One-Point Collocation

The orthogonal collocation method, as we have attempted to illustrate in previous examples, sustains an aecuracy, which will increase with the number of points used. Occasionally, one is interested in the approximate behavior of the system instead of the computer intensive exact behavior. To this end, we simply use only one collocation point, and the result is a simplified equation, which allows us to quickly investigate the behavior of solutions, for example, to see how the solution would change when a particular parameter is changed, or to determine whether the solution exhibits multiplicity. Once this is done, detailed analysis can be carried out with more collocation points. 

We illuminate these attractive features by considering the difficult problems of diffusion and reaction in a slab catalyst sustaining highly nonlinear Hinshelwood kinetics. The mass balance equations written in nondimensional form are taken to be 

Noting the symmetry of this problem, we make the usual substitution u = x, and the mass balance equations become 

Now we choose one collocation point, i, in the domain [0,1], and since we know the value of y at the surface of the catalyst, we will use it as the second interpolation point, that is, M2 = 1- For these two interpolation points, we have two Lagrangian interpolation polynomials, li(u) and l2(u), given as 

Using the Lagrangian interpolation polynomials li(u) and l2(u), the approximate solution for y(u) can be written as 

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Use rough and ready methods but do not carry them beyond their point of usefulness. A one-point collocation solution, for example, may not be very accurate, but may give an insight. A sketched phase plane may be qualitatively correct even though its numerical value is way off (see The Phase Plane in Chapter 4). 

Boundary value problems are encountered so frequently in modelling of engineering problems that they deserve special treatment because of their importance. To handle such problems, we have devoted this chapter exclusively to the methods of weighted residual, with special emphasis on orthogonal collocation. The one-point collocation method is often used as the first step to quickly assess the behavior of the system. Other methods can also be used to treat boundary value problems, such as the finite difference method. This technique is considered in Chapter 12, where we use this method to solve boundary value problems and partial differential equations. 

The first step in the solution procedure is discretization in the radial dimension, which involves writing the three-dimensional differential equations as an enlarged set of two-dimensional equations at the radial collocation points with the assumed profile identically satisfying the radial boundary conditions. An examination of experimental measurements (Valstar et al., 1975) and typical radial profiles in packed beds (Finlayson, 1971) indicates that radial temperature profiles can be represented adequately by a quadratic function of radial position. The quadratic representation is preferable to one of higher order since only one interior collocation point is then necessary,6 thus not increasing the dimensionality of the system. The assumed radial temperature profile for either the gas or solid is of the form... 

Since more than one axial collocation point is generally necessary. 

Then starting from the most general form of the dimensionless equations after the one-point radial collocation, we can apply the OCFE procedure. As... 

The model discretization or the number of collocation points necessary for accurate representation of the profiles within the reactor bed has a major effect on the dimensionality and thus the solution time of the resulting model. As previously discussed, radial collocation with one interior collocation point generally adequately accounts for radial thermal gradients without increasing the dimensionality of the system. However, multipoint radial collocation may be necessary to describe radial concentration profiles. The analysis of Section VI,E shows that, even with very high radial mass Peclet numbers, the radial concentration is nearly uniform and that the axial bulk concentration and radial and axial temperatures are nearly unaffected by assuming uniform radial concentration. Thus model dimensionality can be kept to a minimum by also performing the radial concentration collocation with one interior collocation point. 

The pellet mass and heat balances are described by second order differential equations of the two point boundary value type. For this case the reaction is neither too fast nor highly exothermic and therefore the concentration and temperature gradients inside the pellet are not very steep. Therefore the orthogonal collocation method with one internal collocation point was found sufficient to transform the differential equation into a set of algebraic equations which were solved numerically using the bisectional method (Rice,... 

The maximization problem is to maximize (7.35) subjected to constraints (7.32-7.34). The derivation of the Pontryagin maximum principle for the system and the optimality conditions are given in Appendix D, for one internal collocation point. The derivation of the optimality conditions for N internal collocation points is given by Elnashaie and El-Rifaie (1978). 

The reaction considered is relatively simple being a single reaction of moderate rate and moderate exothermicity and therefore one interior collocation point, for the solution of the two-point boundary value differential equations describing the catalyst pellets, was sufficient to obtain accurate results. 

Application of the simple case of one internal collocation point (n = 1 and 0) at this collocation point is (u,), the Laplacian at the interior collocation point can be written as ... 

For one interior collocation point, where N = 1, these equations can be simplified and given by ... 

An examination of the Fourier method, which is a special case of an orthogonal collocation representation, elucidates the main considerations of representation theory. It will be shown that by optimizing the representation the quantum limit of one point per unit phase space volume of h can be obtained. Moreover, the Fourier method has great numerical advantages because of the fast nature of the algorithm (22-26). This means that the numerical effort scales semilinearly with the represented volume of phase space (27). 

There are really many different variants of this method based on the selection of the support points and the elements to discretize the interval. Some of them have special names that highlight their approach. For instance, if the points are selected as the roots of an orthogonal polynomial and if the elements have only one point in common, the method is said to be a finite-element orthogonal collocation. On the other hand, if each element consists of three points and the adjacent elements share two points, the method is said a finite-difference method. In some cases, when the elements have common points, the single residual is not zeroed, but the sum of residuals is calculated in the same point using all the elements that are sharing it. The aim of these variants is to find a well-conditioned system of equations with a structure that makes its solution particularly efficient when the number of variables is rather large. 

The finite difference method can be obtained as follows. Define in the domain a number of grid points and replace (approximate) the Laplace equation by a finite difference expression. When a Dirac delta weight function is used at each grid point i, one forces the finite difference trial to be zero at each point. Therefore, the finite difference method (FDM) can be considered as point collocation method. 

If we use the one internal collocation point approximation for the diffusion reaction equation inside the floe as shown in the next section, then the left-hand side of Eq. (6.133) can be approximated using the orthogonal collocation formula (refer to Appendix E for orthogonal collocation method), and, thus, Eq. (6.133) becomes... 

The second step in this equation involves a property called Green s identity. Using either method brings one to the point where the solutions of both require the same basic approaches solving a matrix problem. As in the case of collocation, the L sample points are used to generate the rows of the A matrix and b vector whose elements are written m,k = y) (x, y) dx dy and = b x, y)[Pg.257]

Although additional radial collocation points increase the dimensionality of the resulting model, they may be necessary to accurately express the radial concentration profiles. Preliminary analysis in this section considers only one interior radial concentration collocation point, although a detailed analysis of this assumption is presented in Section VI,E. 

Further analysis shows that, for this particular example, the fastest modes (group I) correspond directly to the gas temperatures at the interior collocation points and those of groups II and III correspond to the concentrations. One should not infer from this conclusion that a general correspondence between the dynamical modes and the physical variables always exists. In this example, the correspondence results from the major differences in the magnitudes of the various groups. 

This problem was solved using a one-dimensional RFM with 200 collocation points, evenly distributed along the x-axis using Algorithm 16. Figure 11.6 compares the analytical solution with computed RBFM solutions up to Pe= 100. As can be seen, even for convection dominated cases, the technique renders excellent results. It is important to point out here that for the radial functions method no up-winding or other special techniques were required. 

For this type of sampling, the collection of additional sample volumes for MS/ MSD analysis is not possible. The liners, however, usually contain enough soil for the analysis of the sample itself and for MS/MSD analysis. In this case, we designate one of the field samples as MS/MSD on the COC form. Collocated field duplicates are usually collected as the middle and the lower liners from the same depth interval or occasionally by placing another borehole next to the location of the primary sampling point. 

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