652
Ind. Eng. Chem. Process Des. Dev. 1981, 20,652-658
New Correlation for the Prediction of Activity Coefficients for Binary Nonpolar Hydrocarbon Mixtures Chung-Ton Lin and Thomas E. Daubert' Depatiment of Chemical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802
A correlation, based on the van Laar theory and the Soave equation of state, has been developed for estimating acthri coefficients from the critical properties and the acentric factors of pure components. For binary systems with zero Interaction coefficients, the acthri coefficients c a n be calculated directly from this correlation. A parameter 4 is required for those systems with interaction coefficlents greater than zero. Values of /,are evaluated from the vapor-liquid equilibrium data for binary mixtwes with a Merence in acentric factor between 0.0052 and 0.14 and cannot be extrapolated beyond this range without prior verification. The accuracy of Kvalues calculated from this method is comparable to those calculated from the two-parameter Wilson equation and is better than those calculated from the Soave equation.
Introduction In a mixture, the activity coefficient of a component, which is a measure of nonideality of the mixture, is defined as the ratio of the activity of that component in the mixture to that of an ideal solution. The activity is the ratio of the fugacity of the component to the fugacity of an arbitrary standard state, usually the pure liquid at system temperature and pressure. When limited experimental vapor-liquid equilibrium (VLE) data are available, the activity coefficients in a binary mixture can often be estimated by using some empirical or semiempiricalcorrelations, such as the van Laar, Wilson (Wilson, 19641, NRTL (Renon and Prausnitz, 1968) or UNIQUAC (Abrams and Prausnitz, 1975) equations. These equations provide a thermodynamically consistent interpolation or extrapolation of the data. When there are no VLE data a t hand, other procedures, such as the Scatchard-Hildebrand theory (Hildebrand, 1929; Scatchard, 1949) or UNIFAC (Fredenslund et al., 1975) method must be employed to predict the activity coefficients directly from other types of data. Although there are many expressions and procedures available for estimatingactivity coefficients, it is interesting to find that none of them can be applied to calculate activity coefficients from the most widely used properties of pure components such as critical temperature, critical pressure, and acentric factor. The purpose of this work was to develop such a Correlation for nonpolar hydrocarbon mixtures. The van Laar-type equation correlates the VLE of the binary nonpolar hydrocarbon mixtures quite well. The strategy utilized was first to calculate activity coefficients from experimental VLE data and then to correlate the activity coefficient with the critical properties and the acentric factors of constituents by a modified van Laartype model. The resulting correlation has been applied successfully to calculate K values for many nonpolar hydrocarbon mixtures. The accuracy of those predicted K values is comparable to those calculated from either the Wilson equation or the Soave equation. However, this correlation is not recommended to be applied to those systems with values of the difference of two acentric factors either larger than 0.14 or less than 0.0052. Modified van Laar Model Development van Laar assumed that a mixture of two liquids-x, moles of liquid i and xj moles of liquid j-is formed by the 0196-4305/81/1120-0652$01.25/0
following three steps (Prausnitz, 1969): (1) Each pure liquid is expanded isothermally to very low pressure, i.e., ideal gas. (2) Two low pressure gases mix. (3) The gas mixture is compressed isothermally to the original liquid pressure. Based on these three steps, one can calculate the excess which is the s u m of internal energy change of mixing, P, these internal energy changes. The internal energy change of each step is evaluated from the P-V-T relation for fluids by the following thermodynamic identity.
The Soave equation (Soave, 1972) was chosen in this investigation to evaluate the internal energy changes for both pure fluids and mixtures.
p = - -RT
V-b
a V(V+b)
The following mixing rules for d and b were applied for mixtures. d,
= X&
+ 2xix1~idj(l- kij) + b, = xibj + xjbj
xj%j
(3) (4)
where (5) and m = mixture. By assuming that the liquid molar volume of fluid is proportional to the constant b of Soave's equation and the mixture is a regular solution, Vi = kbi, one obtains the following equation for the excess Gibbs free energy change of mixing.
