Ind. Eng. Chem. Process Des. Dev. 1980, 19, 205-211
205
Calculation of Phase Equilibria for Mixtures of Ethylene and Low-Density Polyethylene at High Pressures David D. Liu and John M. Prausnitz" Depa.rtment of Chemical Engineering, University of California, Berkeley, California 94720
The perturbed-hard-chain theory is applied to phase-equilibrium calculations for mixtures of ethylene and lowdensity polyethylene. Two binary parameters are required for phase-equilibrium calculation from ambient to 2000 atm. These two parameters are determined from experimental data for the critical locus and for low-pressure Henry's constants. For each phase, the concentration of ethylene and the molecular-weight distribution can be calculated at given temperature and pressure. Such calculations are useful for designing devolatilization equipment. A procedure is also given for reactor design conditions as a function of molecular-weight distribution; these are important for reaclor design.
the solubility is insensitive to both molecular weight and chain-branching of the polymer. (2) Pressure above 1200 atm. The compositions of the ethylene-rich phase and the polymer-rich phase approach each other as the pressure rises to the critical pressure; at that pressure, ethylene and polymer are mutually soluble in all proportions. Phase equilibrium data in this region are important to assure that only one phase is present in the chemical reactor (Bonner et al., 1974; Harmony et al., 1977). A homogeneous, clear polymer-solvent mixture may become cloudy when either the temperature or pressure changes. The point in temperature-pressure-composition space where the solution turns cloudy is called the cloud point. Cloud-point data for the region above 1200 atm have been reported by Ehrlich (1965), Swelheim et al. (1965), Cernia and Mancini (19651, and Steiner and Hod6 (1972). However, since the cloud-point curve is sensitive to the molecular-weight distribution of the polymer (Koningsveld, 1970) and since the molecular-weight distributions are not reported in any of these cloud-point measurements, it is difficult to compare calculated with experimental results. The cloud-point data consistently show that the mutual solubility between ethylene and polyethylene increases with rising temperature when the pressure is above 1200 atm. The maximum pressure in the isothermal cloud-point curve is the threshold pressure. Ehrlich (1965) reported threshold pressures from 115 to 200 "C using fractionated low-density polyethylene; that pressure increases with rising molecular weight and falling temperature. Since the threshold pressure is close to the critical pressure when the molecular-weight distribution is narrow, Ehrlich's data indicate how the critical pressure is influenced by changes in temperature and molecular weight. (3) Intermediate Region. Pressures above 100 atm and below the Critical Pressure. Luft and Lindler (1976) measured solubility curves for ethylene-polyethylene mixtures at 130 "C through flash experiments. The solubility of polyethylene in the ethylene-rich phase
Introduction Low-density polyethylene is manufactured at pressures near 2000 atm and temperatures in the region 2OC-300 "C. Ethylene-polyethylene mixture must be kept in one phase under reactor conditions. Since the reaction does not go to completion, unreacted ethylene must be recovered and recompressed for recycle. Rational process design for ethylene recovery separators requires optimum conditions of temperature and pressure so that significant energy saving can be achieved. Further, to avoid explosion hazards, residual ethylene must be completely removed from polyethylene in the storage bins (Beret et al., 1977). Toward these ends, it is necessary to know the phase equilibria of ethylene-polyethylene mixtures as a function of temperature and pressure. This work describes a method for calculating these equilibria. The method is based on the perturbed-hard-chain theory for fluid mixtures containing both small and large molecules (Donohue and Prausnitz, 1978). 'The method is similar to but more up to date than that reported by Bonner et al. (1974). Phase Equilibria: Qualitative Features Figure 1 shows a schematic three-dimensional phase diagram for the ethylene-polyethylene system. The phase diagram can be subdivided into three distinct pressure regions. (1) Pressure from Ambient to 100 atm. The ethylene-rich (gas) phase contains only an extremely small amount of polymer. Ethylene in the polymer-rich (liquid) phase is in equilibrium with essentially pure ethylene. Solubility of ethylene in polyethylene is small; therefore, Henry's law describes the solubility well. Phase equilibrium data in this region are important for process design of multistage devolatilization equipment used in lowdensity polyethylene plants. Due to experimental difficulties, low-pressure solubility data for ethylene in polyethylene have become available only recently (Maloney and Prausnitz, 1976; Cheng and Bonner, 1977, 1978; Beret et al., 1977). These data show that the solubility of ethylene in polyethylene decreases with rising temperature and that 0196-4305/80/1119-0205$01.00/0
