Ind. Eng. Chem. Res. 1989,28, 127-130 Grilc, V. Vestn. Slov. Kem. Drus. 1979, 26, 123-135. Lamb, H. Hydrodynamics, 6th ed.; Cambridge University Press: New York, 1945. Lang, S. B.; Wilke, C. R. Znd. Eng. Chem. Fundam. 1971,10, 341. Mekasut, L.; Molinier, J.; Angelino, H. Can. J. Chem. Eng. 1979,57, 688. Moeti, L. T. M.S. Thesis in Chemical Engineering, Georgia Institute of Technology, 1984. Nagata, S. Trans. SOC.Chem. Eng. Jpn. 1950,8, 43. Pavlushenko, I. S.; Kostin, N. M.; Matveev, S. F. Zh. Prikl. Khim. 1957, 30, 1160. Quinn, J. A.; Sigloh, D. B. Can. J. Chem. Eng. 1963, 41,15. Selker, A. H.; Sleicher, C. A., Jr. Can. J. Chem. Eng. 1965,43, 298. Skelland, A. H.P.; Ramsay, G. G. Znd. Eng. Chem. Res. 1987,26,77. Skelland, A. H. P.; Seksaria, R. Ind. Eng. Chem. Process Des. Deu. 1978, 17, 56.
127
van Heuven, J. W.; Beek, W. J. Paper No. 51, Proc. Int. Solvnt. Extract. Confer., 1971. Vermeulen, T.; Williams, A. M.; Langlois, G. E. Chem. Eng. h o g . 1955, 51, 85f. Wellek, R. M.; Agrawal, A. K.; Skelland, A. H. P. AZChE J. 1968,2, 854. Zwietering, T. N. Chem. Eng. Sci. 1958,8, 244.
A. H. P. Skelland,* L. T.Moeti School of Chemical Engineering Georgia Institute of Technology Atlanta, Georgia 30332 Received for reuiew January 16, 1987 Reuised manuscript received August 22, 1988 Accepted October 16, 1988
Correlation for the Second Virial Coefficient for Nonpolar Compounds Using Cubic Equation of State A correlation for the prediction of the second virial coefficient of nonpolar compounds has been developed by using a cubic equation of state and requires the availability of the critical pressure, critical temperature, and Pitzer acentric factor a. Predictions are in excellent agreement with the experimental data and compare well with the values obtained by means of Tsonopoulos correlation and four cubic equations of state: Soave-Redlich-Kwong, Peng-Robinson, Kubic, and LielmezsMerriman modification of Peng-Robinson equation. The expanded form of the pressure explicit virial equation of state may be written as follows:
P=
q 1 +v+ V
B -
C
D
v2
v3
- + - + ...
where B is the second, C the third, D the fourth virial coefficient, etc. Often, however, at moderate pressures, density is less than half the critical and the virial equation truncated after the second term provides an excellent estimate of the vapor-phase fugacity coefficient. While there are several methods for predicting the second virial coefficient (Pitzer and Curl, 1957; Tsonopoulos, 1974; Hayden and O’Connell, 1975; Martin, 1967, 1979, 19841, for our use in this work, we have adopted the general form of the Tsonopoulos (1974) modification of the three-parameter corresponding states correlation of Pitzer and Curl (1957):
3=
f‘0’ + ,f(l) (2) R TC where f ( O ) represents the reduced second virial coefficient of simple fluid which has zero acentricity, f ( l )is the correction term for normal fluids, and w is the Pitzer acentric factor. Recent work (Adachi et al., 1983; Yu et al., 1985) discloses that none of the currently popular cubic equations of state give accurate values of second virial coefficients except the equation of Kubic (1982). The Kubic equation uses Tsonopoulos’ correlation to predict one of its constants. However, even the Kubic equation yields not very accurate values for vapor pressures and densities. The noted shortcomings of cubic equations of state to accurately predict the values of second virial coefficients may be better understood if we consider the generalized form of the cubic equation of state (Schmidt and Wenzel, 1980): a(T) p = - -RT (3) V - b V 2+ ubV+ wb2 If eq 3 is expanded in inverse molar volume similar to eq
0888-5885/89/2628-0127$01.50/0
1,the second virial coefficient is given as
(4) where
a ( T ) = a(TC)a (5) The reduced form of the second virial coefficient is
where
F = a/T,
(7)
Shaw and Lielmezs (1985) showed that for most cubic equations of state instead of eq 7 the F function could be written as a power series in inverse reduced temperature. For instance, the F function for Soave-Redlich-Kwong (Soave, 1972,1980) and Peng-Robinson (1976) equations would assume the form of
F
=
c 2 c3 c1+ + T,0.5 Tr
