Activity Coefficient Relations in Miscible and Partially Miscible Multicomponent Systems Solke Bruin’ Laboratory for Physical Technology, Eindhoven Uniuersity of Technology, Eindhoven, Netherlands
Based on a quasilattice model of a multicomponent solution, three equations are derived relating component activity coefficients to the composition of a liquid mixture. The equations also apply to systems showing limited miscibility. Temperature dependence of activity coefficients i s built in. The equations were tested for 16 binary systems, 5 of which show limited miscibility, and for 4 ternary systems. The enthalpic Wilson equation, one of the three, gives the best prediction of binary vapor-liquid equilibrium. Both the enthalpic Wilson equation and the Orye equations give good results of comparable accuracy for the ternary systems.
PREDICTION
of thermodynamic properties of mixtures from pure component properties is the main goal of thermodynamics of liquid mixtures. Progress in this field has made it possible to calculate the properties of a mixture of n components from the pure component properties supplemented with experimental vapor-liquid equilibrium (VLE) data for all the binary systems that can be constructed out of the n components: a total of l / 2 n ( n - 1) (I’rausnitz et al., 1967). The starting points of these computational methods are thermodynamic models of multicomponent mixtures, which provide calculational procedures to predict the vapor-liquid equilibrium of the mixture from data on the binary systems. From such models relations between the activity coefficients, y and the composition of the liquid are derived. I n such relations, parameters characteristic of interactions between pairs of molecules of different kind appear. Formally, such activity coefficient relations have the form
Values for the interaction parameters, Pi, and P 3 ( , can be calculated from VLE data on the binary system of components i and j . The best known activity coefficient equations are, undoubtedly, the van Laar equations. These equations have proved t o be very useful for binary systems but less satisfactory for multicomponent systems (Neretnieks, 1968; Orye, 1965; Orye and Prausniti, 1965; Prausnitz et al., 1967). The two-parameter Wilson equation gives better results than the van Laar equations for many binary and multicomponent mixtures (Nagel and Sinn, 1966, 1967; Prausnitz et al., 1967; Wilson, 1964a, b). No ternary or higher parameters are needed in the generalization to multicomponent systems. Temperature dependence is t o some extent built into the Wilson equations, making them suited to, for example, the isobaric vapor-liquid equilibrium calculations commonly encountered in distillation. An undesirable feature of the Wilson equations is their inapplicability to partially miscible systems. The modification of the equations that Wilson proposed to overcome this difficulty means introduction of a n extra parameter. I n the present paper three activity coefficient equations are discussed. These equations follow from the quasilattice
model of multicomponent liquid mixtures developed by Guggenheim (1935, 1944a, b, 1952). This model yields a rather general relation for the excess Gibbs f~inction,in which the enthalpic and entropic contributions appear as separate terms. Introduct>ion of van Laar or Wilson parameters into this relation leads to a number of possible activity coefficient equations including the van Laar and Wilson equations themselves. I n Table I all poasible combinations of Wilson paramet,ers (Aij), van Laar parameters (Aij), and Gibbs function are given. The Orye equations (Orye, 1965) result on introduction of Wilson parameters in the general relation for the excess Gibbs function. Retaining only the enthalpy part assuming the excess entropy to be ncgligible (regulm solutions), yields “enthalpic Kilson” equations. The “extended van Laar” equations (EVL equations) result from introduction of van Laar parameters into the general Gibbs function, as indicated in Table I. Introduction of van Laar parameters int’o the entropy part, assuming the ent,halpy to be negligible (athermal solutions), gives equations unable to predict partial miscibility (Bruin, 1969), and therefore is not discussed. The Orye, enthalpic Wilson, and EVL equations are derived and discussed. The derivation of the Orye equations here differs from that given originally by Orye. Subsequently, the equations are generalized to multicomponent systems. The equations were tested by application to 16 binary and four ternary systems, prediction of parameters from infinite dilution activity coefficients, and prediction of temperature dependence of activity coefficients. Binary Systems
The activity coefficient, yl, in a binary liquid mixture of components 1 and 2 is given by the exact thermodynamic relationship In y1
947 10
[2-
(1)
dnl ( N g E / R T ) ]T , P , n %
+
where N is the total number of moles: N = nl nz. The molar excess Gibbs free energy of mixing, g E , in Equation 1 is related to the excess molar entropy and enthalpy according t o =
Present address, Western Utilization Research and Development Division, Agricultural Research Service, .4lbany, Calif.
=
hE
-
TsE =
UM
+ PvM
-
TsE
(2)
Adopting the quasilattice model for liquid mixtures of molecules of different size, a s developed by Guggenheim (1935, Ind. Eng. Chem. Fundam., Vol. 9, No. 3, 1970
305
Table I.
