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Ind. Eng. Chem. Res. 2004, 43, 5380-5388
Nonideal Solution Limitations to the Use of Quadrature in Some Differential Phase Equilibrium Computations Robert P. Holderman and Kraemer D. Luks* Department of Chemical Engineering, University of Tulsa, 600 South College Avenue, Tulsa, Oklahoma 74104-3189
This paper demonstrates that the use of quadrature to designate pseudocomponents of continuous systems in solid-fluid phase equilibrium computations can lead to results that would be inconsistent with what one would expect from actual (more continuous) descriptions. Differential distillation (liquid + vapor) and differential crystallization (liquid + solid) computations are performed using two compositional representations: (1) a 94-component integer mixture of n-paraffinic species and (2) a 12-component quadrature representation of the same n-paraffinic mixture. Results for the differential distillation process do not vary significantly between the two representations; however, corresponding results for the differential crystallization process are dramatically different. Computations suggest that strong nonideality in a mixture phase can render its pseudocomponent description inadequate, leading to unrealistic results when that phase equilibrates with another. Introduction It has been shown by previous investigators1-8 that quadrature can be a very effective way of specifying the pseudocomponents of a continuous mixture. An oftensited example of a continuous mixture is the C7+ portion of a crude oil, which has been studied in the context of fluid-phase equilibrium problems by these investigators. One can represent the distribution of the many species as a continuous function of some parameter(s) [say, carbon number] and then pseudoize the mixture using quadrature integration mathematics. One can then opt to model the resulting pseudocomponent mixture with an equation of state. Computational convergence of phase equilibrium results usually occurs at a relatively low order/level of quadrature. Such asymptotic convergence formally ensures that one’s computed result will be equivalent to the result that would be obtained if many (n f ∞) pseudocomponents were employed. Additionally, quadrature discretization can be shown to avoid certain conservation of mass inconsistencies that are inherent in the continuous formalism of phase equilibrium computations, including saturation computations.9 The solid-fluid equilibrium problem has been of interest to us for some time. Our work has focused on a prototype n-paraffinic C7+ model system whose pseudocomponent compositional characterization is determined by finite Laguerre-Gauss quadrature. Earlier studies demonstrated that employing the assumption that the solid phases formed are “pure pseudocomponents” led to results for crystal-point temperatures that failed to display asymptotic convergence as the quadrature level increased. However, satisfactory quadrature convergence of crystal-point temperature calculations can be obtained if the solid-fluid phase equilibrium description permits solid solution formation. One might expect that this formal flexibility would enable execution of solid-fluid phase equilibrium computations with results converging as the * To whom correspondence should be addressed. Tel.: (918) 631-2974. Fax: (918) 631-3268. E-mail: kraemer-luks@ utulsa.edu.
quadrature level n f ∞, as is seen in fluid phase equilibrium computations. This paper addresses the prospect of convergence of solid-fluid phase equilibrium computations. Continuous mixtures are composed of a large number of species spanning a range of properties, making them candidates for compositional characterization using quadrature. Consequently, any particular species in a continuous mixture would have neighbors of very similar molecular nature and would likely form solid solutions with them upon crystallization. In his review of n-paraffin phase equilibrium behavior, Turner10 pointed out that n-paraffins of neighboring carbon number will form solid solutions at carbon numbers as low as ca. 30, the crystalline nature of which can be complex and variable. Solid solutions, nonideal in nature, would therefore be an expected phenomenon in continuous mixtures such as the homologous series of n-paraffins addressed in the case calculation presented herein. Solid-solution nonideality is strong enough that it can realistically lead to the formation of more than one solid phase coexisting with the liquid phase. We previously performed solid-fluid phase equilibrium calculations for both an equilibrium and a sequential scenario11 for a quadrature-characterized mixture (the same as described herein in Appendix A) of n-paraffinic hydrocarbons extending from carbon number 7 to carbon number 100, having a molecular weight of 200. The degree of nonideality introduced into the solid-phase description was empirically based on what was observed by Turner. Multiple solid phases were present at equilibrium at temperatures below the mixture’s crystal point. Removing solid phases from the equilibrium system at 0.5 K intervals as the system cooled (the sequential scenario) reduced the occurrence of the formation of multiple solid phases relative to the equilibrium scenario. In fact, it can be argued that, in a truly differential crystallization process, wherein the solid phase is continuously removed as it forms with decreasing temperature, there will be no detectable solid-multiphase state. In Figure 1a, the path of a differential crystallization process by its nature tracks within the two-phase L-S region close to the liquid fraction L ) 1 locus, as indicated by the
