Gibbs Energy Analysis of Phase Equilibria Lee E. Baker, SPE. Amoco Production Co. Alan C. Pierce, SPE, Amoco Production Co. Kraemer D. Luks, SPE, U. of Tulsa

Abstract Equations of state are used to predict or to match equilibrium fluid phase behavior for systems as diverse as distillation columns and miscible gas floods of oil reservoirs. The success of such simulations depends on correct predictions of the number and the compositions of phases present at a given temperature, pressure, and overall fluid composition. For example, recent research has shown that three or more phases may exist in equilibrium in CO 2 floods.

This paper shows why an equation of state can predict the incorrect number of phases or incorrect phase com-positions. The incorrect phase descriptions still satisfy the usual restrictions on equality of chemical potentials of components in each phase and conservation of moles in the system. A new method and its mathematical proof are presented for determining when a phase equilibrium solution is incorrect.

Examples of instances where incorrect predictions may be made are described. These include a binary system in which a two-phase solution may be predicted for a single-phase fluid and a multi component CO 2 /reservoir oil system in which three or more phases may coexist.

Introduction Advances in reservoir oil recovery methods have necessitated advances in methods for prediction of phase equilibria associated with those methods. It was long considered sufficient to approximate the reservoir behavior of oil and gas systems with models in which compositions of the phases in equilibrium were unimpor-tant. In such a model, the amounts and properties of the phases are dependent on pressure and temperature only. Later, experience in production from condensate and volatile oil reservoirs showed that models incorporating compositional effects were required to simulate the phase equilibria adequately. This led to the use of con-0197-7520/82/0010-9806$00.25 Copyright 1982 Society of Petroleum Engineers of AIME

OCTOBER 1982

vergence pressure correlations and subsequently to the development of more sophisticated equation of state methods for modeling and predicting phase equilibria.

For adequate description of the compositional effects that occur in enhanced oil recovery processes such as CO 2 and rich gas flooding, an equation-of-state ap-proach is a virtual necessity.

The use of equations of state for phase equilibrium prediction is not limited to the petroleum industry. Such equations also find wide use in basic chemical and physical research, and in the refining and chemical proc-essing industries.

Solution techniques for phase equilibrium problems are varied and depend to some extent on the application and equation of state used; however, there are three restrictions that all phase equilibrium solutions must satisfy.

First, material balance must be preserved. Second, for phases in equilibrium there must be no driving force to cause a net movement of any component from one phase to any other phase. In thermodynamic parlance, the chemical potentials for each component must be the same in all phases. Third, the system of predicted phases at the equilibrium state must have the lowest possible Gibbs energy at the system temperature and pressure.

The requirement that the Gibbs energy of a system, at a given temperature and pressure, must be a minimum is a statement of the second law of thermodynamics, equivalent to the more common version requiring the en-tropy of an isolated system to be a maximum. The equivalence is demonstrated formally in Ref. I, for ex-ample. If the Gibbs energy of a predicted equilibrium state is greater than that of another state that also satisfies Requirements I and 2, the state with the greater Gibbs energy is not thermodynamically stable.

Requirements I and 2, material balance and equality of chemical potentials, are used commonly as the sole criteria for solution of phase equilibrium problems. In many cases, there is no problem with this usage;

731

PB

P2

4 P3fl

a..

PI

PA

0 XB--

Fig. 1-Pressure/composition diagram for AS system.

however, there are important instances when the Gibbs energy minimization requirement also must be con-sidered. As discussed later, equality of chemical poten-tials is necessary but not sufficient for minimization of the Gibbs energy.

The iterative solution techniques usually used in find-ing equilibrium states may lead to a trivial situation (all phases present having the same properties) that always will satisfy the first two requirements but not necessarily the third. Of more importance perhaps, particularly in enhanced oil recovery, is the possibility of predicting false two-phase states in or near a three-phase region. These states can satisfy material balance and chemical potential restrictions while failing to minimize the Gibbs energy. This leads to predictions of incorrect phase volumes and properties (density, viscosity, etc.), which can adversely affect the results of an oil recovery simulation.

Several authors 2-7 have reported the occurrence of multiple (three or more) phases in rich gas/oil and CO 2 /oil reservoir systems. Others S- IO have shown that when there are multiple phases, it is possible to predict equilibria at states that do not minimize the Gibbs energy of a system.

This paper describes such failures in solving the phase equilibria problem in terms of Gibbs energy analysis. It also presents a method, and a mathematical proof sup-portive of the method, for identifying false solutions. Examples of the technique are given. For an elegant treatment of some properties of a Gibbs energy surface, the reader is directed to Coleman, II Dunn and Fosdick, 12 and references in those papers.

