Theoretical prediction of phase behaviour at high temperatures and pressures for non-polar mixtures. Part 1.—Computer solution techniques and stability tests

Abstract

The classical mathematical description of critical points of binary mixtures and the computer technique used to solve the relevant equations are described. The techniques used here are applicable to any closed equation of state and one fluid model prescription.

The critical points, in a given range of temperature and volume are located as the solution of two simultaneous equations: (∂p/∂v)_T,x(∂²G/∂x²₂)_T,p= 0 (∂p/∂v)_T,x(∂³G/∂x³₂)_T,p= 0 with reduced volume v and reduced temperature T as independent variables. The search procedure consists of three parts. Firstly the zeros of the first quantity are established around the perimeter of an area in the v, T plane defined by the range of v and T. Secondly the locus of the zeros of this quantity are tracked by a novel stepping procedure. The sign of the second quantity is determined at each step, a change in sign indicating that a zero and hence a critical point has just been passed. Thirdly, the critical point is accurately located by a bisection technique.

Examples of the loci for mixtures are given and the critical points obtained compared with those obtained by Scott and van Konynenburg for the van der Waals equation of state.

Tests for the stability of critical phases which can be applied to analytic expressions for derivatives of the free energy with respect to composition of a mixture are described.

The tests are applied so that if a firm decision is made at any point between material stability, metastability and unstability the sequence of tests ends. Initially tests for mechanical stability are applied and points which are mechanically stable are subjected to further tests to determine whether the critical phase is materially stable, metastable or unstable.

If (∂⁴G/∂x⁴₂)_p,T is negative the critical phase is materially unstable otherwise a detailed study of the (∂²G/∂x²₂)_p,T against composition diagram is needed. In the more difficult cases numerical integration of the area under the (∂²G/∂x²₂)_p,T against composition curve has to be undertaken. The determination of stability for systems in which (∂²G/∂x²₂)_p,T against composition diagram has multiple interacting loops is also described.