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Ind. Eng. Chem. Fundam. 1985, 2 4 , 246-257
Ng, S.;Harris, H. G.; Prausnitz, J. M. J . Chem. Eng. Data 1969, 14, 482. Nicolaides, G. L. Ph.D. Dissertation, University of Illinois, Urbana, IL. 1977. Null, H. R . "Phase Equilibrium in Process Design"; Wiley: New York, 1970. Pecsok, R . L.; Gump, 8.H. J . Phys. Chem. 1967, 7 1 , 2202. Pecsar, P. E.; Martin, J. J . Anal. Chem. 1966, 38, 1681. Petsev. N.; Dimitrov, C. J . Chrcnatogr. 1965, 20, 15. Petsev, N.; Dlmitrov, C. J . Chromatogr. 1966, 2 3 , 382. Petsev, N.; Dimitrov, C. J . Chromatogr. 1969, 4 2 , 31 1. Pierotti. G. J.; Deal, C. H.; Derr, E. L.; Porter, P. E. J . Am. Chem. SOC. 1956, 78, 2989. Pierotti, G. J.: Deai, C. H.; Derr, E. L. Ind. Eng. Chem. 1954, 5 1 , 95. Porter, P. E.; Deal, C. H.; Stross, F. H. J . Am. Chem. SOC. 1956, 78, 2999. Prausnitz, J. M. "Molecular Thermodynamic of Phase Equilibria"; Prentice Hail: New York. 1969. Purnell, J. H. Endeavour 1964, 2 3 , 142. Reid, R. C.: Prausnitz, J. M.; Sherwood, T. K. "The Properties of Gases and Liquids"; McGraw-Hili: New York, 1977. Santacesaria, E.; Berlendis, D.; Carra, S.Fluid Phase Equilib. 1979, 3 , 167. Scatchard, G. Chem. Rev. 1931. 8 , 321. Scheller, W. A.; Petricek. J. L.; Young, G. C. Ind. Eng. Chem. Fundam. 1972, 1 1 , 53. Sheehan, R. J.; Langer, S. H. Ind. Eng. Chem. Process Des. Develop. 1971, IO, 44. Snyder. F. S.; Thomas, J. F. J . Chem. Eng. Data 1968, 13, 527. Stalkup, F. I.; Kobayashi, R . AIChE J . 1963, 9 , 121. Staverman, A. J. Recueil 1950, 69, 163.
Stock, R.; Rice, C. B. F. "Chromatographic Methods"; Wiiey: New York, 1974. Summers, W. R.; Tewari, Y. 8.; Schreiber, H. P. Macromolecules 1972, 5 , 12. Takahashi, Y.; Urone, P.; Kennedy, G. H. J . fhys. Chem. 1970, 7 4 , 2333. Tassios, D. P. Ind. Eng. Chem. Process Des. Dev. 1972, 1 1 , 43. Turek, E. 8.; Arnold, D. W.; Greenkorn, R. A,; Chao. K. C. Ind. Eng. Chem. Fundem. 1979, 18, 426. Van Deempter, J. J.; Zuiderweg, F. F.; Kiinkenberg. A. Chem. Eng, Sci. 1956, 5 , 271. Van Horn, L. D., Ph.D. Dissertation, Rice University, Houston, TX, 1966. Van Ness, H. C.; Abbott, M. M. "Classical Thermodynamics of Nonelectrolyte Solutions"; McGraw-Hill: New York. 1982. Venkatachalam, V. R. M.S. Thesis, Clarkson College, Potsdam, NY, 1978. Windsor, M. L.; Young, C. L. J . Chromatogr. 1967, 27, 355. Wong, K. F.; Eckert. C. A. Ind. Eng. Chem. Fundam. 1971, 10, 20. Yodovich, A.; Robinson, R. L.; Chao, K. C. AIChE J . 1971, 1 7 , 1152. Young, C. L. Chromatogr. Rev. 1968, 10, 129. Zarkarlan, J. A,; Anderson, F. E.; Boyd, J. A,: Prausnitz, J. M. Ind. Eng. Chem. Process Des. Dev. 1979, 18, 657.
Received for review September 3, 1982 Revised manuscript received June 27, 1984 Accepted August 6, 1984
Thermodynamics of Quadrupolar Molecules: The Perturbed-Anisotropic-Chain Theory Pannalal Vlmalchand and Marc D. Donohue' Department of Chemical Engineering, The Johns Hopkins University, Baltimore, Maryland 2 1218
The pertrurbed-hard-chain theory (PHCT! for molecules interacting with spherically symmetric forces has been modified and extended to include anlsotropic (multipolar) interactions. The anisotropic perturbation expansion of Gubbins and Twu is used in combination with lattice theory, with the assumption that the molecules are effectively linear. The original fourth-order perturbation with square-well potential used in the PHCT is replaced by a second-order perturbation of the LennardJones potential. The resulting equation of state, the perturbed-anisotropicchain theory, is applicable to simple as well as large polymeric molecules with or without anisotropic interactions. The new equation has been applied to pure fluids having substantial quadrupole moments and their mixtures. Mixture properties can be predicted accurately from pure-component properties alone and can be fitted within experimental uncertainty with small values of a binary interaction parameter.
