I n d . E n g . Chem. Res. 1987, 26, 2532-2542
2532
ethylene is a more important coke precursor than propylene. For instance, the relative rate of the two steps is 264:l a t 850 OC. Brown and Albright (1976) and Ghaly and Crynes (1976) have the same point of view also.
R = gas constant rc = coke deposition rate, mg/(cm2 min) T = pyrolysis temperature, K T , = reference temperature, K x = conversion
Conclusion We designed and assembled an experimental system for studying coke deposition in hydrocarbon pyrolysis. The system is reliable and feasible for a wide purpose of research work. The surface effects on coke deposition and gas product yield have been studied. The mechanism of coke deposition is proposed as where the coke precursors, ethylene and propylene, adsorb on the surface first and then the adsorbed coke precursors react on the surface via dehydrogenation, condensation, and other reactions to form coke. Relying on the mechanism, we developed the model including two parallel reactions of two coke precursors for propane pyrolysis:
Greek Symbol 6 = expansion factor Registry No. C, 7440-44-0; propane, 74-98-6; ethylene, 74-85-1; propylene, 115-07-1.
-
CzH4
k l = 5.89
X
X
coke
1O1O exp(-230.29/RT)
l/&H6
k2 = 2.21
kl
k2
coke
lo8 exp(-165.22/RT)
Nomenclature A = frequency factor d = reactor diameter, m E = activation energy, kJ/mol F = molar flow rate of hydrocarbons, mol/s k = specific reaction rate coefficient 1, = equivalent reactor length, m n, = dilution factor, mol/mol P, = reference pressure, Pa
Literature Cited Albright, L. F.; Tsai, T. C. In Pyrolysis: Theory and Industrial Practice; Albright, L. F., Crynes, B. L., Corcoran, W. H., Eds.; Academic: New York, 1983; Chapter 10. Brown, S. M.; Albright, L. F. In Industrial and Laboratory Pyrolysis; Albright, L. F., Crynes, B. L., Eds.; ACS Symposium Series 32; American Chemical Society: Washington, DC, 1976; pp 296-310. Dunkleman, J. J.; Albright, L. F. In Industrial and Laboratory Pyrolysis; Albright, L. F., Crynes, B. L., Eds.; Acs Symposium Series 32; American Chemical Society: Washington, DC, 1976; pp 241-273. Ghaly, M. A.; Cryunes, B. L. In Industrial and Laboratory Pyrolysis; Albright, L. F., Crynes, B. L., Eds.; ACS Symposium Series 32; American Chemical Society: Washington, DC, 1976; pp 218-240. Holmen, A.; Lindvaag, 0. A. ACS Symp. Ser. 1982,202,45. Lahaye, J.; Badie, P.; Ducret, J. Carbon 1977, 15, 87. Pramanlk, M.; Kunzru, D. Ind. Eng. Chem. Process Des. Deu. 1985, 24, 1275. Shah, Y. T.; Stuart, E. B.; Sketh, K. D. Ind. Eng. Chem. Process Des. Deu. 1976, 15, 518. Shen, G.; Lou, Q.;Niu, F. J. Hebei Inst. Technol. 1985, 4, 48. Sundaram, K. M.; Froment, G. F. Chem. Eng. Sci. 1979, 34, 635. Zou Renjun Principles and Techniques of Pyrolysis in Petrochemical Industry; Chemical Industry: Beijing, 1982.
Received for review April 1, 1987 Revised manuscript received August 19, 1987 Accepted September 4, 1987
A Statistical Mechanics Based Lattice Model Equation of State Sanat K. Kumar,*t Ulrich W.S u t e r , * and R o b e r t C. Reid Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
A statistical mechanics based lattice model equation of state (EOS) in a closed analytical form is developed for the modeling of the phase behavior of multicomponent mixtures of molecules of disparate sizes. The EOS is compared to others in the same genre. The lattice EOS is used to model quantitatively experimental VLE data across a broad spectrum of binary mixtures of molecules, using a single binary interaction parameter, which is temperature-independent for many systems. Results obtained also demonstrate that this EOS is successful in reproducing quantitatively trends observed in polymer-supercritical fluid equilibria across variations in temperature, pressure, and polymer chain length, again with one interaction parameter that is independent of operating conditions. The phase behavior of mixtures of molecules of disparate sizes, specifically polymer-supercritical fluid mixtures, is of considerable theoretical and practical interest. To model such systems, one needs an appropriate equation of state (EOS) that is applicable over a large range of densities. During the last decade, EOS's have been used increasingly to correlate and model the complex phase behavior of molecular mixtures under a variety of conditions. Cubic EOS's have been employed extensively (Soave, 1972; +Currentaddress: IBM Almaden Research Center, San Jose, CA 95120-6099. 0888-5885/87/ 2626-2532$0~50/0
Carnahan and Starling, 1972; Peng and Robinson, 1976), although others have also been used [see for instance Dieters (198111. This genre of EOS, however, proves inadequate when the size differences between component molecules become large. Modeling of mixtures of molecules of dissimilar sizes has in the past been performed by the use of either of two general techniques. The first one is based on perturbation theory; an example of such an EOS is the modified perturbed hard chain EOS (PHCT) (Donohue and Prausnitz, 1975; Vimalchand and Donohue, 1985; Sandler, 1986) which has been used successfully to model mixtures of 1987 American Chemical Society
Ind. Eng. Chem. Res., Vol. 26, No. 12, 1987 2533 small and large hydrocarbons. The apparent complexity and the large number of adjustable parameters in PHCT models makes them unattractive for our purposes. The second genre of models used for polymer-solvent mixtures has been the lattice model approach. The earliest attempts to model polymer-solvent systems were made by Flory (1941, 1942), Huggins (1941, 1942), and others (Staverman, 1950; Tompa, 1956) who developed activity coefficient models through the use of a mean-field analysis of molecules placed on a three-dimensional lattice. Most later developments in this field are attempts to alleviate the shortcomings inherent in these pioneering attempts. These improved models in their turn can be broadly classified into three categories which are discussed below. Kleintjens (1983) and Kleintjens and Koningsveld (1982a) use the Flory-Huggins entropy term in their expression for the Helmholtz energy of the polymeric system. They depart from the older work by allowing their “enthalpic” contribution to be a function of density and composition and by introducing empty sites in the lattice, termed “holes”. The resulting EOS has been used successfully to model the phase behavior of binary mixtures of water, carbon dioxide, and decane (Nitsche et al., 1983) and ethylene and naphthalene (Kleintjens and Koningsveld, 1982b). This model, however, is characterized by the a priori assumption of a functional form for the canonical partition function in order to derive the EOS and other thermodynamic quantities. Flory (1970), 30 years after his initial contribution, in an attempt to obtain an EOS rather than activity coefficients, resorted to the use of a cell-theory model (Hirschfelder et al., 1954) to formulate the canonical partition function for the ensemble of pure polymer chains. This approach uses the concept of the c parameter as suggested by Prigogine (1957) to account for the external degrees of freedom of a chain molecule. The resultant EOS is a two-parameter corresponding-states model that has been extended to mixtures. Patterson and eo-workers (Patterson and Delmas, 1969; Zeman et al., 1972; Zeman and Patterson, 1972) have used this formulation to predict critical curves and lower critical solution temperatures (LCST) for polymer solutions. Simha and co-workers (Simha and Somcynsky, 1969; Jain and Simha, 1980) have adopted the same technique, viz., the cell-theory approach, along with a different expression for the free volume for spherical and chain molecule fluids. These models, however, suffer from a 2-fold disadvantage: first, one needs to assume a functional form for the free-volume term, which cannot be assigned unequivocally from theory; second, an extra parameter, c, which is unknown and has to be obtained by fitting experimental data to the model is introduced. Also, one needs to know the volume dependence of this parameter in order that the EOS may be applicable to gaslike and liquidlike phases. If the model were extended to multicomponent mixtures, one needs to assign a mixing rule for the c parameter, which introduces an extra degree of uncertainty to mixture predictions. Some of the difficulties encountered in the free-volume theories have been overcome by the “lattice fluid” theories. Sanchez and Lacombe (1974,1976,1977) applied the more rigorous Guggenheim-Huggins-Miller approximation (Guggenheim, 1954) to enumerate the number of possible configurations when polymer chains were placed on a lattice. They also included empty sites in their lattice which they termed holes. The explicit consideration of the number of external degrees of freedom and the flexibility of a chain molecule, used to determine the functional form of the free volume, is unnecessary in this formulation.
Okada and Nose (1981a,b) have accounted for the nonrandom distribution of free volumes (or holes) and thus improved on the work of Sanchez and Lacombe. These models assume that a lattice cell holds exactly one small molecule or one chain segment. While this assumption will have no effect on the problem formulation for pure components, in the extension to mixtures of molecules of unequal chain segment sizes, it introduces additional complexity since one now has to account for the mixing of two quasi-lattices of unequal cell sizes. [This difficulty can be somewhat overcome if one resorts to Staverman’s (1937) use of a characteristic interaction area for each type of site.] Also, in the formulation of Sanchez and Lacombe, it is (unphysically) assumed that a lattice site has an infinite coordination number. Panayiotou and Vera (1982) improved on the lattice fluid model by a priori setting the lattice cell size and coordination number. The lattice therefore is a convenient division of three-dimensional space. A unit cell does not have the function of holding exactly one small molecule or chain segment; a molecular segment can occupy a fractional number of sites. The advantage of using this method is that the extension of a pure-component model to mixtures is, in principle, straightforward since we no longer have to concern ourselves with the mixing of lattices of unequal cell size or coordination number. It is the objective of this work to establish an EOS and related thermodynamic quantities for a pure component and, subsequently, closed form expressions for mixtures. Panayiotou and Vera (1982) have derived an expression for pure components and, with some approximations, extended it to binary mixtures. Their formulation, however, cannot be extended to multicomponent mixtures in a closed form. We use their pure-component EOS in our modeling and provide new, analytical expressions for the multicomponent EOS and chemical potentials. The utility of lattice EOS’s in modeling the VLE properties of mixtures of molecules (both small and large), not demonstrated to date, is also examined.
Pure Components Theoretical Development. Consider a three-dimensional cubic lattice of coordination number z and of unit cell size uH. Each molecule is assumed to occupy r, lattice cells or sites (where r, can be fractional), and the lattice has empty sites called holes. There are No holes and N 1 molecules. To account for the connectivity of the segments of a molecule, an “effective chain length, q,, is defined, so that zq, now represents the effective number of external contacts per molecule. In eq 1it has been assumed that zq, = zrl - 2rl + 2 (1) chains are not cyclic but that they may be branched. The effective interaction energy between segments of molecules (species 1) is a square well potential of depth --ell, while the interaction energy of any species with a hole (species 0) is zero. Only nearest-neighbor interactions are accounted for, and pairwise additivity is assumed. The canonical partition function for this ensemble can be formally represented as =
C
all possible states In1
exp(-PE{,$ =
fik&
(2)
where = l/(kBT). a k e is the contribution of the kinetic energy terms to the canonical partition function and is assumed to be only temperature-dependent; 0, is the configurational part of the partition function. In the following, we avoid taking temperature derivatives of
2534 Ind. Eng. Chem. Res., Vol. 26, No. 12, 1987
thermodynamic quantities and can thus employ Q and O, interchangeably. Following the approach of Panayiotou and Vera (1982) and assuming the random mixing of holes and molecules, we obtain an expression for O, which is valid outside the critical region of the pure component; i.e., Qc
= MT)Nlg(No,Nl,rl,z)e x ~ [ - ~ E ( N ~ , N ~ , r (3) ~ , z ) l y( T ) represents a temperature-dependent function which
