Prediction of Vapor-Liquid Equilibrium Data I. H. SPINNER, BENJAMIN C.-Y. LU,
AND
W. F. GRAYDON
Department of Chemical Engineering, University of Toronto, Toronto, Canada
A
NUMBER of methods for the representation of vaporliquid equilibrium data have been reviewed ( 6 , 1 7 ) . Some of these methods have been useful for predicting or extrapolating z - y data. These predictions require as primary information measured 5 - y data or temperature-composition data for the binary system in question. The empirical method described in this paper permits the prediction of the vapor-liquid equilibrium data for the binary system, component 3-component 2, using as primary information the measured 2 - I/ data for the two binary systems, component 2-component 1 and component 1-component 3. In the literature pertaining to copolymerization there have been derived equations relating instantaneous monomer composition to polymer composition (16, 66). These authors have indicated the formal similarity between copolymer-monomer composition plots and vapor-liquid equilibrium diagrams. The analogous equation for vapor-liquid equilibrium is given as follows:
If Equation 1 and hence the binary constants a12 and are known for a system, Equations 2 and 2a permit the calculation of constants A2 and BZfor component 2 on the basis of some assumed values of A1 and B1 for component 1. Similarly, for the binary system, component I-component 3, values of a13 and a31 may be determined. Using Equations 2 and 2a and the same arbitrary values of A1 and B1,values for As and Bs may be calculated. By substitution of Az, Bz, Aa, and B3, in Equations 2 and 2a constants a25 and aaz for the binary system, component 2-component 3, may be computed. Thus a prediction for the vapor-liquid equilibrium data for the system 2-3 is obtained.
(1)
(3)
EVALUATION OF BINARY CONSTANTS
As an example of the method of evaluation of constants a12 and a21 in Equation 1 the binary system acetone-ethanol has been used in the sample calculation below. Equation 1 may be rearranged to give
By the substitution in Equation 3 of the x - y data for the system acetone-ethanol as given in Table I, four equations are obtained, which are plotted in Figure 1.
or, rearranged,
where z and y refer to mole fractions in the liquid and vapor, respectively, and a12 and a21 are arbitrary constants. Subscripts 1 and 2 refer to components. . Equation 1 has been found to be an excellent representation of the vapor-liquid equilibrium data for a number of binary systems. The constants a12 and azl, characteristic of a binary system, have been expressed in terms of constants characteristic of the two components.
a12=
4 e-Bi(Bi A2
- Bz)
Table I.
Data for System Acetone XI
0.05 0.20
0.40
o.eo a
(2)
- Ethanola
Yl 0.155 0.417 0.605 0.739
From Compilation of Chu (8).
Although the solution is not unique, the area in which the values of a12 and a21 must lie is small. From this plot the solution has been taken as a12= 1.55 and aZ1= 0.24. These values of the binary constants are suitable for the prediction of 2 - y data for other binary systems. Somewhat greater precision for curve fitting and extrapolation of data may be obtained by slight adjustment of these constants within the area defined by the intersections-for example, in this system the best fit for the four compositions used as primary information was given by a12 = 1.55 and a21 = 0.256. These values were used for the extension of data within the acetone-ethanol system. Table I1 illustrates the precision with which Equation 1 may be used to represent vapor-liquid equilibrium data, The precision indicated by the values in Table I1 is typical of that obtained for systems listed in Table I11 and numbered 1 to 20. These binary systems, chosen from the compilation of Chu (8), were those for which data were given a t 1 atmosphere for sets of three binary systems involving only three components -that is, sets of systems of the type 1-2, 2-3, 1-3. Twentyone such binary systems were found. Three of these binary systems (21, 23, and 24 of Table 111) were not fitted by Equation I with the precision indicated in Table 11. It was found that the data for systems 21 to 24 did not give plots of the sort shown
where A and B are constants for the component indicated by the subscripts. Analogous semiempirical equations have been suggested in the polymerization field by Alfrey and Price ( 2 ) . Equation 1 is similar in form to the three-constant equations proposed for vapor-liquid equilibrium data by Prahl (19). Algebraic equations of this kind have also been proposed by Clark ( 7 ) and by Kretschmer and Wiebe (16). Equation 1 has an advantage in that it requires only two constants to give a good representation of the data. The equation may be combined with the relationship given by Redlich and Kister ( g 2 ) . This combination permits the evaluation of the constants in Equation 1 using as primary information a single liquid-vapor equilibrium composition determination and the vapor pressure data for the pure components. I n addition, Equation 1 provides a simple algebraic representation for the relative volatility, The major advantage of Equation 1is that by using Equations 2 and 2a constants a12 and a21 may be used for the prediction of other binary constants. 147
