Calculation of Phase Envelopes and Critical Points for ...

Fluid Phase Equilibria, 4 (1980) l-10 1

0 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

CALCULATION OF PHASE ENVELOPES AND CRITICAL POINTS FOR

MULTICOMPONENT MIXTURES

MICHAEL L. MICHELSEN

Instituttet for Kemiteknik, Technical University of Denmark, Bygning 229,

Lundtofteuej 100, DK-2800 Lyngby (Denmark)

(Received July 7th, 1979; accepted in revised form September 25th. 1979)

ABSTRACT

Michelsen, M.L., 1980. Calculation of phase envelopes and critical points for multicompo-

nent mixtures. Fluid Phase Equilibria, 4: l-10.

An algorithm for the construction of complete vapour-liquid phase envelopes, capable

of accurately determining the critical points and maxima in temperature and pressure is

described.

Recently developed methods for direct calculation of critical points are critically exam-

ined, and a computationally more convenient formulation is suggested.

INTRODUCTION

Calculation of vapourliquid equilibria for multicomponent mixtures using

a single equation of state is difficult in the near critical region, as it may not

be known in advance whether a given set of specifications corresponds to a

single solution, to multiple solutions or to no solution (Erbar, 1975; Heide-

mar-m and Khalil, 1979). It is therefore of interest to be able to locate certain

key-points on the phase envelope, such as the critical point, the temperature

maximum and the pressure maximum.

In this paper a method for rapid and accurate construction of the com-

plete phase envelope for a specified total composition and phase ratio is

described. In addition a significant simplification in a recently developed

method for direct determination of the critical point is discussed.

The present work is based entirely on the Soave-Redlich-Kwong equation

of state for representing vapour-liquid equilibria (Soave, 1972; Christiansen

et al., 1979), but the methods may equally well be applied to any other equa-

tion of state provided an algorithm for evaluation of component fugacities

and their first order partial derivatives with respect to temperature, pressure

and composition for specified temperature, pressure, composition and fluid

state. The algorithm used in the present work is described fully in Christiansen

et al. (1979).

2 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

CALCULATION OF PHASE ENVELOPE

Assume that one mole of composition (zi. 2s. . . . . z,) is split into phase 1

containing 1 - F moles and of composition (x1, x2, . . . . x,) and phase II con-

taining zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBAF moles of composition ( yi, y2, . . . . y,). The following set of equations

must be satisfied at equilibrium:

f;’ - fi’ = 0 i = 1, 2, . . . . n (1)

&~Yi~xi~=O (2)

(yi/zi)-Ki/[l + (Ki-l)F] =O i = 1, 2, . . . . n (3)

(xi/zi) - l/[l + (Ki - l)F] = 0 i = 1, 2, . . . . n (4)

Elimination of the xi and yi using (3) and (4) reduces this set to (n + 1)

equations in (n + 2) variables:

g&,8) = f?(a,B)--_fi’(a,B) = 0, i=1,2 , . ..) n (5)

gn+l(a, B) = gl bita, B)--“i(a,B)l = 0 (6)

where zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBAa is the vector of dependent variables,

a T = (K,, K2, . . ..L T, P) (7)

and p is the vector of fixed specifications,

BT = (Zlr 22, '--,Zrtr 0 (8)

In the following the complete set of solutions a is called the phase envelope,

and a particular solution corresponds to one point on the phase envelope.

A particular solution is obtained, adding a specification equation for one

of the dependent variables

gn+2(a, b)=ak-s= 0 (9)

and solving the resulting (n + 2) equations

& a, 8 ,s) = 0 (10)

by Newton-Raphson iteration: given an estimate a(“‘) of the solution vector,

an improved estimate acm+l) is found from

J(m)Aa +&m) = 0 (11)

a(m+l) =(x(m) + Aa (12)

where J is the Jacobian matrix at acm), Jii = (dgi/aai).

The effectiveness of this approach depends mainly on two factors: the


choice of specification variable and the quality of the initial estimate, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBAa (O).

The method to be described is intended to yield automatically a specification

that leads to a unique solution of (10) and a very accurate initial estimate

corresponding to this specification.

In conventional flash calculations zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBAP is specified. This works well in the low

pressure region, in particular for bubble point calculations (F = 0), since the

vapour phase fugacity coefficients depend only weakly on vapour phase com-

position and temperature.