Introduction Design and operation of processing equipment requires a detailed knowledge of vapor-liquid equilibria. While such phase equilibria can be measured experimentally, determination of phase behavior is time consuming. Obtaining accurate phase equilibrium properties for fluid mixtures containing heavy aromatic hydrocarbons involved in coal gasification and liquefaction is especially complex. Therefore, predictive techniques, particularly those valid over a wide density range, would be useful in predicting equilibrium properties. Though there are numerous equations of state and activity coefficient correlations in the literature, no theory satisfactorily predicts the phase behavior of a wide variety of fluids over a wide range of density and temperature. In general, prediction of phase behavior is done in one of three ways: by empirical correlations, by semitheoretical equations, or by theoretical equations based on molecular models. Both empirical correlations and semitheoretical equations are easier to use, but they are restricted to either simple systems or to specific temperature, pressure, and 0196-4313/85/1024-0246$01.50/0
composition ranges. While theoretically based equations are usually preferred because they allow interpolation and extrapolation of limited data with more confidence, such equations are not without limitations. They are often restricted for calculations of thermodynamic properties to either vapor phases at low and moderate densities or to dense gases and liquids. Often, theoretical equations are restricted to certain classes of fluids containing simple argon-like molecules or complex polyatomic molecules (i.e., polymers). For example, the virial equation of state, when truncated after the second or third term, can be used only for low or moderate vapor densities. The equations of state of Prigogine (1957) and Flory (1970) are useful both for small molecules and for polymers, but they are accurate only at high densities. Hard-sphere theories (Zwanzig, 1954; Alder et al., 1972) and soft-sphere theories (Rowlinson, 1964; Barker and Henderson, 1967) are applicable only to simple (spherical) molecules. The perturbedhard-chain theory (PHCT) of Beret and Prausnitz (1975) and Donohue and Prausnitz (1978),which can be used at all fluid densities for simple and polymeric fluids, is restricted to isotropic molecules. The purpose of this work 0 1985 American
Chemical Society
Ind. Eng. Chem. Fundam., Vol. 24, No. 2, 1985
is to establish a theoretically sound equation for the entire fluid density range valid for a wide variety of molecules, including systems where the molecules interact with anisotropic (polar) forces. The central problem in the development of a theory of fluids is to constrtict a general partition function capable of representing equilibrium properties over a wide range of temperature, density, and composition. To be useful, the partition function should contain only a few physically significant molecular parameters which can be determined from limited experimental data. A reasonable approximation of such a partition function for nonpolar fluids was proposed by Beret and Prausnitz (1975) and was used to develop the perturbed-hard-chain theory of Donohue and Prausnitz (1978). PHCT is applicable to nonpolar molecules of any size and is valid for calculations of both gas and liquid properties. It bridges the gap between hardsphere theories for argon-like molecules and lattice theories for polymeric molecules; it works well for molecules of all shapes and sizes and predicts accurately the thermodynamic properties of fluids encountered in natural-gas and petroleum refining operations. In this work, the PHCT partition function is modified so that molecular anisotropies are taken into account. Perturbed-Anisotropic-Chain Theory The perturbation theories of Barker and Henderson (1967), Gubbins and Twu (1978), and of others (which are discussed in a comprehensive review by Barker and Henderson, 1976) accurately describe the properties of simple methane-like molecules. The perturbation terms contain integrals over two-, three-, and four-body distribution functions. These are evaluated using computer simulation data for the radial distribution function and the superposition approximation of Kirkwood (1935). Unfortunately, these theories are not useful for larger molecules (such as benzene, hexadecane, or polystyrene), since they do not take into account the nonspherical nature of these molecules and hence the effect of density on molecular vibrations and rotations. On the other hand, Prigogine’s theory of fluids (1957), which treats these density-dependent molecular rotations and vibrations as equivalent translations, adequately describes the liquid-phase properties of small and large molecules but fails at low densities. The perturbed-anisotropic-chaintheory (PACT) corrects these deficiencies to a large extent. Its derivation is presented in the following sections. The extension of the theory to mixtures is based on the PHCT of Donohue and Prausnitz (1978). The PHCT is essentially a one-fluid theory but it eliminates the random mixing assumption. By combining the PHCT with perturbation terms for molecular anisotropies and ideas from Prigogine’s theory for chain molecules, we have developed a three-parameter equation of state which can be used to calculate the thermodynamic properties of fluids and their mixtures even when the components differ in size, shape, or in the nature of their attractive forces. Theory of Dense Polar Fluids Perturbation theory is an extension of van der Waals’ ideas. van der Waals realized that deviations from ideal gas behavior are caused by molecular attractions and repulsions. Bulk fluid properties, which differ from one fluid to another, can be explained by considering differences in molecular structure, the link between bulk fluid properties and molecular properties being provided by statistical mechanics. As shown by Vera and Prausnitz (1972), the essential ideas of van der Waals are expressed easily in a
247
partition function. For a pure fluid, the generalized van der Waals partition function is factored to account separately for molecular attractions, repulsions, rotations, and vibrations as
where A = h/(2rrnkT)li2
For a monatomic ideal gas the last three terms are unity. For real fluids, each of the last three terms depends on temperature and density, In the generalized van der Waals theory, the attractive and repulsive partition functions are assumed to be of the form qrep
