Computation of Vapor-Liquid Equilibrium Data from Binary and Ternary Vapor Pressure and Boiling Point Measurements lsamu Nagata* and Tatsuhiko Ohta Department of Chemical Engineering. Kanazawa University. Kanazawa. 920. Japan
A computational procedure is presented for the determination of binary and ternary vapor-liquid equilibrium data from the total pressure method which requires only P-x or T-x data. The method involves the determination of excess Gibbs free energy by successive approximations to the total pressure surface according to the method of Mixon, et a/. (1965). The total pressure or boiling point data were correlated with a set of orthogonal polynomials described by Hutchison and Fletcher (1970). Calculated results for binary and ternary systems at isothermal or isobaric condition are in good agreement with experimental data
Introduction Van Ness, et al. (1973), have presented a critical discussion of all of the factors that play a part in the various reduction methods for isothermal binary vapor-liquid equilibrium data. They suggested that the preferred procedure for data reduction is the one based on P-x data out of three different procedures which may be used to reduce a set of P-x-y data, because the measurements of y are usually the least reliable. Furthermore, they recommended the following step to produce numerical values of the functions, y1,y2, and GE. (1) Use the least-square spline fit of Klaus and Van Ness (1967) to smooth the P us. x data. (2) Process the smoothed data by the method of Mixon, et al. (1965). The method of Mixon, et al., has an advantage that it is not necessary from the start to assume a particular model for the GE function. Numerical integration procedures of the coexistence equation have been discussed for isothermal binary P-x data (Ljunglin and Van Ness, 1962; Nagata and Ohta, 1972; Sadler, et al., 1971) and for isobaric binary data (Barner and Adler, 1973; Ljunglin and Van Ness, 1962) to obtain vapor compositions. Use of the coexistence equation often involves a more troublesome process from a computational point of view than the method of Mixon, et al. (Mixon, et al., 1965; Van Ness, 1970). The spline fit method of Klaus and Van Ness, which is especially useful for binary data smoothing, has not been applied to ternary systems. Hutchison and Fletcher (1970) presented a method developed using orthogonal polynomials in two dimensions for representing T and y surfaces as a function of two liquid compositions for isobaric, ternary vapor-liquid equilibrium data. The method is fast, noniterative, and can be extended to hyper-surfaces in three or more dimensions. We found that the method is also more flexible in binary and ternary data fitting than the polynomials used by Mixon and others. Mixon, e t al., assumed vapor phase ideality. However, Prausnitz, e t al. (1967), recommended that vapor-phase corrections should always be included in vapor-liquid equilibrium calculations, because in some cases the error introduced by neglecting vapor phase nonideality is quite appreciable. Under the assumption of vapor phase nonideality, the present work shows that combination of the method of Mixon, et al., and that of Hutchison and Fletcher leads to accurate calculations of various thermodynamic functions, and that the method is applicable to isobaric systems, which were not attempted by Mixon, et al. 304
Ind. Eng. Chem., Process Des. Develop., Vol. 13, No. 3, 1974
Experimental Section Heat of mixing data were obtained for the 2-propanolcyclohexane system a t 35 and 4 5 T , because the present work requires an estimate of the heats of mixing a t boiling points for this system. Guaranteed reagent grade cyclohexane was passed through a 1.2-m 30-mm i.d. glass column packed with silica gel of spectroscopic quality. 2-Propanol was fractionally distilled twice in a glass column packed with McMahon packing after drying over anhydrous copper sulfate for 2 days. Refractive indices (n25D) of the purified materials are compared with the published values of Riddick and Bunger (1970) (RB) as follows: cyclohexane, 1.42355 (exptl), 1.42354 (RB); 2-propanol, 1.3748 (exptl), 1.3752 (RB). The apparatus and experimental procedure were described in a previous paper (Nagata, et al., 1970). Table I lists the results. Calculation Procedure From the thermodynamic relations (Klaus, 1967), the liquid-phase activity coefficients for an N - component system are given by
