Convergence of the Continuous Fluid−Solid Phase Equilibrium

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Ind. Eng. Chem. Res. 2000, 39, 790-796

Convergence of the Continuous Fluid-Solid Phase Equilibrium Problem Using Quadrature Compositional Characterizations Joseph A. Labadie and Kraemer D. Luks* Department of Chemical Engineering, University of Tulsa, 600 S. College Avenue, Tulsa, Oklahoma 74104-3189

The solid-fluid phase equilibrium problem is solved for an n-paraffinic C7+ model fluid whose pseudocomponent compositional characterization is determined by finite Laguerre-Gauss quadrature, under the assumption that the initial solid phase formed is a “pure” pseudocomponent. Of interest is how rapidly the crystal point temperature result converges with an increase in the level of quadrature. It is concluded that the highly nonlinear nature of the fluid phase’s Gibbs energy of mixing presents a serious challenge to obtaining a satisfactory asymptotic result. Furthermore, it is suggested that the convergence of fluid-phase equilibrium results with increasing quadrature level may likewise be affected, though probably to a lesser extent. Introduction

related to n-paraffinic molecular weight through

Crude oil systems are composed of so many components that their specific identity defies precise analysis. Consequently, it can be useful to view the C7+ portion of an oil as a continuous mixture. A live oil would then be a semicontinuous mixture of well-defined gaseous species plus a continuous C7+ fraction which may exceed 60 mol % of the system; in turn, a stock tank oil might be viewed as a single continuous system. With respect to such continuous and semicontinuous systems, it has been shown that quadrature can be a very effective way of specifying the pseudocomponents of the C7+ portion of an oil, and subsequently determining oil properties, provided one can accurately represent the distribution of the many species as a continuous function of some parameter(s), say, carbon number.1-8 Convergence for relatively low order of quadrature usually occurs in the phase equilibrium problem e.g., using an equation of state, and appears to ensure that one’s computed result will formally be equivalent to that which 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 a continuous formalism of phase equilibrium computations, including saturation computations.9 Given the simple exponential distribution

F(I) ) Re-RI/(e-RA - e-RB)

(1)

) 0, A g I, I g B for a continuous fluid with its carbon number I limited to the range A ) 6.5 to B ) 100.5, one can accurately represent that portion by as few as four pseudocomponents when performing a flash calculation. (Du and Mansoori10,11 were the first, to our knowledge, to use the simple exponential distribution in phase equilibrium calculations, although Shibata et al.8 point out that Vogel et al.12 earlier represented experimental evidence supporting the use of the exponential form for C7+ mole fraction distributions.) The carbon number I can be * To whom correspondence should be addressed. Telephone: (918) 631-2974. Fax: (918) 631-3268. E-mail: [email protected].

MW ) 14.0I + 2.0

(2)

The average carbon number for a given carbon number range (A, B) can be expressed in terms of the exponential decay parameter R, through

〈I〉 ) [(RA + 1)e-RA - (RB + 1)e-RB]/[R(e-RA - e-RB)] (3) In the limit B f ∞, 〈I〉 f A + 1/R. The higher the molecular weight MW of the continuous mixture, the smaller R is or, equivalently, the flatter the exponential distribution F(I) becomes. In earlier work, we have shown that pseudocomponent characterization of continuous [crude oil] mixtures using finite Laguerre-Gauss quadrature is able to generate dew-point calculations that are asymptotically the same, to a reasonable degree, precision for n ) 4 or more pseudocomponents. [The system in question was a live oil with a C7+ fraction of about 30 mol % having a molecular weight of 178. A variant of the RedlichKwong equation of state was used to model the thermodynamics.] This result is appealing because for such a system the molecular nature of the heaviest C7+ pseudocomponent depends on the value of n. In this problem, however, the effect of an increase in the molecular weight of the heaviest C7+ pseudocomponent on the computed dew-point pressure is offset by the accompanying decrease in its mole fraction. Equivalently, one may argue that the mathematical complexity of that dew-point problem is essentially captured when n g 4. In this present study, we examine the solid-fluid phase equilibrium problem wherein a fluid phase description using a classical equation of state is coupled to a simple mathematical artifice for introducing the solid phase fugacity function (because such equations of state do not admit a solid phase directly). The practical interest in such computations lies in the prediction of solids formation in pipelines, wellbores, and near-wellbore reservoir formations, especially in the offshore scenario. The broad solid-formation problem often includes both asphaltene and wax formation.

