Ind. Eng. Chem. Res. 1989,28,445-454
445
MATERIALS AND INTERFACES Estimation of the Solubilities of Organic Compounds in Polymers by Group-Contribution Methods Rosemary Goydan* Arthur D. Little, Inc., Acorn Park, Cambridge, Massachusetts 02140
Robert C. Reid Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Hsiao-Show Tseng 3M Corporate Research Laboratory, St. Paul, Minnesota 55144
An important step in evaluating the suitability of protective clothing against chemical challenges is to determine the equilibrium solubility of the chemical in the polymer comprising the clothing. Experiments are usually employed to obtain such information; however computational methods have been recently proposed. Three estimation methods by Oishi and Prausnitz, Holten-Andersen et al., and Ilyas and Doherty are evaluated in this paper. All involve a molecular group-contribution approach. T o compute the solubility, or the activity of the challenge chemical as a function of weight fraction, only the molecular structure of the chemical and of the polymer need be known. Six common polymers were studied with a wide variety of challenge chemicals. In general, there was good agreement between experimental solubilities (and activities) and those computed from the three methods. The Holten-Andersen et al. approach was found to be the most accurate, while the Oishi-Prausnitz method was also fairly accurate and more widely applicable. Workers in the chemical and allied industries must be protected against potential health hazards resulting from contact with chemicals. Protective clothing is often specified, but few reliable data exist to indicate the degree of protection actually provided by any given material to specific chemicals. There is no ideal barrier that ensures complete protection. Neglecting obvious problems such as cracks or pin holes in the material, chemicals migrate through protective clothing by processes that are described as solution-diffusion mechanisms. In other words, the external surface of the protective clothing absorbs chemicals (i.e., a chemical-polymer solution results), and following this step, the chemical moves through the protective clothing barrier by diffusion caused by the concentration gradient. To analyze and predict such events, one needs (a) to estimate the solubility of a chemical in a specified protective clothing polymer and (b) to characterize the diffusion coefficient. In the present paper, we address only the first problem, i.e, the calculation of the solubility of a chemical in a specified protective clothing polymer. Most of the previous work on this problem has invoked solubility parameters of the chemical and polymer as the important correlating variables (Barton, 1975; Hansen and Beerbower, 1971; Lloyd and Meluch, 1985). While success has been achieved in some instances, this approach is less promising than some of the newer, more theoretical methods. Also, these recent advances allow for the a priori estimation of key parameters via a group-contribution approach. In this paper, we consider three such estimation methods and evaluate their accuracy and generality. In each of the three techniques, the activity of a solute dissolved in a polymer is estimated as a function of solute 0888-5885/89/2628-0445$01.50/0
mass fraction and temperature. To obtain the solubility of the solute in the polymer, we make the assumption that pure solute contacts and dissolves in the polymer, but no polymer dissolves in the solute. Thus, the “solubility” is defined as the concentration where there is unit activity of solute. The molecular structure of both the solute and polymer must be known, and in some cases, the densities of the pure components are required. In one technique, an estimation of the polymer molecular weight is needed, but for large molecular weights, the final result is insensitive to the actual value selected. The calculation of the solute activity is usually broken down to estimate the combinatorial and residual parts-and, often, there is a separate free-volume contribution.
Oishi and Prausnitz Method Employing the activity coefficient correlation UNIQUAC (Abrams and Prausnitz, 1975) as a base, Fredenslund et al. (1975) proposed a group-contribution method for activity coefficients in liquid mixtures. The procedure, termed UNIFAC, was widely adopted to estimate vaporliquid equilibria and even liquid-liquid equilibria. Tabulations of group parameters have been published (Fredenslund et al., 1977a,b; Gmehling et al., 1982; Macedo et al., 1983; Magnussen et al., 1981; Reid et al., 1987; Skjold-Jerrgensenet al., 1979) and recently updated (Tiegs et al., 1987). Oishi and Prausnitz (1978) extended the application of UNIFAC to include what was termed by the authors as polymer solutions. We have applied their proposed methodology to the case of a solute “dissolved” in a solid polymer. The activity of the solute (al) is separated into three components for calculational convenience: 0 1989 American Chemical Society
446
Ind. Eng. Chem. Res., Vol. 28, No. 4, 1989 In ul = In ulc + In ulR + In ulFV
(1)
The estimation of each of the three components is discussed below. In ulc is the configurational contribution and accounts for the difference in the sizes and shapes of the constituent molecules. It depends upon the overall makeup of the mixture. In its determination, we require two different types of properties, Q, a group area parameter, and R, a group volume parameter, for each component [solute (1) and polymer (2)]: G
Q1
=
Cv,(l)Q(j)
,=l
(2)
G
Qz =
Xv,'2m'Q(j)
J=1
(3)
The residual activity coefficient of group j in the solute molecule, In ri(l),is calculated in an identical manner to In rj except that w1 is set equal to unity. Thus, In ulR is assured to be equal to zero as w1 approaches unity. To determine the free-volume contribution to In ul, In ulFV = 3c In [(u11/3- 1)/(umix1/3 - I)] c([(ul/umix) - 11/[1 - ( 1 / ~ 1 ) ~ / ~(15) 1} With p1 and pz, the densities of the pure solute and polymer, respectively (in g/cm3), the reduced, dimensionless volumes, u1 and umix, are u1 = M1/(15.17bRlpl) (16) umix
G
Rl = Cv,(')R(j)
(4)
J=1
G
Rz =
C
,=1
j)
(5)
To simplify the equations, we define, for binary mixtures Q* = wiQi/Mi + ~ z Q z / M z ( ~ )
(6)
R * = wlR1/M1 +
(7)
In these equations, is the number of times that group j occurs in the solute molecule, while v,(2m) represents the number of times group j occurs in each repeat unit of the polymer. The summations are over all groups G. MI and Mz(m) are the molecular weights of the solute and repeat unit of the polymer. w1 and wz are the weight fractions solute and polymer. The properties Q ( j )and R(j) represent contributions to the molecular area and volume due to group j and are given in Table I of the supplementary material for a large number of groups. This table is also appliable for use in UNIFAC computations. To determine the configurational term, In ulc = In d1 + 42
+ (Z/2)Ql[ln ( W h )- 1 + $1/&1 (8)