Discussion Equations of state often are used to predict phase equilibrium solutions. Gibbs energy analysis is the use of an equation of state to calculate a Gibbs energy surface and to determine whether a predicted equilibrium state has the lowest possible Gibbs energy. We first illustrate the use of a Gibbs energy surface to determine phase equilibria and show how false solutions can be obtained.

732

91 I I 12

9z 1---

I I "" I I

I I I I I I I I

o X Ll FHO Xv XB--

Fig. 2-Gibbs energy diagram for A-S system at p 1 .

Then we describe a method for detecting false solutions and illustrate why it succeeds when other methods fail. Finally, we give specific examples of the utility of the method, using a version of the Redlich-Kwong equation of state.

Illustration of the Problem Fig. 1 shows the pressure/composition diagram for a system composed of hypothetical Components A and B at a fixed temperature. The system exhibits two-phase (L I + V) behavior at low pressures; a three-phase region (L I + L2 + V) limited to a single pressure, P 3, for a binary system; and regions of L I + L2 and L2 + V two-phase equilibrium at pressures greater than P 3' In some pressure and composition ranges, the system is single-phase L I , L2 , or V.

The purpose of using an equation of state is to predict accurately the phase behavior of such a system. The first step in solving a phase equilibria problem can be de-scribed mathematically as finding a plane tangent to the Gibbs energy surface (g surface), with material balance restrictions. (See Appendix for a more detailed descrip-tion.) The slope of the tangent plane corresponds to the component chemical potentials (related to the fugacities). The points oftangency of the plane and the g surface correspond to the compositions of the predicted equilibrium phases. For binary systems, as illustrated here, the g surface is a curve and the tangent plane is a straight line. For multicomponent systems, the g surface is a hypersurface and the tangent plane is a hyperplane. The material balance restrictions require the overall com-position of a multiphase system to lie within the region bounded by the points of tangency of the g surface and the tangent plane.

These concepts are illustrated in some detail in Figs. 2 through 5, which represent the Gibbs energy surface (curve) for the A-B system at increasing pressures.

Fig. 2 shows the Gibbs energy diagram (Gibbs energy vs. composition) for the A-B system at pressure PI

Fig. 3-Gibbs energy diagram for A-8 system at p', .

for a hypothetical homogeneous (single) phase at all compositions and does not represent the actual Gibbs energy except in the single-phase region and at the equilibrium phase compositions. The g surface is con-cave upward except in the two-phase composition region. A tangent to the g surface indicates the equilibrium liquid and vapor compositions (points of tangency). For a given feed composition (as indicated in Fig. 2), the liquid- and vapor-phase fractions can be determined by application of the lever rule. The Gibbs energy, g 2, is the molar sum of the phase Gibbs energies (g's at the point of tangency) and'lies on the tangent line at the feed composition. The two-phase Gibbs energy, g 2, is less than the Gibbs energy, g 1 , of the hypothetical homogeneous phase of this composition, indicating that the two-phase system is more stable than a single phase.

For all feed compositions outside of the two-phase region, a tangent to the g surface does not intersect the surface at any other point; however, for any composition within the two-phase region, a tangent to the hypothetical g surface at that point does intersect (and lie above) the g surface at some point in the overall com-position range. In the next section and in the Appendix we show that this is a necessary and sufficient condition for stability of a predicted phase equilibrium solution. For a stable solution, the tangent corresponding to the solution does not lie above the g surface at any composition.

Fig. 3 shows the Gibbs energy surface for A-B at a higher pressure, P'I , which is still less than P 3. In this case, the g surface resembles that at PI, but an additional lobe (the incipient L2 phase) has appeared at a composi-tion between that of the liquid and vapor phases. A tangent to the g surface locates equilibrium phase com-positions xLI and Xv for any feed composition in the two-phase region. An additional tangent to the g surface at XLI 'and xv' locates a false solution. For the feed in-dicated, either one of the tangent lines corresponds to a

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t ""

I 91 ~-93 -J.---

I I I I I I I I I I I

0 Xu FEED XL2 Xv XB--

Fig. 4-Gibbs energy diagram for A-8 system at p 3'

pair of phases satisfying material balance and equality of chemical potentials, but only the lower tangent gives a minimum Gibbs energy (g 2 < g 2 ' < g 1 ). Consistent with our earlier observation, only the lower tangent does not lie above the g surface at any point.

