EMPIRICAL CORRELATION OF SECOND VIRIAL COEFFICIENTS FOR VAPOR-LIQUID
EQ U I LI BR I U M CA LCU LAT IO NS J.
P. O ' C O N N E L L ' A N D
M. PRAUSNITZ
J.
Department of Chemical Engineering, University of California, Berkeley, Calif.
At moderate pressures, second virial coefficients of pure and mixed vapors provide good estimates for vapor-phase fugacity coefficients as required in multicomponent vapor-liquid calculations. When only purecomponent parameters are used-e.g., critical constants-agreement with experimental data is satisfactory for pure components and for most mixtures; for mixtures which exhibit specific interactions between unlike components a single binary association parameter represents the data over a substantial temperature range. Brief consideration is given to the magnitude of errors in calculated vapor-liquid equilibria caused by errors in vapor-phase fugacity coefficients,
accurate prediction of multicomponent vapor-liquid equilibria, it is usually necessary to take vapor-phase nonideality into account. This may be done through the use of a n equation of state as discussed in numerous references (3, 5). IVe present here a correlation for a simple equation of state which is applicable up to moderate densities, remote from the critical, such as those prevailing in most typical separation operations. For low or moderate vapor densities a useful equation of state is the virial equation terminated after the second virial coefficient: OR
-Pu_
RT
- 1
+ B/u
where u is the molar volume and B is the second virial coefficient, a function of temperature and composition only. T h e composition dependence of B for a n n'component mixture is given exactly by N
'$7
where Bit and B,, are the pure-component second virial coefficients, and where B i j (i # j ) is the second virial cross coefficient. These coefficients depend on the intermolecular forces between the species involved and are functions of temperature only. For a binary mixture, Equation 2 becomes
+
+
Bmix = ~12B11 2~1~2B12 ?e2B22
(24
For vapor-liquid equilibrium, the vapor-phase fugacity, f r v , of a component may be expressed in terms of its mole
where u may be found from Equations 1 and 2 . ponent 1 in a binary mixture, Equation 4 becomes
For com
Several correlations have been reported for second virial coefficients of nonpolar substances and a few for polar substances (9, 72, 76). However, to simplify matters for application to vapor-liquid equilibria, we have developed a new correlation in a form suitable for computer application and requiring a minimum of experimental information (73). Stockmayer's (77) model for polar substances indicates that in addition to the parameters expressing the effects of size and nonpolar (dispersion) forces, the intermolecular potential energy function includes a term to account for the electrostatic forces between molecules; this term contains the square of the electric dipole moment. In addition, it seems reasonable to use other characteristic parameters where hydrogen-bonding or other "chemical" associations may be important. IVe have established a correlation for the reduced second virial coefficient which consists of three generalized functions : one for the nonpolar contributions to the second virial coefficient, one for polar interactions based on the dipole moment, and finally a n "association" function for substances which exhibit specific forces such as hydrogen bonds. Pure Gases
For pure, nonpolar gases a n excellent correlation for the second virial coefficient has been given by Pitzer and Curl (72) based on a three-parameter theory of corresponding states. I t has the form
fraction, y i , the pressure, P, and the vapor-phase fugacity coefficient, pi, by
From the virial equation of state, truncated after the second term, the fugacity coefficient is 2 .v In p i =
-
u
C y B. =I 3
z3
-
Po In -
RT
(5) where P,, = critical pressure of i
T,,
= critical temperature of
wz =
(4)
i
acentric factor of z
T R = reduced temperature, T / T , , T h e acentric factor is defined by
Present address, Department of Chemical Engineering, University of Florida, Gainesville, Fla.