where $1 and 4zrepresent volume fractions of solute and of polymer in the mixture, while 1 9 ~and Oz simulate the solute and polymer area fractions. Z is the coordination number and is usually set equal to 10: ~$1= wlRl/MlR * = 1 - 42 (9) 01
= wlQi/MIQ* = 1 - 6 2
(10)
In ulR is termed the residual contribution; it is related to group area fractions S ( j ) and to terms $ ( j , k ) , which are functions of the interaction energies between various groups B ( j , k ) : S(i) =
Q(.i)[(ei/Qi)v,(')
+ (~~/Q~)v,'~"'I
$ ( i , k ) = exp[-B(j,k)/TI
(11)
(12)
where B ( j , k ) # B ( k , j ) . Values of B ( j , k ) are provided in Table I1 of the supplementary material. With eq 11 and 12, we can calculate the group residual activity coefficient rl as G
In
ri = Q(j)(l- In [kC= l b ( k ) $ ( k , j ) l G
and
+ (wz/~z)]/(l5.17bR*)
(17)
= 1.1 and b = 1.28. The free-volume contribution is normally negligible for mixtures of low molecular weight components but must be introduced when solute-polymer solutions are involved. Thus, in the Oishi and Prausnitz method, there are two pure-component group-contribution properties, Q(j) and R(j), and one interaction group-contribution term, B( j , k ) . In addition, the densities of the two pure components must be specified at the system temperature. c
Ilyas and Doherty Method Ilyas and Doherty (Doherty, 1986; Ilyas, 1982) developed a group-contribution method to estimate phase equilibrium properties for systems containing small molecules as well as polymers. Their approach involved selecting an equation of state (EOS) believed to be applicable for the wide diversity of systems of interest and then employing this EOS to calculate phase equilibrium concentrations with the EOS parameters obtained from group contributions. Their choice for an EOS was that proposed by Sanchez and Lacombe (1976, 1978),which is based on lattice fluid theory. In selecting this EOS, they were able to place some physical interpretation on the parameters in order to judge which might be more appropriate to estimate from group contributions. As applied to the cases of interest here, for a binary mixture of solute (1) and polymer (2), they express the activity of the solute, ul, as In a, = In ulc + In ulR
(18)
In this treatment, vacant lattice sites are allowed. Thus, the free-volumecontribution is included within the residual term. The configurational contribution is In ulc = In c $ + ~ 42 (19)
d1and & are volume fractions of solute and polymer and, as in the Oishi and Prausnitz method, are computed from eq 4, 5, 7, and 9. However, for this method, the closepacked group molar volume values R ( j ) are from Table I11 of the supplementary material. The residual term In ulR describes the extent of the energetic interactions between different constituent groups in the mixture and is determined by a relation identical in form with that in the Oishi and Prausnitz method (eq 14), but where the group activity coefficient for group j is found from G
G
C S ( m ) $ ( j , m ) /C O(n)$(n,m)l(13)
m=l
= [(wl/pl)
n=l
In
rl = (1/2RTu)R(j)[2C@(k)MN(j , k ) k=l
G
G
C C @(m)Wn)MN(m,n)l(20)
m = l n=l
Ind. Eng. Chem. Res., Vol. 28, No. 4,1989 447 The term @G) is the volume fraction of group j in the mixture, i.e., @ ( j )= R(jH(41/R&o)
+ (42/R2)v/2”]
(21)
The term MNG,k) is found from the P( j ) and P ( k ) contributions in Table I11 of the supplementary material with
MN( j , k ) = P ( j ) + P ( k ) - 2D(j,k)[P(j)P(k)]1/2 (22) Here, D( j , k ) is a binary interaction parameter between groups j and k as shown in Table IV of the supplementary material. Note D ( j , k ) = D ( k ,j ) . The parameter P ( j )has the dimensions of pressure or energy/volume. Finally, to determine u for use in eq 20, G
v=
G
G
E @(k)RR(k)- C C @ ( m ) @ ( n ) D V T ( m , n )
k-1
(23)
m = l n=l
with
D V T ( j , k ) = [RR(j) + RR(k)][l - Y ( j , k ) ] (24) Reduced group molar volume parameters RR(j ) are given in Table I11 of the supplementary material and are dimensionless. Y(j , k ) is a binary excess volume interaction parameter and is found in Table IV of the supplementary material. To find I’j(l),eq 20-24 are employed with w2, the polymer weight fraction, set equal to zero and the solute weight fraction, wl, set equal to one. Then, u becomes vl, the dimensionless volume for the pure solute. In summary, the Ilyas-Doherty method requires three pure-component group contributions, R( j ) , P(j ) , and RR( j ) ,given in Table I11 of the supplementary material, and two binary group interaction parameters, D( j , k ) and Y( j , k ) , in Table IV of the supplementary material, to calculate solute activity. Holten-Andersen et al. Method Holten-Andersen and his colleagues at the Instituttet for Kemiteknik have developed a group-contribution method applicable for estimating phase equilibria in polymer-solute systems (Andersen et al., l984,1986a,b, 1987). In essence, they began with the Flory-Huggins model (Flory, 1942; Huggins, 1942) for the Gibbs energy of mixing but proposed several major changes. Two of the more important deal with nonrandom effects and with the density differences in the components. In the Flory-Huggins formulation, the calculation of the residual contribution to the solute activity is based on the assumption 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 leads to serious error. Various attempts have been made to correct the concept. Holten-Andersen et al. chose to allow local compositions to be different from the bulk composition. As this same concept is embodied in the UNIFAC method, Holten-Andersen et al. began with the residual term of UNIFAC and modified it as noted later. Density effects are not taken into account in conventional theories of liquid mixtures, which assume that all solutions have the same configurational structure. In mixtures involving a polymer, however, there are significant density effects. To treat solute-polymer systems, a model is required in which density enters as a variable; i.e., an equation of state is implied. Often the terms involving volumes are separated and treated as a free-volume contribution as in the Oishi and Prausnitz method. HoltenAnderson et al. developed a free-volume expression that would lead to the appropriate ideal gas limit at low den-
sities and, also, would separate the translational energies from rotational energies, with the latter being affected by the system density. In addition to the changes noted above, two other modifications were proposed to treat attractive energies. These modifications involve random-nonrandom energetics and nonrandom UNIFAC energies. Holten-Andersen et al. assumed that the attractive potential between two molecules could be represented as a sum of energies of random orientation and of favorable orientation. This distinction was found advantageous in describing a number of phenomena met with in polymer solutions. Also, it would explain the fact that cohesive energy densities of long-chain components are larger than expected on the basis of shorter chain homologues; i.e., the packing of long-chain compounds is more pronounced than for short-chain compounds. This type of treatment introduces two molecular energies, an energy from random packing and an “extra” contribution from favorable configurations. In addition to the energy divisions noted above, for mixtures, it was assumed that the energetic contribution for nonrandom mixing was similar in form to that used in UNIFAC. This forces the interaction parameter to be related to a Helmholtz energy, as will be seen later. The final equations proposed by Holten-Andersen et al., rearranged for a binary system of a solute (1) and a polymer (2), lead to an equation for In al identical in form with eq 1. Also, area and volume fractions are determined as in eq 2-7 but with R( j ) and Q(j)group values given in Table V of the supplementary material. +1 and Ol are calculated as in eq 9 and 10. The configurational contribution to In a, is found from a modification of eq 8, In alc = In