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Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 2, 1980
where N is the number of molecules, Vis the total volume, and T is the absolute temperature; V, is the free volume, b is the thermal de Broglie wavelength, 4 stands for the potential field, and k is Boltzmann's constant. As suggested by Prigogine (1957), the rotation-vibration contribution to the partition function is factored into an external part (qr,v)ext and an internal part (qr,v)int; the latter is independent of density. The external rotational-vibrational degrees of freedom are treated as equivalent translational degrees of freedom. The total number of external degrees of freedom per molecule is 3c where c reflects the size and flexibility of the molecule. Beret and Prausnitz proposed that 260
I80
150
110
(qr,v)ext
Temperature, Celslus
Figure 1. Temperature-pressure-composition diagram for ethylene-low-density polyethylene (after Ehrlich).
is sensitive to the molecular-weight distribution and to the polymer weight fraction of the flash feed entering the flash chamber. However, the solubility of ethylene in the polymer-rich phase is much less sensitive to these variables. Maloney and Prausnitz (1976) indicated that a projection of the polymer-rich phase boundary onto the pressure-composition plane shows a temperature inversion in this region. Statistical Thermodynamics of Polymer-Monomer Equilibria It is experimentally tedious and difficult to measure phase equilibria for polymer-monomer mixtures over a wide range of temperatures, pressures, and molecular weight distributions. It is desirable, therefore, to establish a physical model, using statistical thermodynamics, so that design engineers may interpolate and extrapolate limited experimental data with confidence. The well-known Flory-Huggins lattice theory of polymer solutions is not suitable for mixtures containing polymer and a supercritical gas. Maloney and Prausnitz (1976) have shown that calculated ethylene solubilities based on the Flory-Huggins theory may be in error by 50% or more when the pressure is above 200 atm and when the interaction parameter is estimated from low-pressure data. The large error arises because, unfortunately, the HugginsFlory x parameter depends on temperature, pressure, and composition (Patterson, 1969; Flory, 1970). An alternate theoretical method, based on Flory's partition function and subsequent equation of state (Flory, 1970), is useful for ethylene-polyethylene phase equilibria when the pressure is above 200 atm (Maloney and Prausnitz, 1976; Bonner et al., 1974). However, since Flory's equation of state fails for dilute gases (Bonner et al., 1973), any correlation method based on that equation, without modification, is limited to the high-pressure region. Flory's partition function has been modified to attain better low-pressure results (Beret and Prausnitz, 1975a,b; Cheng and Bonner, 1978). The partition function proposed by Beret and Prausnitz has been extended to mixtures (Donohue and Prausnitz, 1978; Kaul, 1977; Liu and Prausnitz, 1979; Gmehling et al., 1979). Here we use that partition function to represent phase equilibria for ethylene-low-density polyethylene mixtures. Perturbed-Hard-Chain Theory for Pure Fluids The canonical partition function Q is of the general van der Waals form. For a pure fluid
= (vf/V)'-'
(2)
for the entire fluid-density range. When reasonable expressions are used for the effect of density on Vf and 4 , the partition function goes to the correct ideal-gas limit as the density approaches zero; also, the partition function approaches that of Prigogine for liquids when the density is high. For Vf, Beret and Prausnitz used the expression of Carnahan-Starling (1969) for hard spheres. For the field 4, Beret and Prausnitz used the computer simulation results of Alder et al. (1972). The equation of state is derived by differentiation.
