Method for Predicting. . .
Vapor-Liquid Equilibrium Relationship in Multicomponent Systems Generalized correlations permit accurate and rapid estimation of standard-state liquid fugacity and ideal K data WILLIAM E. EHRETT’ and JAMES
H. WEBER
Department of Chemical Engineering, University of Nebraska, Lincoln 8, Neb.
DWIGHT S. HOFFMAN Department of Chemical Engineering, University of Idaho, MOSCOW, Idaho
T H E R E are a number of ways to predict vapor-liquid equilibrium relationship in multicomponent systems. One method. which is more rigorous but more difficult than most, involves the ratio of the standard-state fugacities of the pure substances. This is often referred to as the “ideal” K . or vaporization equilibrium constant. and mathematically is:
K = fi/f: (1) The relationship between the ideal K and the y x ratio ( K ’ ) for a given substance in a mixture of vapor and liquid at equilibrium conditions i s :
where y, and (0%represent the liquid and vapor phase activity coefficients, respectively, and reflect departures from the Lemis and Randall fugacity rule. It is apparent from Equation 2 that prediction of the y / x ratio requires knowledge of the ideal R and activity coefficients. This study \vas undertaken to develop generalized correlations for the prediction of ideal K‘s and to compare the results obtained by these correlations with literature data. T o determine the value of the ideal K for a substance, volumetric data for the saturated liquid and vapor phases as well as the superheated vapor are required. \-olumetric data for the subcooled liquid are desirable but not necessary. The equations used for calculation of the fugacity data and, in turn, ideal 17’s are 1 Present address, E. I. d u Pont de Nemows & Co.. Inc., Clinton. Iowa
1’1 ET (P-P,) ,f;=
(4)
fPne
No assumptions are required in the derivation of Equation 3 ; to derive Equation 4 the effect of pressure on the molal volume of the liquid is assumed to be negligible. In addition to the fugacity data calculated by Equations 3 and 4, certain assumptions are necessary for calculation of ideal K values. For example, under conditions of temperature and pressure at which the pure substance does not exist as a vapor, the value o f 3 cannot be calculated rigorously, but must be estimated. Similarly, under conditions at which the pure substance does not exist as a liquid, the value of ho must be estimated. In the latter case, the molal volume of the liquid may be assumed to be independent of pressure and equal to the volume at the saturation ternperature; in the former case, extrapolations of generalized f ’ P charts are used. Hoffman and Weber ( 2 - 4 ) have shown and that these methods of calculating f f for the hypothetical vapor and liquid
ft
states give results which appear to be consistent with the fugacity values for the real states. Regardless of the assumptions needed to calculate fugacity values in the hypothetical liquid and vapor states, volumetric data are necessary for calculation of one half of the standard-state fugacity data. Because volumetric data are relatively scarce, the need for a generalized method to predict ideal K values is apparent. I n developing such a method it is only necessary to develop a generalized plot of E; a number of authors have presented generalized f / P plots. .4recent correlation of this type is that of Lydersen, Greenkorn, and Hougen (5) in which parameter 2, is used in addition to the reduced temperature and pressure. The correlation presented by these authors was used in this work for the determination off: data. No attempt was made to improve its accuracy, as it is already good. Hoffman and Weber (2) have shown that if the logarithm of E / P 0 is plotted against the total pressure with temperature parameters, a series of straight lines results. This plot can be generalized
Figure 1. A generalized correlation of ;f data for substances with similar critical compressibility factors can be obtained by plotting f;/Pv v5. PR with parameters of reduced temperature z. = 0.275
3= 0.002 VOL. 51, NO. 5
M A Y 1959
71 1
for substances whose critical compressibility factors are equal by plotting logarithm fe/Pv us. the reduced pressure with parameters of reduced temperature. Such plots were made as a part of this study; a plot of f f / P , , us. P R on rectangular coordinates yielded better results than a semilogarithmic plot, Plots of fe/P. us. PR are included as Figures 1 and 2 ; the former is for substances with critical compressibility factors of approximately 0.275 and the latter is for substances with critical compressibility factors of 0.288 The ff values obtained from Figures 1 and 2 and the fi values determined from the appropriate f / P chart of Lydersen and others ( 5 ) were used to determine ideal K data for benzene, 1-butene, and methane. These K's, those of Lydersen, Greenkorn, and Hougen, who gave generalized K charts in which K is a function of the reduced temperature and pressure and the critical compressibility factor, and the K data determined from the nomograph of Scheibel ( 9 )were compared with literature data (table). The proposed method is more accurate than that of Lydersen, Greenkorn, and Hougen. Results obtained by it and from the nomograph of Scheibel(9) are of comparable accuracy. However, very few comparisons were possible because of the limited scope of the nomograph. Because a generalized correlation can be obtained by plotting fp/P. against the reduced pressure with parameters of reduced temperature and f i / P data can
'
Mbthon. 2 s .0.290 Ethane h.0285
+ /e)
I
K
v.
