Ind. Eng. Chem. Res. 1990,29, 875-882
seen that there is no marked difference in the various crystal size distributions. The reason for the deviation of the data from the unilinear distribution predicted by the ideal MSMPR model is not clear. Such deviations are often observed in analyzing MSMPR data and have been explained as indicative of the occurrence of agglomeration (Budz et al., 1986) or size-dependent growth (Jancic and Garside, 1976). Table VI summarizes all published data on the apparent kinetics of gypsum and hemihydrate crystallization, derived assuming ideal MSMPR conditions (unilinear distribution). The results obtained in this study using a bilinear distribution yield somewhat larger values for the apparent growth rate and nucleation rate. Registry No. PO(OH)3, 7664-38-2; Fe3+, 20074-52-6; A P , 22537-23-1; gypsum, 13397-24-5.
Literature Cited Adami, A.; Ridge, M. J. Observations on Calcium Sulphate Dihydrate Formed in Media Rich in Phosphoric Acid: I. Precipitation of Calcium Sulphate Dihydrate. J . Appl. Chem. 1968, 18, 361-365. Addai-Mensah, J. Effect of Impurities of the Filterability of Gypsum Formed in Phosphoric Acid Solution. MSc. Dissertation, Technion, Haifa, Israel, 1987. Amin, A. B.; Larson, M. A. Crystallization of Calcium Sulfate from Phosphoric Acid. Ind. Eng. Chem. Process Des. Deu. 1968, 7, 133-137. Becker, P. Phosphates and Phosphoric Acid; Fertilizer Science and Technoloty Series; Marcel Decker Inc.: New York, 1983. Ben-Yosef, E.; Holdengraber, C.; Metcalfe, J.; Gryc, S. Factors Affecting the Filterability of Gypsum Obrained in a Bench Scale Continuous Unit for WPA. Abstract of Papers, 50th Israel Chemical Society Meeting; Jerusalem, Israel, 1984; p 198.
875
Budz, J.; Jonas, A. G.; Mullin, J. W. Effect of Selected Impurities on the Continuous Precipitation of Calcium Sulphate (Gypsum). J . Chem. Technol. Biotechnol. 1986, 36, 153-161. Gilbert, R. L., Jr. Crystallization of Gypsum in Wet Process Phosphoric Acid. Ind. Eng. Chem. Process Des. Deu. 1966,5, 388-391. Jancic, S.; Garside, J. A. New Technique for Accurate Crystal Size Distribution in an MSMPR Crystallizer. In Industrial Crystallization; Mullin, J. W., Ed.; Plenum Press: New York, 1976; pp 363-372. Monaldi,, R.; Barbera, A.; Socci, F.; Venturino, G. Revamping and Energy Cost Reduction Obtained in one of Montedison's Phosphoric Acid Plants with Low Investment Cost. Presented at the IFA Technical Conference, 1982; Paper TA/82/8, 32 pp. Mullin, J. W. Crystallization, 2nd ed.; Butterworth: London, 1972. Orenga, M. Production of Phosphoric Acid from Phaloborwa Rock from the Pilot Studies to Experience of the Industrial Plants of Rhone Poulenc. Presented a t the IFA Seminar, Raw Materials in South Africa, Johannesburg, 1983; Part 111, pp 45-55. Robinson, N. Fisons' Experience on the Effect of Phosphate Rock Impurities on Phosphoric Acid Plant Performance. Presented a t the ISMA Technical and Economic Conference, Orlando, FL, 1978; Paper TA/78/8, 16 pp. Sarig, S.; Mullin, J. W. Effect of Trace Impurities on Calcium Sulphate Precipitation. J . Chem. Technol. Biotechnol. 1982, 32, 525-531. Sarig, S.; Kahana, F.; Epstein, J. A.; Stern, S. The Effect of Aluminum and Silicate Ions on Calcium Sulphate Dihydrate Precipitation in Steady State Systems. Scanning Electron Microsc. 1981, 4 , 253-257. Sikdar, K. S.; Ore, F.; Moore, J. H. Crystallization of Calcium Sulfate Hemihydrate in Reagent Grade Phosphoric Acid. AIChE Symp. Ser. 1980, 76 (No. 193), 82-89. Slack, A. V. Phosphoric Acid; Fertilizer Science and Technology Series; Marcel Decker: New York, 1968; Part I. Received for review October 24, 1989 Accepted December 1, 1989
Group-Contribution Flory Equation of State for Vapor-Liquid Equilibria in Mixtures with Polymers Fei Chen, Aage Fredenslund,* and Peter Rasmussen Znstitut for Kemiteknik, The Technical University of Denmark, DK-2800 Lyngby, Denmark
T h e equation of state presented in this work is a group-contribution extension of a slightly modified form of the Flory equation. The equation is similar to, but simpler than, the Holten-Andersen model. A new correlation for the degree of freedom parameter, C, has been introduced. T h e energy interaction parameters are based on group-group interactions, and also the C parameters are calculated by using the group-contribution approach. Results from correlation and prediction of pure-component liquid-phase properties and of vapor-liquid equilibria of mixtures with only solvents and of mixtures with solvents and polymers are presented. T h e results show that, although the parameters have been determined mostly from experimental information on normal boiling components and mixtures of these, the new equation of state is able to predict vapor-liquid equilibria for a large variety of mixtures of polymers and solvents over a wide range of temperatures with good accuracy.
