Global orthogonal collocation

The model was solved using orthogonal collocation on finite elements (OCFE). Orthogonal collocation on finite elements was developed by Carey and Finlayson (26) for solution of boundary layer problems. Carey and Finlayson used OCFE to solve the simultaneous heat and mass transfer equations describing a catalyst pellet and found the new method to be more efficient than finite difference techniques. They also showed that OCFE was applicable to boundary layer problems that could not be solved by global orthogonal collocation. Jain and Schultz (27)... 

All these factors are functions of the concentration of the chemical species, temperature and pressure of the system. At constant diffu-sionai resistance, the increase in the rate of chemical reaction decreases the effectiveness factor while al a constant intrinsic rate of reaction, the increase of the diffusional resistances decreases the effectiveness factor. Elnashaie et al. (1989a) showed that the effect of the diffusional resistances and the intrinsic rate of reactions are not sufficient to explain the behaviour of the effectiveness factor for reversible reactions and that the effect of the equilibrium constant should be introduced. They found that the effectiveness factor increases with the increase of the equilibrium constants and hence the behaviour of the effectiveness factor should be explained by the interaction of the effective diffusivities, intrinsic rates of reaction as well as the equilibrium constants. The equations of the dusty gas model for the steam reforming of methane in the porous catalyst pellet, are solved accurately using the global orthogonal collocation technique given in Appendix B. Kinetics and other physico-chemical parameters for the steam reforming case are summarized in Appendix A. 

The effectiveness factors at each point along the length of the reactors are calculated for the key components methane and carbon dioxide, using the dusty gas model and simplified models I and II. The catalyst equations resulting from the use of the dusty gas model are complicated two-point boundary value differential equations and are solved by global orthogonal collocation technique (Villadsen and Michelsen, 1978 Kaza and Jackson, 1979). The solution of the catalyst pellet equations of the simplified models 1 and 2 at each point... 

The global orthogonal collocation technique was used for solving the equations described above. 

B.2 APPLICATION OF THE GLOBAL ORTHOGONAL COLLOCATION TO THE DUSTY GAS MODEL EQUATIONS OF THE POROUS CATALYST PELLETS FOR THE STEAM REFORMING OF METHANE... 

B.2 Application of the Global Orthogonal Collocation to the Dusty Gas Model Equations of the Porous Catalyst Pellets for the Steam Reforming of Methane 442... 

FIGURE B.l Schematic diagram for the solution of the dusty gas model using the global orthogonal collocation technique. 

The method taught in Chapter 8 (as well as in Section 12.4) can be applied over the whole domain of interest [0,1] (any domain [a, b] can be easily transformed into [0,1]), and it is called the global orthogonal collocation method. A variation of this is the situation where the domain is split into many subdomains and the orthogonal collocation is then applied on each subdomain. This is particularly useful when dealing with sharp profiles and, as well, it leads to reduction in storage for efficient computer programming. 

Fig. 12.15. Also shown in the figure are plots of the numerical solution using the global orthogonal collocation method (Example 8.4), shown as a dashed line. The exact solution for the nondimensional reaction rate is tanh = 0.01. It...

PARSIM optionally provides the method of Finite Differences (FD) for space discretization. An advantage of this method is the lower bandwidth of the Jacobian matrix. Nevertheless, much more node points are needed to achieve the same accuracy compared to the OCFE method as demonstrated below. The method of global Orthogonal Collocation (OC) is provided additionally by PARSIM but should be used only for systems without steep gradients. 

Fig. 13. Orthogonal collocation on finite elements global indexing system.

Keywords orthogonal collocation, finite elements, dynamic optimisation, global optimisation, CNMPC... 

The collocation points are chosen as the zeros of the global orthogonal polynomial (Jacobi polynominal) / ) " (m) for slab geometry, where u is defined by ... 

This type of ordinary differential equation is conveniently solved by a global method where the dependent variables are approximated in the whole interval by polynominals. The method used here is the orthogonal collocation method, where orthogonal polynominals are used in the approximation. 

The global approach uses an interpolation based on a family of global functions which span all the sampled space with appropriate boundary conditions. This approach which is due to Gauss, is termed collocation (Sec. III.A). In a more elaborate form, based on orthogonal functions it is termed pseudospectral representation (Sec. III.B) (16). Since any local method is global within a small interval we will start by analyzing global approaches. 

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