10.1021/ie990542t CCC: $19.00 © 2000 American Chemical Society Published on Web 02/03/2000

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Herein we are addressing only the wax-formation problem in which certain n-paraffins and some of their isomers may crystallize out of the fluid oil phase upon cooling. Of special interest here is the thermodynamic crystal point, which is similar to, but not the same as, the cloud point, which marks the visible precipitation of solids often from a metastable dissolved state. Employing equilibrium thermodynamics to describe the thermodynamic crystal point necessarily precludes metastable phenomena; it should produce a useful upper bound to the temperature at which solid formation would occur. We will hereafter refer to the highest temperature at a given pressure at which a solid may appear in the equilibrium fluid phase as the “crystal point temperature”. The Solid-Fluid Problem We characterize the fluid phase as a continuous mixture of C7+ species having (estimated) n-paraffinic properties. The phase is modeled as a mixture of pseudocomponents using finite Laguerre-Gauss quadrature. These pseudocomponents have the potential to crystallize as “pseudopure” solid phases. We seek the crystal point temperature, that temperature at a pressure of 1 atm absolute at which the first crystal of one of the pseudocomponents forms. The equilibrium equation is

fil(P0,T,{xi}) ) fis°(P0,T)

(4)

for any pseudocomponent that forms a (pseudopure) solid phase. The liquid phase fugacities are determined from the Soave-Redlich-Kwong equation-of-state model:13

P ) RT/(v - b) - a(T)/[v(v + b)] xibi ∑ i)1

n

a)

)

Tti -1 T

(6)

Equation 6 is a simplification of the form in Prausnitz et al.14 Tti is the triple point temperature of pseudocomponent i, and ∆hifus is the heat of fusion of pseudocomponent i at the triple point. The pure liquid (hypothetical) fugacity fis° is readily calculated from the SRK equation of state. To employ this model, one must have the parameters Tc, Pc, ω, Tt, and ∆hfus as a function of the carbon number for the pseudocomponents. It was assumed that these pseudocomponents are n-paraffins for the sake of the calculation demonstration. The following correlations are used for the parameters:

Critical temperature (K):15 Tc ) 959.98 + exp(6.81536 - 0.211145nC2/3) (7) Critical pressure (bar) (this paper): Pc ) 191.8823nC-0.9657

(8)

Acentric factor (this paper): ω ) 1.85 - exp(0.68 - 0.05nC0.88)

(9)

Ti ) 374.5 + 0.02617[12.01115nC + 1.00797(2nC + 2)] - 20172/[12.01115nC + 1.00797(2nC + 2)] (10) (5a)

n

xibi ∑ i)1

bi ) 0.08664 n

RTci Pci

n

∑ ∑ xixjaij i)1 j)1

aij ) xaiiajj aii ) Riaci R2Tci2 aci(Tci) ) 0.42747 Pci Ri1/2 ) 1 + mi(1 - TRi1/2) mi ) 0.480 + 1.574ωi - 0.176ωi2

Enthalpy of fusion at the triple point temperature (cal/gmol):16 ∆hfus ) 0.1426[12.01115nC + 1.00797(2nC + 2)]Tt (11)

where

a(T) )

(

n

∑ ∑xixjaij i)1 j)1

b)

ln(fil°/fis°) ) (∆hifus/RTti)

Triple point temperature (K):16

n

b)

Often, a binary interaction parameter is used for aij, i * j; in this solely computational study, wherein no comparisons are made with experimental data, no binary interaction parameters (corrections) are used. The solid phase fugacities, which are for “pure” pseudocomponents, are determined using the artifice:

(5b)