P = kT
(9) TIN
(3)
It is well known that a van der Waals type equation of state cannot represent well the second virial coefficient at lower temperatures. Therefore, an empirical correction term was proposed by Kaul (1977). When that is included, the partition function is In Q = In QBeret+ In Qsv (4) where In QBeretis given by eq 1 and 2 1.7(q/c - 1.45)4(3.65- q / c ) In Qsv = (5) T%(i+ 1 0 / 6 ) ~ Here = TIT*, 6 = u / u * , and T* = ( q / c ) ( e / k ) . Parameter t / k represents the characteristic potential energy for two (nonbonded) molecular segments. Parameter q (not to be confused with q in eq 1 and 2) represents the external surface area of a molecule. At high densities, the second-virial-coefficientcorrection (eq 5) is negligible compared to the leading term in eq 4. Equation 4 is applicable to all fluid densities for fluids containing small or large molecules as shown schematically in Figure 2. There are three parameters in the pure-fluid equation of state; the specific hard-core volume u * ~ the ~ , characteristic temperature P,and the characteristic pressure Pr. Parameter P* is related to the degree-of-freedom parameter c by (c / r ) RT* p* = u*seg
where R is the gas constant. The number of segments per is the hard-core molar molecule is designated by r and volume per segment. Following Donohue and Prausnitz (1978), we assume that u* = 0.0096 L/mol, corresponding to a -cH~- segment in a7arge normal paraffin. Three pure-component parameters are determined from experimental PVT and vapor-pressure data. They are Pr, T*, and u * , ~ . u*,,~is related to u * , ~by u*sp =
u*segr
(molecular weight)
(7)
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 2, 1980
IDEAL
GAS
DENSITY
-
The functions p)are extensions of functions given by Alder et al. (1972). The_procedure for calculating the composition average of T'" depends on superscript i. The procedures used here eliminate the conventional approximation that the mixture is random. The averaging procedures used ensure that the partition function satisfies all four boundary conditions. Details are given by Donohue and Prausnitz (1978). In addition to u*,~,P ,and Pll for each pure component, we need binary parameters to calculate thermodynamic properties of mixtures. For two segments, one from molecule i and the other from molecule j , the interaction energy parameter kij is defined by CLOSE-PACKED LIQUID
Figure 2. Perturbed-hard-chain theory provides an equation of state for fluids containing simple or complex molecules, covering all fluid densities.
Parameters for ethylene were obtained from the P V T data of Benzler and Koch (1955) and from vapor-pressure data compiled by Zwolinski and Wilhoit (1971). Parameters for low-density polyethylene were obtained from high-pressure P V T data of Beret (1975). Perturbed-Hard-Chain Theory for Mixtures When extended to mixtures, the partition function given by eq 4 must meet the following boundary conditions: (1) The second virial coefficient must be a quadratic function of the mole fractions. (2) At high densities, the athermal entropy of mixing must reduce to the Flory-Huggins entropy of mixing when the reduced volume is independent of composition. (3) We define an excess quantity as that relative to the ideal gas a t the same volume, temperature, and composition. The excess chemical potential of a monomer in a concentrated polymer solution must remain finite as the polymer chain becomes infinitely long. (4) T o take into account molecular clustering when the mixture contains molecules with significantly different potential energies, eq 4 must be consistent with the perturbation-theory results of Henderson (1974) for spherical molecules. The Helmholtz energy A is related to the partition function by A = -kT In Q (8) Following Alder et al. (1972), we expand the Helmholtz energy in a power series in reciprocal reduced temperature A - A(ideaJgas) = A'C')+ A(l)/'Z' + A ( Z ) / p+ A(3)/1'3 + A(4)/T"(9) The leading term. in the right-hand side of eq 9, A", is the repulsive term. I t follows from the hard-sphere equation of Carnahan and Starling (1969)
7
