6262
Ind. Eng. Chem. Res. 2004, 43, 6262
CORRESPONDENCE Comments on “Generalized Procedure for Estimating the Fractions of Nonbonded Associating Molecules and Their Derivatives in Thermodynamic Pertubation Theory” Michael L. Michelsen IVC-SEP, Department of Chemical Engineering, Technical University of Denmark, DK2800 Lyngby, Denmark
Sir: Tan et al.1 in a recent article in this journal described a procedure for calculating thermodynamic properties from an association model. In particular, they are concerned with calculating the derivatives of the fraction of nonbonded associating molecules with respect to process variables such as temperature, density, and composition with the aim of using these derivatives to calculate mixture fugacity coefficients and their temperature and composition derivatives. As the motivation for this work, it is stated that a previous development by Michelsen and Hendriks,2 which simplifies the calculation of physical properties in association models, does not extend to properties that require second derivatives of the Helmholz energy. This statement must be based on a misunderstanding of the work by Michelsen and Hendriks, where even explicit expressions for these second derivatives are provided. To cite from their paper: “The evaluation of second derivatives of the Helmholz function with respect to temperature, volume, and composition does involve the first derivatives of X with respect to these variables. The matrix of second derivatives with respect to the vector of variables, r, can be written compactly as follows
( ) Aassoc RT
rr
) Qrr - QrXQXX-1QXr”
(14)
Clearly, second derivatives of the Helmholz function require that first derivatives of the fraction of nonbonded molecules be calculated. In the above formula-
tion, the last term contains the derivative of the X vector with respect to the vector of variables, r, given by
∂X ) -QXX-1QXr ∂r This expression is again equivalent to eq 6 of Tan et al.1 It should be equally clear that second derivatives of the fraction of nonbonded molecules are not needed for caluclation of properties such as composition derivatives of fugacity coefficients or residual heat capacities. Earlier, we presented3 computing times for multicomponent mixtures that include all derivatives. Typical times for 15-component mixtures, calculating all derivatives and including solving the equation for the density, are less than half those reported by Tan et al. for calculating the nonbonded mole fractions only in binary and ternary mixtures. Literature Cited (1) Tan, S. P.; Adidharma, A.; Radosz, M. Generalized Procedure for Estimating the Fractions of Nonbonded Associating Molecules and Their Derivatives in Thermodynamic Perturbation Theory. Ind. Eng. Chem. Res. 2004, 43, 203-208. (2) Michelsen, M. L.; Hendriks, E. M. Physical Properties from association models. Fluid Phase Equilib. 2001, 180, 165-174. (3) von Solms, N.; Michelsen, M. L.; Kontogeorgis, G. M. Computational and Physical Performance of a Modified PC-SAFT Equation of State for Highly Asymmentric and Associating Mixtures. Ind. Eng. Chem. Res. 2003, 42, 1098-1105.
IE049697D
10.1021/ie049697d CCC: $27.50 © 2004 American Chemical Society Published on Web 08/04/2004