wi
- log,, (Pi"/P,,) - 1.000 VOL. 6
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245
where Pis is the saturation (vapor) pressure of i a t T / T c i= 0.7. T h e empirically determined functionsfB(0)(T R ) andf,(')( TR) are
B ( l l ( T R= )
0.073
0.46 + __ - 0.50 -- 0.097 - 0.0073
T,
TR2
Tx3
(7)
TR8
For pure polar gases we have developed a new empirical correlation based on a n extended corresponding states theory. I t has the form:
where fB(0) and f B ( l ) are given by Equations 6 and 7. The symbol w H i stands for the acentric factor of the polar component's homomorph. (A homomorph of a polar molecule is a nonpolar molecule having approximately the same size and shape as those of the polar molecule-for example, the homomorph of acetone is iscbutane.) The empirical function f,(pR: T R ) depends on the reduced temperature, TR, and on p R , the reduced dipole moment, defined by
where pi is the dipole moment of component i in Debye units, the critical pressure is in atmospheres, and the critical temperature is in degrees Kelvin. The function f, was determined from experimental data for 1 7 polar fluids which do not exhibit specific chemical forces. For p R > 4, it is given by
f,,(pR,TR)= -5.237220 -I- 5.665807 (In pn) 2.133816 (In p R ) 2 4- 0.2525373 (In , L L ~ )f ~
1 [5.769770
TR
- 6.181427 (In p R ) f 2.283270 (In p ~ ) ' 0.2649074 (In
pR)3]
(9)
The unusual form of Equation 9 arises from the way in which the experimental second virial coefficients for polar gases deviate from those calculated using Equations 6 and 7. Lfhen p R is small, the nonpolar equations predict second virial coefficients which are too negative and the contribution to be given by Equation 9 must be positive. This incongruous behavior is apparently due to the inadequacy of our homomorph method, which tends to cvercorrect for small polar effects, LVhile homomorph critical constants might be used, this would mean that more input data must be made available, and, for simplicity, we have chosen to use only the actual critical constants. With increasing p R , the polar contribution passes through a positive maximum and then decreases very rapidly, becoming negative a t p R approximately equal to 50, and very negative a t the largest values of pR of about 250. The form of Equation 9 was the only one of several investigated which adequately represented the data over the whole range of c ( ~ . Because of the logarithmic terms, for molecules of very low polarity (pR < 4), Equation 9 is not applicable. For such molecules one should usef,(pR, TR) = 0. The association constant q i is an empirically determined quantity which reflects the tendency of a component-e.g., an alcohol-to associate with itself to form dimers. The association function fa(TR) is given by 246
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PROCESS DESIGN A N D DEVELOPMENT
~ , ( T R= ) exp [6.6 (0.7
- TR)]
The value of 7%is determined from the difference between experimental data and results obtained from the sum of Equations 6, 7, and 9. Only a single vapor density is necessary to determine 7 % . The form of Equation 1 0 was dictated by the behavior of most '.associating" substances. In general, the contribution of the association function is a significant portion of the total second virial coefficient a t temperatures near and beloiv the normal boiling point. Table I presents w H , pR, and q L . Some substances which kvould be expected to associate-e.g., HLOand "3-do not have association constants. Again, this indicates that the critical constants account for much of the contribution of association to the second virial coefficient. Some substances for which we give values of q i might not actually '.associate"e g , rhloroform-but the appearance of the constant significantly improves the accuracy of calculated cross coefficients. Although the association factor is essentially empirical, definite trends can be found within chemical classes such as alcohols. esters, and amines; as the location of the associating
