Numerical Methods for Simulation and Optimization of Chemically Reacting Flows in Catalytic Monoliths

Abstract

The aim of this work is to develop numerical methods and software for simulation and optimization of complex processes in catalytic monoliths to achieve better understanding of the physic-chemical processes in catalytic reactors. The fluid dynamics are modelled by the boundary layer equations (BLEs), which are a large system of parabolic partial differential equations (PDEs) with highly nonlinear boundary conditions arising from the coupling of surface processes with the flow field inside the channel. The BLEs are obtained by simplifying the comprehensive model described by the Navier-Stokes equations and applying the boundary approximation theory. The surface and gas-phase chemical reactions are described by detailed models. The PDEs are semi-discretized using the method of lines leading to a structured system of differential-algebraic equations (DAEs). The DAEs are integrated by an implicit method, based on backward differentiation formulas (BDF). We develop a new BDF code with tailored efficient and robust numerical methods by exploiting the structure, and by an appropriate scaling for ill-conditioned iteration matrices, and by computing consistent initial values. Efficient methods for computation of partial derivatives in the framework of automatic differentiation and of finite differences are introduced and compared. Our newly developed simulation tool is more stable than the existing simulation tool, and faster than by a factor of ten to more than 60, depending on the applications. To improve the performance of catalytic reactors (e.g., maximizing gas conversion or selectivity) we can control certain process conditions, such as temperature at the catalyst wall or the ratio of catalytic active surface area to the geometric surface area or the gas composition, the temperature, or the velocity at the inlet of the catalyst. It is the first time that this problem is generally formulated as an optimal control problem constrained by a system of PDEs describing the chemical fluid dynamics process and additional constraints. The direct shooting approach in combination with sequential quadratic programming (SQP) method is used for solving the resulting optimal control problem. An efficient numerical method for computation of the derivatives required by the SQP method is introduced. In addition, error analysis for the numerical Newton method is investigated in detail. We introduce a new error model. Based on our error model and analysis, the limiting accuracy of the solution of nonlinear equations by the numerical Newton method can be obtained. Our newly developed software package for simulation and optimization can be applied to different reaction mechanisms and channel settings with different initial/boundary conditions. This software is applied to two practical applications: catalytic combustion of methane and conversion of ethane to ethylene. The numerical results are presented. The simulation software provides a useful tool for the validation of reactions mechanisms. The software package allows, e.g., for a better design and operation of the conversion of natural gas to higher hydrocarbons or the improvement of exhaust treatment in cars.