where c is a constant equal to In [(k + l)/k]. The activity coefficient of component i in a mixture is related directly to the excess Gibbs free energy by the equation 0 1981 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 20, No. 4, 1981 853
(7) Substituting eq 6 into eq 7 and differentiating
where
lij
= 2ckij
Equation 8 is the van Laar expression for the activity coefficient. In this work, predicting constants Ai and A, from the critical properties was attempted. The constant c was correlated from the VLE data for those systems with ki,equal to zero and then was applied to evaluate the value od lij for those systems with nonzero interaction coefficients. Calculation of Activity Coefficients from VLE Data The condition which must be satisfied for equilibrium between a liquid, at a specific temperature T and pressure p , and a vapor at the same conditions, is that fugacity of each component in liquid phase is equal to that in vapor phase fiV = f i L (10) In order to calculate fivfrom measurable PVT data, the fugacity coefficient, $Ii, is introduced. fiV = $I&P (11) For calculating the fugacity of the liquid phase, eq 12 is used f i L = xirifiO (12) where f? is the fugacity a t a reference state which is arbitrarily chosen as the pure liquid state at T and P. The activity coefficient, yi, is a function of pressure, temperature, and composition. Partial molar volume is introduced in the following equation to account for pressure effects (Prausnitz, 1969) and 1 atm is chosen as a reference pressure pr
where Vi is the partial molar volume of component i in the mixture. The fugacity at the reference state is calculated from data on saturated vapor and liquid by the equation P' ViL fio = pi"4i" (14)
E
where the exponential term is usually called the Poynting Correction. Substituting eq 11,12,13, and 14 into eq 10 and rearranging
Table I. Comparison of the Calculated and Experimentally Derived Activity Coefficients for Benzene(1) and Cyclopentane(2) Mixtures at Two Different Temperatures XI
YIRa
71ca
7 2
R
Y *c
0.1417 0.2945 0.4362 0.5166 0.5625 0.8465
Temperature, 25 "C 1.408 1.4180 1.010 1.253 1.2605 1.043 1.151 1.1580 1.095 1.135 1.108 1.1158 1.087 1.0943 1.160 1.010 1.0149 1.380
1.0229 1.0558 1.1082 1.1495 1.1751 1.3951
0.1417 0.2945 0.4362 0.5166 0.5625 0.8465
Temperature, 45 "C 1.353 1.3342 1.009 1.219 1.2005 1.039 1.130 1.1135 1.085 1.092 1.0772 1.119 1.074 1.0606 1.141 1.008 0.9919 1.325
1.0167 1.0459 1.0921 1.1278 1.1515 1.3323
Superscript R indicates reported values while c means calculated values from eq 15.
Equation 15 is applied to calculate the activity coefficients from VLE data. The Lee-Kesler correlation (Lee and Kesler, 1975), which has been found to accurately predict saturated vapor pressure (Reid et al., 19771, was chosen in this work. To calculate the saturated liquid density, the revised Rackett equation (Spencer and Danner, 1972) with a constant determined from experimental data is utilized. The fugacity coefficients of a saturated vapor are calculated from either the virial equation which is truncated after the second virial coefficient or the Soave Equation of State (Lin and Daubert, 1978). Lin and Daubert (1978, 1980) have compared the estimated fugacity coefficients of a component in a mixture from different equations of state and found that both the Soave and Peng-Robinson equations (1976) with interaction coefficient, kij, gave accurate estimations. The Soave equation of state was chosen in this work. Partial molar volumes were calculated from the Peng-Robinson equation as discussed by Lin and Daubert (1980). The accuracy of the estimated thermodynamic properties, especially fugacity coefficients and saturated vapor pressure is essential for the calculation of activity coefficients. The effect of liquid molar volume is less significant. The estimated activity coefficients obtained from eq 15 were fairly close to those reported in the literature as shown in Table I for benzene and cyclopentane mixtures. At low pressures the effect of partial molar volume on calculated activity coefficient is negligible. In high pressure systems the accuracy of the partial molar volume becomes important. Data Correlation a. Evaluation of van Laar Constants from VLE Data. The following equation relates the excess Gibbs free energy to activity coefficients for a binary mixture.
-/2E- - x i In yi + x j In yj U
RT Equation 17 is easily established by eliminating yi and y, from eq 8 and 16.
If xixj/GE/RTL plotted VS. xi, [ ( l / A j )- ( l / A i ) ]and 1/Ai can be obtained directly from the slope and intercept of the plot, .respectively. As a result, values of Ai and Aj can be easily evaluated.