Background In general, pure fluids have different free volumes, i.e., different degrees of thermal expansion compared with the hard-core liquid volumes. When liquids with different free volumes are mixed, these differences contribute to the excess functions. Differences in free volumes are not taken into account in conventional theories of liquid mixtures. They explicitly or implicitly assume all liquids to have the same configurational structure. For mixtures with components of low molar mass, this assumption apparently leads to acceptable results. In mixtures involving a polymer, however, the free volume dissimilarities may be significant. A thorough discussion of this point is found in Elbro et al. (1989). It has been shown by Delmas et al.
(1962) that the free volume differences are responsible for the occurrence of lower critical solution temperatures (LCSTs) of polymer solutions at sufficiently high temperatures. Treatment of these effects requires a model where density, besides temperature and composition, enters as a variable; in other words, an equation of state is needed. To develop an equation of state for liquids and liquid mixtures, it is convenient to start with the generalized van der Waals canonical partition function (Sandler, 1985). For a pure fluid, it is, 2 = (l/n!)(Vr/A3)"(q,,,)n exp(-E/PkT)n (1) where n is the number of molecules in the total volume
0888-5885/90/ 2629-0875$O2.50/0 0 1990 American Chemical Society
876
Ind. Eng. Chem. Res., Vol. 29, No. 5 , 1990
V a t temperature T, k is the Boltzmann constant, E is the mean intermolecular (attractive) energy, and '1 is the de Broglie wavelength depending only on temperature and molecular mass. V, is the free volume. The term qr," represents the contribution (per molecule) from rotational and vibrational degrees of freedom. Prigogine (1957) suggested that the rotational and vibrational degrees of freedom be factored into two parts: an external contribution that depends on volume and temperature and an internal contribution that depends only on temperature. Prigogine suggested further that, for long-chain molecules, the external rotational and vibrational degrees of freedom can be considered as equivalent to translational degrees of freedom. This treatment was adopted by Flory et al. (1964), who proposed the following simple configurational integral for a pure component:
orientations and ordered orientations, respectively. w is a measure of the relative number of favorable configurations, and b is a measure of the number of segment units engaged in the favorable orientations. The values chosen for w and b were w = 0.0025 and b = 2.0. By use of this partition function and eq 3, the resulting equation of state is
-(v -)
P = nRT 61f3 + C 6113
+ Ev
In these expressions, qi is the segment-surface area of component i, ni is the moles, Bi is the segment-surface area fraction, Vi* is the hard core volume, and Ciis the external degree of freedom parameter. E is given by
where where 6 = v / V *; V * is the molar hard core volume, and t is the average configurational potential energy per segment-surface area, q. The relationship between the canonical partition function and the equation of state is P = bT( d In 2
-1
to"i
ell
=
T
+
[+ 1
w
exp(-bt,,,,/RTO)
w[exp(-bt,,ji/RTii) - 11
7) T,n
Applying eq 3 yields the following form of the Flory equation of state: p ~ ? 5113 1 -=---
i;
$13
- 1
ET
(4)
where = p / p * , L? = u/V*, = TIT*. p * , V*, and T * are characteristic parameters that can be determined from volumetric data in several ways (Flory et al., 1964; Lichtenthaler et al., 1978). The Flory equation of state has been applied for correlating phase equilibria in mixtures with nonpolar and slightly polar compounds, including polymer-solvent mixtures. Experimental data are required to determine the parameters in the Flory equation of state, so it is not a predictive method. Taking the Flory equation of state as a starting point, Holten-Andersen and co-workers (Holten-Andersen, 1985; Holten-Andersen et al., 1986, 1987) have developed a group-contribution method applicable for predicting phase equilibria in polymer solutions. Three major changes have been introduced: changes in the free volume term, changes in the attractive energy term, and the introduction of group-contribution approach. Holten-Andersen (1985) showed that the Flory equation of state does not reduce to the ideal gas limit at low density. In an attempt to remedy these difficulties, the free volume term was changed to
The attractive energy term of the partition function was derived as
random orientations
where
cg
and
t_
favorable orientations
are interaction energies from random
l a J , = a,, - a,, - TAS:P
(12)
A S ?b is an extra binary entropic interaction parameter introduced to correct for additional interactions due to hydrogen bonding of OH groups. The Holten-Andersen equation of state (EOS) appears to be a powerful technique for predicting polymer solution thermodynamic properties. However, unfortunately, as High and Danner (1988) have demonstrated, the HoltenAndersen EOS is rather complicated to use and to understand. The purpose of this work is to develop a new equation of state that is similar to that of Holten-Andersen and overcomes the complications of the Holten-Andersen model.
