22
Ind. Eng. Chem. Process Des. Dev. 1966, 25, 22-31
Prediction of Enthalpies of Mixing with a UNIFAC Model Dlnh Dangt and DlmHrlos P. Tassios” New Jersey Institute of Technology, Newark. New Jersey 07102
A rnodkd UNIFAC model is used to correlate and predict the enthalpies of mixing of organic mixtures. This model requires two interaction parameters per group pair, and values for 172 such pairs are presented here. Excellent predictlons for both binary and mutticomponent systems are obtained. Typical errors, in the temperature range of 273-400 K, are less than 15 % . For systems containing cyclic compounds, somewhat poorer results may be encountered.
Knowledge of enthalpies of mixing ( W is ) important in several chemical engineering applications such as distillation or heat-exchanger design. Even though such data are available for a variety of systems, most likely binary mixtures (Christensen et al., 1982), a prediction scheme is required in the typical case. Several studies have attempted to predict enthalpies of mixing, from vapor-liquid equilibrium data (Nicolaides and Eckert, 1978; Nagata and Yamada, 1972) or from group contribution methods (Nguyen and Ratcliff, 1974; Siman and Vera, 1979). The second approach is of course, much more attractive because it can be applied to a large variety of binary and multicomponent mixtures. Rupp et al. (1984) compared three group contribution models in the correlation and prediction of enthalpies of mixing: the AGSM model of Den and Deal (1969) adopted for enthalpies of mixing by Ratcliff and his co-workers (1974), the UNIFAC model of Fredenslund et al. (1975), and the modified UNIFAC model of Skjold-Jorgensen et al. (1980). The first two models used temperature-dependent interaction parameters, while the third model contains temperature-independent interaction parameters with a universal temperature dependency for the coordination number 2. Rupp et al. concluded that the third model, referred to hereafter as the UNIFAC model, provides the best correlation and prediction results. They presented the values for 19 pairs of primary (CH,/G) interaction parameters. In the temperature range of 0-100 OC, the typical correlation and prediction results are between 5% and 15%. Poorer results, however, were observed for systems where strong hydrogen bonding is present (alcohols, organic acids). Typical errors are in the 10-30% range. For systems containing alcohols, Stathis and Tassios (1984) proposed the UNIFAC/Association model by including a “chemical contribution” term that accounts for the alcohol polymerization. With this model, they improved significantly the correlation and prediction results for systems containing alcohol compounds. Typical errors are in the range 5-15%. In this study, we present the interaction parameters for 136 new group pairs. Finally, to make this contribution self-contained, we also include the interaction parameter values of Rupp et al. and Stathis and Tazlsios. A brief description of the UNIFAC/Association model is presented in the Appendix section. The Model The enthalpy of mixing, AHm,can be calculated from the excess Gibbs free energy, GE,by using the relationship Engelhard Specialty Chemicals Division, Menlo Park CN28, Edison, N J 08818. 0196-4305/86/1125-0022$01.50/0
Equation 1,in conjunction with the UNIFAC model, yields the following expressions for the enthalpy of mixing:
Exi=
AHm =
i
(3)
a
where x i = liquid mole fraction of component i, = partial molar excess enthalpy of component i, Nki = number of groups of type k in component i, Hk = excess enthalpy of group k, & = same as Hk but in a reference solution containing only molecules of type i, Qk = area parameter of group k, Om = area fraction of group m, om = Q m x m / ( C Q n x n ) (5) n
X , = mole fraction of group m in the mixture CXiNmi
(7)
2 = temperature-dependent coordination number given
by Z = Z(T) = 35.2 - 0.1272T
+ 0.00014P
(9)
amn = temperature-independent interaction parameter between groups m and n. Results and Discussion Table I lists the group classification and their corresponding surface area parameters used in this study. Subgroups within a main group usually have various degrees of substitution at the carbon atom. For example, the CH2group is divided into four subgroups: CH,, CH,, CH, and C. The interaction parameters of the subgroup with 0 1985 American
Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 25,