+1
+ 1 - dl/xl
(25)
where x1 is the mole fraction of solute in the mixture, x i = (wi/Mi)/[(wi/MJ
+ (wz/Mz‘”H2)]
(26)
H 2 is the average number of repeat units in the polymer; i.e., MJm)Hzis the polymer molecular weight. For large values of H2,x1 = 1and eq 25 reduces to eq 19 of the Ilyas and Doherty method. Before the Holten-Andersen et al. relations for calculating In alRand In alFVare presented, several important parameters are introduced. There are three potential energies of random packing: E O l l , between molecules of solute (1): E022, between molecules of polymer (2); and E012, between solute and polymer molecules. G
EOll =
G
E022 =
G
Q1-’C C ~j‘’)Q(j) Vk(’)Q(k) E ( j , k ) (27) j=1 k = l G
Q2-2C
vj(2m)Q( j ) uk(2m)Q(k)E ( j , k ) (28)
j=l k=l
G
E012 =
G
(QIQ2)-lC u/’)Q(j)
vk(2m)Q(k)E( j , k )
(29)
j=l k = l
Q1 and Q2 are found with eq 2 and 3 with values of Q( j ) from Table V of the supplementary material, while E ( j , k ) is an interaction energy between groups j and k . Values of E ( j , k ) are given in Table VI of the supplementary material; note that E ( j , k ) = E ( k , j ) . In addition, there are interaction energies. As originally proposed by Holten-Andersen et al., expressions for such energies contained adjustable parameters, but at the present state of development, they have been set at definite values. The equations shown below have incorporated these numerical constants. Consider first the interaction
448 Ind. Eng. Chem. Res., Vol. 28, No. 4, 1989
energy E l l for molecules of pure solute: E l l = ul-'jEOll - 2720/[1 + 400 exp(-654.3/ulT)]) (30) T i s in degrees kelvin, and u1 is a dimensionless, reduced volume for pure component 1 as determined from an equation of state described later. In some instances, however, the interaction energy for the pure solute must be evaluated a t the reduced volume of the mixture, u. (u is also found from the above-mentioned equation of state.) In this case, eq 30 is rewritten as E l l M = u-'jEOll - 2720/[1
G
C1 = -0.64
+ 400 exp(-956.2/u2T)])
= -0.64
E12 = u-l(E012 - 3288/[1
+ 400 exp(-791.0/uT)])
where C ( j ) values are shown in Table V of the supplementary material and VHC, = 21.24R1
(49)
VHC2 = 21.24HzRz
(50)
VHCM = xlVHC1 + (1 - x1)VHCZ
(51)
The next step is to employ these parameters to determine the reduced volumes u l , u2, and u. A t low pressures, the Holten-Andersen et al. equation of state is
+ (u11/3 + C1)/(u11/3 - 1) (52) 0 = (SE2/8.314T) + (uZ1I3+ C ~ ) / ( U ~- '1) / ~ (53) 0 = (SEM/8.314T) + ( u ' / ~+ C ) / ( U ~- /1) ~ (54) 0 = (SE1/8.314T)
(34) In addition to energetic terms, there are comparable Helmholtz energies for the constituents. Their definitions expressed in a manner comparable to eq 30-34 are A l l = (EOll/ul) (4.157/T) In [0.9975 + 0.0025 exp(654.3/ulT)]
+ H ~ C V , ( ~ " ) C ( ~ ) R ( ~ )(48) ]=1
+ 400 exp(-956.2/uT)]]
(33) and finally, the interaction energy between solute and polymer, which is always evaluated at the mixture reduced volume, u, i s
(47)
G
C2
(32) E22M = u-ljE022 - 3975/[1
+ Cv,'l)C(j)R(j ) ,=1
+ 400 exp(-654.3/uT)])
(31) In a similar manner, two interaction energies are found for the polymer molecules, E22 = u2-'{E022 - 3975/[1
with 81 and O2 as calculated earlier by using eq 10 with eq 2, 3, and 6 and constants from Table V of the supplementary material. The final parameters relate to the external degrees of freedom of the solute, C1, and polymer, C2, as well as the hard-core volumes, VHC1, VHC2,and VHCM. Assuming no branching, they are obtained from (Holten-Andersen et al. have a computer program that determines C, values given the thermal expansion coefficient)
with
c = x1C1 + (1- xJC2
(55)
In eq 52-54, the system energies are functions of the reduced volumes, so an iterative solution is necessary. With these definitions, the residual and free-volume contributions to the solute activity may be calculated:
A l l M = (EOll/u) (4.157/T) In [0.9975 + 0.0025 exp(654.3/uT)] A22 = (E022/~2)(4.157/T) In [0.9975 + 0.0025 exp(956.2/uzT)]
In ulR = 5Ql([(A11M - A11)/8.3147'] + 1 - ln[fll + O2 exp(-(A21 - AllM)/8.314T)] - 8,/[01 + O2 exp(-(A21 - AllM)/8.314T)] 02/[01 + d2 exp((A12 - A22M)/8.314T)]] (56)
A22M = ( E 0 2 2 / ~ -) (4.157/T) In [0.9975 + 0.0025 exp(956.2/uT)] A12 = A21 = ( E 0 1 2 / ~ ) i4.157/Ti In [0.9975 + 0.0025 exp(791.0/uT)]
and, for the free-volume term,
One then combines the interaction energies and the Helmholtz energies to determine the so-called system energies:
In ulFV = 3(1 + Cl) In [(ul1i3 -
SEI = 5Q1Ell SE2 = 5Q2H,E22 SEM = 5Qlxl[E11M + (Yl/YYl)] + 5QzH2(1 - xi)[E22M + (Y2/YYJI
The Holten-Andersen et al. method as described is not applicable for systems where the solute contains an OH group. In such cases, the terms (A21 - A11M) and (A12 - A22M) in eq 43-46 and in eq 56 must be modified to (A21 - A l l M - TS21) and (A12 - A22M - 7312). To compute S21 and S12,
(40) (41)
(42)
Q1 and Q2 are found, as before, with eq 2 and 3 and the values from Table V of the supplementary material; H2 is the degree of polymerization, and xl, the solute mole fraction, is calculated from eq 26. The interaction energies E l l , E22, E l l M , and E22M were given in eq 30-33. The remaining terms are defined below: Y , = &(E12 - E11M) exp[-(A21 - AllM)/8.314T] (43) YY, = 81 + 8, exp[-(A21 - AllM)/8.314T] (44)
Y2 = 8,(E12 - E22M) exp[-(A12 - A22M)/8.314T] (45) YY2 = O1 exp[-(A12 - A22M)/8.314T] + 82 (46)
G
i ) / ( ~l /I)] ~
C, In ( u l / u ) (57)
G
S21 = (QlQZ)-l[CvJ'"Q(j) 2( J=1
-
k=l
~ k " ~- ) uk'l')Q(k)
S(j,k)l (58)
G
G
S12 = (Q1Q2)-1[C~,'2"'Q(j)C ( ~ k " ) - ~k'~"')Q(k) S(j,k)] I=,
k=l
(59) where S ( j , k ) # S ( k , j ) . Values are given for the OH interactions in Table VI of the supplementary material. Unless either the solute (1)or the polymer (2) contains an OH group, S12 = S21 = 0. In the Holten-Andersen et al. method, there are three pure-component group contribution properties, Q(j ) , R( j ) ,
Ind. Eng. Chem. Res., Vol. 28, No. 4, 1989 449 and the C parameter, as well as one binary interaction parameter. For molecules containing an OH group, an additional binary entropy group-contribution parameter is required. In addition, the degree of polymerization must be specified.
