Ind. Eng. Chem. Res. 1998, 37, 4889-4892
4889
RESEARCH NOTES Analytic and Fast Numerical Solutions and Approximations for Cross-Association Models within Statistical Association Fluid Theory Thomas Kraska† Institut fu¨ r Physikalische Chemie, Universita¨ t zu Ko¨ ln, Luxemburger Str. 116, D-50939 Ko¨ ln, Germany
Analytic expressions for the fractions of nonbonded association sites have been derived for different cross-association models in the framework of the statistical association fluid theory (SAFT). The comparison with solutions of a chemical theory for association shows that the solutions of both theories are equivalent. Therefore, simple approximations obtained from the chemical theory can be applied to cross-association models of SAFT. Analytic solutions simplify the application of SAFT in chemical engineering modeling. For other cross-association models, for which the fraction of nonbonded sites cannot be obtained analytically, an efficient alternative numerical solving procedure is described. The statistical association fluid theory (SAFT) is a physically sound theory for modeling associating fluids. It is based on the thermodynamic perturbation theory of Wertheim1 which provides the basic relation between the fraction of nonbonded association sites and the free energy due to association. SAFT has been applied to different reference model fluids such as the hard-sphere or the Lennard-Jones fluid. Calculations of the phase behavior with the resulting equations of state show good agreement with molecular simulations2 and experimental data.3-9 In SAFT the fraction XA of nonbonded sites A is related to the association strength ∆AB and the fractions XB of all other kind of association sites B by eq 1:
XAj )
1
∑ixi ∑B XB ∆A B
1+F
j i
i
(1)
pair correlation function. The complete expression for ∆AiBj can be looked up in the papers on SAFT.3-9 However, this temperature dependence of AiBj does not agree with the physical picture. The use of the geometric mean for the attraction parameter AiBj itself is physically reasonable, but it does not lead to simplifications of the mathematical structure of SAFT.12 In this paper analytic solutions of eq 1 for several cross-association site models have been derived without simplification. Furthermore, approximations obtained within a chemical theory of association are shown to be applicable to SAFT. In addition, an alternative procedure of reducing the numerical effort is described, which can be applied to any cross-association site model. Model I
i
Here, F is the density of the fluid and xi the mole fraction of substance i. The association strength ∆AjBi is a function of the pair correlation function of the reference fluid. For pure fluids, eq 1 can be solved analytically for the different association site models.10 For mixtures, eq 1 is usually obtained numerically by solving a system of nonlinear equations. That often causes complications especially if one is interested in derivatives of the free energy. In order to obtain analytic solutions for XAi the crossassociation strength has been approximated by the geometric mean of the self-association strengths11 ∆AiBj
) x∆AiBi∆AjBj. It should be noted that the geometric mean for the cross-association strength implies a temperature-dependent cross-association parameter AiBj because ∆AiBj is given by cAiBj(exp(AiBj/kBT) - 1). Here, cAiBj is an abbreviation collecting some variables and the † Phone: +49-221-470-4553. Fax: +49-221-470-4900. Email:
[email protected].
For a mixture of substance 1 with one association site A+ and substance 2 with one association site B- forming, for example, a donor-acceptor or Lewis acid-base complex and any number of substances without association sites, eq 1 leads to a 2 × 2 system of nonlinear equations (two equations and two variables):
XA )
1 1 + Fx2XB∆AB
(2)
XB )
1 1 + Fx1XA∆AB
(3)
The sign characterizes an association site similar to a charge. Association sites with unequal signs attract each other with ∆ > 0 while all other ∆ are set to zero. Rearranging and combining these equations leads to a quadratic polynomial in XB which can be solved analytically (eq 4). With the result for XB one can calculate XA using eq 2.