Table I. Pure Component Parameters ( I n order of increasing critical temperature) Substance P, Debye WH Methyl fluoride 0.105 1.82 0.010 0.55 Phosphine 0.201 Dichlorodifluoromethane 0.55 0.152 1.30 Dimethyl ether 0.010 1.47 Ammonia 0.105 Methyl chloride 1.86 0.105 1.25 h.lethylamine 0.105 1.61 Sulfur dioxide 0.187 0.65 Trimethylamine 0.152 1.03 Dimethylamine 1.29 0,187 Dichloromonofluoromethane 0.152 1.22 Ethylarnine 0.152 Acetaldehyde 2.70 0.105 1.80 Methvl bromide 0.152 2.05 Ethyl'chloride 0.252 1.16 Ethyl ether 0.201 0.50 Trichloromonofluoromethane 0.201 1.37 Methyl formate 1,1,2-Trichloro-l,2,2-trifluoro0,240 1.40 ethane 0.201 0.92 Diethylamine 0.152 1.58 Ethyl mercaptan 0,152 2.03 Ethyl bromide 0,201 1.60 tert-Butyl alcohol 0.215 1.72 Methyl acetate 0.187 1.60 2-Propanol 0.252 Ethyl formate 1.93 2.88 Acetone 0.187 0.152 1.54 Dichloromethane 0,105 1.66 Methanol 0.152 1.69 Ethanol Ethyl acetate 0,278 1.78 Methyl iodide 0.105 1.60 2.80 Methyl n-propyl ketone 0.278 1.69 Methyl propionate 0.326 2.70 Methyl ethyl ketone 0.187 Triethylamine 0.66 0,310 sec-Butyl alcohol 0.215 1.60 Chloroform 1.02 0.187 0.297 1.89 Propyl formate 1-Propanol 0,201 1.68 Acetonitrile 0.152 3.94 Isobutyl alcohol 0.215 1.64 0.233 1.58 Fluorobenzene 0.252 1.65 1-Butanol Methyl isobutyl ketone 0.302 1.65 Nitromethane 0.187 3.44 0.215 2.20 Pyridine 2-Picoline 0.233 1.95 0.233 2.40 3-Picoline 0.233 2.60 4-Picoline '12'ater 0,010 1.84
9
0.00 0,30 0.00 0.55 0.00 0.00 0.72 0.00 0.20 0.56 0.30 0.62 0.00 0.00 0.00 0.28 0.00 0.58 0.00 0.28 0.31 0.00 0.54 0.62 0.65 0.36 0.00 0.32 1.21 1 .00 0.50 0,00
3.55 0.49 0.30 0.27 0.58 0.28 0.39 0.57 0.00 0.49 0.00 0.45 0.50 0.00 0.20 0.36 0.34 0.28 0.00
part of the molecule becomes surrounded and covered up by larger side groups, qt decreases as expected. T h e polar contributions of Equations 9 and 10 are useful below a reduced temperature T R of 0.95 and for a reduced dipole p R greater than 4 but should be neglected when the polar interactions are small. Pitzer (72) has indicated the accuracy with which his correlation predicts second virial coefficients for pure nonpolar substances. I n our calculations, with all of the normal paraffins through heptane, carbon tetrachloride, cyclohexane, and benzene, the errors were never larger than 100 cc. per gram mole or lOy0 and were usually within 35 cc. per gram mole over ranges of reduced temperature from 0.5 to 0.8. For pure polar substances, Table I1 gives the deviations of the calculated values for several classes of compounds. Complete listings of the data are available. I n general, the errors were less than 75 cc. per gram mole, and were usually within the experimental rrror, except for a few substances. For some of these, such as ethanol and acetone, there is some uncertainty or internal inconsistency in the experimental data, but for others-e g., nitromethane, acetonitrile, and some of the fluorocarbons-the correlation is not very good a t low reduced temperatures; for these cases the errors may be of the order of 200 cc. per gram mole and occasionally more. Fortunately, under these conditions, large errors in the virial coefficients rarely cause substantial errors in the fugacity coefficients. As expected, in those instances where a correction for association was included, agreement with experiment is considerably better. Cross Coefficient for Mixtures
To estimate cross coefficients B,,, the previously presented correlating equations for pure components are used with suitable mixing rules for the various parameters. For the case where z a n d ] are both nonpolar gases, Equations 5, 6, and 7 give RiI where P,,, replaces P,&; T,,, replaces T c L ;and w t j replaces w ~ . There are no “universal” mixing rules, but in the absence of experimental data for a given z-j mixture, reasonable approximations are given by (74)
TCij= (TciTcj)l’z
(1 1)
where Vc is the critical volume. If reliable critical volume data are not available, they should be estimated from the relation
V,
RT
= -! (0.293
PC
-
0.08
W)
where w is the acentric factor of the substance, not of its homomorph. For nonpolar mixtures whose components are of very different size, corrections should be employed (2) to lower the geometric mean of critical temperatures, Equation 11, and the cube-root mean of the critical volumes, Equation 13. For the case where i is a polar substance and j is nonpolar, Equations 5, 6, and 7 are used for Bij where Equation 11 is used for Tcij. For wij, we use L
Equation 13 is used for P,,,. For the case where both i and j are polar, Equation 8 is used for BI,. I n Equation 8, T,, is replaced by T,,, from Equation 11, wHi is replaced by w X L J: WHLl
=
12
(WHz
+
(16)
WH,)
P,, is replaced by P,,, from Equation 13, while the reduced dipole moment, 118,is given by
Finally,
vi
is replaced by
vij
according to
L