Summary of Relations for g E Resulting from Different Combinations of Quasilattice Equations and Parameters Type of Parameter Aii Aij
(Wilson)
included
(van Laar)
++ +-
1 2 3 4 5 6
gE
hE-term
++ 4++
-
+++
-
-
1944a, b, 1952), we may write the following equation for the energy of mixing, uM, and the excess entropy: sE = -R(zl In {rl/(rlzl
+
r24)
+ 22
UM
=
ln{rz/(rlzl+
rs2)
1)
(3) (4)
x12u12
I n Equation 3 rl is the number of sites in the quasilattice structure occupied by a molecule of type 1. u12 represents a n interaction energy: UIZ =
‘/2~N(2~12 -
-
(5)
~ 1 1 WZZ)
N is the Avogadro number, w12 denotes the contribution to the potential energy by a pair of sites, one of which is occupied by a n element of a molecule of type 1, the other by an element of a different molecule of type 2; and z is the number of nearest neighbors to a site, the “coordination number.” I n Equation 4 XI2 is defined as the number of pairs of neighboring sites occupied by different molecules, one of which is of type 1 and the other of type 2, divided by the total number of nearest neighbor sites. T o calculate X ~ ZGuggenheim , introduced the quasichemical approximation
which is the mass-action law for a “reaction” where a molecule 1 is brought from a pure liquid 1to a (1-2) liquid mixture and simultaneously a molecule 2 is brought from pure 2-liquid to the (1-2) liquid mixture. I n this reaction (zX11) and (zX,) bonds are broken up, while 2 ( 2 X 1 2 ) bonds are formed. The number of 1-1 pairs and 2-2 pairs can be expressed as 22x11 = z
(rlz1
-
XIS)
(64
22x22 = z
(rs2 -
X12)
(6b)
=
(r14(rza)/(r1x1
=
(rlzJ(rs2)ud(rlz1
rE-term included
Name of Resulting Equation for y i
++ + +
Wilson van Laar Orye Enthalpic Wilson Extended Van Laar (EVL)
t
Renon and Prausnitz (1968) recently developed a relation for g E in which the condition u12+ 0 on Equation 5a was relaxed. Moreover, distinction was made between the transfer of a molecule 1 into the (1-2) liquid mixture [giving (zX12’) bonds] and the transfer of a molecule 2 into the (1-2) liquid mixture [giving (zX2,’) bonds]. The resulting relations for the activity coefficients’ (NRTL equations) were shown to give excellent representation of a wide variety of binary and ternary liquid mixtures. I n the present paper Equation 9 is used as a starting point to derive activity coefficient equations. Guggenheim (1944b) showed that the inaccuracy of Equation 9 is theoretically of the order of ( u ~ ~ / R T ) ~ . If one assumes in Equation 2 that h E = 0 (athermal solutions; Flory, 1942; Huggins, 1942) one obtains for g” gE/RT
=
In
(
rlxl
+r1 rsr2) +
+
TZXZ)
(7)
+
When Equations 3 and 8 are substituted in 2 and assumed to be zero, one obtains for g E
(8) UM
(
+ ra2) “
(10)
~1x1
-
(12)
(VI/VZ)
Another assumption is due to Carlson and Colburn, in their modification of the van Laar parameters:
m/RT
= A12
(13)
rluzl/RT = m/RT = A 2 1 (rl/r2)
equals
(A21/A12)
:
(rl/rd + ( A d A d
(14)
is Parameters A12 and A 2 1 are sometimes called “effective molar volumes” (Hildebrand and Scott, 1950). Wilson suggested another expression for (n/n), taking into account nonrandomness effects by weighting (uI/u2) with a Boltzmann factor containing interaction energies X : (r1/r2)-+ (u1/v2) exp
306 Ind. Eng. Chern. Fundarn., Vol. 9, No. 3, 1970
In
Equations 9 to 11 give three relations for QE as a function of liquid composition in terms of the quasilattice parameters, rl and r2, and the interaction energy, u12. To obtain twoparameter equations for y1 and y2 from Equations 9 to 11, rl, rz, and u12 have to be combined to two parameters. Different suggestions have been made. The most obvious assumes the ratio (rl/r2)to be equal in a first approximation t o the ratio of molar volumes (ul/u2) :
while u12 = u21. The implication is that rg2)
xz
On the other hand, if s E = 0 (regular solutions, van Laar; Black, 1959; Black and Derr, 1963a, b; Wohl, 1946) Q E is given by
rZu12/RT f
For UM one obtains UM
-T~E
(rl/rJ
I n the particular case where all energies of mixing are zercall u12 -c 0 (all molecules distributed a t random)-combination of 5a and 6 gives X12
= hE
{-
(Al2
- hm)/RT) = Apl
(15a)
-
and similarly for (r2/r1)
(r2/T1)
- All)/RT]
exp { -(Alz
(v2/u1)
The EVL equations (17), enthalpic Wilson equations (20), and Orye equations (21) were tested by
: 5
A12
(15b)
One can expect A12, All, and A22 to be proportional to zw12, zwl1, and zws2 in Equation 5 . Therefore
u12/RT + (2A12 - All - An)/RT
=
-In
(A12h)
(16)
Equations 14 and 15 provide two reasonable choices (van Laar and Wilson parameters, respectively) to introduce parameters into Equations 9 to 11, giving six combinations. I n Table I the combinations are indicated. Combining Equation 10 with Wilson parameters gives the Wilson equation. Introducing van Laar parameters into Equation 11 gives the van Laar equations. When van Laar parameters are introduced in Equation 10, a relation obtained is unable to predict phase separation, in much the same way as the Wilson equation. Extended van Laar (EVL) Equations. Substituting A12parameters into Equation 9 and differentiating the resulting equation for gE according to Equation 1 gives for the activity coefficient, yl In y1 = 1
- In (a
+
--
A1222/A21)
21
+
1
In
(22
+ A ~ ~ Z I / A I Z(17)) ~
(A1&)/(A21n
+
22)
r1
-
exp (-AdRT) exp ( - A d R T ) xzvz exp (-A12lRT)
~1
+
-
y1 =
(21
+
22
In
(11121121)
A1252)(22
f
(19)
X Azi~i)
For In y z the analogous expression results by rotation of indices (1 + 2 + 1). Orye Equations. Substitution of Kilson parameters in Equation 9 gives equations for y derived by Orye (1965), who used a different approach. In
y1 =
1
Again, In
- In (zl + ~ ~ 2 x + 2 ) zz{ ~12/(zl+ A ~ ~z ~ )
y2
follows from rotation 1 + 2 + 1.