=i~fi/6
207
(11)
where 17 = u / ( u * ) . Here u is the molar volume and u* is the molar hard-core volume. Expressions for calculating the composition average are designed to satisfy the first three boundary conditions. The higher-order terms in eq 9 represent the attractive contributions. They have the form A(') -N k-T - fco ( 6 , W , ( C ) ) (i = 1, 2, 3, 4) (12)
Parameters tii and cjj are characteristic segment-segment potentials for i-i and j-j molecule contacts, respectively. Partmeters cij, tii, and cjj are used in averaging procedures for T")'s. These parameters depend only on the chemical nature of i and j ; they do not depend on temperature, density, or composition. Parameter cii cannot be determined from pure-component P V T and vapor-pressure data alone because data reduction always gives the product tq. Following Donohue and Prausnitz (1978), we set cii/k = 105 K for normal alkanes. For other substances, qi parameters are estimated by optimizing agreement between calculated and experimental properties of binary mixtures with normal alkanes. Parameter q / r is determined from Pll according to
Data reduction for ethylene-polyethylene indicates that a hard-core-volume binary parameter 6, is needed to give the correct temperature dependence of the critical pressure. Parameter 6, corrects the excluded volume through (V*P
( c u * ) = -( c / r )
u*seg
(15)
where ( c / r ) is given by
( c / r ) = C $j(c/r)j - C aij$i$j{(c/r)i + ( c / r ) j ) I
i>j
(16)
where ?jij is a binary parameter. The segment-fraction $ j is defined by
1
For pure components
Binary parameters kij and 6, are determined from two sources of binary data: Henry's constants and the temperature dependence of the critical pressure.
Phase Equilibrium Calculations The chemical potential of component k in a mixture is
where m refers to all the other components.
208
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 2, 1980
Table I. Pure-Component Properties
PVT data
vapor pressure
substance
range
ethylene
273-423 K 100-2500 bar 413-573 K 1-1000 bar
LDPE
av dev, %
2
av dev, %
159-280 K
0.8
1.7
The conditions for phase equilibrium between two phases containing ethylene (1)and polydispersed polymer (2) are Fla
(204
= PlB
(1 = 1, 2, 3, ...)
= PZB
&la
(20b)
where superscripts CY and /3 represent polymer-rich and monomer-rich phases, respectively. Subscript 1 refers to a monodisperse polymer species with r21segments per molecule. The chemical potential can be separated into an ideal-gas contribution and a contribution due to intermolecular forces pk /.lkIC
=
=
pko
Fk1'
+ /.&kIF
(21)
$k + RT In 7
(22)
urku*wg
where &,IG is the chemical potential of an ideal gas at the same temperature, density, and composition; P ~ Ois a function of temperature only. Superscript IF denotes intermolecular force. For each polymer, yIF is a linear function of the number of segments provided three assumptions are made: (1)The parameters for the polymer are independent of chain length. (2) Binary parameters kij and 6, are zero when i and j refer to polyethylene segments, regardless of molecular weight, i.e. k21,21, =0 (for 1'= 1, 2, 3, ...I (234 (for E' = 1, 2, 3, ...)
821,21'= 0
(2%)
(3) Binary parameters kij and 6ij are equal to constants k l z and 612, respectively, when i and j stand, respectively, for ethylene and polyethylene segment, regardless of molecular weight, i.e. (for 1 = 1, 2, 3, ...) (23~) k l , z l = k12 = 612
(for I = 1, 2, 3, ...)
(23d)
+ Pzs"'F
(24)
For a polymer, then &IF
=
rZ1'K2SiF
where pZsIFis the contribution of intermolecular forces to the chemical potential per polymer segment. Functions pZSIFand pzsvIFdo not depend on a polymer chain length. Equation 20b can be rearranged to a more useful form after substituting eq 21, 22, and 24 into eq 20b. The result is
VZl _ -- &exp(rzru)
parameters
range
( 1 = 1, 2, 3,
...)
$szl
(25)
P*, bar 884
u*,,, c m 3 / g
T*,K
0.8659
210
750
0.6717
503
Ehrlich 5 D o t 0 e M, = 17,000
E 2000
P 'y
4
1600
I
I200t
u
1000I20
140
160
180
200
220
240
260
280
C r t f i c o l Temperature, Celsws
Figure 3. Critical loci for ethylene-low-density polyethylene mixtures.