Table I. G r o w Classification and Q Values main subgroup Qk example group CH3 0.848 ethane: 2CH3 CHZ CHZ 0.540 propane: 2CH3, lCHz CH 0.228 isobutane: 3CH3, 1CH C 0.000 2,2-dimethylpropane: 4CH3, 1C C=C CHz=CH 1.176 propene: lCHz=CH, lCH, CHz=C 0.988 2-methylpropene: lCHz=C, 2CH3 CH=CH 0.867 2-butene: lCH=CH, 2CH3 CH=C 0.676 2-methyl-2-butene: lCH=C, 3CH3 C=C 0.485 2,3-dimethyl-2-butene: 1C=C, 4CH3 c=c CHsC 1.088 propyne: lCH=C, lCH3 CEC 0.784 2-butyne: 1C=C, 2CH3 ACH ACH 0.400 benzene: 6ACH OH OH 1.200 propanol: lCH3, 2CHz, 1 0 H CHZCO CH3C0 1.488 acetone: lCH,CO, lCH, CHzCO 1.180 3-pentanone: lCHzCO, 2CH3, lCHz CHO CHO 0.948 acetaldehyde: lCHO, 1CH3 COOC CH3CO0 1.728 methyl acetate: 1CH3C00, lCH3 CHzCOO 1.420 methyl propanoate: lCHzCOO, 2CH1 CHzO CH30 1.088 dimethil ether: 1CH30, lCH, CHpO 0.780 diethyl ether: lCHzCO, 2CH3, lCHz CHO 0.468 diisopropyl ether: lCHO, 4CH3, 1CH COOH COOH 1.224 acetic acid: lCOOH, lCH3 HCOOH 1.532 formic acid: lHCOOH CHzNHz CHSNHZ 1.544 methylamine: 1CH3NHz CHzNHz 1.236 ethylamine: lCHzNHz,lCH3 CHNHp 0.924 isopropylamine: lCHNH,, 2CH3 CzNH CHBNH 1.244 dimethylamine: lCH,NH, lCH3 CHzNH 0.936 diethylamine: lCHzNH, 2CH3, lCHz CHNH 0.624 diisopropylamine: lCHNH, 4CH3, 1CH CH3N 0.940 trimethylamine: 1CH3N, 2CH3 C3N CHZN 0.632 triethylamine: lCHzN, 3CH3, 2CHz CHZCN CH3CN 1.724 acetonitrile: 1CH3CN CHzCN 1.416 propionitrile: lCHzCN, lCH3 CHzNOz CH3NOZ 1.868 nitromethane: 1CH3NOz CHzNOz 1.560 nitroethane: 1CHzNO2,lCH, CHNOz 1.248 2-nitropropane: 1CHNOZ,2CH3 c1 c1-1 0.720 1-chloropropane: 1Cl-1, 1CH3, 2CHz c1-2 0.728 2-chloropropane: 1C1-2, 2CH3, 1CH CHzClz CHzClz 1.988 dichloromethane: 1CH2& CHCl, CHC1, 2.410 chloroform: lCHCl, CCl, CCl, 2.910 tetrachloromethane: 1CC1, Br Br 0.832 bromoethane: lBr, lCH,, lCHz MezSO MezSO 2.472 dimethyl sulfoxide: lMezSO pyridine pyridine 2.113 pyridine: lpyridine ACF ACF 0.524 hexafluorobenzene: 6ACF ACHz ACCH3 0.968 toluene: 1ACCH3, 5ACH ACCHz 0.660 ethylbenzene: lACCHz, lCH,, 5ACH ACCH 0.348 isopropvlbenzene: 1ACCH. 2CH1. 5ACH ACNOz ACNOz 1.104 nitrobenzene: lACNO,. 5ACH ACCl ACCl 0.844 chlorobenzene: lACC1:’5ACH ACNHZ ACNHz 0.816 aniline: lACNHz, 5ACH ACBr ACBr 0.952 bromobenzene: lACBr, 5ACH
the other functional groups are equal to those of the main group. The interaction parameters among subgroups are assumed equal to zero. To enhance the reliability of the interaction parameters values, we used all available experimental data for a given interaction parameter pair in their determination. The interaction parameters are obtained by using the minimization function
No. 1, 1986 23
Figure 1. Evaluated interaction parameters. 0
rl a,
P
-1000
\
a,
3 2
7
m
.x.
.d
uVI . i
a
ri
2
-2000
Y
W
-3000 x1
Figure 2. Correlation results for the system (1)CHC13 + (2) dibutyl ether: T = 25 OC; (-1 correlation; ( 0 )experimental data (Beath and Williamson, 1969).
where V is the number of data points. Figure 1 shows a matrix of the available interaction parameters including those of Rupp et al. (1984) and of Stathis and Tassios (1984) for a total of 172 pairs. The corresponding values for these interaction parameters are presented in Table 11. Water is not, unfortunately, included because it requires a model that accounts for both association and solvation effects. Correlation results are very good with errors less than 10% in most cases. Detailed results are given by Dang (1984), and typical cases are shown in Figures 2 and 3 and in Table 111. In the latter, the values of the average absolute error, S ,
24
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 1, 1986
Table 11. Group Interaction Parametersa n
m CHz C=C
c=c
ACH OH CHzCO CHO COOC CHzO COOH CHZNHZ CzNH C3N CH&N CHzNOz c1 CHzClz CHC13 CCl, Br
MezSO pyridine ACCHz ACNOz ACCl ACF ACNHz ACBr CHZ
c=c c=c
ACH OH CHzCO CHO COOC CHzO COOH CHZNHp CpNH C3N CH,CN CHZNOp c1 CHpC12 CHC1, CCl, Br MezSO pyridine ACCHz ACNOp ACCl ACF ACNHz ACBr CHZ
c=c c=c
ACH OH CHZCO CHO COOC CHZO COOh CH,NH, CzNH C3N CHZCN CHZNO,
c1
CH2C12 CHC13 CCl,
CHZ 0.00 -1.47 6.75 1.90 99.57 12.35 -19.35 114.10 14.99 17.25 -5.88 8.16 '66.91 4.54 3.01 -5.70 -0.39 -2.78 1.79 433.90 88.35 -3.24 -6.77 0.44 40.68 8.35 89.81 -10.29 81.31 n.a. n.a. 53.44 47.97 n.a. n.a. n.a. n.a. n.a. 0.00 -0.96 10.29 -17.78 n.a. n.a. n.a. 19.02 n.a. n.a. n.a. n.a. n.a. n.a. 145.80 n.a. n.a. n.a. 65.92 n.a. n.a. 8.26 n.a. n.a. n.a. n.a. n.a. n.a. n.8. n.a. n.a. -4.00 n.a. n.a. -2.20 21.32 -12.80
C=C 7.67 0.00 n.a. -20.51 70.81 n.a. 31.84 n.a. -2.97 n.a. n.a. n.a. n.a. n.a. n.a. n.a. 7.89 -3.37 -17.27 n.a. n.a. n.a. -19.22 n.a. n.a. n.a. n.a. n.a. 50.02 n.a. n.a. 5.49 -65.03 n.a. n.a. n.a. n.a. n.a. -1.04 0.00 n.a. n.a. n.a. n.a. n.a. -82.98 n.a. ma. n.a. -12.72 n.a. n.a. 1.06 n.a. n.a. n.a. 38.83 n.a. n.a. 11.78 n.a. n.a. n.a. n.a. -15.34 -24.09 n.a. 35.72 n.a. n.a. n.a. n.a. -7.33 -13.10 42.57
CzC 51.53 n.a. 0.00 -6.05 n.a. n.a. n.a. n.a. -25.08 n.a. n.a. n.a. n.a. n.a. n.a. n.a. n.a. n.a. 17.71 n.a. n.a. n.a. n.a. n.a. n.8. n.a. n.a. n.a. 22.65 n.a. n.a. -5.95 -41.07 n.a. n.a. n.a. -12.82 n.a. 0.96 n.a. 0.00 n.a. n.a. n.a. n.a. -63.88 12.55 n.a. n.a. n.a. -41.80 n.a. -14.89 31.05 n.a. n.a. 19.48 22.21 n.a. -0.70 -26.88 -15.96 n.a. 100.59 75.10 ma. n.a. n.a. 385.30 17.19 3.90 32.47 n.a. ma. ma.
