Collocation technique

The resulting set of model partial differential equations (PDEs) were solved numerically according to the method of lines, applying orthogonal collocation techniques to the discretization of the unknown variables along both the z and x coordinates and integrating the resulting ordinary differential equation (ODE) system in time. 

Of the various methods of weighted residuals, the collocation method and, in particular, the orthogonal collocation technique have proved to be quite effective in the solution of complex, nonlinear problems of the type typically encountered in chemical reactors. The basic procedure was used by Stewart and Villadsen (1969) for the prediction of multiple steady states in catalyst particles, by Ferguson and Finlayson (1970) for the study of the transient heat and mass transfer in a catalyst pellet, and by McGowin and Perlmutter (1971) for local stability analysis of a nonadiabatic tubular reactor with axial mixing. Finlayson (1971, 1972, 1974) showed the importance of the orthogonal collocation technique for packed bed reactors. 

A drawback of the orthogonal collocation technique is its inability to accurately define profiles with sharp gradients or abrupt changes, since the... 

For the nonlinear case, the nonlinear two-point boundary value differential equation(s) for the catalyst pellet can be solved using the same method as used for the axial dispersion model in Section 5.1, i.e., by the orthogonal collocation technique of MATLAB s bvp4c. m boundary value solver. 

Fig. 6-2. Fractional change in current with dimensionless time r for a step to higher rotation rate, calculated as an expansion of exponentials by the collocation technique. From the bottom to the top curve 2, 4, 6, 8 and 10 exponential terms were utilized in the calculation with Sc = 1500 and 6 = 0.3. After [12].

This technique was further improved by Albery et al. [11] and later by Blauch and Anson [12] who used the orthogonal collocation technique for numerical resolution of Eq. (3-1). In this last case, the effects of hydrodynamic relaxation and of imperfect motor response were taken into account, but the unsteady hydrodynamic flow was basically that described by Sparrow and Gregg [36]. 

In Chapter 11 of this book we will use the thin spline radial function to develop the radial basis functions collocation method (RBFCM). A well known property of radial interpolation is that it renders a convenient way to calculate derivatives of the interpolated function. This is an advantage over other interpolation functions and it is used in other methods such us the dual reciprocity boundary elements [43], collocation techniques [24], RBFCM, etc. For an interpolated function u,... 

Collocation techniques are based on the fact that a field variable in a continuous space can be approximated with linear interpolation coefficients and basic functions located on discreet points sprinkled on the domain of interest, as schematically presented in Fig. 11.1. 

Numerical Method. Both the isothermal FFB and the adiabatic MAT models are very stiff due to the coke deactivation terms >j. The spline orthogonal collocation technique was used to solve the above models (19). Typically, the distance x was divided into two regions (0collocation points in each region. At the interface between the two regions, both the concentration and the mass flux were taken as continuous. The value of y varied with the degree of stiffness. 

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. 

Useful discussions with F. J. Krambeck are sincerely appreciated. We thank D. M. Nace for obtaining the experimental data. Also, D. H. Anderson s effort in programming the spline orthogonal collocation technique is appreciated. 

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 approach, the ODE or DAE process models are discretised into a set of algebraic equations (AEs) using collocation or other suitable methods and are solved simultaneously with the optimisation problem. Application of the collocation techniques to ODEs or DAEs results in a large system of algebraic equations which appear as constraints in the optimisation problem. This approach results in a large sparse optimisation problem. 

The model (12.15) to (12.17), (12.19) to (12.30) is solved in Matlab using the bvp4c function, which implements a collocation technique for solving partial-difFerential equations [16]. 

Another method employed to study the boundary effects on electrophoresis of a single particle is the boundary collocation technique. Investigations have been conducted for the particle migration along the axis of a circular orifice, normal to a conducting circular disk [41], and along the... 

Electrophoretic interactions between spherical particles with infinitely thin double layers can also be examined using the boundary collocation technique [16,54]. This method enables one not only to calculate the interactions among more than two particles, but also to deal with the case of particles in contact, for which the bispherical coordinate solution becomes singular. Analogous to the result for a pair of spheres, no interaction arises among the particles in electrophoresis as long as all the particles have an equal zeta potential. This important result is also confirmed by a potential-flow reasoning [10,55]. 