To initiate calculations a pressure specification an+a = S1 (= e.g. 20 atm) is

used here. As suggested by Mollerup (1979) the K-factors in predominantly

hydrocarbon systems are reasonably well approximated by

Kd = (PCi/P) exp{5.42[1- zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA(TCj/T)] j (13)

in the low pressure region. Solving (6) for T using (13) for the K-factors

yields an initial estimate of temperature and phase compositions, from which

convergence of the full set of eqns. (10) normally takes place in 3--5 itera-

tions. One point on the phase envelope, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBAas1 is hence available.

The derivative of the solution vector with respect to the specification S is

easily determined by differentiation of (12):

J d(a)/dS + zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBAa&)/as = 0

where

(14)

a(g)/aS = (0, 0, .., 0, -I)

since S enters only in gn+a.

(15)

J is known from the Newton-Raphson iteration. Provided (11) is solved for

Aa by triangular decomposition only an extra back substitution is required to

provide d(a)/dS.

The pressure specification is now changed to a,+s = Sa (= e.g. 25 atm), and

(10) is resolved using as initial estimate the linear extrapolation.

a$:) = a s1 + 6% -%) d(ah,/dS (16)

The components of a and d(a)/dS are now known at points S = S, and

S = S,. These values are fitted to a third degree polynomial in S

a,(S) = Dlie + CujlS + (Yi2S2 + (Yi3S3 (17)

which is used to provide initial estimates corresponding to the subsequent

specification S = Ss.

If the calculations were to proceed in this manner with specifications of

still higher pressures, one would rapidly enter a region where the elements

dai/dS became large in magnitude, making the extrapolations (17) inaccurate

and finally inapplicable. To avoid this the specification variable is automati-

cally selected in each step as that element of the e-vector with the numeri-

cally largest rate of change, that is, the numerically largest value of dai/dS.


To put the elements of a on a comparable basis these elements are here taken

as

aT = (ln(K1), .._, ln(K,), In T, In P) (13)

For the continued construction of solutions to (10) the extrapolation

polynomials (17) are always based on the two last points on the phase enve-

lope. The stepsize, that is, the difference between two subsequent specifica-

tions, is chosen within prescribed limits such that an increase takes place if

less than 3 iterations are used to solve (lo), and a decrease if more than 4 itera-

tions are used. The convergence criterion used is that the norm of Aa is less

than lo-‘.

The automatically chosen step variable is normally the K-factor of the

least volatile component on the bubble point side and the K-factor of the

most volatile component on the dew point side.

For F = 0 the phase envelope is constructed at increasing P on the bubble

point side, through the critical point and back at decreasing P on the dew

point curve. For F = 1 the identical phase envelope is traversed in the oppo-

site direction provided the phase envelope is continuous. Finally, for F = 0.5

the “bubble point” and “dew point” curves are identical, and the critical

point coincides with a maximum in temperature and pressure.

No difficulties are encountered in passing the critical point, and the critical

temperature and pressure, usually accurate to 0.01 K/0.01 atm, are easily

found from the interpolation polynomials (17), based on points on each side

of the critical. Extrema in T or P are indicated by sign changes of dT/dS or

dP/dS and their values are evaluated by differentiation of (17) and solving for

dT/dS = 0 or dP/dS = 0.

A wide variety of examples have been tested, including the 32 systems

analyzed by Peng and Robinson (1977). Here we shall only present two sets

of results, the first being of the type encountered in the majority of cases and

the second a rather unusual type with two separate branches of the phase

envelope.

The P-T curves for F = 0 and F = 0.5 for a ‘I-component system investi-

gated by Heidemann and Khalil (1979) are shown in Fig. 1. For the F = 0

curve 15 points were constructed, the total computing time on the IBM 3033

being 0.2 sec. A similar typical natural gas mixture with 13 components

required 0.6 sec.

A 5-component mixture rich in HzS, very similar to that described by

Heidemann and Khalil (1979) is shown in Fig. 2. The phase envelope consists

of two separate branches, one extending to infinite pressure, and the mixture

has no critical points.

The bubble point branch in Fig. 2 consists of part (a) corresponding to

vapour-liquid equilibrium and part (b) corresponding to liquid-liquid equi-

librium, judging from the compressibility factors of the two phases. For sys-

tems with a single critical point a clearcut distinction between vapour and

liquid phases can be based on the critical compressibility, but for systems

Fig. 1. Nalural gas mixture, 7 components. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

with no critical point (or with multiple critical points) no clear distinction is

available. In the present case the compressibilities for the point at 30 atm on

part (a) are 0.09 and 0.67, respectively, indicating a liquid and a vapour

phase, while the compressibilities at the corresponding pressure on part (b)

are 0.09 and 0.15. The dew point branch exhibits a minimum in compressibil-

ity factors for both phases near the pressure minimum, and the compressibil-

ity factors increase zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBAto infinity along the branch extending to infinite pressure.