Table
II. literature Data for Systems Selected to Test Activity Coefficient-Composition Equations (Isobaric at 1 Atm) Temp. Range,
Comparison of Equations 19 and 15 illustrates that Equation 19 is exact only in the limit xl + 1. Substitution of Equation 19 in Equation 18 and differentiating gives In
with m the number of data points. A nonlinear multiple regression subroutine which adjusts the parameters of a function being fitted to experimental data in such a manner as to yield a least squares fit was written, following the method of hfarquardt (1959). Details have been given (Bruin, 1969). The results are summarized in Tables I11 to VII. Values
(18)
Postulating proportionality between the number of sites occupied by a molecule of type 1 and the ratio between its molar volume and a mean molar volume (weighted by interaction energies), one can write VI
The EVL equations (17), enthalpic Wilson equations (20), and Orye equations (21) were tested for 16 binary systems listed in Table 11. All equilibria were isobaric (1 atm), in order to test the built-in temperature dependence of the equations; five systems show partial miscibility. The fitting procedure proposed by Prausnitz et al. (1967) was used. I n this procedure a correction for vapor-phase nonideality is incorporated. The objective function, f, to be minimized was defined as
Az1~2~
A122dA21
-r121zz
Comparison of Equations by Fitting Parameters to Experimental Data on Binary Systems
-
The relation for In y~ results when indices are rotated in the sequence 1 2 1. Equation 13 roughly predicts temperature dependence of A12 and A21, Enthalpic Wilson Equations. Substituting Kilson parameters in Equation 11 gives
qE/RT =
Representation of 16 experimental binary vapor-liquid equilibrium data from literature (T - 2 data a t constant P ) . Prediction of parameter values for binary systems from activity coefficients a t infinite dilution. Prediction of temperature dependence of activity coefficients a t infinite dilution (an extremely severe test).
System
"K
Reference
1. hIethano1-water
2. Ethanol-water 3. 1-Butanol-water 4. 2-Butanol-water 5. Acetone-water 6. Butanone-mater 7. Methyl acetatewater 8. Furfural-water 9. 2-Propanol-water 10. Icetone-methanol 11. Acetone-ethanol 12. Ethanol-2-propanol 13. Ethanol-benzene ( P = 750 mm Hg) 14. Ethanol-methylc yclopentane 15. Methylcyclopentane benzene 16. hlethanol-ethanol
Uchida, 1934 Carey and Lewis, 1932 Stockhardt and Hull, 1931 Altsybeeva and Belousov, 1964 Othmer et al., 1952 Othmer et al., 1952 hlarshall, 1906
338-69 351-68 372-84 364-66 330-61 330-61 329-60
International Critical Tables, 1928 Wilson and Simons, 1952 Uchida et al., 1950 Hellwig and Van Winkle, 1953 Ballard and Van Winkle, 1952 Tyrer, 1912
371-432
342-47
Sinor and Weber, 1960
339-49
Griswold and Ludwig 1943 hmer et al., 1956
344-52
354-68 329-37 330-48 351-55
338-49
Ind. Eng. Chem. Fundam., Vol. 9, No. 3, 1970
307
Table Ill.
Fitting Results for Wilson Equations to Binary Systems Parameters, col/mole
System
A12
Methanol-water Ethanol-water 2-Propanol-water Acetone-methanol (isothermal) Acetone-methanol Acetone-ethanol Acetone-water llet'hanol-ethanol Ethanol-2-propanol Ethanol-benzene ( P = 750 mm Hg) Ethanol-methylcyclopentane Methylcyclopentane-benzene
Table IV.