ACH 12.09 54.76 130.54 0.00 168.60 -8.59 -21.09 41.95 -33.63 1.65 16.97 11.43 10.11 15.58 -18.33 521.01 12.86 4.81 -10.62 3.16 -9.72 16.39 11.83 2.57 13.42 65.50 1.01 68.87 n.a. n.a. 1.62 10.30 9.74 n.a. n.a. n.a. 5.08 36.32 ma. n.a. 0.00 -0.26 n.a. n.a. -25.28 46.48 -23.53 501.17 n.a. 1.56 n.a. 31.20 n.a. n.a. n.a. 72.71 n.a. n.a. 20.04 n.a. n.a. n.a. n.a. n.a. n.a. n.a. n.a. n.a. n.a. ma. n.a. n.a. n.a. n.a.
OH 545.77 253.99 n.a. 365.40 0.00 343.15 38.98 291.46 2.10 n.a. -16.87 -59.25 -23.76 9.23 n.a. 261.28 n.a. 124.10 67.18 240.30 n.a. n.a. 45.19 n.a. 52.83 n.a. n.a. 52.10 80.05 n.a. n.a. 4.42 n.a. 17.40 n.a. ma. n.a. n.a. n.a. n.a. n.a. -2.99 0.00 n.a. n.a. n.a. 338.02 n.a. n.a. n.a. 27.93 n.a. 2.28 n.a. n.a. n.a. -16.95 n.a. n.a. 6.21 -39.04 142.20 n.a. n.a. 144.20 n.a. -31.70 -7.96 28.77 -3.37 143.57 126.51 n.a. 0.55 9.96
CHZCO 52.25 n.a. n.a. 23.28 350.93 0.00 n.a. 23.72 11.54 n.a. -37.20 n.a. n.a. -12.24 -28.20 -4.81 1.13 33.14 64.17 75.95 n.a. n.a. 43.50 n.a. -9.58 n.8. -41.53 n.a. 74.52 n.a. n.a. 27.19 64.12 -4.73 -11.19 ma. -20.10 n.a. n.a. n.a. n.a. n.a. n.a. 0.00 -7.27 n.a. 24.75 -14.99 n.a. n.a. -3.51 n.a. 14.20 n.a. n.a. n.a. 2.23 n.a. n.a. -15.54 n.a. ma. n.a. n.a. 6.75 n.a. n.a. n.a. 194.20 n.a. n.a. n.a. n.a. n.a. 26.31
CHO 106.88 245.79 n.a. 19.79 34.60 n.a. 0.00 n.a. 13.08 n.a. n.a. n.a. n.a. n.a. n.a. 1.89 n.a. 41.95 n.a. n.a. ma. n.a. ma. n.a. n.a. n.a. n.a. n.a. 26.60 -6.98 n.a. -1.59 n.a. -16.75 n.a. 1315.10 -6.19 n.a. n.a. n.a. n.a. n.a. n.a. 23.01 0.00 0.24 -4.10 n.a. -18.64 1.66 n.a. n.a. ma. n.a. 104.20 n.a. 156.26 n.a. n.a. 96.21 n.a. 41.06 n.a. 45.15 10.38 57.42 ma. n.a. n.a. n.a. n.a. n.a. 38.20 64.33 118.05
COOC 44.98 n.a. n.a. 4.16 249.07 -9.98 n.a. 0.00 -13.01 -19.52 n.a. n.a. n.a. n.a. n.a. n.a. -5.53 27.24 2.04 25.53 n.a. ma. 10.68 n.a. n.a. n.a. -38.86 n.a. 15.27 -5.24 n.a. -16.37 -39.40 -42.96 -48.37 -32.32 -44.49 -19.34 -31.90 -9.72 -68.27 211.23 n.a. n.a. 0.11 0.00 9.03 -10.33 -46.29 -19.47 n.a. n.a. 38.97 n.a. 56.74 n.a. 20.20 n.a. n.a. 4.80 -34.48 n.a. n.a. n.a. n.a. n.a. n.a. n.a. n.a. n.a. n.a. n.a. n.a. n.a. n.a.
CHzO 21.92 17.63 18.11 36.81 95.28 7.70 34.83 20.67 0.00 -31.43 n.a. n.a. 21.92 n.a. n.a. -2.30 -35.97 -40.74 -6.21 0.72 n.a. 19.51 75.39 n.a. -4.69 13.58 115.75 n.a. 2.75 130.64 -18.33 -2.72 -30.06 -24.75 ma. 47.78 -4.04 14.78 n.a. n.a. -33.81 -12.44 24.95 -0.98 13.61 -4.59 0.00 127.54 -21.89 -25.65 n.a. n.a. -3.72 -12.68 118.05 n.a.
COOA 20.22 n.a. n.a. 14.43 n.a. n.a. n.a. 59.55 -17.81 0.00 n.a. n.a. n.a. 11.37 n.a. n.a. ma. 14.46 5.37 n.a. n.a. -66.65 ma. n.a. n.a. n.a. -94.44 n.a. 62.78 n.a. n.a. 16.93 83.00 -27.58 n.a. -9.01 4.92 n.a. n.a. n.a. n.a. 45.86 n.a. 175.14 n.a. 16.63 5.69 0.00 n.a. ma. n.a. n.a. n.a. n.a. n.a. n.a.
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 1, 1986 25 Table I1 (Continued) CHz n.a. 0.00 n.a. 39.28 n.a. n.a. 2.55 n.a. n.a.
m Br MezSO pyridine ACCHz ACNOz ACCl ACF ACNHz ACBr
C=C n.a. n.a. 0.00 169.05 n.a. n.a. -16.10 -46.29 n.a.
C=C n.a. 0.92 1.67 0.00 n.a. 146.91 -15.05 47.32 n.a.
Table 111. Typical Correlation Results system benzene + pyridine CC14 + I-hexane CCl, + 2-pentanone benzene + triethylamine CC14 + dipentyl ether acetic acid + methyl acetate benzene + acetone CHzClz + l-heptene CC14 ethyl acetate benzene + I-octene benzene + acetic acid benzene + dibutylamine CHzClz diethyl ether CHC13 + acetone CHC13 + triethylamine
+
+
Table IV. Tmical Prediction Results system benzene + pyridine benzene + nitromethane CCl, + diethyl ether CHC13 + dibutyl ether CHzClz + l-decene CHC13 + 1,7-octadiene CHzClz + methyl acetate l-octyne + dibutyl ether CHzClz + acetone CHzClz + acetone CHC13 + butyl ethyl ether CHC13 + dipropylamine dichloroethyl ether + bromohexane ethyl acetate + bromohexane CzHzCHBrCOOCzH5 bromohexane
+
ACH n.a. n.a. n.a. n.a. 0.00 n.a. n.a. -12.66 n.a.
temp, "C 25 25 25 30 25 35 30 25 25 25 35 30 25 25 25
OH n.a. n.a. n.a. 020.12 n.a. 0.00 n.a. 70.03 n.a.