A stability analysis as described in the next section shows that phase I of

part (a) and both phases of part (b) are thermodynamically unstable, while

the stability criterion is not violated on the zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBAF = 1 branch.

Fig. 2. 5-component mixture, rich in H2S.

Fig. 3. 5-component mixture, rich in H,S. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

A slight change in composition yields a system with two critical points

and two intersecting branches as shown in Fig. 3. The construction of part (a)

is initiated easily with a low pressure bubble or dew point calculation as de-

scribed above, but a point on part (b) cannot be obtained in this manner.

Here the construction is started, specifying both phases as liquid and using as

initial estimate for the dependent variables those obtained for a correspond-

ing point on Fig. 2. The remaining portion of the curve is constructed with-

out problems. The presence of this branch is suggested by the results ob-

tained in Fig. 2, and it cannot be excluded that other isolated branches may

exist.

On part (a), phase I is unstable on the bubble point branch below 30 atm,

while both phases of part (b) are unstable up to the temperature and pressure

maxima. The critical point of this branch is hence located in an unstable

region. On the remaining portion the stability criterion is only violated for

phase II on a short section extending from the maximum to close to the inter-

section point.

DIRECT DETERMINATION OF CRITICAL POINTS

If only a determination of critical properties is required, the procedure used

in the previous section may appear unnecessarily complicated since repeated

solution of n + 2 equations rather than a single solution of two simultaneous

non-linear equations as formulated by Gibbs, is used. Until recently, however,

the effort involved in setting up, evaluating and solving these two equations

far exceeded that of constructing the phase envelope.

The first major attempt to develop a general procedure for direct deter-

mination of Tc and PC was that of Peng and Robinson (1977) who used the

criterion of Gibbs in the form: zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

F,(T, P) = det(U) = 0

F,(T, P) = det(M) = 0

the matrices U and M being (n - 1) X (n - 1) with elements

Uij = (a’G/aXiaXj)T,r (i, j) = 1, 2, . . . . n - 1

Mlj = a(det( U))/axj

Mij = Uij i = 2, .., n - 1

(19)

(20)

(21)

(22)

(23)

where G is the Gibbs molar free energy.

The computationally expensive part of using (19, 20) is that of evaluating

the elements Mlj. For analytic evaluation as used by Peng and Robinson,

analytic expressions for (a3G/axiaxjax ) k T,e (or, equivalently, for (a2(ln fi)/

aNjaNk),,e) are required, and about (n - 1)2 determinants of order (n - 1)

must be computed.

The derivation of analytic expressions for the second order partial com-

position derivatives of the fugacity coefficients of constant temperature and

pressure is unattractive and tedious, and errors are easily made. For simple

two-constant equations of state, this procedure is certainly possible. However,

the large number of determinant evaluations makes the method extremely

expensive when n is large, since the computational effort increases as n5.

The recent modification by Heidemann and Khalil(l979) is far superior.

Their criterion is formulated in terms of the Helmholtz free energy A: at the

critical point, a mixture of total composition N must satisfy

QAN=O, MhN#o (24)

and

C= F cc ~iANjANk(a3A/aNiaNjaNk)T,v = 0 (25) i k

where

Qij = (a2AIaNiaNj)T,V =RT(a(h-~f~)/aNj)~,~ (26)

Evaluation of the elements of Q and the cubic form C hence requires ex-

pressions for the first order and second order partial composition derivatives

of the fugacities at constant temperature T and total volume V. These deriva-

tives are more easily evaluated than those required by Peng and Robinson

and expressions for all derivatives needed are given by Heidemann and

Khalil. It is easily shown that the cubic form C corresponding to a given AN

can be reduced to very simple terms, the evaluation requiring essentially only

two double summations. The main computational effort required in solving

(24, 25) is that of determining conditions where a homogeneous solution to

the set of linear algebraic eqns. (24) exists.

To evaluate zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBATc and PC (or rather Vc), Ileidemann and Khalil used nested

iterations. In an inner loop at fixed V, Newton iteration is used to determine

a value of T such that det(Q) = 0. The AN vector is determined and normal-

ized, and C is evaluated to correct V in the outer loop. The procedure is

repeated until convergence is achieved.