654
Ind. Eng. Chem. Process Des. Dev., Vol. 20, No. 4, 1981
Table 11. A List of the Systems with Zero Interaction Coefficient range system
T,K
ethene( 1)-ethane( 2) n-pentane( 1)-cyclohexane( 2) n-hexane( l)-cyclohexane(2) cyclohexane( 1)-n-heptane( 2) benzene( 1)-cyclopentane( 2) benzene( 1)-cyclopentane( 2) benzene( 1)-cyclopentane( 2) benzene( 1)-cyclopentane( 2) benzene( 1)-cyclopentane( 2) benzene( l)-cyclopentane( 2) n-hexane( 1)-methylcyclohexane( 2) 1-hexene(1)-n-hexane( 2 ) 1,4-dimethylbenzene( 1)-1,3-dimethylbenzene( 2) methylcyclopentane( 1)-methylbenzene( 2)
200.0 320.0 345.0 360.0 298.0 308.0 318.0 324.0 338.0 348.0 355.0 333.0 412.0 348.0
no. 1 2 3
4 5 6 7 8 9 10 11 12 13 14
no. 1
2 3 4 5 6 7 8 9 10 11 12 13 14 a
value of c of eq 20
range of liquid (xi) compn
values of k i j
0.1-0.9 0.453-0.73 0.4 15-0.969 0.05 1-0.97 3 0.1417-0.8465 0.1417-0.4362 0.1417-0.8465 0.022-0.21 0.661-0.730 0.831-0.994 0.184-0.837 0.248-0.877 0.10-0.90 0.4125-0.929
0.0 0.0 0.0 0.0 0.79 x 0.79 x 0.79 x 0.79 x 0.79 x 0.79 x 0.17 x 0.56 X 0.11 x 0.13 x
10-7 10-7 10-7 10-7 10-7 10-7 10-5 10-5 10-5
comp 1
comp 2
1613.95 1.2399 0.2684 0.1487 6.1527 6.0649 5.3057 4.7406 7.1430 11.2403 1.0251 10.7536 35.7656 5.9498
1122.50 1.3262 0.5389 0.2529 6.1986 6.9434 5.1284 6.1884 5.3219 5.6474 1.5574 5.6188 39.9216 4.6816
of P, atm
N
2.0-4 .O 9 1.0 3 1.0 6 1.0 12 0.38 4 0.19-0.45 3 0.41-0.81 4 1.0 6 1.0 4 1.0 8 1.o 9 0.79-0.88 5 1.0 5 1.0 9 coefficient of correlation I ~ i - ~ j i comp 1
0.0052 0.0366 0.0813 0.1362 0.0177 0.0177 0.0177 0.0177 0.0177 0.0177 0.0624 0.0109 0.0068 0.0171
0.986 0.989 0.997 0.556 0.999 0.999 0.997 0.980 0.996 0.614 0.927 0.696 0.541 0.842
comp2 0.988 0.996 0.987 0.776 0.996 0.989 0.997 0.819 0.999 0.982 0.962 0.985 0.526 0.987
N = number of data points; T = temperature; P = pressure.
The following two equations, which are derived based on eq 8, are recommended (Black, 1958) for the calculation of van Laar constants directly from activity coefficients.
Values of Ai and A, are easily acquired by plotting either (log yi)0.5vs. (log yj)0.5or (log yj)0.5vs. (log yi)0.5. As depicted in Figure l, the van Laar constanh calculated from different equations have essentially the same values. Any one set of three is chosen for the correlation as discussed in the next section. b. Correlation of Constant C. Equation 9 can be rewritten as Ai = c F + ~ LijG, (20) where Fi and Gi are functions of critical properties and acentric factors of both component i and j and are given by eq 21 and 22, respectively. Tci pi = ---[(I + M(@~)(I - ~ , ~ 5 ) ) 1 / 2 ( 1+ ~ ( ~ ~ ) ) 1 / 2 ~-~ 1 PciT (1 + ~ ( ~ -~~~p5))1/2(1 ) ( i + ~ ( ~ ~ ) ) 1 / 2 ~ ~ j ' (21) / 2 1 2
Since the value of lij is the interaction coefficient, k,, times a constant, those systems with k,'s equal to zero are chosen to calculate c. Chueh and Prausnitz (1967) recommended some values of kij for many binary systems. For those systems where k i / s are not available, eq 24 was adopted (Chueh and Prausnitz, 1967; Teja, 1978) for estimating kij of paraffin-paraffin mixtures.