, h # !. .Y 1
+ + \ I
where 4 = GEjRT. Equation 1 can be approximated by its finite difference representation as follows. For binary systems the independent composition X I is given by XI = 1 1 , where 1 is an integer and 1 is a lattice interval. The finite difference representation for the partial derivatives of Q can be written as
Then the finite difference form of eq 1 becomes
Table I. Experimental Heat of Mixing Data for the 2-Propanol(l)-Cyclohexane(2) System a t 35 and 45"
where
35 O x1
Similarly for ternary systems = lA;X, =
X
6Q-- Q i + i as,
-
Qv-i
a$ _ - Qi
/ ,
~
2A
0.1030 0.1177 0.1636 0.2126 0,2861 0.3132 0,3832 0.4758 0.4887 0,5172 0.5933 0.6489 0.7301 0.8860 0.8912
JA
' ax,
r+i
-
Qi 1 - 1
.a
45
HE, cal/mol 147.5 160.7 180.3 199.8 222.5 227 .O 234.4 229.7 227.3 225.8 214.4 191.1 160.4 79.8 76 .O
XI
0.0849 0.1040 0.1450 0.1816 0,2425 0.3000 0.3174 0.4133 0,4919 0.5599 0.6743 0.7128 0.7309 0,7343 0.7967 0.8520 0.8545 0 ,8582 0.9439
HE, cal/mol 144.8 166.1 204.5 221.6 246 . O 262.1 262.3 270.3 263 .O 245.2 204.5 185.9 177.8 178.8 146.4 112.3 109.2 106.1 47.5
positions are calculated from eq 7 . where
The total pressure of an N-component system can be calculated by
' 7tx!f!1 P=CI =
1
(c,
(1)
where pf is the vapor-phase fugacity coefficient and f L O is the fugacity of pure component i in the liquid phase (Prausnitz, e t al.. 1967).f t o is given by
where pts is the fugacity coefficient of pure vapor i at temperature T and saturation pressure P , S , and u,L is the liquid molar volume of pure component i. Fugacity coefficient pLsis estimated from the correlation of Lyckman, e t al. (1965). Mixon, et al. (1965), utilized relaxation techniques to establish the GE-x relation from P-x data. The procedure consists of iterative improvements in an initially assumed GE-x relation until a precise fit to the smoothed P-x data is obtained. In this work the P-x or T-x data are correlated by the orthogonal polynomials described by Hutchison and Fletcher (1970).
The numerical procedure for determinating the vaporliquid equilibrium data may be summarized as follows. 1. Fit the P-x or T-x data from eq 6. 2 . The initial GE-x relation is guessed and boundary information is given. For binary systems, the @-x relation is assumed to be expressed by G E = 4GE,,, x l ( l - XI), where GEeit is an arbitrary extremum set by a programmer at equimolar concentration ( = 0 -300 cal/mol). For ternary systems, the initial values of @ are assumed to take constant values ( = 50-200 cal/mol) over the entire concentration range. 3. Calculate y, by the finite difference representation of eq 1 using a point relaxation technique and substitute the results into eq 4 for all lattice points (in the case of isobaric calculations, the values of a T / d x , and P,S can be obtained by using the fitted T-x data from step 1). Lattice interval is set to be 0.005mole fraction for binary systems and to be 0.05 mole fraction for ternary systems. 4. Compare the fitted value of P obtained from step 1 with the calculated value derived from step 3 at the lattice point and if difference between these two values is beyond a range of allowance. then add the correction term
x 1 ? . f 5 = X 1 X 2 . f 6 = X??.f 7 = x '.... (for ternary systems)
to GE a t each lattice point to improve a degree of approximation. o is a multiplying factor and was set to be 0.011 to 0.012 for binary systems and to be 0.03 to 0.04 for ternary systems. (For the isobaric calculations, the computational procedure is the same except that the value of P is equal to that of isobaric condition ( e . g . ,760 mm Hg).) 5 . The process is repeated until the deviations in the total pressure fall within the tolerance of the desired precision of results. 6. Calculate vapor compositions from eq 7 using the final results.
The coefficient D m p or D m is ~ calculated from the P-x or T-x data in accordance with the article of Hutchison and Fletcher (1970). From the final GE-x values, the equilibrium vapor com-
Calculated Results The present method is applied for eight binary and two ternary systems a t isothermal or isobaric conditions. At
.\I
I1
P = CD,rii'fnl;T ..,=:
=
CDn,,,fm
(6)
m=1
where f m is defined as follows
f, = 1.f 2
= XI. f
.>
, = xl-.
