Calculation of critical points using a reduction method
Résumé
Efficient and accurate calculation of critical points is an important aspect of general phase equilibrium calculations. When calculating critical points, the computationally expensive part is to solve the spinodal equation. For Heidemann and Khalil formulation the dimensionality of the problem is equal to the number of components (nc). In this paper we describe a new procedure enabling the reduction of problem dimensionality. The reduction is effectively achieved by spectral decomposition of the matrix C̄ with elements (1 - Cij), where Cij denotes the binary interaction parameters (BIP). The dimensionality of the problem is given by the number of reduction parameters, M = m + 1, where m is the number of nonzero (or non-negligible) eigenvalues of C̄. The spinodal equation consists in equating to zero a determinant of order M (for naturally occurring hydrocarbon mixtures, M is much less than nc). The procedure suggested by Michelsen and Heidemann for zero BIP is a particular case of our approach. For the spectral decomposition of C̄, the method is particularly efficient for mixtures with many components and few nonzero BIP. The proposed method proved its reliability when tested for several mixtures having various phase diagram shapes. Finally, we show that critical points of mixtures with more than 50 components, with a detailed paraffin-naphthene-aromatic (PNA) distribution of the heavy fraction, can be calculated by only performing linear algebra operations with a small dimensionality. Application of the reduction method leads to important savings in computer time.