w = capacity ratio in batch stirred-tank experiments, VIVRK (-) z = dimensionless diffusivity parameter, Equation 9 (-) y = dimensionless time parameter, Equation 10 (-) 2
= column length, em
GRCCKLCTTCRS P = variable of Equation 13 (-) t =
bed voidage (-) time parameter, Equation 4 ; meclianism parameter (-) superficial residence time, Equation 2, see or mill
.r= dimensionless BR =
x= P = lJ= 7
=
4
liquor viscosity, g/cm/sec dimeiisionless resistance parameter, Equation 11 (-) dimensionless time parameter, Equation 5 (-)
literature Cited
Andrus, G. AI.,Sugar Azucar, 62 ( 5 ) ,54 (1967). Anzelius, A., A . Angew. Moth. Mech., 6 , 291 (1926). Bagster, D. F , Znt. Sugar J . , 72, 134 (1970a). Bagster, L). F., Znf. Sugar J . , 72, 200 (1970b). Bird, li. B., Stewart, W. E., Lightfoot, E. N., “Transport Phenomena,” Wiley, New York, X.Y.) 1960, p 707. Brown, R. L., Richards, J. C., “Principles of Powder Alechanics,” Pergamon, Oxford, England, 1970, p 16. Carberry, J. J., AZChE J., 6,460 (1960). Dietz, V. It., Pennington, N. L., Hoffman, II. L., J . Res. -Tat. Bur. Stand., 49 (6), 365 (1952). Farber, L., LlcDonald, E. J., Carpenter, F. G., Proc. Tech. Session Cane Sugar Refining Research, p 85, 1969.
Gaffney, B. J., Drew, T. B., Znd. Eng. Chon., 42, 1120 (1950). Hiester, N. K., €{adding, S.B., Nelson, 11. L., Vermeulen, T., AZChE J . , 2 , 404 (1956). IIelfferich, F., “Ion Exchange,” McGraw-Hill, New York, N.Y., 1962. n 385. Hougen,‘O. A , , Watson, K. A I . , “Chemical Process Principles,” Part 3, Wiley, Xew York, N.Y., 1966, p 1087. Johnstone, 11. E., Thriiig, 11. W., “Pilot Plants, Models and Scale-up Methods in Chemical ‘Engineering,” lIcGraw-Hill, S e w Tork, S.Y.,1957, p 65. Kuiiin, I Chcm. Eng. Progr. Sfyrnp. Scr., 5 5 124). 71 (19.59). Oliiey, I-
. 2.4
h
f 1.7
c
5 .-.u
=
.-a
t
y3
si 1.6 1 V
2.2
:
0
2.0 .-La .a
g
4
Benzene(2)f n-Heptane(3)at 45°C
t
I
.-.-.a 1.5 5
5
1
1.8
2 1.4
1.6
1.3
1.4
1.2
1.2
1.1
1.0
1.0 0 I
I
I
I
I
.I
.2
.3
.4
.5
I
.6
I
.7
I
.a
I
.9
I
t
. I .2
1
.3
l
.4
XI
Figure 1. Correlation of acetonitrile coefficients
I
I
.5
.6
,
.7
I
.a
,
.9
x2
+ benzene activity
Figure 2. Correlation of benzene coefficients
+ n-heptane activity
and Table 1. Entropy Weighting Function Parameters
-
Pure-component parameters
where
and
Component
Pi/, atm
anii“,cal/mol
Acetonitrile Benzene n-Heptane
0 2748 0 2940 0 1502
7691 7817 8429
Mixture
Acetonitrile benzene 0 6644 Acetonitrile n-heptane 0 5004 Benzene n-heptane 0 2974 Obtained from a fit of the GEvalues only.
+
1Iaking the further definitioiis that
+ +
Mixture parametersa PI*’, atm AHlt’, cal/mol
6678 9 8165 3 7604 2
Activity Coefficient Prediction
and
the binary GE equation becomes
The biliary activity coefficient equations are
a nd
In these equations, there are tn-o parameters rvhich must be found from mixture data. They are P12’ = Pz1’and AH1?’ 3 A H2i”. 1 16 Ind. Eng. Chem. Process Des. Develop., Vol. 1 1, No. 1 , 1972
The act,ivity coefficient equations were obtained by differentiating Equation 19 with respect to the mole numbers. The correct,ness of the GE vs. composition relationship in Equation 19 can be checked by fitting GE data to obtain Pi?‘ and AHl2’ values and then using those values to calculate the activity coefficient,s with Equations 20 and 21. Table I lists both the pure-component parameters, Pii‘ and aHiiV, and the mixture parameters, PB’ and AHla”. The latter were obtained from a fit of the GE values for each binary. I n the GEfits for t,hetwo miscible binaries, zero weights were assigned to points below x = 0.15 and above 2 = 0.85 where the experimental errors are the largest. The predicted activity coefficient curves, using the parameters from Table I, are compared with the experimental values in Figures 1-3. A similar prediction using the Van Laar equations is shown in Figure 3 by the dashed curves. The reproduction of the y values was excellent for t,he tn-o miscible binaries. Xlthough Equations 20 and 21 did not reproduce the esperimental values exactly for the partially miscible syst’ems,they performed much better than t’heVan Laar equations.