r4 = x , :, f-= "
xi', ... (for binary systems)
f l = 1,f, = x , , f j = x,, f 4
=
Ind. Eng. Chern., Process Des. Develop., Vol. 13, No. 3,1974
305
f
306
Ind. Eng. Chem., Process Des. Develop., Vol. 13, No. 3,1974
Table 111. Calculated Results for the Benzeneil)-2-Propanol12)System a t 760 mm HgCL Xi
Exptl Calcd ( I ) " Calcd ( I I ) c Exptl Calcd ( I ) Calcd (11) Exptl Calcd ( I ) Calcd (11) Exptl Calcd (1) Calcd (11) Exptl Calcd (I) Calcd (11) Exptl Calcd (I) Calcd (11) Exptl Calcd (I) Calcd (11) Exptl Calcd ( I ) C a k d (11) Exptl Calcd (I) Calcd (11) Exptl Calcd ( I ) Calcd (11) Exptl Calcd ( I ) Calcd (11) Exptl Calcd ( I ) Calcd (11) Exptl C a l d (I) C a l c d (111 Exllt I CalLcl (1, Calcd (11)
Ex1)f.l Calcd Calcd Expi 1 Calcd Calrd Exptl Calcd Calad
(I) (11)
(I) (11) (I) (11)
0.084 0.085 0.085 0.126 0,125 0,125 0,153 0.155 0,155 0.199 0.200 0.200 0.240 0.240 0.240 0.291 0.290 0.290 0.357 0,355 0.355 0.440 Os, 440 0.440 0,492 0.490 0.490 0.556 0.555 0.555 0.624 0.625 0.625 0.685 0.685 0.685 0.762 0,760 0 , 760 0 ,836
u ,835 0 ,835 0,887 0 ,885 0.885 0.936 0 ,935 n ,935 0.972 0 ,970 0.970
Yl
0.208 0.186 0.186 0.276 0.259 0.260 0.316 0.306 0.308 0.371 0.364 0.367 0.410 0.406 0.410 0,451 0.447 0.451 0.493 0.488 0.493 0.535 0.530 0.535 0.558 0.552 0.557 0.583 0.381 0.585 0.612 0.620 0.620 0.638 0.650 0.646 0.673 n 677 0,672 0 717
0,714 0.710 0 ,760 0.755 0,752 0 ,825 0 ,822
0.820 0.901 0.896 0.896
T,"C
Yl
78.5 78.45 78.45 77.1 77.16 77.16 76.3 76.29 76.29 75.3 75.19 75.19 74.4 74.40 74.40 73.6 73.66 73.66 73 .O 72.99 72.99 72.4 72.46 72.46 72.3 72.27 72.27 72.2 72,12 72.12 72 . O 72.07 72.07 72,l 72.13 72.13 72.4 72 36 72 36 73 0 72.95 72.95 73.8 73.81 73.81 75.5 75.44 75.44 77 . 5 77.36 77.36
2.6280 2.3210 2.3310 2 ,4229 2 ,2840 2.2975 2 ,3395 2 ,2345 2 ,2497 2.1760 2.1338 2,1507 2.0491 2 ,0289 2.0464 1.9048 1.8919 1 ,9093 1 ,7284 1 ,7216 1.7381 1 ,5500 1.5326 1.5468 1 ,45001 1.4431 1.4545 1 ,3444 1.3464 1.3545 1.2651 1.2772 1,2757 1.1973 1.2191 1.2104 1.1243 1.1360 1 1277 1 0711 1 ,0703 1 ,0639 1 ,0434 1 ,0385 1.0341 1,0180 1.0171 1.0150 1 ,0070 1 ,0077 1 . 0071
Y?
0.9835 1.0145 1.0135 0.9973 1 .01'T8 1.0157 1.0046 1.0227 1.0196 1s.0181 1.0350 1 ,0303 1.0448 1.0516 1.0454 1.0777 1.0808 1.0727 1.1257 1.1330 1,1227 1.2162 1 ,2262 1.2134 1.2801 1 ,2933 1 ,2804~ 1 ,3881 1.3962 1 ,3845 1 ,5387 1 ,5051 1.5079 1.7071 1,6471 1 ,6689 2.0166 1.9768 2 O(J70 ,
2 4719 2 ,4846 2.5216 2 ,9450 2 ,9545 2.9935 3.5387 3.55557 3 ,5898 4 2236 4.1621 4.1836
GI,cal mol 46.09 59.20 58.83 75.97 82.61 81.87 93.01 99.69 98.66 117 .06 123.97 122.54 141.92 143.71 142.03 165.75 165.43 163 .60 186.75 188 .07 186 33 207 .69 207 ,47 206 .22 211.62 213.40 212 ,56 212.87 21,5.17 214.88 211.79 210 08 210.05 200.22 200.96 200 46 175 .97 17s 8 3 177 52 1 1 I 39 142 28
140 53 110.12 108 .97 107 38 67.62 68.12
67.21 32.83 34.97 34.67
Data of Nagata (1964). Calcd ( I ) : heats of mixing were estimated using Wilson's equation. Calcd (11):heats of mixing were neglected.
