Ind. Eng. Chem. Res. 1987,26, 1382-1390
1382
Phase Equilibria of Polymer Solutions by Group Contribution. 1. Vapor-Liquid Equilibria John Holten-Andersen, Peter Rasmussen, and Aage Fredenslund* Instituttet f o r Kemiteknik, The Technical University of Denmark, DK-2800Lyngby, Denmark
A new group-contribution model for predicting phase equilibria in polymer solutions is presented. T h e model has the form of a liquid-phase equation of state and is a n extension of the work of Prigogine, Patterson, and Flory. T h e equation of state contains a free volume term, which leads to the correct ideal gas limit. In addition, the equation has an attractive term which takes into account both random intermolecular orientations and favored intermolecular configurations. The molecular model parameters are obtained by using the group-contribution approach. Results from the correlation and prediction of pure-component liquid-phase properties and of vapor-liquid equilibria of solvent mixtures and polymer solutions are presented. T h e results show that the new model is able t o correlate and predict with good accuracy both pure-component PVT properties and vapor-liquid equilibria of mixtures containing polymers. Phase equilibria play an important role in the processing and application of polymers. Knowledge of the equilibrium behavior of specific systems is often necessary in order to design polymer manufacturing processes or to predict process performance. In the area of polymer solutions, empirical extensions of the solubility parameter concept (Hansen, 1967) have provided qualitative information regarding phase equilibria in polymer solutions. This is still the most widely used method in industry. However, for the last 20 years the understanding of phase equilibria in polymer solutions has been considerably extended, both from an experimental and a modeling point of view. In the same period new, predictive models have been developed for describing phase equilibria in mixtures of low molar mass compounds. Group-contribution models such as UNIFAC and ASOG (see, for example, Fredenslund et al., 1977) have gained extensive application in the chemical processing industry. This work describes a new method for semiquantitative prediction of phase equilibria in polymer solutions. The method combines modern theories of polymer solutions with the group-contribution approach. The method is primarily aimed a t solutions rather concentrated in polymers, and it is limited to the normal pressure range and temperatures between approximately 275 and 425 K. It is thus well suited for applications within, e.g., the coatings industry.
Model Development The Flory-Huggins expression, based on lattice theory, for the Gibbs function of mixing (Flory, 1942; Huggins, 1942) of nz moles of a polymer with nl moles of a solvent is
AG/RT = n, In cpl
+ n2 In cpz + Xplcpz(nlr, + nzrz)
(1)
combinatorial term residual term where ri is the volume of component i expressed as the number of segments in a lattice (segment volume), pi is the segment-volume fraction of component i, and x is conceived of as a parameter describing the energetic interactions in the lattice. In practice, the x parameter exhibits considerable variation with temperature, pressure, and even composition. Various empirical schemes have been proposed for correlating these variations. However, such procedures are not 0888-58851 87 12626-1382$01.50/0
