2rpP rpE

The harmonic mean to estimate effective diffusion coefficients (second expression on RHS of Eq. (6.30)) can handle abrupt changes in values of rp without requiring an excessively fine grid in the vicinity of the change (see Patankar, 1980 for more details).

The foregoing discussion allows one to formulate a set of algebraic equations comprising one algebraic equation for each CV (per variable). However, CVs having one or more faces coinciding with the boundaries of the solution domain may require special treatment. The discussion on implementation of boundary conditions within the framework of the finite volume method is postponed until Section 6.3.3. Methods for the solution of algebraic equations resulting from the discretization process are discussed below.

6.2.2. Solution of Algebraic Equations

After implementing all the boundary conditions, one obtains one algebraic equation per node (per variable), which relates the variable value at the node to the values of variables at several neighboring nodes. The numbers of equations and unknowns are equal and, therefore, the system is well defined and closed. The algebraic equation for any CV has the following form:

i where P denotes the node at which the governing equation is approximated, nb denotes neighboring nodes and the index i covers all the neighboring nodes involved in dis-cretized approximation. The coefficients (aP and anb) and the source term, SpC, are estimated using the initial guess or previous iteration values. The node P and its neighbors appearing in the above equation form the so-called computational molecule. Examples of computational molecules are shown in Fig. 6.7. Before discussing the solution of algebraic equations of the form (6.31), it will be useful to introduce the concept of under-relaxation.

It must be remembered that linear algebraic equations of the form (6.31) are approximate forms of the original non-linear discretized equations. The overall iterative procedure of repeatedly solving linearized equations to obtain solutions of non-linear equations is susceptible to divergence, especially when these are coupled with equations of other variables. In order to control the magnitude of change during each iteration, an under-relaxation parameter, a$ is introduced:

where $pew is the solution of Eq. (6.31) and * indicates the guess value or the previous iteration value. The under-relaxation parameter, , takes a value between zero and one. Lower values of under-relaxation parameter lead to more stable but slower convergence. The optimum value of under-relaxation parameter is problem dependent. Generally, small values are used during the early iterations, which are gradually increased as convergence is approached. Converged results are independent of the values of under-relaxation parameters. Instead of explicitly applying under-relaxation as shown in Eq. (6.32), it is more efficient to combine Eqs (6.31) and (6.32) to form the modified equation (Patankar, 1980):

The diagonal dominance of such a modified discretized equation set is increased since the coefficient of in the modified equation is larger than that in Eq. (6.31) while other coefficients remain the same. This formulation has a positive effect on many iterative solution methods and is, therefore, recommended.

The system of linear algebraic equations (Eq. (6.31) or (6.33)) can be written in matrix notation:

where A is a square matrix (since the number of equations and unknowns must be equal), <p is a vector of unknown variable values at the grid nodes and B is the vector containing the remaining terms. The matrix A is usually a sparse matrix (most

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