where g(NoJVl,rl,z)represents a combinatorial term which we obtain following Guggenheim (1954), and the exponential term is a Boltzmann probability contribution of the maximum term to the canonical partition function (i.e., the so-called mean-field approximation has been used). X(T) is a temperature-dependent function that accounts for the conformations of a molecule (Sanchez and Lacombe, 1977). Writing the terms in eq 3 explicitly yields
P
N1ql
exp[ p l q l ~ lNo l + Nlql
]
(4)
where the exclamation mark (!) stands for a generalized factorial, i.e., a! = r ( a + 1). X(T) is assumed to be independent of density; i.e., the internal configurations of a chain molecule shall change only with changes in temperature. It must be emphasized here that we do not need to use the c parameter that was suggested by Prigogine (1957) to account for the external degrees of freedom for a chain molecule. Since this class of lattice models are not based on a cell-model-typepartition function, the question of how the external and internal degrees of freedom are separated is never encountered (Sanchez and Lacombe, 1974). The bridge from the canonical partition function to thermodynamics is through the use of standard statistical mechanics (Huang, 1963):
A = -k,T In
(5)
where A is the Helmholtz energy of the system. The EOS and related thermodynamic quantities can then be obtained through familiar techniques (Model1 and Reid, 1983). Using the reducing parameters z
~ € 1= 1
PCUH= R T *
(6)
and defining V, the total volume of the system, as
v = UH(N0 + r,NJ
(7)
we obtain an EOS for a pure component; i.e.,
Here 29 is the effective surface area fraction of molecules in the lattice, 29
E
qlNl/(NO+ q1NJ = (ql/rl)/[c
+ (ql/rl) - 11 (8b)
and the tilde ( - ) denotes a reduced variable. All quantities, except u, in the EOS are reduced by the parameters in eq 6. The specific volume, u, is nondimensionalized by u*, the molecular hard-core volume, which is defined as u* = NlrluH or
rl =
U*/(NluH)
(9)
Expressions for the chemical potential of a pure component can also be derived from eq 5,
is constant for a species distributed between phases at equilibrium. Detailed derivations of the expressions for the EOS and the chemical potential are given in Panayiotou and Vera (1982). In this context, it must be noted that the EOS (eq 8) is identical with Panayiotou and Vera's result (1982);we include its derivation here because it is used implicitly in the derivation of the mixture EOS, presented below. Determination of Pure-Component Parameters. In the pure-component EOS (eq 8), there are apparently four independent, unknown parameters: we chose z, uH, u*, and ell. As stated earlier, a lattice cell here does not have the function usually assigned it in the lattice fluid theories, namely, to hold one chain segment. In this theory, the lattice merely serves to describe the number of ways neighbors in space can mutually be arranged. It is, however, mandatory that the cells are neighter excessively small nor large since eq 1only has meaning if the cells are large enough that a small, compact molecule such as water or carbon dioxide does not occupy many adjacent sites (no "internal surfaces"). Also the cells must be small enough to allow only one small molecule or chain segment to claim a cell for itself. These arguments lead to a choice of a cell volume in the range 5-30 A3 (3 X 104-18 X lo* m3 mol-l). A related question is the value to be chosen for the lattice coordination number, z. The lattice, having lost its literal meaning in the lattice fluid theories, nevertheless imposes its connectivity on the fluid phase. In three dimensions, the only case of interest here, a lower bound is probably the value for the simple cubic (or von Neumann) lattice with z = 6; it is unlikely that a small, compact molecule in a dense packing would be surrounded by fewer neighbors. An upper bound is clearly the number of nearest neighbors a closest packing of identical spherical objects provides, i.e., z = 12. Hence, z must fall in the range 6-12. We set z = 10 and U H = 9.75 X lo+ m3mol-l (corresponding to 16.2 ~ 3 1 ~ ~ 1 1 ) . To model real substances, we need to determine ell and u*. For a pure component below its critical point, the technique of Joffe et al. (1970) was employed. This involves the matching of chemical potentials of the component across the liquid and vapor phases a t the vapor pressure of the substance. Also the actual and predicted saturated liquid densities were matched, PL = Pv
(Ila)
uL(actual) = uL(predicted)
(1lb)
This set of equations was solved by the use of Newton's method to yield the pure-component parameters. In Table I, we list values of € l l / k B and u* for some common chemicals. It must be e m p h i z e d here that the values of ell/kB and u* are temperature-dependent functions. Also, the specific choice of lattice parameters ( z and uH) was found to have no effect on the capability of the EOS to reproduce pure-component vapor-liquid equilibrium (VLE) in the limits stated above (3 X lo4 5 uH 5 18 X lo+, 6 Iz I12). Pure-component vapor pressure data for polymeric substances are not available. Instead, we used the parameters given by Panayiotou and Vera (1982), obtained by fitting the EOS to the thermal expansion coefficient, isothermal compressibility, and thermal pressure coeffi-
Ind. Eng. Chem. Res., Vol. 26, No. 12, 1987 2535 Table I. Pure-Component Parameters for Some Common Chemicals:“ z = 10, v m = 9.75 X 10“ ma mol-’ chemical T,K u*, ms mol-’ ~ l l / kK~ , 432.6 283 17.28 X 10“ water 422.9 17.82 X 10“ 293 328.4 19.45 X 10“ 413 190.5 56.97 X 10“ 293 methanol 55.28 X 10” 163.3 283 ethanol 55.27 X 10” 163.0 293 136.5 413 59.75 x 10” 71.51 X 10“ 147.5 293 1-propanol 140.9 86.82 X 10“ 293 1-butanol 129.9 148.73 X 10” 1-octanol 293 123.6 87.59 X lo4 tert-butyl alcohol 293 98.9 128.56 X 10” n-heptane 293 101.0 184.60 X lo4 293 n-decane 106.3 257.40 X lo4 293 n-tetradecane 123.9 68.44 X 10” 293 acetone 119.9 82.68 X 10” benzene 293 158.0 88.68 X lod 327 aniline 146.9 106.07 X 10” naphthalene 308 188.7 113.65 X 10” 323 anthracene 202.2 94.93 x 10” 313 benzoic acid 37.07 X 10” 219.5 294 formic acid 177.4 55.83 X 10” 293 acetic acid 124.4 98.89 X 10” 293 chlorobenzene 108.6 97.40 X 10” 293 cyclohexane 120.1 111.86 X lo4 293 o-xylene 117.0 113.68 X 10” 293 p-xylene
Table 11. Pure-Component Parameters for Some Common Chemicals above Their Critical Temperaturesf z = 10, v H = 9.75 x 10” ms mol-’ chemical T,K u * , m3 mol-’ cll/kB, K 5.553 X 73.732 ethane 290-370 3.872 X 82.026 carbon dioxide 300-370 ~
“Carbon dioxide P-u data from Angus et al. (1976). Ethane data from Goodwin et al. (1976).