- A11
A12
42.11 63.98 82.48 76.83
18.79 18.78 18,77 41.24
369.55 368.65 368.32 323.15
0.3612 1.1578 2.155 0.3442
. - 93.454
489.719 440.50 1489.07 -264.02 -241.331 119.80
78.47 79.82 81.42 42.93 62.825 61.82
42,07 62.27 18.67 63.39 83.11 94.61
337.15 348,25 361.0 349.75 355.05 342.6
0.6474 0.804 2.700 1.279 1.246 0.627
118.82 95.89
339.45 352.79
1.884 0.5890
- 64,236
469.02 198.85 215.889 1540.28 2230.23 13.68
236.08 253.694
Error in P Total, %
Ai2
Azi
0.8649 1,6601 2.4260 3.8772 3.5447 0.6166
0.5210 0.8643 1.1625 1,1759 1.1387 0.5442
0.5713 0,4549 2.3880 3,5045 3.0565 4,6895 -0.5629 -0.01381 1.8341
0.3521 0.7246 1,2026 3.3491 1.4795 0,3083
0,6410 0.5170 0,7340 0.5871 4.4151 1 ,4555 5.1996 1 ,6525 5,5212 1,8050 27.5862 1.1008 -0.0580 0.5691 -0.006945 1.462 1.2814 0,3565
2.7404
1.6889
2.771
0.3259
0.3505
0.5796
for (XIZ - AH), ( A ~ z - XZZ), €12, and €21 are computed a t the temperatures indicated in the tables. As a reference, the Wilson and van Laar equations were also checked. General trends are that EVL equations (17) give poor results when compared to the van Laar equations. A remarkable result is that the enthalpic Wilson equation (20) gives very good represen-
Table V. System
Methanol-water Ethanol-water 1-Butanol-water 2-Butanol-water Acetone-water Butanone-water Methyl acetate-water Furfural-water
308
Error in P Total, %
472.368 910.615 1358.35 472.74
Parameters
Methanol-water Ethanol-a ater 2-Propanol-water 1-Butanol-water 2-Butanol-water Acetone-methanol (isothermal) Acetone-methanol Acetone-ethanol Acetone-water Butanone-water Methyl acetate-water Furfural-water Methanol-ethanol Ethanol-2-propanol Ethanol-benzene ( P = 750 mm Hg) E t hanol-methylcylopentane hfethylcyclopentanebenzene
F, O K
198.113 380.315 543.845 -54.175
Fitting Results for van Laar Equations to Binary Systems
System
Molar Volumes, Cm3/Mole v1 V2
- A22
61.58 120.66
tation for most of the systems, in many cases bett,er than the van Laar or Orye equations (21). Prediction of Parameter Values from Infinite Dilution Coefficients
For In yl a t infinite dilution of component 1 in a binary mixture one can derive For EVL Equations In
ylm =
+
1 - In (AIz/AzI) - A Z ~ A I ZAZI
For Enthalpic Wilson Equations In
ylm =
-In
(24)
(A12h)/An
For Orye Equations In
ylm =
1 - In
111-2
-
AZI - In
(hlzA21)/Az1
Ail
0.6021 0.9875 1.5696 1.5093 1.4392 1.8041 1.6971 1 ,7366
1 ,0429 1.9225 4.6182 4.2917 2.5208 3.7793 3.1399 5.4891
Ind. Eng. Chem. Fundom., Vol. 9, No. 3, 1970
(25)
Similar relations follow for In yz" by letting 1 + 2 + 1. If In ylmand In yZmare known-e.g., from Pierotti-Deal-Derr correlations (Pierotti et al., 1959) or Helpinstill and Van Winkle correlations (1968)-parameter values can be calculated. Solving Equation 25 and the similar relation for yZmfor A12 and A21 is tedious, requiring a double iteration scheme. Therefore the Orye equations were not tested. For the EVL and enthalpic Wilson equations convenient computational methods have been discussed (Bruin, 1969). Some results for the EVL equation are summarized in Table VIII. I n Figures 1 to 3 y's and activities, a, are plotted for acetone and water, 1-butanol and water, and 1-octanol and
Fitting Results for EVL Equations to Binary Systems
Ai2
(23)
Parameters €21, cal/mole
ell, col/mole
442.194 723.459 1162.336 1093.13 1032.47 1282.593 1214,279 1282.762
765,917 1408,405 3419.983 3108,21 1808.370 2686.815 2246.585 4054.630
r,
.
OK
369.55 368.65 372.65 364.45 361.00 361.00 360.05 371.71
Error in P Total, %
0.3616 0,6503 4.0834 1.7815 5.2401 5.5858 6.3476 24.1892
Fitting Results for Enthalpic Wilson Equations
Table VI.