isothermal condition the excess volume for a mixture was considered negligible. Isothermal equilibrium and its pure component vapor pressure data were taken from the same literature. Vapor pressure data for use in isobaric calculations for the following materials were obtained from the Antoine equation (benzene, cyclohexane, and 2-propanol, (MI, 1964);methanol (Riddick and Bunger, 1970);water (Dreisbach, 1961)). Binary Systems. Table I1 presents calculated vaporliquid equilibrium data obtained in this study and the results of Mixon and others for the chloroform-ethanol system a t 55" to illustrate the capability of both methods. The table clearly demonstrates that the present approach gives better agreement with experimental data than that of Mixon, et al. This may occur because we got a better f i t of the P-x data and because we took into consideration vapor phase nonideality. Calculated results (Table 111) for the benzene-'-propanol system a t 760 mm Hg were ob-
tained in two cases to study whether the heat of mixing term is significant in data reduction. (I) Heat of mixing data were extrapolated to boiling points of a mixture using the Wilson equation whose energy parameter differences were assumed to vary linearly with temperature (Nagata and Yamada, 1972). (11) Heats of mixing were negligible. The table shows that the effect of heats of mixing on final results is not striking. Results for the methanol-water system at 760 mm Hg were obtained on the assumption that the heats of mixing were negligible (Table IV). The results of five component binary systems for two ternary systems described below agreed well with experimental data as shown in Tables V-IX (see paragraph at end of paper regarding supplementary material). Ternary Systems. Vapor-liquid equilibrium data were calculated for the acetone-chloroform-n-hexane system at 55" from the P-x data of component binary systems and ternary system. Calculated results at selected points are Ind. Eng. Chem., Process Des. Develop., Vol. 13, No. 3 , 1974
307
Table IV. Calculated Results for the Methanol(1)-Water (2)System a t 760 mm Hg
T,"C
Yl Xl
ExptP 0.000 0.261 0.405
0.000 0.050 0.100 0.150 0.200 0.575 0.250 ... 0.300 0.660 0.350 .., 0.400 0.720 0.450 ... 0.500 0.772 0,550 ... 0.600 0.820 0.650 ... 0.700 0.867 0.750 ... 0.800 0.912 0.850 ... 0.900 0.955 0.950 ... 1.000 1.000 Root mean square deviation = 0
Calcdc 0.0000 0.2741 0.4236 0.5163 0.5802 0.6276 0.6653 0.6970 0.7253 0.7515 0.7763 0.8000 0.8227 0.8448 0.8669 0.8896 0,9131 0.9365 0.9585 0.9790 1.0000
Exptlfl 100.00 92.39 87.53 84.01 81.48 79.48 77.90 76.56 75.36 74.19 73.16 72.20 71.29 70.45 69.58 68.69 67.83 66.97 66.14 65.31 64.50
Y2
'i I
Calcd 99.99 92.42 87.48 84.03 81.48 79.50 77.90 76.54 75.33 74.22 73.18 72.20 71.29 70.42 69.57 68.72 67.84 66.96 66.11 65.33 64.49
ExptP
Calc&
-
-
1.9515 1.7639 , . .
1.5323 ...
1.3233 ...
1.1870 ...
1.0984 ...
1,0394
...
1,0051 ... 0,9849
...
0,9764 . . .
Exptlh
1.0000
2.0543 1.8567 1.6888 1.5497 1.4342 1.3386 1 ,2602 1.1969 1.1465 1.1061 1.0731 1.0456 1.0229 1.0053 0.9934 0.9873 0.9849 0.9828 0.9794 1.0000
-
Calcd
1 .oooo 1.0180 1.0367 ... 1.0566
1 .oooo 1.0029 1.0120 1.0267 1.0466 1.0714 1.'1136 1.1008 . . . 1.1343 1.1919 1.1709 . . . 1.2098 1.2730 1.2508 1 ,2946 1.3586 1.3421 1.3934 1.4441 1.4458 ... 1.4927 1.5423 1.5258 ... 1 ,5444 1.6982 1.5707 ... 1.6457 ,
.