amenable for generalization, and therefore theoretically based arguments for these variations have been given much attention (see, for example, Bonner, 1975). Two different kinds of arguments have been proposed: arguments concerned with the effects of nonrandomness and arguments concerned with component density differences. Nonrandomness. The residual term of the FloryHuggins expression assumes that the molecular segments are randomly mixed in the solution. However, for mixtures where the attractive potentials between the components differ appreciably, this zero-order approximation is incorrect. Guggenheim (1952) has developed a procedure for a first-order approximation to remedy this situation. In this approximation, each configuration of molecular contacts is assigned a statistical weight according to its configurational energy. The procedure does, however, not lead to explicit equations for the thermodynamic potentials and is therefore numerically inconvenient. The same concept of nonrandomness is less rigorously embodied in the “local Composition” concept, according to which the local composition of species departs from the overall composition according to an expression involving local interaction energies. This concept leads to an expression for the residual e x c w Gibbs function which has two parameters per binary interaction and which is explicit. In the UNIQUAC model (Abrams and Prausnitz, 1975), the residual Gibbs function for a binary mixture is given by AGreS/RT= -qlnl In (6, + OZ exp(-Af,,/RT)) q2nz In (6, + 8, exp(-At12/RTl) (2) where qi is the segment-surface area and Oi the segmentand Atlz are surface area fraction of component i. interaction energy parameters which must be obtained from experimental data. A major advantage of the UNIQUAC model is its ability to correlate nonsymmetric mixing functions, which is an essential feature of especially hydrogen-bonding systems. Density Differences. Density effects 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 most often leads to acceptable results. In mixtures involving a polymer, however, there are considerable density effects which cause special contributions to the mixing functions. It has 0 1987 American Chemical Society
Ind. Eng. Chem. Res., Vol. 26, No. 7, 1987 1383 Table I. Correlation of Pure-Component Properties thermal pressure heat of coeff, vaporizaJ/m3/K X tion, J/mol 10" x 10-4 compd temp, K exptl calcd exptl calcd ref. 293 0.84 0.84 a n-Cs n-C1l3 Marlex 50 PIB CIS-polyisoprene toluene ethylbenzene PS acetone MEK MIBK ethyl acetate butyl acetate PVAC PMMA diethyl ether dibutyl ether dioxane dimethoxyethane PEO PPO 1,2-dichloroethane 1,1,2-trichloroethane PVC ethanol 2-propanol 1-butanol
1-hexanol
353 293 393 413 463 273 423 293 323 293 353 283 333 273 373 273 323 273 333 273 353 273 343 273 333 308 367 393 410 273 308 273 333 293 353 293 353 319 343 273 333 273 313 273 313 355 369 273 333 293 353 293 353 293 353
0.54 1.05 0.63 0.71 0.59 1.30 0.67 1.26 1.09 1.21 0.88 1.30 0.96 1.59 0.96
0.58 1.09 0.63 0.71 0.59 1.30 0.63 1.26 1.09 1.21 0.88 1.30 0.96 1.59 0.96
1.42 1.09 1.00 0.92 0.96 0.71
1.44 1.33 1.32 0.99
1.35 1.24
1
main group CHo
cc
a a
C
CHZCO
coo
d
4.26 4.38 d, g 4.05 4.15 1.42 i 1.09 1.00 k 0.92 0.96 d, g 0.75 3.23 3.31 d, g 2.95 3.07 3.75 3.62 d , g 3.54 3.42 3.56 3.90 d, g 3.31 3.20 1.43 C 1.30 C 1.34 1.00 3.64 3.52 d , g 3.43 3.44 3.91 4.08 d, g 3.51 3.94 1.16 k 1.11 4.37 4.37 d , g 4.06 4.10 4.52 4.58 d , g 4.03 4.16 5.23 5.02 d, g 4.73 4.68 6.07 5.83 d, g 5.46 5.42
a Orwoll and Flory, 1967. * Eichinger and Flory, 1968. Allen et al., 1960. dLandolt-Bornstein, 1971. e Hacker and Flory, 1971. fHocker etal., 1971. BDanner and Daubert, 1983. hFlory and Hocker, 1971. 'McKinfley and Goldstein, 1974. kHellwege et al.,
1962.
been shown by Delmas and others (Delmas et al., 1962) that the density difference is responsible for the occurrence of a lower critical solution temperature (LCST) of all polymer solutions a t sufficieatly high temperatures. The treatknent 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. For engineering purposes, the most common approach
OH CHzCl CHClz CHC13 CC13 CC4
0.7000 1.4644 2.0269 2.8169 2.5895 3.3795
coo
CHZO
f 3.11 d , g 2.97 3.50 h 3.31 4.35 d , g 4.08
CHBO CHzO CHO
ACH AC ACCH3 ACCH2 ACCH CHCO CHzCO CHSCO
d
e