appropriate for phase equilibrium calculations since they ensure that model predictions will always converge to the correct pressure limits when the mixture is essentially a pure component (i.e., when the mixture is infinitely dilute in the other components). Discussion. The EOS given by eq 8 differs from the one suggested by Sanchez and Lacombe (1974) since the latter was developed for lattices with large coordination numbers. While this approximation simplifies calculations and frees the model from the arbitrariness of a chosen value of z , a large value of z is physically unrealistic, since this means that a molecule may have an infinite number of neighbors. This situation is infeasible if a molecular segment is to have a finite size. If z is set to infinity in eq 8, we recover Sanchez and Lacombe’s (1974)lattice fluid EOS,
aAll vapor pressure and liquid density data from Reid et al. (1977). Additional data on ell and u* for ethanol and water as a function of temperature are published in Kumar et al. (1985).
cient data. However, it must be noted that, in general, the parameters obtained from vapor pressure and liquid density data are more reliable for use in an EOS for phase equilibrium calculations than parameters deduced from properties less associated with phase equilibrium. In the vicinity of the critical point, the EOS must not be used since the mean-field approximation was evoked in its derivation. However, as for all classical EOS’s, the standard criteria for the critical point can be written and solved to yield the parametric equations
Z, = (Pu/RT),
= f(PUH/RT),
(tll/RT)c = g(PUH/RT)c
Equation 8, therefore, is simply the lattice field EOS for finite values of z. The EOS fails in the high z limit when used to model mixtures (see section on mixtures in Results and Discussion). In the dilute limit, eq 13 is identical with the result one obtains from the Flory approximation (Flory, 1942) for the configurational part of the canonical partition function, 0, (see eq 4). This is clear if one terms the “holes” as “solvent” (with E~~~~~~~~ - tpoly-solv= 0) and makes the assumption of a dilute “solution” and the following definitions:
(124
fj =
A derivation of eq 12a and 12b is provided in the Appendix section. The point to be made here is that the predicted critical compressibility is not constant but varies according to the critical point of the pure substance under consideration and is typically between 0.23 and 0.35. This is in contrast to classical two-parameter EOS’s (Soave, 1972; Peng and Robinson, 1976) wherein a constant value of the critical compressibility is predicted. For supercritical components, pure-component parameters were obtained by fitting experimental P-u data on an isotherm by using eq 8. Data for the substances tested (ethane, carbon dioxide) suggest that the computed u* is only a weak function of temperature and q1is a constant within regression error. For both supercritical systems modeled, therefore, u* and ell are assumed to be constant. In Table 11, we present pure-component parameters for ethane and carbon dioxide above their critical temperatures. In summary, we have determined pure-component properties in most instances (except polymers) by employing P-u data, rather than by fitting the EOS’s (eq 8) to properties obtained by taking temperature derivatives of thermodynamic functions (like thermal expansion coefficients). Parameters determined in the former fashion (i.e., by fitting to P-u data), as stated earlier, are more
4pof-l
p/T
(12b)
.i;= x-’ r1 = npo1 = POsmuJRT
(14)
Here 4polis the volume fraction of the polymer, x is the Flory-Huggins interaction parameter, nPl is the degree of polymerization, P- is the osmotic pressure of the solution, and u, is the molar volume of the solvent. One then recovers the Flory (1942) equation for the osmotic pressure from eq 13, posm
= -(RT/uJ[ln (1- 4poJ
+ (1 -
l/npol)cbpol+ xcbpo121
z
- m,
fj
m
(15)
It is clear that under appropriate approximations and assumptions, the lattice EOS (eq 8) does reduce to the older models. Under the relaxation of these constraints, we recover in turn the Sanchez-Lacombe (1977) lattice fluid EOS and eq 15. Equation 8 represents a general lattice model EOS that incorporates a smaller level of arbitrariness than most existing models. In order to understand qualitatively the behavior of the pure-component EOS, it is examined in the limiting case of small molecules, i.e., when q,r 1. Then, eq 8 reduces to the form
-
We write eq 16a in a compressibility form as
2536 Ind. Eng. Chem. Res., Vol. 26, No. 12, 1987
The second term in eq 16b, Za,can be rewritten as
where 2, is termed the ”attractive” contribution to the compressibility, 2 (=Pv/(RT)),since it has the same form as the attractive term in the van der Waals EOS (van der Waals, 1873). On examining the data for P* and u* for ethanol and water and computing the parameter a in eq 17, it was found that for temperature variations in a range of 150 K this parameter always changed by less than 370, themselves although the computed values of u* and showed a 7% variation (Kumar et al., 1985). In the limit of small molecules, therefore, the lattice EOS has an attractive term that closely approximates the characteristic of the van der Waals type attractive term. On expanding the “repulsive” term in eq 16 in a Taylor series, we obtain ZR = PRu/(RT) = -17 In (1- p ) = 1 + p / 2 + p2/3 + p3/4 + ... (18) where 3 = 116, and the series is value for p N 0. Here ZR represents the contribution of the “repulsive” term to the compressibility. Now expanding the van der Waals repulsion term [=RT/(u - b), where b plays the same role as u* in eq 81 around p = 0, after defining /3 = 2b/u, we obtain ZR = U / ( U - b) = 1 + p / 2
+ p2/4 + p3/8 + ...