Equation 25 makes it possible to use Flory's method (1944) to solve simultaneously the mass balance equations and the equations of phase equilibrium. Details concerning the calculation of P~~~~ and pZsVIFmay be found in the Supplementary Material. (See the paragraph at the end of the paper regarding supplementary material.) The numerical method described by Bonner et al. (1974) is used for the multicomponent phase-equilibrium flash calculation. In this calculation, the polydisperse polyethylene is divided into a number of pseudo-components (cuts) as suggested by the molecular-weight distribution. Typically, about ten cuts are used. Results Table I presents pure-component parameters. As proposed by Lichtenthaler et al. (1978), polyethylene parameters are obtained by fitting high-pressure PVT data directly, instead of fitting the linear expansion coefficients and thermo-pressure coefficients for each temperature. For ethylene, vapor-pressure and PVT data are used to determine ethylene parameters. All parameters in Table I are independent of temperature and density. To determine binary parameters, we choose experimental Henry's constants at low pressure and experimental projections of the critical locus onto the P-T surface. Henry's constants in molten low-density polyethylene have been measured by Maloney and Prausnitz (1976) and by Cheng and Bonner (1977); the latter are somewhat higher than the former. We make no attempt here to resolve this disagreement because our primary purpose is to demonstrate the application of perturbed-hard-chain theory to phase equilibrium calculations for supercritical gas-polydispersed polymer mixtures over a wide pressure range. In this work we use the Henry's constants reported by Maloney and Prausnitz (1976). Binary data reduction yields k1z = 0.0093 (26)
where
I Y
0 = - exp[(p2SvIFJ- p z s " I F 9 / R T j 08
and
where T is in kelvins. Figure 3 shows the critical locus using eq 26 and 27 for two monodispersed polymers. The estimated error for calculated critical pressures between 115 and 220 "C is 4%. Both the magnitude and the temperature dependence of critical pressure are sensitive to hI2. Experimental critical
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 2, 1980
209
2oool-----l
-Experimental --Calculated Phase Boundary
,Fitted , to Low/ \ Pressure Phast
1400 “OOt
i/’ 1 h 60,000 20,000 c 40,000 13,333
Data of L u f f and Lindler h”, :23,000, Initial
4 Welqht Fercent
2oo 00
Weight Percent
F i g u r e 4. Calculated two-phase boundary depends on some of the experimental data for obtaining binary parameters. ‘(Maloney a n d P r o u s n i t z )
--
( 8 a s g l v e n in text for t w o - p h a s e r e g i o n )
T
- _ _ - - I------x -
I
1
2
4
6
8
Weight Percent Polymer
I
I
1
This modification retains the same dPc/dTc;therefore, the mutual solubilities slightly below the critical pressure increase with temperature. When eq 28 is used, the calculated critical pressures are higher; however, as shown in Figure 4,the calculated solubilities are much more accurate. Figure 5 shows calculated Henry’s constants for ethylene in low-density polyethylene. Calculations are shown for h12 = 0 and for h,, from eq 28. The vertical bars in Figure 5 represent the estimated experimental error. Due to the large uncertainties of experimental Henry’s constants, we detect no influence of h12 on the calculated Henry’s constants. On the other hand, we have shown elsewhere (Liu and Prausnitz, 1979) that calculated Henry’s constants are
Figure 6. Solubility of low-density polyethylene in ethylene-rich phase at 130 “C.
sensitive to h12. T o make reliable phase-equilibrium calculations over a wide pressure range, it is important to use first, the experimental slope of the critical locus to determine b12 and second, experimental Henry’s constants to determine hlz.
