Critical States of Ternary Mixtures and Equations of State Robert R. Spear,' Robert 1. Robinson, Jr., and Kwang-Chu Chao"2 School of Chemical Engineering, Oklahonia State Cniversity, Stillwater, Okla. 7&07&
The vapor-liquid critical states of ternary mixtures are determined b y a procedure that follows the rigorous classical definition derived from consideration of incipient material instability. The defining equations are solved based on the use of the Redlich and Kwong equation of state. The results are only slightly less accurate than those previously reported for binary mixtures. The calculated loci of constant critical temperatures are remarkably linear in male fractions, while the loci of constant critical pressures vary in a complex manner.
I n a previous investigation (Spear, et al., 1969), the thermodynamic definition of the critical state of binary mixtures was shown to provide a basis for the calculation of the critical state with the use of a suitable equation of state. In the present work, the critical-state criteria and calculational techniques required for ternary mixtures are investigated. Criiical State of Ternary Mixtures Defined
equations of st'ate because of the appearance of T , P , x2, and x3 as the independent variables. Use of the Helmholtz free energy arid the independent variables T , V , x2, and x3 are required for most practical calculations. The crit'ical-state determinant M has been expressed in eq 2 in ternis of the critical state determinant for convenience. By performing the iiidicat,ed derivations, the determinant JP equat,ion becomes
The classical definition of the critical state of multicomponent mixtures was established by Gibbs (1928) from considerations of incipient material instability i n a homogeneous phase. For ternary mixtures, the definition can be expressed concisely in the following determinant equations.
("">
I(%) 1
dx2ax3
T,P
dx32
I
\-,
T,P,a,
Prigogine aiid Defay (1954) showed that eq 1 defines the material instability boundary that separates the unstable states from t'he stable and metastable st,at'esof a homogeneous phase. Equation 2 indicates that the critical stat'e is a unique point on the material instability boundary. In our investigations (Spear, et al., 1969) of critical states of binary mixtures, we obtained a simplified calculat'ional procedure by observing that' the second equation is equivalent to a pressure extremum condition. Unfortunately, a simplifying transformation of the second aiid more complex critical-state equat'ion made for binary mixtures is not possible for ternary and higher multicomponent mixtures. Therefore, both critical-state equations for ternary m i s h r e s must be evaluated with a suitable equation of state. The critical-state equations expressed in terms of the Gibbs free energy are not convenient for use with pressure explicit Present address, International Petro Data I n c . , Calgary, Alberta, Canada. Present address, School of Chemical Engineering, Pmdue Cniversity, Lafayette, Ind. 47907. 1
588
Ind. Eng. Chem. Fundam., Vol. 10, No. 4, 1971
The various partial derivatives required to evaluate the critical state determinants C and JP have been expressed in terms of the molar Helmholtz free energy and pressure by z making use of the thermodynamic relationship ( b A / d V ) T ,= - P and can be evaluated using a pressure explicit equation of stat.e and the corresponding equation defininiiig the molar Helmholtz free energy for that equation of state. The complete expressions were given by Spear (1969). The relative complexity of the critical state criteria for biliary and ternary mixtures can be compared a t this point,. The t\vo critical state determinants L- and Jf for ternary mixtures require a t,otal of sixteen derivatives, four third order, nine second order, and three first' order, to calculate the critical state. The two simplified critical state equations for binary mixtures required only four derivatives, one second order aiid three first-order derivatives. Thus, although each term in t'he critical state determinants C aiid JI can be evaluat'ed without difficulty, t'he expressions for the crit'ical state criteria of ternary mixtures are more complex than t,hose for binary mixtures. Calculations Using the Redlich and Kwong Equation of State
The thermodynamic theory of the critical state of multicomponent mixtures has shown that only two critical state relations, the determinants C arid Jf equal to zero, exist regardless of the number 0'' components in the mixture.