Fig. 4 shows the g surface at the three-phase pressure, P 3 The phase equilibrium solution for the multiphase region (XLI to xv) is given by a tangent at XLI, XL2, and Xv. As in Figs. 2 and 3, in the composition range XLI to Xv the upper curve is the (hypothetical) Gibbs energy for a homogeneous phase (g 1 at the indicated feed), while the actual Gibbs energy g3 is a mole-weighted sum of the phase molar Gibbs energies at the points of tangency, XLI,xL2, andxv

The relative amounts of phases L I, L2, and V are not fixed by the compositional lever rule in this binary case, but they are bounded in that as pressure is increased, L2 is formed from L 1 and V until one or the other of these phases is consumed.

Fig. 5 presents the Gibbs energy surface atp2 >P3' There are two two-phase regions: XLI -x L2, and X L2' -Xv For the indicated feed, three possible states satisfying material balance and equality of chemical potentials are indicated. Only one of these (xLI-XL2, Tangent Line 2) has the minimum Gibbs energy, The solutions cor-responding to 2' and 2" both have Gibbs energies less than g 1 of the hypothetical single phase but greater than the Gibbs energy of the true solution.

A feed with composition between XL2 and xL2' will be single-phase at the temperature and pressure specified. However, Tangent Line 2' corresponds to a two-phase (false) solution for a feed in this composition range, This is an exception to the heuristic rule of thumb that, for a given feed, the more phases present at equilibrium, the lower the Gibbs energy will be. 13

An observation may be made here concerning the utili-ty of second derivative tests 9 for determining stability of a phase equilibrium solution. Such a test indicates the

733

+ ""

Fig. 5-Gibbs energy diagram for A-B system at P2'

direction of curvature of the g surface. For a stable solu-tion, the g surface must be concave upward at the points of tangency. Downward curvature indicates definite in-stability. The solution (tangent) labeled as 2" is unstable by this criterion because the g surface at the point of tangency near Xv is concave downward. The g surface is concave upward for both tangency points of Solution 2', however, and this solution would be considered stable by the second-derivative criterion.

At pressure P3 shown in Fig. I, the lobe of the g sur-face corresponding to the V phase will have vanished. The Gibbs energy curve will resemble that of Fig. 2, with L I and L2 phases forming from a feed.

We have shown schematically, and for very simple systems, that false solutions (not corresponding to the minimum Gibbs energy) to phase equilibrium problems can be generated for compositions in and around multi phase regions. The nature of the g surface and

~.-----.------.-----.------r-----.

6000

L

is:! 4000 ::> Vl Vl .....

Q:: 0..

2000

OL-____ ~ ____ ~ ____ ~ ______ L-____ ~ o

Fig. 6-Pressure/composition diagram for C0 2 /toluene at 38.1C.

associated tangents must be examined concurrently with the solution of mass flow equilibria equations (material balance and equality of chemical potentials) to ensure that the minimum Gibbs energy has been found.

We also have indicated that the tangent corresponding to an equilibrium solution qnnot lie above the Gibbs energy surface for any composition. This is described in more detail and extended to multicomponent systems in the next section and in the Appendix.

The Gibbs Energy Analysis The general phase equilibria problem for a feed of I com-ponents consists of finding a stationary state that is an equilibrium state. A stationary state corresponds to an extremum or saddle point in the total Gibbs energy of a system (Appendix, Definition 2), where the system is a set of phase compositions satisfying the mass balance re-quirement. An equilibrium state is a stationary state cor-

TABLE 1-EQUATION OF STATE AND PARAMETERS

nRT An2 P=----- .

V -nB T'h V(V +nB)

I I

A = L; L; X,XJCl'.;j' 1= 1 j= 1

f

B= L; x/3;. i= 1

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Component CO 2

toluene n-decane iodobenzene

Fluid Properties Tc Pc T (K) (kPa) (0C) na nb

304.16 7398.1 -25.0 0.42564 0.08003 35.0 0.43776 0.08886 38.1 0.43776 0.08886

591.77 4067.9 38.1 0.44485 0.07986 617.55 2096.0 -25.0 0.46432 0.07042 721.11 4529.9 35.0 0.41931 0.08213

Interaction Parameters, C Ii C0 2 /toluene 0.080 C0 2 /n-decane 0.095 CO 2liodobenzene 0.1365

SOCIETY OF PETROLEUM ENGINEERS JOURNAL

2000 ,-----,-------,----y---,-------,

"0 E

1500

:::; 1000 >-- a::: UJ z UJ 500 Vl

Fig. 9-Gibbs energy of mixing of CO 2 In-decane at - 25C, 1610.6 kPa.

