Orthogonal collocation example

Parabolic Equations m One Dimension By combining the techniques apphed to initial value problems and boundary value problems it is possible to easily solve parabolic equations in one dimension. The method is often called the method of lines. It is illustrated here using the finite difference method, but the Galerldn finite element method and the orthogonal collocation method can also be combined with initial value methods in similar ways. The analysis is done by example. 

The simultaneous solution strategy offers several advantages over the sequential approach. A wide range of constraints may be easily incorporated and the solution of the optimization problem provides useful sensitivity information at little additional cost. On the other hand, the sequential approach is straightforward to implement and also has the advantage of well-developed error control. Error control for numerical integrators (used in the sequential approach) is relatively mature when compared, for example, to that of orthogonal collocation on finite elements (a possible technique for a simultaneous approach). 

It should be noted that since the mathematical description of the packed bed reactor consists of three dimensions, one does not need to select a single technique suitable for the entire solution but can choose the best technique for reduction of the model in each of the separate dimensions. Thus, for example, orthogonal collocation could be used in the radial dimension where the... 

Although in this chapter we have chosen to linearize the mathematical system after reduction to a system of ordinary differential equations, the linearization can be performed prior to or after the reduction of the partial differential equations to ordinary differential equations. The numerical problem is identical in either case. For example, linearization of the nonlinear partial differential equations to linear partial differential equations followed by application of orthogonal collocation results in the same linear ordinary differential equation system as application of orthogonal collocation to the nonlinear partial differential equations followed by linearization of the resulting nonlinear ordinary differential equations. The two processes are shown ... 

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. 

FD methods are point approximations, because they focus on discrete points. In contrast, finite element methods focus on the concentration profile inside one grid element. As an example of these segment methods, orthogonal collocation on finite elements (OCFE) is briefly discussed below. 

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 will illustrate the application of orthogonal collocation to a number of examples. Elliptic PDEs will be dealt with first and typical parabolic equations occurring in chemical engineering will be considered next. 

We saw in the last example for the elliptic PDE that the orthogonal collocation was applied on two spatial domains (sometime called double collocation). Here, we wish to apply it to a parabolic PDE. The heat or mass balance equation used in Example 11.3 (Eq. 11.55) is used to demonstrate the technique. The difference between the treatment of parabolic and elliptic equations is significant. The collocation analysis of parabolic equations leads to coupled ODEs, in contrast to the algebraic result for the elliptic equations. 

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...

The orthogonal collocation on finite elements can also be applied to partial differential equations as straightforward as we did in the last example for ODE. If the partial differential equations are linear, the resulting set of equations will be a set of coupled linear ordinary differential equations. On the other hand, if the equations are nonlinear, the discretized equations are coupled nonlinear ordinary differential equations. In either case, these sets of coupled ordinary differential equations can be solved effectively with integration solvers taught in Chapter 7. More of this can be found in Finlayson (1980). 

This cooling of fluid in a pipe with wall heat transfer resistance was solved by Michelsen (1979) using the method of orthogonal collocation. This problem without the wall resistance is a special case of the situation dealt with by Michelsen, and is often referred to as the Graetz problem (see Example 10.3 and Problem 3.4). 

Example 10.2 considers the modeling of a CYD process in a parallel fiat plate system. The entrance length problem was analytically dealt with by the method of combination of variables. Here, we assume that the chemical reaction at the plate follows a nonlinear reaction, and apply the orthogonal collocation to solve this problem numerically. 

This chapter discusses finite-difference techniques for the solution of partial differential equations. Techniques are presented for pure convection problems, pure diffusion or dispersion problems, and mixed convection-diffusion problems. Each case is illustrated with common physical examples. Special techniques are introduced for one- and two-dimensional flow through porous media. The method of weighted residuals is also introduced with special emphasis given to orthogonal collocation. 

For nonlinear reaction kinetics, a numerical solution of the balance Equation 4.121 is carried out. For example, for second-order kinetics, R = kcACB, with an arbitrary stoichiometry, the generation rate expressions, ta = —va CaCb and tb = —vb caCb, are inserted into the mass balance expression, which is solved numerically using, for example, a polynomial approximation (orthogonal collocation method). The performances of the normal dispersion model and its segregated or maximum-mixed variants are compared in Figure 4.34. The symbols are explained in the figure. The comparison reveals that the differences between the segregated, maximum-mixed, and normal axial dispersion models are notable at moderate Damkohler numbers R = Damkohler number). 

The balance equations for column reactors that operate in a concurrent mode as well as for semibatch reactors are mathematically described by ordinary differential equations. Basically, it is an initial value problem, which can be solved by, for example, Runge-Kutta, Adams-Moulton, or BD methods (Appendix 2). Countercurrent column reactor models result in boundary value problems, and they can be solved, for example, by orthogonal collocation [3]. The backmixed model consists of an algebraic equation system that is solved by the Newton-Raphson method (Appendix 1). 

With this technique, the partial differential equations can be reduced to a set of ordinary differential equations (or a set of algebraic equations). The following example will illustrate the orthogonal collocation technique for the initial-value problem. 

Example 2.1 Solve the differential Equation 2.15 using the orthogonal collocation technique for different numbers of collocation points and compare it with the analytical solution in the range 0 to 0.125. 

Example 5.5 Solution of the Optimal Temperature Profile for Penicillin Fermentation. Apply the orthogonal collocation method to solve the two-point boundary-value problem arising from the application of the maximum principle ofPontryagin to a batch penicillin fermentation. Obtain the solution of this problem, and show the profdes of the state variables, the adjoint variables, and the optimal temperature. The equations that describe the state of the system in a batch penicillin fermentation, developed by Constantinides et al.(6], are ... 

The DVR is related to, but distinct from pseudo-spectral and collocation methods of solving differential equations. For the DVR there is an orthogonal transformation which defines die relation of die DVR to the finite basis representation (FBR). > Thus, for example, the Hermidan character of operators remains obvious in the DVR. Both pseudo-spect and collocation methods, however, use a "mixed" representation operators and, as such, do not display the Hermitian character of operators such as H. Thus the advantages of the DVR are that the accuracy is that of a Gaussian quadrature and it is a true representation, while the collocation methods permit more freedom in the choice of points, a distinct advantage in some multidimensional problems. 

There are a number of different kinds of orthogonal polynomials one can use, including continuous polynomials like Lagrange or Legendre polynomials [22], [21], and discrete ones, such as Hahn s polynomial [23]. The orthogonality property allows one to obtain the roots of the polynomial Xi, i =, 2,..., m — 1. Since orthogonal polynomials are also formed by linear combination of x or (for simplicity we can take the example of polynomials in x), Equation 2.30 can be rewritten at each collocation point in terms of new coefficients di... 

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