Parameters, CaI/Mole
- xi1
XlZ
System
Methanol-water Ethanol-water 2-Propanol-water 1-Butanol-water 2-Butanol-water Acetone-methanol Acetone-ethanol Acetone-water Butanone-water Methyl acetate-water Furfural-n-ater Methanol-ethanol Ethanol-2-propanol Ethanol-benzene ( P = 750 mm Hg) Ethanol-methylcyclopentane Methylcyclopentane-benzene
Molar Volumes, Crn3/Mole
- A22
XlZ
-208.7681 - 325,7964 - 423.065 -357.8748 - 279.327 -216.966 -116.696 -436.468 -396.248 -419.1093 -202.367 188.7053 -1329.7304 745.400
571.3078 897.1059 1150.147 1229.9084 1098,612 506.093 399.150 1185.436 1307.900 1290.6923 1158.676 -3290.622 1267.75 - 76.970
1006.585 - 35.740
- 182.718 238.011
Parameters, CaI/Mole A12
Methanol-Tyater Ethanol-water 2-Propanol-water 1-Butanol-water 2-Butanol-water Acetone-methanol Acetone-ethanol Acetone-water Butanone-water Methyl acetate-water Furfural-water Methanol-ethanol Ethanol-2-propanol Ethanol-benzene ( P = 750 mm Hg) Ethanol-methylcyclopentane Methylcyclopentane-benzene
- XI1
A12
450.860 747,2929 962.733 1043.374 921.84 408.528 - 84,774 1023.049 1122.348 1290.692 970.618 1,2133 213,857 - 197.128
-211.440 -348.018 -435.954 -369.779 - 279,327 - 235.9306 243.84 -479.035 -435.140 -419.1093 -231.3797 - 7.4205 - 179.812 654.954
-330.439
931 ,4907
- 53,107
167.237
~~
1, O K
System
Table IX.
44,548 63.977 82.48 98.366 85.668 78.47 79.82 81 ,420 89.533 87.373 82.899 42.94 62.82 61.82 61.58 120.66
18.792 18,779 18.77 18.837 18.721 42.07 62.27 18.675 18,633 18.663 18.823 62.39 81.43 94.61 118.82 95.89
1, O K
Error in P Total, %
369.55 368.65 368.32 372.65 364.45 337.15 348,25 361.00 361.00 329.63 371.71 349.75 355,05 342.6
0.3473 0.6506 1.1245 4.1245 1.6216 0.654 0.8046 3,2695 4.1933 3.4580 23.0276 0,4790 0.3871 0.3938
339.45 352.79
2.3874 0.5921
Molar Volumes, Cm3/Mole
- A22
v1
r,
v2
18.791 18.779 18.77 18.837 18.721 42,07 62.27 18.675 18.633 18,663 18.823 62.392 81.43 94,61
44,548 63,977 82,47 98.366 85.688 78.47 79.82 81,420 89.533 87.373 82.899 42,935 62.82 61.23 61,58 120.66
118.82 95.89
OK
Error in P Total, %
369.55 368.65 368.32 372.65 364.45 337.15 348.25 361.00 357.75 329.63 371,71 349.75 355.05 342.60
0.3479 0.7323 1.1650 4,4161 1,7331 0.6566 3.380 4.0037 4.9331 4.4630 23,2258 1.3748 1.372 0.4385
339.45 352.79
2.7651 0.6113
~~
Table VIII. A-Parameters Estimated from Infinite Dilution Values for Activity Coefficients
Acetone-\\ atera 373 15 2-Heptanone-waterb 298 15 1-Butanol-water* 298 15 1-Octanol-waterb 298 15 Acetone-1-octanolb 373 15 l-Butanol-1373 15 octanolc 2-Heptanone-l373 15 octanolb Othmer, 1952* Pierrotti, 1959. c Wilson and Deal, 1962.
V2
Fitting Results for Orye Equations to Binary Systems
Table VII. System
v1
An
Ai2
1 5745 2 1590 1 6545 2 2763 3 6702 3 5548
5228 8669 6425 6137 8224 6639
8 7 7 7 6 6
0 8354
6
2 9 4 11 0 1
1 8025
No. of Iterations
water. Results for acetone and water are compared with the data of Othmer (1952). Phase separation is predicted for 1butanol and water and 1-octanol and water a t approximately the right liquid compositions, as can be seen from the activitycomposition plots. The enthalpic Wilson equation was tested using methanol and water, ethanol and water, and 1-butanol and water. Table IX gives values for (Alz - All) and (A12 - h 2 2 ) computed from ylmand y 2 " . Reasonable agreement exists between values computed from Equation 24 and those given in Table VI. I n Figure 4 a graph is given for binary systems with positive deviations from ideality, from which A 1 2 and Azl follow directly when ylm and 7 2 " are known. The procedure is illustrated for methanol (1) water (2): In ylm = 0.521, In yz" = 0.865. FromFigure 4: A12 = 0.60 (point 1). Interchange of indices gives coordinates of point 2: A21 = 1-00.
+
and Derr
Estimated A Parameters for Enthalpic Wilson Equations (20) from Activity Coefficients at Infinite Dilution (ENTLAM program) System
Aiz
Azi
(A12
- Azz)
(Xi2
1. Methanol-water. (338-72°K) 0.5990 0.9944 - 257.553 2. Ethanol-water. (351-72'K) 0.4832 0.9280 - 364.712 3. 1-Butanol-waterb (372-80°K) 0.3049 1.0054 - 344.979 a Stockhardt and Hull, 1931. Pierrotti, Deal, and Derr correlations, 1959.
