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A comparison of solution methods for the chemical equilibrium problem Ruda, Margarita M.

Abstract

This thesis deals with computing the equilibrium composition of a multiple species reacting mixture. When pressure and temperature are constant and the system is ideal, this is the chemical equilibrium problem. It is possible to approach this problem as the minimization of a non-linear objective function subject to linear equality constraints. The objective function represents the Gibbs' free energy of the system; the constraints refer to the conservation of the elements. Such a formulation corresponds to a "dual geometric program" which is related to another optimization problem known as the "primal geometric program". In those chemical equilibrium problems with many species, the "primal geometric programming" formulation includes less variables and constraints (inequality ones) than the dual formulation. We first compared the primal and dual formulations of the chemical equilibrium problem. Both formulations were solved with a Generalized Reduced Gradient code on seven examples. The primal formulation proved to be 30% faster than the dual for middle-sized problems (up to six simultaneous reactions). The code failed when trying to solve a dual problem of 24 species and 4 elements; but this same problem was easily solved when formulated as a primal geometric program. As the geometric programming theory includes sensitivity analysis, and we were, also interested in the effects of small changes of pressure and temperature on the optimal solution, we compared sensitivity analysis with re-optimization of the problem. Sensitivity analysis proved to be between 30% to 50% faster than re-optimization. It also yielded accurate results for the more abundant species when relatively small changes of temperature and pressure were operated. However, the equilibrium concentrations of trace species hardly matched those calculated by re-optimization. From these results we recommend the use of the primal formulation and of re-optimization to solve the chemical equilibrium problem, and we present a computer code that has been tested on a variety of examples.

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