For second virial cross coefficients of nonpolar substances the correlation based on the above mixing rules works well if the components are similar in size. Hoivever, for extremely different molecules such as nitrogen and decane, the predicted values are somewhat too negative, sometimes approaching 300 cc. per gram mole, but more frequently are of the order 100 cc. per gram mole. Unfortunately, these mixtures are often under high pressure a t equilibrium and substantial errors in vapor-phase fugacity coefficients of the very heavy components can result. O n the other hand, such errors are serious only in calculating liquid concentrations from vapor composition data, a calculation which is only rarely encountered in engineering practice and for which large errors can also result from small inaccuracies in the vapor data. Therefore, this limitation is probably not extremely serious for practical applications. Also, more accurate second virial coefficients may often be obtained by using the empirical corrections of Benson (2). For cross coefficients containing polar compounds, Table I11 indicates the deviations of the calculated values from experimental data for various kinds of pairs. Table IV compares some of the experimental and calculated results for many of the systems studied. It also illustrates the differences in experimental results betlieen different investigators, particularly for polar gases. T h e experimental data used for comparison are subject to large errors, since the quantity B i z
Deviation of Calculated from Experimental Second Virial Coefficients for Pure Polar Fluids N o . of No. of Data Range of Reduced R M S Deviations Class Substances Pointsa Temperature Cc./g. mole 76 13 101 Halocarbons 0.42-1.33 151.5 18.6 10 59 hlcohols 0.56-0.84 63.9 18.1 Ethers, aldehydes, ketones 6 61 0.59-1.03 68.7 9.6 6 34 Esters 0 . 6 1-0.81 39.8 3.8 Amines 7 69 0.61-0.93 30.5 6.2 Heterocyclics 4 18 0.56-0,71 46.0 3.5 Others ( S O ? ,PH3, CHaN02, CHsCN) 4 36 0.54-0,92 414.6 12.5 Table II.
a
Experimental and calculated values available from the American Documentation Institute.
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is obtained by subtracting from one another two experimental quantities of very similar magnitude. As a result, experimental uncertainties of the order of 100 cc. per gram mole are not uncommon and, in most cases, the correlation is probably nearly as accurate as the data.
E T H E R I I I - ACETONE (21 SYSTEM FROM CORRELATION FROM DATA OF ZAALISHVILI
---
'\
1.000
T = 333.2 " K
'=
'
ATM'
Prediction of Vapor-Phase Fugacity Coefficients
The purpose of developing a correlation for second virial coefficients is to make reliable estimates of vapor-phase fugacity coefficients as given by Equation 4. To illustrate, Figure 1 gives fugacity coefficients for two binary systems: ether-acetone and propane-methyl chloride. Even a t 1 atm., the deviations from ideal-gas behavior (p = 1) are significant and we see that p has a substantial composition dependence. Thus, assuming validity of either the ideal-gas law or the Lewis fugacity rule for the ether-acetone system at only 1 atm.
0.975 -
-
1.00
---
PROPANE-METHYL CHLORIDE 12) SYSTEM FROM CORRELATION FROM DATA OF KAPPALLO, E T A L
T = 2 3 3 'K Table 111. Experimental and Calculated Second Cross Virial Coefficients for Mixtures Containing Polar Components
'2
I
I
of
RIMS Deoiations Data System Class Pointsa Cc./g. male %H ydrocarbon-halocarbon 34 87.8 9.1 Hydrocarbon-ether, ketone, nitromethane 19 152.5 23.2 Other nonpolar-polar-eg., 5.0 CC14, Xr 5 63.9 Polar-polar Nonassociating 5 77.2 12.7 Associating. No binary correction 21 371 35.5 .Associating. Correction based on binary data 21 45.9 3.8 Experimental and calculated values aoailable from the American Docunientation Institute.
Figure 1 . Vapor-phase fugacity coefficients for binary mixtures from experimental and calculated second virial coefficients
Experimental and Calculated Second Virial Coefficients for B I I ,Cc./G. Mole T, K . Exptl. Calcd. System - 1243 -1254 333 Acetone-benzene -1024 -1047 353.2 - 620 -809 383.2 -2733 -1946 282.3 Acetone-n-Butane -1680 - 1376 321 - 990 -1049 353 Acetone-cyclohexane -1093 -1142 343 Acetone-n-hexane -1439 -1351 323.2 Acetone-nitromethane - 1370 -1313 333,2 Acetone-ether -1200 - 1203 343.2 -1450 - 1479 326.2 n-Hexane-ether -1210 -1219 352 -915 -915 333.7 Chloroform-benzene -1100 -1086 315.7 Chloroform-carbon tetrachloride - 840 -841 343.2 -780 - 781 352 Chloroform-n-heptane -1716 -1410 193 Methyl chloride-propane -1169 - 1083 21 3 -1678 - 1646 193 Methyl bromide-propane - 1348 -1183 213 -1084 -911 233 -1040 - 804 244 -718 - 603 273 - 572 - 503 293 - 546 - 487 297 -459 - 398 321 -1040 - 804 244 Methyl bromide-n-butane -459 -398 321 - 402 - 425 313 Methyl bromide-n-pentane Table IV.