10.1021/ie040035y CCC: $27.50 © 2004 American Chemical Society Published on Web 07/22/2004
Ind. Eng. Chem. Res., Vol. 43, No. 17, 2004 5381 Table 1. Liquid Mole Fraction for the n-Paraffin System When Flashed at 500 K and 1 bar, as a Function of the Pseudoization Method number of quadrature components, n 2 3 4 5 6 7 8 9 10 11 12 number of integer components 94
Figure 1. Schematic diagram of the differential crystallization process. (a) Process follows the locus denoted by the heavy arrows within the two-phase L-S region with xL f 1, across the three phase L-S-S′ region (III) near the L vertex (xL f 1), and then within the two-phase L-S′ region with xL f 1. (b) Magnification of the L vertex of region III. Shown are arrows denoting differential crystallization steps with δT f 0. With each temperature-lowering step δT, an infinitesimal amount of solid phase is formed and then removed, returning the system to xL ) 1, prior to the next step. The extent of region III is infinitesimal as xL f 1, and region III is consequently “stepped over” by a single temperature-lowering step δT during the process.
bold arrows. It passes through the three-phase L-SS′ region near the L vertex (as shown in the magnification in Figure 1b) and then continues within the twophase L-S′ region, again close to the liquid fraction L ) 1 locus. The extent of the L-S-S′ region (III) is infinitesimal along the process path when L f 1, thereby rendering solid-multiphase L-S-S′ occurrence nonexistent or inconsequential. In this present study, we perform calculations for the same model n-paraffinic system as used in our earlier studies.11,12 Differential distillation and crystallization calculations are performed for the continuous system when characterized as (1) a mixture of 94 n-paraffinic hydrocarbons of integer carbon number (intended to mimic a continuous mixture) and (2) a 12-component quadrature-characterized pseudoized n-paraffinic mixture, extending from carbon number 7 to carbon number 100, having a molecular weight of 200. Details of the 12-component quadrature and 94-component discretizations are presented in Appendix A. The fluid-phase and solid-phase mixture formalisms and the component and pseudocomponent parameter correlations are presented in Appendix B. If the quadrature characterization is valid, the computational results for approaches 1 and 2 should be very similar. Differential Distillation Computations Differential distillation computations were performed for the n-paraffin system at 1 bar, starting at the bubble-point temperature and ascending 50 K by (differential) steps of size δT ) 0.1 K. The conceptual prototype continuous n-paraffinic mixture extends between carbon numbers 6.5 and 100.5, with the mole fraction curve for the mixture exponentially decaying with increasing carbon number. The descriptions used for the n-paraffin system were respectively (1) 12 pseudocomponents as provided by Laguerre-Gauss quadrature with n ) 12 and (2) 94 integer components
liquid-phase mole fraction, L
pseudoization
0.3041 0.4034 0.4835 0.5205 0.5168 0.4982 0.4914 0.4941 0.4975 0.4984 0.4977
quadrature quadrature quadrature quadrature quadrature quadrature quadrature quadrature quadrature quadrature quadrature
0.4982
integer
reflecting the exponential distribution. (See Appendix A.) It is assumed throughout this study that the 94component description physically represents the continuous mixture. It is against computations for this 94component mixture that computations employing more abbreviated pseudoized descriptions are evaluated in terms of their predictive success. Strictly speaking, the 94-component integer description, although by necessity realistic if the mixture were solely n-paraffinic, is not continuous, in that it does not have an unlimited number of components. It can be argued, however, that this description does mimic a truly continuous mixture. For example, consider flashing the 94-component integer system at a given temperature and pressure and comparing the resulting predicted liquid fraction with those predicted by Laguerre-Gauss quadrature for levels n ) 2-12. Historically, investigators have found that continuous systems can be reasonably mimicked by quadrature descriptions at levels of no more than 4-6. Of course, additional precision and system complexity or molecular parameter range, such as carbon number, would logically require a higher level of quadrature for predictive success. Table 1 shows the liquid