A Systematic Procedure for Analysis and Modification of Chemical Equilibrium Related Equation Sets to Enable Convergence to Physically Meaningful Solutions

Mordechai Shacham, Chemical Engineering, Ben-Gurion University of the Negev, Beer-Sheva, Israel
Neima Brauner, School Of Mechanical Engineering, Tel-aviv University, Tel-aviv, Israel

Computation of the equilibrium composition in complex reactions gives rise to the need to solve systems of non-linear algebraic equations (NLE). These systems of equations are notoriously difficult to solve. Common causes of the difficulties include: 1. Presence of discontinuities and/or regions where some of the functions are undefined. 2. While chemical equilibrium problems have only one physically meaningful solution (with only positive and real components), several additional solutions with negative and/or complex components may be obtained. Because of the difficulties involved special techniques and programs were developed for solving such problems. However, the general purpose NLE solvers available in widely used software packages
(see, for example,https://en.wikipedia.org/wiki/Comparison_of_numerical_analysis_software) may not be able to yield the physically meaningful solution.
We propose a systematic procedure for analysis and modification of NLE sets representing chemical equilibrium compositions, so that the general purpose solvers will be able to find the relevant physically meaningful solution.
The procedure includes Identification of expressions with discontinuities (like logarithm, square root etc.) and modification of such expressions (whenever possible) as to eliminate the discontinuities (replacing log by exp., for example), reduction of the dimension of the problem by converting some of the implicit equations to explicit ones (in terms of an "output" variable). The reduction of the dimension enables assignment of initial estimates only for variables for which such estimates can be obtained based on physical considerations.
In the presentation the proposed algorithm will be described in detail and several examples for solving equilibrium composition related problems, before and after applying the proposed procedure will be presented.