248
l&EC PROCESS DESIGN A N D DEVELOPMENT
Nonassociating Mixtures Containing Polar Components B1?,Cc./G. Mole B2?,Cc./G. Mole Exptl. Calcd. Exptl. Calcd. Ref. - 962 -885 -1100 -1126 (6) -910 -769 -991 - 971 (20) -710 -814 - 638 -799 - 927 - 805 -862 -825 ( 70) - 684 - 635 - 504 -616 - 830 -1045 -595 -1118 (7) - 934 - 1290 -785 -1301 ( 7) -1937 -2866 -2679 -2592 (4) - 640 - 850 -726 -836 ( 78) -587 - 760 - 632 - 772 -1126 - 860 -1130 - 888 (7) -935 - 700 - 940 - 722 -996 - 1090 -1030 -1119 (8) -1157 -1445 -1195 -1316 (8) - 940 -1120 - 1005 - 1064 -1184 -1780 - 960 -1760 (7) -1015 -1123 -1039 -1101 (77) - 794 - 823 - 842 -850 -1015 -1205 -1015 -1101 (7) - 899 - 823 -850 - 823 -713 - 678 - 678 - 687 - 639 -610 -617 -619 (8) - 497 - 477 - 477 - 484 -410 - 427 -410 -416 (7) -415 - 394 - 394 (8) - 353 -540 - 404 342 - 340 - 878 - 1230 -915 -1182 (8) -635 -465 -616 -459 -615 -1187 -765 -1071 (75)
results in errors of the order of 3%. O n the other hand, results based on our correlation are accurate to within 1%. I n this case the differences in calculated and experimental Bll, B I Z ,and Bzz are 14, 8 6 , and 57 cc. per gram mole, respectively. For the propane-methyl chloride system there is good agreement between calculated and experimental results; even a t 5 atm. the maximum errror is only 1.67,. At 233' K., the errors in B11, B12, and B22 are 9, 13, and 8 cc. per gram mole, respectively; and at 293' K. they are 6, 12, and 38 cc. per gram mole, respectively. Figure 2 shows results for a system exhibiting specific interactions. I n this case the composition dependence of q i s opposite in direction to that shown in Figure 1. Although the correlation is excellent for pure components, chloroform and ether, the predicted cross coefficient is not sufficiently negative, because of association (solvation) of the two substances. T h e predicted fugacity coefficients are in error by 5% for either component when dilute. T o fit the experimental data, a specific association constant of VI?* = 0.33 was added to that obtained from Equation 18. All the experimental cross-
1.0-
coefficient data for this system are adequately represented from 326' to 363' K. and the fugacity coefficients are predicted to within 1% a t 326.2' K. Table V also shows calculations for five other systems where a specific association constant was added; all 20 calculated values are within 50 cc. per gram mole of the experimental data over reduced temperature ranges of 0.6 to 0.725. I n order to assess the utility of our correlation, it is appropriate to consider the effects of errors in the fugacity coefficients on the prediction of multicomponent vapor-liquid equilibria. For any component i, the vapor-liquid equilibrium equation is
where for the liquid phase, xi is the mole fraction, 7 t is the activity coefficient, and ftoL is the reference fugacity, all a t the temperature and pressure of the system. At constant composition and a t conditions remote from critical, the fugacity of a component in the liquid phase is relatively insensitive to pressure, but is usually strongly dependent on temperature.
I
CHLOROFORM ( I ) - E T H E R ( 2 ) S Y S T E M F R O M C O R R E L A T I O N WITHOUT BINARY CORRECTION PARAMETER F R O M C O R R E L A T I O N WITH B I N A R Y CORRECTION P A R A M E T E R FOUND FROM DATA AT A L L A V A I L A B L E T E M P E R A T U R E S F R O M D A T A O F FOX A N D L A M B E R T
-----
T = 326.2 O p = I ATM.