Effect of Molecular-Weight Distribution on Phase Diagram For the polydispersed polymer, the molecular-weight distribution is given by the log-normal distribution function (Lansig and Kraemer, 1935). Figure 6 shows the effect of molecular-weight distribution on solubility in the ethylene-rich phase at 130 “C. Solubility of polyethylene in ethylene is extremely small when the pressure is below 100 atm. As the pressure rises, the low molecular-weight fractions dissolve in the ethylene-rich phase. Most of the higher molecular-weight fractions remain in the polymer rich phase until the critical pressure is approached. Therefore, the shape of the isothermal solubility curve of the ethylene-rich phase is highly sensitive to the molecular weight distribution of low-density polyethylene, as illustrated in Figure 6. The “stepping” shape of the data reported by Luft and Lindler suggests the presence of a low-molecular-weight “hump” in the molecular-weight distribution of their low-density polyethylene sample. The shape of the isothermal solubility curve for the polymer-rich phase is less sensitive to the polymer molecular-weight distribution than that of the ethylene-rich phase. Figure 7 shows the ethylene-low-density polyethylene phase diagram at 130 “C. Calculated polymer-rich phase compositions below 600 atm are identical for all three model low-density polyethylene samples using a log-normal molecular-weight distribution. Phase boundary a, which is calculated with the lowest number-average and weight-average molecular weight, shows the highest mutual solubility between polymer and ethylene. Phase boundaries b and c are calculated with the same number-average molecular weight. The polymer-rich sides of b and c overlap from zero to 1500 atm, although b has a wider molecular-weight distribution than c. This indicates that at very high pressures, near the critical region, the number-average molecular weight has an important influence on the solubility of ethylene in polyethylene. When the number-average molecular weight of lowdensity polyethylene is high, the solubility of ethylene in
210
Ind. Eng.
Chem. Process Des. Dev., Vol. 19, No. 2,
1980 Calculated, A s s u m l r g M o r o Polyethylene M = 17,000
f 0
p
0
c
29,000 6,666
200'C
06
/ ,
5 40,00010,000 c 20,000 10,000
1400
5
1200
0
0
,
200
400
1
1
600
800
1
,
1000 1200
Pressure, o t m
800
Figure 9. Solubility of ethylene in polyethylene a t two temperatures. Note crossing of the two curves.
6ool
important that only one phase be present. At a given temperature in the region 115-220 "C, the estimated error in the calculated critical pressure for a polymer with known molecular-weight distribution is about f4%.
400
0
20 40 60 80 Weight Percent Polymer
I00
Figure 7. Effect of molecular-weight distribution on ethylenepolyethylene phase equilibria a t 130 "C. o Estimated. L u f t and L i r d l e r (1976) al
5
1
b Calculoted, T h i s Work
0.8- c E s t i m o t e d ., Molonev 0.8C Moioney o r d Prausnitz (1976)
-
d Measured, Cherq and Bonrer
-
04
C
0
C
Acknowledgment The authors are grateful to Dr. Dennis P. Maloney for helpful discussions, to Drs. Luft and Lindler for sending us their experimental data, and to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for financial support. Nomenclature A = Helmholtz energy c = degree-of-freedom parameter k = Boltzmann's constant ki.= interaction energy binary parameter = total number of molecules P = pressure P, = critical pressure P* = characteristic pressure for pure substance Q = canonical partition function qr,"= molecular partition function, describes the rotational and vibrational motions q = surface area parameter of a molecule r = number of segments in a molecule T = absolute temperature T* = characteristic temperature for a pure substance Tc = critical temperature T = reduced temperature V = total volume i; = reduced volume Vf = free volume L ' * , , ~ = hard-core molar volume for a segment u*sp = specific hard-core volume u* = hard-core molar volume for a molecule u = molar volume for a fluid
d
200 400 600 800 1000 1200 1400 1600 Pressure, o t m
Figure 8. Solubility of ethylene in high-molecular-weight low-density polyethylene at 103 "C.
polyethylene (at constant temperature and pressure) approaches a minimum. The fugacity of ethylene a t that minimum composition is equal to the fugacity of pure ethylene a t the same temperature and pressure. Figure 8 shows the solubility of ethylene in high-molecular-weight, low-density polyethylene in 130 "C. At low pressure, the solubility of ethylene estimated by Luft and Lindler is significantly higher than that reported by all other workers. However, our calculated value agrees well with the data of Luft and Lindler a t pressures above 1000 atm. Figure 9 shows the calculated ethylene solubility in low-density polyethylene a t two temperatures. The two solubility lines cross a t ca. 800 atm. This unexpected result, discussed by Ehrlich (1965) and by Maloney and Prausnitz (1976), is correctly predicted by our calculations.