0 .00 36.62 63.95 ...
91.22 . . .
111.22 ..
120.45
...
115.36, ...
99.81 ...
I
77.48 .,.
50.47 ... 21.21
-
-
G ' ; ,callmol -~___
Exptl"
...
0 .oo
0,0080 0.02 Data of Kojima, et al. (1968). Data of Ochi and Kojima (1971). Heats of mixing were neglected.
Cald 0 .oo
28.16 52.03 71.71 87.43 99.44 107.94 113.16 115.35 114.78 111.72 106.37 98.87 89.31 77.83 64.68 50.34 35.35 19.90 3.44 0.oo 5.30
Table X. Calculated Results for the Acetone(l)-Chloroform(2)-n-Hexane(3) System a t 55"
Exptl Calcd Exptl Calcd Exptl Calcd Exptl Calcd Exptl Calcd Exptl Calcd Exptl Calcd Exptl Calcd fc
0.199 0,200 0.298 0,300 0.400
0.313 0.306 0.215 0.217
0,500 0,402 0,400
0,400
0,401 0.400
0,300
0,121 0,442 0,433 0,353 0,351 0,391 0.395 0,559 0,553 0,357 0.353
0.300 0,300 0,200 0,200 0.099
0.501 0,500 0,502 0.500
0.100 0.401
0.400
0.299 0.300 0,299 0,300 0.499 0,500
0.197 0,200
0,501
0.301 0.300 0,201 0,200 0,201 0,200
0.197 0.200 0,300 0,400 0,400 0,300 0,300 0.602 0.600
0.114
0.403 0,410
0.442 0.452 0.493 0.498 0,273 0,280 0.290 0,305 0.209 0.198 0,097 0.095 0.163 0.168
0.283 0.284 0.343 0.331 0.393 0.381 0.285 0.287 0,357
0.344 0.400 0.407
0,344 0.352 0.480
0.479
641.2 641.2 641.2 640.9 632.5 633.0 672.6 673.6 672.7 673.4 705.8 707.5 781.2 782.8 703.0 706.5
0.920 0.900 0.951 0.959 1,009 1.063 1.013 0.997 1.088
1.078 1.260 1.271 1.184 1.171 1,740 1.700
0.841 0.857 0.919 0,943 1.012 1.025 0,746 0.771 0.793 0.839 0.800
0.764 0.617 0.608 0.930 0,971
1.871 1.866 1.514 1.450
1.275 1.237 1.994 1.981 1.639 1.581 1.442 1.469 1.825 1.872 1.145 1.152
8.4 10.5 46.1 48.1 67.8 67.4 18.9 20.2 52.5 58.3 96.6 94.4 107.8 109.2 116.3 120.7
Data of Kudryavtseva and Susarev (1963).
Table XI. Calculated Results for the Benzeneil)-Cyclohexanei2)-2-Propanoli3)System a t 760 mm Hg'
T. XI
X:!
X7
0,049
YI 0.198 0.202 0,202 0,292 0.283 0,282 0.692
Y?