26
240
24
220
22
m
I
Acetonitrile (1)+ n-Heptane (3) at 45°C
4
180
20
-a -mu 160
18
\
-140
16
I 'FI
14
Yi
120
W
u
12
100
10
80
8
60
-
Local Composition Predictions
40
Van Laar Predictions
20
Y
I
1
I
I
I
- 0 2 .04
.M .08
-
1
. 10 "
1
.92
.96
1
,98
XI
Figure 3. Comparison of local-composition and Van Laar correlations of acetonitrile n-heptane activity coefficients
Figure 5. Correlation of n-heptane at 45°C
+
GX
and
HE
d In Plz' dPlz'/dT AHIS" dT P12' RT2 ~
200-
-
HE
= 21x2 [S2i(AHiiY
x1
W
60
40 20 ~~
.1
.2
.3
.4
.5
.6
.7
.8
.9
X1 Figure 4. Correlation of benzene a t 4 5 ° C
G E
and
HE
(22)
Equation 22 has assumed t h a t the vapor is a n ideal gas and that t,he liquid volume is negligible in comparisoii to the vapor volume. T h e resulting H E equation has three unknown parameters, P12', AHl2', and (dhH12i')ldT:
Acetonitrile (1) + Benzene (2) at 45°C
180
+
data for acetonitrile
data for acetonitrile
+
- AHn") + On(AH22" x2
X2ez1
+ -zlelzAHIz?
-
(23) If Equation 19 relates GE to the temperature correctly, i t should be possible to determine PI*', AHlz", and cl(AH12")ldT values from GE data a t two or more temperatures and then predict the H E data. This possibility is under investigation, and the results will be reported a t some later date if they are significant. It is not possible to fit H E dat'a a t one temperature and use the P12'and aHl2' thus obtained to predict the GE data. T h e GE and HE data can be correlated simultaneously using Equations 19 and 23. The parameters were found by 11011linear regression analysis minimizing the objective function
Excess Enthalpy and Volume Equations
The equation for the excess enthalpy is derived from the GE expression by the appropriate differentiation. T h e temperature derivatives of 2 1 2 and zZ1are found b y differentiation of Equations 13 and 14. This introduces the temperature derivatives of P12' and ilpplication of the Clausius-Clapeyron equation t o the hypothetical mixture permits substitution for d P l z ' / d T in terms of AHlzV.
+
2n (24) The simultaneous correlation of the GE and H E data for the acetonitrile benzene system is shown in Figure 4. Thia system is difficult to fit because the GE data are almost symmetrical while the HE data are very skewed. T h e simultaneous correlation of the G E and HE data for the acetonitrile n-heptaiie system is shown in Figure 5.
+
+
Ind. Eng. Chem. Process Des. Develop., Vol. 1 1, No. 1 , 1972
1 17
Benzene (2) t n-Heptane (31 at 45°C
Acetonitrile (11 + Benzene (2) + n-Heptane (3) at 45°C
I
w
,I
.*------
-L
--- - -_
\
I-
Figure data
7. Prediction of solubility envelope from binary
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 x2 Figure 6. Correlation of n-heptane at 45°C
G E
and
HE
data for benzene
+
Data such a s these are very difficult t o fit, and to obtain a perfect fit, i t would be necessary to allow the pure-component parameters t o be variables obtained from the mixture data. The benzene n-heptane data correlation was good as shown in Figure 6. Recent published attempts to correlate, simultaneously, H E and GE (or activity coefficient) data include those of Orye (1965) and Lowell and Van Winkle (1970). Orye attempted the prediction of H E from GE data using the Wilson equation and a modification of it. His prediction of the H E curve for the acetonitrile benzene system at 45°C was not very good. The predicted curve was almost symmetrical whereas the data are very skewed. Lowell and Van Winkle used a twosuffix Rlargules equation with one temperature coefficient to correlate activity coefficients a t two temperatures and predict the excess enthalpy for alcohol ethyl acetate systems. No data reduction procedure directly analogous to the simultaneous correlation of GE and H E data shown in Figures 4-6 is available for comparative purposes. An equation for the excess volume can be obtained by differentiation of the GE equation with respect to pressure. However, two new unknown parameters, the molar volume and the thermal expansivity of the hypothetical 1-2 fluid, are introduced. The simultaneous correlation of GE and V E data involves four unknown parameters, and the simultaneous correlation of H E and V Edata involves five. The probability of accurately predicting GP data from H E and V E data a t one temperature is small.
+
+
+
Multicomponent Equations
The biggest advantage of local composition equations over most of the common algebraic expansions is that they can be extended t o multicomponent properties without introducing new constants dependent on multicomponent data. 1 18
Ind.