Group-Contribution Flory Equation of State To make the model more readily applicable, the Holten-Andersen model has been modified on the following three points: Energy Term of Favorable Orientations. In the Holten-Andersen EOS, the attractive potential between two molecules was represented as a sum of energy of random orientations, to, and of favorable orientations, c ~ It is the intention that this distinction would be advantageous in describing a number of phenomena met in polymer solutions, and for pure compounds, the distinction would explain the fact that the packing of long-chain compounds is more pronounced than for short-chain compounds. This type of treatment appears to be physically reasonable, but it causes the model to be complex to use. The terms of eq 6 are analyzed below. The energy of favorable orientation is given by the second, bracketed term. T o estimate the significance of this term, the value of the term -w + w exp(-t,b/RTii) for some n-alkanes and for polymers may be compared with unity. The comparison is given in Table I. It may be seen that the values of this term for all the listed compounds are much smaller than unity. As the molecular weight increases, the values
.
Ind. Eng. Chem. Res., Vol. 29, No. 5, 1990 877 Table I. Influence of Ordered Orientations in Equation 6 compd C6H14
C8H18 C10H22 C12H26 C14H30 C16H34 C18H38
C20H12 CZ4H50 C36H74
C40H82 C1BH98 C50H102
CSOH122
-w
+w
cut
T, K
exp(-c,b/RTii)
cal/mol
313 313 313 313 313 313 313 313 313 313 313 313 313 313
0.009 344 0.013 69 0.01664 0.01868 0.020 14 0.021 20 0.022 01 0.022 64 0.023 54 0.024 92 0.025 17 0.025 52 0.025 59 0.025 86
-650 -752 -802 -830 -848 -860 -868 -874 -882 -892 -893 -895 -896 -897
313
0.026 86
-900
313 373 423 473
0.026 93 0.015 89 0.011 29 0.008 505
-900 -900 -900 -900
313
0.028 66
-900
313 373 423
0.028 68 0.017 00 0.012 13
-900 -900 -900
[
\
k i In EOj ex.(
V* = CniVi*/n
13
1 co
I
1
E = x - ~ q i n i tii i 2
tu
i
+
I
EOj exp(-(Atji - TAS!f)/RT)Atji I
x 8 k eXp(-(Atki - TAskF)/RT) k
(15)
C-Parameter Correlation. A new correlation for the C parameter of component i has been introduced instead of the simple, linear temperature dependency of the C parameter in the Holten-Andersen model:
loco
CARBON NUMBER
Figure 1. Orientational energy
(14)
Equation 8 is now changed as follows:
I
-330
2)]))
C = CniCj/n
i
PIB ( n = 714) MW = 40000 PIB (n = 71428) MW = 4000 000
-
Applying the new partition function yields the equation of state in the same analytical form as eq 7. For mixtures, the linear mixing rule is used for the hard core volume V* and the C parameter:
PE (n = 357) MW = 10000 PE ( n = 6428) MW = 180000
pression has been adopted for mixtures instead of the random mixing expression of the Flory model:
vs carbon number.
