

p, (k) J(k)p(k)] T

. [f(k)

+p,(k)J(k)P(k)]

(C-7)

is given by

. . . . . . . . . . . . . . . . . . . . . C-8)

Although the steepest-descent method avoids

divergence, it converges very slowly near the solution.

Powell's hybrid method is a combination of a Newton

like method and the steepest-descent method. It has the

divergence-avoiding characteristic of steepest-descent

method and the rapidly converging property of the

Newton-like method near the solution.

Powell's algorithm computes a scalar e k), o ~ e k )

1, that is used to define the step (k) =x(k+ 1) - x (k ) as

follows.

where elk)

=

1 corresponds to the Newton-like method

and elk) =0 corresponds to the steepest-descent method.

The Jacobian matrix J and its inverse H are computed

numerically at the beginning of the iteration and updated

at each step using rank-one matrices according to

Broyden.

I2

[

f (k+I) _f k ) _J(k) (k) T(k)

J(k+I) =J(k)

+ .

T(k) (k)

C-IO)

530

H(k+I )

=H(k)

[ (k+

1)

_ H(k)[f(k+ I - f (k )J j

T(k)Hlk)

~ ~ ~ ~ ~

T(k)H(k)[f(k+I) _f lk ) ]

C-ll)

A method using this updating scheme is called quasi

Newton, and the final convergence is superlinear rather

than quadratic .

The computation

of

e

n

Eq. C-9 is given in the follow

ing. Let be the maximum norm of the correction vector

. Therefore, if

I l a k ) I I ~ ~ , ........................... C-12)

e k) is set to unity (a quasi-Newton step). Otherwise a

multiple of p is included in in order to satisfy

I l e a + l - e ) P I I = ~

..................... C-13)

Straightforward algebra gives for j the expression

e ~

2

-IIPI1

2

/[ra-p) Tp + a Tp _ ~ 2 ) 2

+ llaI1

2

_ ~ 2 ) ~ 2

-IIPI12)]'12).

.

.......

C-14)

The = 5 in all examples.

SI

Metric onversion Factors

O OF-32)/1.8

psi x 6.894 757

onversion

factor

is

exact

C

E+OO = kPa

SP J

Original manuscript received in Society of Petroleum Engineers office July 31,1979

Paper accepted for publication Aug. 19, 1982. Revised manuscripl received Feb. 7,

1983. Paper SPE 8285) first presented at the 1979 SPE Annual Technical Can

ference and Exhibition held in Las Vegas, Sept. 23-26.

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