CHzCO n.a. 16.45 30.88 -2.39 n.a. n.a. 0.00 n.a. n.a.
AKL, J / mol
temp, "C 40 45 25 25 25 25 45 25 0 45 25 25 25 30 30
3 4 5 11 11 13 16 22 24 38 41 52 130 162 253
ea, J/mol 6 12 42 78 59 33 43 23 36 90 143 193 194 66 120
CHO n.a. n.a. 50.34 112.47 68.35 107.44 n.a. 0.00 n.a.
error % 17.5 4.0 4.2 2.5 2.9 25.2 8.2 1.8 22.5 5.9 13.0 11.4 6.8 6.8 8.6
error % 25.0 1.3 7.5 2.8 3.5 15.7 8.6 15.2 6.2 7.8 6.0 4.9 15.6 13.2 9.4
COOC n.a. n.a. n.a. n.a. n.a. n.a. n.a. n.a. 0.00
CHzO
COOA
source Christensen et al., 1982 Grolier, 1976 Kiyohara et al., 1977 Gutierrez Losa, 1979 Williamson, 1974 Nagata et al., 1975 Christensen et al., 1982 Woycicky, 1980 Christensen et al., 1982 Marsh, 1975 Nagata et al., 1975 Gutierrez Losa, 1981 Findlay, 1974 Tamura and Nagata, 1981 Christensen et al.. 1982
source Christensen et al., 1982 Brown, 1973 Williamson, 1974 Christensen et al., 1982 Woycicky, 1980 Woycicky, 1980 Van Ness and Abbot, 1974 Inglese et al., 1979 Van Ness and Abbot, 1974 Van Ness and Abbot, 1974 Christensen et al., 1982 Christensen et al., 1982 Christensen et al., 1982 0th et al., 1980 0th et al., 1980
Table V. Prediction Results from Single vs. Multiple Correlation: Experimental Data, Woycicky (1980) singleb multiplec system temp, "C eJ/mol , error % status" AH'&, J/mol error % CHzClz + l-hexene 25 103 11.3 P 73 7.8 25 38 4.4 P 49 4.0 CHzClz + 3-hexene 25 48 12.7 P 17 4.0 CHzClz + 1,5-hexadiene 25 22 1.8 C 28 3.0 CHiClz + 1-heptene 25 40 3.4 P 64 6.0 CHzClz + I-octene CHzClz + l-nonene 25 33 1.8 P 57 3.2 CHzClz + l-decene 25 59 3.5 P 82 4.0 av 5.6 4.6
status" C C C C P P P
"C, correlation; P, prediction. system used to evaluate interaction parameters (amn= 7.89; unm = -6.98). cFour systems used to evaluate interaction parameters (amn= 6.00; anm= -4.97).
are presented along with the values of the maximum error observed, AH&. Prediction results are also very good with errors below 15% in mwt cases (Dang, 1984). Typical results are shown in Figures 4 and 5 and in Table IV. In some cases, due to the limit of the experimental data, the interaction parameters were evaluated from one set of data only. To examine the reliability of these interaction parameter values, a series of "single" correlations was compared to "multiple" correlations. If one set of exper-
imental data is used to evaluate the interaction parameter values, then the correlation is called single. On the other hand, if more than one set of data is used, the correlation is called multiple. A typical case for single vs. multiple correlation results is presented in Table V. The results show that no significant loss of accuracy is observed when only one set of data is used. This is further supported by the results presented in Table VI and in Figure 6 for the prediction of a variety of ketone + alkane systems as a function of temperature
26
Id.Eng. Chem. Process Des. Dev., Vol. 25, No. 1, 1986 0
30
i &I
4 a,
P
\
E"
o
-1000
\
a,
3 3
3
7
0
r?
F
m
.rl .rl X
.rl
x
SI
.I 3 :
Q
,+-
v1 VI
.3 Q
4
;" -2000
4 -30 5
42 1
W
W
-3000
-60
U
0
x1
x1
Figure 3. Correlation results for the system (1)benzene + (2) butan& T = 25 OC; (-) correlation; ( 0 )experimental data (Marongiu et al., 1974).
loo
-200
IA
ti.
L
1
I
1
0
x1 ~
~
Figure 4. Prediction results for the sy-atem (1) CHC13 + (2) ethers: T = 25 OC; (-) prediction; (0,A,D) experimental data (Beath and Williamson, 1969).
in the range of -20-80 "C. The accuracy of the prediction results is remarkable considering that the interaction parameters (CH2CO/CH2)were obtained from a single system, acetone + n-hexane at 20 OC. Prediction reliability with respect to temperature is also demonstrated again in Figure 7 for benzene + cyclohexane from 7 to 120 "C.
