61 Introduction

Mathematical models of flow processes are non-linear, coupled partial differential equations. Analytical solutions are possible only for some simple cases. For most flow processes which are of interest to a reactor engineer, the governing equations need to be solved numerically. A brief overview of basic steps involved in the numerical solution of model equations is given in Section 1.2. In this chapter, details of the numerical solution of model equations are discussed.

In general, numerical solution of the governing transport equation replaces continuous information contained in the exact solution of partial differential equations by discrete information available at a finite number of locations (grid points). The values of all the dependent variables at these finite numbers of grid points are considered as basic unknowns. The task of a numerical method then becomes one of providing a set of algebraic equations for these unknowns and prescribing an algorithm to solve these algebraic equations. The algebraic equations (called discretized equations) involving the unknown values of dependent variables are derived from the governing partial differential equations. Some assumptions about how the unknown dependent variables change between grid points are necessary for such derivation. Generally, piecewise profiles are assumed, which describe variation over a small region around the grid point in terms of values at that grid point and the surrounding grid points. To facilitate this, the solution domain is divided into a number of sub-domains or computational cells (the process is called grid generation), so that a separate profile assumption can be associated with each computational cell.

For a given differential equation, there can be several different ways to derive the discretized equations (finite difference, finite volume, finite element). A brief introduction to these three methods is given in Chapter 1 (Section 1.2). As mentioned therein, finite volume methods ensure integral conservation of mass, momentum and energy over any group of control volumes and, of course, over the whole solution domain. This characteristic exists for any number of grid points (and not only for the limiting case of a large number of grid points) and is the most attractive feature from the reactor engineer's point of view. Thus, as a limiting case of one computational cell, the finite volume equations become equivalent to those written for the ideal, completely mixed reactor (with which most reactor engineers are quite familiar). This book, therefore, discusses details of the finite volume method. This chapter is restricted to discussing application of the finite volume method to solve the general conservation equations discussed in Chapter 2. Applications to solve more complex model equations (like those governing multiphase flows or reactive mixing) are discussed in the next chapter.

Before discussing the finite volume method, it is worthwhile to examine the desired properties of the numerical solution method, which are summarized below:

(a) Consistency: In a consistent method, the error between the discretized equation and the exact equation (called the truncation error) tends to zero, as the grid spacing tends to zero. Truncation error is usually proportional to a power of the grid spacing, Ax and/or the time step At. It is usually estimated by employing Taylor series expansions to recover the original equation plus the remainder, which represents the truncation error. If the most important term in such a remainder is proportional to (Ax)" or (At)n, the method is termed an nth order approximation. For any consistent method, n should be greater than zero.

(b) Stability: Having consistent approximations does not guarantee that the solution of the discretized equations system will become the solution of an exact equation in the limit of small step size. In any numerical method, errors appear in the course of solution process. It is essential to ensure that the numerical method does not magnify these errors (such a method is said to be stable). Stability of the numerical method is difficult to determine especially for nonlinear problems. To ensure stability, many numerical methods need to impose limits on the time step or need to employ under-relaxation practices. Some of these issues are discussed later.

(c) Convergence: If the solution of the discretized equations tends to the exact solution of the differential equations as the grid spacing tends to zero, the numerical method is said to be convergent. For linear problems, consistency and stability are the necessary and sufficient conditions for convergence. For non-linear problems, however, convergence is usually checked by carrying out numerical solutions for a number of successively refined grids. Usually a consistent and stable numerical method leads to a grid-independent solution.

Besides examining these properties of numerical methods, specific efforts need to be made to assess the accuracy of numerical solutions of flow processes. Various types of errors and possible ways of estimating and controlling these errors are discussed in Section 6.5. Application of a finite volume method to solve partial differential equations and specific algorithms to treat pressure-velocity coupling are discussed in Sections 6.2 and 6.3, respectively.

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