where gR represents the number of configurations in a case where molecules mix randomly and gNR is a factor that corrects the combinatorial term for the effects of the nonrandom distribution of molecules. [It must be emphasized, that, in all cases, summation indexes (i,j,k,l) go from 1 to n, the number of components. If holes were assume to mix nonrandomly, then the indexes would go from 0 to n, where 0 would then refer to holes.] gR can be represented, following Guggenheim (1954), as
while gNR is represented following Panayiotou and Vera (1982) as
(19)
On comparing eq 18 and 19, we see that the repulsion 0) are identical. terms in the low-density range ( p Differences do appear a t high densities, however; they are caused by the presence of the logarithmic functionality in this lattice model’s repulsive term. In summary, therefore, the lattice EOS behaves similar to the van der Waals EOS a t low densities in the case of small molecules. From the above discussion, it is clear that the EOS in eq 8 is a simple extension of well-known EOS (Kleintjens, 1983; Flory, 1970; Sanchez and Lacombe, 1976, 1977; Okada and Nose, 1981a). It reduces in some limits to well-known EOS both for polymer molecule and small molecule EOS’s, thus suggesting that it has the correct functional form for applications to systems of interest. It also has the merit that it has only two adjustable parameters per pure component. It is expected that the physical realism built into this EOS will yield a maximum of fit to experiment and predictive power with a minimum number of adjustable parameters when extended to mixtures.
Mixtures Theory. Consider a mixture of No holes and Nimolecules of type i (i = 1, n) placed on a lattice having the same values of the coordination number (z) and the cell size (uH) as the pure-component lattices. Molecules of type i have ri segments and interact with nearest-neighbor segments of j molecules with an interaction energy -q, (with eo; 0, for all j ) . Pairwise additivity of interaction energies is assumed. The canonical partition function for this ensemble can be formally represented as in eq 2. We again make the mean-field approximation and write QC
where eq 20 is analogous to eq 3 and the braces, ( ),denote the sets of all values of the enclosed symbol. The Xi factors, accounting for the internal degrees of freedom in the chains, are again assumed to be functions of temperature only. E(No,(Ni),(ri),z,(tij)) represents the total potential energy of the system. To be consistent with the purecomponent EOS, in the following derivation, we assume that holes mix randomly. [The extension to the case where holes mix nonrandomly is simple, in principle. For pure components, this has been done by Panayiotou and Vera (1982).] Molecular segments, however, are allowed to mix nonrandomly, and the partition function, therefore, has to incorporate terms that account for this effect. For convenience we can rewrite eq 20 in the form
nXiNlg(No,(Ni),(riJ,z,(€ij)) exp[-~E(No,iNi~,Iri),z,I€i;))I (20)
where
Ni?= Nir/t9 Nir = Nir/t9
f
(z/2)(Niqi)ai (i = j )
(23a)
2(z/2)(Niqi)8, (i # j )
(23b)
Here Ni? represents the number of i-j segment contacts, on a “hole free” basis, in the case where all molecular segments mix randomly and Ni, represents the number of i-j contacts in the nonrandom case. We introduce the nonrandomness correction ri,for the distribution of the segments of species i about the segments of species j . It is defined by
Ni, = NijOri,
(24)
where the Ni;s must satisfy the mass balance constraints (Panayiotou and Vera, 1981)
Nii + ZNi, = Nizqi I
for all i
(25)
Since all the unlike interactions Ni, (i # j ) must equal N,i, r.. L1 = rJ l i#j (26) The potential energy of the system, E(No,{Ni~,(ri))z,{€i;}), under the assumption of the pairwise additivity of interaction energies is
E = tYCCNi,(-ti;)
(27)
where 19 represents the total surface area fraction of the lattice occupied by molecular segments [=xNiqi/(No + C N i q i ) ] .To determine the nonrandomness corrections, therefore, we must substitute eq 22-27 in eq 21 and maximize 9, with respect to all the r,’s (i # j ) . [The
Ind. Eng. Chem. Res., Vol. 26, No. 12, 1987 2537 remaining rii(i = j ) can then be determined from the mass balance constraint, eq 25.1 The resulting equations are also known as the quasi-chemical equations (Guggenheim, 1954) and can be represented, following the notation of Kehiaian et al. (1978),as Pi? =
riirjje x p ( 9 A c i j / k B T ) r.? LJ I r.,r..t.. 11 41
i #j
(28a)
(28b)
12
Reducing parameters for this derivation are defined in a manner analogous to eq 6 and the system volume defined as in eq 7: ( 2 / 2 ) =~ R~T * = P U H (38)
v = (NO+ CNir,)UH
(39) We present here the final expressions for the EOS and the chemical potentials of component i in the mixture,
where A t41. . =
E., 11
5
+ EI1. . - 2e..11
(29)
For a multicomponent system, the resulting equations are a coupled set of quadratic equations. Clearly, these allow for an analytically closed form solution only in the case of a two-component system. However, if it is recognized that a value of tijdifferent from unity represents a departure from randomness, then we can expand eq 28a as a multidimensional Taylor series in terms of the small parameters (tu- 1; i # j ) . The general first-order solution to the equation set (eq 28a) is
-P T= I n [ u’rl]+3n[
f
(qM/rM) - 1 u’
1-
92 + ~ 93[ ~ ~ ~ ~ Q I g l ~ + k Q ~ ( ~ , l A ~ l T tklAckl - t k j A E k j - t c k A E ~ k ) / k B T l ( z / ~ ) 9 2 C ~ 3 c c ~ , 9 , ~ k ~ l [ ~-~t b j, A - cf k ~l ) j+( 2 ~ j l
( 1 - tl1)(25,iA~,i- tLIAclJ - h & k i ) ] / k ~ T (40)
rij= 1 + ( 1 - t i j ) / 2 + yZxc(i- tlm)919m 1/2C(1 - t i m > a m - 1/2C(1- 6 m j ) a m (30)
where ai is the surface area fraction of the segments of i molecules, on a hole free basis (=Niqi/CNiqi).This approximate solution to the quasi-chemical equations has been tested against exact numerical solutions, and it was found that it is good to within 5% over most of the ranges tested for three and four-componentsystems. Higher order solutions for these equations exist in the literature (Mickeleit and Lacmann, 1983), and these can be used if greater accuracy is desired. On using eq 30 in the equation for the total energy of the system (eq 271, one obtains
E = -(2/2)[CNiqi]9~CCQjgk€jjk +
‘/zCCCC9j9k919mcjk(Ekm+ t j m - t1m - f j k ) ) (31) We now define the mixing rule for the mixture parameter CM in terms of the pure-component parameters, Q.I EM
= ECQj8kcjk
= (-E) / ( 8 ( z / 2 ) [ CNiqiI)
(32)
+
f/zCCCC9j9k919mEjk(tkm+ t j m - t l m - t j k ) (33) Mixing rules for qM, rM,and uM* are defined as qM = Cqixi
(34)
rM = Crixi
(35)
uM* = c u i * x i
(36)
where x i is the mole fraction of component i in the relevant phase. The only unknown quantities in the set of equations 32-36 are the “unlike segment” interaction energies, tip A suitable combining rule for this parameter is suggested by eq 29, cij = tij
= 0.5(cii
Eii
i =j
+ e j j ) ( l - &j)
(37)
i
# j
6, is thus a measure of the departure of the mixture from randomness; i.e., when 6U.s are zero, the mixture is random since the appropriate t i . is then identically equal to unity. The mixture EOS and related thermodynamic quantities can now be derived by the use of standard techniques: the detailed derivations are provided elsewhere (Kumar, 1986).