R Q 0.9011 0.848 0.6744 0.540 0.4469 0.228 0.2195 -0.086 1.3454 1.176 1.1167 0.867 1.1173 0.988 0.676 0.8886 0.6605 0.485 0.5313 0.400 0.3652 0.120 1.2663 0.968 1.0396 0.660 0.8121 0.348 1.2157 0.990 1.310 1.4457 1.6724 1.488 1.0020 0.880 1.1450 1.088 0.9183 0.780 0.6908 0.468
subgroup CH, CH; CH C CHZ=CH CH=CH CHz=C CH=C
c=c
ACH
b
3.13 3.01 3.57 3.24 4.40 3.84 1.30 1.30 0.88 0.88
Table 11. Group R , Q , and C Values
OCHzCHzO OH CHCl
CKc
0.146
0.304
0.192
0.284 0.385 0.080
0.207 1.200 0.54 1.260 0.212 1.668 2.388 2.074 2.794
is to apply a generalized van der Waals equation of state (see, for example, Vera and Prausnitz, 1972), where the configurational partition function for a pure fluid of n molecules is factored into a free volume and an average potential, or attractive, term as 2 = (PVJ"{exp(-E/VRT)Jn
(3)
free volume attractive where VFv is the free volume of the system, V is the volume, and E the configurational potential energy of the system if all the centers of mass of the segments are at rest in the centers of the lattice cells. In van der Waals theory, VFv is given by V - nb. For long-chain molecules, Prigogine (1957) proposed to treat the density-dependent rotational motions of the molecules as equivalent to the translational motions of the center of mass and to quantify the total number of density-dependent motions (degrees of freedom) through a molecular parameter C. Instead of 3 independent translational motions per molecule, a chain molecule has 3C independent and density-dependent motions. This treatment was adapted by Flory and co-workers (Flory et al., 1964), who proposed the following simple configurational integfal for a pure component:
= (nV*)n(gl/3- 1)3Cne-nqc/RTfi (4) where v' = V/ V*, V* is the molar hard-core volume, and t is the average configurational potential energy per segment-surface area q. For a mixture, Flory et al. (1964) propose a simple linear mixing rule for the hard-core volume and the C parameter and a quadratic mixing expression for the energy: V* = CniVi*/n 1
C = CniCi/n 1
(5)
where ej is the segment-surface area fraction of segment
1384 Ind. Eng. Chem. Res., Vol. 26, No. 7, 1987 Table 111. Grow-Interaction Parameters, J/q-unit CH2 c=c ACH C Hg -1958 0 54.0
c=c
-3946
ACH
CHZCO 402
COO 753
CH2O 259
OCHZCHZO 126
0
251
536
63
42
-3096
251
536
63
42
-4226
84
-234
126
-6569
-636
285
CHZCO
coo
OH 837 -10.8 879 -8.95 879 -8.95 992 -2.47 1774 3.72 423 13.4 720 -3.01 -7992
-1987
CH20
-3126
OCHZCHZO OH 5.48
4.35
4.35
1.97
0.80
-7.28
0.94
CHCl
CHCl 631 -4.6 -4.6 100 209 397 10.0
1234 3.31 -3213
-4.84 nThe values in the diagonal are
t,,,,.
The off-diagonal top values are Atmn. The off-diagonal bottom values are
Ask;.
Table IV. Representation of Solvent Mixture Infinite Dilution Activity Coefficients, y m a component 1 toluene MEK MEK butyl acetate ethyl acetate ethyl acetate dibutyl ether diisopropyl ether diisopropyl ether diethyl ether dioxane 1,2-dimethoxyethane dioxane dioxane ethanol 1-hexanol 1-propanol 1-butanol ethanol 1-butanol 1,2-dichloroethane 1,2-dichloroethane chloroethane 1,2-dichloroethane
TIP,K/Pa 298 323 348 348 323 340 308 323 1.013 X lo5 293 353 343 323 323303 1.013 X lo5 1.013 X lo5 2.20 x 104 273 1.013 X lo5 343 1.013 X lo5 298 313
component 2 heptane heptane toluene heptane ethylbenzene acetone hexane toluene MEK ethyl acetate heptane benzene acetone ethyl acetate heptane ethylbenzene MEK butyl acetate diethyl ether dioxane heptane MIBK diethvl ether ethanol
Yy(obsd)
Yylcalcd)
1.6 3.7 1.5 2.0 1.2 1.1 1.1 1.3 2.2
1.6 3.5 1.4 1.9 1.2 1.2
1.5 3.9 1.3 1.8 1.2 1.18
1.1
1.1 1.4
1.1
2.8 1.0 1.3 1.2 22.0 2.0 1.7 1.8 4.8 1.1 2.2 0.85 1.3 5.5
1.3 2.4 1.0 3.0 1.0 1.3 1.1
20.0 2.5 2.0 2.1 5.0
?';lobsd)
2.6 1.0 4.0 1.0 1.5 1.2 7.5 1.3 1.7 2.2 2.8
1.1
1.1
2.0 0.85 1.5 5.6
3.2 0.78 1.7 8.0
Y;lcaJcd)
1.5 4.1 1.3 1.8 1.2 1.18 1.1 1.3 2.2 1.0 5.0 1.0 1.4 1.2 6.0 1.2 1.6 1.9 2.8 1.1 1.8 0.82 1.6 8.6
"The observed data are taken from Gmehling and Onken (1977).