- A221
638.017 952.309 1220.01
Temp.,
OK
369.55 368.65 372.65
~~
Ind. Eng. Chem. Fundam., Vol. 9, No. 3, 1970
309
I-Butanol (1)-Water (2) T = 298 O K
A 12" 1.6545
XI
Figure 2. system
-
X,=aons
i,=0.540
xI+
Activity and activity coefficients for the 1-butanol (1)-water
Prediction of Temperature Dependence of ylmand yzm
The temperature dependence of activity coefficients is given by the exact thermodynamic relation
(2
For the EVL equations one obtains by differentiation of Equation 17 and taking the limit zl + 0,
assuming e12 and to be independent of temperature. The enthalpic Wilson equation gives where Hio is the enthalpy of component i in the standard state and Riis the partial molar enthalpy of mixing. At infinite dilution of component i,
310 Ind. Eng. Chem. Fundam., Vol. 9, No. 3, 1970
I\
I
I
1
2 k -t-
12Or-
*
AIZ= 2.2763
I I6137 T = 298'K
A21' ~
XI
Figure 3. system
/
I
System
loo--
\
1
System I-Octanol(l)+Water(2)
-
XI
Activity and activity coefficients for 1 -octanol (1)-water (2)
2.: 8
-x" C
I*.( 1.5
I .c
0.5
0.0 In Yla, for enthalpic Wilson equation from In ylm and In y2m __c
Figure
4. Nomogram giving
.211
and
Ind. Eng. Chem. Fundom., Vol. 9, No. 3, 1970
31 1
The Orye equation yields
gives the predicted temperature dependence of In ylm and water (2) for the EVL, In yz" in the system methanol (1) enthalpic Wilson, and Orye equations. Experimental data from Perry (1963) are also given. It is evident that the predicted temperature dependence is poor for all equations. The enthalpic Wilson equation appears to give results closer to the experimentally observed values for the temperature dependence of In ylm than the Orye equation. It must be admitted that the test of predicting temperature dependence of activity coefficients is extremely severe. The conclusion is that ( ~ I Z- A l l ) and (A12 - A l l ) must have some dependence on temperature.
+
-
Rotation of indices in the sequence 1 2 1 gives similar relations for the temperature dependence of In 72". Figure 5 -)r
WILSON EWATIONS FOR FOUR TERNARY SYSTEMS I' + acetone ; x methanol; w t e r II: acetone ; * methand;. ethonol o ethanol ; 1%-popanol : water E: uethanol ; imethylcyclopentane: o benzene
m:
/
"1I
+
0.6
Yexp.
0.5 -
-
0.4
0.3-
B
I
-
3.0
2.5
4.0
3.5 ( '/T)XIO~
(OK-')
Figure 5. Predicted temperature dependence of In yim for system methanol (1)-water (2)
Ycolc
Figure 7.
-- Enthalpic Wilson equation -*
- O r y e equation
-
Fit of Wilson equation to ternary VLE data
0 Experimental d a t a
_ - _ -EVL equations
ENTHALRC WILSON EOUATONS FOR FOUE TE3hARY SYTER'S 1: + acetone: x methanol; e water II: acetone : tr methanol; ethanol E:o ethanol ; iso-propanol;* water E: ethonol ; * methylcyclopentane : o benzene
li
VAN LA& EQJATIOK FOR FOUR TERNARY SYSTEMS I: + ccetone; x methanol; e w t e r II: acetone ; li methanol: ethanol a: oethanol ; is0 ptopanol;* water o,8- E:o ethanol ; * methylcyclopentone; o benzene @"-
/
0.6
Yexp. 0
0.6 0'7:
0.5 -
0
0.4
// 0 0 6 .
*f 0,
0.2
312
-
*
02
03
0.4
0.5
06
07
08 YCOlC
Figure 6.
-
0.3-
0
0 0
I
Yexp.
09
-
Fit of van Laar equations to ternary VLE data
Ind. Chem. Fundam., Vol. 9, No. 3, 1970
0.1
I0
Figure 8. data
6
I
I
0.2
0.3
0.4
0.5
I
!
06
0.7
I
0.8
I
0.9
I!