+
C C ~ C 2 Q ~ g k g l g m ~ j k ( ~ tk] m m - t1m
- Elk)
+
+ t]1 + t k l + t j 1 - 2hl - 2 f j k ) +
~ ~ ~ s j s k a l ( c ] k ( ~ k ~
+ f j k - t l k - t j i ) + EkL(fk, + E,] - El] - t k r ) ] + ‘J(1- ~ ) C C C C 9 , 9 k g l ~ m ~ ~ k ( ~ +~ fl~Acjm m A ~ k mCji(ttk
- t]kAcjk)/kBn+ ( z / 8 ) q , C C C c 9 j 9 k s ~ 9 m ((1- f]k)(2f]m - 261k - f i m + 2 f ] c - ‘$1,) (1- f , k ) ( % i m - t l k - t l m ) - (1- t ~ , ) ( 2 t ] m- f ] k - t1m)t + (2/8)q,0(1- 9)CCCCs,9kg19m/kBT(tjkA€]k(2t]m f j k - Elm) - (1- t]k)(2t]mAt]m - t ] k A E ] k - flmAclm)j (41)
tlmAElm
I
We note that the first three terms in the mixture EOS are identical with the terms in the pure-component EOS (eq 8). The additional terms in eq 40 account for the nonrandom distribution of molecule segments. Panayiotou and Vera’s (1982) EOS for binary mixtures is identical with eq 8. They state that they had explicitly accounted for the nonrandom distribution of molecular segments. However, it is seen in eq 40 that the nonrandomness correction terms introduce two extra terms as compared to eq 8. An approximation must thus have been made by Panayiotou and Vera (1982) for them to have obtained the same formal expression for both the pure-component and binary mixture EOS’s. In eq 41, y , ( T ) is a temperaturedependent function (same as the y ( T ) for the appropriate pure component i in eq 9), and the last two terms account for the nonrandom distribution of molecular segments. For all binary mixtures examined, a simpler form of eq 40 and 41 may be used with little loss of accuracy. These approximations, shown below, have been tested for binary mixtures only, however, and may not be sufficiently accurate for mixtures with more than two components:
2538 Ind. Eng. Chem. Res., Vol. 26, No. 12, 1987
I
z -__
----
I
-_-i-___-i
02
p -
3
“3
cz
36 $IOl€
Ice ?cc o r -cetane
Figure 1. Comparison of (-) lattice model predictions and experimental data of Weisphart (1975) for the acetone-benzene binary 303, (A) 313, and (v)323 K with 6, = 0.022. mixture at (0)
06
+OC’ 0“
q*s
;a
Figure 3. Comparison of (-) lattice model predictions and experimental data of Ng et al. (1980) at (0) 310 and (A)352 K with a,, = 0.09 for the H2S-n-heptane system. g ,
i
E -
i 0
C2
Figure 2. Comparison of the experimental data of Pemberton and Mash (1978) for the ethanol-water binary mixture at (A) 323, (0) 343, and (V) 363 K with (-) lattice model predictions with b,, = 0.085.
Results and Discussion. The expressions derived for the mixture EOS and the chemical potential of component i in a multicomponent mixture were used to model the phase equilibria of binary mixtures. A set of nonlinear equations were obtained
w& = p:
a t constant T , P
(44)
for all components i across all phases s and t. These equations were solved by the use of Newton’s method to yield the equilibrium-phase compositions. Mixtures of small molecules (acetone-benzene, ethanol-water) were considered first. In Figures 1 and 2, a comparison is made between the fitted and experimental low-pressure VLE data (Weisphart, 1975; Pemberton and Mash, 1978) for these systems. An excellent fit to the data is obtained in both cases, with the use of one temperature-independent parameter (tiij) per binary. The interesting aspect of this modeling is the temperature independence of .6, I t has been shown above that for pure components of small molecules, the lattice EOS has an attractive term with an essentially temperatureindependent van der Walls parameter, a. Extending this argument to mixtures results in the prediction of the temperature independence of bij for binary mixtures of small molecules, which could make the modeling technique predictive. In examining the sensitivity of the model to the assumed value of z, it was found for the ethanol-water system that the model results were insensitive to z values in the vicinity of z = 10. For unrealistic values of z (z > 15), however, it was found that the model was incapabie of even qualitatively fitting mixture VLE behavior. It is speculated that the reason for the failure of the model in the limit of large z lies in the fact that a molecular segment is presumed to have more nearest neighbors than possible even
34
06
CB
‘
Vole ?act 01ZC2
Figure 4. Comparison of the experimental data for the acetoneCOz binary mixture (Panagiotopoulos,1986; Katayama et al., 1985) at (A) 313 and (0) 333 K with (-) lattice model predictions with 6,, = 0.0.