With their recommendations for facilitating the solution of (24, 25) we

found this approach very efficient. A single determination of the critical

point by this procedure typically requires 30-50% of the computing time

needed for the phase envelope construction.

It is, however, of interest to point out that the cubic form C and hence the

need to evaluate all second order partial composition derivatives can be

avoided completely, since C can be rewritten:

C = C C ANiANj[q ANk(a3A/aNiaNjaNh)] i i

= FE ANiANjQii = ANr Q‘A N i

(27)

where zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

Q* = ; (Q(N + SAN, T, V)),,, (28)

The matrix Q’ is a partial composition derivative of Q in the AN-direction.

Since only a single derivative is required, numerical differentiation becomes

much more attractive, e.g.

Q* =(lle)[QW + e AN, T, VI - QW, T, VI + O(E)

or

(29)

Q* = (1/2e)[Q(N + EAN, T, V) - Q(N- EAN, T, V)] + O(E’)

where e is chosen suitably small.

(30)

This modification is particularly attractive for more complicated equations

of state, where it may not be possible to derive compact expressions for C, or

where the evaluation of the second order composition derivatives of the

fugacities is too cumbersome.

The approach described above can also be used in connection with partial

derivatives of fugacities, taken at constant T and P (which are the derivatives

normally required for phase equilibrium calculations), the only difference

being that Q has dimension n - 1 rather than n, that is zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

Qij = RT(a In filaNj),,p (i, j) = 1 , . . . . n - 1 (31)

and

A@ = (AN,, . . . . AN,_,, 0) (32)

In the present work this Q-matrix wad used to test the fluid phases found

in the phase envelope construction for thermodynamic stability, a necessary

(but not sufficient) criterion for stability being that Q is positive definite.


CONCLUSION zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

An algorithm for rapid construction vapour-liquid phase envelopes has

been developed. The algorithm is based on the Newton-Raphson solution of the

non-linear equation, automatic selection of the most convenient specification

variables and the use of previously calculated solutions to obtain initial esti-

mates. No difficulties are encountered in the critical region, and the critical

point as well as regions of retrograde behaviour are determined accurately.

Recent methods for direct determination of the critical point are discussed,

and a modification, in which second order partial derivatives of the compo-

nent fugacities are not needed, is presented.

NOTATION zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

A

c

: c gi J Ki M

Ni

i: PC

ii* R

S S T

TC

u zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBAV

xi Yi zi

Helmholtz free energy

cubic form

fraction of phase II

fugacity of component i

Gibbs free energy

see eqns. (5), (6), W, (10) Jacobian matrix, Jij = (ag,/ aej)

equilibrium factor for component i matrix of partial derivatives, eqns. (22), (23)

amount of component i in mixture

number of components in mixture

pressure

critical pressure

matrix of partial derivatives, eqn. (26)

directional derivative of Q

gas constant

specification

parameter of eqn. (28)

temperature

critical temperature

matrix of partial derivatives, eqn. (21)

volume

molar fraction of component i, phase I

molar fraction of component i, phase II

molar fraction in combined mixture

Superscripts

I, II phase I or phase II

m m-th iteration

10

Subscripts

i, i, k

Greek zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

component numbers

i-th dependent variable, eqn. (7)

i-th specification variable

increment for numerical differentiation

REFERENCES

Christiansen, L.J., Fredenslund, Aa. and Michelsen, M.L., 1979. Successive approximation

distillation calculations using the SRK equation of state. Computers and Chemical

Engineering (in press).

Erbar, J.H., 1975. Thermodynamic property predictions in process design, Vapor-Liquid

Equilibria in Multicomponent Mixtures, Jablonna, Poland.

Heidemann, R.A. and Khalil, A.M., 1979. The calculation of critical points, Paper

presented at 86th AIChE meeting, Houston, USA.

Mollerup, J., 1979. Private communication, Instituttet for Kemiteknik.

Peng, D.Y. and Robinson, D.B., 1977. A rigorous method for predicting the critical prop-

erties of multicomponent systems from an equation of state. AIChE J., 23: 137.

Soave, G., 1972. Equilibrium constants from a modified Redlich-Kwong equation of

state. Chem. Eng. Sci., 27: 1197.

Calculation of Phase Envelopes and Critical Points for ...

Documents