Fig. 10-Gibbs energy of mixing of C0 2 /n-decane at - 25C, 1613.4 kPa.

energy curve, where there are multiple real com-pressibility solutions of the equation of state, only the solution giving the lowest Gibbs energy should be con-sidered, and that is the one shown in Fig. 7. We em-phasize again that the solid Gibbs energy curves plotted in the two-phase region (12 to 99% CO 2 ) do not repre-sent the equilibrium Gibbs energy of the system; the dashed line tangent to the Gibbs energy surface represents the true equilibrium Gibbs energy in the two-phase region. The solid curves are the Gibbs energies calculated as if the system were a single homogeneous phase at the pressure, temperature, and composition specified.

Example 2: CO 2 /n-decane at -l3F (-25C). The pressure/composition diagram for this system is shown in Fig. 8. Both CO 2 and decane are subcritical in this case. In the pressure range considered, the system ex-hibits one-phase, two-phase (L, + V, L, + L 2 , L2 + V) and three-phase (L, + L2 + V) behavior. The three-phase region for a binary is limited to a single pressure (at fixed temperature), in this case 234 psia (1613.4 kPa). The solid lines are the equation of state predictions, and they closely mimic experimental data. '8 Gibbs energy of mixing predictions are shown in Figs. 9 through 12. In Fig. 9, atp=233.6 psia (1610.6 kPa) (P

L I + V. The case is analogous to that of a pure liquid at its boiling point, where liquid and vapor may coexist in any proportion.

Fig. 11 shows the calculated Gibbs energy curve for p=235 psia (1620.3 kPa). There is an LI +L2 region, an L 2 ' + V region, and an intervening single phase (L2 ) region (as seen in Fig. 8). The tangent lines for the cor-rect solutions are shown. Also shown is a false L I + V prediction obtained for a feed composition in the single-phase region. At the points of tangency of this unstable solution, the Gibbs energy curve is concave upward; se-cond derivative tests could not determine stability of the solution. However, the tangent line intersects a lobe of the Gibbs energy curve, as theory predicts it should for an unstable solution. This is also a prime example con-tradicting the heuristic rule proposed by Gautam and Seider 13 that a P-phase solution, if it exists, has a lower Gibbs energy than a P-I phase solution.

Fig. 12 presents the Gibbs energy curve for CO 2 /n-decane atp=250 psia (1723.7 kPa) (p> >P3q,). In this case, the vapor lobe of the curve has disappeared, leav-ing only an L I + L2 two-phase region and L I and L2 single-phase regions.

Example 3: CO2 /Iodobenzene at 95F (35C). The CO 2 /iodobenzene system is interesting because it demonstrates the possibility of falsely predicting a critical point. Experimental data on this system are not available, so the specific description of the phase behavior may be regarded as somewhat conjectural. However, we also have observed the illustrated phenomenon in other, more complex, systems in which the presence of a predicted critical point was not sup-ported by experimental data.

The predicted pressure/composition diagram for CO 2 /iodobenzene at 95F (35C) is shown in Fig. 13. There is a liquid/vapor (L I + V) region, a liquid/liquid (L I + L 2) region, a liquid/vapor (L2 + V) region bound-ed above by a critical point, and a three-phase pressure. At a slightly higher temperature [98.4F (36.9C)], the L2 + V region disappears, with a two-phase K-point (L I + L2 = V critical fluid) solution.

The most interesting aspect of this diagram is the critical point solution indicated at 96.8% CO 2 , 993.1 psia (6847.2 kPa).

The Gibbs energy curve in the region of this critical point is shown in Fig. 14. The mathematical criteria for a binary system critical point,

(::~) T,p.n2 = (:~~) T.p.n2 =0, ......... (2) are satisfied. However, the inequality (a 4 G/an I 4h.P.n, > 0 that ensures stability is not satisfied. Consequently: the dashed line tangent to the curve at the predicted critical point lies above the curve almost everywhere out-side the immediate vicinity of the point. The curvature of the Gibbs energy surface at this point is caused by the in-cipient L2 lobe, which grows downward as pressure is increased. The true liquid/vapor phase equilibrium is in-dicated by the dashed line tangent to the g surface at L I and V of Fig. 14.

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10000

9500

900l

8500

L1 .

I I

L1 + L2 ~-

CRITICAL POINT- V L1+~+V,\

iii

6500

6000

5500

I FALSE CRITICAL POINT .--/" ~L1+V

I

0.4 0.5 0.6 0.7 0.8 0.9 MOLE FRACTION CO2

Fig. 13-Pressure!composition diagram for CO 2!iodo-benzene at 35C.