The correlation for ω was obtained by fitting acentric factor data for n-paraffins of carbon number nC ) 7-14 and imposing an asymptotic value on the ω such that ω f 1.85 as nC f ∞. The critical pressure correlation was devised such that, given Tc in eq 7, the behavior of b in eq 5b would be linear with respect to nC. The necessity of modifying Pc from, say, that proposed by Tsonopoulos15 is as follows: if Pc is allowed to decrease too rapidly with nC, then the fugacity function fil can become extremely large (that is, fil/pxi . 1) for the higher carbon numbers (through its dependence on bi), which is an unacceptable nonphysical result at P ) 1.01325 bar. Other correlations for Pc might also avoid this computational problem; eq 8 is an arbitrary but effective way to avoid this problem. (It could also be argued that the Soave-Redlich-Kwong equation of state was never intended to describe species of carbon numbers approaching 100. However, the choice of model used herein, computational difficulties aside, is one of con-

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Ind. Eng. Chem. Res., Vol. 39, No. 3, 2000 Table 1. Pseudocritical Temperature of a C7+ Fluid, Extending from Carbon Number 7 to 100 with MW of 200, as a Function of the Number n of Components Determined by Finite Laguerre-Gauss Quadrature

Figure 1. Difference in pseudocritical temperature, -∆Tc, between quadrature levels n and n - 1 as a function of the level n (from Table 1). The straight line is an exponential fit of the results.

venience and not crucial to demonstrating the quadrature convergence of the solid-fluid phase equilibrium problem.)

n

Tc (K)

-∆Tc (K)

1 2 3 4 5 6 7 8 9 10 11 12

694.766 662 670.204 965 667.211 740 666.649 123 666.518 271 666.485 238 666.476 472 666.474 053 666.473 365 666.473 164 666.473 105 666.473 087

24.561 697 2.993 226 0.562 616 0.130 852 0.033 033 0.008 766 0.002 419 0.000 688 0.000 201 0.000 060 0.000 018

Table 2. Carbon Numbers and Mole Fractions for a C7+ Mixture of Molecular Weight 200 (A ) 6.5, B ) 100.5), As Determined by Finite Laguerre-Gauss Quadrature, for the Cases of n ) 4, 8, and 12 Components

Finite Laguerre-Gauss Quadrature Finite Laguerre-Gauss quadrature was employed, as it qualitatively represents the nature of typical crude oil C7+ portions, with their composition functions exponentially decaying with carbon number (molecular weight). 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)4

n)8

n

∫0Cf(X) e-X dX ) ∑wif(Xi)

(12)

i)1

The sets {wi} and {Xi} are determined by maximizing the polynomial order that f(X) can rigorously be. For a chosen n, the integral on the left-hand side is represented precisely to order 2n - 1.17 This particular quadrature is attractive for representing functions and processes where the continuous distribution is exponentially decaying. The use of the quadrature is relatively straightforward. Given a C7+ portion of an oil extending from, say, carbon number 7 to 100 with a molecular weight 200, one specifies A ) 6.5 and B ) 100.5 and determines the parameter R using eqs 2 and 3. The upper limit of the finite integral in eq 12 is C ) R(B - A). A choice of n is made (depending on how precise one wishes the integral representation to be), and the resultant sets {wi} and {Xi} are translated into sets of mole fraction and carbon numbers. For example, the normalized {wi} are the mole fractions for the C7+ portion, which correspondingly have the carbon numbers {Ii}. This discrete description is used to perform the phase equilibrium computation. The precision with which one can calculate the sets of mole fractions and carbon numbers can be progressively challenging as n is increased. These sets must be calculated accurately in order to evaluate the convergence of a problem’s answer with increasing n. Using double precision, the authors were able to generate reliable values of these sets for (A, B, MW) ) (6.5, 100.5, 200) for values of n to at least 12. Shortening the carbon number range of the problem to (6.5, 40.5, 200) led to degradation of the description’s precision for n > 9. (Upgrading the programming language to quadruple precision can relieve this degradation.) A demonstration of the precision of the quadrature used herein is given in Table 1 and Figure 1, wherein the property of

n ) 12

species

carbon number

mole fraction

1 2 3 4 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 9 10 11 12

8.865 414 19.252 258 39.298 370 72.088 773 7.513 400 11.843 770 19.644 440 30.897 814 45.457 347 62.776 195 81.161 861 96.116 327 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