Constants C1-C3 are functions of the acentric factor, w. Figures 1 and 2 show that below T, = 0.8 the slope of the second virial coefficient increases rapidly with decreasing temperature. If we compare the functional dependence of F on reduced temperature T,, we see that the F function developed from the generalized cubic equation of state (eq 7 and 8) and, depending on reduced temperatue T, with the power of -1, does not adequately describe the swift change of the second virial coefficient in the low-temperature region. Replacing the simple a function as given by Soave (1972) or Peng-Robinson (1976) by a more complex function (eq 7 and 8), however, would correct for this inadequacy, permitting us to follow the second virial coefficient change in the low-temperature range with a relatively high degree of accuracy. These observations and the availability of new experimental data (Dymond 0 1989 American Chemical Society
128
Tnd. Eng. Chem. Res., Vol. 28, No. 1, 1989 0.0
where
II -- Hu ta
Following the development of Pitzer-Curl (1957) and Tsonopoulos (1974) of the three-parameter theorem of corresponding states with the acentric factor, w , as the third parameter, we write the a function for a normal fluid as
nc
a(Tr,u)= aC0)(Tr) + a(''(Tr) p i
0 8
rlfi
10
1.2
(16)
where
Reduced Temp?r,%ture,Tr
CY(0)=
m
CcrO Tr (1-1)
(17)
i=l
and a(l)
= i=l
Benzene 2 0
I
L P 4
I O
08
0 6
1 7
Reduced Temperature 'k
il
Tr (1-L)
(18)
Equation 17 or do)represents a of simple fluids, while eq 18 or a(') corrects for the deviation of normal fluids from the simple fluids. In this work, all acentric factors of simple fluids are assigned zero value even if the actual literature values (Reid et al., 1977) differ from zero. To determine the numerical values of coefficients cL0and the number of terms appearing in eq 17, the general criterion of minimum variance of the curve fit was used. Once an expression for the do)function (eq 17) was obtained, then was calculated from
Figure 1. Second virial coefficients of n-butane and benzene versus reduced temperature T,. ( 0 )Experimental, Soave-RedlichKwong, ( -) Peng-Robinson, (- - -) Tsonopoulos, (-1 this work, eq
(19)
(-e)
:3-21
and Smith, 1980) prompted us to develop the correlation presented in this work. The proposed relation has been compared with the results obtained from Tsonopoulos' correlation (1974) and the equation of state of Kubic 11982),Soave-Redlich-Kwong (Soave, 1972), Peng-Robinson (1976) and the Lielmezs-Merriman (1986) modifi-ation of the Peng-Robinson equation.
w
where CY in turn was calculated from eq 14. To obtain the number of terms appearing in eq 18, again the general criterion of minimum variance of the curve fit was used. The final proposed expressions in the expanded form for do)(eq 17) and (eq 18) are as follows: 14.360017 45.000285 + CY(') = -1.4524905 + -__ TI Tr2 78.907097 - 79.449258 45.841959 - 14.078304 +
Proposed Correlation The proposed correlation has been developed by using the Peng-Robinson equation of state reproduced from eq :? Fv setting u = 2 and w = -1; that is, a(T) p = - -RT (9) V - b V2+2bV-b2 Equation 4, this work, represents the second virial coefficient for a cubic equation of state, where for the PengRobinson form of the cubic equation of state (eq 9) we have a(T 1 = a(Tc)a(Tr,w) (10) a(Tc)=
R2T: Qa-
R2Tc2
= 0.45724-
P C
pc
(11)
Combining eq 4 and 10-12 and simplifying results in
Solving eq 13 for
CY
we obtain
Tr3
CY")
= -4.3816022
Tr5
T,4
+ 15.205023 - 20.874489 Tr
T,"
1.7835426 Tr7 +
T,2 12.697209 2.5851848
T,3 TI4 The do)function curve-fit final form (eq 20) was determined by means of linear least-squares regression methods from a set of 57 experimental data points representing neon, argon, krypton, and xenon with a curve fit variance of 0.0003175. The final expression for the a(l)function (eq 21) was obtained similarly from a set of 124 experimental data points representing propane, n-butane, n-pentane, isopentane, neopentane, n-hexane, n-heptane, n-octane, benzene, ethylene, and propylene with a variance of 0.010 41. All experimental data used were taken from the most recent compilation of second virial coefficients by Dymond and Smith (1980). As the data used were thought to be of sufficient reliability, no further evaluation of their accuracy was made.