=
Vf
7
(2)
and (3)
where the free volume, Vf, is the volume available to the center of mass of a single molecule as it moves about the system holding the positions of all the other molecules fixed, and $12 is the intermolecular potential energy of one molecule due to the presence of all other molecules. Vf and $ are discussed in detail later. The last term in eq 1 accounts for molecular rotations and vibrations. In van der Waals’ equation of state and in most other perturbation theories, rotational and vibrational motions are ignored. It is incorrect to do this, however, since these motions depend on density and hence they affect the equation of state, particularly for polymers. In the PACT, as in the PHCT, the density dependence of the rotational and vibrational contributions to the partition function are accounted for by using a modification of Prigogine’s treatment. The rotational and vibrational partition function is factored into external (density-dependent) and internal (density-independent) terms as qr,v
=
[~r,vlint[qr,vlext
(4)
The internal term is characteristic of an isolated molecule and is a function of temperature alone. This affects only the ideal-gas heat capacity. Except for the amount of momentum transferred, the effects of external rotational and vibrational degrees of freedom are treated as equivalent to those of translational degrees of freedom. In the PACT, we treat the nonidealities in the equation of state (i.e., attractive and repulsive forces) caused by rotational and vibrational motions as equivalent to the nonidealities caused by the translational motion of a molecule. Each translational degree of freedom contributes (5)
to the partition function. In this equation, the attractive contribution to translational motion is given by a Boltzmann factor whose argument is the ratio of the attractive potential energy, 412, to the kinetic energy, kT. For polyatomic molecules, kinetic energy arises from translational, rotational, and vibrational degrees of freedom. However, not all of these degrees of freedom affect the equation of state. For example, many vibrations are of such small amplitude and high frequency that at normal
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fluid densities they do not affect the intermolecular interactions. Of the 3n degrees of freedom for a molecule with n atoms, 3c degrees of freedom are defiied (Prigogine, 1957) as those which affect the intermolecular interactions, and therefore the kinetic energy associated with all these degrees of freedom must appear in t,he Boltzmann factor (i.e., k T becomes ckT). The three translational degrees of freedom are density dependent and 3c is defined as the total number of density-dependent degrees of freedom. Therefore, the density dependence of each of the (3c - 3) external rotational and vibrational degrees of freedom is then assumed to be given by eq 5 and hence
repulsions. For molecular attractions, isotropic molecules are assumed to interact with Lennard-Jones potential. For such molecules, the attractive energy is calculated by modifying the perturbation expansion of Barker and Henderson (1967) to take into account segmental interactions. Using their Monte Carlo simulation studies, the integral appearing in the first-order perturbation term is fitted to a polynomial in close-packed reduced volume, ijd, which is based on the hard-sphere volume, ud*. Similarly, the integrals for the second-order perturbation term are fitted to a polynomial in iid, using the calculations of Smith et al. (1970) with Percus-Yevick values of the hard-sphere radial distribution function. A PadG approximation is used to account for higher-order terms (see Appendix A). Results for spherical molecules are then extended to chainlike molecules with
By use of the above equations, the generalized van der Waals partition function given by eq 1 can be written as
(12) and
For anisotropic molecules, we assume that the potential can be factored so that 4 = @so + @ani (8)
where qPois the potential due to the isotropic nature of the molecule. The isotropic potential is taken as an unweighted average over all orientations. The anisotropic potential of a molecule, 4mi,arises due to multipolar forces, anisotropic induction forces, anisotropic dispersion forces, etc. (Gubbins and Twu, 1978). In this work, we are considering only molecules in which quadrupolar forces predominate and all other anisotropic interactions are assumed to be negligible. Molecules which interact with dipolar and other anisotropic forces will be treated in a subsequent publication. For polyatomic molecules we assume +iso Eiso$iso[p,ij = tq$so[p,Q (9) ~
where r is the number of segments in a molecule. The quadrupolar interaction energy between molecules is calculated with the perturbation expansion of Gubbins and Twu (1978), assuming the molecules to be effectively linear. The anisotropic forces are treated as a perturbation over isotropic (Lennard-Jones) molecules. Their results for small quadrupol? molecules are extended to chain-like molecules using v', T, and a TQwith u- = - u= u*
U
NArg3/(W2)
(14)
and
where
and 4 a n i oc E a n i
$ani[p,ij = fQq$ani[p,
(10)
where t is the energy per unit area due to the isotropic part of the molecule, tQ is the quadrupolar energy per unit area, and q is the normalized external surface area of the molecule (relative to the surface area of a sphere of the size of a segment). Therefore, for polyatomic molecules, $1 2ckT is proportional to the sum of EqlckT and cQq/ckT. For a monomer, this reduces to (t/kT + EQ/kT). The attractive potential energy, 4, appearing in eq 7 can thus be written as a sum of contributions due to isotropic, and multipolar anisotropic, interactions. The modified partition function for a multipolar molecule is Vf (11)
For central-force molecules, with @ani = 0, the above partition function reduces to a form analogous to that proposed by Donohue and Prausnitz (1978). A detailed derivation of thermodynamic properties using the modified partition function is given in the Appendices. In this work, the free volume expression of Carnahan and Starling (1972) is used with a temperature dependent hard-core diameter, d, to represent the effect of molecular