K
Figure 2. Vapor-phase fugacity coefficients for a strongly associating mixture-
Table V.
Experimental and Calculated Second Virial Coefficients for Associating Mixtures
Acetone-chloroform
Acetone-methylene chloride
C hloroform-ether
Ether-ethyl bromide Ether-methyl iodide
Ethyl bromide-methyl bromide a
K. 333.2 343.2 353.2 363.2 323 333 343 353 363 326.2 338.2 352 363 293.1 313.2 313.2 328.2 343.2 358.2 293.1 313.2
SpeciJic association correction added using
Biz
Bii
Temp., System
Exptl.
Calcd.
-1375 -1200 -1024 -879 -1340 -1210 -1095 - 990 -900 -965 -890 -780 -740 -1386 -1107 -1046 -900 - 768 - 682 - 795 - 669
- 1240
-1140 -1047 - 962 -1353 -1242 -1142 -1049 - 963 -981 -881 -781 -715 -1224 -1001 -999 - 872 -772 -691 -759 - 636
Ex@. -910 -855 -775 -710 -1260 -1130 -1020 -915 -820 -1520 -1290 - 1030 - 870 -1356 -1160 -1046 - 900 - 768 -682 -1047 -881
B22
Calcd.
Ca1cd.a
1112
-659 - 604 -556 -514 - 842 -768 -703 - 645 -594 - 948 -855 -762 - 701 - 770 - 632 -560 -493 - 440 - 398 -617 -520
-941 -852 -775 -707 -1256 -1131 - 1023 -925 -840 - 1366 -1214 -1060 -959 -1403 -1141 -1048 -893 - 768 - 668
0.16
-1088
-878
...
... ... 0.21
... ... ... ...
0.33
... ... ...
0.30
...
0.24
...
... ...
0.15
...
Exptl. -2025 -1300 - 995 - 930 -672 -614 -565 -523 - 480 -860 -790 - 700 - 650 -795 - 669 -706 -612 -517 - 462 -478 - 402
Calcd.
Ref.
-919 -841 - 773 -714 -702 - 636 -580 -531 -488 -888 - 804 - 722 -669 -759 - 636 -370 - 333 - 303 -278 -503 -425
(79)
(7)
(7)
(75) (78)
(75)
1112.
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From the N - 1 independent mole fractions in a given phase and one intensive variable, pressure or temperature, the N Equations 19 may be solved to give the mole fractions in the other phase and the unknown intensive variable, temperature or pressure. The effect of errors in (az on the predicted variables depends on the directions of the errors for all the components. If all the calculated fugacity coefficients are too high or too low by the same percentage, the mole fractions will be close to the correct ones, but the error in pressure or temperature will be determined by the mole fraction average of the deviations in the fugacity coefficients. If some of the deviations of the fugacity coefficients are positive, while others are negative, the error in the composition, y z (or x i ) , will be essentially determined by a factor of (1 - y z ) [or (1 - x i ) ] times the difference of the absolute values of the average positive and average negative deviations. Thus the compositions of dilute components will be affected most. O n the other hand, the error in pressure or temperature will be determined by the mole fraction average of all of the deviations in fugacity coefficients, a quantity which can be small. By way of illustration, let us consider the problem of predicting the pressure and the vapor-phase composition of a binary mixture from liquid-phase mole fractions and temperature. Let us suppose that values of (a for the two components are too high by 1%. Since (a and the liquid-phase variables are not sensitive to small changes in pressure, these errors will not significantly affect the composition, but they will give a calculated pressure which is low by essentially 1%. If the fugacity coefficients are in error by 1% in opposite directions, the composition of a component will be changed by about 2% of the calculated mole fraction of the other component. The percentage error in the pressure will be about 1% of the difference of the correct mole fractions. T h e effect on the predicted temperature at a given pressure is more subtle, since temperature does not appear explicitly in Equation 19 and since the liquid-phase variables are also sensitive to temperature, but the same general trends occur with smaller percentage errors in the absolute temperature. Thus, while the errors of using a correlation for fugacity coefficients such as that presented here may occasionally be large, the effect on some of the yariables of interest can often be small. O n the other hand, since vapor-phase corrections based on our correlation are considerably better than concentration-independent corrections (Lewis rule), and very much better than none a t all, it is