For other hydrocarbon mixtures, Chaudron et al. (1973) proposed the equation
/ 2
For nonpolar hydrocarbons, values of kij usually increase with the increasing absolute values of the difference in acentric factors, i.e. Iwi - wjl. Systems with values of kij of or less are assumed to have zero interaction coefficients. Table I1 lists these systems together with the coefficients of correlation for the van Laar model and the values of c which are evaluated directly from the calculated activity coefficients. The c values are then correlated with the absolute difference of the two acentric factors. In c = 0.287 - 14.691~;- w j ( + O.O3071/l~i- W j (
(26)
Equation 26 has a coefficient of correlation of about 0.9 and enables predictions of the activity coefficient from
Ind. Eng. Chem. Process Des. Dev., Vol. 20, No. 4, 1981 655 Table 111. A List of the Systems with Nonzero Interaction Coefficient ~~
no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
system T, K no. system T,K 399.0 ethylbenzene( 1)-vinylbenzene( 2) 315.0 23 2,2,54rimethylhexane( l)-ethylbenzene( 2) 354.0 ethylbenzene( 1)-vinylbenzene( 2) 335.0 24 benzene( 1)-methylcyclohexane( 2) 368.0 ethylbenzene(1)-vinylbenzene( 2) 350.0 25 benzene( 1)-methylcyclohexane( 2) ethylbenzene(1)-isopropylbenzene( 2) 420.0 26 benzene( 1)-1,3-dimethylbenzene( 2) 310.0 323.0 n-hexane( 1)-methylcyclopentane( 2) 343.0 27 benzene( 1)-1,3-dimethylbenzene( 2) isopropylbenzene( 1)-n-butylbenzene( 2) 430.0 28 2,2,44rimethylpentane( 1)-methylbenzene( 2) 376.0 350.0 methylcyclohexane( 1)-toluene( 2) 341.0 29 benzene( 1)-2,2,44rimethylbenzene( 2) 289.0 methylcyclohexane( 1)-toluene( 2) 355.0 30 n-pentane( 1)-benzene( 2) 346.0 methylcyclohexane( 1)-toluene( 2) 373.0 31 n-hexane( 1)-benzene( 2) 351.0 methylcyclohexane( 1)-toluene( 2) 378.0 32 n-hexane( 1)-methylbenzene( 2) 360.0 n-heptane(1)-methylcyclohexane(2) 373.0 33 methylbenzene( 1)-n-octane( 2) 391.0 ethylcyclohexane( l)-ethylbenzene( 2) 384.0 34 methylbenzene( 1)-n-octane( 2) 255.0 316.0 35 ethane( 1)-n-butane( 2) n-pentane(1)-methylcyclohexane( 2) 233.0 methylcyclopentane(1)-benzene( 2) 348.0 36 benzene( 1)-n-heptane( 2) 340.0 2,2,44rimethylhexane(1)-methylcyclohexane( 2) 372.0 37 benzene( 1)-n-heptane(2) 353.0 n-octane( 1)-1,4-dimethylbenzene(2) 405.0 38 benzene( 1)-n-heptane( 2) 366.0 n-octane( l)-ethylbenzene(2) 324.0 39 benzene( 1)-n-heptane(2) 358.0 n-octane( l)-ethylbenzene( 2) 359.0 40 cyclohexane( 1)-methylbenzene( 2) 370.0 n-octane( l)-ethylbenzene( 2) 385.0 41 cyclohexane( 1)-methylbenzene( 2) 375.0 n-octane( 1)-ethylbenzene( 2) 400.0 42 n-heptane( 1)-methylbenzene( 2) 356.0 ethylbenzene(1)-n-butylbenzene(2) 420.0 43 benzene( 1)-2,2,44rimethylpentane(2) 400.0 1-octene( l)-ethylbenzene( 2) 400.0 44 naphthalene( 1)-n-tetradecane( 2) % error of predicted % error of predicted activity coefficient activity coefficient no. of no. of by van Laar model by van Laar model data values data values points no. points of kij comp 1 comp 2 comp 1 comp 2 no. of kij 1 2 3 4 5 6 7 8 9 10 11 12
13 14 15 16 17 18 19 20 21 22
8 10 8 5 7 5 6 7 9 9 10 9 4 16 12 4 4 6 9 8 4 7
0.00006 0.00006 0.00006 0.00023 0.00092 0.00031 0.00032 0.00032 0.00032 0.00032 0.00109 0.00043 0.00151 0.00060 0.00080 0.00094 0.00101 0.00101 0.00101 0.00101 0.00109 0.00132
2.9341 2.7665 3.9015 0.2512 0.2528 2.4790 14.9857 0.8230 0.4291 0.9140 1.2025 0.8129 0.4837 2.9902 1.2491 0.6233 2.1628 0.7983 0.7800 0.6855 0.5629 0.6569
3.6040 1.9821 3.3140 2.2647 2.0988 1.8719 0.7770 2.7010 0.1990 0.2899 0.5194 0.8326 2.5484 1.4318 0.8009 1.9136 2.4857 1.0143 0.8537 1.0395 2.1785 0.5891
critical properties and acentric factors for those systems with kij equal to zero. A plot of the constant c vs. Iwi - wjl is shown in Figure 2. Values of c obtained from eq 26 were then applied for those systems with kiis greater than zero using eq 20. There is one parameter, 1.. to be determined from the vapor-liquid equilibrium %ita of those sytems. To find the optimal 1, for K value prediction, one has to minimize the sum of the square of errors of predicted K values of both components 1 and 2. Application of the New Correlation to K Value Prediction Systems with zero interaction coefficient, which were used to correlate eq 26, were listed in Table 11. Forty four other nonpolar hydrocarbon binary mixtures with nonzero interaction coefficient are shown in Table 111. As shown in these two tables, the original van Laar model correlates the activity coefficients of such mixtures quite well. Figure 3 compares the estimated K values of this modified van Laar model for activity coefficients with experimental K values for six different systems. These estimations are as good as those of the original van Laar model.
