A reduction method for phase equilibrium calculations with cubic equations of state

Abstract

-In this work we propose a new reduction method for phase equilibrium calculations using a general form of cubic equations of state (CEOS). The energy term in the CEOS is a quadratic form, which is diagonalized by applying a linear transformation. The number of the reduction parameters is related to the rank of the matrix C with elements (1-C ij ), where C ij denotes the binary interaction parameters (BIPs). The dimensionality of the problem depends only on the number of reduction parameters, and is independent of the number of components in the mixture.

A comparison of conventional and reduction approaches for phase equilibrium calculations

“…(A7), as in conventional methods. Any procedure [10,13,20,21] to decompose the quadratic form A (Eq. A3) can be used to obtain the reduction parameters and the reduction matrix [8].…”

“…The reduction methods have been developed by Jensen and Fredenslund [9], Hendriks and van Bergen [10], Kaul and Trasher [11], Pan and Firoozabadi [12], Nichita and Minescu [13], Li and Johns [14], Nichita et al [15], and more recently Nichita and Graciaa [16] for flash calculations, and by Firoozabadi and Pan [17], Nichita et al [18], Hoteit and Firoozabadi [19], Nichita and Petitfrere [6] for phase stability testing. Different procedures for the decomposition of the quadratic form (which appears in the mixing rule of the energy parameter of an EoS) into linear forms have been proposed [10,13,14,20,21].…”

Isochoric Phase Stability Testing for Hydrocarbon Mixtures

“…The M (with M usually much smaller than nc) reduction parameters Q˛D P i q˛i x i (where q˛i ,˛D 1, M , are the elements of the reduction matrix; Hendriks, 1988) are simply replaced by Q˛D P i q˛i d i , for any set of reduction parameters (Nichita, 2006b;Nichita et al, 2007a;Nichita and Minescu, 2004).…”

Multiphase equilibrium calculations using a reduction method

“…Starting with the first proposed reduction method (Michelsen's three-equation flash [10], applicable for all BIPs equal to zero) and the enunciation by Hendriks of the "reduction theorem" [9] (stating the circumstances under which the dimensionality of phase equilibrium problems can be reduced), many applications of the reduction method have been reported for two-phase flash calculations [11][12][13][14][15][16][17][18][19][20][21][22][23][24] and for phase stability analysis [25][26][27][28][29]. The application of the reduction method in phase equilibrium problems is restricted by the form of the mixing rules in the EoS, that is, reduction requires that the EoS parameters (presented in Appendix A) be linear forms (the case of the covolume B) or decomposable into linear forms (such as the energy term A).…”

“…Several procedures to decompose the quadratic form A (Eq. (A3)) have been proposed: by spectral decomposition [12], by completing the square [11,15], by using linear transformations [16], or lowrank approximations [17], or by minimizing the approximation error of the parameter A [21], etc.…”