When the gap width between two particles becomes very small, numerical calculations involved in both the bispherical coordinate method and the boundary collocation technique are computationally intensive because the number of terms in the series required to be retained to achieve a desired accuracy increases tremendously. To solve this near-contact motion more effectively and accurately, Loewenberg and Davis [43] developed a lubrication solution for the electrophoretic motion of two spherical particles in near contact along their line of centers with the assumption of infinitely thin ion cloud. The axisymmetric motion of the two particles in near contact can be approximated as the pairwise motion of the spheres in point contact plus a deviation stemming from their relative motion caused by the contact force. The lubrication results agree very well with those obtained from the collocation method. It is shown that near contact electrophoretic interparticle... 

The CNMMR model with laminar flow liquid stream in the annular region consists of three ordinary differential equations for the gas in the tube core and two partial differential equations for the liquid in the annular region. These equations are coupled through the diffusion-reaction equations inside the membrane and boundary conditions. The model can be solved by first discretizing the liquid-phase mass balance equations in the radial direction by the orthogonal collocation technique. The resulting equations are then solved by a semi-implicit integration procedure [Harold etal., 1989]. 

The problem of a porous catalyst pellet, which had been addressed in the paper of Ray and Hastings, was later treated extensively by Jensen and Ray (242,297). They used surface coverage equations and mass and heat balances for the whole pellet, all of which, except for the heat balance, were solved for the nonlumped case. The solutions of the resultant partial differential equation set were obtained by collocation techniques. The surface reaction was assumed to be unimolecular and slightly more complex than the mechanism analyzed by Ray and Hastings in that the adsorption step was permitted to be reversible ... 

Sorensen, J. P., Simulation, Regression and Control of Chemical Reactors by Collocation Techniques, Doctoral Thesis, Danmarks tekniske Hpjskole, Lyngby (1982). 

The following is the resulting NLP problem P2, derived from PI after discretization of the differential equations by using the orthogonal collocation technique ... 

The integrals over the required micropore size distribution is evaluated by the orthogonal collocation technique and the adsorption energy is found from gas-solid potential energy minimum inside the pore by a univariate minimization routine DUMING [13]. 

Since the model equation are coupled partial differential equations, they are solved numerically by using a combination of the orthogonal collocation technique [14] and an ODE integrator [15]. 

For example the heterogeneous equivalent approximation (12), the unequal box size method (13), the orthogonal collocation technique (14) and the aforementioned implicit scheme all will lead to decreased computer time, although a price is usually paid in generality and/or accuracy. 

A number of nonequilibrium models fall into the general framework described above. The differences between models are due primarily to the models of flow and mass transfer on a tray (or within a section of packed column). Young and Stewart (1990), for example, use collocation techniques to solve a boundary layer model of cross-flow on a tray. An alternative approach that builds on the models of mass and energy transfer described in Chapters 11 and 12 has been developed in a series of papers by Taylor and co-workers (Krishnamurthy and Taylor, 1985a-c, 1986 Taylor et al., 1992). The latter model and some illustrations of its use are presented in this chapter. 

The appendices present the parameters and empirical correlations necessary for the models discussed in the book. They also give basic information on the use of the orthogonal collocation technique for the solution of non-linear two-point boundary value differential equations which arise in the modelling of porous catalyst pellets and the estimation of clFectiveness factors. The application of orthogonal collocation techniques to equations resulting from the Fickian type model as well as models based on the more rigorous Stefan-Maxwell equations are presented. 

In this section, a model similar to that used by Cardoso and Luss (1969) is considered, with the exception that the assumption of solid isothermality is relaxed. A finite difference solution for the transient equations with symmetrical boundary conditions is presented using the Crank-Nicholson method (Lapidus, 1962). Another more efficient, method of solution is considered which is based on the orthogonal collocation technique first used by Villadsen and Stewart (1967), Finlayson (1972) and Villadsen and Michelsen (1978). Several assumptions for model reductions are investigated. 

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. 

For the catalyst pellet diffusion-reaction problem, which is the most frequently encountered in modelling industrial fixed bed catalytic reactors, the orthogonal collocation technique (Villadsen and Stewart, 1967 Villadsen and Michelsen, 1978) is certainly the most efficient. 

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