23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44
5 5 10 6 3 18 13 7 25 3 7 4 3 20 9 21 5 8 10 21 7 5
0.00145 0.00171 0.00171 0.00192 0.00192 0.00214 0.00256 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.00484 0.00695
0.9362 0.7236 7.7932 0.9169 1.2052 3.0211 0.9584 1.8246 2.8006 0.2913 15.0884 2.4619 1.7393 1.0542 0.9497 1.4625 1.3605 1.4230 2.1325 1.8232 1.0077 2.1134
1.5998 2.0406 0.6100 8.0347 7.2031 2.4561 1.2798 5.5624 1.1873 1.1443 3.8557 0.2801 1.4299 3.3374 3.5743 1.7831 1.6563 0.5129 1.3712 4.8871 2.7654 8.8747
A comparison of the estimated K values of this modified van Lam model with those of the Wilson equation is shown in Table IV. The accuracy of the estimation of this one adjustable parameter model is comparable to that of the two-parameter Wilson equation. In addition, the K values estimated from this model are better than these of the Soave equation, which is most ofter recommended for nonpolar hydrocarbon VLE prediction. As listed in Tables V and VI, the overall error in the K values predicted from this model is about half the error obtained when usi,ng the Soave equation. Generally speaking, the predicted K values from the Soave equation are better than that of this one-parameter van Laar type model for aromatic-aromatic mixtures, but poorer in paraffin-aromatic systems. For paraffin-paraffin solutions, the differences between the estimates of the two prbcedures are very small. Limitation of This van Laar Type Model Due to the assumption of regular solution, this model is not applicable to those mixtures with significant excess volume and excess entropy of mixing.
656
Ind. Eng. Chem. Process Des. Dev., Vol. 20, No. 4, 1981
Table IV. Comparison of the Errors in K Value Predictiona
no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
systems ethylene( l)-ethane( 2) n-pentane(1)-cyclohexane( 2) n-hexane(l)-cyclohexane( 2) methylcyclopentane(1)-methylbenzene( 2) methylcyclohexane(1)-methylbenzene( 2) methylcyclohexane(1)-methylbenzene( 2) ethylcyclohexane( 1 )-ethylbenzene( 2) methylcyclopentane(1)-benzene( 2) n-octane(1 )-ethylbenzene( 2) n-octane(l)-ethylbenzene( 2) n-octane(1)-ethylbenzene( 2) n-octane( 1)-ethylbenzene( 2) 2,2,5-trimethylhexane(1)-ethylbenzene( 2) benzene(1)-methylcyclohexane( 2) benzene(1)-methylcyclohexane( 2) benzene(1)-2,2,3-trimethylbutane(2) n-hexane(1 )-benzene( 2) n-hexane(1)-methylbenzene( 2) methylbenzene(1 )+-octane( 2) benzene( 1)-2,2,4-trimethylpentane( 2) n-heptane(1)-methylbenzene( 2) n-hexane(1 )-methylcyclohexane( 2) n-hexane(1)-methylcyclopentane( 2) cyclohexane(1 )-heptane( 2)
T, K 200 320 345 348 373 378 384 348 324 359 385 400 399 354 368 350 346 351 391 356 375 373 343 360
N 9 3 6 9 9 9 9 16 4 6 9 8 5 5 10 13 25 3 4 7 21 10 7 12
proposed model
Wilson equation
error, %
error, %
K1
.
K2 3.7717 3.8411 1.1748 6.6738 0.8536 0.6440 2.8838 0.9633 1.9713 1.1199 0.9788 0.9034 1.7503 3.5712 0.6433 5.5585 4.2197 1.0490 0.9486 5.2504 1.8636 0.5930 2.1936 1.9294
K1 2.6398 0.9872 0.9998 3.0196 0.4267 0.6122 1.8876 6.1021 1.9691 1.1587 0.8322 0.5928 0.9409 1.0692 7.4465 3.6749 5.3520 0.7849 2.9927 2.1798 2.6290 0.7681 0.5536 2.7 167
K2 1.3702 0.9197 2.4215 8.5977 0.5420 0.2610 2.7661 2.1749 2.2417 2.3485 2.4797 2.4038 3.6907 6.9674 4.9365 5.8337 1.0211 1.1342 0.8836 2.0283 1.8498 1.4702 3.3699 3.4177
2.8675 2.4337 N = number of data points; K 1 = K value of component 1;K2 = K value of component 2.
2.1769
2.7137
6.5305 3.5052 2.1864 3.6099 0.8810 1.0048 2.4435 4.3815 5.9973 1.8697 2.7492 1.1058 2.0384 1.0780 9.0401 2.6172 3.5252 1.1530 2.7811 1.8500 2.8668 1.7176 0.2531 2.5491
overall error
-;.k
.