-
kale Fit of enthalpic Wilson equations to ternary VLE
I I -
EQUA';!OP.IS
CRYE
FOR FOJR TERNARY SYSTEMS
Table X. Ternary Systems Used to Test Enthalpic Wilson and Orye Equations (Isobaric at 1 Atm)
I. + acetzne; x retnono ; e wo:er G.9 II. ccetoie ; tr rneinmol; ethanol m:oetccnc! : + is0 D ~ O ~ G C O ~wafer :+ O,s-E. oehcnol ; * rnethylcyclopentane; o benzene
System
1. Acetone-methanol-water 2. hcetone-methanolethanol 3. Ethanol-2-propanolwater 4. Ethanol-methylcyclopentane-benzene
0.7tt
G
C.,
G2
0.3
04
05
G.7
06
0.8
0.9
kaic
Figure
9. Fit of Orye equations to ternary VLE data
Multicomponent Systems
The EVL equations (17), enthalpic Wilson equations (20), and Orye equations (21) can be readily extended t o multicomponent systems. I n fact, the quasilattice model of Guggenheim (1944a, b, 1952) was developed for multicomponent systems. I n the zeroth approximation only two-body interactions are taken into account and therefore the Gibbs free energy is obtained by a summation procedure. The EVL equation is not discussed here, as it is clear from Table V that this equation gives rather poor results. The enthalpic Wilson equation for a multicomponent mixture becomes 25 111 (A15Aji) ff4j
xjxm
Reference
-
1n (AjrnArnj) aiim
(31)
a#j#m
Griswold arid Buford, 1949 hmer, Paxton, and Van Winkle, 1956 Kojima, Ochi, and Sakawaza, 1969 Sinor and Weber, 1960
eters A i j and A j i (Tables I11 to VII), and the values obtairied were used in bubble temperature calculatioiis using the programs described by Prausnitz et al. (1967). Pure component properties were t.aken from the literature. Critical pressures and temperatures were taken from the literature or in some cases calculated with Lyderseri correlations (Reid and Sherwood, 1958). Acentric factors, dipoles, and vapor pressure data were taken froni Prausnitz et nl. (1967) and O'Connell and Prausnitz (1964). I n Figures 6 to 9 the computed vapor compositions for a number of liquid compositions for each syst.ern are compared wit,h t,he experimental compositions as reported in the references given in Table IX. As a reference, the results for the van Laar and Wilson equations for the same four ternaries are also given. Conclusions are that the enthalpic Wilsori and the Orye equations give a considerably better fit of vapor-liquid equilibrium in ternary systems than the van Laar equat'ions. The van Laar equation gives a good fit for t'he acetone-methanolwater system, but poor results for the ethanol-methylcyclopentane-benzene and ethanol-2-propanol-water syst.ems. The Wilson equation gives a very good fit of VLE data for the ethanolmethylcyclopentane-benzene system. The results for the ethanol-2-propanol-water syst.ern, however, are approximately of the same accuracy as t'he enthalpic Wilson and Orye equations. For syst'ems near to phase separation or for systems showing phase separation, the enthalpic or Orye equations are recommended. For other systems, the Wilson equations are attractive to use. Acknowledgment
The Orye equations for a multicomponent mixture are In Y i
=
1 - In
(,
)-Z($q-
Aipxp
The author is grateful to the Computer Center of the Eindhoven University of Technology for the use of its facilities. Especially the assistance of Marijke ter Morsche is gratefully acknowledged. Literature Cited
Amer, H. H., Paxton, R. R., Van Winkle, RI., Znd. Eng. Chem. 48. 142 f18.56). \ - - - - , -
Altsib&va, A. I., Belousov, V. P., Russian J . Phys. Chem. 38 ( 5 ) , 676-8 (1964). Ballard, L. H., Van Winkle, >I., Znd. Eng. Chem. 44, 2450 (1952). Black, C., Znd. Eng. Chem. 51, 211 (1959). Black, C., Derr, E. L., Papadopoulos, 31.N., Znd. Eng. Chem. 55 f81. 40 ~.,, ~.il863a1. ~~~..~.,.
Equat>ions31 and 32 were tested on four ternary systems listed in Table IX. Binary VLE data were reduced to param-
Black, C., Derr, E. L., Papadopoulos, hI. N., Znd. Eng. Chem. 55 (9), 38 (1963b). Bruin, S., doctoral thesis, Agric. University Wageriirigen; Publication 48, L. E. B. Foundation, 1960. Carey, J. S., Lewis, W. K., Znd. Eng. Chem 23, 882-83 (1932). Flory, P. J.,J. Chem. Phys. 10, 51-61 (1942). Griswold, J., Buford, C. B., Znd. Eng. Chern. 41, 2347 (1949). Griswold, J., Ludwig, E. E., Ind. Eng. Chem. 35, 117 (1043). Guggenheim, E. A., Proc. Roy. SOC.A 148,304-12 (t935). Guggenheim, E. A., Proc. Roy. SOC.A 183,203-12 (1944a). Ind. Eng. Chem. Fundam., Vol. 9, No. 3, 1970
313
Guggenheim, E. A,, Proc. Roy. SOC.A 183,213-27 (1944b). Guggenheim, E. A., “Mixtures,” Chap. XI, Clarendon Press, London, 1952. Hellwig, L. R., Van Winkle, hl., Znd. Eng. Chem. 45, 624 (1953). Helpinstill, J. G., Van Winkle, M., Ind. Eng. Chem. Process
Pierrotti, G. J., Deal, C. H., Derr, E. L., Znd. Eng. Chem. 51 ( l ) ,
Hildebrand, J. H., Scott, R. L., “Solubility of Non-electrolytes,” 3rd ed. 133, Dover Publications, New York, 1950. Huggins, I%., Ann. N . Y . Acad. Sci. 43, art. 1, 1-32 (1942). International Critical Tables, McGraw-Hill, New York, 1928. Kojima, K., Ochi, K., Nakawaza, Y., Intern. Chem. Eng. 9 (2),
Rejdy R , C., Sherwood, T. K., “Properties of Gases and Liquids,” pp. 7-25, PvlcGraw-Hill, New York, 1958. Renon. H.. Prausnitz, J. hI., A.Z.Ch.E. J . 14 ( l ) ,135-44 (1968). Sinor, J. E’., Weber, J: H., J : Chem. Eng. Data 5,’244 (1960). Stockhardt, J. S., Hull, C. M., Znd. Eng. Chem. 23, 1438-40
Design Develop. 7, 213-20 (1968).