with a, being the surface area of a segment (defined as unity) and p is the soft-core diameter of a segment. The quadrupole moment, Q is related to the quadru olar interaction energy per segment, :Q by Q: = Q2,!r(u*)5/3, where r is the number of segments per molecule. Complete equations for the configurational Helmholtz energy and the equation of state for pure fluids are given in Appendix A. All other configurational thermodynamic properties can be derived from it. Mixtures
The pure-component partition function is extended to mixtures using a one-fluid approximation, however, without the usual random-mixing assumption. For a mixture containing N , molecules
(16) where N = ENi i
Ind. Eng. Chem. Fundam., Vol. 24, No. 2, 1985 249
100.0
tc
-I
-
-
-
-
500
20.0 10.0
/
5.0
2.0
-
i 1 1 1.0
10
CARBON
0.5
-
0.2
-
0.1
-
-
0.05
-
0.02
-
-
12
NUMBER
Figure 1. Attractive energy per molecule a a function of carbon number for aromatic compounds benzene, naphthalene, and 1methylnaphthalene. Table I. Pure-Component Parameters P , K 100u*,L/mol ethane 227.34 3.2065 ethylene 204.36 2.9568 carbon dioxide 192.77 2.2272 benzene 379.94 5.6353 naphthalene 493.22 8.4040 1-methylnaphthalene 494.50 9.1822
1
2
3
I/TPK)
x
4
5
6
1000
Figure 2. Comparison of experimental and calculated vapor pressures. c
TQ*, K
1.1383 1.1801 1.1788 1.3362 1.5234 1.7742
2.003 20.94 155.40 157.78 204.75 151.59
Following Barker and Henderson (1976) and Gubbins and Twu (1978), the Helmholtz energy can be expanded as a power series in reciprocal temperature as A=
Table 11. Binary Interaction Parameters 100 0.28 0.81 -0.92 0.95 -1.29 1.55 2.26
kij X
benzene-1-methylnaphthalene carbon dioxidebenzene ethylenecarbon dioxide carbon dioxide-ethane ethane-benzene ethylene-ethane carbon dioxide-1-methylnaphthalene
For dissimilar shaped molecules, the quadrupolar energy of the mixture is given by [ C Q q l M = EQZXiqi
(19)
1
(17)
where the higher-order terms, A?, Aqnni,etc., are taken into account with a PadG approximation as shown in Appendix B. The one-fluid theory is applied differently to each term of the perturbation expansion. This eliminates the approximation that ',he mixture is completely random (Henderson and Leonard, 1971; Donohue, 1977). The mixing rules of Donohue and Prausnitz (1978), which are based on the lattice theory model, are used for calculations of the isotropic terms. The mixing rule for anisotropic terms is based on the following two conditions. First, for mixtures of both small and large molecules, the mixture properties should be based on surface and volume fractions rather than on mole fractions. Second, the anisotropic part of the second virial coefficient should have a quadratic dependence on mole fraction. Complex molecules can be divided into r segments, with a as the soft-core diameter of each segment. Each molecule has a surface area q and a quadrupolar interaction energy of fQ per unit surface area. For mixtures of similar nonspherical molecules (like a mixture of benzene and naphthalene), the soft-core volume of the mixture is [u*]M
a:
[rdIM= dCxjrj j
(18)
The product of these two mixing rules can be generalized to ~ x i X , ~ Q v q i r ~ ' J , 3l r
(20)
1 1
where the molecular quadrupole is replaced by a segmental quadrupole. The cross term, Q,, is calculated from purecomponent parameters using the geometric mixing rule as EQ~J
= (EQ,~~Q,~)~'~
(21)
The mixing rule given by eq 20 represents the interaction between segmental anisotropies and should give better representation of mixtures containing large molecules than the mole fraction averaged interaction between molecular quadrupoles. The mixing rules used in the perturbation expansion, complete equations for the configurational Helmholtz energy, the equation of state for a mixture, and the chemical potential of component i in a mixture are all given in Appendix B. All other configurational thermodynamic properties can be derived from these equations.
Data Reduction Pure-component parameters have been obtained for six
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Ind. Eng. Chem. Fundam., Vol. 24, No. 2, 1985
1.4
-
1.2
-
1.0
-
\______I ____I-: C6H6
2 Y
T
i
’,1
hb
0.8
i
- 8
8
-
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263 K
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I
223 K 8
A
io
20
30
PRESSURE
40
eip: 29815 98 A 2
T
expt
- --- ccllc
50
.3
60
(bars)
Figure 3. K factors for ethylene-carbon dioxide system ( k , , = -0.0092). Experimental data are from Mollerup (1975) and Bae et al. (1982).
fluids (ethane, ethylene, carbon dioxide, benzene, naphthalene, and 1-methylnaphthalene) with appreciable quadrupole moments. There are a total of nine parameters ( P ,u*, c , E , q, r, u, Q, TQ*)for each pure component. Of these only six are independent because of the relations T* = t q / c k , u* = NAr u3/&, and TQ*= tQq/ck. The parameters t and u are taken from Donohue (1977). These were found by correlating the parameters for a large number of similar fluids. For pure fluids with reliable quadrupole moment data, €Qqis found from the relation, C Q q = Q 2 / ( d / 2 v * / N ~ ) 5For / 3 . other fluids, CQ is found by comparison. For example, since naphthalene consists of two benzene rings fused together, the segmental quadrupolar energy of naphthalene can be taken as that of benzene. Figure 1shows correlation of the total attractive energy with carbon number for aromatic systems. Since the quadrupole moment, Q, is available or can be estimated, the remaining three parameters ( P ,u*, and c ) are neccessary to calculate pure-component properties; these were determined from experimental vapor-pressure and liquid-density data. The pure component parameters r and q are always relative to that for a -CH2- segment whose r and q are chosen to be 1. Fluid-mixture properties are calculated with pure-component parameters and a single binary interaction parameter defined by Experimental K-factor data or Henry’s constant data are used to determine the binary interaction parameter. All parameters are independent of temperature, density, and composition.
Results The interactions of isotropic molecules are calculated with the Barker-Henderson second-order perturbation expansion using the Lennard-Jones potential. This has
I
8
-
K
A2315 K A23
-
colc
-4
I
I
I
I
I
1
10
20
30
40
50
60
I
P R E S S U R E (bars)
Figure 4. K factors for ethane-benzene system ( k , = -0.0129). Experimental data are from Kay and Nevens (1957) and Ohgaki et al. (1976).