Discussion and Comparison of the Methods All three techniques employ similar expressions to compute the activity of the solute in the polymer, i.e., eq 1for Oishi and Prausnitz and Holten-Andersen et al. and eq 18 for Ilyas and Doherty. In the latter, the free-volume contribution is included within their residual term. The configurational term in each method yields a negative number for In ulc, which is large at low values of w1 and approaches zero as w1 1. This term reflects the variation of the entropy of mixing with composition and originally was derived by Flory (1941,1942) and Huggins (1941) for a model liquid mixture with no heat of solution (athermal) and for which the molecular sizes of the components were greatly different. As Prausnitz (1969) has shown, this treatment leads to the Holten-Andersen et al. form (eq 25), which, for high polymer molecular weights, reduces to that of Ilyas and Doherty (eq 19). The configurational term for the Oishi and Prausnitz method is the same as that derived for the original UNIQUAC correlation (Abrams and Prausnitz, 1975) and employs the Staverman (1950) treatment. This last approach introduces both component area and volume fractions, whereas the other two methods utilize only volume fractions (41, 42). The residual contribution to the solute activity involves the concept of group activity coefficients in all three methods. Equations 11-14 are the key relations for the Oishi and Prausnitz procedure and are identical in form with those used in UNIFAC (Fredenslund et al., 1975). As described earlier, Holten-Andersen et al. modified the UNIFAC residual term to introduce density effects, entropic effects, and nonrandom energies. Their final form is given in eq 56 and involves Helmholtz energies of interaction. For the Ilyas and Doherty procedure, the Sanchez-Lacombe (1976,1978) equation of state was used to develop the Gibbs energy expression. This equation can be used to calculate phase equilibrium properties with the EOS parameters obtained from group contributions. Since vacant lattice sites are allowed in the Sanchez-Lacombe treatment, the resulting residual activity expression contains the free-volume contribution. The free-volume contribution accounts for changes in volume during mixing of the components. This effect is normally neglected for systems comprised of small molecules, but it can be important when one component is a polymer. Prausnitz et al. (1986) discuss this point in some detail. In the free-volume terms used here (Oishi and Prausnitz, eq 15; and Holten-Andersen et al., eq 57), In ulm is positive (aim > 1). The introduction of the free-volume concept implies that an equation of state (EOS) was employed to estimate volumes of the pure components as well as for the mixture. Oishi and Prausnitz used the Flory EOS, while Holten-Andersen et al. employed an EOS developed in their study. To calculate the free-volume contribution, the C parameter representing the external degrees of freedom is required. It is a fixed constant in the Oishi and Prausnitz method but was expressed as a parameter to be found from group contributions in the Holten-Andersen et al. procedure. In summary, all three methods attempt to formulate expressions for the excess Helmholtz energy of mixing of the solute with the polymer. Then, by appropriate differentiation, one obtains a relation for the solute activity.
-
Entropy contributions are largely accounted for in the combinatorial term and volume changes in mixing by the free-volume term. Energetic effects resulting in nonrandom molecular orientations are treated in the residual term. While each method has a theoretical base, simplifications and approximations have been introduced to make the procedures tractable.
Applications The range of polymer and solute systems to which these three methods can be applied to estimate solute activities depends upon the availability of two different types of group-contribution parameters. First, there are pure group parameters to characterize such properties as group volume [ R ( j ) ]or group area [ Q ( j ) ] . Second, there are binary interaction parameters to account for energetic, volumetric, and entropic effects between groups. Available group parameters are shown in Tables I-VI of the supplementary material; clearly, the tables are incomplete. For the Oishi and Prausnitz and Holten-Andersen et al. methods, additional values of pure-component parameters R( j ) and &(j ) could be estimated from sources such as the van der Waals group constants. Similarly for the Ilyas and Doherty method, R ( j ) and RRG) could be found by multiple regression of their EOS parameters, which, themselves, would be fitted to pure-component P-V-T data. While there is reasonable expectation that new p u r e component groups could be incorporated in all three methods, this is not true for missing binary group parameters. Accurate vapor-liquid equilibrium data involving components with the missing groups must be available, and a suitable regression program would be necessary. The most broadly applicable method is that of Oishi and Prausnitz, who drew upon the extensive results of UNIFAC. The other two methods, at the present time, can treat only a limited number of chemicals and polymers. As an illustration, we show the applicability of all methods to a few common polymers in Table VII. As our principal concern is focused on protective clothing applications, the first six polymers in Table VI1 were of primary interest in our predictive model validation efforts. However, we also included polystyrene because of the extensive literature available for this polymer. As noted in Table VII, the Ilyas and Doherty procedure can only treat natural rubber, polyethylene, and polystyrene, while the Holten-Andersen et al. method is applicable to these and also to butyl rubber. All seven polymers can be treated by the Oishi and Prausnitz approach. We computed activity-composition curves for a large number of solutes dissolved in the polymers of interest. When necessary, pure-solute densities were obtained from Timmermans (1950) and Wilhoit and Zwolinski (1973). The polymer densities used are shown in Table VII. Some representative results are presented in Figures 1and 2 for the polystyrene systems and in Figures 3-8 for examples with natural rubber. In these figures, the computed activities from the Holten-Andersen et al., Oishi and Prausnitz, and Ilyas and Doherty methods are indicated by the solid, dashed, and dash-dot lines, respectively. The open triangular symbols represent the experimentally measured solubilities (i.e., weight fractions at unit solute activity), and these were generally obtained from equilibrium immersion or vapor sorption tests. The open circles and squares show activities from literature data that were obtained from gravimetric vapor sorption or by vaporpressure-lowering techniques. As shown in Figure 1 for the methyl ethyl ketone/ polystyrene case and in Figure 2 for the propyl acetate/
450 Ind. Eng. Chem. Res., Vol. 28, No. 4, 1989 Table VII. Availability of Pure-Component Group Parameters for Commonly Used Polymers
polymer natural rubber butyl rubber neoprene rubber nitrile rubber Viton rubber polyethylene polypropylene polystyrene poly(viny1 chloride) poly(viny1 acetate) poly(ethy1ene oxide) poly(propy1ene oxide) poly(viny1 alcohol) polymethacrylate poly(methy1 acrylate)
chem structure [CHzC (CH&CHCHZ] , [CH&(CH3)21n [CH2C(Cl)=CHCHZ], [CH2CH=CHCHZCH2C(CN)Hln [CFzCH2CF2C(CF3)Fln ~CHzCHzIfl [CHzC(CH3)HIn [CH&(Ph)Hl, [CH2C(Cl)H] , [CH2C(OOCCH3)H], [CH2CH20], [CH2C(CH3)HO], [CH,C(OH)H], [CHzC(CH3)(COOCH3)In [CH2C(COOCH3)H],
,
density, Oishig/cm3 Prausnitz 0.93 Y 0.93 Y 1.32 Y Y 0.98 Y 1.79 0.92 Y 1.05
Y Y Y Y Y Y Y Y Y
HoltenAndersen et al.