do not change very much. For the same compound, the value of the -w + w exp(-t,b/RTu') term decreases with increasing temperature. Figure 1 shows t, vs carbon number for n-alkanes. It may be seen that the absolute value of t, increases with the chain length and reaches a constant value after n (number of carbon atoms) is greater than 50. We conclude that the term describing the energy of favorable orientations has a negligible effect compared to the introduction of the group-contribution approach. This conclusion should be especially correct for the mixtures with polymers where the chain lengths are much longer than those of small solvents. As pointed out by Holten-Andersen (1985), there are systems where this term may be significant when the properties of mixtures are correlated by fitting molecular interaction parameters. In this work, an attractive energy partition function that is similar to the Flory expression has been adopted: (13) In addition, by considering the appreciable differences of attractive potential between the compounds in polymer-solvent systems, a nonrandom UNIQUAC-like ex-
where v,,,~ is the number of group n in molecule i, the reference temperature To is taken as 298.15K, R, is the normalized van der Waals volume for group n given as in the UNIFAC model (Fredenslund et al., 1977), and CT n, CT,n,and C,O are the temperature-independent term, tke temperature coefficient, and a constant term for group n taking into account special effects. Note that the division of molecules into groups is performed in a similar way as in the UNIFAC model. The introduction of "branching" as in the Holten-Andersen model is thereby avoided. The temperature dependence of the C parameter is rigorously introduced into the Helmholtz free energy, and a new energy expression is obtained, replacing eq 15: 1 E = C-zqini i2
I
tii
+
l&]
EOj exp(-(Atji - TAS$b)/RT)Atji i
XOk k
eXp(-(A€ki - ThSkb)/RT)
3R In (-)?ni[
(17)
The new model is based on the group-contribution approach. The group-contribution expressions for the molecular parameters, Vi*,qi, to,+ and AS;b are Vi* = C21.238Rm m
qi =
CQm m
(20) (21)
878 Ind. Eng. Chem. Res., Vol. 29, No. 5 , 1990
m
n
In these expressions, indexes m and n refer to groups m and n and i and J refer to molecules i and j. Expressions for the activity coefficients may now be derived by using classical thermodynamics; if we write the three contributions to the activity coefficients separately, In 1, = In
yicomb
+
combinatorial
In yJf'
+
free volume
In
fratt
(25)
attractive
then
Figure 2. X-Y diagram for acetone (1)-decane (2) a t 333 K.
where pi is a segment volume fraction, y-1
-
CX]vj*
Figure 3. X-Y diagram for chloroform (1)-4-methyl-Z-pentanone (2) a t 760 mmHg.
I
Inclusion of Pressure. Holten-Andersen (1985) simplified the equation of state by assuming the pressure of the system to be zero. This approximation is good for dense liquids, since pressure does not significantly influence liquid-phase properties. However, this approximation will cause some difficulties when the temperature of the system is raised to a point where the equation of state does not have a liquid root for the solvent. We distinguish between three different cases concerning liquid roots. If the liquid root for the solvent can be obtained a t the state of mixtures, the simplified zero-pressure equation of state is used directly. When the liquid root for the solvent cannot be obtained, a sufficiently high pressure (Pmin) is imposed to produce a liquidlike root for the solvent. But when the temperature is over the critical temperature of the solvent predicted by the equation of state, the model cannot be used. Our computer program corresponding to the group-contribution Flory equation of state automatically distinguishes between these three cases. In case 3, no computations can be made. Parameter Estimation The model presented has parameters related to pure and parameters related component properties (Ciand to properties of binary mixtures (eou and ASjb). As shown above, these parameters are in turn calculated from the group parameters C , , , CT,,, Cn0, 6, , At,,,, and, for hydrogen bonding systems, also from AS%. The information used to determine these group parameters are thermal expansivities ( a = (1/v)(aV/877,) and thermal pressure coefficients (0= (dP/aT),) of the pure liquids and VLE data for binarv mixtures of components with low molar
//
I
Figure 4. X-Y diagram for 1-hexene (1)-vinyl acetate (2) a t 323 K.
mass (Gmehling and Onken, 1977). Where thermal pressure coefficients are not available, heats of vaporization have been used. The parameters so obtained are given in Tables I1 and 111. In Table 11, the values of C , ,for some of the groups are negative, which reflects the somewhat empirical nature of the model.