.
.
.
. .
.
8
.
I
!
. .
-. . -
. .
I I
,
,
I
.
..
I
N
.
.!.
. , . . . . . . . . ! . -, ! . , . ' . , . . ,. ,
.!.I
,
'
,
,
!
. -: , . :.:
,
I
! ,
I
,
' 1
I.
.
1.:
: ! .;-I.
.
...... - . . . . .
.., ..!_.
..:
: - -: : : . . ._
____ -. .
,
.:
..
,
! !-
. . I
.
.
.
I
I
,
:. '
!
.
;
. I
c-!
-+a
1
'
.
,
I,
.
A-C::!:
i- j -
-e--
-. ..,.... . . . . . . . . , . . . . . . . . . ,. , . . . . . . ,. . . . . . . . . . . ." I . , . . . I , . !
..-,.:
-!
,
!.J
I,
. . . . . . . . . . . . . . . . . . , . , . . - i,. .- . . . . _ r . / . , _ . . . . _1 , _j, . :'
. . . . ..
,
,
1
. . .1 . .. . . . . . ,: , .*.
.
I
'
.
I
........... .
.:
.
,
-___
.
.
.. ,. '. . . . .
- . . ., _ ; _ . _-1 ! !
...
......
.I . , ,
,
.
. . . . . ., . I' . - ... , .
. . ,. .. . .,..
. ,
.
. ,
'
-.,- ,-!-,.
I
*
.
-
. . , . I .. .
..... I
. .
... . . . . . .
__
-.
,
.
8
'
,
-1
.
. . . .
,. . . . . . I. .'
..
.- i
,
.... '
..., I
.
. I
,
.- .. - 1
0 .. ,l I
a 0.5
(In y,)
2
X 10
Figure 1. Evaluation of van Laar constants for methylcyclohexane(1) and toluene(2) at 373 K.
In the critical region since large deviations exist in predicted saturated liquid molar volume using the Rackett equation, this procedure should not be applied to predict VLE when the reduced temperature of any component in
l
. . .:, I,
80.
'
I
-
I"
:
,. . .. . . !
loo
120
: '-!.. I
,
'
._0 ; ..: !I!. I I
.
140
IO
Iwi-wjl x i o 3
Figure 2. Relation between C and difference in acentric factors.
the mixture is greater than 0.99. When the saturated vapor pressure is less than 10 mmHg or the temperature of the system is below the boiling point of any constituent, the accuracy of the predicted vapor pressure from the LeeKesler correlation is reduced. As a result, poor agreement
Ind. Eng. Chem. Process Des. Dev., Vol. 20, No. 4, 1981 857 Table V. Comparison of Errors of Predicted K Values between the Proposed Model and the Soave Equation of State for Systems with Zero Interaction Coefficient a % error of proposed model % error of Soave ~
no.
system
T,K
N
1 2 3 4 5 6 7 8 9 10 11
ethene( l)-ethane( 2) n-pentane( l)-cyclohexane( 2) n-hexane( l)-cyclohexane( 2) cyclohexane( 1)-n-heptane( 2) benzene( l)-cyclopentane( 2) benzene( l)-cyclopentane( 2) benzene( l)-cyclopentane( 2) n-hexane( 1)-methylcyclohexane( 2) lhexene( 1)-n-hexane( 2) 1,4-dimethylbenzene( 1)-1,3-dimethylbenzene( 2) methylcyclopentane( 1)-methylbenzene( 2 )
200 320 345 360 303 322 343 355 333 412 348
9 3 6 12 7 10 12 9 5 5 9
overall error a
K value, compl
K value,
K value,
comp2
compl
6.5305 3.5052 2.1864 2.5491 3.2674 3.3155 0.9405 0.9365 1.4980
3.7717 3.8411 1.1748 1.9294 3.8096 1.8352 5.3610 0.6514 5.3803
3.6099
6.6738
8.1457 0.5666 2.2062 4.0254 16.8449 17.2164 1.7069 1.4588 1.0868 0.2097 5.9543
K value, comp2 5.9379 1.6670 0.9119 3.1944 10.6318 7.4153 21.1087 0.4440 0.7648 0.1524 12.7822
2.8339
3.4428
5.4020
5.9100
N = number of data points; % error = [ E N (estimates - experimental value)/experimental value) 1002/N]’”.