5.858 49 × 10-1 3.671 02 × 10-1 4.605 57 × 10-2 9.933 02 × 10-4 2.980 73 × 10-1 3.943 11 × 10-1 2.234 64 × 10-1 6.961 66 × 10-2 1.287 25 × 10-2 1.518 56 × 10-3 1.330 16 × 10-4 1.132 81 × 10-5 1.745 99 × 10-1 2.964 07 × 10-1 2.645 62 × 10-1 1.603 84 × 10-1 7.130 45 × 10-2 2.428 04 × 10-2 6.587 95 × 10-3 1.495 47 × 10-3 3.038 37 × 10-4 6.048 41 × 10-5 1.294 40 × 10-5 2.782 12 × 10-6

pseudocritical temperature is calculated, using the correlation in eq 7, which is a very nonlinear function. n

Tc(pseudocritical) )

xiTci(Ii) ∑ i)1

(13)

It can be seen that the convergence is well behaved to n ) 12 and that Tc converges to better than 0.05 K with five terms. Example Calculations of Solid-Fluid Phase Equilibria Using the Soave-Redlich-Kwong model, eq 5, with the parameter correlations in eqs 7-11, the solid-fluid equilibrium problem in eq 4 was solved for choices of n from 2 to 12, for a pressure of 1.013 25 bar. The model C7+ system was one of molecular weight 200, with A ) 6.5 and B ) 100.5 (for which the value of R ) 0.130 834). Table 2 gives the descriptions of C7+ for n ) 4, 8, and 12. In every calculation, the heaviest species of the C7+ mixture was the first to precipitate and therefore its appearance as a “pure” solid phase marked the crystal point temperature. Table 3 and Figure 2 give the crystal point temperature as a function of n. These points

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Figure 2. Crystal point temperature as a function of the number of finite Laguerre-Gauss quadrature pseudocomponents used, for a C7+ fluid extending from carbon number 7 to 100 with a molecular weight of 200. Table 3. Crystal Point Temperature and Pseudocomponent Carbon Number of the Solid Phase as a Function of the Number of Finite Laguerre-Gauss Quadrature Pseudocomponents Used, for a C7+ Fluid Extending from Carbon Number 7 to 100 with a Molecular Weight of 200 components n

crystal point T (K)

Cn, solid phase

2 3 4 5 6 7 8 9 10 11 12

327.506 353.741 371.798 381.430 385.642 387.301 387.846 387.894 387.712 387.424 387.088

32.5472 53.6121 72.0888 84.0983 90.6357 94.1352 96.1163 97.3176 98.0922 98.6179 98.9899

Table 4. Crystal Point Temperature and Pseudocomponent Carbon Number of the Solid Phase as a Function of the Number of Finite Laguerre-Gauss Quadrature Pseudocomponents Used, for a C7+ Fluid Extending from Carbon Number 7 to 40 with a Molecular Weight of 200 components n

crystal point T (K)

Cn, solid phase

2 3 4 5 6 7 8 9

316.381 323.738 324.505 323.534 322.126 320.662 319.259 317.944

27.0004 33.7828 36.7390 38.1462 38.9005 39.3462 39.6298 39.8217

Table 5. Crystal Point Temperature and Pseudocomponent Carbon Number of the Solid Phase as a Function of the Number of Finite Laguerre-Gauss Quadrature Pseudocomponents Used, for a C7+ Fluid Extending from Carbon Number 7 to 1000 with a Molecular Weight of 200

mathematically satisfy the relations

fil(P0,T,{xi}) ) fis°(P0,T), i ) n fil(P0,T,{xi}) ) fis°(P0,T), i ) 1, 2, ..., n - 1

Figure 3. Crystal point temperature as a function of the number of finite Laguerre-Gauss quadrature pseudocomponents used, for a C7+ fluid with a molecular weight of 200 for B ) 40.5, 100.5, and 1000.5.