Figure 5. Prediction results for the system (1) 2-butanone + (2) CCl,: (-) prediction; ( 0 )experimental data (Kiyohara et al., 1977). Table VI. Prediction Results for Ketone-Alkane Systems Using Parameters Obtained from a Single System (Acetondn -Hexane at 20 "C, acH,/CH&O = 59.93; a c a s o j c ~ , = 3.67) temperature, "C systems -20 -10 0 20 25 40 80 acetone + n-C4 12.5 7.3 acetone + n-C 8.5 3.8 0.6 acetone + n-C6 8.4 5.4 acetone + n-C7 5.4 acetone + n-CB 7.7 2.3 8.7 acetone + n-Clo acetone + n-Cl3 8.1 1.0 6.7 10.7 acetone + n-C18 1.0 6.3 acetone + n-C17 19.5 5.1 2-butanone + n-C6 17.3 7.1 4.5 7.8 2-butenone + n-C7 2-butanone + n-Cs 8.0 2-butanone + n-Clo 8.2 2-butanone + n-Clz 13.1 2-butanone + n-Cl3 17.8 2.4 9.3 2-pentanone + n-C5 7.2 2-pentanone + n-C8 8.8 8.3 2-pentanone + n-C7 8.5 2-pentanone + n-C8 2-pentanone + n-Clo 8.3 3-pentanone + n-C5 8.1 25.8 3-pentanone + n-C7 7.6 11.4 4-heptanone + n-C5 12.0 30.0 4-heptanone + n-C9 8.3 26.3 4-heptanone + n-C13 2.1 av abs error, % 12.5 7.3 10.5 4.6 8.7 9.9 7.5
In applying the original UNIFAC model to calculate activity coefficients, no differentiation is made between cyclic and noncyclic compounds. As a result, the interaction parameters between cyclic CH2 group (CH2C)and any other group G are equal to those of the CH2group with group G: aCH2E/G
= aCH1/G
aG/CHzC= aG/CH2
(12)
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 1, 1986 27 Table VII. Prediction Results for Systems Containing Cyclic Compounds without CHz/CHze system temp, O C J/mol error % cyclohexane 1-hexene 25 178 76.9 212 85.1 cyclohexane + 1-octene 25 10 95 ethylcyclohexane + benzene 10.9 25 ethylcyclohexane + benzene 132 15.0 methylcyclohexane + benzene 25 14.3 117 methylcyclohexane + benzene 14.1 35 113 cyclohexane + acetone 25 8.1 93 cyclohexane + 2-butanone 8.5 30 118 cyclohexane + 2-pentanone 8.4 25 90 cyclohexane + 2-hexanone 167 15.5 30 cyclohexane + 1-butanal 12.9 25 163 cyclohexane + ethyl acetate 25 348 26.9 cyclohexane + methyl acetate 25 222 15.7 258 cyclohexane + methyl acetate 15.8 35 cyclohexane + methyl acetate 14.6 274 45 cyclohexane + pentanonitrile 25 76 5.8 cyclohexane diethyl ether 25 58.4 257 cyclohexane + acetic acid 35 39.7 349 cyclohexane + hexylamine 173 25 15.8 cyclohexene + hexane 25 61.3 84 cyclohexene + heptane 25 77 60.8 tetrahydrofuran + hexane 30 477 65.5 tetrahydrofuran + heptane 25 63.8 507 tetrahydrofuran + decane 25 64.3 590 tetrahydrofuran + tetradecane 25 64.5 665 tetrahydrofuran + diethyl ether 25 116 92.8 tetrahydropyran + hexane 30 65.1 369 tetrahydropyran + heptane 25 382 64.0 tetrahydropyran + decane 422 25 63.0 tetrahydropyran + tetradecane 25 495 65.6
e-,
+
+
with CH2/CHzC J/mol error % 28 8.4 11 4.8 72 8.0 96 11.0 72 8.7 67 8.4 76 4.0 77 5.7 67 5.2 75 5.6 56 4.9 259 18.4 163 8.9 183 9.2 223 8.4 75 6.8 90 17.8 303 32.3 45 3.0 45 31.2 46 28.2 45.8 336 361 43.2 421 44.1 489 44.5 11.7 18 34.3 190 31.7 198 29.8 220 35.6 273
e=,
source Christensen et al., 1982 Christensen et al., 1982 Woycicky, 1972 Woycicky, 1972 Nagata et al., 1978 Nagata et al., 1978 Christensen et al., 1982 Kiyohara et al., 1974 Christensen et al., 1982 De Torre et al., 1980 Marsh, 1975 Ratnam et al., 1962 Nagata et al., 1973 Nagata et al., 1973 Nagata et al., 1973 Christensen et al., 1982 Christensen et al., 1982 Nagata et al., 1975 Christensen et al., 1982 Christensen et al., 1982 Christensen et al., 1982 Guillen et al., 1978 Inglese et al., 1980 Inglese et al., 1980 Inglese et al., 1980 Christensen et al., 1982 Guillen et al., 1978 Inglese et al., 1980 Inglese et al., 1980 Inglese et al., 1980
Table VIII. Prediction Results for Multicomponent Systems: Experimental Data, Christensen et al. (1982) system benzene + tetradecane + heptane benzene + tetradecane + decane benzene + acetonitrile + CCl, benzene + acetone t CHC13 acetic acid + pyridine + CHC1, benzene + cyclohexane + ethyl acetate benzene + cyclohexane + hexane benzene + cyclohexane + heptane benzene + cyclohexane + CCl, benzene + cyclohexane + 2-butanone benzene + cyclohexane + toluene benzene + cyclohexane + hexane + heptane benzene + cyclohexane + hexane + heptane + toluene
temp, "C
e a l ,
J/mol
error %
25
130
10.1
25 45 25 20 35
117 117 326 523 87
6.7 8.7 18.7 10.6 4.6
25 25
98 59
6.0 1.9
25 20
44 80
6.2 5.6
20 20
47 56
4.6 3.6
20
109
16.4
group. We recommend the following interaction parameters for this case: I
-40
-29
I
I
I
I
I
0
20
40
60
80
Temperature,
100
OC
Figure 6. Prediction results of ketones with alkanes as a function of temperature with the interaction parameters obtained from a single system (Table VI): (*) number of systems predicted a t each temperature.
Typical prediction results using this approach are presented in Table VII. They show that poorer predictions are seen for systems containing cyclic compounds. Significant improvements, however, are observed by maintaining eq 12 except for the case where G is the CH2
The values for these parameters were evaluated from a data base of alkanes with cyclohexanes and typical prediction results are also presented in Table VII. Even though the use of the CHZC/CH2interaction parameters improves significantly the prediction results for systems containing cyclic compounds, the performance is still not in par with those of noncyclic systems especially with the polar cyclic compounds (Table VII). A special treatment for cyclic compounds is, therefore, under consideration. Very good prediction results are also obtained for multicomponent systems as shown in Table VI11 and Figure 8. This table contains 11 ternary, 1 quaternary, and 1 quinary mixtures. The results for the two ternary
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 1, 1988
28
Table IX. Prediction Results for External and 1n)ernel Group Assignments error, % system temp, OC ext 2-pentanone + ethyl acetate 25 2.9 2-pentanone + diethyl ether 25 1.8 25 %pentanone + CCl, 4.2 25 2-butanone + CCl, 10.5 45 2-hexanone + bromohexane 2.9 25 dimethoxymethane + CCl, 15.1 25 1,l-dimethoxyethane + CCl, 12.1
int 49.7 36.3 48.3 59.3 87.5 31.0 10.6
source Christensen et al., 1982 Christensen et al., 1982 Kiyohara et al., 1977 Kiyohara et al., 1977 De Torre et al., 1980 Meyer and Giusty, 1977 Meyer and Giusty, 1977
1200
1000
800 900 a,
3 a,
rl
P
P
\
\
3
rl
3
v
a
600
0
m
m . i
.d
.z 600
. 5. d 4
2
4
400
Lo
.r(
.3
+ .a m Y S
3 0.