Calculation of Pure Component and Solvent Mixture Properties Table IV presents a selection of experimental and correlated pure component properties. It may be seen that the present model is able to correlate the data for both pure polymers and pure solvents with good accuracy and
Ind. Eng. Chem. Res., Vol. 29, No. 5, 1990 879 Table 11. Group R , Q ,and C Values of the GC Flory EOS" main group subgroup no. 1
name CHZ
no. 1 2 3 4 5
2
ACH
6 n
3
CHZCO
8
4
coo
9
5
CHzO
10 11
12
6
OH
13 14
7
c=c
15 16 17 18
8
cc1
19 20
9
CCl2
21 22
10
cc13
23 24
11
CCl,
25
name R Q C," CH, 0.9011 0.848 -0.073 82 CH, 0.6744 0.540 0.1080 CH 0.4469 0.228 0.344 2 C 0.2195 0.000 0.477 9 c-CHZ 0.6744 0.540 0.007 814 2,2,4-trimethylpentane: 5(1), 1(2), 1(3), l(4) cyclohexane: 6(5) ACH 0.5313 0.400 0.007 046 AC 0.3652 0.120 0.287 4 toluene: 1(1),5(6), l(7) c=o 0.7713 0.640 0.356 2 2-butanone: 2(1), 1(2), l(8) coo 1.0020 0.880 0.368 2 butyl acetate: 2(1), 3(2), l(9) CHzO 0.9183 0.780 0.3180 CHO 0.6908 0.468 0.318 0 CH,OCHz 1.5927 1.320 0.094 49 diisopropyl ether: 4(1), 1(3), l(11) diethyl ether: 2(1), 1(2), l(10) 1,4-dioxane: 2(12) OH 0.7000 1.200 0.036 41 CH30H 1.0000 1.000 0.031 72 ethanol: 1(1),1(2), l(13) methanol: l(14) CHz=CH 1.3454 1.176 0.1503 CH=CH 1.1167 0.867 0.176 2 CH2=C 1.1173 0.988 0.496 1 CH=C 0.8886 0.676 0.321 0 isoprene: 1(1),1(15), l(17) CHzCl 1.4654 1.264 -0.233 4 CHCl 1.2380 0.952 0.709 5 1,2,3-trichloropropane: 2(19), l(22) CHzClz 2.2564 1.988 -0.358 7 CHClz 2.0606 1.684 -0.072 92 1,1,2-trichloroethane: 1(19), l(22) CHC13 2.8700 2.410 -0.004 503 CC13 2.6410 2.184 0.194 5 l,l,l-trichloroethane: 1(1), l(24) CCl, 3.3900 2.910 -0.064 53 tetrachloromethane: l(25)
c,o
CT,*
-3.570 -3.570 -3.570 -3.570 -3.570
0 0 0 0 0
-2.020 -2.020
0.200 0 0.2000
-6.647
0
6.139
0
3.383 3.383 -6.719
0 0 0
6.901 6.901
-0.002 66 -0.002 66
35.95 35.95 35.95 35.95
0 0 0 0
-7.668 -7.668
0 0
-7.668 -7.668
0 0
6.199 6.199
0 0
-15.07
0
a Here and in other tables. GC Florv EOS is an abbreviation for group-contribution Flory equation of state. "Cyclohexane: 6(5)" means that one molecule of cyclohexane is composed of six "c-CHz* grou&.
that the new C-parameter correlation gives a correct temperature dependence. Figures 2-5 show predicted vapor-liquid equilibria for four different solvent mixtures. Note that none of the mixtures shown were included a t all in the data base for estimating parameters. Such results are in general found to be of similar quality as those predicted by UNIFAC. Chlorinated compounds have a very varying behavior, where a considerable difference is noted between the behavior of monosubstituted and trisubstituted components. In this work, it has been chosen to represent their behavior through four different groups (CC1, CC12, CCl,, CC1,). The result for the chloroform-4-methyl-2-pentanonesystem in Figure 3 indicates that the behavior of this type of system has been well represented.
00
02
04
OE
08
10
MCiLE F?ACTION, X '
Prediction of Vapor-Liquid Equilibria in Mixtures Containing Polymers The parameters determined from low molar mass compounds, as shown in Tables I1 and 111, have been used to predict the vapor-liquid equilibria in mixtures with polymers.
Figure 5. X - Y diagram for 1-hexene (1)-ethanol (2) a t 333 K.
Figures 6-9 show predicted and experimental solvent activities in PS, PIB, and PPO. In these figures, the computed activities from the group-contribution Flory EOS and the Holten-Andersen model are indicated by the
880 Ind. Eng. Chem. Res., Vol. 29, No. 5 , 1990 Table 111. Group-Interaction Parameters, Caloriedq-unit" CHp ACH CHpCO COO CHpO CHpOCHp CH2 -544 2.73 420 126 110 29 ACH
-840
CHzCO
C=C 1.57
CCl 21.3
CC12 21.3
CC13 7.71
CCI, 5.10
360
83
52
4.42
-1.86
-19.6
-19.6
-9.92
-4.80
-2320
-87
171
na
371
82.5
82.5
145
317
-1660
-10.7
-31
180
-16.6
-16.6
12.8
123
-840
0
70
89.6
-72.6
-130
79.4
-838
-6.17
na
na
-24.9
na
-714
42.4
-5.08
-2.54
-14
-633
na
na
na
-633
na
na
-778
na
coo CHpO CHpOCHp
c=c CCI CClp CClB
cc1,
-622
OH 1.70
2.12
"The values in the diagonal are
-1.60 6".
1.84
"n "- , ci
C L
1.96
The off-diagonal top values are Ac,,.