(14)

The amount of solid phase of species n, the heaviest species, is infinitesimal; thus, the fluid composition is unchanged. The crystal point temperature result appears to stabilize with increasing n at values of n g 8. However, the modest but consistent downward trend of the crystal point temperature from a maximum at n ) 9 is disturbing. One does not see such behavior with the pseudocritical temperature in Table 1. To elucidate this trend further, computations of crystal point temperatures were performed for two other values of B: (1) B ) 40.5, a value substantially less than 100.5, for which one would expect model difficulties caused by the use of eq 8 for Pc with the SRKEOS to be minimal. (2) B ) 1000.5, wherein model difficulties caused by the use of eq 8 for Pc with the SRKEOS would offer a worse case scenario. Table 4 gives crystal point temperature results for B ) 40.5 for a C7+ fluid with a molecular weight of 200, while Table 5 gives the same for B ) 1000.5. Figure 3 compares the crystal point temperature results for B ) 40.5, 100.5, and 1000.5. What is seen is that asymptotic convergence of the crystal point temperature is not achieved over the n range of the calculations shown, even for the case of B ) 40.5. The results at B ) 1000.5 are suspect because of model considerations a priori quadrature. Suffice it to

components n

crystal point T (K)

Cn, solid phase

Tfus (K)

2 3 4 5 6 7 8 9 10 11 12

327.592 354.945 378.158 398.192 415.514 430.681 444.186 456.421 467.676 478.154 485.467

32.5943 54.5732 78.3052 103.1118 128.6548 154.7388 181.2396 208.0733 235.1796 262.5143 290.0434

342.591 368.304 384.966 398.477 410.615 422.071 433.155 444.028 454.774 465.444 476.069

say that the model formulation (equation of state and correlations for Tc, Pc, and ω) itself must, of course, be sensible in order to generate reasonable fugacity descriptions with which the phase equilibrium computation is carried out. This requirement of model sensibility is dramatically illustrated by the results in Table 5, wherein the crystal point temperatures actually are greater than the Tfus of the solid phase species, which is, of course, a nonsensical result, given the equation of state model chosen. The fluid phase fugacity functions for the species of carbon numbers . 100 tended to blow up to large values which, in turn, dictated solid phase formation at temperatures greater than the species melting temperature, despite their presence in the fluid phase at extremely low concentrations. The blame is probably attributable to the correlation for Pc. Calculations for B . 100 will therefore not be discussed further.

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Table 6. Compressibility Factor z as a Function of the Number of Finite Laguerre-Gauss Quadrature Pseudocomponents Used, for a C7+ Fluid Extending from Carbon Number 7 to 100 with a Molecular Weight of 200 components n

z ) Pv/RT

2 3 4 5 6 7 8 9 10 11 12

0.012 230 75 0.012 265 66 0.012 271 20 0.012 272 31 0.012 272 55 0.012 272 60 0.012 272 62 0.012 272 62 0.012 272 62 0.012 272 62 0.012 272 62

Table 7. Total, Ideal, and Excess Reduced Gibbs Energy of Mixing as a Function of the Number of Finite Laguerre-Gauss Quadrature Pseudocomponents Used, for a C7+ Fluid Extending from Carbon Number 7 to 100 with a Molecular Weight of 200 G/RT components n

total

ideal

excess

2 3 4 5 6 7 8 9 10 11 12

-4.230 728 -4.484 719 -4.662 604 -4.809 399 -4.936 284 -5.048 087 -5.148 056 -5.238 517 -5.321 168 -5.397 283 -5.467 833

-0.417 497 -0.653 911 -0.829 751 -0.976 573 -1.103 522 -1.215 344 -1.315 318 -1.405 780 -1.488 432 -1.564 547 -1.635 097

-3.813 231 -3.830 809 -3.832 852 -3.832 825 -3.832 762 -3.832 743 -3.832 738 -3.832 736 -3.832 736 -3.832 736 -3.832 736