INDUSTRIAL AND ENGINEERING CHEMISTRY
148 Table 11.
Vapor-Liquid Equilibrium Data
System Acetone(l)-Ethanol(2), 0.10 0.30 0.50
Xi
a12
::;:: 0":;'
ti
1.55, mi = 0.256 0.70 0.80
=
0.90
:':::
;:::"0
System Benzene(l)-Cyclohexane(Z), a18 = 0.698,a21 = 0.697 Xi 0.118 0.254 __ 0,502 0.738 0.900 0,290 0.502 0.698 0.948 yi expt. 0.145 V I calcd. 0.149 0,289 0.501 0.703 0.946
where CY represents the relative volatility, I'r and me the vapor pressures of the pure components, and y1 and y2 are activity coefficients. If the data are assumed to be correct, the vapor phase is assumed to be an ideal gas mixture and the activit,y coefficients are independent of temperature. Then
-
System Ethanol(l)-Water(2), a12 = 0.89,an = 0.035 0.5079 0.6763 0.0966 0.2608 0.558 0.656 0.738 gi expt. 0.437 0.545 0.620 0,737 y: calcd. 0.453
J1 log
01
0.895
in Figure 1. The intersections were not localized and the uI2, a21 values were therefore doubtful. Thus it was known prior to trial that the data could not be extrapolated with precision using Equation 1. The product of the constants in Equation 1 (aQlX a l a )provides a useful guide to the precision that may be expected. As shown in Table 111, the data that were accurately represented by Equation 1 are characterized by an ala x ual product greater than 0.1. The data that were poorly fitted are characterized by an ala X a21 less than 0.1. The ethanol-water system occupies an intermediate position, because it has a low u12 X a21 product but is represented with fair precision by Equation 1as shown in Table 11. An example of the poor representation obtained for a system of low a12 X a21 product is shown in Table IV. 9 s noted above, the relationship of Redlich and Kister ($3) may be combined with Equation 1 to solve for constants and uZ1in Equation 1 using only one experimental point. dxl
=
jl 5 log
dxl
+
s:
log
Table 111.
1
2
;: 5
6
7 8 9 100 110
12 13 14 15 16 17 18'
19 20 e 21
22c 23 c 24 All
a
5 C
d e
Systeiri 4cetone-ethanol ( 8 ) Acetone-ethylene dichloride (f0) Acetone-methanol ( 8 ) Benzene-cvclohexane (83) Benzene-ithylene dichloride (8) Benzene-toluene ( 8 ) Carbon tetrachloride-benzene (f8) Carbon tetrachloride-toluene (8) Chloroform-benzene (I)) Cyclohexane-ethylene dichloride (10) Ethanol-water (60 Ethvlene dichloride-toluene (13) n-Hiptane-methylcyclohexane (4) n-Heptane-toluene ( 8 4 ) Methanol-ethanol (8) Methanol-water ( 8 )
=
+1 +
XI(U,2
- 1)
21(1 -
a211
~~~~
a21
Hence
This equation provides a line on the a12 X aZ1plot v-hich xilay be used in conjunction with one set of x - y data to establish the values of a12 and aZ1. An example of a favorable case is given in Figure 2. In many instances the agreement is less striking. The equilibrium lines themselves may intersect over an area rather than a t a point. I n such cases it may well be that the Redlich and Kister line should be a w r d e d greater weight than an equilibrium line in the estimation of uI2and aQ1. The values of uI2 and az1listed in Table I11 have been determined from liquid-vapor equilibrium-composition data only. In columns 4, 5, axid 6 the values obtained have been checked against the Redlich and Kister relationship. The generally good agreement indicates the intrinsic consistency of the calculations and the precision of the analytical representation of the relative volatility in terms of a12 X ~ 2 1 . The values given in columns 4 and 5 would be identical if the three assumptions listed above were completely valid. PREDICTION OF BINARY CONSTANTS
As it was apparent from Table I11 that low values of the a12 x ail product resulted in a poor representation of the data, such
dxl y2
Constants for Prediction and Representation 1
NO.