Ind. Eng. Chem. Res., Vol. 28, No. 1, 1989 129 Table I. Second Virial Coefficient: compd SRK methane 6.6 5.9 ethane 18.1 propane 74.1 n-butane 41.2 isobutane 78.2 n-pentane 103.3 isopentane neopentane 33.6 n-hexane 205.3 212.1 n-heptane 376.3 n-octane 164.3 benzene 3.9 carbon monoxide 154.0 carbon disulfide 15.8 hydrogen sulfide 1.7 ethylene 6.1 propylene 13.6 1-butene 4.8 nitrogen 5.7 oxygen 1.6 neon 8.0 argon 16.1 krypton 12.1 xenon
Average Absolute Deviationa Value (Centimeters/Mole) Comparisonb PR LM K this work T devest T,range 11.2 15.4 24.6 69.0 43.9 67.0 72.9 37.3 176.5 180.7 341.0 147.9 5.5 135.4 21.4 15.9 18.8 23.0 8.3 9.2 2.7 8.9 15.7 15.8
12.1 16.6 25.0 67.2 44.2 66.6 72.0 38.1 177.2 180.9 339.9 144.8 7.6 131.3 21.9 17.0 19.4 23.6 8.0 9.9 4.1 9.9 15.8 16.2
1.6 8.6 10.3 9.2 22.4 14.2 15.2 9.9 28.4 46.4 84.4 61.1 1.1 26.6 16.5 2.6 5.8 14.7 4.9 3.1 1.5 2.8 2.2 3.8
1.4 5.8 6.1 11.8 24.7 10.9 20.0 3.5 19.0 15.0 30.5 17.3 1.5 39.0 16.1 0.4 1.5 6.6 3.9 2.1 0.5 0.7 1.9 1.4
0.8 5.7 5.8 9.6 24.5 11.9 21.3 3.0 14.1 29.1 51.5 60.1 0.5 28.8 16.2 0.4 1.1 7.3 3.9 2.2 1.5 1.7 2.0 2.2
2.6 2.4 12.3 18.2 28.3 27.1 17.0 27.2 16.9 20.0 1.2 4.8 2.4 3.8 1.0 1.4 3.2 4.8
0.58-3.15 0.65-1.96 0.65-1.49 0.59-1.32 0.67-1.25 0.64-1.17 0.61-0.98 0.69-1.27 0.59-0.89 0.56-1.30 0.53-1.23 0.52-1.07 2.05-3.18 0.51-0.79 1.00-1.32 0.85-1.59 0.77-1.37 0.72-1.00 0.60-5.56 0.58-2.58 1.35-13.51 0.54-6.63 0.53-3.34 0.55-2.24
Nd 16 ~. 15 15 17 11 12 7 10 9 12 12 13 7 9 7 9 8 12 14 11 10 18 14 16
"AAD = Xlexptl- calcd(/N. bSRK = Soave-Redlich-Kwong (Soave, 1972, 1980); P R = Peng-Robinson (1976); LM = Lielmezs-Merriman (1986); K = Kubic (1982); T = Tsonopoulos (1974). cDeviation estimates given by Dymond and Smith (1980). dAll experimental data for second virial coefficient values taken from Dymond and Smith (1980); physical property data from Reid et al. (1977). 00
-0 5
0 + E
-1 0
2m
5
-1 0
-2 5
Reduced Temperature, Tr
0 5
0.0
8
e: 3
-05
!a m -1.0
Argon -1 5
0 0
10
2 0
30
40
5.0
6 0
7 0
Reduced Temperature, Tr
Figure 2. Second virial coefficient of n-octane and argon versus Experimental, Soave-Redlichreduced temperature T,. (0) Kwong, (-. -) Peng-Robinson, (- - -) Tsonopoulos, (-) this work, eq (e..)
14-21.
Discussion The second virial coefficients (eq 4 and 13) have been calculated from the proposed correlation (eq 16-21) for 24 normal and simple fluids. Table I presents a comparison between the second virial coefficient values obtained in this
work and those calculated by means of the Tsonopoulos (1974) correlation and the equations of Soave-RedlichKwong (Soave, 1972,1980), Peng-Robinson (1976), Kubic (1982), and Lielmezs-Merriman (1986). The results presented show that none of these cubic equations of state can represent the second virial coefficient behavior satisfactory for a wide range of temperatures except the Kubic equation for reasons already noted. The second virial coefficient values as predicted by the proposed correlation, this work (eq 16-21); Tsonopoulos' correlation (1974); and Soave-Redlich-Kwong (Soave, 1972, 1980) and PengRobinson (1976) equations of state are plotted against the reduced temperature in Figures 1 and 2 for n-butane, benzene, n-octane, and argon. Since the Lielmezs-Merriman equation gives almost identical results as the Peng-Robinson equation (Table I), it has not been included in the figures. Similarly, the Kubic (1982) equation, which is partly derived from the Tsonopoulos correlation, has not been included. Considering all the data present (Table I, Figures 1 and 21, all the equations of state selected here give satisfactory results for T,> 0.8 but below this reduced temperature deviate from the experimental second virial coefficient values. On the other hand, the proposed correlation shows an excellent agreement with the experimental data over the entire range of temperatures studied. For example, the average abolute deviation of benzene from the proposed correlation is about one-third that of Tsonopoulos' correlation, and for n-heptane, it is about one-half. For the majority of compounds studied, the deviations from both the Tsonopoulos and the present proposed correlations are well within the experimental uncertainties (Dymond and Smith, 1980). Studies to extend the range of application of the proposed correlation to include polar compounds and mixtures have been started. The results obtained and comparisons made (Table I, Figures 1 and 2) strongly support the proposed method. Acknowledgment The financial assistance of the Natural Sciences and Engineering Research Council of Canada is gratefully ac-
130 Ind. Eng. Chem. Res., Vol. 28, No. 1, 1989
knowledged. P. C. N. Mak also thanks the University of British Columbia for the award of a Graduate Fellowship.
