Simultaneous equations solution with matrix algebra

In the previous chapter we saw how determinants are used to tackle problems involving the solution of systems of linear equations. In general, the branch of mathematics which deals with linear systems is known as linear algebra, in which matrices and vectors play a dominant role. In this chapter we shall explore how matrices and matrix algebra are used to address problems involving coordinate transformations, as well as revisiting the solution of sets of simultaneous linear equations. Vectors are explored in Chapter 5. 

For each elementary volume, then, four equations like (9) plus the three equations (10),(11) and (12) are written. The reac tor zones can be separately computed in sequence for each, 7N nonlinear algebraic equations are to be simultaneously solved. This is performed with the aid of a general program for the solution of large, sparse matrix, nonlinear equations systems, already employed (12) for the simulation of the LDPE vessel reactor. More details on the program are given elsewhere (13). 

Putting in n = 0 and i = 0, we have three unknown temperatures Tj 1, T] i, and T o- There are only two algebraic equations so there is one more unknown variable than algebraic equation. Therefore, we cannot explicitly solve for the shell and tube temperatures at the new time level. However, if we formulate the entire set of equations for all distance grid points with the time level i = 0, we get 40 equations in 40 unknowns. The equations are linear so we can use matrix techniques to solve for all the shell and tube temperatures at the new time level simultaneously. This simultaneous solution strategy is called an implicit method. The matrix formulation is shown in Figure 8.8, where... 

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