Polynomials finite-difference techniques

Typically, these methods arrive at the same finite difference representation for a given problem. However, we feel that Taylor-series expansions are easy to illustrate and we will therefore use them here in the derivation of finite difference equations. We encourage the student of polymer processing to look up the other techniques in the literature, for instance, integral methods and polynomial fitting from Tannehill, Anderson and Pletcher [26] or from Milne [16] and finite volume approach from Patankar [18], Versteeg and Malalasekera [27] or from Roache [20]. 

Wajge et al. (1997) attempted to develop rigorous PDAE model for packed batch distillation with and without chemical reaction and used finite difference and orthogonal collocation techniques to solve such model. The main purpose of the study was to investigate the efficiencies of the numerical methods employed. The authors observed that the collocation techniques are computationally more efficient compared to the finite difference method, however the order of approximating polynomial needs to be carefully chosen to achieve a right balance between accuracy and efficiency. See the original reference for further details. 

In this chapter, we will present several alternatives, including polynomial approximations, singular perturbation methods, finite difference solutions and orthogonal collocation techniques. To successfully apply the polynomial approximation, it is useful to know something about the behavior of the exact solution. Next, we illustrate how perturbation methods, similar in scope to Chapter 6, can be applied to partial differential equations. Finally, finite difference and orthogonal collocation techniques are discussed since these are becoming standardized for many classic chemical engineering problems. 

Differential Method In order to use the differential method of data analysis, it is necessary to differentiate the reactant concentration versus space-time data obtained in a plug-flow PBR. There are three methods of differentiation that are commonly used (i) graphical equal-area differentiation, (ii) numerical differentiation or finite difference formulas, and (iii) polynomial fit to the data followed by analytical differentiation. The aim of differentiation is to obtain point values of the reaction rate ( Ra)p at each reactant concentration Q4 or conversion xa or space time (.W/Fao), as required. All three differentiation methods can introduce some error to the evaluation of -Ra)p- Information on and illustration of the various differentiation techniques are available in the literature [23, 26]. 

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