Discussion The crystal point temperature computational results in Tables 3 and 4 are intriguing because the quadrature descriptions are sufficiently precise at the higher values of n employed. See Figure 1 and Table 1. The computation is definitely not asymptotically converged as n f 12; thus, one cannot equate the crystal point temperature result at some finite value of n with that at n f ∞. The failure of the solid-fluid computation to asymptotically converge for any of the values of B employed in the example computations above suggests that quadrature at levels of n f 12 does not satisfactorily capture the nonlinearity of the problem. Given the mathematical foundation of quadrature,17 one can rigorously demonstrate that f(X) in eq 12 can be represented precisely by a quadrature level n if its polynomial order is e(2n - 1). The troublesome nonlinearity exists in the Gibbs energy of mixing function, therefore, within the fugacity formalism itself. It does not come from the SRK EOS model used. Table 6 displays the compressibility factor z of the liquid phase as a function of n, using the SoaveRedlich-Kwong equation of state combined with the correlations for Tc, Pc, and ω in eqs 7-9. This property successfully approaches an asymptotic value valid for n f ∞. However, not all thermodynamic properties of the fluid phase respond as favorably. Table 7 demonstrates that the Gibbs energy of mixing at best converges very slowly with increasing n. This behavior is attributable to its ideal contribution, a model-independent thermodynamic result; the excess Gibbs energy converges successfully, like the compressibility factor z in Table 6. Cursory analysis of the ideal Gibbs energy of mixing as a function of 1/n suggests that it is still

not well converged for n ∼ 50. Given this inherent functional difficulty with the Gibbs energy of mixing in the context of quadrature convergence, that difficulty should extend to the set of species fugacity functions, because the natural logarithm of the fugacity is the partial molar reduced Gibbs energy of mixing: n

∆Gmix/RT )

xi ln(fi) ∑ i)

(15)

Thus, it can be concluded that, before model considerations, the mathematical nature of the ideal Gibbs energy of mixing should challenge the convergence of a phase equilibrium problem using the quadrature approach. The solid-fluid phase equilibrium computations presented herein suggest that it is perilous to assume that the use of quadrature to carry out solid-fluid equilibrium computations will approach a convergent answer for a reasonably finite value of n, regardless of the value of B, or the sensibility of the model. This computational intractability supersedes resolution of finer details: for example, which equation of state should be employed; whether the solid phase is pure, a pseudocomponent, or a mixture of pseudocomponents (although it can be argued that the assumption of a pure solid phase may exacerbate the asymptotic convergence difficulty). The assumption of a pure solid phase is not infallible. It has been documented18 that there will be a limited degree of solid solution formation when cooling narrow mixtures of the higher molecular weight n-paraffins. However, it is the authors’ contention that this is not the primary difficulty with quadrature convergence of the solid-fluid phase equilibrium problem. (Lira-Galeana et al.19 have earlier proposed in the context of wax deposition using pseudocomponents both for the liquid phase mixture and as candidate “pure” solid phases.) Furthermore, the computations suggest that all phase equilibrium problems will be plagued by this lack of quadrature convergence because of the mathematical nature of the Gibbs energy of mixing function. Fugacity functions are formally related to intercepts of tangent hyperplanes to the Gibbs energy of mixing hypersurface with the pure-component vertexes. Therefore, it seems obvious that the slow convergence of the Gibbs energy of mixing must correspondingly affect the individual and collective behavior of the family of fugacity functions of the liquid phase pseudocomponents. However, it is the impression of the authors that fluid phase equilibrium problems appear to exhibit quadrature convergence. One can perhaps offer some insight into why the fluid phase equilibrium problem is better behaved in terms of quadrature convergence than the solid-fluid phase equilibrium problem by comparing the dew-point problem9 with this present crystal point problem. There is an obvious difference between the two problems. In the dew-point problem, the infinitesimal phase is a mixture of pseudocomponents, albeit weighted toward the very heaviest ones, whereas the infinitesimal phase herein is a single pseudocomponent. More important, in our opinion, is the fact that the infinitesimal liquid phase in the dew-point problem is located on the same Gibbs surface as the dominant vapor phase. Figure 4 schematically illustrates that the common tangent hyperplane for the fluid-fluid phase equilibrium problem touches two points on the Gibbs hypersurface for the dew-point problem. In contrast, the common