Table VI. Comparison of Errors of Predicted K Values between the Proposed Model a and the h a v e Equation of State for Systems with kij # 0 Soave procedure
error, %
error, %
no.
systems
T, K
K1
K2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44
ethylbenzene(1)-vinylbenzene( 2) ethylbenzene( 1)-vinylbenzene( 2) ethylbenzene( 1)-vinylbenzene( 2) ethylbenzene( 1)-isopropylbenzene( 2) n-hexane( 1)-methylcyclopentane( 2) isopropylbenzene( 1)-n-butylbenzene( 2) methylcyclohexane( 1)-toluene( 2) methylcyclohexane( 1)-toluene( 2) methylcyclohexane( 1)-toluene( 2) methylcyclohexane( 1)-toluene( 2) n-heptane( 1)-methylcyclohexane( 2) ethylcy clohexane( 1)-ethylbenzene( 2) n-pentane( 1)-methylcyclohexane(2) methylcyclopentane( 1)-benzene( 2) 2,2,44rimethylhexane(1)-methylcyclohexane( 2) n-octane( 1)-1,4-dimethylbenzene( 2) n-octane( l)-ethylbenzene( 2) n-octane( l)-ethylbenzene( 2) n-octane( l)-ethylbenzene( 2) n-octane( l)-ethylbenzene( 2) ethylbenzene(1)-n-butylbenzene( 2) 1-octane( l)-ethylbenzene( 2) 2,2,54rimethylhexane( l)-ethylbenzene( 2) benzene( 1)-methylcyclohexane(2) benzene( 1)-methylcyclohexane( 2) benzene( 1)-1,3dimethylbenzene(2) benzene( 1)-1 ,a-dimethylbenzene( 2) 2,2,4-trimethylpentane(1)-methylbenzene( 2) benzene( 1)-2,2,44rimethylbenzene(2) n-pentane( 1)-benzene( 2) n-hexane( 1)-benzene( 2) n-hexane( 1)-methylbenzene( 2) methylbenzene( 1)-n-octane(2) methylbenzene( 1)-n-octane( 2) ethane( 1)-n-butane( 2) benzene( 1)-n-heptane(2) benzene( 1)-n-heptane( 2) benzene( 1)-n-heptane(2) benzene( 1)-n-heptane(2) cyclohexane( 1)-methylbenzene( 2) cyclohexane( 1)-methylbenzene(2) n-heptane( 1)-methylbenzene(2) benzene( 1)-2,2,44rimethylpentane(2) naphthalene( 1)-n-tetradecane( 2)
315.0 335.0 350.0 420.0 343.0 430.0 341.0 355.0 373.0 378.0 373.0 384.0 316.0 348.0 372.0 405.0 324.0 359.0 385.0 400.0 420.0 400.0 399.0 354.0 268.0 310.0 323.0 376.0 350.0 289.0 346.0 351.0 360.0 391.0 255.0 233.0 340.0 353.0 366.0 358.0 370.0 375.0 356.0 400.0
5.4179 2.7705 4.9836 2.8383 0.2531 4.1337 16.9150 0.8214 0.8810 1.0048 1.7176 2.4435 0.5612 4.3815 2.0058 3.4506 5.9973 1.8697 2.7492 1.1058 0.5914 0.6947 2.0384 1.0780 9.0401 7.1671 4.4654 4.3777 2.6172 2.9991 3.5252 1.1530 27.5498 2.7811 4.4150 2.3985 1.5510 3.0046 3.8745 3.8207 2.7890 2.8668 1.8500 2.1374
3.2637 3.3806 6.8302 8.8753 2.1936 2.4087 0.6439 3.2195 0.8536 0.6440 0.5930 2.8838 3.4202 0.9633 0.5289 1.7515 1.9713 1.1199 0.9788 0.9034 7.0840 2.0590 1.7503 3.5712 0,6433 20.0092 15.7178 3.0503 5.5585 5.8434 4.2197 1.0490 4.2897 0.9486 7.0061 3.0070 3.8053 3.4076 2.0660 2.0821 0.8512 1.8636 5.2504 10.2099
3.7941 1.6898 1.4421 1.0010 1.5381 3.2767 31.6676 3.3384 3.2495 6.4335 2.1579 2.9686 0.7143 14.2157 2.9559 4.2553 1.7489 0.5590 1.8970 0.3381 0.7547 0.6905 0.597 1 0.9629 11.9463 1.3613 0.4142 2.4882 3.0130 4.9213 8.9622 3.1687 20.9079 3.0726 2.5230 3.0667 1.8252 3.0596 4.8377 4.0410 17.1051 3.7899 1.7336 50.4691
1.2838 2.5869 2.6318 3.0130 1.6861 7.2854 2.1841 4.3080 3.8637 3.1746 1.2101 2.6371 3.9975 4.9247 2.1131 1.6604 3.2726 1.3321 1.1115 1.3493 4.6633 2.2693 1.7893 18.4097 0.3860 11.5410 12.5309 2.0686 8.3027 19.3507 3.6856 4.9225 4.9167 1.3891 5.7572 7.1525 4.8296 4.5293 1.4664 12.3801 1.0783 6.1315 3.3105 101.6570
3.7145
3.9016
5.6678
8.6791
overall error a
proposed model
K 1 = K values of component 1;K2 = K values of component 2.