Conclusion The perturbed-hard-chain theory is applied to mixtures containing a supercritical gas and a polydispersed polymer. Phase-equilibrium calculations from ambient to 2000 atm requires two binary parameters. These two binary parameters (eq 26 and 28) are determined from Ehrlich's critical locus data and from Maloney's data for Henry's constants at low pressures. A computer program has been developed to implement this application of perturbedhard-chain theory. The convergence method is similar to that reported earlier (Bonner et al., 1974). The concentration of ethylene and the molecular-weight distribution can be calculated for each phase at a given temperature and pressure. Such calculations are useful for designing devolatilization equipment. A procedure is also given for estimating critical conditions; these are important for reactor design where it is
Greek Letters a = heavy phase /3 = light phase 6,j = binary parameter eij = segment-segment interaction energy = segment fraction
+
'i = thermal deBroglie wavelength 4 = potential field for a fluid p = chemical potential Subscripts 1 = ethylene 2 = polyethylene
i,j = segment of type i,j 1 = monodisperse polyethylene k,m = components S = segment SV = second virial coefficient Superscripts IG = ideal gas IF = intermolecular forces
Ind. Eng. Chem. Process Des. Dev. 1980, 19, 21 1-219
SV = second virial coefficient Literature Cited
211
Lichtenthaler, R. N., Liu, D. D.. Prausnitz, J. M., Macromolecules, 11, 192 (1978). Liu, D. D., Prausnitz, J. M., J . Appi. Polym. Sci., 24(3), 725 (1979). Luft, G.,Lindler, A., Angew. Makromol. Chem., 56, 99 (1976). Maloney, D. P., Prausnitz. J. M., Ind. Eng. Chem. Process Des. Dev., 15, 216 (1976). Patterson, D., Macromolecules. 2, 672 (1969). "Molecular Theory of Solutions", Chapter XVI, NorthHolland Press, Prigogine, I., Amsterdam, 1957. Steiner, R., Horl6. K., Chem. lng. Tech., 44, 1010 (1972). Swelheim, T., De Swaan Arons, J., Diepen, G. A. M., Red. Trav. Chim. Pays-Bas, 64, 261 (1965). Zwolinski, B. J., Wilhoit, R. C., "Handbook of Vapor Pressures and Heat of Vaporization of Hydrocarbons and Related Compounds", Thermodynamics Research Center and the American Petroleum Institute. Texas A&M University, College Station, Texas, 1971
Alder, B. J., Young, D. A., Mark, M. A,, J . Chem. Phys., 56,3013 (1972). Benzler, H., Koch, A. V., Chem. Zng. Tech. 27, 71 (1955). Beret, S.,Prausnitz, J. M., AIChE J., 21, 1123 (1975a). Beret, S.,Prausnitz, J. M., Macromolecules, 8, 878 (1975b). Beret, S.,Muhle, M. E., Villamil, I.A., Chem. Eng. Prog., 73, 44 (1977). Beret, S.,Ph.D. Thesis, Appendix C, University of California, 1975. Bonner, D. C., Bazua, E. H., Prausnitz, J. M., Ind. Eng. Chem. Fundam., 12, 254 (1973). Bonner. D. C., Maloney, D. P., Prausnitz, J. M., Ind. Eng. Chem. Process Des. Dev., 13, 91 (1974). Carnahan, N. F., Starling, K. E., J . Chem. Phys., 51, 639 (1969). Cernia, E. M., Mancini, C., Kobunshi Kagaku, 22, 797 (1965). Cheng, Y. L., Bonner, D. C., J. folym. Sci., Polym. f h y s . Ed.. 15,593 (1977); 16, 319 (1978). Donohue, M. D., Prausniti!, J. M., AIChE J., 24, 849 (1978). Ehrlich, P., J . Polym. S a . , A3, 131 (1965). Flory, P. J.. J . Chem. Phys., 12,425 (1944). Flory, P. J., Discuss. Fariiday SOC.,49, 7 (1970). Gmehling, J., Liu, D. D., Prausnitz, J. M., J. Chem. Eng. Sci., 34, 951 (1979). Henderson, D., J . Chem. Phys., 61, 926 (1974). Harmony, S. C., Bonner, D. C., Heichelheim, H. R.. AICHE J., 23, 758 (1977). Kaul, 6 . K., Ph.D. Thesis, University of California, Berkeley, 1977. 49, 164 (1970). Koningsveld, R., Discuss. Faraday SOC., Lansig, W. D., Kraemer, E. D., J . Am. Chem. SOC.,57, 1369 (1935).