j and represents the potential energy between segments j and i. The Flory model has been shown to account for both the temperature and the composition dependence of the residual free-energy term in nonassociating mixtures. Other models have been proposed based on the same lines of thought (Beret and Prausnitz, 1975; Sanchez and Lacombe, 1978). In the present work, the Flory model has been chosen as a starting point. Modified Flory-Prigogine Model. The model by Flory et al. has been modified on three points, the purpose of which is to render the model applicable for associating mixtures and to enable the introduction of the groupcontribution approach. The free volume term has been changed to
-
-
This expression, unlike the Flory expression, leads to the It is correct ideal gas limit, i.e., ZFV V" as V constructed by taking into account that the internal motions of the molecules are in fact not equivalent to the
translational motions of the center of mass, because these motions are restricted in space around the center of mass, while the motions of the center of mass are restricted by the total volume of the system a t hand. The modified expression leads to a somewhat less temperature-dependent C parameter, when fitted to experimental pure-component data. Equation 6 is also used for mixtures with the mixing rules of eq 5. The attractive term has been modified on two points. The first modification considers the attractive potential of two molecules to be the sum of two different contributions: (1)an energy of random orientations of two molecules and (2) an energy of favorable orientations or favorable packing configurations of the two molecules. This distinction has been found advantageous in describing a number of phenomena met in polymer solution thermodynamics. For the pure components, the distinction explains the fact that the cohesive energy density of long-chain components is larger than is expected on the basis of the shorter chain homologues. The packing of the long-chain compounds is more pronounced than for the short-chain compounds. For mixtures, the packing concept explains the phenomena of "destruction of orientational
Ind. Eng. Chem. Res., Vol. 26, No. 7, 1987 1385 1
f
Ethylacetate Polyisoprene
Benzene PIB 298 K
w1
w1 1
0
Figure 1. Solvent activities in polymer solutions:. Eichinger and Flory (1968). In all the figures, the abscissa is the weight fraction of the solvent, ul,and the ordinate is the solvent activity, a,.
1
I
Figure 3. Solvent activities in polymer solutions: Booth et al. ( 1964). 1.
1 al
MEK Polyisoprene Benzene Polyisoprene 298 K
w1 0 1
1
I
1
1
I1
Figure 2. Solvent activities in polymer solutions: Eichinger and Flory (1968).
Figure 4. Solvent activities in polymer solutions: Booth et al. (1964). 1
order" investigated by, among others, Patterson and Barbe (Barbe and Patterson, 1980). This separation of the attractive energy leads to the following form of the attractive part of the partition function: ZAW = ex.[
"'1
- 2R Tij
[(l- w
Diethylketone Polypropylene 298 K
+ w e x p ( - ~ , b / R T ~ ) ) ] (7) ~~~/~~
In this expression, eo is the potential energy per segment-surface area of random packing configurations,while E , is the extra energy contribution from packing into more favorable configurations. 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 favored orientations. In the subsequent applications, the w and b parameters are considered to be constants, while E , is considered a molecular property depending upon the structure of the molecule. The proposed constancy of the w and b parameters is a purely practical measure to reduce the number of adjustable parameters. The values chosen for w and b are w = 0.0025 and b = 2.00. A short derivation of eq 7 is shown in the Appendix section. The second modification of the attractive term concerns its extension to mixtures, where a nonrandom UNIQUAC-like expression has been adopted instead of the random mixing expression of the Flory model Z mix A P =. - R T In
Cej exp(-Aaji/RT))/RT I
(8)
wl 1
Figure 5. Solvent activities in polymer solutions: Brown et al. (1964).
where aji is considered to be a Helmholtz energy of interaction and given by aji = t o j i / i j - (RT/b) In (1 + w(exp(-beuji/RTu') - l)]
where s represents one segment unit. In this expression, only one binary parameter (aji = ail) is used for systems with no hydrogen bonding. For hydrogen-bonding systems, however, one parameter is not enough to adequately describe the systems. It has been found that binary vapor-liquid equilibrium data may be correlated by introducing an extra binary entropic interaction parameter, AS>b,into the attractive or residual term:
1386 Ind. Eng. Chem. Res., Vol. 26, No. 7, 1987 1
1
Toluene Polystyrene
Ace tone Polystyrcne
W1
Figure 6. Solvent activities in polymer solutions: Bawn et al. (1950).
I
I
1
0
0
1
Figure 8. Solvent activities in polymer solutions: Bawn and Wajid (1956).