reduced computation times significantly without any loss in accuracy when compared to the fourth-order perturbation expansion with the square-well potential used in the PHCT (Donohue, 1978). Pure-component parameters calculated from experimental liquid-density and vaporpressure data are tabulated in Table I. With these parameters, errors in calculated vapor pressures and liquid densities are typically within experimental uncertainty. Figure 2 shows the experimental and calculated vapor pressures for six fluids. The average error is less than 2%. Liquid densities are calculated over a wide range of temperature and pressure with similar accuracy. Saturated liquid and vapor volumes usually are predicted within experimental uncertainty up to 0.95 of the critical temperature. One difficulty in using this theory is that the molecule’s quadrupole moment must be known. Vrbancich and Ritchie (1980), by direct experimental measurements using the electric field gradient birefringence method, determined a value of -10.0 f 0.6 X esu for the quadrupole moment of benzene. This value, which is comparable to results of several recent molecular-orbital calculations, is used to calculate the properties of benzene. Calvert and Ritchie (1980) experimentally measured the quadrupole moment of naphthalene by use of the electric field gradient birefringence method and report a value of 13.7 f 1.5 X esu, with which the properties of naphthalene are calculated. However, if the segmental quadrupolar energy (Q) of naphthalene is taken to be that of benzene, then the quadrupole moment of naphthalene, calculated with the pure-component parameters of naphthalene, is 16.9 x lovz6esu. Even though this calculated value of the quadrupole moment for naphthalene is not within the experimental uncertainty of the reported value, this method of calculating Q from pure-component parameters can be used to yield a reasonable estimate of Q for complex
Ind. Eng. Chem. Fundam., Vol. 24, No. 2, 1985
251
K CO, KC2H6 A 4.0
I-
- ---
.
expt calc
3.0
A '463.05 K 543.45 K calc
-
2.0
0:
2U 2
1.0
y
0.01
I I
I 40
I 30
I 20
I 10
I
I 60
I
50
COP-
Figure 5. K factors for carbon dioxide-1-methylnaphthalene system (kij = 0.0226). Experimental data are from Sebastian et al. (1980). I
I
I
-
A
I
I
I
I
PRESSURE (borr)
I
2
ETHANE SYSTEM
expt
- COIC
ik
A 0)
-
I
expt A 199.82 K 263.15 K 293.15 K
+
- calc
1 6
0
c
14
f Y
1 2
PRESSURE (bars)
I
. . I
10
I
I
I
I
20
30
40
50
PRESSURE (bars)
Figure 6. K factors for ethylene-ethane system (kij = 0.0155). Experimental data are from Hanson et al. (1953) and Fredenslund et al. (1976).
molecules. Since the presence of a methyl group in 1methylnaphthalene will have little effect on the delocalized x orbitals of the naphthalene ring, the quadrupole moment of 1-methylnaphthalene is taken as that of naphthalene. There is no reported value of the quadrupole moment of 1-methylnaphthalene in the literature to check the accuracy of this assumption. Binary interaction parameters for the different mixtures studied are given in Table 11. These interaction parameters are evaluated by fitting the mixture partition function
Figure 8. Comparison of K factors for ethane-carbon dioxide system. Experimental data are from Fredenslund and Mollerup (1974). K factors are calculated with the binary interaction parameter, kij = 0 and using the Peng-Robinson equation, PR, the perturbedhard-chain theory, PHCT, and the perturbed-anisotropic-chain theory, PACT.
given by eq 16 to either experimental K factor data or Henry's constant data. With these parameters, the calculated K factors are within experimental uncertainty. The calculated and experimental K factors are compared in Figures 3 to 7. Figures 8 to 11show the experimental K factors and the calculated K factors with kij = 0. In these figures only pure-component parameters are used in evaluating the K factors by using the PACT. For comparison, the K factors calculated with the Peng-Robinson (1976) equation of state (with kij = 0) and with the PHCT and the square-well potential (wth kij = 0) are also shown. It is apparent from these figures that the inclusion of anisotropic terms sig-
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Ind. Eng. Chem. Fundam., Vol. 24, No. 2, 1985
I
I
I
‘
I
‘
I
I
1I
’0°
I
i ,
\
- - _ _--
,,/PR ,
I
06
-__
I
263 13
16
’7
coc ( k , 0 )
‘K 18
r9
20
P R E S S U R E (bars)
Figure 9. Comparison of K factors for ethylene-carbon dioxide system. Experimental data are from Mollerup et al. (1975). Legend is as given in Figure 8. 001
I
I
0
I 10
1
20
1 30
I
40
1 1
P R E S S U R E (bars)
nificantly improves the predictions of the PACT over the PHCT. The K factors predicted by the PACT are much closer to experimental values than either the PHCT or the Peng-Robinson equation, even though all three equations of state predict pure-component vapor pressures to approximately the same degree of accuracy. Most equations of state contain pure-component parameters which can be adjusted (often resulting in physically unreasonable parameters) for accurate prediction of pure-component properties. However, rigorous test of an equation of state lies in its ability to predict mixture properties accurately when using pure-component parameters alone. The mixing rules used in PHCT and PACT take into account the differences in molecular size and shape. In binary systems where the molecules are of approximately the same size and shape (as in the COz- ethane system in Figure 8 and the C02-ethylene system in Figure 9), both the PHCT and the Peng-Robinson equation predict similar K factors. The K factors for the carbon dioxide-ethane system and the carbon dioxide-ethylene system, calculated by using the PHCT with the square-well potential and the Peng-Robinson equation, end at K = 1. That is, for the respective conditions, both the PHCT and the Peng-Robinson equation do not show either an azeotropic point or the phase inversion that actually occurs. On the other hand, the perturbed-anisotropic-chain theory follows the trend of the experimental points and shows both the azeotropic point and the phase inversion. In binary systems such as ethane-benzene (Figure 10) and COz-1-methylnaphthalene (Figure ll),where molecules differ in size as well as in shape, the PHCT predicts K factors which are in less error than those predicted by the Peng-Robinson equation. Again, the PACT does much better than the other equations. Figure 12 shows an experimental Henry’s constant for benzene in 1-methylnaphthalene (Donohue et al., 1985) and calculated values using the PACT with the binary interaction parameter set to zero. For comparison, Henry’s
Figure 10. Comparison of K factors for ethane-benzene system. Experimental data are from Ohgaki et al. (1976). Legend is as given in Figure 8.