dxl = 0
Froin Equation 1 _0.8943 ~__
System Methyl Ethyl Ketone(l)-Toluene(2), a12 = 2.05,azi = 0.262 0.701 0 861 0.468 0.119 0.313 Xi vi expt. 0.304 0,552 0.685 0.840 0 928 0.684 0.839 0 929 0.304 0 551 y1 calcd. Experimental data from compilation of Chu ( 8 ) .
a:
E
0.8943
5:
J1 log
VoP. 48, No. 1
1.55 2.75 0.85 0.70 1.11 2.40 0.995 2.42
2
0.24 0.33 0.33 0.70 0.90 0.413 0.81 0.41
3
1.55 2.70 0.83 0.698 1.11 2.50 0,995
4
0.256 0.372 0.380 0.330 0.31 0.915 0 394 0.458 0.30 0,249 0.148 0.171 0.490 0 697 0.008 0.000 0.90 0,999 0,042 0.040 0.44 1.000 0.383 0.403 0.81 0.805 0 056 0,049 2.50 0.43 0.435d 0,992 0.374 2.70 0.87 ... 0 . 2Gj 2.36 0.274 0:ii 0.41 0.285 0.115 0,39 0.036 0.051 0.93 0.035 0.032 0.89 0.035 0.357 0.356 0.350 2.20 1,000 0,454 2.20 0.454 0.355 1.083 0.028 1.000 0.924 1.09 0.945 0.037 1.14 Data scattered 0.621 0.545 0.152 0.148 1.72 1.72 0,578 0 994 0.233d 0.578 0.223 2.44 0 103 0 220 0,090 2.49 0.590 0.546 1.10 1.05 0.625 0.630 0.118 Methylcyclohexane-toluene ( $ 0 ) 0 693 0.107 0,705 0,843 0.84 0,835 0.85 Methyl ethyl ketone-benzene (84) 0,024 0 009 2.10 0.800 2.05 0.262 0.418 Methyl ethyl ketone-toluene (84) 0.285 0 399 ... ... 1.68 3.96 2.36 .4cetone-chloroform if 0.068 0.073 . 0) . 5.0 0.00 Acetone-water ( 8 ) ,.. 0 758 0.05 .. 0.01 Methanol-benzene ( 8 ) , . . ,.. 0.28 0.01 0.003 0.208 0 191 0.01 0.80 0.08 Methanol-ethylene dichloride (10) ... ... 0.266 0.1 0 81 Methyl ethyl ketone-n-heptane ($4) , , . 0.70 0.07 0.05 0,272 0.268 vapor pressure data for pure components were taken from Perry (18) except methylcyclohexane, taken f r o m Jordan (14).
ij
0.331 0,458 0 I71 0 000 0.043 0.388
0.044 0.387 0 280 0 051 0 35lj
0.349
0 030 0.148 0 236
0.549 0 105 0.002 0 401 0 OY1
. I .
Values in column 5 computed b y addition of quantity
I . .
.c
log ri d x t o ralues in column 4. Y2
These Bystems have an azeotrope. Values calculated for these systems Ti-ere from vapor pressures of pure components a t boiling points only. This system has a high boiling azeotrope.
0,195 kl
27i
INDUSTRIAL AND ENGINEERING CHEMISTRY
January 1956
System Characterized by Low a 2 1 X
Table IV.
a12Product
(Equation 1 is a less precise representation) System Methanol-Benzene, a12 = 0.28,an = 0.01
0.050 0.420 0.454
dl
g~ expt.
yxcalod.