Y Y N N N Y
Y Y N Y Y Y Y Y Y
unavailable group values HoltenAndersen Ilyaset al. Doherty
IlyasDoherty
Y >C< C(Cl)=CH CN CN CF, CF2, CF3 CF, CF2, CF3
N N N N
C(Cl)=CH
Y Y Y N Y Y Y
c1
C1
OH
N
Y Y
1
oO '.!
Figure 1. Calculated and measured activities of methyl ethyl ketone in polystyrene at 25 "C: (0) Bawn et al. (1950); (-1 Holten-Andersen et al. calculation; (- - -) Oishi and Prausnitz calculation; Ilyas and Doherty calculation. (-e-)
c , , , , 1 0.2 0.4 0.6 0.8 10 .
o'oO.O
W e i p Fisc #on
Figure 2. Calculated and measured activities of propyl acetate in Bawn and Pate1 (1956); (-) Holten-Anpolystyrene at 25 "C: (0) dersen et al. calculation; (- - -1 Oishi and Prausnitz calculation; (-. -) Ilyas and Doherty calculation.
polystyrene case, predictions of solute activities using the Oishi and Prausnitz and Holten-Andersen et al. methods are fairly accurate, while the Ilyas and Doherty method overestimates in the former case and underestimates in the latter one. For the benzene and acetone/natural rubber systems given in Figures 3 and 4, similar results were obtained. For solutes such as methanol and pentanol in natural rubber, only experimental data a t solute activ-
0:2
0:4
0:6
0.8
5
Figure 3. Calculated and measured activities of benzene in natural Eichinger and Flory (1968a-c); (0) Saeki et al. rubber at 25 O C : (0) (1982); (A) average value from Paul and Ebra-Lima (1970) and Weeks and McLeod (1982); (-1 Holten-Andersen et al. calculation, (- - -1 Oishi and Prausnitz calculation; (-. -) Ilyas and Doherty calculation.
0.21 o'ok.O
4'2
44
0.6
0.8
1.L
Weight Fiacf~on
Figure 4. Calculated and measured activities of acetone in natural Booth et al. (1964); (A)average value from Paul rubber at 25 OC: (0) et al. (1976) and Bhown et al. (1986); (-) Holten-Andersen et al. calculation; (- - -1 Oishi and Prausnitz calculation; (- -1 Ilyas and Doherty calculation. +
ities equal to unity are available, and these are shown in Figures 5 and 6. For methanol/natural rubber, the prediction of activities by the Ilyas and Doherty method is poor, as indicated in Figure 5. Similar results were also
Ind. Eng. Chem. Res., Vol. 28, No. 4, 1989 451
2 . 0 p
'
\
1. B 1. 6
I
I d
1. 4 1. 2
1.0 W e ~ g rF m t o i
0.
Figure 7. Calculated and measured activities of cyclohexane in Eichinger and Flory (1968a-c); (0) polyisobutylene at 25 OC: (0) Bawn and Pate1 (1956); (A)Bhown et al. (1986); (-) Holten-Andersen et al. calculation; (- - -) Oishi and Prausnitz calculation.
e
0. 6
7
0. 4
0. 2 0.0
0. 0
0.8 '.OI
I
,
I
0. 2
0. 4
0. 6
0. B
1.0
f
i.""21
0.6//
Weight Fraction
0.4
Figure 5. Calculated and measured activities of methanol in natural rubber at 25 "C: (A)Paul et al. (1976); (-) Holten-Andersen et al. calculation; (- - -) Oishi and Prausnitz calculation; (---) Ilyas and Doherty calculation. 2.4
1
0.2f
1
O*'!.O
I
0:2
Ok
0:6
Ol8
l:O
We ghf Flact 01
Figure 8. Calculated and measured activities of benzene in polyEichinger and Flory (1968a-c); (-) Holisobutylene a t 25 "C: (0) ten-Andersen et d. calculation; (- -) Oishi and Prausnitz calculation.
2. 2
-
2. 0 1.8 1. 6 1. 4
1.2
1.0 0.
e
0. 6 0. 4 0.2
n. n -.
the Oishi and Prausnitz and the Holten-Andersen et al. proceedures agree well with experimental data as can be seen in Figures 5 and 6. The calculated and experimental activities of cyclohexane and benzene in polyisobutylene are given in Figures 7 and 8, respectively. Due to the lack of group parameters for the Ilyas and Doherty method, polyisobutylene cannot be treated. Predictions of activities for cyclohexane and benzene in polyisobutylene by the Holten-Andersen et al. method are excellent when compared to literature data. Estimations using the method of Oishi and Prausnitz are also quite satisfactory. It is interesting to note that the estimation methods often predict a skewed van der Waals loop. However, since any domain where dal/dwl < 0 is thermodynamically unstable, the predicted curves with al > 1 should be ignored. A comparison of results for predicted solute solubilities in natural rubber is given in Table VI11 and displayed graphically in Figures 9-11. In Table VIII, we have actually .defined the "solubility" as the predicted weight fraction solute when al = 0.99. This definition was adopted since the slopes of the al versus w 1curves were often quite small as al 1, and very large errors were occasionally found if one insisted that the predicted al become unity. In general, the predictions from all of the three techniques are accurate to within an order of magnitude. At the present stage of theoretical development for the estimation of solution thermodynamic properties, the Holten-Andersen et al. and the Ilyas and Doherty methods are less developed and can treat a limited number of chemical classes. These include ketones, alcohols, esters, ethers, simple chlorinated compounds, and most hydrocarbons. The approach suggested by Oishi and Prausnitz is well
8 8
0.0
0.2
0. 4
0. 6
0.