1
MEK Polystyrene 298 K
Propylacetate Polystyrene 298 K
w1 0
I
1
I
I
w1
11
I
Figure 7. Solvent activities in polymer solutions: Flory and Hijcker (1971).
This extra entropy parameter for hydrogen-bonding components is assumed independent of density, and therefore it does not affect the equation of state. From the expressions given above, all necessary thermodynamic quantities may be derived. The main equations are given below. The Gibbs mixing function AG/RT = AA/RT = E n , In cp, + combinatorial Gill3 - 1 3(1 + Ci)In 7 ,113 -1
1
I
I
Figure 9. Solvent activities in polymer solutions: Bawn and Wajid (1956).
t/
7
I
Diprouvlether Poiysi;rene 298 K
I
u
free volume
Figure 10. Solvent activities in polymer solutions: Baughm (1948).
E;z(q,n,/RT)[a,,(fi) 1 - alr(fir) attractive R T In
EO, exp(-Aa,,/RT)]
(11)
I
where 0,is the pure-component reduced volume and v' the reduced volume of the mixture. The chemical potential
they are of no importance at low pressures. The pressure
P
-( -) +
= nRT filla
V
0113
C
+E
-1
where, as in eq 5,
C = GniCi/n i
V* = CniVi*/n i
1 - In
EOj exp(-Aaji/R7')
-
J
c(@j exp(-Aai,/RT)/E& I
k
eXP(-AUkj/RT))
The terms involving (dfi/dni)p,Tc,have been neglected since
and E is given by
nV*0
(13)
Ind. Eng. Chem. Res., Vol. 26, No. 7, 1987 1387
1
Acetone
Benzene PVAC 303 K
w1
w1 0.
Figure 11. Solvent activities in polymer solutions: Nakajima et al. (1959).
o exp(-bEUji/RTii)
Eji
cWji
(15) + 1 + w(exp(-bEUji/RTi7) - 1) 0
= EOji/V'
A ~ j i=
tji
-~ i i
(16)
GroupContribution Approach. The model requires the following molecular parameters: Vi*, qi,hfi, €0 'i, E, 'i; for hydrogen-bonding systems additionally ASij and ASjifib. The following group-contribution expressions are adopted for the calculation of these parameters: Vi* = C21.238Rm m
qi =
CQm
(18)
m
Ci = -0.640 + ZCgcRgc m
= E,,
Astb =
m
Ce;)CepE,, m
= -[C,,E,,]~'~
= -[E
E,ji
+ ECEGREG
n
+ AE,, W J.I. EW .l y
/2
CeE)Z:(@) - @)&b, m
n
I
I
1
,1
Figure 12. Solvent activities in polymer solutions: Matsumara and Katayama (1977).
Parameter Estimation The group parameters Czc, em,, and, for hydrogenbonding components, AS",", need to be determined from experimental information. The information used are thermal expansivities ([ 1/VI (dV/dT)p) and, if available, thermal pressure coefficients ((dP/6TjV) of the pure liquids. Where thermal pressure coefficients are not available, heats of vaporization have been used. The binary parameters have been determined from vapor-liquid equilibrium data and heats of mixing for binary mixtures of low molar mass compounds. The parameters so obtained are compiled in the Appendix section.
Correlation of Pure-Component and Solvent Mixture Properties
(19) (20)
(21) (22) (23)
In these expressions (1) indexes m and n refer to groups m and n; i and j refer to molecules i and j . (2) Rn and Q, are the normalized van der Waals volumes and surface areas for group n given as in the UNIFAC model (Gmehling et al., 1982). (3) C"; is a degree of freedom group parameter for groups situated in the main chain. (4) The quantity CkGis the degree of freedom parameter for branch point groups (e.g., the CH group in propane, 2methyl). It is calculated as CEG = Czc - 0.015. We use the relevant group volume parameters from UNIFAC (R,) for RZC and RZG. (5) E,,,,and AS:\ are group-interaction parameters. (6) tUjj is a pure-component molecular parameter which has been assigned the value -2720 J/q-unit (a q-unit is the arbitrary surface area unit used in the UNIFAC model; see, for example, Fredenslund et al., 1977) for solvents and -3975 J/q-unit for polymers. These average values may, however, in specific instances be adjusted if required. The model in this work has the following parameters: nonadjustable, R and Q, (from UNIFAC); adjustable, C:", and A S g . All other quantities entering into eq 17-23, such as Vi*, qi,Ci, CEG,RZC,REG,and eoji are obtained from these parameters.
