# INTEGRATION OF STIFF DIFFERENTIAL EQUATIONS IN CHEMICAL REACTOR MODELLING

#### Abstract

Several techniques to improve the reliability and efficiency of generalized stiff integrators in solving the mass and energy balances for plug-flow tubular reactors are considered. These are tested for the combustion of hexane in a complex reaction system. First, three model simplifications are considered. The steady-state approximation (SSA) and pseudo-steady-state approximation (PSSA) are analyzed to assess the errors incurred and the computational savings achieved. Both procedures are found to have little value in improving the efficiency of algorithms to accurately solve the mass and energy balances. The equilibrium approximation (EA), in which the forward and reverse rates of a reaction are equated, reduces the number of ODE mass balances, but adds an algebraic constraint. This is shown to significantly reduce stiffness and sensitivity when applied for the fastest reactions. A technique to compute the approach to chemical equilibrium is presented and combined with numerical integration to solve the EA model. Tests in the post-flame region of the hexane combustion model indicate accurate solutions can be obtained, but a reduction in computation time requires a more efficient minimization of Gibbs free energy. Two possible improvements are proposed. Next, two asymptotic solutions which significantly reduce the integration time are discussed. For the entrance region, where stiff integrators can be inefficient, low-order asymptotic expansions can be derived with few operations. Algorithms to compute the polynomial coefficients and estimate the range of applicability are presented. For decaying species in reactions modelled by sub-first-order kinetics, a simple asymptotic solution avoids the integrator stall that results when using relative error control. Finally, several transformations are discussed. Integration of linearly independent ODEs for a subset of "basic" species improves efficiency when the loss of precision is small. An effective procedure to detect loss of precision is presented. Scaling is shown to have no effect on stiff integrators using relative error control. With absolute error tolerances, scaling gives more uniform error weighting for trace species. A preliminary algorithm is applied to the hexane combustion system with good results. Two independent variable transformations that eliminate steep gradients are discussed. Both increase the complexity and nonlinearity of the model, severely limiting their utility.

#### Subject Area

Chemical engineering

#### Recommended Citation

WHITE, CHARLES WALTER, "INTEGRATION OF STIFF DIFFERENTIAL EQUATIONS IN CHEMICAL REACTOR MODELLING" (1983). *Dissertations available from ProQuest*. AAI8406734.

https://repository.upenn.edu/dissertations/AAI8406734