0.620 0.61.0 0.608
Y .I
0.182
o c
73.5 72.9 72.9 73.4 72.7 72.7 74.9 74.5 74.5 70.4 70.0 70.0 71.9 71.7 71.7 70.5 70.4 70.4 71.5 71.5 71.5
5
I
1.216 1.263 1.263 1.161 1.189 1.185 1.011
^ i'. .,
1.027 1.031 1.027 1.050 1.056 1.053 1.326 1.359 1.365
-(a
5.223 5.414 5.478 5.482 5.204 5,279 3.803 3.975 4.026 1.984 2.049 2.094 2.003 2.008 2.042 1.821 1.715 1.736 1.467 1.478 1.474
cal 'mol 96.5 105.7 104.3 102.9 116.9 115 3 82.0 90.3 89.0 229.9 238.7 237.0 188.5 192.1 190.3 233.6 237.8 237 0 225.8 224.8 221.4
0.751 0.200 Exptl 0.188 Calcd (I)" 0,200 0,750 0 . 0 5 0 0.190 Calcd ( I I ) r 0,200 0,750 0,050 0.163 0.545 0.310 0,648 0,042 Exptl 0.538 0.179 0,300 0.650 0 , 0 5 0 Calcd (I) 0.181 0.537 Calcd (11) 0,300 0,650 0,050 0.159 0.149 0.052 0,143 0.805 Exptl 1.020 0.684 0.169 0.147 0.150 0,050 0.800 Calcd (I) 1.016 0,169 0.149 0.150 '0,050 0.682 Calcd (11) 0 , 8 0 0 1.170 1.408 0,313 0,252 0,428 0.259 0.252 0.496 Exptl 1.403 1.188 0,253 0.315 0,250 0.250 0.432 0,500 Calcd (I) 1.399 1.171 0.322 Calcd (11) 0.500 0,250 0,250 0.426 0 . 2 5 2 1.561 1.124 0.322 0,057 0,241 0,610 0.068 0.702 Exptl 1.642 1.123 0.062 0,333 0,250 0.605 Calcd (I) 0,700 0,050 1.665 1.111 Calcd (11) 0,700 0,050 0.250 0.598 0.064 0.338 1.557 1.195 0.332 0,180 0,290 0.488 Exptl 0,552 0.158 1.648 1.225 0,323 0,497 0.180 0.300 0,550 0,150 Calcd (I) 1.663 1.211 0.182 0,327 0,300 0.491 Calcd (11) 0,550 0.150 1,310 1.778 0.393 0,548 0.059 0,044 0.408 0.548 Exptl 1.866 1.291 0.388 0.070 Calcd (I) 0,550 0,050 0,400 0.542 1.914 1,290 0.079 0.387 0,400 0.541 Calcd (11) 0.550 0,050 Data of Nagata (1964).h Calcd (I): heats of mixing were estimated using Wilson's equation. Calcd (11):heats of mixing were neglected. 11
308
Ind. Eng. Chem., Process
Des. Develop., Vol. 13, No. 3, 1974
compared with the experimental data in Table X. The agreement is satisfactory. Similarly, calculations were done for the isobaric benzene-cyclohexane-2-propanol system a t 760 mm Hg. Ternary heats of mixing a t boiling temperatures were estimated using the Wilson equation whose binary temperature dependent parameters for the benzene-cyclohexane and benzene-2-propanol systems were taken from a previous paper (Nagata and Yamada, 1972). Table XI shows that calculated results for this system are in good agreement with the experimental data. This table suggests that heats of mixing correction has a slight effect on ternary data reduction as observed in binary systems.
Conclusions The method of Mixon, et al., which was proposed for calculations of isothermal binary and ternary vapor-liquid equilibrium data from the total pressure data, was combined with the method of Hutchison and Fletcher, which uses orthogonal polynomials in fitting the P-x or T-x data points. The calculated results for eight binary and two ternary sytems a t isothermal or isobaric conditions are in good agreement with t h e experimental data. The heat of mixing data a t boiling points are found to affect only slightly the isobaric data calculations. The numerical procedure presented here can give reliable vapor-liquid equilibrium data from the precise P-x or T-x data. Acknowledgment The computer facilities were provided by the Data Processing Centers, Kyoto and Kanazawa Universities. Shuhei Yasuda assisted with the computer work. Kenji Kazuma, Isamu Doi, Shigeru Kabuta, and Shigeo Tsuchida assisted with the experimental work. Nomenclature Dmp, D m ~ = coefficients of eq 6 fm = simple polynomial terms f L o = fugacity of pure component i, atm GE = excess Gibbs free energy, cal/mol HE = heat of mixing, cal/mol M = maximum number of polynomial terms P = total pressure, atm P,S = saturation pressure of pure component i, atm Q = Q function as defined by CFIRT R = gasconstant T = absolute temperature, "K vLL = molar liquid volume of pure component i, cm3/mol VE = excess volume of mixing, cm3/mol xl = liquid phase mole fraction of component i y L = vapor phase mole fraction of component i
