Collocation orthogonal

The orthogonal collocation method uses a series representation for the potential and current and solves for the coefficients of the series by satisfying the governing equations with boundary conditions at each fitting point. Again, treatment of nonconnected regions is difficult, and some judgment and experience is usually required of the user. 

Comparison between common techniques for solving the Laplace 

Applicability to arbitrary boundaries Poor Good Best 

FDM finite-difference method, FEM finite-element method, BEM boundary-element method 

Numerical solutions for the current distribution that have been presented in the literature may be divided into three classes (1) dedicated code that has been developed to simulate a specific class of problems, (ii) commercially available software, dedicated to simulating electrochemical systems, and (iii) general scientific CAD software that can be customized for solving electrochemical systems. Each class is briefly discussed below  

When using the method of weighted residuals, the most important factor is the choice of the trial function. Important guidelines come from the boundary conditions and the symmetrj of the problem. A further extension of the method is choosing collocation points as roots of orthogonal polynomials. This is known as the method of orthogonal collocation. The basic property of any orthogonal polynomial is 

This method involves solving the diffusion equation  

In fact, this sets the weighted residual equal to zero  

The choice of the weighting function determines the method. In the collocation method, the weighting function is chosen to be a displaced dirac function  

SO the residual is zero at specified points The choice of these points can make 

These equations can be evaluated at all AM-2 collocation points. Equations (58) and (59) can be expressed in a simpler matrix notation, where the — superscript represents a one-dimensional matrix and the = superscript represents a two-dimensional matrix. 

---

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. 

Other methods can be used in space, such as the finite element method, the orthogonal collocation method, or the method of orthogonal collocation on finite elements (see Ref. 106). Spectral methods employ Chebyshev polynomials and the Fast Fourier Transform and are quite useful for nyperbohc or parabohc problems on rec tangular domains (Ref. 125). 

Packages exist that use various discretizations in the spatial direction and an integration routine in the time variable. PDECOL uses B-sphnes for the spatial direction and various GEAR methods in time (Ref. 247). PDEPACK and DSS (Ref. 247) use finite differences in the spatial direction and GEARB in time (Ref. 66). REACOL (Ref. 106) uses orthogonal collocation in the radial direction and LSODE in the axial direction, while REACFD uses finite difference in the radial direction both codes are restricted to modeling chemical reactors. 

Boundary value methods provide a description of the solution either by providing values at specific locations or by an expansion in a series of functions. Thus, the key issues are the method of representing the solution, the number of points or terms in the series, and how the approximation converges to the exact answer, i.e., how the error changes with the number of points or number of terms in the series. These issues are discussed for each of the methods finite difference, orthogonal collocation, and Galerkin finite element methods. 

Orthogonal Collocation The orthogonal collocation method has found widespread application in chemical engineering, particularly for chemical reaction engineering. In the collocation method, the dependent variable is expanded in a series of orthogonal polynomials. See "Interpolation and Finite Differences Lagrange Interpolation Formulas. ... 

This problem can be solved using a combined optimization and constraint model solution strategy (Muske and Edgar, 1998) by converting the differential equations to algebraic constraints using orthogonal collocation or some other model discretization approach. 

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). 

Villadsen, J., and Stewart, W. E. (1967). Solution of boundary-value problems by orthogonal collocation. Chem. Eng. Sci. 22, 1483-1501. 

In case an analytical solution of Eqs. (6) and (7) is not available, which is normally the case for non-linear isotherms, a solution for the equations with the proper boundary conditions can nevertheless be obtained numerically by the method of orthogonal collocation [38,39]. 

Using piecewise constant control profiles and orthogonal collocation on finite elements, this approach was further developed by Renfro (Renfro, 1986 Renfro et al, 1987) to deal with much larger problems. More recent simultaneous applications that involve SQP, orthogonal collocation, and piecewise constant control profiles have been presented by Patwardhan et al (1988) for online control, and by Eaton and Rawlings (1988) for optimization of batch crystallizers. These studies have shown that simultaneous approaches can be applied successfully to small-scale applications with complex constraints. 