Eng. Chem. Process Des. Develop., Vol. 1 1,
NO. 1, 1972
The multicomponellt form for the excess free energy is
.shere $t, arid 7i1 are as originally defined. The activity coefficient equatioii is In y r
=
These equations have the same form as those derived by Renoii and Prausnitz (1968), but the parameters are evaluated differently. Prediction of Ternary GE Values. An excess free energy prediction was made for each of the 51 ternary data points measured by Palmer and Smith (1971) for the acetonitrile benzene n-heptane system. The parameters were obtained in various ways to study the effect of the data base 011 the predict ion. When tie lines alone were used to obtain the six u n k i i o m parameters, the RNSD for the ternary G E values was 82.1 cal which was definitely unacceptable. When the binary GE data were used to obtain the sis correlation constants shown in Table I, the RAISD for the predicted ternary GE values was 9.2 cal/mol. This is roughly 3% of the average GE value. The same binary data were used to find the NRTL equation parameters. The parameter cy was allowed to be a variable, so there rvere nine constants for the ternary system. The prediction of ternary GEvalues was not as accurate as with the entropy a eighting function approach, the RAISD being 25.7 cal. When solubility data were used to obtain the entropy weighting function correlation Constants for the partially miscible binary, the ternary prediction gave the same RNSD of 9.2 cal as when the binary G E data were used. This is an important result because solubility data are far more easily measured than GEdata for partially miscible mixtures.
+
+
16
i
m
J41
E
.12
Acetonitrile (1) + Benzene (2) + n-Heptane (3) at 45°C
0 Palmer (1971)
Nomenclature
G H
Gibbs free energy (cal/mol) enthalpy (cal/mol) AHV enthalpy change of vaporization P‘ vapor pressure, a t m R gasconstant RMSD root mean squared deviation S = entropy, cal/mol “K T = temp,OK V = volume, cc/mol 2 % = mole fraction in liquid = local composition of i in fluidj
/ 4 I
.UL
Figure 8.
in the region of the plait point. This prediction XLS made with the same information used to predict the solubility envelope in Figure 7 .
.04
I
I
I
I
I
I
I
.06 .08 ,IO .12 ,I4 ,I6 .18 x2 (Heptane P hasel
Prediction of distribution curve from binary d a t a
When both the GEand the H E data were used, as in Figures 4-6, the ternary predictions had a n RMSD of 12.7 cal. This result was not quite as good as when the binary parameters were based only on G Edata. However, these results were very encouraging because they indicated that it may be possible t o predict multicomponent activity coefficients in highly nonideal systems a s a function of both temperature and composition. For a ternary mixture, six binary constants and three temperature derivatives must be found from the binary data t o permit such predictions. Prediction of T i e Lines. An algorithm developed by Xu11 (1970) has been used with the multicomponent activity coefficient correlation t o predict the solubility envelope and tie lines for the acetonitrile benzene n-heptane system. T h e prediction of the solubility envelope shown in Figure 7 was based only on the solubility data for the partially miscible binary, and GE and H E d a t a for t h e miscible binaries. There was very little change when only GE data were used t o specify the parameters for the miscible binaries. T h e prediction of a solubility envelope is often difficult because of the flatness of the activity surfaces in t h a t region, particularly around the plait point. Small errors in the predicted y z values can cause a large horizontal displacement in the calculated solubility envelope. T h e predicted distribution of benzene between the extract and raffinate phases, shown in Figure 8, was accurate except
+
+
= = = = = =
GREEKLETTERS O(
=
N R T L equation parameter
y = activity coefficient
e
= = @ =
T~~
local-composition weighting function local-composition energy parameter local-composition weighting function
SUPERSCRIPTS
’ = denotes a configurational property
E
=
denotes an excess property
literature Cited
Holmes. 31. J.. Van Winkle, IT.,Ind. Eng. Chem., 62 (I), 21 (1970). Lowell, P. S., Van Winkle, hI., Ind. Eng. Chem. Process Des. Develop., 9 (2), 289 (1970). Null, H . R., “Phase Equilibrium in Process Design,” p 211, Wiley, New York, N.Y., 1970. Orye, R. V., PhD dissertation, University of California (Berkeley), (1965). Orye, R. ST., Prausnitz, J. Ind. Eng. Chem., 57 ( 5 ) , 18 (1967). Palmer, D. A., DSc dissertatlon, Washington University, (1971). Palmer, D. h.,Smith, B. D., J . Chem. Eng.Data, 17, 71 (1972). Renon, H., Prausnitz, J. SI., AIChE J . , 14, 133 (1968). Scott, R. L. J., J . Chem. Phys., 2 5 , 193 (1956). Wilson, G. >I., J . Amer. Chem. Soc., 8 6 , 127 (1964). RECEIVED for review March 26, 1971 ACCEPTEDJuly 2, 1971 Dr. Palmer was the American Oil Foundation Design Fellow at Washington University for three years. Research was sup-
ported by the Fellowship, by National Science Foundation Grant GK-1971, and by industrial participation fees in the laboratory. Assistance was also received from the Washington University Computing Facility through National Science Foundation Grant G22296.
Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 1 , 1972
1 19