FALSE CRITICAL

o __ f!'..L..S._~~~!!~~~_~'?'!l!_!~~!-- ___ P_~I~~_-

-~~~~~~~--~~ o 0.2 0.4 0.6 0.8 MOLE FRACTION CO2

Fig. 14-Gibbs energy of mixing of CO 2!iodobenzene at 35C, 6847.2 kPa.

26000

24500

21000

;;:, 17500 "'"

l5!' 14000 L1 :::>

'" ~ 10500 c..

3500

O~~L-~L-~ __ ~ __ ~ o 0.2 0.4 0.6 O. 8

MOLE FRACTION CO2

UPPER BO~D OF LJ + V SOLUTIONS

LJ + ~ + V LOWER BO~D OF LJ ~ SOLUTIONS

Fig. 1S-C02!Leveliand oil phase behavior at 41.1 C.

737

Similar false critical points for this system can be calculated (with the equation of state and parameters shown in Table I) over a temperature range of 70 to 116F (21 to 47C).

Example 4: CO 2 /Levelland Oil at 106F (41.1C). The pressure/composition diagram for the CO 2/ Levelland oil system 7 is shown in Fig. 15. We cannot represent rigorously either the pressure/composition sur-face or the Gibbs energy surface for this system because such a representation would require N dimensions, where N is the number of components in the system. However, the regions of multiphase (L I + V, L I + L 2, L I + L2 + V) solutions are shown, with an indication of the regions where unstable solutions also may be found. In general, with complex systems such as these, L I + V solutions may be obtained not only in the actual L I + V region but also in the L I + L2 region and the three-phase (L I + L z + V) region. Similarly, L I + L z solutions also may be found in all three regions. Three-phase solutions are found only in the region marked L I + L z + V. Gibbs energy analysis is as successful in detecting false solu-tions for this system as it is in the binaries of Examples I through 3.

It is usually possible to obtain L I + V solutions in the L I + L z region for only a (relatively) small pressure in-terval above the three-phase region, while L I + L2 solu-tions often may be found in the L I + V region at pressures far below the three-phase region. This results from the variation in shape of the Gibbs energy curve, with an L2 lobe persisting at quite low pressures, while the vapor lobe disappears rapidly with pressure increas-ing above the three-phase pressure. This is not surpris-ing, because the properties of a liquid phase are more in-sensitive to pressure variation than the properties of a vapor phase.

Since false phase equilibrium predictions can be ob-tained over a wide range of pressure and composition, it is important to confirm that the correct solution is found. For the CO 2/Levelland oil system, the properties (densi-ty, viscosity) of the L I liquid phase are similar in the L I + V and the L I + L2 regions. However, the proper-ties of the vapor (V) and the L2 liquid phases are strik-ingly different. For example, the L z phase density is commonly two or more times the density of the vapor phase. The use of a false L I + V solution for a fluid in the L I + L2 region of the phase diagram could lead to a much different performance in a reservoir model of a COz flood. Similarly, the third phase present in the L I + L2 + V region could have a marked effect on the fluid flow behavior and relative permeabilities. Unless the model recognizes the possibility of a third phase, a two-phase (L I + V or L 1 + L 2) equilibrium would be predicted, with possibly adverse consequences.

Conclusions For fluid systems that exhibit multiple phases, an equa-tion of state may predict false phase equilibrium solutions.

This paper presents a self-consistent method for deter-mining whether a predicted equilibrium state is false. The method makes use of the equation of state to calculate the Gibbs energy surface and the tangent plane corresponding to the predicted equilibrium solution. If 738

the tangent plane lies above the Gibbs energy surface at any point, the predicted equilibrium solution is false. Conversely, if the plane lies entirely below or tangent to the Gibbs energy surface, the solution does describe the equilibrium state.