10.1021/ie980206x CCC: $15.00 © 1998 American Chemical Society Published on Web 10/30/1998
4890 Ind. Eng. Chem. Res., Vol. 37, No. 12, 1998
XB ) [(1 - 2x1)F∆AB - 1 + [1 + 2F∆AB(x1 + x2) + (F∆AB)2(1 - 2x1)2)]1/2]/(2x2F∆AB) (4) The latter expression is equivalent to the solutions of SAFT and the associated perturbed-anisotropic-chain theory (APACT) presented by Economou and Donohue,13 but it is generalized for the addition of any number of nonassociating substances. APACT is a chemical association theory which employs the mass action law in order to describe the association of ideal gas monomers. Model II
XA )
1 1 + F(x1XB∆AB + x2XC∆AC)
(5)
XB )
1 1 + Fx1XA∆AB
(6)
XC )
1 1 + Fx1XA∆AC
(7)
Rearranging and combining these equations yields a cubic polynomial in XC:
XC3 + pXC2 + qXC + r ) 0
q)
XA )
(8)
(∆AB - ∆AC) - x1F(∆AC)2 + x2F∆AC(∆AC - 2∆AB) x2F∆AC(∆AB - ∆AC)
∆AC - 2∆AB + x2F∆AB∆AC x2F∆AC(∆AB - ∆AC) r)
∆AB (9) x2F∆ (∆AB - ∆AC) AC
XC can be obtained analytically with the Cardanos formula which can be found in the literature.14,15 Usually, there is only one physically reasonable solution with XC ∈ [0, 1] of the three possible solutions. From XC the other fractions of nonbonded sites can be calculated using eq 7 and then 6. The comparison of this solution with the solution of APACT for the corresponding association model16 again shows that SAFT and APACT yield equivalent solutions. This can be verified by setting W1 ) x1XAXB, W2 ) x2XC, a1 ) F∆AB, and a12 ) F∆AC where W1, W2, a1, and a12 are variables used in APACT.16 The Wi are the fraction of free molecules of species i. Model III Model III describes a mixture of one substance with two association sites (A+, B-, nB ) 1) or three association sites (A+, two B-, nB ) 2) and a second substance
1 1 + F(x1nBXB∆AB + x2XD∆AD)
XB ) XC )
Model II describes a mixture of one self-associating substance with one non-self-associating but cross-associating substance plus any number of nonassociating substances. Substance 1 contains two association sites A+ and B- while substance 2 contains only one association site C-. This is a model for mixtures of amphoteric/ acid or base mixtures with the possibility of adding inert substances as alkanol/ketone(/alkanes) mixtures. For this model a 3 × 3 system of nonlinear equations has to be solved:
p)
with two association sites (C+, D-) plus any number of nonassociating substances. Applying this model of two self- and cross-associating plus inert substances requires the solution of a 4 × 4 system of nonlinear equations:
1 AB
1 + F(x1XA∆
+ x2XC∆BC)
1 1 + F(x1nBXB∆BC + x2XD∆CD)
XD )
1 AD
1 + F(x1XA∆
+ x2XC∆CD)
(10) (11) (12) (13)
By combining these equations it is possible to transform the 4 × 4 system to a 1 × 1 system of equations. As a result, a polynomial of the 6th order in one of the fractions of nonbonded sites can be obtained. It is not possible to calculate the roots of this polynomial analytically, but a one-dimensional root finder is much faster than a corresponding multidimensional numerical procedure. The resulting equations are lengthy and not listed here. The transformation can easily be accomplished by the use of a computer algebra program. Below, a source code for the program MAPLE17 is listed. It calculates the expressions for the fractions XY. One of these solutions is the above-mentioned 6thorder polynomial which can be found in the RootOf(...) statement in the MAPLE output. One can solve this polynomial by a numerical algorithm such as the regulafalsi method in the interval [0, 1]. The root of this polynomial is XA. All other fractions can then be calculated from XA analytically. The meaning of the variables a, b, ..., g can be obtained by comparing with those of eqs 10-13: # define the system of equations fa :) XA + a * XA*XB + b * XA*XD -1: fb :) XB + g * XA*XB + c * XB*XC -1: fc :) XC + d * XC*XB + e * XC*XD -1: fd :) XD + f * XA*XD + e * XC*XD -1: # solve this system hh :) solve({fa)0,fb)0,fc)0,fd)0},{XA,XB,XC, XD}): ww :) (hh): sa :) op(1,ww): sb :) op(2,ww): sc :) op(3,ww): sd :) op(4,ww): # extract the solutions of the fraction XA ) XAsolution XAsolution :) op(2,sa); XBsolution :) op(2,sb); XCsolution :) op(2,sc); XDsolution :) op(2,sd); # leave MAPLE quit; Model IV Model IV is similar to model III, but the two crossassociation strengths are set equal
∆AD ) ∆BC ≡ ∆cross
(14)
This model describes a mixture of two self-associating
Ind. Eng. Chem. Res., Vol. 37, No. 12, 1998 4891
substances similar to model III (nB ) 1) with one effective cross-association interaction and any number of inert substances. Examples for those mixtures are amphoteric/amphoteric(/inert) systems such as alkanol/ alkanol(/alkanes) systems. With this assumption the 4 × 4 system of equations for model IV reduces to a 2 × 2 system because of XC ) XD and XB ) XA:18