worthwhile to include them in a thermodynamic framework for vapor-liquid equilibria calculations at moderate pressures. The required fugacity coefficients can be readily calculated with an electronic computer and, therefore, can easily be incorporated into computer programs for calculation of multicomponent vapor-liquid equilibria as described elsewhere ( 73). Acknowledgment
The authors are grateful to the National Science Foundation for financial support and to the Berkeley Computer Center for the use of its facilities. literature Cited
(1) Abbott, M., Ph. D. thesis, Rensselaer Polytechnic Institute, 1965. ( 2 ) Benson, P. R., Prausnitz, J. M., A . I. Ch. E. J . 5 , 301 (1959). ( 3 ) Black, C., Ind. Eng. Chem. 50,403 (1958). (4) Bottomley, G. A., Coopes, I. H., J . Chem. Sod. London 1961, p. 2247. ( 5 ) Chao, K. C., Seader, J. D., A . I. Ch. E. J . 7,588 (1961). ( 6 ) Di Zio, S.,Ph.D. thesis, Rensselaer Polytechnic Institute, 1964. (7) Fox, J. H. P., Lambert, J. D., Proc. Roy. SOC.London A210, 557 (1952). ( 8 ) Francis, P. G., McGlashan, M., Trans. Faraday Soc. 51, 593 (1955). ( 9 ) Guggenheim, E. A., Rev. Pure Appl. Chem. 3, 1 (1953). (IO) Kappallo, W., Lund, N., Schafer, K., 2. Physik. Chem. (Neue Folge) 37, 196 (1963). (11 ) Kappallo, LV., Schafer, K., Z. Elektrochem. 66, 508 (1 962). (12) Pitzer, K. S., Curl, R. F., J . A m . Chem. SOC. 79, 2369 (1957). (13) Prausnitz, J. M., Eckert, C . A., Orye, R. V., O’Connell, J. P., “Computer Calculations for Multicomponent VaporLiquid Equilibria,” Prentice-Hall, Englewood Cliffs, N. J., 1967. (14) Prausnitz, J. M., G u m , R. D., A.Z.CI1.E. J . 4, 430 (1958). (15) Ratsch, M., Bittrich, H. T., Z . Phyzk. Chem. 228, 81 (1965). (16) Rowlinson, J. S., Trans. Faraday SOC. 45, 974 (1949). (17) Stockmaver. W.. J . Chem. Phvs. 9. 798 - - - 11941). -,(18) Zaalishvh, S . D:,-Zhur. Fir. k h i m . 34, 2596 (1960). (19) Zaalishvili, S. D., Kolysko, L. E., Ibid., 35, 3613 (1961). (20) Ibtd., 38, 503 (1964). ~
- 7
\ - -
RECEIVED for review July 22, 1966 ACCEPTED January 13, 1967 Material supplementary to this article has been deposited as Document No. 9362 with the AD1 Auxiliary Publications Project, Photoduplication Service, Library of Congress, Washington 25, D. C. A copy may be secured by citing the document number and by remitting $13.75 for photoprints or $4.50 for 35-mm. microfilm. Advance payment is required. Make checks or money orders payable to Chief, Photoduplication Service, Library of Congress.
SOLVENT MIXTURES FOR SEPARATION
PROCESSES R. A. G R I E G E R A N D C. A. E C K E R T Department of Chemidtry and Chemical Engineering, University of Illinois, Urbana, Ill. NE
of the major problems in the design of industrial separa-
0 tion and recovery equipment is often the proper choice of that solvent with thermodynamic properties which will optimize the yield of desired product. I n this paper we show that a mixture of two solvents, if properly chosen, will frequently yield much better results for such a process than either of the component solvents alone. For example, in a typical process let us suppose that we are recovering some solute A in a solvent, and we naturally wish to maximize the concentration of A in the solvent. Let us assume 250
I&EC PROCESS DESIGN A N D DEVELOPMENT
that the process will approach thermodynamic equilibrium, and a t equilibrium the fugacity of A is a t some fixed value, determined by the properties of the phase from which it is being recovered. Expfessing the fugacity in the usual manner, fA
=
XAYAPA’
(1)
where xA is the mole fraction, y A is the activity coefficient, and PAo is the reference state fugacity (or vapor pressure), a function of temperature only. Since f A and PA’ are considered fixed, to maximize the concentration xA of desired product, we