Nomenclature A = temperature-, pressure,- and composition-

dependent phase parameter (Table I) B = temperature-, pressure-, and composition-

dependent phase parameter (Table I) C ij = interaction parameter for component i with

component j (Table I) D = G-L, the difference between the Gibbs

energy surface and a tangent plane (function of composition)

F = function defined in Lemma 2 (Appendix) G = Gibbs energy surface g = molar Gibbs energy I = number of components

J, K = number of phases e = hypothetical phase vector L = liquid phase (subscripted) L = plane tangent to Gibbs energy surface, G

m ij = composition variable, composition of component i in phase j

mj = vector (mlj' m2j . .. mlj), composition of phase j

m = state, set of phase composition vectors mj n = number of moles n = state, set of phase composition vectors p = pressure, psia (kPa) r = phase composition vector (r I , r 2 . . r I) R = gas law constant

S7 = standard state molar entropy of component i T = temperature

U7 = standard state molar internal energy of component i

V = vapor x = mole fraction IX ij = temperature-dependent component

parameter {3 i = temperature-dependent component

parameter o = negative number

Ej = positive number T = variable parameterizing a curve in com-

position space 11- = chemical potential e = variable defined in Lemma 2 (Appendix)

n a, n b = generalized component parameters, func-tion of the reduced temperature and the component Pitzer acentric factor

Subscripts

c = critical i = component i j = component j (Table I); phase j elsewhere

SOCIETY OF PETROLEUM ENGINEERS JOURNAL

k phase k total

v = component v cf> phase

1,2,3 phase identification

References

I. Callen, H.B.: Thermodynamics, John Wiley and Sons Inc., New York City (1960).

2. Huang, E.T.S. and Tracht, J.H.: "The Displacement of Residual Oil by Carbon Dioxide," paper SPE 4735 presented at the SPE Third Symposium on Improved Oil Recovery, Tulsa, April 22-24, 1974.

3. Shelton, J.L. and Yarborough, L.: "Multiple Phase Behavior in Porous Media During CO z or Rich-Gas Flooding," J. Pet. Tech. (Sept. 1977) 1171-78.

4. Gardner, J.W., Orr, F.M., and Patel, P.D.: "The Effect of Phase Behavior on CO) Flood Displacement Efficiency," J. Pet. Tech. (Nov. 1981) 206-7-81.

5. Henry, R.L. and Metcalfe, R.S.: "Multiple Phase Generation During CO 2 Flooding," paper SPE 8812, presented at the First Joint SPE/DOE Symposium on Enhanced Oil Recovery, Tulsa, April 20-23, 1980.

6. Orr, F.M., Yu, A.D., and Lien, c.L.: "Phase BehaviorofCO z and Crude Oil in Low Temperature Reservoirs," Soc. Pet. Eng. J. (Aug. 1981) 480-92.

7. Turek, E.A. et al.: "Phase Equilibria in Carbon Dioxide-Multi-component Hydrocarbon Systems: Experimental Data and Im-proved Prediction Technique," paper SPE 9231 presented at the SPE 1980 Technical Conference and Exhibition, Dallas, Sept. 21-24.

8. Heidemann, R.A.: "Three Phase Equilibria Using Equations of State," AIChE J. (1974) 20,847-55.

9. Evelein, K.A., Moore, R.G., and Heidemann, R.A.: "Correla-tion of the Phase Behavior in the Systems Hydrogen Sulfide-Water and Carbon Dioxide-Water," I&EC Proc. Des. Dev. (1976) 15, 423-28.

10. Sorensen, J.M. et al.: "Liquid-Liquid Equilibrium Data: Their Retrieval, Correlation and Prediction," Fluid Phase Equilibria (1979) 3, 47-82.

II. Coleman, B.D.: "On the Stability of Equilibrium States of General Fluids," Arch. Rational Mech. Anal. (1979) 36, 1-32.

12. Dunn, J.E. and Fosdick, R.L.: "Morphology and Stability of Phases," Arch. Rational Mech. Anal. (1980) 74, 1-99.

13. Gautam, R. and Seider, W.D.: "Computation of Phase and Chemical Equilibrium: Part II-Phase Splitting," AIChE J. (1979) 25, 999-1006.

14. Yarborough, L.: "Application of a Generalized Equation of State to Petroleum Reservoir Fluids," Equations of State in Engineering and Research, K.C. Chao and R.L. Robinson Jr. (eds.) ACS, Washington, D.C. (1979) 385-439.

15. Baker, L.E. and Luks, K.D.: "Critical Point and Saturation Pressure Calculations for Multicomponent Systems," Soc. Pet. Eng. J. (Feb. 1980) 15-24.

16. Peng, D.Y. and Robinson, D.B.: "A Rigo~ous Method for Predicting the Critical Properties of MUlticomponent Systems from an Equation of State," AIChE 1. (1977) 23, 137-44.

17. Ng, H.-J. and Robinson, D.B.: "Equilibrium Phase Properties of the Toluene-Carbon Dioxide System," 1. Chern. Eng. Data (1978) 23, 325-27.

18. Kulkarni, A.A. et al.: "Phase Equilibrium Behavior of System Carbon Dioxide-n-Decane at Low Temperatures," J. Chern. Eng. Data (1974) 19, 92-94.