From a calculational standpoint these two relat8ionspermit, the determination of two variables a t specified values of the independent variables. The choice of independent, variables is arbitrary but dictated by convenience for the intended purposes. I n t'liis exploratory study of ternary systems, we chose to specify temperatu.re and pressure as the independent variables, thus leaving: the mole fractions of two components to be determined. T h e development of the search procedure used in a computer program t,o calculate the critical point of a ternary mixture a t a specified critical temperature and pressure required a knowledge of the variation of the values of the critical determinants U aiid .li with the mixture composition. T o determine these variatioiis the determinants U and Ai were evaluated in terms of the Redlich and Kwong equation of state over a wide range of compositions at a specified critical temperature and presuure. The results of these ca1culat)ions for the methane-ethane-n-butalie ternary mixture a t a critical temperature of 700"R and a critical pressure of 905 psia are presented in Figure 1. The most interesting features of Figure 1 are t h a t the curves of constant values of Jf are nearly linear in t h e critical region aiid the U = 0 and ilf = 0 loci intersect a t only one point in the critical region. A search procedure based on the general characteristics of the constant U and .li loci of Figure 1 was developed as follows. The critical temperature and pressure of the mixture are specified. The search procedure begins by est,imating the initial composition of the critical point. The first step is to increment t'he value of x 3 continuously from t'he iiiit'ial compositiori estimate, point A in Figure 1, in t.he direction of decreasirig absolute values of 211 until Ji = 0, point B. The value of U is calculated a t this point. If U is greater t,haii zero, the value of x 2 ic; decreased by some arbitrary amount. Again 2 3 is continuously iiicremeiit'ed in t,he direction of decreasing absolute values of AI until A 1 = 0, point D. If C is still greater than zero, the value of xz is decreased again. If U is less than zero, :is is the case in Figure 1, x Zis increased by a smaller increment. This procedure is repeated along the path E F G H I K uiitil both C and d l equal zero a t the critical point K. The search proc,edure is straightforward and does riot, in general, require a large number of iterations t o find the critical point. However, care must be taken to determine if t'he initial composition estimate is in the region where the curves of constant Ji are no longer linear. I n this region the search procedure does not converge to the critical point. ,Ippropriate safeguards must be incorporated into the computer program to eliminate this possible divergence or to terminate the program if the search procedure does not converge quickly. Execution of the ternary mixture critical-st,ate calculation program was carried out on an I B M 7040 digital computer. Average computation time for one ternary mixture crit'ical point was 25 sec. The complexity of the critical st'ate deterininant's C and Af for ternary mixtures makes the choice of a suitable equat'ion of state much more important than is the case for binary mixtures. I n previous investmigations the tm-0-parameter Redlich-Kwong equat'ion of st,at,e was used t o calculate the critical properties of a variety of binary mixtures of interest. (Spear, et al., 1969). The average error level in the prediction of the critical properties was comparable t o those obtained from the best empirical correlations. The same equation of state was used iu this initial investigation of the equation of state approach in predicting the critical properties of ternary mixtures.
Legend Constant M Lines -.-. Constant U Lines
--_-
0.6
- Calc Path
m
A
c 01
8
5
\ \
0.56
I
L
-e
\
.-
E
\ \
0
\
IL
2-
\
01
0.52
C
m
X
\
I I
\ \
\
.
