Simple orthogonal collocation

Two major forms of the OCFE procedure are common and differ only in the trial functions used. One uses the Lagrangian functions and adds conditions to make the first derivatives continuous across the element boundaries, and the other uses Hermite polynomials, which automatically have continuous first derivatives between elements. Difficulties in the numerical integration of the resulting system of equations occur with the use of both types of trial functions, and personal preference must then dictate which is to be used. The final equations that need to be integrated after application of the OCFE method in the axial dimension to the reactor equations (radial collocation is performed using simple orthogonal collocation) can be expressed in the form... 

When the Fickian diffusion model is used many reaction-diffusion problems in porous catalyst pellet can be reduced to a two-point boundary value differential equation of the form of equation B.l. This is not a necessary condition for the application of this simple orthogonal collocation technique. The technique in principle, can be applied to any number of simultaneous two-point boundary value differential equations. 

For the solution of sophisticated mathematical models of adsorption cycles including complex multicomponent equilibrium and rate expressions, two numerical methods are popular. These are finite difference methods and orthogonal collocation. The former vary in the manner in which distance variables are discretized, ranging from simple backward difference stage models (akin to the plate theory of chromatography) to more involved schemes exhibiting little numerical dispersion. Collocation methods are often thought to be faster computationally, but oscillations in the polynomial trial function can be a problem. The choice of best method is often the preference of the user. 

The same complex kinetics gives multiple steady states isothermally in thin or thick porous electrocatalysts (418). A simple, graphical orthogonal collocation method (422) can show the existence of multiple solutions for concentration within a certain potential range (418). If ohmic losses in the pores cause a potential change within the electrode structure, multiplicity can also arise with respect to potential, even with simpler rate expressions... 

B.l SIMPLE APPLICATION OF THE ORTHOGONAL COLLOCATION TECHNIQUES FOR A FICKIAN-TYPE DIFFUSION-REACTION MODEL FOR POROUS CATALYST PELLETS... 

The orthogonal collocation technique is a simple numerical method which is easy to program for a computer and which converges rapidly. Therefore it is useful for the solution of many types of second order boundary value problems. This method in its simplest form as presented in this section was developed by Villadsen and Stewart (1967) as a modification of the collocation methods. In collocation methods, trial function expansion coefficients are typically determined by variational principles or by weighted residual methods (Finlayson, 1972). The orthogonal collocation method has the advantage of ease of computation. This method is based on the choice of suitable trial series to represent the solution. The coefficients of trial series are determined by making the residual equation vanish at a set of points called collocation points , in the solution domain. 

Song et al (2006) proposed a multivariable purity control scheme using the m-parameters as manipulated variables and a model predictive control scheme based on linear models that are identified from nonlinear simulations. The approach proposed by Schramm, Griiner, and Kienle (2003) for purity control has been modified by several authors (Kleinert and Lunze, 2008 Fiitterer, 2008). It gives rise to relatively simple, decentralized controllers for the front positions, but an additional purity control layer is needed to cope with plant-model mismatch and sensor errors. Vilas and Van de Wouwer (2011) augmented it by an MPG controller based on a POD (proper orthogonal collocation) model of the plant for parameter tuning of the local PI controllers to cope with the process nonlinearity. 

More recently, numerical simulations of the simple PSA process have been developed by Chihara, who used conventional finite difference methods and Raghavan, Hassan, and Ruthven who used the method of orthogonal collocation. For solutions of comparable accuracy the collocation method was shown to require considerably less computer time. Brief details of Raghavan s model are given in Table 11.4. Computed curves showing the approach to steady-state operation are given in Figure 11.21 and a comparison of the... 

Fletcher (1974) introduced unequal 8x intervals Whiting and Carr (1977) applied orthogonal collocation to electrochemistry Shoup and Szabo (1982) applied Gourlay s (1970) hopscotch method to electrochemistry and Heinze et al (1984) showed how to include the boundary value c in the implicit equations of the Crank-Nicolson method, thereby removing a major problem with that method. Britz (1988) applied simple explicit... 

Orthogonal collocation in two dimensions has been used to simulate microdisk edge effects. The first paper in a series (5 up till now), by Speiser and Pons (1982) is a formidable tour de force. A two-dimensional set of polynomials is fitted to the grid and it leads, as in one dimension, to an "easily solved" set of ordinary differential equations. In the fifth part of this series of papers, Cassidy et al (1985), applied the method to electrode ensembles. This is obviously not for the occasional simulator, who is advised to use a simple technique and put up with the long computational times or use someone else s program but the method undoubtedly makes two-dimensional simulations efficient and accurate. 

In the collocation method, we define the weight functions to be Dirac delta functions centered on each node, so that the residual must be zero at each node. Unfortimately, even tiiough the residual is zero at each node, it may be hnge between nodes, especially with strong convection. In simple geometries, accuracy is improved when the nodes are placed at the zeros of orthogonal polynonuals (see Chapter 4). This orthogonal collocation method is discussed in Villadsen Michelsen (1978). 

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