fraction as a function of quadrature level n, as well as the liquid fraction that results using the 94-component integer description when flashing at 500 K and 1 bar. It can be concluded that, for this particular mixture, at a quadrature level of about 10 [mathematically the same as a 19th-order (2n - 1) polynomial13], the liquid fraction calculation is converged to three significant figures and using a quadrature level of g10 is equivalent to employing higher quadrature levels n (n f ∞). The liquid fraction result for the 94-component integer description is consistent with this outcome. Throughout this study, a quadrature representation with n ) 12 is used that is deemed a successful, converged pseudoization of the system for purposes of liquid-vapor equilibrium computations. To demonstrate this equivalence, the computational results using this quadrature description are compared with those obtained using the 94-component integer description. To (10-5 K, the bubble-point temperatures are 430.00203 and 431.15887 K, respectively, for the quadrature and integer descriptions. A differential distillation process was then performed with each of these descriptions using step temperature increases (T f T + δT) with δT ) 0.1 K between flashes, for a total increase of 50 K. The vapor is permanently removed from the flash calculation after each δT step. The small δT step is employed in an attempt to attribute time- and compo-
5382 Ind. Eng. Chem. Res., Vol. 43, No. 17, 2004
Figure 2. Instantaneous and cumulative vapor molecular weights as a function of the differential distillation temperature T for the 12-component quadrature and 94-component integer descriptions for the n-paraffinic mixtures detailed in Appendix A.
Figure 3. Cumulative mole fraction of vapor as a function of the differential distillation temperature T for the 12-component quadrature and 94-component integer descriptions for the nparaffinic mixtures detailed in Appendix A.
sitionwise continuity to the differential distillation process computation. For these two pseudoizations, Figure 2 compares the cumulative mole fraction of the system vaporized as a function of temperature, and Figure 3 compares the instantaneous and cumulative vapor molecular weights. The results are very similar between descriptions, which is not entirely unexpected, given the data in Table 1. The behavior with respect to the individual components is analogous. For the 94-component integer description, the vapor mole fraction of the lightest, most volatile component C7 decreases, whereas the fraction of C8 passes through a maximum in the temperature range T < 480 (Figure 4). The C9-C13 species, being less volatile, increase in mole fraction in the vapor phase gradually with increasing temperature both (a) directly, because the temperature increase enhances their volatility, and (b) indirectly, because the more volatile species C7 and C8 (which have the highest initial feed concentrations) are more rapidly depleted from the system as temperature increases. The liquid-phase compositions (Figure 5) of the more volatile components in turn decrease more rapidly. For the 94-component integer description, component C10 passes through a liquid mole fraction maximum in the temperature range computed, whereas the liquid mole fractions of species
Figure 4. Instantaneous vapor mole fractions of the seven lightest components (labeled by carbon number) as a function of the differential distillation temperature T for the 94-component integer mixture detailed in Appendix A.
Figure 5. Liquid mole fractions of the seven lightest components (labeled by carbon number) as a function of the differential distillation temperature T for the 94-component integer mixture detailed in Appendix A.
C11 and higher increase as, masswise, they mainly stay in the liquid phase, the total number of moles of which have decreased from loss of the lightest components C7C9. One would expect the curves for the C11 and heavier species eventually to exhibit liquid mole fraction maxima as temperature increases further (T > 480 K). Figures 6 and 7 for the 12-component LaguerreGauss quadrature description correspond to Figures 4 and 5. The three most volatile pseudocomponents are considered here: S1 (Cn ≈ 7.06), S2 (Cn ≈ 9.44), and S3 (Cn ≈ 13.71). Their behavior corresponds to that of the 94-component integer description shown in Figures 4 and 5, with S1 behaving like C7, S2 like C9 or C10, and S3 like C13 in terms of increasing or decreasing with temperature. The relative positioning of the curves in Figure 7 is not the same as that in Figure 5 because, in the original feed for the quadrature description, xFEED,2 > xFEED,1, as dictated by the Laguerre-Gauss quadrature mathematics (Appendix A). Figure 8 is a plot of the percentage of each species that is removed from the system by the differential distillation process in the two descriptions used in Figures 4-7. The data for the three most volatile quadrature components fall very close to the data for the seven most volatile components in the 94-component integer description.