* S
w
W
300
200
0
0
1
0
0
1
0 x1
X1
Figure 7. Effect of temperature on prediction results (1)benzene + (2) cyclohexane: (-) prediction; (0, A, m) experimental data (Elliott and Wormald, 1976).
Figure 8. Prediction results for the system (1)benzene + (2) cyclohexane + (3) hexane a t x 2 = 0.1: T = 25 OC; (-) prediction; ( 0 ) experimental data (Christensen et al., 1982).
mixtures containing chloroform are impressive considering that solvation effects are not accounted for. In certain molecules, more than one group assignment is possible as demonstrated, for example, with 2-pentanone: 1CH3C0,2CH2,and lCH3 (external); 1CH2C0,1CH2, and 2CH3(internal). In such cases, significantly different results can be obtained as shown by the prediction results in Table IX,and the external group assignment is recommended. Actually, the use of external group assignment involves a larger surface area parameter Qkvalue for the group, Q k = 1.488 for CH3C0 vs. 1.180 for CH2C0 in ketone, which leads to a larger deviation from the ideal enthalpy of mixing. This should be expected because the closer a group is to the end of the molecule, the less it is hindered by the other group in interacting with the other species. For example, the enthalpies of mixing for 2-pentanone n-pentane are greater than those for 3-pentanone + n-pentane. The introduction of the coordination number 2 in front of amnin eq 7 is considered unfortunate (Rasmussen, 19831, and, consequently, the expression of Z(7') in eq 9 must be viewed as a universal temperature-dependent coefficient for the interaction parameters. The success of this "universal coefficient" in accounting for the temperature dependency of the interaction parameters, especially as demonstrated in Figure 6 where a single system was used
in their evaluation, is rather remarkable. Conclusions The results in this study demonstrate that the modified UNIFAC model of Skjold-Jorgensen et al. provides a reliable method to estimate the enthalpies of mixing for a large variety of organic liquid mixtures. A total of 172 pairs of interaction parameter values are presented. Prediction results, in the temperature range 273-400 K, are typically below 15%. Somewhat poorer results, however, may be encountered for systems containing cyclic compounds in spite of the improvement realized by including the interaction parameters for the CH2"CH2 pair. Acknowledgment We thank Pericles Lagonicos, an undergraduate student at New Jersey Institute of Technology, for his assistance in evaluating some of the interaction parameters. D. Dang also thanks Dr. Glen A. Hemstock, L. Alan Jarnagin, and Robert L. Kolesar of Engelhard Specialty Chemical Division for the use of company facilities and for their encouragement. Nomenclature umn = UNIFAC interaction parameter, temperature independent
+
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 1, 1986 29
A = component, non-alcohol B = component, alcohol D = minimization function, eq 10 GE = excess Gibbs free energy H = excess enthalpy of group k H!! = same as H k in a reference solution containingonly type 1. molecules AH = excess enthalpy, J/mol AHi = partial molar excess enthalpy of component i Aho = enthalpy of hydrogen bond formation,-7.0 kcal/g-mol K = association constant, a function of temperature and alcohol N , = number of carbon atoms in alcohol Nki = number of groups of type k in component i Qi = area parameter of group i R = gas constant S = average absolute error, eq 11 T = absolute temperature To = reference temperature, 323 K V = number of data points x i = liquid mole fraction of component i X i = mole fraction of group i in the mixture 2 = lattice coordination number Subscripts
A = non-alcohol B = alcohol c = chemical contribution calcd = calculated exptl = experimental p = physical contribution
@B, @Bl, and $Ll are the overall volume fraction of the alcohol, of the alcohol monomer, and of the alcohol monomer in the pure alcohol state, respectively, 1 + 2K@B- 1/Y* (-4-6) @Bl = 2P@B 1+2K-l/Y $il= lim = (A-7) XA-4) 2P where X A is the mole fraction of non-alcohol A in the solution. Stathis and Tassios developed the following expression for the association constant KOof alcohols at To= 50 "C:
KO= a ( N J b M-8) N , is the number of carbon atoms in alcohol (N, > 2), and b = -0.435 for all alcohols. a = 145 for primary alcohols. For methanol and ethanol, KOis 450 and 190, respectively. a = 97 for all secondary alcohols. a = 82 for all tertiary alcohols. They also recommended a value of Aho = 7.0 kcal/g-mol for all alcohols. B. Illustrative Calculation. The following example is used to illustrate the calculation of enthalpy of mixing for (1)CHC1, + (2) acetone at 25 OC, and x1 = 0.2352. Let Ck
=
Eemknk m
Pk
=
xem#mk
m
Superscripts m = property of mixing
* = reference solution containing molecules of only one type
Greek Letters Om = area fraction of group m @B = volume fraction of the alcohol @pl = volume fraction of alcohol monomer