in a cubic close-packed lattice. The capability of the lattice EOS in the modeling of the VLE of mixtures of molecules of different sizes, a t moderate pressures, was examined next. The results for the H2S-n-heptane system a t 310 and 352 K (Ng et al., 1980) are shown in Figure 3. For the temperatures considered, it is seen that there is good agreement between the correlation and the experimental data. The lattice model thus provides the capability to obtain good, quantitative fits to experimental VLE data for binary mixtures of molecules below their critical point. The model was then applied to two-phase compositionally symmetric binaries with a supercritical component and a liquid solute. In Figure 4, the model calculations are compared to experimental data for the carbon dioxide-acetone system a t two temperatures [313 and 333 K,see Panagiotopoulos (1986) and Katayama et al. (1985)l. A single, temperature-independent binary interaction parameter was employed in the fitting of the model to data. Fits to the data are in good agreement with the measured data outside the critical region. Since the model formulation makes use of the mean-field approximation, it is not expected to perform well in the critical region. It has been suggested that a lattice model can be made to fit the behavior of substances in the vicinity of the critical point with the use of a “decorated lattice model” (Gilbert and Eckert, 1986). This genre of models builds the critical exponents into the problem and thus can model the critical region with greater accuracy. For out purpose, however, the near-critical region (i.e., IT - T J / T , < 0.1, IP - P,I/P, < 0.1) is of little interest. In Figure 5, the model behavior is compared to experimental data for the benzoic acid-carbon dioxide binary at three temperatures [318, 328, and 343 K;see Schmitt (1984)l. Since the solid benzoic acid phase is crystalline, it is pure. In our modeling of this system, it was therefore
Ind. Eng. Chem. Res., Vol. 26, No. 12,1987 2539 c
g
F
P 1
3
.~
e\i
, t 91 Si5
250
lW5
2 - X
C l a i r MolecJlar Vass
Figure 5. Lattice model predictions (-) with experimental data on the benzoic acid-CO2 binary mixture (Schmitt, 1984) a t (V)318, (A) 328, and (0) 343 K with 6ij = 0.12 (318 K), 0.10 (328 K), and 0.092 (343 K).
Figure 8. Experimental data for the partition coefficient of different styrene oligomers between a polymer-rich phase and an ethane-rich phase at 25 MPa and 333 K: ( 0 ) data obtained with a polymer of average molecular weight of 2000 and (A)from a polymer of molecular weight 800.
F ’Y -
c
sc
25
3
5
_‘_ _
?e&ceb ?erst’,
Figure 6. Solubility of acridine in COz, plotted against the reduced 328 and (A)343 K compared to (-3 density of the COPphase a t (0) lattice model predictions with 6,, = 0.135. Experimental data are of Schmitt (1984).
Pfessue (LylFc
Figure 7. Comparison of the experimental data for the solubility 328 and (A)343 K with (-) of acridine in COz (Schmitt, 1984) at (0) lattice model predictions with 6,, = 0.135.
assumed that the solid phase was essentially pure. (The two phases in equilibrium are, therefore, compositionally asymmetric.) Outside the critical region, the lattice EOS provides a good fit to the measured data. In the critical region, however, at one pressure, the equation of state predicts the presence of a three-phase region. This is the cause for the discontinuity of the curves in the region of sharp rise in solubility. Outside of the critical region (approximately 7% away from T, and Pc), the model correlates the data within the accuracy of the experimental measurements. The model also provided a good representation of the volumetric properties of other solid-supercritical systems. In Figure 6, we show a plot of the solubility of acridine in carbon dioxide as a function of density a t 328 and 343 K. The goodness of fit displayed by this plot along with Figure 7 suggests that the lattice model EOS has the correct functional form to reproduce the P-u properties of dilute binaries in the supercritical region. In summary, for solid-supercritical systems, the lattice model can reproduce experimental data to within experimental accuracy outside the critical region. Volumetric properties of these systems are also well predicted.
Figure 9. Comparison of (-) lattice model predictions with experimental partition coefficient data for the polystyrene-ethane 34 MPa with 6, system a t 333 K and (A) 10, (+) 15, (v)25, and (0) = 0.018.
For the six different systems examined, covering a range of pressures and phase behavior, the capability of the lattice EOS to correlate phase equilibria for mixtures of “small”molecules is comparable to traditionally employed techniques (for example, the single-parameter PengRobinson EOS). While we present details of this comparison elsewhere (Kumar, 1986), the point to be emphasized is the utility of the lattice EOS, developed mainly for the correlation of mixtures of molecules of disparate sizes, to model mixtures of small molecules of essentially the same size. In Figure 8 we present a plot of the experimental equilibrium solubility of polystyrene chains in ethane at 25 MPa and 333 K as a function of oligomer molecular mass (Kumar et al., 1986, 1987). The plotted ordinate is a partition coefficient (rather than the equilibrium solubility) which is defined as
K = w:/w?
(45)
wherein w: represents the mass concentration of the ith oligomer in the ethane phase (in units of kg of oligomer/kg of ethane), while w! is the mass concentration of the same oligomer in the solid phase on an ethane-free basis (in units of kg of oligomer/ kg of ethane-free solid). An important implication of the linearity of the log K vs molecular mass plot is that the equilibrium solubility of an oligomer from a polymer into a SCF phase occurs as if the condensed phase were an ideal solution. In essence, therefore, we are dealing with a linear combination of independent n-mer ethane quasi-binaries. We use this deduction as the starting point in our modeling. Parameters for polystyrene above its glass transition temperature (-380 K) are available (Panayiotou and Vera, 1982). These were employed in our modeling in the temperature range 323-343 K, with the following two assumptions. (i) As the presence of compressed gases reduces the glass transition temperature of a polymer (Wang et al.,
2540 Ind. Eng. Chem. Res., Vol. 26, No. 12, 1987
1 i
c
_
I
Figure 10. Comparison of (-) lattice model predictions with experimental partition coefficient data for the polystyrene-ethane system at 25 MPa and (A)313, (+) 323, and (v)333 K with JI2 = 0.018.