Ind. Eng. Chem. Res., Vol. 43, No. 17, 2004 5383
taneous trace of solid phase s for each component present in the mixture description
µil(P,T,{xi}) ) µis(P,T,{si}),
i ) 1, ..., n
(1)
An equivalent statement of these mass flow equilibria equates the well-known fugacity functions
ˆfil(P,T,{xi}) ) ˆfis(P,T,{si}),
i ) 1, ..., n
(2)
Closure of the associated flash problem, with solution for the unknowns (P, T, {xi}, {si}) is traditionally achieved by fixing T ) To and P ) Po and coupling eqs 2 with n mass balance equations
Lxi + Ssi ) zi, Figure 6. Instantaneous vapor mole fractions of the three lightest components as a function of the differential distillation temperature T for the 12-component quadrature mixture detailed in Appendix A.
i ) 1, ..., n
which yields a problem of 2n equations in 2n unknowns. The solid-phase fugacity is described by an activity coefficient model that is detailed in Appendix B
fis(P,T,{si}) ) fiso(P,T)γi({si})si
Figure 7. Liquid mole fractions of the three lightest components as a function of the differential distillation temperature T for the 12-component quadrature mixture detailed in Appendix A.
Figure 8. Percent of original batch species vaporized by 50 K differential distillation for the lightest components (labeled by carbon number) of the 12-component quadrature and 94-component integer descriptions for the n-paraffinic mixtures detailed in Appendix A.
Differential Crystallization Computations The solid-fluid two-phase equilibrium differential crystallization problem formally requires one to find the solution to the set of equations balancing chemical potentials between the liquid phase l and the instan-
(3)
(B3)
If the {γi} were unity, then the solid phase would be an ideal solution. The {γi} functions account for the nonideal solution behavior of the solids, including their tendency to form more than one equilibrium crystalline phase in the presence of a liquid phase during a flash. This possibility was discussed and computationally demonstrated in an earlier paper, but as pointed out in the Introduction, it is irrelevant to the differential crystallization process if the solid-phase increments removed are truly infinitesimal. To support this claim, we employed stationary-point analysis in concert with the differential crystallization process when using the 12-component quadrature description. To (10-5 K, the crystal-point temperatures are 387.25679 and 388.38712 K, respectively, for the quadrature and integer descriptions. Adjacent pseudocomponents are different enough in that molecular description that more than one mathematically competitive solution to the solid-fluid equilibrium problem can occur at some flash temperatures. Stationary-point analysis,14 also known as Gibbs energy analysis15 determines the correct, most stable solution. Stationary-point analysis can also determine when the differential crystallization process steps over a potential liquid-solid-solid region, as in Figure 1, in essence switching from one solidfluid solution to another as the incremental (nearly infinitesimal) solid phase is removed from the process with decreasing temperature. For the temperature increment of δT ) 0.1 K employed, stable three-phase solutions were never found, i.e., the differential crystallization process did step over the three-phase regions schematically illustrated in Figure 1. Similar attention was not required for the 94-component integer description because of the molecular similarity of neighboring integer species. Figure 9 presents the instantaneous and cumulative solid-phase molecular weights as a function of the differential crystallization temperature T for the 12component quadrature and 94-component integer descriptions. The contrast in behavior between the results of the two descriptions is dramatic. Although both differential crystallization processes are two-phase solidliquid in nature throughout, the 12-component quadrature description leads to solidification of the pseudocom-
5384 Ind. Eng. Chem. Res., Vol. 43, No. 17, 2004
Figure 9. Instantaneous and cumulative solid molecular weights as a function of the differential crystallization temperature T, for the 12-component quadrature and 94-component integer descriptions for the n-paraffinic mixtures detailed in Appendix A.
Figure 11. Cumulative solid-phase mole numbers of the three heaviest components as a function of the differential crystallization temperature T for the 12-component quadrature mixture detailed in Appendix A.
