Numerical Modeling Of Incomplete Combustion In A Liquid Chemical Rocket Engine

Chemical rocket engines require very accurate modeling of chemical combustion to be optimized and safely developed. One may solve for chemical equilibrium by using the thermodynamic technique of minimizing the free energy of the system subject to mass-balance constraints. This is computationally accomplished by using Lagrange multipliers to constrain the system and using Newton’s method for multivariate root finding to identify free-energy minima within the constraints. In a manner suitable for a mathematical audience loosely familiar with chemistry, we will introduce the chemical problem which produces the constrained optimization problem Gordon and McBride seek to solve. We provide an overview of the concept of Lagrange multipliers and how they are used to transform constrained optimization problems into simpler unconstrained optimization problems, for both singly- and multiply-constrained systems. Further, we build upon the familiar Newton’s method for single-variate root finding to explain how Newton’s method for multivariate rootfinding works. From these concepts and first thermodynamic principles, we derive a model of the chemical problem which may be solved by numerical methods. We present this algorithm which will solve for the equilibrium state of arbitrary incomplete combustion problems and demonstrate it as an operational Python program. We evaluate its accuracy and capability by comparing it to the algorithm of Gordon and McBride, which has been made publicly available as the program NASA Chemical Equilibrium with Applications (CEA).