120
I
I
I
I
,
h (T. Parameters)
pr
I
, I
I
I
R
Figure 2. A generalized correlation of f: data for substances with similar critical compressibility factors can b e obtained b y plotting RIP, vs. PR with parameters of reduced temperature zc = 0.288
be similarly correlated, logically the ratio of f i / P to f e / P v can likewise be correlated in a generalized manner. T h e ratio of these two quantities, or Pn/KP, has been used by Mertes and Colburn (7) and Scheibel (9) and referred to as Z. The same conclusion can be reached by starting with the equation of state P ( V - 0)
=
RT
Equation 3, the expression for standard-state vapor fugacity is
f."
=
BP Pe RT
the
(6)
Similarly, an expression for f p c may be developed and the standard-state liquid fugacity obtained by substitution in Equation 4. The result is
(5)
TT(P,+ VI P - VIP.) 1
fP
If this relationship is substituted in ~
~~
Comparison of
tl -
(1)
P,e
=
~
(7)
~
Values Obtained from the Proposed Correlations with K Values Obtained from Previously Presented Correlations Shows Advantage o f Proposed Correlations Results obtained by the proposed method are accurate Proposed Method Lydersen, Others and .f: Plots Z Plot (6) A v . ahs. Mas. Av. abs. Max. Av. abs. hlax. error, error, error, error, error, error,
fp
PR
Substance Benzene
TR
Range
0.7
0.02091.42 0.02091.42 0.02091.42 0.02091.42
0.8 0.9 1.0
No. of Points 5
%
%
%
%
%
%
2.1
5.9
1.04
2.2
13.6
18.2
5
1.9
4.0
1.40
3.8
9.9
15.2
5
1.0
1.9
0.9
2.5
6.2
11.8
5
1.8
3.4
1.40
3.4
3.8
6.0
5
3.0
8.5
5.0
9.8
10.9
17.6
5
3.1
8.0
3.0
6.6
7.0
15.4
5
1.3
3.8
0.7
1.2
5.1
7.5
5
1.95
3.1
1.8
4.5
1.3
3.8
1.7
Summation 1-Butene
0.74 0.82 0.90 0.98
0.092 10.343 0.10290.686 0.1711.029 0.10291.029
2.4
Summation Methane
1.2
0.76 0.90 1.00
0.03861.235 0.03861.235 0.03861.235
2.6
2.2
1
2.1
2.1
0.855
1
1.9
1.9
0.092 10.1372 0.10290.238 0.1710.470 0.7721.029
3
0.0
0.0
3
0.0
0.0
3
0.4
1.3
2
0.5
1.0
0.2
1.0
2.3
12.9
29.9
5
2.3
6.0
1.8
3.4
9.0
14.4
2.1
1.0
6.1
1.9
2.9
3
2.1
1.6
1.2
0.02090.142 0.02090.285 0.285
8.4
5
5
PR
Range
Scheibel (9) Av. abs. Max. No. of error, error, points % % 2 3.3 6.3
4.2
1.4
22.5
0.0386 0.154 0.3090.617
2
0.4
0.8
2
2.8
5.0
0
...
...