\
\
\
\
\
\ \
\
0.4 0
\
\
\
\
\ \
\
\
M:O
\ \ \
0.2
M=t30
0.3
xp
0.4
0.5
,Mole Fraction of Component 2
Figure 1. Calculation path of ternary mixture critical state
search
procedure for
The equation of state developed by Redlich and Kwong (1949) is a two-parameter equation of the form
p = - -RT
a
v -b
TO.5V(V
+ b)
(4)
For pure substances the equation of state parameters a and b are usually expressed a s a = 0.4278R2TC2 "P,
b
=
0.0867RTC/P,
(5) (6)
The constants a and b of the Redlich-Kwoiig equation of state for ternary mixtures were determined from the following mixing rules
where a,, and b,, are binary cross-interaction parameters and can be related t o the pure component parameters by aij= e i j 4 a i a j
bij
=
4ijdbibj
(9) (10)
Redlich and Kwong used the geometric average for a i j , equivalent to e I j = 1, and the arit,hmetic average for b i j , equivalent to qiij = ( b i b , ) / 2 ( b l b j ) ' / s , The partial derivatives required for the evaluation of the critical-state determinants are evaluated using the RedlichKwong equation of stat,e. The complet,e expressions Tvere given by Spear (1969). Joffe and Zudkevitch (1966) showed that substant'ial improvement in t,he representation of fugacity of gas mix-
+
Ind. Eng. Chem. Fundam., Vol. 10, No. 4, 1971
589
Ethane
LEGEND
0
EXPERIMENTAL DATA BY EKINER (1966)
- - - -- R - K e -R - K
EQUATION WITH = IO EQUATION WITH
eI2 = 0 . m 813
0.075
eZ3= 1.000
n- H e p t a n e (973O R 1
Figure 2.
940'
9200
900'
8800
860°
Critical temperature-composition diagram for the ethane-n-pentane-n-heptane
tures could be obtained by treating 0 and 4 as empirical parameters. The results of the critical-state calculations for binary mixtures indicated t h a t adjusting the value of binary interaction parameters in the mixing rules of the constant a of the Redlich-Kwong equation of state could significantly reduce the average error levels in the predicted critical properties for most binary mixtures.
(845'R)
system
corresponding binary pairs were used in the mixing rules. All +,Iwere kept constant a t a value of 1. Use of the optimum value of o!, in binary mixture critical-state calculatioiis significantly reduced the errors in the predictions of the critical properties of 38 binaries forming four general classes of mixtures. Results and Discussion
Ternary Mixture Critical State Calculations
Critical-state conditions were calculated in this work for seven ternary systems which have been investigated experimentally. Even though only a few (from one to six) mixture compositions were reported for each ternary system, our calculations covered wide ranges of compositions to permit comparisons with the experimental values and to reveal the contours on the composition diagrams. I n all the systems studied our calculated ternary contour curves merge correctly with our calculated binary mixture values previously reported by Spear, et al. (1969). Two separate sets of calculations to determine the critical properties of seven ternary mixtures were made using the Redlich-Kwong equation of state. I n the first set of calculations the simple mixing rules with e,, = 1 and 4 1 1= 1 were used. The variation of the critical temperature and critical pressure with composition was investigated for the seven mixture systems. Qualitative and semiquantitative agreement with experimental critical properties was observed. I n the second set of calculations, the optimum previously determined b y Spear, et al. (1969), to give the best representation of the critical pressure-composition relationship for the 590 Ind.
Eng. Chem. Fundom., Vol. 10,
No. 4, 1971
The general behavior of the calculated and experimental critical temperature and pressure is illustrated in Figures 2 and 3 with the system ethane-n-pentane-n-heptane. Comparisons of the experimental and calculated critical temperatures and pressures for seven ternary systems appear in Table I. I n general, semiquantitative results were obtained for the prediction of critical properties with no empirical adjustment of the interaction parameters e)1 and @ 2 1 . The average absolute error in the critical temperature predictions was 4.3% for the seven ternary systems investigated. As shown in Figure 2, each calculated critical isotherm was linear across the entire composition range of the teriiary system and terminated a t the calculated compositions of the two binary systems having the same critical temperature. The complexity of the ternary mixture critical-state deterused t o calculate the critical properties minants II and makes this linear relationship somewhat surprising. However, a linear relationship of this type has been reported for similar hydrocarbon systems and is the basis for a simple method to predict critical temperatures of ternary mixtures (Grieves, et al., 1962). The magnitude of the average error in predicting the critical temperature of ternary mixtures was only slightly
Ethan@
LEGEM
0
EXPERIMENTAL OATA BY O(INER
-- - -- R - K
-R
EQUATION WITH
el,
=
e12 = 0975 813 823
n - Heptane [ 397 p5iO)
e
Figure 3.