Received for review October 30, 1978 Accepted October 11, 1979
Supplementary Material Available: Calculation of chemical potentials in ethylene-low-density polyethylene mixtures (9 pages). Ordering information is given on any current masthead page.
Electrochemical Gasification of Coal-Simultaneous Production of Hydrogen and Carbon Dioxide by a Single Reaction Involving Coal, Water, and Electrons Robert W. Coughlln' and Mohammad Farooque Department of Chemical Engineering, The University of Connecticut, Storrs, Connecticut 06268
Coals and other forms of solid carbonaceous fossil fuel are oxidized to oxides of carbon at the anode of an electrochemical cell and hydrogen is produced at the cathode. These gases are thereby produced in relatively pure states. The reaction proceeds at very mild temperatures and at operating electrical potentials lower than 1 V, i.e., significantly lower than the thermodynamic potential of water electrolysis. The process may be viewed as driven simultaneously by energy supplied at low temperatures in approximately equal proportions by the coal and by an external electrical source. I t is expected that coal can supply a larger proportion of the energy if the process is operated at higher temperature for which the required electrical potential will be lower.
Introduction Intense scientific and engineering activity is presently focused on the conversion of the solid fossil fuel coal into clean, fluid synthetic fuels and important chemical intermediates. The degree of fluidity of such a fuel is strongly influenced by its hydrogen content, ranging from a 4:l hydrogen-to-carbon atomic ratio for methane (the major component of natural gas) to the hard coals which are predominantly carbon. Accordingly, almost all coal conversion processes whether they involve liquefaction to an oil or conversion to methane gas require a t least some hydrogasification of coal to produce hydrogen as an intermediate. The chemistry of these processes has been discussed extensively by Mills (1972), and the technology and chemistry are thoroughly adumbrated in the encyclopedia by Kirk and Othmer (1966). Focusing only on the carbon in coal, we can represent the hydrogasification of coal by the well-known steamcarbon reaction 0196-4305/80/1119-0211$01.00/0
This reaction is strongly endothermic and its equilibrium is highly unfavorable a t ordinary temperatures, as indicated by the standard enthalpy and Gibbs free energy changes for reaction I: AH" = +31.4 kcal/mol and AGO = +21.8 kcal/mol. In order to conduct reaction I a t temperatures sufficiently high (-800 "C) to assure favorable equilibrium, and also to supply the endothermic heat of reaction, coal is gasified in practice by treating it with a mixture containing both steam and oxygen. In this way a part of the coal is combusted C(s) + Ozk) COz(g) (11) and the heat released by reaction I1 (So = -94.1 kcal/ mol, AGO = -94.3 kcal/mol) provides the thermal energy and assures the high temperatures required by reaction I. The detailed chemistry and technology of coal gasification are complex and also involve, for example, reactions between COz and coal, between CO and HzO, and between
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0 1980
American Chemical Society