Note that state variable profiles are one order higher than the controls because they have explicit interpolation coefficients defined at the beginning of each element. With this representation of Z(t) and U(t), we can extend this approach to piecewise polynomials and apply orthogonal collocation on NE finite elements (of length Aoc,). This leads to the following nonlinear algebraic equations ... 

For simultaneous solution of (16), however, the equivalent set of DAEs (and the problem index) changes over the time domain as different constraints are active. Therefore, reformulation strategies cannot be applied since the active sets are unknown a priori. Instead, we need to determine a maximum index for (16) and apply a suitable discretization, if it exists. Moreover, BDF and other linear multistep methods are also not appropriate for (16), since they are not self-starting. Therefore, implicit Runge-Kutta (IRK) methods, including orthogonal collocation, need to be considered. 

Index 1 problems two-point orthogonal collocation (4-stable) ... 

Biegler, L. T., Solution of dynamic optimization problems by successive quadratic programming and orthogonal collocation, Comp, and Chem. Eng. 8(3/4), 243-248 (1984). 

One of the most populax numerical methods for this class of problems is the method of weighted residuals (MWR) (7,8). For a complete discussion of these schemes several good numerical analysis texts are available (9,10,11). Orthogonal collocation on finite elements was used in this work to solve the model as detailed by Witkowski (12). 

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. 

A significant step in the numerical solution of packed bed reactor models was taken with the introduction of the method of orthogonal collocation to this class of problems (Finlayson, 1971). Although Finlayson showed the method to be much faster and more accurate than that based on finite differences and to be easily applicable to two-dimensional models with both radial temperature and concentration gradients, the finite difference technique remained the generally accepted procedure for packed bed reactor model solution until about 1977, when the analysis by Jutan et al. (1977) of a complex butane hydrogenolysis reactor demonstrated the real potential of the collocation procedure. 

Orthogonal collocation Leads directly to state-space representation specified over the domain May lead to control modeling difficulties since inputs affect all states immediately. 

Orthogonal collocation Can account for sharp Can have problems with... 

Another potential solution technique appropriate for the packed bed reactor model is the method of characteristics. This procedure is suitable for hyperbolic partial differential equations of the form obtained from the energy balance for the gas and catalyst and the mass balances if axial dispersion is neglected and if the radial dimension is first discretized by a technique such as orthogonal collocation. The thermal well energy balance would still require a numerical technique that is not limited to hyperbolic systems since axial conduction in the well is expected to be significant. 

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... 

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. 

The orthogonal collocation method has several important differences from other reduction procedures. Jn collocation, it is only necessary to evaluate the residual at the collocation points. The orthogonal collocation scheme developed by Villadsen and Stewart (1967) for boundary value problems has the further advantage that the collocation points are picked optimally and automatically so that the error decreases quickly as the number of terms increases. The trial functions are taken as a series of orthogonal polynomials which satisfy the boundary conditions and the roots of the polynomials are taken as the collocation points. A major simplification that arises with this method is that the solution can be derived in terms of its value at the collocation points instead of in terms of the coefficients in the trial functions and that at these points the solution is exact. 

Since the resulting system after radial collocation is still too complex for direct mathematical solution, the next step in the solution process is discretization of the two-dimensional system by orthogonal collocation in the axial direction. Although elimination of the spatial derivatives by axial collocation increases the number of equations,8 they become ordinary differential equations and are easily solved using traditional techniques. Since the position and number of points are the only factors affecting the solution obtained by collocation, any set of linearly independent polynomials may be used as trial functions. The Lagrangian polynomials of degree N based on the collocation points... 

Figure 10b shows that CO conversion is much higher under adiabatic operation due to the higher bed temperatures. Note that the conversion of the C02 becomes important as soon as the CO is nearly depleted. The rippling in the C02 curve is a result of the axial orthogonal collocation.14 Numerical solution problems such as this will be discussed in Section VII. 

---