Table IV. Properties of Pure Components component T,K Pex, &, PIB (MW = 40000) 273 0.3026 0.3248 298 0.2715 0.2858 0.2435 0.2535 323 0.2140 0.2213 353 0.1967 0.2030 373 0.1620 0.1654 425 0.3578 0.3608 PS (MW = 3524) 283 0.2537 0.2522 C16H34 293 0.2274 0.2263 313 0.2044 0.2037 333 0.1841 0.1838 353 0.1660 0.1661 373 0.1496 0.1502 393 0.3076 0.3187 ethyl acetate 273 0.2836 0.2772 293 0.2516 0.2407 313 0.2365 0.2240 323 0.2212 0.2081 333 0.2050 0.1931 343
:;'
-2.47
c 5
0.3
WCiGdT FSACTICN OF BENZCNC
4.65
-1.61
-1.61
2.07
2.03
OH 413 -2.13 453 -2.34 -17.3 1.41 476 -0.79 -457 1.41 482 -1.47 1340 -1.62 275 1.33 275 1.33 330 -2.63 504 -2.10 - 1950 0
The off-diagonal bottom values are AS':\. 1:
1
% error
7.3 5.3 4.1
3.4 3.2 2.1 0.8 0.6 0.5 0.3 0.2 0.0 0.4 3.6 2.2 4.3 5.3 5.9 5.8
i o
Figure 6. Solvent activities in polymers.
solid and dashed lines, respectively. The stars represent the experimental data (Wen et al., 1989). For all four systems, the predictions of solvent activities using the group-contribution Flory EOS are satisfactory and as good as or often even better than those resulting from the Holten-Andersen model.
I
^ : i
1
d
? n i
d
02
0 4
05
WEIGHT 'RACTION
08
C
OF C 6 H 1 2
Figure 7. Solvent activities in polymers.
Figure 8. Solvent activities in polymers.
The Holten-Andersen model may not be used for systems with cyclohexane. In this work, a c-CH2 subgroup has been included into CH, main group. Figure 9 shows that this treatment is successful. Table V presents observed and predicted solvent activity coefficients at infinite dilution of solvents in polymers. The results of the Holten-Andersen EOS and the UNIFAC-FV
Ind. Eng. Chem. Res., Vol. 29, No. 5, 1990 881 Table V. Predictions of Infinite Activity Coefficient no.
svstem PE (MW = 35000) + hexane hexane heptane octane PIB (MW = 53000) + hexane cyclohexane benzene toluene PS (MW = 3524) + benzene PS (MW = 1780000) + toluene PS (MW = 76000) + decane PS (MW = 20000) + MEK cyclohexane butyl acetate octanol PEO (MW = 7500) + hexane cyclohexane toluene MEK acetone PVA (MW = 83350) + chloroform benzene ethyl acetate PMA (MW = 63200) + 2,2,4-CsHi~ cyclohexane PVC (MW = 41000) + acetone MEK chloroform heptane benzene ethyl acetate
1
2 3 4 5 6 7 8 9 10 11
12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 total
T. K
exp data R"
n-
GC-FLORY
398 423 398 398
5.73 5.91 5.10 4.74
6.21 6.54 5.56 5.12
423 423 323 323
7.59 5.40 6.56 5.86
395
JHA EOS
UNIFAC-FV R" %
n-
%
8.4 10.6 9.0 8.0
5.04 5.81 4.59 4.28
12.0 1.72 10.0 9.7
7.26 4.25 6.53 6.01
26.7 28.1 28.0 26.8
6.87 5.29 6.85 7.39
9.5 2.0 4.4 26.1
6.62 na 8.14 9.40
12.8 na 24.1 60.4
6.17 4.64 6.92 6.32
18.7 14.0 5.6 7.8
5.34
6.06
13.5
4.83
9.6
4.16
22.1
395
5.50
5.78
5.1
4.67
15.1
4.32
21.4
473
11.29
11.20
0.8
19.09
69.1
10.58
6.3
435 445 445 445
14.67 8.75 9.92 14.68
15.82 8.81 4.38 10.20
7.8 0.7 55.8 30.5
na na 7.83 14.50
naa nab 21.1 1.2
7.83 6.61 6.34 14.20
46.6 24.4 36.1 3.3
344 372 344 363 344
33.87 18.31 5.47 6.38 7.44
32.S' 18.82 5.55 6.50 6.99
3.1 2.8 1.5 1.9 6.0
40.87 na 7.33 7.74 7.59
20.7 na 34.0 21.3 2.0
na na na na na
nac na na na na
398 373 423
1.97 5.56 5.92
1.73 6.94 5.88
12.2 24.8 0.7
na 7.14 6.76
na 28.4 14.2
na na na
na na na
383 383
35.49 18.27
33.91 17.50
4.4 4.2
25.32 na
28.7 na
na na
na na
393 393 393 393 393 393
11.82 10.87 6.06 44.98 9.88 12.37
12.86 10.78 7.25 44.00 9.92 12.20
8.8 0.8 19.6 2.2 0.4 1.4 9.2
na na na na na na
na na na na na na 20.8
na na na na na na
na na na na na na 21.1
%d
OTemperature is higher than the critical temperature. bThe group parameters are not available. CThedensity of the polymer a t the system temperature is not available. % means percent error, 100(lyexp- ycdl/yexP). ''O
GC
predictions are of the same general quality as those shown in this work.