1.0
Weight Fraction
Figure 6. Calculated and measured activities of pentanol in natural rubber a t 25 "C: (A)Bhown et al. (1986); (-) Holten-Andersen et al. calculation; (- - -) Oishi and Prausnitz calculation; (-. -) Ilyas and Doherty calculation.
observed with ethanol as a solute. For pentanol/natural rubber, the three techniques follow the same general trend and show a maximum in solute activity; however, the Ilyas and Doherty method tends to underestimate the weight fraction solute at unit activity. The computed weight fractions of methanol and pentanol in natural rubber by
-
452 Ind. Eng. Chem. Res., Vol. 28, No. 4, 1989
t I t
001 0 001
01
1
1
110
Experimental Solubility
Figure 9. Comparison of solubility predictions using Oishi and Prausnitz approach with experimental data for natural rubber/solute systems at 25 "C (g/cm3).
{
00 100'
10
01
Experimental Solubility
Figure 10. Comparison of solubility predictions using Holten-Andersen et al. approach with experimental data for natural rubber/ solute systems at 25 "C (g/cmS).
Table VIII. Comparison of Predicted Solubilities of Organic Compounds in Natural Rubber with Experimentally Measured Values __ __ exptl solub," calcd solub: g/cm3 compd g/cm3 UNIFAC Holt-And Ily-Doh acetic acid 0.07@ 0.062 na 0.002 acetic anhydride 0.03gd 0.012 0.009 0.009 acetone 0.095 0.10 0.13 0.090 benzene 3.2e 1.6 1.; 6.5 benzyl alcohol 0.15 0.028 0.019 na butylamine 1.4d 0.76 na 0.33 carbon tetrachloride 8.2 7.3 2.4 na cyclohexane 2.8 4.1 0.93 5.8 cyclohexanone 2.4 2.4 0.62 9.8 (dimethylamino)l.ld 0.74 na na propylamine dimethylethanol0.17d 0.020 na 0.010 amine dimethylformamide 0.039' 0.052 na na ethanol 0.007 0.019 0.019 6.2 ethyl acetate 0.43 0.34 0.75 0.52 2-ethyl-1-butanol 0.42 0.11 0.068 0.007 ethylene dichloride 2.2f 0.21 2.9 na ethylenediamine O.ONd 0.038 na 6.5 n-heptane 1.6 3.5 2.0 4.5 n-hexane 1.3 3.4 1.8 4.5 2-propanol 0.040 0.048 0.035 0.014 methanol 0.002 0.008 0.008 5.7 methyl acrylate 0.52d 0.15 0.14 na methylchloroform 4.g 4.4 5.0 na methyl ethyl ketone 0.51 0.18 0.24 0.14 methyl isobutyl 1.2e 1.1 0.55 0.26 ketone methyl methacrylate l.ld 0.28 0.37 0.27 1-pentanol 0.12 0.088 0.085 0.17 tert-pentanol 0.39 0.082 0.085 0.028 1-propanol 0.088 0.047 0.036 0.015 n-propyl acetate 1.3 0.40 0.22 1.4 tetrachloroethylene 7.0 4.2 na na tetrahydro4.4 4.6 1.6 3.8 naphthalene toluene 3.6 3.7 1.2 1.4 trichloroethylene 7.5 4.8 na na o-xylene 3.8e 5.0 0.79 1.5 "All values from Paul et al. (1976) unless marked otherwise. *na indicates that the required group interaction parameters are not available for this solute/polymer system. Pioneer Industrial Products (1984). dConoley et al. (1987). "Paul and Ebra-Lima (1970). fWeeks and McLeod (1980).
include nitriles, tertiary amines, phosphorus-containing compounds, and chemicals containing fluorine. P
1
l r
.01
Experimental Solubility
Figure 11. Comparison of solubility predictions using Ilyas and Doherty approach with experimental data for natural rubber/solute systems at 25 "C (g/cm3).
developed for a large number of chemicals. The chemical functional groups that cannot be treated by this method
Conclusions
In general, where applicable, the Holten-Andersen et al. method proved to be the most accurate. However, the Oishi and Prausnitz procedure is also quite satisfactory and has an advantage in its broader applicability. The results from the Ilyas and Doherty method are less accurate and, as shown, should not be used for methanol or ethanol systems. In any case, since the estimations were all carried out with no other input than molecular structures (and the pure-component densities in the Oishi and Prausnitz procedure), the predicted activities are remarkably close to the experimental values. Such good fits provide encouragement to extend the methods to encompass more types cf groups so as to increase the applicability of the techniques. Acknowledgment
We sincerely appreciate the help and guidance of Professor M. Doherty (University of Massachusetts, Amherst)
Ind. Eng. Chem. Res., Vol. 28, No. 4, 1989 453
and Professor A. Fredenslund and Dr. J. Holten-Andersen ( I n s t i t u t t e t for Kemiteknik, Lyngby, D e n m a r k ) in preparing this paper. Although the research described in this article has been funded wholly or in part by the U.S.Environmental Protection Agency t h r o u g h Contract 68-033293 to Arthur D. Little, Inc., it has not been subjected to Agency review, and, therefore, does not necessarily reflect the views of the Agency; no official endorsement should be inferred. Registry No. Polystyrene, 9003-53-6; polyisobutylene, 900327-4; acetic acid, 64-19-7; acetic anhydride, 108-24-7; acetone, 67-64-1; benzene, 71-43-2; benzyl alcohol, 100-51-6; butylamine, 109-73-9; carbon tetrachloride, 56-23-5; cyclohexane, 110-82-7; cyclohexanone, 108-94-1; (dimethylamino)propylamine,109-55-7; dimethylethanolamine, 108-01-0; dimethylformamide, 68-12-2; ethanol, 64-17-5; ethyl acetate, 141-78-6; 2-ethyl-1-butanol, 97-95-0; ethylene dichloride, 107-06-2; ethylenediamine, 107-15-3;heptane, 142-82-5;hexane, 110-54-3;2-propanol, 67-63-0; methanol, 67-56-1; methyl acrylate, 96-33-3; methylchloroform, 71-55-6; methyl ethyl ketone, 78-93-3; methyl isobutyl ketone, 108-10-1; methyl methacrylate, 80-62-6; 1-pentanol, 71-41-0; tert-pentanol, 75-85-4; 1-propanol, 71-23-8; n-propyl acetate, 109-60-4; tetrachloroethylene, 127-18-4; tetrahydronaphthalene, 119-64-2; toluene, 108-88-3; trichloroethylene, 79-01-6; o-xylene, 95-47-6.