constants calculated with the Peng-Robinson equation, UNIFAC, and the PHCT with kij = 0 are also shown. These Henry’s constants can be fitted within experimental error by using the PACT with a value of the binary interaction parameter of 0.0028. The values of hi. for this and the other mixtures containing 1-methylnaphthalene suggest that the estimated value of the quadrupole moment of l-methylnaphthalene is reasonable. In the PHCT, quadrupolar interactions are treated implicitly as equivalent dispersion interactions. However in the PACT, the explicit inclusion of quadrupolar interactions yields pure-component parameters that reflect more accurately the actual forces with which molecules interact and results in better prediction of mixture properties. Both PACT and PHCT can be used to predict mixture properties accurately with binary interaction parameters. However, because the pure-component parameters used in the PHCT have less physical significance, the PHCT requires large values of the binary interaction parameter compared to the PACT. The fairly accurate prediction of mixture properties by the PACT using pure-component parameters alone and within experimental uncertainty with low values of a binary interaction parameter suggests that the PACT could be used to predict, to a fair degree of accuracy, mixture properties of systems involving complex molecules for which no experimental data exist. We are extending the partition function to compounds and mixtures with other multipolar anisotropies such as methanol (dipole), phenol (both dipole and quadrupole), etc. We hope that this will result in a theory useful in predicting the thermodynamic properties of a wide range of pure fluids and mixtures. Extension of the partition function to ternary (or higher) systems is straightforward
A erpt -cdc
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320
PRESSURE ( b a r s )
Figure 11. Comparison of K factors for carbon dioxide-l-methylnaphthalene system. Experimental data are from Sebastian et al. (1980). Legend is as given in Figure 8.
and only pure-component and binary parameters are required.
Acknowledgment Support of this research by the Office of Basic Energy Sciences, US. Department of Energy, under Contract No. DE-AC02-81ER10982 is gratefully acknowledged. The authors wish to thank The Johns Hopkins University Computing Center and the Engineering Computing facility for the use of their facilities.
Appendix A Pure Fluids. In the PACT, two variables are used to describe the size of a segment. The first is the segment's soft-core diameter, a, which is the intermolecular distance where the Lennard-Jones potential is zero. The second is the segment's hard-core diameter, d, which depends on temperature. Two different segmental diameters are needed since the Lennard-Jones terms are calculated as perturbations about the hard-sphere potential (and hence depend on the hard-sphere diameter), while the anisotropic terms are calculated as a perturbation about the Lennard-Jones potential (and therefore depends on the Lennard-Jones diameter, a). The ratio of hard-cofe diameter, d, to soft-core diameter, a, is evaluated (up to T = 10) from the integral given by Barker and Henderson (1967) and fitted to a polynomial in reduced temperature as
a12= 7.029 X where alo= 0.9976, all = -3.0554 X a13 = 1.1149 X and a14 = 7.403 X Two similar Pad6 approximations are used to account for contributions due to the higher-order terms in the perturbation expansion. One is for the higher-order terms
TEMPERATURE
OK
Figure 12. Comparison of Henry's law constant for benzene in 1-methylnaphthalene, Experimental data are from Donohue et al. (1985). Legend is as given in Figure 8.
in the perturbation expansion for isotropic interactions (Lennard-Jones potential); the other is for the expansion for multipolar anisotropic interactions about the Lennard-Jones potential. The Helmholtz free energy for molecules interacting with Lennard-Jones potential is given by
where AIU and AZUare the first- and second-order terms in the perturbation expansion and Arep is the Helmholtz energy of the reference, hard-sphere molecules given by Carnahan-Starling expression as
where 7 = r (2lI2)/6. Values of the integral for the firstorder perturbation term were calculated at several densities by Henderson (1982) with Monte Carlo simulation results for the radial distribution function of hard-sphere molecules. These were then fitted to a sixth-order polynomial in reduced volume, i i d , and the first-order term for attractive Helmholtz energy is then
with the constants All = -8.538, A12= -5.276, AI3 = 3.73, A14 = -7.54, A15 = 23.307, and A16 = -11.2. The integrals in the second-order perturbation term are similarly fitted to second- and third-order polynomials in Bd with the data of Smith et al. (1970). They evaluated the integrals with Percus-Yevick values of the hard-sphere
254
Ind. Eng. Chem. Fundam., Vol. 24, No. 2, 1985
radial distribution function. The second-order attractive Helmholtz energy is
The equation of state for anisotropic quadrupolar molecules can be written as Z= 1--A3ani)2
]/, (
with the constants Cll = -3.938, C12= -3.193, C13 = -4.93, and C14 = 10.03; C11 = 11.703, C22 = -3.092, C23 = 4.01; and C24 = -20.025; C,, = -37.02, C32 = 26.93, and C33 = 26.673. In effect we have a total of 17 constants compared to 36 constants of Alder et al. (1972) or 24 constants in PHCT with square-well potential of Donohue and Prausnitz (1978). The computation time, which is approximately proportional to the number of constants, is significantly reduced without any loss in accuracy for calculations of properties of fluids and their mixtures by use of the perturbed-anisotropic-chain theory. The equation of state for pure fluids interacting with Lennard-Jones potential is given by
AzBni
where
and
Appendix B Mixtures. The results obtained for pure components are generalized to multicomponent mixtures as shown by Donohue and Prausnitz (1978) using mixing rules derived from lattice theory. The mixing rule for isotropic molecules (eq 22 of Donohue and Prausnitz, 1978) and for quadrupolar anisotropic molecules (eq 20) can be rewritten by use of the definition of T* (= E q l c k ) and TQ* (= Egqlck) as
where
and
The expression used for the attractive term in the equation of state is derived by differentiating the Pad6 approximation for the attractive Helmholtz energy with volume. The Helmholtz energy for molecules with anisotropic interactions is given by c
where (...) denotes a mixture property. The Helmholtz energy for a mixture is calculated from eq A7 and A2 and the equation of state is calculated with eq A10 and A6 and with the mixing rules based on eq B1 and B2. The first-order perturbation term for isotropic interactions of fluid mixtures corresponding to eq A2 for pure components becomes Alm(CT*Ud*)(Ud*)"-' x NkT ,,, TU"
-where A" is given by eq A2. The second- and third-order perturbation terms for anisotropic interactions are calculated with expressions given by Gubbins and Twu (1978) assuming the molecules to be effectively linear. For quadrupolar interactions Azmi - - - 1 2 . 4,$lo) 47
Nk T
TQ20
and Asani cK - - 2 . 6 1cJ(15) 1 7 + 77.716Nk T
TQ3i7
?Q3fi2
(A91
where the redsced temperature based on the quadrupolar interactions, TQ = TITB*. The J and K integrals appearing in this perturbation expansion are functions of reduced temperature and reduced volume. They are calculated with functions given by Gubbins and Twu (1978) with the constants modified slightly (multiplied by d2 wherever reduced density appears) since the reduced volume, defined here, is based on the close-packed molecular volume.