0.270 0.575 0.519
0.586
0.817 0.655 0,691
0.610
0.581
0.902 0.730 0.782
149
There remains the choice of sign for the Bz and B3values. The signs are chosen so that the three B values will be consistent with increasing or decreasing polarity of the components. For example, the dielectric constants of these three components are in the order methanol, ethanol, acetone. Hence the values of BZ (acetone) and B3 (methanol) would be chosen with opposite signs, as BI (ethanol) has been assumed to be zero. B2 could be chosen either positive or negative. In this example BSis taken as 4-0.994 and Bsas -0.0705. These values of A2, A3, Bz,and Ba are resubstituted in Equations 2 and 2a and eonstants a82 and ~ 2 for a Equation l are calculated to be a23 = 0.831 and aaz 0.385. These values of a28 and a82 are independent of the values of A I and B1 used in their calculation. The calculation may be summarized by the equations below: i=
0.6
-
04
a23
=
a32
=
N
0
0.2
0 The sign in the square bracket is taken to be the sign of the -0 2
0
2
I
4
3
5
at2
Figure 1. Sample solution for a12 and acetone ethanol
-
a21
quantity
(DzD -T
~ where ) D1,Dz, and D nrepresent
dielectric
constants for the components.
for system
data were not suitable for prediction. Data characterized by an a12X a21 product greater than unity, although wellrepresented, were not suitable for prediction. From the data in Table I11 good predictions would be expected when a12 X a21 products were large, and poor predictions would be expected if low a12 x az1 products characterized the primary information or the predicted data. a12 X a21 products between 0.2 and 0.1 are regarded as doubtful. The prediction of constants in Equation 1 for the binary system acetone-methanol is given below as a sample calculation. The primary information for the prediction is provided by the vapor-liquid equilibrium data for the binary systems acetoneethanol and methanol-ethanol. The constants for Equation 1 for these systems are as follows: a12
Component
Component Ethanol Acetone Methanol
NO.
1
2 3 Assumed
Constants = 0.24 a n = 1.55
(112
BI
A I = 1.00
=
ais = 0.678 as1 =
1.72
0.00
Figure 2. Sample solution for all and a21 for system benzene- cyclohexane Dotted line obtained using Redlioh and Kister relationship ( 2 2 )
These values are substituted in Equations 2 and 2a. System 1-2 A2 =
-
= 4.16
a12
B ; = 2.3 log
~
a12
1 X
= a21
2.3 X 0.429 = 0.986
Bz = Jrr0.994 System 1-3
1 A3 = - = 1.73 a13
B:
=
1 2.3 log ___ = 2.3 X 0.00217 = 0.0050 a13
X
a31
I33
= zk0.0705
The precision with which the vapor-liquid equilibrium data may be predicted is illustrated in Figures 3 to 8. These five predictions are those possible using as primary information the data listed by Chu and eliminating predictions that involve binaries of low a12 X a21 product. As may be seen from the figures, sets of three binary systems in which there wa,e no azeotrope or one azeotrope gave good predictions. There was available only one set of three binary systems involving tn-o azeotropes. The prediction for this set is shown in Figure 8. In this case the existence and composition of the azeotrope are predicted, but the prediction is otherwise of low accuracy. Figures 9 and 10 are examples of the lowered precision of the prediction when one of the binary systems is Characterized
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
150 1 .o
10
0.8
08
0'6
06
*
Vol. 48, No. 1
h
0.4
04
0.2
0 2
0
0
0
0.2
0.4
0.6
0.8
0 2
0
1.0
04
X
10
08
06
x
Figure 3. Predicted vapor-liquid equilibrium curve and data (13) for system ethylene dichloride-toluene at 1 atmosphere
Figure 4. Predicted vapor-liquid equilibrium curve and data (24) for system n-heptane-toluene at 1 atmosphere
Predicted constants. an = 2.18, a21 = 0.46. Product = 1.0. Primary information: Benzene-toluene ai2 X azi = 1.000, benzeneethylene dichloride a i 2 X a z I = 0.999. As all a12 X a21 products are above 0.15, a good prediction is expected
Predicted constants. a12 = 1.19, ail = 0.581. Product = 0.68. Primary information: n-Heptane-meth) lcyclohevane a12 x an1 = 1.000, methjlcyclohexane-toluene a12 X 0 2 1 = 0.693. A s all a12 x a21 products are abme 0.15, a good prediction ifi expected
PREDICTION OF TERNARY DATA
1.0
The ternary data for three of the sets of three components which were used above are available ( 7 ) . One set involved binary a12 x uz1products all greater than 0.2 and was thus expected to provide a good ternary prediction. Equations have been given in the copolymerization field for three ( 1 ) and for n ( 1 1 , b6) components, The equations for three components are as follows:
0.8
0.6
Y,:Y,:Y3
=
21
[" + a31 a21
a21 a32
+
1"
a31 a23
[x, + a4 12
+
"1
a13
0.4
9.2
where Y 1 :Y,: Y,is the ratio of mole numbers.