0.875
1.000
00
-
n Pentane (490 pria)
Critical pressure-composition diagram for the ethane-n-pentane-n-heptane
Table 1. Average Absglute Percentage Errors in the Prediction Critical Temperature a n d Pressure
% error System
Bij
= 1
in To
Optimum
eij
eij =
1
Optimum B i j
7.4
4.8
17.7
15 2
Methane-ethane-n-pentane Methane-propane-n-butane Methane-propane%-pent ane
6.2 6.1 2.8
1.5 5.2 2.1
10.8 5.9 5.3
7 2 6 1 1.5
Ethane-propane-n-pent ane Ethane*-pentane*-heptane Propane-n-but ane-n-pentane
3.2 3.4 0.8
2.2 0.7 0.6
2.5 3.5 1.1
2.5 1.o 1.0
4.3
2.4
6.7
4.9
greater than the corresponding 1.0 to 1.5’% error level encountered in binary piixture critical calculations. The range of average absolute errors in the prediction of the critical pressure for ternary systems was only slightly greater than the corresponding binary mixture calculations. The variation of the critical pressure with mixture composition is shown in the triangular composition diagrams of Figure 3. The curves of constant critical pressure are nonlinear. As was the case for the critical temperature predictions, the curves of constant critical pressure terminate a t the compositions determined in the binary mixture critical-state calculations.
system
of
% error in po
Methane-ethane%-butane
Totals
1.0
K EQUATION WITH
Reference
Forman and Thodos (1962) Cota and Thodos (1962) Billman, et al. (1948) Rigas, et QZ. (1959) Dourson, et al. (1943) Mehra and Thodos (1962) E t t e r and Kay (1961) Ekiner and Thodos (1966) E t t e r and Kay (1961) Nelson and Holcomb (1954)
The magnitude of the errors in the calculated critical properties for a particular ternary mixture system seemed t o be dependent on the similarity of the molecules of the components. Figure 4 shows the definite trend in the relationship between the magnitude of the error in the predicted critical pressure and the total molecular weight of the components of the ternary system. The accuracy of the critical temperature and pressure predictions was generally improved by using the optimum value of the binary interaction parameter elj. Use of the optimum value of e,, resulted in a shift of the curves of constant Ind. Eng. Chem. Fundam., Vol. 10, No. 4, 1971
591
fitting of the binary d a t a and can be expected to improve as the representation of the binaries is improved with better equations of state. The ternary mixture calculations qualitatively differ from the binary because of a n important simplifying condition applying to binaries but not to ternaries. From this viewpoint, the ternaries are the same as the higher multicomponent systems. The equations remain t h e same for three and more components. The present calculations mark a significant step in the direction of a general procedure for multicomponent mixtures.
+
20.0
Acknowledgment
The Computing Center of Oklahoma State University generously donated computer time which made this work possible.