Figure 12. Instantaneous solid mole fractions of the three heaviest components as a function of the differential crystallization temperature T for the 12-component quadrature mixture detailed in Appendix A.
nonideality that promotes pseudocomponent separation, essentially purification, by crystallization.
Figure 10. (a) Cumulative mole fraction of solid crystallized as a function of the differential crystallization temperature T for the 12-component quadrature and 94-component integer descriptions for the n-paraffinic mixtures detailed in Appendix A. The temperature range where the heaviest pseudocomponents are crystallized is featured here, with the subregions of crystallization labeled for species 10-12 for the 12-component quadrature description. (b) Same as Figure 10a. The temperature range where the species 7-9 for the 12-component quadrature description are crystallized is featured here.
ponents selectively. The molecular differences between adjacent heavier pseudocomponents generate a level of
The selective crystallization of the pseudocomponents in the case of the 12-component quadrature description in turn affects the appearance of the cumulative-solidsformed curve, offering a cascade of crystal formations corresponding to the heavier pseudocomponents. By comparison, the same curve for the 94-component integer description is smooth. The total amount of solids formed during the differential crystallization process is not that different between descriptions, as illustrated in Figure 10a,b. Figures 11 and 12 further elucidate the pseudocomponent separation that occurs with the 12component description. The solid-phase compositions are relatively pure pseudocomponents, as the heavier pseudocomponents are removed sequentially by decreasing carbon number. The removal of the heavier species in the 94-comonent integer description also happens sequentially (Figure 13) but the solid-phase mole fractions have pseudocomponent maxima typically of only 0.1 for Cn < 98 (Figure 14).
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Examining eq B7 for the case Akj ) 2, the ethane + n-paraffin and propane + n-paraffin liquid mixtures suggest that Ao ≈ 5/4. Quadrature at the level n ) 4 for the continuous n-paraffin mixture described in Appendix A yields pseudocomponents with carbon numbers 8.865, 19.252, 39.298, and 72.089 respectively.11 Adjacent pseudocomponent pairs correspondingly yield liquidphase values of Akj of 0.736, 0.856, and 0.924 (in ascending order), all well below the immiscibility limit. Only pseudoization of mixtures undergoing solid-fluid phase equilibria appears to be cause for concern. Nomenclature
Figure 13. Cumulative solid-phase mole numbers of the nine heaviest components (labeled by carbon number) as a function of the differential crystallization temperature T for the 94-component integer mixture detailed in Appendix A.
Figure 14. Instantaneous solid mole fractions of the nine heaviest components (labeled by carbon number) as a function of the differential crystallization temperature T for the 94-component integer mixture detailed in Appendix A.
Discussion The computations performed demonstrate that, although popular pseudoization schemes, such as the Laguerre-Gauss quadrature method herein, can be satisfactorily used for traditional vapor-liquid equilibrium calculations, they might not accurately produce the behavior of continuous mixtures of solids in solidfluid equilibrium even at higher quadrature levels, say, n ) 12. The reason for the shortcoming is the high level of nonideality one sees between species in solid-phase solutions as compared to liquid-phase solutions. The high level of solid-phase nonideality has traditionally supported the use of crystallization as a method of purification. One would not normally expect such a pronounced contrast between the vapor-liquid behavior of a continuous mixture vis-a`-vis a quadrature pseudoization of the same. As examples, the binary mixture C2 + nC18 is the first ethane + n-paraffin mixture to exhibit liquid-liquid-vapor immiscibility.16 Lighter n-paraffins are fully miscible with ethane. Likewise, the binary mixture C3 + nC30 is the first propane + n-paraffin mixture to exhibit liquid-liquid-vapor immiscibility.17