0 0
0.2
0.4
0.6
0.8
X
Figure 5 . Predicted vapor-liquid equilibrium curve and data (24) for system methyl ethyl Iretone-toluene, at 1 atmosphere
-
Predicted constants. (112 = 2.06, a21 = 0.345. Product = 0.70. Primary information: Methyl ethyl ketone-benzene ais X a21 8.705, benzene-toluene an X azi = 1.000. As all alz X azi producta are above 0.15, a good prediction i s expected. System methyl ethyl ketone-benzene has a n azeotrope
by a low ul2 x uplproduct. Figure 9 is an example of the prediction from primary information involving a low u12 x a21 product. Figure 10 is an example of a prediction of low a12 X a21 product obtained from acceptable primary information. The method of prediction seems promising for sets of binary systems involving no more than one azeotrope and for which the aI2 X aZ1product of the primary data and the prediction exceed a value of about 0.15.
The primary information required for the ternary prediction is the data for the vapor-liquid equilibria of the three binary systems involving the three components of the ternary system. Because, as shown above, the data for one of the three binary systems may be predicted, the minimum primary information for the prediction of the ternary data is the data for two of binary systems of the three components. The prediction for a ternary system is given in Table V. A good prediction was expected in this case, as all of the binary constant products (alz x azl) were greater than 0.20. Two other predictions were possible from the data given by Chu (8). Both of these systems, methanol-ethanol-water and methyl ethyl ketone-n-heptane-toluene, involved low binary constant products (UI,X a d for the primary information. The predictions in these cases are less precise, as shown by the examples in Table VI. DISTILLATION EQUATIONS
I n addition to its use for the extension and prediction of data, Equation 1is a simple correlation of x - y data which gives a good representation for many binary vapor-liquid equilibria. It
January 1956
151
INDUSTRIAL AND ENGINEERING CHEMISTRY 1.0
0.8
0.6 >r 0.4
0.2
-
0 0
0.6
0.4
0.2
1.0
0.8
0
X
0.2
0.4
x
0.6
0.0
1.0
Figure 6. Predicted vapor-liquid equilibrium curve and data (8) for system carbon tetrachloride-toluene at 1 atmosphere
Figure 7. Predicted vapor-liquid equilibrium curve and data (8) for system acetone-methanol at 1 atmosphere
Predicted constants. ala = 2.40, aa1 = 0.336. Product = 0.8. Primary information: Carbon tetrachloride-benzene a 1 2 X aai = 0.805, benzene-toluene a l p X (121 = 1.000. As all an X aal products are above 0.15, a good prediction is expected
Predicted constants. au = 0.831, a31 = 0.385. Product = 0.32. Primary information: Acetone-ethanol ala X azl = 0.372, methanol-ethanol a n X an = 0.994. A s all a 1 2 X aai products are above 0.15, a good prediction is expected
may thus be used to provide rapid algebraic solutions to a number of useful equations which otherwise require graphical treatment. For example, Rayleigh’s equation for sample batch distillation ($1) may be integrated after substitution of Equation 1 to yield
L = ___ a21 a12a21 - 1 In--’ ln-:2’ L2 1 - a21 2, (1 - a d ( l - a211 lnx:(a12 a21 - 2) - (a21 - 1) x:(a,, a21 - 2) - (a21 - 1)
+
0.8
~
+ +
+
0.6
a12
1-
a12
lz 1 1-
2:
A
2:
0.4
where moles of original charge moles of residual charge = mole fraction of more volatile component in original 2; charge = mole fraction of more volatile component in residual 2: charge alS,a21 = constants in Equation 1 for binary system
Ll L2
1.0
=
=
Similarly, in the Bogart equation (3) for batch rectification ‘the usual graphical solution may be avoided by integration after substitution of Equation 1. The integrated equation may [be written as follows:
0.2
0 0
0.2
0.4
0.6
0.8
1.0
X
Figure 8. Predicted vapor-liquid equilibrium curve and data (10) for system cyclohexane-ethylene dichloride at 1 atmosphere Predicted constants. ala = 0.759, 021 = 0.612. Product = 0.47. Primary information: Benzene-cyclohexane ala X aai = 0.49, benzene-ethylene dichloride a11 X a21 = 1.0. This system provides an example of precision obtained for a set of three systems involving two azeotropes
Table V. xexpt.