1982), we accept that polystyrene in the temperature range 323-343 K, in the presence of ethane, is above its glass transition temperature. (ii) ell and u* (in units of m3 kg-') were assumed independent of chain length. In Figure 9 , we compare experimental partition coefficients and the results obtained from our lattice model for these two-phase, quasi-binary systems. A single, adjustable parameter (612 = 0.018) was employed a t the three pressures 15,25, and 35 MPa a t 333 K in Figure 9 and at the three temperatures 323, 333, and 343 K at 25 MPa in Figure 10. The value of a12 was obtained by fitting the model to data at 25 MPa and 333 K. Results at the other two pressures and temperatures are, therefore, predictions and are in good agreement with experimental data beyond a molecular weight of approximately 1200 g mol-'. The reason for the lack of agreement between the model and experiment below this molecular mass is speculated to be due to two factors. First, we assume that segments of all oligomers interact with the same energy, ell. Clearly, this assumption is not valid for the chain ends, and the model breaks down for shorter chains. Second, the results shown in Figure 9 are valid only as long as there are two phases at equilibrium. It is known that at these temperature and pressure conditions, pure styrene monomer is completely soluble in ethane. Results predicted with a two-phase quasi-binary model are, therefore, unreliable for the smallest oligomers. In summary, a lattice model has been used to correlate partition coefficients with variations in chain lengths, pressures, and temperatures using a constant value of the interaction parameter, a12. An estimate of this parameter can be obtained from partition coefficient data of any oligomer above the cutoff value of 1200 g mol-' for polystyrene, a t any pressure. Knowledge of the interaction parameter (6'J will thus make the model predictive for all other operating conditions. In this context, we point to the fact that the lattice EOS has, to date, been tested only for binary mixtures. The applicability of the model to ternary mixtures is currently being examined.
Conclusions A statistical mechanics based lattice model equation of state and related thermodynamic quantities for multicomponent mixtures have been presented, and a simplified version for binaries (eq 42 and 43) has also been derived. It is shown that this model is a logical development and improvement over the work in the literature in this area. Results obtained to date demonstrate that this lattice EOS is successful in quantitatively modeling polymer-SCF equilibria, across variations in temperature, pressure, and polymer chain length with the use of one apparently temperature-independent parameter. For other SCF
systems, outside the critical region, it performs satisfactorily. To our knowledge, this is the first attempt to employ lattice model EOS's of this genre for the modeling of the VLE of real molecular mixtures. The results, to date, on binary mixtures suggest that this model is a useful tool in correlating and modeling data on systems of molecules, both large and small. In summary, we developed this model for a specific purpose: to model the phase behavior of polymer-SCF systems. We wanted to develop an EOS that would be simple and have the least number of adjustable parameters. It is gratifying to note that we can satisfy these needs by using a mean-field lattice model EOS.
Acknowledgment We thank Prof. Prausnitz, Prof. Koningsveld, and Dr. Dieters for helpful comments. Dr. D. T. Wu and Greg Noll of Du Pont provided us with invaluable support with their careful analysis of the document and also by subsequently verifying our calculations. The authors gratefully acknowledge support from the National Science Foundation, Division of Chemical, Biochemical and Thermal Engineering under Grant No. CBT 85-09945.
Nomenclature a = attractive parameter in the van der Waals EOS, J m3mol-2 A = total Helmholtz energy of system, J b = volume parameter in the van der Waals EOS, m3 mol-' E = potential energy of the system of molecules g = combinatorial term in canonical partition function kB = Boltzmann's constant N = number of molecular species or holes P = pressure, N m-2 Po,, = osmotic pressure, N m-2 q = effective chain length of a chain molecule, accounting for the connectivities of the segments r = chain length of a chain molecule R = universal gas constant, J mol-' K-' T = temperature, K u = specific volume of system, m3 mol-' uH = lattice unit cell size, m3 mol-' V = total volume of system, m3 x = mole fraction w = weight fraction of polymer species, kg of polymer/kg of SCF z = lattice coordination number 2 = compressibility ( P v / ( R T ) )dimensionless , Greek Symbols P = l/(kBT) y = term in chemical potential accounting for internal degrees
of freedom of a chain molecule = nonrandomness correction (eq 24) 6, = binary interaction parameter, between segments of i and j molecules t = interaction energy between molecule segments, J mol-' t9 = total surface area fraction of molecular segments h = function accounting for the number of internal degrees of freedom of a chain molecule 1.1 = chemical potential, J mol-' tLJ= term in the quasi-chemical equation (eq 28b) p = density, mol m-3 Q = canonical partition function 4I = volume fraction of species i x = Flory-Huggins parameter Subscripts 1 = species 1 0 = species 0, holes a = attractive property (eq 17) c = critical property c = configurational property (eq 3)
r
Ind. Eng. Chem. Res., Vol. 26, No. 12, 1987 2541 i = species i i j = quantity related to the i-j molecule segment pair ke = kinetic energy contribution L = liquid-phase property M = mixture property NR = nonrandom configuration of molecule segments osm = osmotic term pol = polymer property R = random configuration of molecule segments R = repulsive contribution (eq 18) s = solvent property (Flory-Huggins formulation) sol = solvent property V = vapor-phase property Superscripts * = reducing quantity - = quantities on a hole free basis
- = reduced state variable
s = condensed-phase property s = phase s (eq 44) t = phase t (eq 44) v = vapor-phase property 0 = random configuration of molecule segments and holes
Appendix: Derivation of Critical Point Criteria The critical point of a pure substance can be represented mathematically as the point where its isothermal compressibility and its partial derivative with respect to volume simultaneously diverge. dP/dvIT = d2P/dv2IT= 0 at T = T ,
(Al)
When we use the lattice EOS in this equation set, the critical point predicted by this EOS can be obtained. An analysis of the lattice EOS with the Buckingham pi theorem shows that there must be five dimensionless groups in a reduced EOS. We define four of these groups as Z H = P v H / ( R T ) = P/T' (A24
z = PV/(RT)
(A2b)
Z* = P u * / ( R T )
Wc)
< = tii/(RT)
(A24
The coordination number z is then chosen as the fifth dimensionless group. The EOS can be formally rewritten as 71(ZH,Z* ,Z,