a(T) ) Soave-Redlich-Kwong parameter A ) lower carbon number bound Akj ) activity coefficient parameter for species k and j Ao ) 100/3 ) empirical activity coefficient constant for n-paraffins b ) Soave-Redlich-Kwong parameter B ) upper bound on the carbon number C ) upper bound on the reduced carbon number variable ˆf ) fugacity f ) distribution function g ) molar Gibbs energy ∆hfus ) enthalpy of fusion I ) carbon number L ) number of moles of liquid phase mi ) Soave-Redlich-Kwong parameter MW ) molecular weight n ) number of quadrature pseudocomponents nC ) carbon number Cn, Ci, Cj ) carbon number P, Po ) pressure R ) gas constant S ) number of moles of solid phase si ) mole fraction of solid phase T ) temperature v ) molar volume wi ) quadrature weighting factor X ) quadrature variable x ) quadrature mole fraction xi ) mole fraction of liquid phase zi ) mole fraction of liquid phase Greek Symbols R ) exponential distribution decay parameter Ri ) Soave-Redlich-Kwong parameter γ ) activity coefficient ωi ) acentric factor µ ) chemical potential Subscripts c ) critical point n ) carbon number label fus ) fusion i, j, k ) species l ) liquid phase R ) reduced s ) solid phase t ) triple point Superscripts o ) pure phase mix ) mixing property
Appendix A. Quadrature Formalism and Resulting Mixture Discretization In this study, we use the simple exponential distribution
5386 Ind. Eng. Chem. Res., Vol. 43, No. 17, 2004
F(I) ) Re-RI/(e-RA - e-RB) ) 0,
A g I, I g B
(A1)
to describe a continuous fluid, with its carbon number I limited to the range A ) 6.5 to B ) 100.5. Equation A1 qualitatively represents the nature of typical crude oil C7+ portions, with their composition functions exponentially decaying with carbon number. (Du and Mansoori18,19 were the first, to our knowledge, to use the simple exponential distribution in phase equilibrium calculations, although Shibata et al.8 pointed out that Vogel et al.20 earlier represented experimental evidence supporting the use of the exponential form for the C7+ mole fraction distributions in crude oils.) The carbon number I is related to an n-paraffinic molecular weight through
MW ) 14.0I + 2.0
(A2)
Table 1A. Composition of the Contnuous Mixture of n-Paraffins, Having Molecular Weight 200 with A ) 6.5 and B ) 100.5, Using the Finite Laguerre-Gauss Prescription for n ) 12 component
Cn
mole fraction
1 2 3 4 5 6 7 8 9 10 11 12
7.059 756 9.443 870 13.709 279 19.809 501 27.659 784 37.115 628 47.936 322 59.727 769 71.860 408 83.374 238 92.934 154 98.989 879
0.174 599 0.296 407 0.264 562 0.160 384 0.071 305 0.024 280 0.006 588 0.001 495 0.000 304 6.0484 × 10-5 1.2944 × 10-5 2.7821 × 10-6
Fluid-phase fugacities are determined from the SoaveRedlich-Kwong equation-of-state (SRK EOS) model21
The average carbon number for a given carbon number range (A, B) can be expressed in terms of the exponential decay parameter R, through the expression
P ) RT/(v - b) - a(T)/[v(v + b)] n
b)
〈I〉 ) [(RA + 1)e-RA - (RB + 1)e-RB]/[R(e-RA - e-RB)] (A3) Finite Laguerre-Gauss quadrature was employed, as this particular quadrature is designed for the representation of functions and processes where the continuous distribution is exponentially decaying, as in eq A1. This pseudoization approach has been used before in similar applications.6-9 Briefly, finite Laguerre-Gauss quadrature approximates in an optimal way integrals of the type n
∫0 f(X)e-X dX ) ∑wif(Xi) C
(A4)
n
a)
ln(x) ) -0.13083378595Cn - 1.18269027656 (A5) Appendix B. SRK Equation-of-State Formalism and Pseudocomponent Parameter Correlations The continuous C7+ portion is discretized as a mixture of pseudocomponents using finite Laguerre-Gauss quadrature, as in Appendix A. The resulting C7+ pseudocomponents comprise part or all of the fluid and solid phases in a particular phase equilibrium problem.