(B3)
where
The first-order term has a simple quadratic dependence in composition and is rigorously random with respect to composition. The second-order term for isotropic interactions has a complex composition dependence. Nonrandomness becomes important in the higher-order terms and this is accounted for by using different mixing rules for (V) as shown by Donohue and Prausnitz (1978). In the form given by eq A5, the second-order term for isotropic interaction of fluid mixtures becomes
Ind. Eng. Chem. Fundam., Vol. 24, No. 2, 1985
where (cT*ud*), ( T* )L, and ( P ) ( 2 ) are defined by Donohue and Prausnitz (1978). For anisotropic quadrupolar interactions, the perturbation expansion of Gubbins and Twu (1978) for linear moIecules is used. Their expression for the second-order term is combined with the mixing rule derived from lattice theory (eq 20) to give
Azmi w T= -12.44 cTQ*~u*)J(~") 7% (
035)
where
(B11) The first-order perturbation term for chemical potential of component i for molecules interacting with LennardJones potential is given by
-PIU- kT m
The third-order term in the perturbation expansion for quadrupolar interactions is
255
+--)
Alm(cT*)( l + - - N a(CT*) mN aN(ud*) T(fid)m (CT*) dNi (Ud*)N aNi (BW
where
A3mi ( CTQ*(~)U*(~))K -NkT - - 2.611 ( CTQ*~U*)J(~') + 77.716 7%
ToV2
036)
where
The J and K integrals are evaluated at average reduced density, ( u* ) / u and average reduced temperature T / ( T* ) , where
The equation of state for mixtures is given by eq A10 with the perturbation terms in the expansion for mixtures given by
0313) The second-order perturbation term of chemical potential of component i for a mixture of Lennard-Jones molecules is
-1
( m + l ) N dN(ud*) ( ~ d )*N
Z3ani= 2.611
dNi
( CTQ*~U*)J(~')
2%
where The Pad6 approximation for residual chemical potential of component i is given by an expression similar to eq A10 with 2 s being replaced by the chemical potential, p. The perturbation terms in the chemical potential expression are then calculated by differentiating the appropriate Helmholtz energy expressions with respect to Ni. The chemical potential of component i in a mixture of hardsphere molecules is
Qi
and
+
256
Ind. Eng. Chem. Fundam., Vol. 24, No. 2, 1985
N ---
a ( T * ) @ )-
The derivative of In J(15)and In K with respect to N , is evaluated in a way similar to d In J('O)/dNi. Other thermodynamic properties of pure fluids and mixtures are obtained by applying appropriate thermodynamic relations to the Helmholtz free-energy expression. These complex equations allow calculation of thermodynamic properties of anisotropic quadrupolar molecules of differing size, shape, or intermolecular potential energy.
Nomenclature
For anisotropic quadrupolar interactions, the secondorder term of chemical potential of component i is
-
-=kT
1 (CTQ*~LJ*)J(~~) N (C T Q * ~ U ) * UP
-12.44
J
and
A = Helmholtz energy Alm,C,, = dimensionless constants used to calculate the first-order perturbation term 3c = number of external degrees of freedom per molecule d = hard-core diameter of a segment E = intermolecular potential energy well depth h = Planck's constant k = Boltzmann's constant k , = binary interaction parameter J,K = integral involving intermolecular potential energy and radial distribution function K = K factor m = atomic mass N = number of molecules N A = Avagadro's number q = normalized surface area per molecule (relative to the surface area of sphere of the size of a segment) q = molecular partition function Q = canonical partition function Q = quadrupole moment r = number of segments per molecule ( r = 1 for a -CH2segment) T =: absolute temperature u = molar volume u* = characteristic (soft-core) volume per mole V, = free volume 2 = compressibility factor
with
Greek Letters
The third-order term is given by
p = l/ckT e = characteristic energy per unit surface area of a molecule p = density a = soft-core diameter of a segment T = a constant equal to 0.7405 = potential field
-(T*)
a
Subscripts
where
att = attractions d = based on hard-core diameter of a segment ext = external (dependent on density) int = internal (independent of density) i, j = component indices used in mixing rules L = linear M = mixture rep = repulsions r, v = rotational and vibrational s = segment 1, 2, 3 = perturbation expansion terms Superscripts ani = anisotropic interactions is0 = isotropic interactions
rep = repulsions LJ = Lennard-Jones * = reducing parameter = reduced property
-
Ind. Eng. Chem. Fundam. 1985, 2 4 , 257-260
Literature Cited Alder, B. J.; Young, D. A,; Mark, M. A. J. Chem. Phys. 1972, 5 6 , 3013. Bae, H. K.; Nagahama. K.; Hlrata, M. J. Chem. Eng. Data 1982, 2 7 , 25. Barker, J. A,; Henderson, D. J. Chem. Phys. 1967, 4 7 , 4714. Barker, J. A.; Henderson, D. Rev. Mod. Phys. 1978, 48, 587. Beret, S . ; Prausnitz. J. M. AIChE J. 1975, 2 1 , 1123. Calvert, R. L.; Rltchie, G. L. D. J . Chem. Soc., Faraday Trans. 2 1980, 76, 1249. Carnahan, N. F.; Starling, K. E. AIChE J. 1972, 78, 1184. Donohue, M. D. Ph.D. Dissertation, University of California, Berkeley, CA, 1977. Donohue, M. D.; Prausnitz, J. M. AIChE J. 1978, 2 4 , 849. Donohue, M. D.; Shah, D. M.; Connaliy, K. G.; Venkatachalam, V. R . Ind. Eng. Chem. Fundam. 1985, In press. Flory. P. J. Discuss. Faraday Soc.1970, 49, 7. Fredenslund, Aa.; Mollerup, J. J. Chem. SOC.,Faraday Trans. 1 1974, 7 0 , 1653. Fredenslund, Aa.; Mollerup, J.; Hall, K. R. J. Chem. Eng. Data 1976, 2 1 , 301. Gubblns, K. E.; Twu, C. H. Chem. Eng. Sci. 1978, 33, 863. Hanson, 0. H.; Hogan, R. J.; Ruehlen, F. N.; Cines, M. R. Chem. Eng. frog. Symp. Ser. 1953. 4 9 , 37. Henderson, D. IBM Research Laboratory, Sen Jose, CA, personal communlcatlon, 1982.