APPENDIX

Theorems Related to Gibbs Energy Analysis The following terminology and definitions are needed in the development of the theorems.

Let a hypothetical phase of I components be represented by r=(rl, r2 ... r/), where ri is the number of moles of species i in the phase. If a feed

OCTOBER 1982

stream is represented by (nl, n2'" n1), we say the phase r is admissible if 0 < r i < n i' This discussion is limited to systems of admissible phases, in which each component is present (at least in a trace amount) in each phase. The results are not strictly applicable to a system in which any component is excluded from one or more phases.

A possible state of a system of I species and J phases resulting from a feed n is designated m=[mj] wheremj is thejth phase. Conservation of mass requires

J

L: mij=ni ............................ (A-I) j=1

for each i, I $. i $. I. Herein, it is assumed that one has a nonreacting system.

At a given pressure and temperature, the Gibbs energy function for a phase is G=G(r). The Gibbs energy is a continuous first-order function in accordance with classical thermodynamics (e.g., see Callen I). The total Gibbs energy for a system of J phases is

J

L: G(mj), ..................... (A-2) j=1

which is the sum of the Gibbs energies of the constituent phases.

An equilibrium state for a system will exhibit a global minimum in the Gibbs energy. The following definitions are made to distinguish between phase states correspond-ing to local and global minima.

Definition 1. A state n of J admissible phases satisfying conservation of mass is an equilibrium state if Gt(n)=min Gr(m) where the minimum is taken over all states m of K admissible phases satisfying conservation of mass (K not necessarily equal to 1).

Definition 2. Let n be a state of J admissible phases satisfying conservation of mass and let G be differen-tiable at each phase in n. Suppose that for every differen-tiable curve m(7)=[mj(7)], which is defined for 7 in some open interval containing 0 and which satisfies con-servation of mass-i.e.,

J

L: mij(7)=ni, ......................... (A-3) j=1

and for which mjCO) = n j, I $.j $.J, we have

d A -Gr[m(7)]I7=O =0. . .................... (A-4) d7

Then we say that n is a stationary state. A stationary state corresponds to an extremum (local

or global minimum or maximum) or saddle point of the total Gibbs energy. These two definitions suggest that out of the stationary states found, the equilibrium state

739

must be identified. The purpose of the theorems presented is to develop criteria for this determination.

We show in Theorem I that the conditions of mass flow equilibria,

fLij=fLib l~i~I, l~(j, k)~J, ............ (A-5)

are equivalent to the condition of stationarity of the system.

Theorem 1. Let n be a state of J admissible phases [n)] satisfying conservation of mass. The state n is a sta-tionary state if and only if G is differentiable at each n) and the chemical potentials, fLij, do not vary with the phase-i.e., fLij =fLib as in Eq. A-5.

Proof. Suppose that the chemical potentials do not vary with phase. Consider nl(T) as in Definition 2. By the chain rule,

...................... (A-6)

Then

J ~ ~ aG[m)(O)] x dmij(O) , LJ LJ ......... (A-7) )=1 i=1 amij dT

but

aG[m)(O)] aG(n) ................ (A-8) =---=fLij,

amu ami} .

and fLij does not depend on the label}. Thus,

~ i=1

J

~ dmij(O) --0 fLi} LJ

)=1 dT ' ............... (A-9)

since ~ mij(T)=ni = fixed feed constraint, and so )=1

J

~ )=1

dmij(O) dni ----''--- = - =0. . ................ (A-IO)

dT dT

740

Thus, n is a stationary state. Conversely, stationarity of a state n implies that the conditions of mass flow equilibria apply. This can be demonstrated by consider-ing special curves, along each of which the only varia-tion is in one component in two different phases.

Central to the development of the Gibbs energy analysis is recognizing that n can be determined by the points of tangency of a hyperplane tangent to the surface G.

Theorem 2. Let n be a state of J admissible phases [n)] satisfying conservation of mass. Then G r is sta-tionary at n if and only if G is differentiable at each n) and the surface G has the same tangent plane at each of the points [n ) ] . Proof. The plane L j tangent to Gat n j is

Also,

since G is a first-order function (Euler's theorem). It follows that

I aG L)(r)= ~ -a (n)ri = ~ fLijri' ....... (A-l3)

i=1 ri i=1

But because fL ij does not depend on) for a stationary state (Theorem I), L) (r) is the same tangent plane for each of the} phases. Conversely, if all L j are the same, differentiation shows that the fL U ' s do not depend on}.

