An algorithm for calculating critical points in multicomponent mixtures which can easily be implemented in existing programs to calculate phase equilibria

This paper describes an add-on procedure for the calculation of critical points in multicomponent mixtures according to the method by Heidemann and Khalil [1]. The benefit of this approach is that all equations of state (EOS) and mixing rules of an existing program for calculating phase equilibria, which can be a commercial product or self-made, are available for the calculation of critical points. The method by Heidemann and Khalil requires the first and second partial derivatives of the fugacities with respect to the male numbers (∂ ln fk/∂ni)T,V,ni≠j(∂2 ln fk/∂nl∂nj)T,V,nl≠i,j. In our work the first partial derivatives are obtained numerically with a four-point differencing scheme and the second partial derivatives are obtained from the first partial derivatives with a numerical directional derivative as suggested by Michelsen [2]. This new approach allows the combination of the method by Heidemann and Khalil with any program for calculating phase equilibria and eliminates the need for the cumbersome determination of the analytical first partial derivatives of the fugacity with respect to the mole numbers. The implementation into the commercially available process simulator ASPEN PLUS Version 9.2 [3] is described and some calculation results are shown.