Data for System n-Heptane(l)-Methylcyclohexane(2)-Toluene(3) yexpt.
Ypred. A
X1 xa xa 1/1 Ya Yl Yl YZ Y3 Y1 0.223 0.550 0.227 0.238 0.568 0.204 0.2040 0,5355 0.2605 0,2265 0.444 0.346 0.210 0.443 0.345 0.212 0.4070 0.3485 0.2445 0,443 0.643 0.137 0.220 0.647 0.134 0.219 0.647 0.6030 0.1360 0.2610 0.3185 0.4750 0.2065 0.334 0.462 0.204 0.2890 0.4705 0.2405 0.328 0.2200 0.2200 0.560 0.218 0.222 0.5600 0.2225 0.2605 0.560 0.5170 Prediction A. The three binary systems were used as primar information. Prediction B. The :wo binary systems and the predicted vahes for n-heptane-toluene, ‘713 0.581, were used a8 primary information.
(8)
ypred. B
Ya Y3 0,549 0.2245 0.351 0.206 0.135 0.218 0.458 0,214 0.2205 0,2195
= 1.19 and
as1
=
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
152
VOI.
48,No. 1
where liquid composition, mole fraction of low-boiling component (component 1) JI = total number of moles of liquid in kettle e = time, hours V = overhead column vapor, moles per hour
x
Table VI.
=
Subscripts = feed or charge stock to kettle = distillate or product = residual kettle liquid = total elapsed time = a12 a21 - 2 kz = a12 2a21 - 3 ka = a12 - 1 kr = a21 - 1
Data for System biethyl Ethyl Ketoiie(1)n-Heptane(2)-Toluene(3) ( 8 )
2
3
0.384 0.208 0.109 0.578 0.743
0.431 0,506 0.316 0.173
0.175 0.286 0.575 0.249 0,032
0.225
yprea.
yerpt.
Xexpt.
1
1 0.579 0.435 0.284 0.700
0.762
._
2
3
1
2
3
0.342 0,418 0.345 0 186 0,227
0.079 0.147 0.371 0.114 0.011
0.541 0,404 0.274 0.692 0.749
0.382
0 087
0,459 0,137 0.321 0 355
0.204 0.239
0.104
0.012
This prediction used as griniary information data from two hinary systems and predicted values for methyl ethyl ketone-n-heptane, a12 = 0.705 and a21
= 0.201.