Table I presents a selection of pure-component, energy-related properties and the correlation of these on the basis of the group parameters given in Tables I1 and 111. The general conclusion that may be drawn from these results is that a variety of different components, including polymeric species, are well correlated by a limited number of group parameters. The quality of the results for both solvents and polymers is achieved through the packing parameter, e,, which for the solvents has the value of -2720 J/q-unit, while for the polymers it has the value of -3975 J/q-unit. Detailed studies reveal some scatter around these average values. It has been found, when fitting the parameters to thermal pressure coefficients of the pure substances, that the geometric regular structure of polyethylene yields a lower packing parameter than that of, for example, polyisobutylene. Likewise, the geometric regularity of the benzene molecule corresponds to a lower packing energy than that of ethylbenzene or xylene. However, for the present purpose, it does not appear justified to make such fine distinctions. Table IV presents a selection of binary vapor-liquid equilibrium data, reduced to infinite dilution activity coefficients by extrapolation to zero concentration. The present group-contribution method is generally able to correlate the data to within a relative error of 10%. The quality of the results naturally depends on the detailed distinctions made with respect to the different groups. Two types of compounds have caused some consideration in this respect: the ethers and the chlorinated compounds. It was not possible to correlate the data of all ether compounds by using just one ether group. It appeared that two ether groups in proximity of one another (as in dioxane, 192-dimethoxyethane) gave rise to quite different properties from the normal, isolated ether groups. This
1388 Ind. Eng. Chem. Res., Vol. 26, No. 7 , 1987 1
I
/
Ethylacetate PVAC
303 K
I
fen 343 K
0
1
0
I
Figure 13. Solvent activities in polymer solutions: Matsumara and Katayama (1977).
I
1
1
w1
1
Figure 15. Solvent activities in polymer solutions: Booth and Devoy (1971). Table V. Observed and Predicted Infinite Dilution Activity Coefficients (Qi = a;/w;) component 1 component 2 T, K QT(obd, R&,, ref n-C6 n-C16 293 2.40 2.40 a n-C, n-C32 346 3.06 3.10 a n-C6 PIB 298 6.4 5.8 b n-C7 PIB 298 5.8 5.3 b n-C8 PIB 298 5.4 5.1 b benzene PEO 361 4.7 4.5 C MEK PEO 361 4.6 5.2 C butyl acetate PEO 361 5.0 5.7 C ethanol PEO 373 7.1 9.1 d 1-propanol PEO 373 5.5 5.8 d octane PEO 373 28.0 44.0 d toluene PS 423 5.2 5.1 toluene PVAC 423 6.4 5.8 f MEK PVAC 423 6.3 6.2 f butyl acetate PVAC 423 6.9 7.2 f 2-propanol PVAC 423 7.5 7.9 f 1-butanol PVAC 423 6.6 6.3 f hexane PBMA 423 10.4 9.8 f benzene PBMA 423 4.6 4.6 f MEK PBMA 423 7.4 6.9 f butyl acetate PBMA 423 5.9 5.6 f 1-butanol PBMA 423 7.8 8.6 f Orwoll and Flory, 1967. *Leung and Eichinger, 1974. Cheng and Bonner, 1974. dRatzch et al., 1980. eCowitz and King, 1972. 'Newmann and Prausnitz, 1973. ~
PEO 343 K
0
1
I
w1
1
Figure 14. Solvent activities in polymer solutions: Booth and Devoy (1971).
proximity effect is a well-known problem in group-contribution methods. Similar effects were exhibited by the chlorinated compounds, where a considerable difference is noted between the behavior of monosubstituted and trisubstituted components. In this work it has been chosen only to represent the mono- and disubstituted compounds, and even this simplification should be regarded with caution.