= volume fraction of alcohol monomer in the pure alcohol state
Appendix A. The UNIFAC/Association Model. For the enthalpy of mixing of an associated solution, Stathis and Tassios (1984), following the approach of Renon and Prausnitz (1967), write A l P as the sum of two contributions: physical (q) and chemical
(w):
AIP=q+w
(-4-1)
The physical part reflects interaction among groups and is described by the UNIFAC model. The chemical part accounts for the chemical equilibria of hydrogen-bonding polymerization reaction and is given by
Then eq 4 can be rewritten as
Calculate Hk. Let us define group 1 as CH,, group 2 as CH3C0, and group 3 as CHCl,. Table I shows that component 1 (CHClJ has only one group, group 3; component 2 has one group 1 and one group 2. On the basis of this information, Nki can be expressed Nl2 = 1 N,, = 0 N,, = 0 N22 = 1 N31 = 1 N32 = 0 The surface area parameters for these groups are obtained from Table I: Q1 0.848 Q2 = 1.488 Q3 = 2.410 Similarly, - . the interaction parameters are shown in Table
11:
where: XB = mole fraction of alcohol, Aho = enthalpy of hydrogen bond formation, K = association constant calculated from its value KOa t some reference temperature TO, In K = In KO-
$( $ $-) -
(A-3)
all = 0
a12
= 52.25
a13
= 15.27
aZ1= 12.35 a22= 0
a23 = -42.96 = 33.14 a33= 0
-2.78 a32 Calculate Z(T) and Z1T) Z = 35.2 - 0.1272T + 0.00014P 2' = -0.1272 + 0.00028T a t 298.15 K, Z = 9.72040, and 2' = -0.04372. Calculate the mole fraction of group 1 in the mixture a31 =
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 1, 1986
30
XlNll
x1=
+ XlN21 + XlN31
xlNll
+ x2N12 + X z N 1 2 + x 2 N 2 2 + x3N32
(0.2352)(0) +(0.7648)(1) = 0.43336 (0.2352)(0 + 0 + 1) + (0.7648)(1 + 1 + 0) similarly X 2 = 0.43336, and X 3 = 0.13327. Calculate the area fraction Om of group 1
-
QiXi
01 =
+ Q2X2 = Q3X3 (0.848)(0.43336) (0.848)(0.43336) + (1.488)(0.433 36)
E1 = 0.000 24 E2 = 0.000 19 E3 = 0.000 31
Calculate Fk m
(0.27558)(0) F1 = 0.922 99
QiXi
+ (2.410)(0.13327) = 0.27558
r m
(0.48356)(0.00285) + 0.74147 (0.240 86)(0.00152) 1.429 78 F1 = 0.002 1 2 FZ = -0.001 48 F3 = 0.00150
+
Calculate Gk
O2 = 0.48356
6 , = 0.24086
Calculate $mn
-T)
(9.720 40)(0)
aT)all
$11
= e-(
= 0.42667
$11
=1
$21
= 0.817 65 = 1.04636
$31
$12
)
= exp( - (2)(298*15)
$22 $32
$13
ci 0.00053 G1 = - =Pi 0.92299 =1
G1 = 0.000 57 G2 = 0.001 86
Calculate $kn
Hk = Q k ( G k - Ek
=
(9.720 40)(0)
(-0.043 72) - 9.720 40 - (2)(298.15) )('I(2(298.15) 2(298.
$i1 = 0 $i2 = 0.002 85 $i3 = 0.00152 = 0.001 29
$;2
=0
$;3
= -0.011 08
$i1 = -0.000 37 $i2 = 0.002 47 $h3
=0
Calculate Ck
Ck = C1
=
01$;1
m
O2$21
+ 83$31
= (0.27558)(1)
+
(0.484 56)(0.81765) + (0.24086)(1.04636) Pi = 0.922 99 Pz 0.74147 P3 = 1.429 78 Calculate Ek
El =
i\H, = CNki(Hk -
+ O2& + 03$gl = (0.27558)(0) +
m
61$11
ai
Calculate The value for is determined by using the same procedure described earlier with the exception of the way for defining Nki. In calculating H k , the group mole fraction is based on the mixture while it is based on the pure compound when calculating z$k+ For example, component 1 (CHClJ has only one group, group CHC1,; therefore Nll = 0, N2, = 0, and N31 = 1. Similarly, component 2 (CH,COCHJ has one CH3group and one CH3C0 group; thus, N12= 0.5, N22 = 0.5, and N32 = 0. The calculated values for are ql= 731.97, 6 ,= 1501.13, 6 1 = -7515.15, 6 2 = 521.13, 6 1 = 0, 6 3 2= -5931.19. Calculate Enthalpy of Mixing AH". Calculate
a
Com$mk
(0.483 56)(0.00129) + (0.240 86)(-0.00037) C1 = 0.000 53 C2 = 0.001 38 C3 = -0.004 94 Calculate Pk pk = csm$mk =
+ Fk)Rp
H, = 0.848(0.00057 - 0.000 24 + 0.002 12)(8.337)(298.1E1)~ Hi = 1535.47 H2 = 208.95 H3 = -4043.18
ai.
$;l
= -0.003 46
Calculate Hk
= 0.77964
= 1 $23 = 2.014 36 = 0.58262 $33 = 1
G3
(0.275 58)(1)(0.00053) + (0.922 99)2 (0.483 56)(0.42667)(0.00318) (0.74147)2 (0.240 86)(0.779 64)(-0.004 94) (1.429 78)2
+
k
m i
= Nii(Hi - Si)+ N21W2 - Si)+ = (Ol(1535.47 - 731.97) (0)(208.95) -
N31(H3 - $ 1 )
(-7515.75)
Similarly,
a
+ + (1)(-4043.18 - 0) = -4043.18
= -277.84. Calculate hK/
=~ x , A H = (0.2352)(-4043.18) ~
+
(0.7648)(-277.84) = -1163.45 J/mol Compared with the experimental value (-1147.75 J/mol), the absolute percent error is 1.4%. Literature Cited Beath, L. A.; Williamson. A. G. J . Chem. ??"odyn. 1988, 1 , 51. Brown, I. Int. DATA Ser.. Sel. Data Mlxhrres, Ser A: 1973, 47. Chrlstensen, J. J.; Hanks, R. W.; Izatt, R. M. "Handbook of Heats of Mixing"; Wlley: 1982. Dang, D. M.S. Thesls, New Jersey Instltute of Technology, Newark, 1984. De Tone, A.; Velasco, I.; Gutierrez, Losa C. J . Chem. Thermodyn. 1980, 12, 87. Derr, E. L.; Deal, C. H. Inst. Chem. Eng. Symp. Ser. 1969, No. 32, 3, 40. Elliott, K.; Wormald, C. J. J . Chem. Thermodyn. 1978, 8 , 881. Flndlay, T. J. V. J . Chem. Thermodyn. 1974, 6 , 387. Fredenslund, Aa.; Jones, R. L.; Prausnitz. J. M. AIChE 1975, 2 1 , 1086. Grdler, J.-P. E. Int. Data Ser., Sei. Data Mixtures, Ser. A 1978, 67. Gulllen, M. D.; Gutlerrez Losa, C. J . Chem. Thermodyn. 1978, 10,567.