1
y GO
I 02
0 4
06
39
12
WEIGHT FRACTION OF BENZENE
Figure 9. Solvent activities in polymers.
activity coefficient model (Oishi and Prausnitz, 1978) are also given for comparison. For most of the systems studied, the group-contribution Flory EOS compares favorably to the more complicated Holten-Andersen model, which again gives results comparable with the UNIFAC-FV model. Chen (1991) shows VLE predictions for a large number of solvent-solvent and polymer-solvent systems. The
Conclusion A group-contribution modified flory equation of state has been developed. A new C-parameter correlation has been introduced. Sample results from the correlation and prediction of pure component liquid-phase properties and of vapor-liquid equilibria for both solvent mixtures and mixtures of solvents and polymers are presented. The results obtained with the new model are as good as and often even better than those of the Holten-Andersen model. The new model may be applied over a wider range of temperature compared to the Holten-Andersen model. The application of the model to liquid-liquid equilibria, including cloud points, and to polymer mutual compatibility prediction is under consideration. Acknowledgment We thank S ~ r e nDah1 for his advice and help in the computations.
Nomenclature MEK = methyl ethyl ketone PE = polyethylene
I n d . Eng. C h e m . Res. 1990.29, 882-891
882
Fredenslund, Aa.; Gmehling, J.; Rasmussen, P. Vapor-Liquid Equilibria using UNIFAC; Elsevier Scientific: New York, 1977. Gmehling, J.; Onken, U. Vapor-Liquid Eyuilibrium Data Collection: DECHEMA Chemistry Data Series: DECHEMA: Frankfurt Ihf., 1977; Vol. I. High, M. S.; Danner, R. P. Prediction of Solvent Activities in Polymer Solutions. Presented at the AIChE Meeting. Washington, DC, Nov 1988. Holten-Andersen, J. Group Contribution Model for Phase Equilibria of Polymer Solutions. Ph.D. Thesis, The Technical C'niversity o: Denmark, Lyngby, 1985. Holten-iZndersen, J.; Fredenslund, Aa.; Rasmussen, P.; Carvoli, G. Phase Equilibria in Polymer Solutions by Group Contribution. Fluid Phase Equilih. 1986, 29, 357. Holten-Andersen, J.; Rasmussen, P.; Fredenslund, Aa. Phase Equi lihria of Polymer Solutions by Group Contribution. 1. VaporLiquid Equilibria. /rid. Eng. Chem. Res. 1987, 26. 1382. Lichtenthaler, R. N.: Liu, D. D.: Prausnitz. J. M. Mar.romolecui,. . 1978. 2 2 . 192. Oishi, T.; Prausnitz, J. M, Estimation of Solvent Activities in Polymer Solutions Using a Group Contribution Method. Ind. Eng. Chem. Process Des. Deu. 1978, 17, 333. Prigogine, I. The Molecular Theory of Solutions; North Holland. Amsterdam, 1967. Sandler, S. I. The Generalized van der Waals Partition Function. I . Basic Theory. Fluid Phase Eyuilih. 1985, 19, 233. Wen, H.; Elbro, H. S.; Alessi, P. Data Collection on Poiymer Containing Solutions and Blends; DECHEMA Chemistry Data Series: DECHEMA: Frankfurt/M.. preliminary version 1989.