++
is obtained. by
In either case the azeotropic composition is given a a12
Equation 1 is an algebraic expression of the relative volatility as a function of the liquid composition for a binary system a t constant pressure YE? =
ff
=
YZXl
(-) + + a12x1
22
a2122
21
-
d =
9
1
x2
The form of Equations 2 and 2a prevents the prediction of data for a system involving a high-boiling azeotrope. Equation 1 may be used to represent such data, as shown in Table I11 (system 20). Systems such as numbers 9 and 20 in Table 111, which are characterized by u12 X az1 > 1, cannot be used for prediction whether the system has a high-boiling azeotrope or not. TABULATION OF A AKD
Equation 1 may be used to extend to nonideal systems the methods of calculation which require constant relative volatility, such as the method of Eshaya (9) for the computation of the number of theoretical plates required for a given separation. 1 It is apparent that when a12 = - in Equation 1, CY = a12 and a21
the relative volatility is independent of composition. This ideal case occurs for those components which are of very similar polarity, Bl = &. When the values of a2, and a12are both less than unity-, a low-boiling azeotrope is obtained. When the values of aZ1 and a12 are both greater than unity, a high-boiling azeotrope
n
The binary and ternary predictions given previouslj 1i:ive been made using sets of three binary systems of three components. Hence the A and B values used in the computations are dependent on the assumed values for one of the three components in each set and on the equilibrium data for two of the three binary systems. It is possible to bring all of the A and B values for the components to a common base. In many cases three or four binary systems are required to provide the link between a given component and the chosen etandard component. The A and B values have been calculated using benzene as standard and taking for this substance -4 = 1
1.0
IO
0.8
08
0.6
06 ZI
r,
0.4
04
0.2
02
0
0 0
0.2
0.4
0.6
0.8
IO
X
0
02
04
06
08
IO
x
Figure 9. Predicted vapor-liquid equilibrium curve and data ( 8 ) for system methanol-water at 1 atmosphere.
Figure 10. Predicted vapor-liquid equilibrium curve and data (24) for system methyl ethyl ketone-n-heptane at 1 atmosphere
Predicted constants. a12 = 1.83, azi = 0.0233. Product = 0.043. Primary information: Ethanol-water ais X azi = 0.032, methanol-ethanol a i 2 X an = 0.994. This system provides a n example of less precise prediction obtained using primary information characterized by low a12 X a n product
Predicted constants. ale = 0.705, azi = 0.21. Product = 0.14. Primary information: n-Heptane-toluene a i 2 X azi = 0.621, methyl ethyl ketone-toluene an X ail = 0.600. This system provides an example of less precise prediction obtained when a low un X a11 product is predicted
INDUSTRIAL AND ENGINEERING CHEMISTRY
January 1956
153
1.0
Table VII. Values of A and B Constants for a Sumber of Components (Benzene assumed A = 1.00,B = 0 , 0 0 0 )
0.8 Component Toluene Methylcyclohexane
0.6
n-Heptane Ethylene dichloride Benzene Methyl ethyl ketone Carbon tetrachloride
2,
0.4
Cyclohexane Ethanol Acetone X I ethanol
0.2
Water
0 0
02
0.6
0.4
0.0
1.0
Water
X Methanol
Figure 11. Predicted vapor-liquid equilibrium curve and data (23) for system cyclohexane-methylcyclohexane at 1 atmosphere Prediction made from A and B values of Table VI
and B = 0. The values calculated on this basis have been listed in Table VII. The normal boiling points and the dielectric constants of the components at 20” C. are included for comparison. It is apparent that the A and B values of Table VI1 may be used to predict constants u12 and uZ1’for any binary system. Thus i t is not essential to have data for two systems in a set of three in order to predict the third binary. Using Table VII, predictions can be made using four- and five-component chains instead of three-for example, the data for the system methylcyclohexane-cyclohexane may be predicted, using Table VII, in two ways. A four-component chain or a five-component chain may be used-namely; methylcyclohexane-toluene, toluene-benzene, benzene-cyclohexane; or methylcyclohexanen-heptane, n-heptane-toluene, toluene-benzene, benzene-cyclohexane. The two predictions were almost identical and are shown plotted in Figure 11 with the data of Richards and Hargreaves (28). These authors state that the scatter in the points resulted from difficulties in the analysis. ACKNOWLEDGMENT
The authors gratefully acknowledge financial aid received through the School of Engineering Research of the University of Toronto. LITERATURE CITED
(1) Alfrey, T., and Goldfinger, G., J . Chen. Phys. 12, 205, 322
(1944). (2) Alfrey, T., and Price, C. C., J . Polymer 9 c i . 2, 101 (1947). (3) Bogart, M. J., Trans. Am. I n s t . Chem. Engrs. 33, 139 (1937). (4) Bromiley, E. C., and Quiggle, D., IND. ENQ. CHEM.25, 1136 (1933). (5) Carey, J. S., and Lewis, W. K., Ibid., 24, 882 (1932). ( 6 ) Carlson, C. H., and Colburn, A . P., “Source Book of Technical Literature on Fractional Distillation,” p. 189, Gulf Research and Development Co., Pittsburgh.