Nomenclature a = parameter in the attraction pressure term of the cubic equation of state defined by eq 10 and 11 b = parameter in the cubic equation defined by eq 12 B = second virial coefficient C = third virial coefficient D = fourth virial coefficient f = function defined by eq 2 F = function defined by eq 7 and 8 N = number of data points Greek Symbols 01 = temperature dependence of parameter a in the cubic equation of state defined by eq 5, 10, 14, and 16 w = acentric factor 0, = coefficient of a defined by eq 11 Qb = coefficient of b defined by eq 12
Literature Cited Adachi, Y.; Lu, B. C.-Y.; Sugie, H. Three-parameter equations of state. Fluid Phase Equilib. 1983, 13, 133-142. Dymond, L. H.; Smith, E. B. The Virial Coefficients of Pure Gases and Mixtures: A Critical Compilation;Oxford University Press: Oxford, 1980. Hayden, J. G.; O'Connell, J. P. A Generalized Method for Predicting Second Virial Coefficients. Ind. Eng. Chem. Process Des. Deu. 1975, 14,209-216. Kubic, W. L. A modification of the Martin equation of state for calculating vapour-liquid equilibria. Fluid Phase Equilib. 1982, 9, 79-97. Lielmezs, J.; Merriman, L. H. Modification of Peng-Robinson equation of state for saturated vapour-liquid equilibrium. Thermochim. Acta 1986, 105, 383-389.
Martin, J. J. Equations of State. Ind. Eng. Chem. 1967,59, 34-52. Martin, J. J. Cubic Equation of State-which? Ind. Eng. Chem. Fundam. 1979, 18, 81-97. Martin, J. J. Correlation of Second Virial Coefficients Using a Modified Cubic Equation of State. Ind. Eng. Chem. Fundam. 1984, 23, 454-459. Peng, D. Y.; Robinson, D. B. A New Two-Constant Equation of State. Ind. Eng. Chem. Fundam. 1976,15, 59-64. Pitzer, K. S.; Curl, R. F., Jr. The Volumetric and Thermodynamic Properties of Fluids. 111. Empirical Equation for the Second Virial Coefficient, J . Am. Chem. SOC.1957, 79, 2369-2370. Reid, R. C.; Prausnitz, J. M.; Sherwood, T. K. The Properties of Gases and Liquids; McGraw-Hill: New York, 1977; pp 629-665. Schmidt, G.; Wenzel, H. A modified Van der Waals type equation of state. Chem. Eng. Sci. 1980, 35, 1503-1512. Shaw, J. M.; Lielmezs, J. A note on the accuracy of departure function predictions. Chem. Eng. Sci. 1985, 40,1793-1797. Soave, G.Equilibrium constants from a modified Redlich-Kwong equation of state. Chem. Eng. Sci. 1972, 27, 1197-1203. Soave, G. A new two-constant equation of state. Chem. Eng. Sci. 1980, 35, 1725-1729. Tsonopoulos, C. An empirical correlation of second virial coefficients. AIChE J . 1974, 20, 263-272. Yu. J.-M.; Adachi, Y.; Lu, B. C.-Y. Selection and design of cubic equations of state. Equations of State: Theories and Applications; Chao, K. C., Robinson, R. L., Eds.; ACS Symposium Series 300; American Chemical Society: Washington, DC, 1985; Chapter 26, pp 537-559.
* Author to whom the correspondence
should be addressed.
Patrick C. N. Mak, Janis Lielmezs* Chemical Engineering Department The University of British Columbia Vancouver, B.C., V6T 1 W5 Canada Received for review April 22, 1988 Accepted September 21, 1988