257
Henderson, D.; Leonard, P. J. I n “Physical Chemistry-An Advanced Treatlse”, Eyrlng, H.; Henderson, D.; Jost, W. Ed. Academlc Press: New York, 1967; Vol. 8, Chapter 7. Kay, W. B.; Nevens, T. D. Chem. Eng. frog. Symp. Ser. 1957, 4 8 , 108. Kirkwood, J. G. J . Chem. Phys. 1935, 3. 300. Mollerup, J. J. J. Chem. Soc., Faraday Trans. 1 1975, 71, 2351. Ohgakl, K.; Sano, F.; Katayama, T. J. Chem. Eng. Data 1976, 2 1 , 55. Peng, D.; Roblnson, D. B. Ind. Eng. Chem. Fundam. 1876, 15, 59. Prlgoglne, 1. “Molecular Theory of Solutions”, North-Holland: Amsterdam, 1957. Rowllnson, J. S. Mol. Phys. 1964, 8 , 107. Sebastiin, H. M.; Nageshwar, G. D.; Lln, H. M.; Chao, K. C. J. Chem. Eng. Data 1980, 2 5 , 145. Smith, W. R.; Henderson, D.; Barker, J. A. J. Chem. Phys. 1970, 53, 508. Vera, J. H.; Prausnitz, J. M. Chem. Eng. J. 1972, 3 , 1. Vrbancich, J.; Ritchie, G. L. D. J . Chem. SOC.,Faraday Trans. 2 1980, 76, 648. Zwanzig, R. W. J. Chem. Phys. 1954, 2 2 , 1420.
Receiued for review December 1, 1982 Revised manuscript receiued June 24, 1984 Accepted July 26, 1984
EXPERIMENTAL TECHNIQUES
Laboratory Reactor System for the Evaluation of Catalysts in Gas-Phase Reactions under Realistic Process Conditions Ruud Snel Chemical Engineerlng Research Group, Council for Scientific and Industrial Engineering,
PO Box 395, Pretoria 0001, Republic of South Africa
A laboratory reactor system has been designed and developed for the safe testing of catalysts at elevated temperatures and pressures and suitable for unattended operation over prolonged periods of time under condltlons kg of powdered catalyst normally prevailing In Industry. The system is suitable for the testing of as ilttle as 5 X uslng a tubular microreactor, or up to 5 X lo-* kg of catalyst pellets In an internal gas recirculation reactor. In research on selective Fischer-Tropsch synthesis, the arrangement has proved to be reliable. It is suitable for appllcatlon In most catalyzed gas-phase reactions. The system is linked to a microcomputer for process data acqulsltion, control of reactor effluent sampling, chromatographic analysis, and the capture, integration, and evaluation of chromatographic data.
Introduction To evaluate the activity, selectivity, and stability of catalysts in a realistic manner, they must be tested under conditions normally employed in industrial practice. Under such conditions pressure increases attributable to laydown of carbonaceous deposits and heavy products or catalyst disintegration are not uncommon. These pressure increases may enhance the rate of reaction, causing further pressure increases and thermal runaway in the case of exothermic reactions. The phenomena normally preclude the use of prolonged operation. Such long-term runs, however, are vital in the search for catalysts that can exhibit suitable stability in operation. Therefore, a reactor system has been developed that includes the necessary safety aspects as well as a simple data acquisition system. The reactor system has 0196-4313/85/1024-0257801 .SO10
been developed for a vapor phase study of Fischel-Tropsch synthesis and is described in this context. However, it can be adapted easily to suit most catalytic studies of gas-phase reactions. Experimental Section Apparatus. A schematic representation of the reactor system is given in Figure 1. All components are made of stainless steel (304or 316), Teflon, or glass, unless stated otherwise. All valves are of the solenoid type unless indicated otherwise and, with the exclusion of the emergency valves, are operated electrically from a flow diagram type control panel. Each valve to vent is in series with a manually operated needle control valve which regulates the venting. Tubing with an outside diameter of 6.4 mm and a wall thickness of 0.89 mm has been used throughout the system. 0 1985 American
Chemical Society