XA ) XD )
1 AB
1 + F(x1XA∆
(15)
+ x2XD∆cross)
1 cross
1 + F(x1XA∆
(16)
+ x2XD∆CD)
This 2 × 2 system can be reduced to a 4th-order polynomial (eq 18) in XD which can be solved analytically.14,15 With XD, the value for XA can be obtained from eq 16. For simplification, the following abbreviations are introduced:
a ) x1F∆AB
b ) x2F∆cross
c ) x1F∆cross
XA ) XD )
y3 )
2ad - cd - cb ad2 - bcd
(18)
2
a - c - 2ad - c + cb ad2 - bcd c - 2a a y0 ) 2 (19) y1 ) 2 ad - bcd ad - bcd y2 )
This model has not been solved for APACT yet. In order to avoid numerical solutions Economou et al.16 and Anderko19 have presented very simple analytic approximations for monomer fractions of the amphoteric/ amphoteric systems within the chemical theory. The equivalence of solutions of the amphoteric/acid for SAFT and APACT suggests the same equivalence for the amphoteric/amphoteric solutions. Therefore, the approximations for APACT have been compared with the exact solution of SAFT presented here. The variables of APACT and SAFT are related by simple equations: W1 ) x1XA2, W2 ) x2XD2, K1RT/v0 ) ∆ABF, K2RT/v0 ) ∆CDF, and K12RT/v0 ) ∆crossF. Here, R is the gas constant, T the temperature, Wi the fraction of free molecules of species i, and Kij the association constants within APACT.16 The fraction of nonbonded sites follows from the APACT approximation of Economou et al.16 written in variables of SAFT:
XA ) XD )
2 1 + x1 + 4F∆AB 2 1 + x1 + 4F∆CD
(20) (21)
These approximations correspond to the solutions of two pure self-associating but not cross-associating substances. The approximation for the chemical theory of Anderko19 is taken from Economou et al.16 and rewritten here in terms of SAFT variables:
1 + x1 + 4F(∆ABx1 + ∆crossx2) 2 1 + x1 + 4F(∆CDx2 + ∆crossx1)
(22)
(23)
These approximations are similar to the solutions of pure self-associating substances but additionally include a cross-association term. They show very good agreement with the analytic solution for SAFT presented here. For the case ∆cross f 0 the Anderko approximation and the SAFT solution are equivalent. The approximation of Economou et al. shows less agreement with the solution of SAFT than the Anderkos expression. However, if the association strengths ∆AB, ∆CD, and ∆cross do not differ too much, the very simple approximations of Economou et al. yield reasonable agreement with the analytic SAFT solution. Setting ∆AB ) ∆CD does not lead to further simplifications. However, one can simplify this model by setting the association strength of all interactions equal. By introducing this additional symmetry, it is possible to reduce the order of the polynomial from 4 to 2.
d ) x2F∆CD (17) XD4 + y3XD3 + y2XD2 + y1XD + y0 ) 0
2
∆AB ) ∆CD ) ∆cross ≡ ∆
(24)
This very restricted model is of interest in theoretical investigations.20 Since ∆ is a function of the molecular diameter, equal ∆-values also imply equal diameters of the molecules. Nevertheless, with some approximations as, for example, the introduction of a mixing rule for the molecule diameter inside ∆, it may also be applied to real systems. The results for XA ) XD are
XA )
x1 + 4F(x1 + x2)∆ - 1 2F(x1 + x2)∆
(25)
Conclusion Analytic solutions for cross-association models in the framework of SAFT have been derived. The analytic solutions are able to describe multicomponent mixtures with two associating compounds such as alkanols or ketones. Mixtures of water with alkanols and inert substances can also be modeled assuming a two-site model for water which is a reasonable assumption.21 The comparison of the cross-SAFT solutions presented here with the solutions for the corresponding model within the chemical theory suggest that also for other association-site models both theories yield equivalent solutions. The general equivalence is very useful because the equations which have to be solved for the chemical theory are simpler than those of SAFT. Since it is easier to verify a solution than deriving it, the chemical theory gives access to solutions of SAFT. This extends the applicability of SAFT for the modeling of associating mixtures enormously. Besides the analytic solutions and approximations, a method of reducing the multidimensional root-finding problem to a one-dimensional root-finding problem has been presented. This method can be applied with minor variations of the listed MAPLE source code (model III) to any association site model in order to obtain the exact values of nonbonded sites. It is a fast and secure method for the calculation of the fractions of nonbonded association sites.