We now sho)V that a stationary state, n, is an equilibrium state if and only if the corresponding com-mon tangent plane of Theorem 2 never lies above the surface G at any point. Let D be the difference between G and the tangent plane L :

D(r)= G(r) -L(r). . .................... (A-14)

D(r)=G(r)- ~ fLijri i=1

I

=G(r)- ~ fLilri ................ (A-15) i=1

for a stationary state n. It is clear from this that D(n )=0, 1s,}s,J. We now show in the next few results that n is an equilibrium state if and only if D is never negative.

Lemma 1. Let n be a stationary state with J admissible phases [n]. Let liz be a collection of K admissible phases [m) i satisfying conservation of mass (K and J are not necessarily equal). Then

SOCIETY OF PETROLEUM ENGINEERS JOURNAL

K

G{(m)-G{(n)= b D(mj)' .............. (A-16) j=i

Proof. From mass balance,

J K

b nij= b mij' 1:51:51. .............. (A-17) j=i j=i

Also, J1.ij =J1.iI for 1:5 i :5/, 1 :5}:51. Now,

so

J

G{(n)= b GUi j) j=i

K

J

= b b J1.iI n ij j=i ;=i

K

b bJ1.;i m ij j=i ;=i K

b L(mj), .................... (A-IS) j=i

b [G(mj)-L(mj)] j=i

K

= b D(mj)' ......................... (A-I9) j=i

Theorem 3. Let n be a stationary state and suppose that G lies on or above the common tangent plane-i.e., D( r) ~ 0 for all admissible phases r. Then n is an equilibrium state.

Proof. Consider any state m containing K phases 1m j J. Then each D(mj) 2: 0, and .

K

b D(mj)2:0 ......................... (A-20) j=i

or

G{(m)-G{(n)2:0 ....................... (A-2I)

by Lemma 1. By Definition 1, n is an equilibrium state. Next we prove that D, as the difference between a

function and its tangent plane, is (roughly speaking) of second order in its argument.

Lemma 2. Let n be a stationary state with J nonzero phases [n jJ. Then, for 1 :5}:5J, there is a functi(),n Fj which approaches 0 as its argument approaches 0 and for which

ID(r)1 :5F/Cr-n/) m;lx Ir;-n;jl . ........ (A-22) . . ISlsf .

OCTOBER 1982

Proof. Recall that

D(r)=G(r)-L(f)

=G(r)-G(n j)-

...................... (A-23)

for each}. Since G is differentiable at nj, the function

D(r)/ b Irv-n'jl ..................... (A-24) v=i

approaches zero as r - n j approaches O-i.e., as r approaches nj. Let

f

F;Cr - n j)=/ID(r) II b Irv -n,jl ......... (A-25) v=i

Then

I f

ID(r)l= b Ir;-nijIID(r)11 b Irv-nvjl ;=i v=i

f

:51 m.ax Ir;-n;;lID(r)11 b I r"

- nVj I ISlsf v=i

=F/(f-n j ) m;lx Ir;-n;/1. ....... (A-26) . ISlsf .

Theorem 4. Let n be an equilibrium state with J ad-missible phases-z at each of which G is differentiable, and suppose that f is an admissible hypothetical phase. Then D(f) 2: O.

Proof. Assume D( f) < O. Note that since n is an equilibrium state, it is also a stationary state.

Choose the set of positive numbers [Ej J so that

nij-E/;>O, ........................... (A-27)

for 1 :5i:5/, 1 :5}:5J. Also, let these numbers be suffi-ciently small that, when Lemma 2 is applied to r= n -E? F(r - n) satisfies j j' j j

Fj(-Ejf)

This state, m, satisfies conservation of mass with respect to the feed composition. Furthermore, each mij is positive and each mj is admissible. It follows that

J+! J J

~ D(mj)= ~ D(mj)+D( ~ E/) ..... (A-31) j=! j=! j=!

J J

~ D(n j -E/)+( ~ Ej)D(f) j=! j=!

J J

~ ~ [D(n j-E/)[+( ~ Ej)D(l\ j=! j=!

. . . . . . . . . . . . . . . . . . . . . . (A-32)

Using Lemma 2,

742

J

+( ~ Ej)D()] j=!

J J

~ F ( - E r)E ( m;lx e i ) + ( ~ EJ )D( ) LJ } } } 1 $[$/ LJ! j=! J=

J J

< ~ [D()[Ej+( ~ Ej)D() j=! j=!

J

=( ~ Ej)[[D()[+D()J j=!

=0, .................................. (A-33)

if D()