Prediction of Vapor-Liquid Equilibria in Mixtures Containing Polymers Figures 1-15 show predicted and experimental solvent activities in a series of different polymers at ambient temperatures. The predicted results are within the 10% relative error which was achieved in the correlation of infinite dilution activity coefficients for binary solvent mixtures. The systems cover polymers with quite different thermal expansivities and solvents with different degrees of polarity. The good agreement, therefore, indicates an essentially correct representation of the volumetric contributions as well as the interaction contributions to the thermodynamic mixing functions. The capability of representing the internal energies of both low and high molar mass compounds as well as their mixtures indicates the success of the group-contribution approach for calculating the interaction energies. The group-contribution method for calculating the number of density-dependent rotational degrees of freedom (the C parameter) should, however, be regarded as provisional. This parameter is most likely highly dependent on the details of the molecular structure, and therefore a group-contribution method should not in general be expected to be very reliable for determining Ci. However, as indicated by the results, the method will in
any case give qualitatively reasonable estimates of vapor-liquid equilibrium compositions. For more quantitative results, experimental thermal expansivities may be used for determining the parameter Ci,or the parameter may be used as a fitting parameter. Table V presents observed and predicted solvent activity coefficients at infinite dilution of the solvent in the polymer. These values are obtained through GLC measurements, the majority of which are performed at elevated temperatures (approximately 50 deg above the glass transition temperature of the polymer). Temperatures above 373 K are clearly outside the range where parameters are estimated. It has been found that predicted activity coefficients are generally too low as the temperature is increased above approximately 373 K. The reason for this is that the negative entropic contributions, caused by the increased difference in densities, are not correctly reproduced a t higher temperatures. The model presented is not capable of representing the density effects over a wide temperature range, which is due to inaccuracies in both the repulsive and the attractive terms with respect to their density dependence. It has, however, been found that a universal temperature function introduced into the attractive potential of the
Ind. Eng. Chem. Res., Vol. 26, No. 7, 1987 1389 pure components extends the range of the model: t0,ii(T)
=
1 - (-)0.07
ii = reduced volume, ii = V/V* wi = weight fraction of i x i = mole fraction of i 2 = configurational integral
- (-Y0.05)
(24)
This function has the effect of decreasing the cohesive energy of the liquids as temperature is increased, thereby increasing their free volume. The results given in Table V are achieved through the adoption of this empirical temperature function. The results so obtained are clearly in qualitative agreement with experimental data, and it appears that the model is able to distinguish quite accurately between the different solvents.
Conclusions A model for prediction of the thermodynamic properties of polymers and mixtures of polymers with solvents has been presented. The model is based on the group-contribution approach and has been shown to yield qualitatively reasonable predictions of pure-component liquidphase properties (thermal pressure coefficients and thermal expansivities) and vapor-liquid equilibrium compositions. The model may be applied to a large variety of mixtures. Vapor-liquid equilibria in mixtures with components ranging from, e.g., hexane to ethanol and from polyisobutylene to polyethylene glycol have been predicted with good results using a limited number of group parameters. The application of the model to liquid-liquid equilibria and to polymer mutual compatibility predictions will be illustrated in a subsequent contribution.
Acknowledgment We are most grateful to Prof. R. C. Reid, MIT,for his constructive comments.
z = coordination number
Greek Symbols
x
= Flory interaction parameter
t12 =
interaction energy
yi = mole fraction activity coefficient ( y i = a i / x i )
= chemical potential Qi= weight fraction activity coefficient (ai = ai/oi) w = model parameter = 0.0025 cpi = segment-volume fraction of i ei = segment-surface area fraction of i pi
Appendix Derivation of Equation 7. Consider the number of segments which are attached to each other in a molecule and which may pack in an oriented fashion to be b. Let us assume that a fraction w of these segments have actually been oriented. The contribution to the attractive part of the partition function from the b segments will be
[ “1
(1- w ) exp -
+ w ex.(
R Ti7
- (eo
+ t,)b R Tii
)
where to is the potential energy of random packing configurations and t, is the extra energy contribution from packing. The exponential terms are of a similar form as in eq 4. The total number of possibilities for fiiding b segments will be znq/2b and hence ZAW =
[
(1- w ) ex.[
-&I
+ w exp[
rnqJ.26
Nomenclature A = Helmholtz function a I 2= interaction Helmholtz energy ai = activity of i b = model parameter = 2.0 C = external degree of freedom parameter E = energy function G = Gibbs function MEK = methyl ethyl ketone (2-butanone) MIBK = methyl isobutyl ketone (4-methyl-2-pentanone) ni = moles of i P = pressure PBMA = polybutyl methacrylate PEO = polyethylene oxide PIB = polyisobutylene PMMA = polymethyl methacrylate PPO = polypropylene oxide PS = polystyrene PVAC = polyvinyl acetate PVC = polyvinyl chloride Q, = normalized segment area of group n q i = segment-surface area of i R = gas constant R , = normalized segment volume of group m ri = segment volume of i AS: = interaction entropy s = one segment unit T = temperature V = volume V* = hard-core volume
z A - = e x p ( - EznQEo )[
l-w+wexp(-&)]
rnq/2b
Registry No. PIB, 9003-27-4;PI, 9003-31-0;PP, 9003-07-0; PS, 9003-53-6; PVAC, 9003-20-7; PEO, 25322-68-3; PPO, 25322-69-4; PE, 9002-88-4;PMMA, 9011-14-7;PVC, 9002-86-2; PBMA, 9003-63-8.