4892 Ind. Eng. Chem. Res., Vol. 37, No. 12, 1998
Acknowledgment The author would like to thank O. Pfohl and the referee for helpful discussions. Financial support of the Deutsche Forschungsgemeinschaft and the Fonds der Chemischen Industrie is acknowledged gratefully. Literature Cited (1) Wertheim, M. S. Thermodynamic perturbation theory of polymerization. J. Chem. Phys. 1987, 87, 7323 and references therein. (2) Chapman, W. G.; Gubbins, K. E.; Joslin, C. G.; Gray, C. G. Theory and simulation of Associating Liquid Mixtures. Fluid Phase Equilib. 1986, 29, 337. (3) Chapman, W. G., Gubbins, K. E.; Jackson, G.; Radosz, M. Equation of State Solution Model for Associating Fluids. Fluid Phase Equilib. 1989, 52, 31. (4) Chapman, W. G.; Gubbins, K. E.; Jackson, G.; Radosz, M. New Reference Equation of State for Associating Liquids. Ind. Eng. Chem. Res. 1990, 29, 1709. (5) Huang, S. H.; Radosz, M. Equation of State for Small Large, Polydisperse and Associating Molecules: Extensions to Mixtures. Ind. Eng. Chem. Res. 1991, 30, 1994. (6) Mu¨ller, E. A.; Gubbins, K. E. Equation of State for Water from a Simplified Intermolecular Potential. Ind. Eng. Chem. Res. 1995, 34, 3662. (7) Kraska, T.; Gubbins, K. E. Phase Equilibria Calculations with a Modified SAFT Equation of State: 2. Binary Mixtures of n-Alkanes, 1-Alkanols, and Water. Ind. Eng. Chem. Res. 1996, 35, 4738. (8) Kontogeorgis, M.; Voutsas, E. C.; Yakoumis, I. V.; Tassios, D. P. An Equation of State for Associating Fluids. Ind. Eng. Chem. Res. 1996, 35, 4310. (9) Pfohl, O.; Brunner, G. Using BACK to modify SAFT Order To Enable Density and Phase Equilibrium Calculations Connected to Gas-Extraction Processes. Ind. Eng. Chem. Res., in press. (10) Huang, S. H.; Radosz, M. Equation of State for Small Large, Polydisperse and Associating Molecules. Ind. Eng. Chem. Res. 1990, 29, 2284.
(11) Elliott, J. R. Efficient Implementation of Wertheim’s Theory for Multicomponent Mixtures of Polysegmented Species. Ind. Eng. Chem. Res. 1996, 35, 1624. (12) Fu, Y.-H.; Sandler, S. I. A. Simplified SAFT Equation of State for Associating Compounds and Mixtures. Ind. Eng. Chem. Res. 1995, 34, 1897. (13) Economou, I. G.; Donohue, M. D. Chemical, Quasi-Chemical and Perturbation Theories for Associating Fluids. AIChE J. 1991, 37, 1875. (14) Bronstein, I. N.; Semendjajew, K. A. Taschenbuch der Mathematik; Teubner Verlagsgesellschaft: Leipzig, 1979. (15) Hacke, J. E., Jr. A simple solution of the general quartic. Am. Math. Monthly 1941, 48, 327. Source codes available at: http://www.uni-koeln.de/math-nat-fak/phchem/deiters/quartic/quartic.html. Title of site: Subroutines for solving cubic, quartic and quintic equations, 1998. (16) Economou, I. G.; Ikonomou, G. D.; Vimalchand, P.; Donohue, M. D. Thermodynamics of Lewis Acid-Base Mixtures. AIChE J. 1990, 36, 1851. (17) MAPLE V Release 4, Waterloo Maple Inc., Waterloo, Canada, 1981-1996. (18) Campbell, S. W. Chemical theory for mixtures containing any number of alcohols. Fluid Phase Equilib. 1994, 102, 61. (19) Anderko, A. Extension of the AEOS model to systems containing any number of associating and inert components. Fluid Phase Equilib. 1989, 50, 21. (20) Kraska, T.; Yelash, L. V. University of Cologne, Ko¨ln, Germany, unpublished results. (21) Gupta, R. B.; Johnston, K. P. Lattice fluid hydrogen bonding model with a local segment density. Fluid Phase Equilib. 1994, 99, 135.
Received for review April 1, 1998 Revised manuscript received August 18, 1998 Accepted August 21, 1998 IE980206X