Ind. Eng. Chem. Process Des. Dev. 1088, 25, 31-35 Gutierrer Losa, C. Int. Data Ser., Sei. Data Mixtures, Ser. A 1981, 62. Inglese. A.; Wllhelm. E.; Grolier, J.-P. E.; Kehiaian, H. V. J . Chem. Thermodyn. 1080, 12, 217. Kiyohara, 0.; Genson, G. C.; Anand, S. C. J . Chem. Thermodyn. 1974, 6 , 355. Kiyohara, 0.; Benson, G. C.; Grolier, J.-P. E. J . Chem. Thermodyn. 1977, 9 , 315. Marongiu, B.; Bros, H.; Coten, M.; Kehiaian, H. V. I n t . Data Ser., Sel. Data Mixtures, Ser. A 1974, 58. Marsh, K. N., Int. Data Ser., Sel. Data Mixtures, Ser. A 1975, 187. Nagata, I.; Asano, H.; Kujiwara, K. Fluid Phase Equilib. 1978, 1 , 21 1. Nagata, I.; Nagashima, M.; Kazuma, K.; Nakagawa, M. J . Chem. Eng. Jpn. 1975, 8 , 261. Nagata. I.; Ohta, T.; Takahashi, T.; Gotoh, K. J . Chem. Eng. Jpn. 1973, 6 , 129. Nagata, I.; Yamada, T. Ind. Eng. Chem. Process Des. Dev. 1972, 1 1 , 574. Nguyen, T. H.; Ratcllff, G. A. Can. J . Chem. Eng. 1974, 52, 641. NlcolaMes, G. L.; Exkert, C. A. Ind. Eng. Chem. Fundam. 1978, 17, 331. Otin, S.; Tomas, G.; Peiro, J. M.; Velasco, I.; Gutlerrez Losa. C. J . Chem. Thermodyn. 1980, 12, 955.
31
Rasmussen, P. Flu@ Phase Equllib. 1983, 13, 213. Ratnam, A. V.; Rao, C. V.; Murti, P. S. Chem. Eng. Sci. 1962, 17, 392. Renon, H.;Prausnitz, J. M. Chem. Eng. Sci. 1967, 22, 299. Rupp, W.; Hetzei, S.; Ojini, I.; Tassios, D. Ind. Eng. Chem. Process Des. D e v . 1984, 23, 391. Siman, J. E.; Vera, J. H. Can. J . Chem. Eng. 1979, 57, 355. SkjoMJorgensen, S . ; Rasmussen, P.; Fredenslund, Aa. Chem. Eng. Sci. 1980, 35, 2389. Stathis, P.; Tassios, D. Ind. Eng. Chem. Process Des. Dev. 1985, 24, 701. Tamura, K.; Nagata, I. Thermchim. Acta 1981, 47, 315. Van Ness, H. C.; Abbott, M. M. Int. Data Ser., Sel. Data Mixtures, Ser. A 1974, 91. Williamson, A. G. Int. Data Ser., Sel. Data Mixtures, Ser. A 1974, 135. Woyclcki, W. J . Chem. Thermodyn. 1972, 4 , 1. Woycicki, W. J . Chem. Thermodyn. 1980. 12, 761. Woycickl, W. J . Chem. Thermodyn. 1980, 12, 767.
Received for review May 17,1984 Accepted March 8, 1985
Control of Water Softening Process by Ion Exchange Using Electrical Conductivity Measurements August Serra’ and Carles Soll Department of Qdmica Tscnica, Fac. Ci&?cles, Universitat Aut6noma de Barcelona. Belleterra, Barcelona, Spain
Manel Adroer CIDA Hidroquimica, S A . , Barcelona, Spain
A control system of water softening by ion exchange is presented. I t has been proved that the porous bed’s electrical conductivity is a convenient variable for controlling this process. A selection of a theoretical model describing the electrical behavior of ion-exchange columns has been made, the “three-wire model” being the best option. An experimental apparatus has been deslgned to measwe the fundamental variables of the system: treated water’s or solution’s electrical conductivity, resin’s electrical conducthrlty, and porous bed’s electrical conductivity. The model parameters have been calculated, and the influence of temperature on them has been studied. Finally, a practical application of the control system is presented.
The softening of water by ion exchange is a very common process in the industry, especially to condition the water for cooling systems and steam generators. In it, the Na+ cations in the resin are exchanged by the Ca2+and Mg2+cations of the water, leaving it hardness-free. In deionization processes, the control can be implemented by measuring the electrical conductivity of the treated water, but in softening, this method cannot be used because that variable does not depend on the type of cation contained in the solution (Visotskii, 1969). Nowadays, the most usual control method is the approximate estimation of the exhaustion time of the column, calculated from the data of the resin exchange capacity (obtained from the manufacturer), volumn of resin contained in the column, and Ca2+-Mg2+mean concentration in the incoming water. Another method actually applied is the measurement of the volumn of water that has passed through the column. In this case, an “impulse meter” is used. When the set point is reached, an alarm is triggered or an automatic regeneration process is started. Those systems, however, lack versatility, since a progressive decrease in the exchange capacity or a sudden change in the incoming water hardness could produce im0196-4305/86/1125-0031$01.50/0
portant problems in the industrial process control. One of them could be the production of inadequate hard water and another the premature regeneration of the resin, wasting the regenerating chemical. Some references (Visotskii, 1969; Helfferich, 1962) point out that the resin’s electrical conductivity depends on the type of cation contained in it. This fact could be used to design a control system. This magnitude, however, cannot be directly measured because the resin beads are constantly immersed in the water flowing through the column. On the other hand, the electrical conductivity of the porous bed (resin beads + water) may be constantly and easily measured. So, our purpose is to prove that its measurement offers a useful way to control the water softening by ion exchange. In order to do that, an apparatus to make the experimental measurements has been designed, a selection of a theoretical model describing the electrical behavior of the column has been carried out, the model parameters for each experimental series have been calculated, and a study of their dependence on the temperature has been made. Finally, we have analyzed the difference between the bed‘s electrical conductivity with the resin in Na+ form and the 0 1985 American
Chemical Society