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Received for review February 27, 1986 Accepted February 19, 1987
Catalytic Tar Conversion in Coal Gasification Systems Eddie G. Baker* and Lyle K. Mudge Battelle Pacific Northwest Laboratories, Richland, Washington 99352
Catalysts for coal tar conversion were tested in a fixed-bed catalytic reactor. Tar and water were mixed with a fuel gas to simulate coal gas from a fixed-bed gasifier. Tests were conducted at 450-690 O C , 1010-1810 kPa, and 1-8-s residence time. Acid cracking catalysts were found t o be the most effective of the catalysts tested. Coke was a primary product, and periodic regeneration was required t o maintain catalyst activity. Hydrocracking catalysts containing CoMo, NiW, and Pd reduced the coke yield from tar cracking but did not reduce the rate of catalyst deactivation.
Background The U.S. Department of Energy's Morgantown Energy Technology Center (METC) is developing technology to economically remove contaminants in hot gas streams produced by both coal gasification or combustion. Hot gas from coal gasifiers could be used in advanced power generation systems such as integrated gasification combined cycles using gas turbines and in fuel cell systems (Cicero and Jain, 1985). Fixed-bed gasification represents the most highly developed of all coal gasification technologies. However, one of its major drawbacks compared to other gasification systems is the production of tars and oils. If not removed from the gas, tars may deposit coke downstream on the high-temperature desulfurization sorbents being developed by METC, on fuel cell electrodes, or in a gas turbine. In addition, organic sulfur compounds in the tar are not removed by high-temperature desulfurization sorbents and may be detrimental to fuel cell systems or gas turbines. As a result, METC was interested in developing a means of removing tars from coal gas a t 500-750 "C and 10002000 kPa (10-20 atm) that could be integrated into advanced power plant systems utilizing fixed-bed gasifiers and hot gas cleanup. The objective of this study a t Battelle-Northwest (BNW) was to evaluate catalysts for removal or conversion of tar in hot coal gas streams. Tars account for only a small percentage of the gasification products, but they are responsible for many of the problems encountered in downstream processing. Therefore, the primary goal of this study was to remove the tar, particularly the organic sulfur compounds, from the gas. Sulfur in the tar should be converted to H2S or COS which could be removed by the high-temperature desulfurization sorbents. Ideally the tar would be converted to a gas product which would re0888-5885/87/2626-1390$01.50/0
cover the heating value of the tar in the gas stream and improve the process efficiency, but this was a secondary objective. I t was envisioned that a fixed bed of catalyst for tar conversion would be placed in front of a high-temperature desulfurization sorbent in the same vessel. The tar cracking catalyst would be operated and regenerated in conjunction with the desulfurization sorbent.
Catalysts for Tar Conversion Acidic cracking and hydrocracking catalysts have been studied extensively for conversion of coal liquids, including tars, to lighter more valuable products, and there are several excellent reviews of this technology (Crynes, 1981; Janardanarao, 1982). One recent study is particularly applicable to our effort. Wen (1984) cracked coal tar from the METC fixed-bed gasifier with a large number of catalysts including synthetic and natural zeolites and several natural clay minerals. A synthetic zeolite, LZ-Y82, was the most effective catalyst tested. The zeolites with wide pores (>0.7 nm) were generally more effective than the smaller pore materials. Coke and gas were the major products in the range of conditions studied by Wen, 400-800 "C, 1-5-s residence (contact) time, and 101 kPa (1 atm) of inert gas. Coal tar contains insufficient hydrogen and oxygen to convert all of the tar to gases by pyrolysis/cracking. Unless hydrogen or oxygen is added from an outside source, production of coke is inevitable. Hydrogen and oxygen (in the form of steam) are available in the raw coal gas. Raw coal gas from the METC gasifier contains about 30% steam on a wet gas basis. The steam, hydrogen, and carbon dioxide in the gas can gasify the coke off the catalyst surface after it forms. The rate of these reactions is too slow to be of significance at temperatures much 0 1987 American Chemical Society