Supplementary Material Available: Table I listing UNIFAC group-contribution parameters for t h e OishiPrausnitz approach, Table I1 listing UNIFAC energetic interaction parameters for Oishi-Prausnitz approach, Table I11 listing t h e Ilyas-Doherty group constants, Table IV listing the Ilyas-Doherty energy and volume interaction parameters, Table V listing the Holten-Andersen et al. group-contribution parameters, and Table VI listing t h e Holten-Andersen et al. interaction and entropy parameters (21 pages). Ordering information is given on a n y current masthead page.
Literature Cited Abrams, D. S.; Prausnitz, J. M. Statistical Thermodynamics of Liquid Mixtures: A New Expression for the Excess Gibbs Energy of Partly or Completely Miscible Systems. AIChE J. 1975,21, 116. Andersen, J. H.; Fredenslund, A.; Rasmussen, P. Towards a Group Contribution Model for Polymer Solutions. SEP 8315, Instituttet for Kemiteknik, Danmarks Tekniske Herjskole, Lyngby, Denmark, Feb 1984. Andersen, J. H.; Fredenslund, A,; Rasmussen, P. A Group-Contribution Model for Polymer Solutions. SEP 8407, Instituttet for Kemiteknik, Danmarks Tekniske H~jskole,Lyngby, Denmark, Aug 1985; revised Dec 1986a. Andersen, J. H.; Fredenslund, A,; Rasmussen, P. Phase Equilibria of Polymer Solutions by Group-Contribution, SEP 8610, Instituttet for Kemiteknik, Danmarks Tekniske H~jskole,Lyngby, Denmark, April 1986b. Andersen, J. H.; Rasmussen, P.; Fredenslund, A. Phase Equilibria of Polymer Solutions by Group Contribution. 1. Vapor-Liquid Equilibria. Ind. Eng. Chem. Res. 1987, 26, 1382. Barton, A. F. M. Solubility Parameters. Chem. Rev. 1975, 75, 731. Bawn, C. E. H.; Patel, R. D. High Polymer Solutions. Part 8-The Vapor Pressure of Polyisobutylene in Toluene and Cyclohexane. Trans. Faraday SOC.1956,52, 1664. Bawn, C. E. H.; Wajid, M. A. High Polymer Solutions. Part 7Vapor Pressure of Polystyrene Solutions in Acetone, Chloroform, and Propyl Acetate. Trans. Faraday SOC.1956,52, 1658. Bawn, C. E. H.; Freeman, R. F. J.; Kamalidelin, A. R. High Polymer Solutions. Part 1-Vapor Pressure of Polystyrene Solutions. Trans. Faraday SOC.1950,46, 677. Bhown, A. S.; Philpot, E. F.; Segers, D. P.; Sides, G. D.; Spafford, R. B. Predicting the Effectiveness of Chemical Protective Clothing: Model and Test Method Development. EPA Project Report EPA/600/S2-86/055, Sept 1986. Booth, C.; Gee, G.; Holden, G.; Williamson, G. R. Studies in the Thermodynamics of Polymer-Liquid Systems. Part 1-Natural Rubber and Polar Liquids. Polymer 1964, 5, 343. Conoley, M.; Prokopetz, A. T.; Walters, D. B. Cumulative Permeation Test Results for the National Toxicology Program. Sub-
mitted for publication in Am. Ind. Hyg. Assoc. J . 1987. Doherty, M. F. Department of Chemical Engineering, University of Massachusetts, Amherst, MA, private communication, 1986. Eichinger, B. E.; Flory, P. J. Thermodynamics of Polymer Solutions. Part 1-Natural Rubber and Benzene. Trans Faraday SOC. 1968a, 64, 2035. Eichinger, B. E.; Flory, P. J. Thermodynamics of Polymer Solutions. Part 2-Polyisobutylene and Benzene. Trans Faraday SOC. 1968b, 64, 2053. Eichinger, B. E.; Flory, P. J. Thermodynamics of Polymer Solutions. Part 3-Polyisobutylene and Cyclohexane. Trans Faraday SOC. 1968c, 64, 2061. Flory, P. J. Thermodynamics of High-Polymer Solutions. J . Chem. Phys. 1941, 9, 660. Flory, P. J. Thermodynamics of High-Polymer Solutions. J . Chem. Phys. 1942, 10, 51. Fredenslund, A.; Jones, R. L.; Prausnitz, J. M. Group-Contribution Estimation of Activity Coefficients in Non-Ideal Liquid Mixtures. AIChE J . 1975,21, 1086. Fredenslund, A.; Gmehling, J.; Michelsen, M. L.; Rasmussen, P.; Prausnitz, J. M. Computer Design of Multicomponent Distillation Columns Using the UNIFAC Group Contribution Method for Calculation of Activity Coefficients. Ind. Eng. Chem. Process Des. Deu. 1977a, 16, 450. Fredenslund, A.; Gmehling, J.; Rasmussen, P. Vapor-Liquid Equilibria Using UNIFAC; Elsevier: Amsterdam, 1977b. Gmehling, J.; Rasmussen, P.; Fredenslund, A. Vapor-Liquid Equilibria by UNIFAC Group Contribution. Revision and Extension. 2. Ind. Eng. Chem. Process Des. Deu. 1982, 21, 118. Hansen, C.; Beerbower, A. Solubility Parameters. In Kirk-Othmer: Encyclopedia of Chemical Technology; 2nd ed.; Wiley: New York, 1971, Suppl., p 889. Huggins, M. L. Solutions of Long Chain Compounds. J. Chem. Phys. 1941, 9, 440. Huggins, M. L. Some Properties of Solutions of Long-chain Compounds. J . Phys. Chem. 1942,46, 151. Ilyas, S. Group Contributions for Fluid Phase Equilibria. Ph.D. Dissertation, University of Massachusetts, Amherst, 1982. Lloyd, D. R.; Meluch, T. B. Selection and Evaluation of Membrane Materials for Liquid Separations. In Material Science of Synthetic Membranes; ACS Symposium Series 269; Lloyd, D. R., Ed.; American Chemical Society: Washington, DC, 1985. Macedo, E.; Weidlich, U.; Gmehling, J.; Rasmussen, P. Vapor-Liquid Equilibria by UNIFAC Group Contribution. Revision and Extension. 3. Ind. Eng. Chem. Process Des. Deu. 1983, 22, 676. Magnussen, T.; Rasmussen, P.; Fredenslund, A. UNIFAC Parameter Table for Prediction of Liquid-Liquid Equilibria. Ind. Eng. Chem. Process Des. Dev. 1981, 20, 331. 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. Paul, D. R.; Ebra-Lima, 0. M. Pressure-induced Diffusion of Organic Liquids through Highly Swollen Polymer Membranes. J . Appl. Polym. Sci. 1970, 14, 2201. Paul, D. R.; Garcin, M.; Garman, W. E. Solute Diffusion through Swollen Polymer Membranes. J . Appl. Polym. Sci. 1976,20,609. Pioneer Industrial Products, Willard, OH, Catalogue M-104-1,1984. Prausnitz, J. M. Molecular Thermodynamics of Fluid-Phase Equilibria; Prentice-Hall: Englewood Cliffs, NJ, 1969; p 293. Prausnitz, J. M.; Lichtenthaler, R. N.; de Azevedo, E. G. Molecular Thermodynamics of Fluid-Phase Equilibria, 2nd ed.; PrenticeHall: Englewood Cliffs, NJ, 1986; Appendix VIII. Reid, R. C.; Prausnitz, J. M.; Poling, B. E. Properties of Gases and Liquids, Their Estimation and Correlation,4th ed.;McGraw-Hik New York, 1987; Chapter VIII. Saeki, S.; Holste, C.; Bonner, D. C. Vapor-Liquid Equilibria for Polybutadiene and Polyisoprene Solutions. J. Polym. Sci., Polym. Phys. Ed. 1982, 20, 793. Sanchez, I. C.; Lacombe, R. H. Statistical Thermodynamics of Fluid Mixtures. J . Phys. Chem. 1976, 80, 2568. Sanchez, I. C.; Lacombe, R. H. Statistical Thermodynamics of Polymer Solutions. Macromolecules 1978,11, 1145. Skjold-Jerrgensen, S.; Kolbe, B.; Gmehling, J.; Rasmussen, P. Vapor Liquid Equilibria by UNIFAC Group Contribution. Revision and Extension. Ind. Eng. Chem. Process Des. Deu. 1979, 18, 714. Staverman, A. J. The Entropy of High Polymer Solutions. Recl. Trau. Chim. Pays-Bas 1950, 69, 163.
