Orthogonal Collocation for Solving PDEs

Up to this point in the text, we have solved exclusively steady-state profiles in tubular reactors. Obviously tubular reactors experience a start-up transient like every other reactor, and this time-dependent behavior is also important and interesting. Calculating the transient tubular-reactor behavior involves solving the partial differential equation (PDE), Equation 8.36, rather than the usual ODE for the steady-state profile. Appendix A describes the method we use for this purpose, which is called orthogonal collocation. 

With reference to item (4), quite often the ODEs generated by separation of variables do not produce easy analytical solutions. Under such conditions, it may be easier to solve the PDE by approximate or numerical methods, such as the orthogonal collocation technique, which is presented in Chapter 12. Also, the separation of variables technique does not easily cope with coupled PDE, or simultaneous equations in general. For such circumstances, transform methods have had great success, notably the Laplace transform. Other transform methods are possible, as we show in the present chapter. 

In Chapters 7 and 8, we presented numerical methods for solving ODEs of initial and boundary value type. The method of orthogonal collocation discussed in Chapter 8 can be also used to solve PDEs. For elliptic PDEs with two spatial domains, the orthogonal collocation is applied on both domains to yield a set of algebraic equations, and for parabolic PDEs the collocation method is applied on the spatial domain (domains if there are more than one) resulting in a set of coupled ODEs of initial value type. This set can then be handled by the methods provided in Chapter 7. 

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