P-version least-squares finite element formulation of a system of convection-reaction nonlinear equations—Fixed bed o-xylene oxidation

Abstract

A steady-state one-dimensional plug-flow heterogeneous model, similar to the one employed by Calderbank et al. [ Chem. Engng Sci. 32, 1435–1443 (1977)]is utilized to simulate o-xylene oxidation in a catalytic fixed bed reactor. The mathematical model describing the reaction consists of five first-order nonlinear differential equations and five nonlinear algebraic equations. The algebraic equations are converted into differential equations using mass and energy balance equations in the gas phase. A p-version least-squares finite element formulation (LSFEF) is constructed for the system of resulting nonlinear coupled differential equations using C ° equal order interpolation for all variables. The least-squares error functional is constructed (integrated sum of squares of the errors or the residuals resulting from the individual equations) without linearization, approximations or assumptions. The minimization of this least-squares error functional results in finding a solution vector {δ} which makes the partial derivatives of the error functional with respect to {δ}, a null vector. This is accomplished by using Newton's method with a line search. The p-version LSFEF results are compared with experimental and numerical results reported by Calderbank et al. and Chandrasekharan and Calderbank [ Chem. Engng Sci. 34, 1323–1331 (1979)]. p-Convergence studies of the least-square error functional and the dependent variables are also presented.