Ethanol Methyl ethyl ketone Acetone Ethylene dichloride
A 0.413 0,655
0.700 0.758 0,901 1.00_ 1.195 1.05 1.23 1.43 3.01 2.40 2.73 4.58 4.75 27.4 28.3 B $3.32 f3.15 f1.380 fl.580 4-1.304 +0.591 4-0.309 f0.0314
Normal, Boilitg Point, C. 110.8 110.3
98.5 83.6 80.0 79.6 76.5
n-Heptane Cyclohexane
Benzene Toluene Benzene Benzene Ethyl dichloride Acetone Ethylene dichloride Ethanol Acetone Methanol Ethanol
81.4 78.5 56.5 64.7 100.0 Dielectric Constant, 200
c.
88 33 ,
0.00 0.00 0.00
Methyloyclohexane
Second Component of Binary Benzene Toluene n-Heptane Toluene Benzene
-0.677 -0.463 -0.690
25 18 21 10 6.0
2.38 2.28 2.24
I . .
-0,600
-0.69 -0.844 -1.45
1.97 2.06
Methanol Ethanol Ethanol Acetone Acetone Benzene Ethylene dichloride Benzene Ethanol Benzene Toluene Benzene n-Heptane Toluene Toluene Benzene Ethyl dichloride
(7) Clark, A. M., Trans. Faraday SOC.41, 718 (1945). (8) Chu, J. C., “Distillation .Equilibrium Data,” Reinhold, New York, 1950. (9) Eshaya, A. M., Chem. Eng. Progr. 43, 555 (1947). (10) Fordvce, C. R., and Simonsen, D. R., ISD. ENG.CHEW.41, 104 (1949). (11) Fordyce, R. G., Chapin, E. C., and Ham, C. E., J . Am. Chem. SOC.70, 2489 (1948). (12) International Critical Tables, vol. 3, pp. 309-17, McGraw-Hill, New York, 1928. (13) Jones, C. A., Schoenberg, E. M., and Colburn, A. P., IND. ENG. CHEM.35, 666 (1943). (14) Jordan, T. E., “Vapor Pressure of Organic Compounds,” Interscience, New York, 1954. (15) Kretschnier, C. B., and Wiebe, R., J . Am. Chem. Soc. 71, 1793 (1949). (16) Mayo, F. R., and Lewis, F. M., Ibid., 66, 1594 (1944). (17) Othmer, D. F., Ricciardi, L. G., and Thaker Rfahesh, S., IND. ENQ.CHEM.45, 1815 (1953). (18) Perry, J. H., “Chemical Engineers’ Handbook,” 3rd ed., pp. 153-64, McGraw-Hill, New York, 1950. (19) Prahl, W. H., IND. ENG.CHEM.43, 1767 (1951). (20) Quiggle, D., and Fenske, M. R., J . Am. Chem. SOC.59, 1829 (1937). , \ - - -
(21) (22) (23) (24) (25) (26)
Rayleigh, Lord, Phil.Mug.4, 521 (1902). Redlich, O., and Kister, A. T., IND. ENG.CHEM.40, 345 (1945). Richards, A. R., and Hargreaves, E., Ibid., 36, 805 (1944). Steinhauser, H. H., and White, R. H., Ibid., 41, 2912 (1949) Wall, F. T., J . Am. Chem. SOC.66, 2050 (1944). Walling, C., and Briggs, E. R., I h i d . , 67, 1774 (1945).
RECEIVED for review October 7, 1954.
ACCEPTED August 12, 1955.