342 (1969).
Merquardt, D. W., Chem. Eng. Progr. 55 (6), 65-70 (1959). Marshall, N., J . Chem. SOC.89, 1350 (1906). Nagel, O., Sinn, R., Chem. Ing. Techn., 38 (lo), 265 (1966). Nagel, O., Sinn, R., Chem. Zng. Techn. 39 ( l l ) , 671-6, 275-82 (1967).
Neretnieks, I., Ind. Eng. Chem. Process Design Develop. 7, 335-9 (1968).
O’Connell, J. P., Prausnitz, J. M., IND. ENG.CHEM.FUNDAM. 3, 347 (1964).
O’g% R. V., Ph.D. dissertation, University of California, erkeley, Calif., 1965. Orye, R. V., Prausnitz, J. M., Znd. Eng. Chem. 57 ( 5 ) , 18 (1965). Othmer, D. F., Znd. Eng. Chem. 44, 1872 (1952). Othmer, D. F., Chudgar, M. M., Levy, S. L., Znd. Eng. Chem. 44, 1872-8 (1952). Perry, J., ed., “Chemical Engineer’s Handbook,” 4th ed., Chap. 13, New York, 1963.
95-102 (1959).
Prausnitz, J. R.I., Eckert, C. A., Orye, R. V., O’Connell, J. P., “Com uter Calculations for Multicomponent Vapor-Liquid Equiligria,” Chap. 1-6, Prentice-Hall, Englewood Cliffs, N. J., 1m
7
’
(1931).
Tyrer, D., J . Chem. SOC.,101, 1104 (1912). Uchida, S., “Chemical Engineer’s Handbook,” J. Perry, ed., Chap. 13, 4th ed., McGraw-Hill, New York, 1934. Jchida, S.,. Ogawa,’S., Yamagusti, M., J CIpan. Sci. Rev. Ser. Z. Eng.’Sci,’ 1,No. 2, 41 (1950):
’
.
Wilson, A., Simons, E. L., Znd. Eng. Chem. 44,2214 (1952). Wilson. G. Pvl.. J . Amer. Chem. SOC.84. 127-33 (1964a). Wilson: G. M.; J . Amer. Chem. SOC.84; 133-7 (1964b).’ Wilson, G. >I., Ileal, C. H., IND.ENG.CHEWFUNDAM. 1, 20-3 (1962).
Wohl, K., Trans. Amer. Inst. Chem. Engrs. 42, 215 (1946). RECEIVED for review December 23, 1968 ACCEPTEDMay 1, 1970 Work supported by the L. E. B. Foundation, Wageningen.
Thermal Reaction of Propylene Kinetics Taiseki Kunugi, Tomoya Sakai, Kazuhiko Soma, and Yaichi Sasaki Department of Synthetic Chemistry, Faculty of Engineering, University of Tokyo, Hongo, Tokyo, J a p a n
Kinetics of the thermal reaction of propylene was studied at temperatures ranging from 703’ to 854’C., atmospheric pressure, and residence times from 0.078 to 3.3 seconds with and without nitrogen dilution. Main primary products were ethylene, methane, hydrogen, butenes, and butadiene in the approximate ratio of 5 : 3 : 1 : 1 : 1 at initial stages of the reaction. Other primary products were methylcyclopqntene, hexadienes, acetylene, and ethane. Secondary products were cyclopentadiene, benzene, polycyclic aromatics, cyclopentene, and toluene. Selectivities of formation of these products, except acetylene and ethane, showed little dependence on temperature. At higher partial pressure of propylene, the selectivities of ethylene and methane formation decreased to some extent. The effect of partial pressure of propylene on the rate of propylene disappearance leads to a three-halves-order equation. The rate constant is given as
k = 10’5*06
of propylene has been studied extensively (Amano and Uchiyama, 1963; Kallend et aZ., 1967; Laidler and Wojciechowski, 1960; Sakakibara, 1964; Wheeler and Wood, 1930). A few experiments at temperatures from 700’ to 850’ and atmospheric pressure cover the conditions of the industrial manufacture of olefins and aromatics by cracking hydrocarbon feedstocks. Analyses of the products have been limited to lighter hydrocarbons below CCor the products formed in narrow ranges of temperatures and conversions. Lack of clear discrimination between the primary and secondary products is due t o these limited analyses of the products. T H E R M A L REACTION
3 14
Ind. Eng. Chem. Fundam., Vol. 9,
No. 3, 1970
cc>/z/(mote’/z sec.).
Reaction products were analyzed in detail to differentiate the primary from the secondary products. The kinetics of the thermal reaction of propylene was discussed in comparison with that of ethylene. I n a following paper mechanisms of the reaction and of formation of higher hydrocarbons above CS are to be discussed. Experimental
Feed propylene was 99.35 mole % pure by gas chromatographic analysis, used without further purification. The impurity was propane, 0.65 mole yo.Oxygen content was less than 1 p.p.m. by weight. Commercially available nitrogen
