Numerical Solution Methods

Abstract

In this chapter several numerical methods frequently employed in reactor engineering are introduced. To simulate the important phenomena determining single-and mul-tiphase reactive flows, mathematical equations with different characteristics have to be solved. The relevant equations considered are the governing equations of single phase fluid mechanics, the multi-fluid model equations for multiphase flows, and the population balance equation. Computers generally make the study of fluid flow easier, more effective and cheaper than using experimental analysis solely. Once the power of numerical simu-lations was recognized, the interest in numerical techniques increased dramatically and the work of developing numerical solution methods for the governing equations of fluid mechanics now constitutes a separate field of research known as computa-tional fluid dynamics (CFD). In engineering practice, the basic conservation equa-tions of fluid mechanics are normally solved by the Finite Difference-and the Finite Volume Methods. Other methods may be more appropriate for equations with partic-ular mathematical characteristics or when more accurate, robust, stable and efficient solutions are required. The alternative spectral methods can be classified as sub-groups of the general approximation technique for solving differential equations named the method of weighted residuals (MWR) [59]. The relevant spectral meth-ods are called the collocation-, Galerkin, Tau-and Least squares methods. These methods can also be applied to subdomains. The subdomain methods are generally divided into two categories, named spectral element-[41, 103], and finite element [42, 89, 98] methods. In the spectral element methods, the solution on each sub-domain (or element) is approximated by a high order polynomial expansion. In the finite element methods, on the other hand, the solution on each element is normally approximated by a first order (low order) polynomial expansion [28].