Orthogonal collocation on finite elements

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

15 Due to the relatively small radius of the well, the thermal capacitance Vpcp of the bed with the thermal well (16 cal/K) is only slightly higher than that without (14 cal/K) and the loss in reaction volume is only about 2%. 

Two major forms of the OCFE procedure are common and differ only in the trial functions used. One uses the Lagrangian functions and adds conditions to make the first derivatives continuous across the element boundaries, and the other uses Hermite polynomials, which automatically have continuous first derivatives between elements. Difficulties in the numerical integration of the resulting system of equations occur with the use of both types of trial functions, and personal preference must then dictate which is to be used. The final equations that need to be integrated after application of the OCFE method in the axial dimension to the reactor equations (radial collocation is performed using simple orthogonal collocation) can be expressed in the form 

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

16 Note that z in this expression is the normalized axial coordinate (equivalent to )  

If we now introduce a new vector Y, a new global indexing scheme evolves for the purpose of programming as 

Equations 12.232 can be solved by any of the numerical schemes presented in Chapter 7. After the vector Y is known, the mean concentration defined in Equation 12.219 can be written as the quadrature 

Plots of the mean concentration versus time for a- = 0.001 and /x = 0.01 and 0.1 are shown in Fig. 12.10 along with a comparison of the singular perturbation solution. 

The previous section showed how straightforward the orthogonal collocation can be when solving partial differential equations, particularly parabolic and elliptic equations. We now present a variation of the orthogonal collocation method, which is useful in solving problems with a sharp variation in the profiles. 

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. 

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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). 

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). 

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. 

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). 

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

Fig. 15. Comparison of profiles computed by orthogonal collocation and orthogonal collocation on finite elements.

Regardless of whether orthogonal collocation or orthogonal collocation on finite elements is used for the discretization, the resulting linear state-space... 

Among the variety of methods which have been proposed for simulation of packed bed dynamics three techniques have been used with success (1) Crank-Nicholson technique [10], (2) transformation to integral equation [11], (3) orthogonal collocation on finite elements [12]. In the following computation, we have used the Crank-Nicholson method with the nonequidistant space steps in the Eigenberger and Butt version [10]. 

A general computational scheme using orthogonal collocation on finite elements has been developed for calculation of rates of mass transfer accompanied by single or multistep reactions. The method can be used to predict enhancement in absorption or desorption rates for a wide class of industrially important situations. 

Approximate vs. Numerical Solution. The accuracy of the approximate reaction factor expression has been tested over wide ranges of parameter values by comparison with numerical solutions of the film-theory model. The methods of orthogonal collocation and orthogonal collocation on finite elements (7,8) were used to obtain the numerical solutions (details are given by Shaikh and Varma (j>)). Comparisons indicate that deviations in the approximate factor are within few percents (< 5%). It should be mentioned that for relatively high values of Hatta number (M >20), the asymptotic form of Equation 7 was used in those comparisons. 

The solution to (P12) gives us the optimal separation profile as a function of age within the reactor. However, except in the case of reactive phase equilibrium, the assumption of a continuous separation profile is not really required. Furthermore, a continuous separation profile may not be implementable in practice. To address this, we take advantage of the structure of a discretization procedure for the differential equation system. In this case, we choose orthogonal collocation on finite elements to discretize the above model. This results... 

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. 

A variety of methods can be used to derive numerical solutions of Eq. 10.61. These methods include mainly finite-difference methods and methods of orthogonal collocation on finite elements. We discuss briefly these methods, the properties of the solutions obtained, and some of the problems of numerical analysis encotmtered in the development and use of algorithms for the computation of solutions of Eq. 10.61 [49,50]. 

Figure 10.12 Comparison of the numerical solutions of the equations of the equilibrium-dispersive model calculated with the forward-backward algorithm (dashed line) and the method of orthogonal collocation on finite elements (dotted line). 250 iL injection of a 15 g/L solution of (+)-Tr6ger s base. Column length, 25 cm column efficiency, Np = 146 plates F = 0.515 u = 0.076 cm/s. Isotherms in Figure 3.25...

In spite of some awkwardness in its formulation, the forward-backward scheme of numerical integration of the ideal model (Eq. 10.79) seems the most efficient way of calculating the band profiles of the equilibrium-dispersive model. It is particularly effective in terms of use of CPU time and is especially suitable for theoretical investigations of optimization strategies [10]. The best alternative procedure is not another finite difference scheme but one using orthogonal collocation on finite elements [9]. This procedure is more accurate but requires a much longer... 

Band Profiles Calculated with Orthogonal Collocation on Finite Elements... 

Many authors have described procedures for the calculation of numerical solutions of the general rate model of chromatography with a variety of initial and boundary conditions corresponding to practically all the modes of chromatography (with the notable exception of system peaks). Orthogonal collocation on finite elements seems to be the most popular approach for these calculations. 

The GRM using the generalized Maxwell-Stefan equations has no closed-form solutions. Numerical solutions were calculated using a computer program based on an implementation of the method of orthogonal collocation on finite elements [29,62,63]. The set of discretized ordinary differential equations was solved with the Adams-Moulton method, implemented in the VODE procedure [64]. The relative and absolute errors of the numerical calculations were 1 x 10 and 1 x 10 , respectively. 

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)... 

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