PEO = poly(ethy1ene oxide) PIB = polyisobutylene PMA = poly(methy1 acrylate) PPO = poly(propy1ene oxide) PS = polystyrene PVA = polyivinyl acetate) PVC = poly(viny1 chloride) Registry No. PE, 9002-88-4;PEO, 25322-68-3; PIB, 9003-27-4; PMA, 9003-21-8;PPO, 25322-69-4;PS, 9003-53-6;PVA, 9003-20-7; PVC. 9002-86-2; MEK. 78-93-3;C&14, 110-54-3;CaH18, 111-65-9; C10H22, 124-18-5; C12H26, 112-40-3; C14H30, 629-59-4; C16H34, 544-76-3; C,,H,,, 593-45-3; C20H42, 112-95-8; C24H50, 646-31-1; C33H74, 630-06-8;C40H82, 4181-95-7;C48H98, 7098-26-2;CSoH102, 6596-40-3: C($,22, 7667-80-3; cyclohexane, 110-82-7; toluene, 108-88-3;acetone, 67-64-1;chloroform, 67-66-3;benzene, 71-43-2; ethyl acetate, 141-78-6;2,2,4-C8HI8, 540-84-1;heptane, 142-82-5; 4-methyl-2-pentanone. 108-10-1;1-hexene, 592-41-6:vinyl acetate, 108-05-3: ethanol, 64-17-5.
L i t e r a t u r e Cited Chen, F. Group-Contribution Model for Mixtures with Polymers. Ph.D. Thesis. The Technical University of Denmark. Lyngby, 1991. Delmas, G.; Patterson, D.; Somcynsky, T. J . Polym. Sci. 1962,57, 79. Elbro: H. S.; Fredenslund, Aa.; Rasmussen, P. A New Simple Equation for A Free Volume Activity Coefficient. Predictions of Solvent Activities in Polymer Solutions. SEP 8913; The Technical Lniversity of Denmark: Lyngby. Denmark. 1989, to he published in ,I.lnrromolecuies. Flory, P. J.,Orwoll. R. .4,; Vrij. pi. J . ilm. Chem. Soc. 1964. 86. 3507.
Rrceiced for reuieu October 10, 1989 Accepted ,January 25, 199Q
Effect of Surfactants on Three-phase Fluidized Bed Hydrodynamics' Rajeev L. Gorowarat and Liang-Shih Fan* D e p a r t m e n t of Chemical Engineering, The Ohio S t a t e 1 iniuersity, Columbus, Ohm 43210
Experiments were conducted t o discern t h e relationship between three-phase fluidized bed hydrodynamics and surfactant solution characteristics. T h e standard characteristic, equilibrium surface tension, is inadequate. A novel method for surface tension evaluation, a dynamic maximum bubble pressure technique, was found to differentiate t h e 12 different solutions studied. T h e surfactant solutions were categorized based upon a combination of the terminal bubble rise velocity reduction, the equilibrium surface tension, and the new bubble surface tension values. These surfactant solution categories were correlated with experimentally observed three-phase fluidized bed and bubble column hydrodynamic behavior. Specifically, empirical correlations for gas holdup are presented Much of the fundamental laboratory work on threephase fluidization hydrodynamics has involved ideal airwater-glass bead systems or, instead of water, another pure fluid. It could be expected that a solution would behave similarly to a pure fluid with the same physical properties. This is not the case for surface tension effects, where behavior has been contrary to that expectation (Kelkar et al., 1983; Saberian-Broudjenni et al., 1984; Tarmy et al., 1984; Fan et al., 1986a). Dramatic changes in system hydrodynamics can occur with the addition of a surfactant. Surfactants affect hydrodynamic behavior of three-phase fluidized beds (and two-phase bubble columns) in three ways. First, the bubble size is generally reduced, increasing the interfacial area. Second, surface tension gradients are
developed around the bubble surface, reducing internal circulation, and hence rise velocity, as well as surface turbulence. Third, bubble coalescence is inhibited. by a variety of proposed mechanisms. Better understanding of the mechanism of surfactant behavior in three-phase fluidized beds may lead to more readily generalized models and correlations developed from air-water-glass bead systems, which can be extended to the complex mixtures in active reactor systems. Ideally. from a sample of desired reactant mixture produced in a laboratory, such as a fermentation bioreactor, the behavior of a three-phase fluidized bed can be predicted. This study provides a methodology for predicting the hydrodynamic behavior and a framework for further study.
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Brief L i t e r a t u r e Review Three-phase fluidized bed fundamentals have been reviewed by Epstein (19811, Muroyama and Fan (19851, Darton (1985), and more recently by Fan (1989). Surface tension of a pure liquid does not Seem to significantly affect the hydrodynamic behavior observed in a three-phase
* To whom all correspondence should be addressed.
'Paper 22i presented at the AIChE Annual Meeting, Washington. DC, Nov 27-Dec 2, 1988. :Current address: E. I. du Pont de Nemours & Co., Inc ,
Chemic& and Pigments Department, Research and Development Division, Edgemoor, DE 19809 0888-5885/90/2629-0882$02.50/0
c
1990 American Chemical Societk