n
∑ ∑xixjaij i)1 j)1
(B1a)
where n
b)
xibi ∑ i)1
RTci bi ) 0.08664 Pci
i)1
The sets {wi} and {Xi} are determined by finding the maximum rigorously possible polynomial order of f(X). For a chosen n, the integral on the left-hand side is represented precisely to order 2n - 1.13 The quadrature pseudocomponent characterization was performed for n ) 12 with carbon number bounds of A ) 6.5 and B ) 100.5 (which determine the exponential decay parameter R in eq A1 using eqs A2 and A3) and a molecular weight of 200. The upper limit of the finite integral in eq A4 is C ) R(B - A), which, for this prototype mixture, is 0.130834. The resultant sets {wi} and {Xi} from eq A4 were translated into sets of mole fraction and carbon numbers and are given in Table A1. The 94-component integer representation employs carbon numbers 7-100. The magnitude of the normalized mole fractions decays exponentially with carbon number. The molecular weight of the mixture is 200. A mathematical representation to 10 significant figures of the integer mole fractions of the 94 components is
xibi ∑ i)1
n
a(T) )
n
∑ ∑xixjaij i)1 j)1
aij ) xaiiajj aii ) Riaci R2Tci2 aci(Tci) ) 0.42748 Pci Ri1/2 ) 1 + mi(1 - TRi1/2) mi ) 0.480 + 1.574ωi - 0.176ωi2
(B1b)
No binary interaction parameters are used for aij, i * j, in this study. Solid-phase fugacities are determined using the following nonideal description
fis(P,T,{si}) ) fiso(P,T)γi({si})si
(B3)
where an artifice relates the pure solid fugacity function to the pure liquid-phase fugacity function as follows
ln(filo/fiso) ) (∆hfus i /RTti)
(
Tti -1 T
)
(B4)
Equation B4 is a simplification of the form in Prausnitz et al.22 Tti is the triple-point temperature of pseudocom-
Ind. Eng. Chem. Res., Vol. 43, No. 17, 2004 5387
ponent i, and ∆hfus is the heat is fusion of pseudocomi ponent i at the triple point. The pure-liquid (hypothetical) fugacity filo is calculated from the SRK EOS. A simple two-suffix Margules approach is used to describe the solid-phase activity coefficients. The reduced excess Gibbs energy function for an n-component mixture is
gexcess
n
)
RT
n
∑ ∑Akjxkxj k)1j)1
(B5)
j>k
Ternary and higher interactions are not considered in the model system calculations. The activity coefficients resulting from eq B5 are n
ln γi )
n
n
Aijxj - ∑ ∑Akjxkxj ∑ j)1 k)1j)1 j*i
(B6)
j>k
The following expression for Akj was used
|Ck - Cj| Akj ) Ao (Ck + Cj)/2
(B7)
Turner10 has pointed out that, within the homologous series of n-paraffins, solid solutions can occur if the difference in molecular length is less than 6%, or roughly (2 carbon numbers for Cn ≈ 35. If Ck ≈ 34 and Cj ≈ 32, then it follows that, according to Turner’s criterion, Ao ) 100/3 for Akj ) 2, the immiscibility onset limit. By contrast, if Ck ) 50 and Cj ) 40, then Akj ≈ 7.4. To carry out phase equilibrium computations that include fluid and/or solid phases containing the quadrature-generated pseudocomponents, one must have the parameters Tc, Pc, ω, Tt, and ∆hfus as functions of the pseudocomponent carbon. The following correlations for n-paraffins were used for the pseudocomponent parameters
Critical temperature (K)23 Tc ) 959.98 + exp(6.81536 - 0.211145nC2/3)
(B8)
(B9)
