Spline collocation

A steady state equivalent of Equations (36) for the ideal plug flow riser reactor, Pe - , can be easily derived, and also will not be shown here. Equations (36) were solved numerically using the spline collocation technique discussed before. The product selectivities in various reactors for the following relative rates, k4/k = 0.5 kg/k = 0.7 k -j = 0.1 kg/k- = 0.05 kg/k-j = 0.035 are shown in Figure 7. 

Figure 1. Concentration profiles generated by spline collocation for absorption with a pseudo-first order reaction, k 2/D1 = 100, Yj = 1000.

Figure 3. Concentration profiles generated by spline collocation for an absorption with two-step reaction. Parameters are same as in figure 2 with k /k =1/...

In all cases, it is essential to solve the model equations efficiently and accurately. Some techniques are discussed in this book and in the appendices, for the solution of the highly non-linear algebraic, differential and integral equations arising in the modelling of fixed bed catalytic reactors. The most difficult equations to solve are usually the equations for diffusion and reaction in the porous catalyst pellets, especially when diffusional limitations are severe. The orthogonal collocation technique has proved to be very efficient in the solution of this problem in most cases. In cases of extremely steep concentration and temperature profiles inside the pellet, the effective reaction zone method and its more advanced generalization, the spline collocation technique, prove to be very efficient. 

Kim, K Lee, K.S., and Lee, J.H. (2010a) Bilevel optimizing control structure for a simulated moving bed process based on a reduced-order model using the cubic spline collocation method. Ind. Eng. Chem. Res., 49, 3689-3699. 

COLNEW A general purpose code for solving mixed-order systems of boundary value problems using spline collocation and Newton s method. 

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]. 

S. M. Mahmoud and M. S. Osman, On a Class of Spline-Collocation Methods for Solving Second-Order Initial-Value Problems, International Journal of Computer Mathematics, 2009, 86(4), 616-630. 

Since Whiting and Carr s 1977 paper, there has been a steady trickle of publications on OC, mostly showing its use in various systems. One might say that this negates the original claim that OC is easily adapted to new systems - if so, why publish it whenever it is done It has been refined for example, Hertl and Speiser (1987) demonstrated "spline collocation" for very fast homogeneous chemical reactions where a very thin reaction layer is formed and the concentration profile is divided into two regions - the reaction layer, and the rest. 

We have, therefore, three main tools for use with very fast reactions spline collocation, the HE variants and the modern packages for multivariable stiff systems. None of these is simple but the problem itself probably precludes simple solutions. 

The simulation of fast chemical equilibrium reactions in cyclic voltammetric reaction-diffusion models with spline collocation. 

In the catalytic reforming of methane for synthesis gas production, Xu and Froment [1989] also applied collocation. The reactions are very fast in that case and the partial pressures of the reacting molecules drop to zero in a very shallow surface layer of the particle. In that case, spline collocation has to be used. Xu and Froment [1989] used 2 subintervals with collocation used as above. Extra equations have to be added to impose continuity of the first derivative in the spline point y. ... 

The essence is that, if the concentration profile simulated is smooth (which it normally is), then the polynomials will be well behaved in between points and no such problems will be encountered. As is seen below, implicit boundary values can easily be accommodated, and by the use of spline collocation [179-181], homogeneous chemical reactions of very high rates can be simulated. This refers to the static placement of the points. Having, for example, the above sequence of points for five internal points, the point closest to the electrode is at 0.047. This will be seen, below, to be in fact further from the electrode than it seems, because of the way that distance X is normalised so that, for very fast reactions that lead to a thin reaction layer, there might not be any points within that layer. Spline collocation thus takes the reaction layer and places another polynomial within it, while the region further out has its own polynomial. The two polynomials are designed such that they join smoothly, both with the same gradient at the join. This will not be described further here. 

Hertl P, Speiser B (1987) Electroanalytical investigations. Part IV. The simulation of fast chemical eqmlibtium reactions in cyclic voltammetric reaction-diffusion models with spline collocation. J Electroanal Chem 217 225-238... 

Pons BS, Schmidt PP (1980) Global spline collocation in the simulation of electrochemical diffusion equations. Electrochim Acta 25 987-993... 

Since the model is based on mechanistic rate relationships, the composition profiles (as well as temperature and pressure) are calculated at positions from the inlet of the catalyst tube to the outlet. The differential reaction rate relationships, as well as the mass balances, heat balances, and pressure drop relationships are solved simultaneously. A global spline collocation algorithm is used to pose the problem, and an SQP algorithm (sequential quadratic programming) solves the relationships. 

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