110
120
130
140
150
160
170
Toto1 Molecular Weight of Pure Components
Figure 4. Average error in calculated critical pressure vs. sum of molecular weights of components of the ternary system
critical temperature and pressure as shown in Figures 2 and 3. The average errors in the critical temperature were reduced to less than 2.0% for those ternary systems not containing the methane-n-butane pair. (Methane-n-butane binary critical behavior was not well described (Spear, et al., 1969).) This 2.0% error level was only slightly greater than the level encountered for binary system critical temperature predictions. The average error in the critial pressure was reduced from 6.7 to 4.9% for the seven ternary systems using the optimum value of o,,. This represents an average loss of approximately 1.3% in the accuracy of the critical pressure predictions in going from binary to ternary mixtures. The general limitation encountered in attempts to improve the critical-state predictions using the optimum o,, becomes clear with an examination of the ethane-n-pentane-n-heptane system. The calculated curves of constant critical pressure are qualitatively correct, but owing to the constituent binary critical pressure predictions not agreeing with experimental values with the use of the optimum o , , , the accuracy of the prediction of critical properties for this ternary mixture cannot be improved. Thus, the present calculations are primarily limited by the accuracy of the binary mixture critical-state calculations. Improvements in the representation of binary mixture critical properties, such as those resulting from the use of a more accurate equation of state, will lead to definite improvements in ternary mixture critical-state predictions. Conclusions
Our calculations based on rigorous derivations with the Redlich-Kwong equation confirm t h e observation of Grieves and Thodos (1962) regarding the linearity of T o contours of ternary systems. The contours are linear t o almost within the precision of our calculations with either set of 0 values. One is tempted to speculate regarding the T , contours of quaternary and higher systems. Are they linear? The accuracy of the equation of state calculations for ternary systems is only slightly less than that for the constituent binaries. The accuracy appears to be limited by the 592
Ind. Eng. C h e m . Fundam., Vol. 10, No. 4, 1971
Nomen c lature
A a
b
G AI P R T U Ti z
Helmholtz free energy per mole parameter of Redlich-Kwong equation of state, (psi) (ft6)(cRO’5)/(lb-mole2) = parameter of Redlich-Kwong equation of state, ft3/lbmole = Gibbs free energy per mole = terlzary mixture critical state determinant = pressure, psia = universal gas constant, 10.71 (psi) (ft3)/(lb-mole) ( O R ) = temperature, “R = ternary mixture critical-state determinant = volume, ft3/lb-mole = mole fraction = =
GREEKLETTERS o = interaction parameter associated with a + = interaction parameter associated with b
SPBSCRIPTS i c
= =
component reference critical-state property
literature Cited
Billman, G. W., Sage, B. H., Lacey, W. N., Trans. A I M E 174, 13 (19481. ~~ - . ,
Coia, H. hl., Thodos, G., J . Chem. Eng. Data 7, 62 (1962). Dourson, R. H., Sage, B. H., Lacey, W. N., Trans. A I M E 151, 206 (1943). Ekiner, 0. Thodos, G., J . Chem. Eng. Data 11, No. 4,457 (1966). Etter, D. 6., Kay, W. B., J . Chem. Eng. Data 6,409 (1961). Forman, J. C., Thodos, G., A .I.CF;E. J . 8 , 209 (1962). Gibbs, J. W., “Collected Works, Vol. 1, p 55, Yale University Press, New Haven, Conn., 1928. 1, 45 Grieves, R. B., Thodos, G., IND.EXG.CHEM.,FUXDAM. (1962). Joffe, J., Zudkevitch, D., IND.ENG. CKEM.,FUNDAM. 5 , 455 (1966). Thodos, G., J . Chem. Eng. Data 7,497 (1962). hIehra, V. S., Nelson, J. M.,Holcomb, D. E., Chem. Eng. Progr. Symp. Ser., 49, No. 7, 93 (1954). Prigogine, I., Defay, R., “Chemical Thermodynamics,” p 250 ff, Longmans, Green, and Co., Ltd., London, 1954. Redlich, O., Kwong, J. N. S., Chem. Rev. 44, 233 (1949). Rigas, T. J., Xason, D. F., Thodos, G., J . Chem. Eng. Data 4, 201 (1gt59). Spear, R . R., “An Equation of State Approach t o the Prediction of Critical States of Mixtures,” Ph.D. Thesis, Oklahoma State University, Stillwater, Okla., Aug 1969. Spear, R. R., Robinson, R. L., Chao, K. C., IND.ENG.CHEM., FUNDAM. 8, 2 (1969). RECEIVED for review June 1, 1970 ACCEPTED July 6, 1971 Support for R. R. Spear came from the NASA Traineeship program and is gratefully acknowledged.