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Conclusions 1. The asymptotic convergence with quadrature level n e 12 of the calculation of either solid-fluid or fluidfluid phase equilibria is intrinsically hampered by the lack of convergence of the ideal Gibbs energy of mixing function. 2. The special nature of the solid-fluid phase equilibrium problem, with its solid fugacity determined from a mathematical artifice separate from the fluid Gibbs energy function, appears to hamper its asymptotic convergence more seriously than that of the analogous fluid phase equilibrium dew-point problem. 3. The asymptotic convergence of the crystal point temperature from solid-fluid phase equilibria appears to be very sensitive to the mathematical assumption of a “pure” pseudocomponent solid phase. An earlier study of dew-point calculations suggests that the assumption of an infinitesimal mixture phase (“solution”) will lessen the convergence problem. Figure 4. Schematic illustration of the difference in the Gibbs analysis geometry of the solid-fluid equilibrium problem (top) and the fluid-fluid equilibrium problem (bottom).

tangent hyperplane for the solid-fluid phase equilibrium problem touches one point on the Gibbs hypersurface and the terminus of a Gibbs energy spike at the vertex of the heaviest pseudocomponent, the spike representing the “pure” pseudocomponent fugacity function. That spike comes from the mathematical artifice (6). As one increases the quadrature level n, the representation of the Gibbs energy of mixing in both problems will continue to change for large values of n (say, >12). For the fluid-fluid phase equilibrium problem, both points of tangency will adjust in concert (so it would seem). However, only one tangent point in the solidfluid phase equilibrium will significantly adjust, that of the fluid phase on the Gibbs hypersurface. The solid phase Gibbs energy spike is a function of temperature only at fixed pressure. It is affected primarily by the nature of the heaviest pseudocomponent in a quadrature description. This nature may be relatively stable for n > 6 when the upper limit B e 100 and less stable for very large B. (It is acknowledged that the size of B will have some effect on the computed result, but it is considered secondary; given a value of B, the primary effect will be the slow convergence of the Gibbs energy of mixing function.) We suggest that results depending on a common tangent hyperplane to the Gibbs hypersurface are much less sensitive to the changes one gets in the Gibbs surface (function), as one increases the quadrature level n. In contrast, results depending on the tangency of the hyperplane to a single point on the Gibbs hypersurface and to the solid phase Gibbs spike appear to be more sensitive to an increase in the quadrature level. While this difference may be the reason for the apparent convergent behavior of the fluid-fluid phase equilibrium problem, closer examination can reveal small deviations in the dew-point computational results9 from what one might consider to be truly asymptotic results as n f large. We point this out as a caveat, to emphasize that the very highly nonlinear nature of the Gibbs energy of mixing function in all phase equilibrium problems may always challenge quadrature mathematics.

Nomenclature a(T) ) Soave-Redlich-Kwong parameter A ) lower carbon number bound b ) Soave-Redlich-Kwong parameter B ) upper carbon number bound C ) upper bound of the reduced carbon number variable f ) fugacity F ) distribution function ∆hfus ) enthalpy of fusion I ) carbon number mi ) Soave-Redlich-Kwong parameter MW ) molecular weight n ) number of quadrature pseudocomponents nC ) carbon number P, P0 ) pressure R ) gas constant SRKEOS ) Soave-Redlich-Kwong equation of state T ) temperature v ) molar volume wi ) quadrature weighting factor X ) quadrature variable xi ) mole fraction R ) exponential distribution decay parameter Ri ) Soave-Redlich-Kwong parameter ωi ) acentric factor Subscripts c ) critical point C ) carbon fus ) fusion i,j ) species l ) liquid phase R ) reduced s ) solid phase t ) triple point Superscripts ° ) pure phase mix ) mixing property

Literature Cited (1) Cotterman, R. L.; Prausnitz, J. M. Flash Calculations for Continuous or Semicontinuous Mixtures Using an Equation of State. Ind. Eng. Chem. Process Des. Dev. 1985, 24, 434. (2) Cotterman, R. L.; Prausnitz, J. M. Continuous Thermodynamics for Phase-Equilibrium Calculations in Chemical Process

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Received for review July 22, 1999 Revised manuscript received December 1, 1999 Accepted December 6, 1999 IE990542T