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Tiegs, D.; Gmehling, J.; Rasmussen, P.; Fredenslund, A. VaporLiquid Equilibria by UNIFAC Group Contributions. 4. Revision and Extension. Ind. Eng. Chem. Res. 1987,26, 159. Timmermans, J. Physico-Chemical Constants of Pure Organic Compounds; Elsevier: Amsterdam, 1950; Vol. I. Weeks, R. W., Jr.; McLeod, M. J. Permeation of Protective Garment Materials by Liquid Halogenated Ethanes and a Polychlorinated Biphenyl. Los Alamos Scientific Laboratory Report LA-8572-MS, 1980.
Weeks, R. W., Jr.; McLeod, M. J. Permeation of Protective Garment Material by Liquid Benzene and by Tritiated Water. Am. Ind. Hygiene Assoc. J. 1982, 201, 43. Wilhoit, R. C.; Zwolinski, B. J. Physical and Thermodynamic Properties of Aliphatic Alcohols; American Chemical Society: Washington, DC, 1973; Vol. 11. Received for review November 12, 1987 Accepted January 17, 1989
PROCESS ENGINEERING AND DESIGN Hydrodynamic Changes and Chemical Reaction in a Transparent Two-Dimensional Cross-Flow Electrofluidized Bed. 2. Theoretical Results Charles V. Wittmann Department of Chemical Engineering, Illinois Institute of Technology, Chicago, Illinois 60616
Theoretical explanations of some of the experimental results of part 1 are given. In the presence of the electric field, caused by a direct current (dc) applied on the fluidized bed and a bone-dry gas, an electric polarization of the particles is assumed. This behavior in turn causes an electric polarization a t the bubble surface. A theory based on potential flow of the emulsion phase around the elliptical bubbles which takes into account the electric work of these surface charges explains the decrease of the rising velocity of the bubbles with increasing field strength. The results show a linear effect of the field strength. This saturation of the polarization with a maximum charge of 8.3 X C kg-l is attributed mainly to triboelectric effects. The bed expansion with increasing field strength a t constant gas flow rate is explained by the additional electric forces a t the electrode walls which support part of the bed weight. In this case, a slightly modified Ergun’s correlation predicts an increase in voidage. Results give a variation of the wall shear stress with power 0.93 of the field strength. The increase in conversion of ozone in the EFB reactor with increasing field strength is explained by the simultaneous variations of the bubble velocity, bed height, and bubble frequency in a three-phase model: bubble, cloud, and emulsion, which are all assumed to be in plug flow. The value for the reaction rate constant K, = 0.346 s-l, which is the single adjustable parameter, agrees with the decrease in exit concentration of about 22%. Both variations of the bubble velocity and the bed height increase the total mass transfer out of the bubble phase. The former also increases the reaction in the cloud phase and the latter the one in the emulsion phase. The drastic decrease in bubble frequency decreases the gas bypassing and substantially increases the reaction in the emulsion phase. Various aspects of the influence of an applied electric field on a fluidized bed have been reviewed, and new quantitative experimental results have been presented in part 1 of this work (Wittmann and Ademoyega, 1987). Theoretical explanations of the peculiar phenomena in such an “electrofluidized b e d (EFB) have been few. Katz and Sears (1969) invoked polarization within the particles caused by the electric field as well as “surface polarization charges”. In this view, the polarized particles are aligned in the field with the positive side of one particle next to the negative side of an adjacent particle, giving rise to attractive interparticle forces. The pressure drop through the bed at “stabilization breakup” was correlated with (A V ) 2supporting this theory. Johnson and Melcher (1975) conje_ctur_edabout-a macroscopic force per unit volume of bed P-VE where P is the polarization density of the particles and E the macroscopic electric field. At the particle scale, these latter authors thought that the electric energy 0888-5885/89/2628-0454$01.50/0
due to this polarization was of a magnitude comparable to the one of the kinetic energy associated with particle fluctuations as well as its potential energy due to gravity and thus was capable of causing the observed hydrodynamic changes in the bed. In a theoretical interpretation of the application of an EFB to aerosol collection, Zahedi and Melcher (1976) considered a bed of electrically polarized particles. Dietz and Melcher (1978a) claimed that electric forces due to polarization of the particles are not large enough to affect the hydrodynamic and gravitational forces acting on the particles. For these forces to dominate, they required the conditions R, < 100 pm and t,*/++ I10. They attributed the hydrodynamic changes in an EFB to interparticular forces resulting from an electric current at the surface of the particles, which constricts at the contacts between two particles. This theory applies to the EFB in the “frozen” or “electropacked” state; i.e., the bed is no C 3 1989 American Chemical Society