Summation
1.7
1.6
11.1
1.6
Over-all summation
1.9
1.8
8.5
1.3
71 2
INDUSTRIAL AND ENGINEERING CHEMISTRY
V A P OR-LlO U I D EC) UIL IB R I UM 2.00
I
1.80
28%
1.60
L GO
I .MI
I40
1.20
1.20
1.00
1.00
0.80
0.80
0.60
0.60
0.40
0 40
0.2 0
0.20
I 0
0.20
0.40
1 0.60 0.80 PI,
I
1
I
1.00
1.20
1.40
Equation 8 can be rearranged in the form
I t can be concluded that a plot of In Z L‘S. P a t constant temperature would result in a straight line, because p and P, are functions of temperature only and V I is assumed independent of pressure. The plot can be generalized by the use of reduced parameters. Such generalized plots for substances with similar values for the compressibility factor were made; but it was found that the use of rectangular coordinates was more satisfactory than a semilogarithmic plot. T o test the accuracy of the correlation presented in Figures 3 and 4, Z and, in turn, K values were determined for benzene, 1-butene, and methane. These K data were then compared with literature values (2, 6, 8) as were the values reported by Lydersen, Greenkorn, and Hougen ( 5 ) and by Scheibel ( 9 ) . A summary of the comparisons (table) shows that the proposed correlation is, in general, more accurate than the method of Lydersen, Greenkorn, and
I
1
I
1
I
I
1.60
Pn
Figure 3. A generalized correlation of Z values for substances with similar critical compressibility factors can be obtained b y plotting Z vs. PR with parameters of reduced temperature zc = 0.275 i 0.002
If Equation 6 is divided by Equation 7 and the result simplified. the relationship for /jfis
0’
Methane-’+ z C ’ 0.290( 6 ) Ethane0 zC* 0.285 ( I )
Figure 4. A generalized correlation of Z values for substances with similar critical compressibility factors can be obtained by plotting Z vs. P R with parameters o f reduced temperature
Hougen and as accurate as that of Scheibel. Again, only a few comparisons were possible in the latter case. The ideal K values obtained by the two proposed methods are of essentially equal accuracy. Lack of P-V-T and ideal K data prevents the construction at this time of charts similar to Figures 1 to 4, for substances whose critical compressibility factors are other than 0.275 and 0.288. No attempt was made to carry the correlation above the critical temperature because of the uncertainty of ff values under this condition. Nomenclature
Any consistent set of units may be used IC = vaporization equilibrium constant, defined in Equation 1 A” = vaporization equilibrium ratio =
P P, R
T V
Z f
7 x y
z a -y
Y/X
= = = = = = = = = = = = = =
total pressure vapor pressure gas law constant absolute temperature molal volume
Pu/PK fugacity partial fugacity-i.e , fugacity of a component in a mixture mole fraction in liquid phase mole fraction in vapor phase compressibility factor = P V / R T residual volume = R T / P - V second virial coefficient liquid phase activiry coefficient =
3 3.
ze =
rp
0.288
= vapor phase activity coefficient =
f”/fiY
SUBSCRIPTS P, c
i
I
= property of substance at saturated
= = =
R = u
=
condition-i.e., at a point o n vapor pressure curve critical condition ith component in a mixture liquid phase reduced conditions vapor phase
SUPERSCRIPT O
= property of pure component in
state standard at temperature and pressure of system literature Cited (1) Barkelew, C . H., Valentine, J. I>., Hurd. C. 0.. Trans. A m . Znst. Chem. Engrs: 43, 25 (i947). (2) Hoffman, D. S., Weber, J. H., P~trol. Rejner 34, No. 2, 137 (1955). (3) Zbid.. 35, No. 3, 213 (1956). (4) Zbid.;NO. 10, 163.
(5) Lydersen, .A. L., Greenkorn, R . A., Hougen, 0. A,, “Generalized Thermodynamic Properties of Pure Fluids,” Univ. Wisconsin, Eng. Expt. Station, Rept. 4 (1955). (6) Matthews, C. S., Hurd, C. O., T r a m . A m . Inst. Chem. Engrs. 42, 55 (1946). 17) Mertes, T. S., Colburn, A . P., IND. ENG.CHEM.39, 787 (1947). (8) Sage, B. H., Lacey, W. iV., Petrol. Refiner 29, No. 10, 123 (1950). 19) Scheibel, E. G.. IND.ENG. CHEM.41, 1076 (1949).
RECEIVED for review July 31, 1958 ACCEPTED February 6, 1959 VOL. 51, NO. 5
M A Y 1959
713