(4) Cotterman, R. L.; Dimitrelis, D.; Prausnitz, J. M. Design of Supercritical Fluid-Extraction Processes Using Continuous Thermodynamics. In Supercritical Fluid Technology; Penninger, J. M. L., McHugh, M. A., Krukonis, V. J., Eds.; Elsevier Science Publishers BV: Amsterdam, The Netherlands, 1985; pp 107120. (5) Cotterman, R. L.; Chou, G. F.; Prausnitz, J. M. Comments on “Flash Calculations for Continuous or Semicontinuous Mixtures Using an Equation of State”. Ind. Eng. Chem. Process Des. Dev. 1985, 25, 840-841. (6) Behrens, R. A.; Sandler, S. I. Use of Semicontinuous Description to Model the C7+ Fraction in Equation of State Calculations. Presented at the SPE/DOE Fifth Symposium on Enhanced Oil Recovery of the Society of Petroleum Engineers, Tulsa, OK, April 20-23, 1986; SPE/DOE 14925. (7) Behrens, R. A.; Sandler, S. I. The Use of Semicontinuous Descriptions to Model the C7+ Fraction in Equations of State Calculations. SPE Reservoir Eng. 1988, 3, 1041-1047. (8) Shibata, S. K.; Sandler, S. I.; Behrens, R. A. Phase Equilibrium Calculations for Continuous and Semicontinuous Mixtures. Chem. Eng. Sci. 1987, 42, 1977-1988. (9) Luks, K. D.; Turek, E. A.; Kragas, T. K. Asymptotic Effects Using Semicontinuous vis-a`-vis Discrete Descriptions in Phase Equilibrium Computations. Ind. Eng. Chem. Res. 1990, 29, 21012106. (10) Turner, W. R. Normal Alkanes. Ind. Eng. Chem. Prod. Res. Dev. 1971, 10 (3), 238-260. (11) Labadie, J. A.; Luks, K. D. Convergence of the Continuous Fluid-Solid Equilibrium Problem Using Quadrature Compositional Characterizations. Ind. Eng. Chem. Res. 2000, 39, 790796. (12) Labadie, J. A.; Luks, K. D. Solid-Fluid Phase Equilibria of Compositionally Complex Mixtures: Contrast of Equilibrium and Process Treatments. Fluid Phase Equilib. 2003, 205, 215232.
(14) Michelsen, M. L. The Isothermal Flash Calculation. Part I. Stability. Fluid Phase Equilib. 1982, 9, 1-9. (15) Baker, L. E.; Pierce, A. C.; Luks, K. D. Gibbs Energy Analysis of Phase Equilibria. Soc. Pet. Eng. J. 1982, 22 (5), 731742.
Acentric factor11 ω ) 1.85 - exp(0.68 - 0.05nC0.88)
(3) Cotterman, R. L.; Bender, R.; Prausnitz, J. M. Phase Equilibria for Mixtures Containing Very Many Components. Development and Application of Continuous Thermodynamics for Chemical Process Design. Ind. Eng. Chem. Process Des. Dev. 1985, 24, 194-203.
(13) Lapidus, L. Digital Computation for Chemical Engineers; McGraw-Hill: New York, 1962; Chapter 2, pp 56-66.
Critical pressure (bar)11 Pc ) 191.8823nC-0.9657
(2) Cotterman, R. L.; Prausnitz, J. M. Continuous Thermodynamics for Phase-Equilibrium Calculations in Chemical Process Design. In Kinetic and Thermodynamic Lumping of Multicomponent Mixtures; Astarita. G., Sandler, S. I., Eds.; Elsevier Science Publishers BV: Amsterdam, The Netherlands, 1991; p 229.
(B10)
Triple-point temperature (K)24 Tt ) 374.5 + 0.02617[12.01115nC + 1.00797(2nC + 2)] - 20172/[12.01115nC + 1.00797(2nC + 2)] (B11) Enthalpy of fusion at the triple point temperature (cal/gmole)24 ∆hfus ) 0.1426[12.01115nC + 1.00797(2nC + 2)]Tt (B12)
(16) Specovius, J.; Leiva, M. A.; Scott, R. L.; Knobler, C. M. Tricritical Phenomena in “Quasi-Binary” Mixtures of Hydrocarbons. 2. Binary Ethane Systems. J. Phys. Chem. 1981, 85, 23132316. (17) Peters, C. J.; van der Kooi, H. J.; de Roo, J. L.; de Swaans Arons, J.; Gallagher, J. S.; Levelt Sengers, J. M. H. The Search for Tricriticality in Binary Mixtures of Near-Critical Propane and Normal Paraffins. Fluid Phase Equilib. 1989, 51, 339-351. (18) Du, P. C.; Mansoori, G. A. Phase Equilibrium Computational Algorithms of Continuous Mixtures. Fluid Phase Equilib. 1986, 25, 840-841.
Literature Cited
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Received for review January 23, 2004 Revised manuscript received May 10, 2004 Accepted May 24, 2004 IE040035Y