K1
K2
Xnd. Eng. Chem. Process
858
Des. Dev. 1981, 20, 658-662 f i = fugacity of component i GE = excess Gibbs free energy kij = interaction coefficient of mixture with components i and j ni = mole number of component i in mixture P, = critical pressure P = pressure Pi" = saturated vapor pressure of component i R = gas constant T = temperature T , = critical temperature V = volume = critical volume Vi = partial molar volume of component i VF = saturated liquid molar volume of component i U = internal energy x i = mole fraction of component i in mixture yi = mole fraction of component i in vapor yi = activity coefficient of component i &' = fugacity coefficient of saturated vapor pressure w = acentric factor L i t e r a t u r e Cited
I
'I COMPOSITION OF CYCLOHEXANE I
lo2
Figure 3. Comparison of K values of cyclohexane-calculated vs.
experimental. exists between experimental and predicted K values from this procedure. In addition, since eq 26 is developed based on data for those systems with values of Iwi - wil between 0.0052 and 0.14, eq 26 should not be used without verification to calculate activity coefficients for any system with an acentric factor difference beyond this range.
Abrams, D. S.; Prausnitz, J. M. AIChE J. 1975, 21, 116. Black, C. I d . fng. Chem. 1958, 50, 403. Chaudron, J.; Asselkeau, L.; Renon, H. chem. Eng. Sc/. 1978, 28, 1991. Chueh, P. L.; Prausnkz, J. M. I d . Eng. Chem. Fundem. 1987, 6 , 492. Fredenslund, A.; Jones, R. L.; Prausnitz, J. M. AICM J. 1975, 27, 1086. Hlldebrand, J. H. J. Am. Chem. Soc.1929, 57, 66. Lee, B. I.; Kesier, M. G. AICM J. 1975, 21, 510. Lin, C. T.; Daubert, T. E. I d . fng. Chem. Process D s . Dev. 1978, 17, 544. Lin, C. T.; Daubert, T. E. I d . Eng. Chem. procesS Des. Dev. lS80, 79, 51. Peng, D. Y.; Robinson, D. B. I d . Eng. chem. fundem. 1976, 75, 59. Prausnitz, J. M. "Molecular Thermodynamics of Fluidghese EqulHbrium"; Prentke-Hell Inc.: Englewood Cllffs, NJ, 1969. Reid, R. C.; Prausnitz. J. M.; sherwood, T. K. "The Properties of Qases and LlquMs", 3rd ed.; McGrawH111: New York, 1977. Renon, H.. Prausnitz. J. M. AICM J. 1968, 14, 135. Scatchard, 0. Chem. Rev. 1949, 44, 7. Soave, G. Chem. Eng. Sci. 1972, 27, 1197. Spencer, C. F.; Danner, R. P. J. Chem. Eng. Lbta 1972, 17, 236. Teja, A. S. Chem. Eng. Sci. 1978, 33, 609. Wilson, G. M. J. Am. Chem. Soc. 1964, 86, 127.
Received for review September 6,1979 Revised manuscript received February 3, 1981 Accepted June 15,1981
Nomenclature Ai = van Laar constant of component i
Applkation of the Modified Potential Theory to the Adsorption of Hydrocarbon Vapors on Silica Gel Taher A. AI-Sahhaf, Earl D. Sloan, and Anthony L. Hines' Department of Chemical and Petroh?un?-Ref/nhg Engineering, Colorado School of Mines, GOMen, Colorado 8040 7
Adsorption isotherms of n-butane, n-pentane, and n-hexane on Davison silica gel were determined gravimetrically at 283,293,and 303 K up to their saturation pressures. The equilibrium adsorption data for all the hydrocarbons studied were correlated by the modified Polanyi potential theory. A generalized adsorption isotherm equation was assumed according to the theory of volume filling of micropores proposed by Dubinin and his co-workers. Using the adsorption isotherms of n-butane at 283 K, constants of the generalized isotherm equation were calculated and were used to predict the other isotherms.
Introduction
Although the adsorption of hydrocarbon gases has been studied extensively, relatively few studies have been made *Department of Chemical Engineering,University of wyominp, Box 3295, University Station, Laramie, WY 82071 0196-4305/81/1120-0658$01.25/0
in which silica gel was used as the adsorbent. Included in the studies using silica gel is the work of Lewis et al. (1950),who measured the adsorption equilibrium for eight hydrocarbons on Davison silica gel at 298 K. Following this, Eberly (1964)measured the adsorption of n-butane on silica gel at temperatures ranging from 303 to 363 K. 0 1981 American Chemical Society