Greek Letters
(€P/RTZ)(aT/dxi) - (VE/RT)(aP/axi)at a lattice point yi = activity coefficient of component i A = lattice interval ( p L = fugacity coefficient of component i, fLv/yiP pis = fugacity coefficient of pure component i, f i S / P i S a,P+,t =
Subscripts i, k, n = component I , j = integers by which a lattice point is designated Superscript E = excess property Literature Cited American Petroleum Institute. Research Project 44, "Selected Values of Properties of Hydrocarbons and Related Compounds, Texas A&M University, College Station, Texas, 1964. Barner. H. E . , Adler, S.B . , Ind. Eng. Chem.. Process Des. Develop.. 12, 71 (1973). Dreisbach. R. R . , Advan. Chern. Ser., No. 29, 474 (1961). Hutchison. H. P., Fletcher, J. P., Chem. Engr. i l o n d o n i . No. 235, CE29 (1970), Klaus, R. L., Ph D. Dissertation, Rensselaer Polytechnic Institute, Troy, N . Y.. 1967 Klaus, R. L., Van Ness, H. C.,AlChEJ.. 13, 1132 (1967). Kojima, K., Tochigi, K., Seki, H . , Watase, K., Kagaku Kogaku. 32, 149 (1968). Kudryavtseva, L. S., Susarev, M. P., Zh. Prikl. K h m , 36, 1710 (1963). Ljunglin, J. J., Van Ness, H. C.. Chem. Eng. Sci.. 17, 531 (1962). Lyckrnan, E W., Eckert, C. A , , Prausnitz. J. M., Chern. Eng. S c i . 20, 685 (1965). Mixon. F. O., Gumowski. E., Carpenter, B. H., l n d Eng. Chem.. f u n d am.. 4, 455 (1965). Nagata. I., Can. J. Chem. Eng . 42, 82 (1964) Nagata, I . , Ohta, T., J . Chem. Eng. Jap.. 5 , 232 (1972) Takahashi, T., Kagaku Kogaku. 34, 1107 (1970) Nagata. I . , Tago, O., Nagata, I . , Yamada, T., lnd. Eng. Chem.. Process Des. Develop.. 11, 574 (1972). Ochi, K., Kojima. K., Kagaku Kogaku. 35, 583 (1971). Prausnitz, J. M., Eckert, C. A . , Orye. R. V.. O'Connell, J. P.. "Computer Calculations for Multicomponent Vapor-Liquid Equilibria." PrenticeHall, Englewood Cliffs, N. J., 1967. Riddick. J. A., Bunger, W. B.. "Organic Solvents," 3rd ed, pp 77, 145, 149, Wiley-lnterscience, New York, N. Y., 1970. Sadler, L. Y., I l l , Luff, D. W., McKinley, M. D., J. Chem. Eng. Data, 16, 446 (1971). Scatchard, G.. Raymond, C. L.. J . Amer. Chem. SOC.. 60, 1278 (1938). Van Ness, H . C . , A / C h E J . . 16, 18 (1970). Van Ness, H . C.. Eyer, S. M . . Gibbs, R. E., AlCh€ J.. 19, 238 (1973).
Received for review J a n u a r y 16,1974 Accepted April 11,1974 S u p p l e m e n t a r y M a t e r i a l A v a i l a b l e . T a b l e s V-IX, c o n t a i n i n g results for five b i n a r y systems, w i l l appear f o l l o w i n g these pages in t h e m i c r o f i l m e d i t i o n of t h i s v o l u m e o f t h e j o u r n a l . Photocopies o f t h e s u p p l e m e n t a r y m a t e r i a l f r o m t h i s p a p e r o n l y o r m i c r o f i c h e (105 X 148 mm, 2 4 X reduction, negatives) c o n t a i n i n g a l l o f t h e supplem e n t a r y m a t e r i a l f o r t h e papers in t h i s issue m a y b e o b t a i n e d f r o m t h e J o u r n a l s D e p a r t m e n t , A m e r i c a n C h e m i c a l Society, 1155 16th St., N.W., Washington, D. C. 20036. R e m i t check o r m o n e y order for $3.00 f o r p h o t o c o p y o r $2.00 f o r microfiche, r e f e r r i n g t o code n u m b e r PROC-74-304.
CORRECTION In the article, "Power Correlations for Close-Clearance Helical Impellers in Non-Newtonian Liquids," by Virendra V. Chavan and Jaromir Ulbrecht [Ind. Eng. C h e m . , Process Des. Develop., 12, 472 (1973)], eq 6, 14, and 16 should be read as follows.
Ind. Eng. Chem.,
Process Des. Develop., Vol. 13,No. 3,1974 309
