Second-order reaction, spline

Figure 2. Concentration profiles generated by spline collocation for absorption with a second order reaction, ki 20 fi2/Di 300 A2o/Alg = 3, Y = 1000.

Data regressions based on the law of mass action are generally adequate for most situations. However, this model only retains validity in liquid-phase reactions at equilibrium without cooperativity. Reactions that involve solid-phase, multiple cooperative binding, and not reaching equilibrium, deviate from the model. Therefore, empirical equations that are not based on the law of mass action have been used for curve fitting also. Among these, polynomial (205) and spline functions are often used (206-209). Polynomial regression can be a second-order (parabolic) or third-order (cubic) function ... 

At the time t = 0 it is known for the batch reactor that the dimensionless reaction rate 0(X) exactly takes the value of one. If the curve of dX/dh over h(t), which has just been obtained with the help of the previous calculation, is now numerically fitted to a spline function of the first or second order, then the initial gradient dX/dh at h(t) = 0 can be determined by extrapolation. This initial gradient exactly corresponds to ... 

The final model hence includes one ordinary second-order differential equation (3.32) with an integral boundary condition at the surface (3.41) for each of the reactions. The numerical solution of the catalyst effectiveness factor can be carried out using the orthogonal collocation method by Villadsen and Michelsen [512]. The steam reforming reaction takes place mainly in the outer shell of the catalyst particle, since large particles are used to limit pressure drop. In this case it is advantageous to divide the catalyst pellet into two sections, an inner section and an outer section, divided by a spline point and with an appropriate coupling between the two sections. The spline collocation method has been used by [525] and [181]. A description of the method, spline collocation, can be found in [512]. 

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