Radial collocation

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. 

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. 

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... 

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... 

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. 

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... 

Note that i = 1, 2,..., N for all of the equations except the last, where i = 1,2,..., N + 1 that the o)t coefficients are the result of radial collocation and that all other dimensionless parameters are defined as shown previously. Furthermore, the reaction terms R M and R s are calculated at the conditions at the collocation points. 

Fig. 8. Gas temperature and CO and C02 mole fractions (a) multipoint radial concentration collocation, type I conditions (b) gas temperatures at the radial temperature collocation point.

Simulations can be performed using additional radial collocation points for the concentration profiles and compared to those using the earlier model. In the simulations, the bulk concentration can then be obtained by integrating the radial profiles ... 

Simulations show that the radial and axial temperature and bulk concentration profiles are effectively not influenced by these modeling differences. Figure 9 shows the radial concentration profiles at = 0.38 and at the reactor outlet. Even with very high Peclet numbers, the differences between the radial concentration profile across the relatively small bed and the assumed uniform profile are minimal. Under typical operating conditions with small Peclet numbers, there is no benefit to increasing the number of radial collocation points, especially in light of the increased dimensionality of the resulting system. 

Fig. 9. Radial mole fraction profiles, multipoint radial collocating, type I conditions ------,...

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... 

All temperatures in this equation are in terms of the conditions at the radial collocation point. 

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 aim of this part of the book is to present the main and current numerical techniques that are used in polymer processesing. This chapter presents basic principles, such as error, interpolation and numerical integration, that serve as a foundation to numerical techniques, such as finite differences, finite elements, boundary elements, and radial basis functions collocation methods. 

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,... 

In collocation methods, different sets of basis functions can also be used, for example radial basis functions, which are functions that depend only on the euclidean distance between collocation points i and j, ri j. 

This allows us to represent partial differential equations as found in the balance equations using the collocation method. Equation (11.47) is a solution to a partial differential equation represented by a system of linear algebraic equations, formed by the interpolation coefficients, oij, and the operated radial functions. The interpolation coefficients are solved for using matrix inversion techniques to approximately satisfy the partial differential equation... 

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. 

O.A. Estrada, I.D. Lopez-Gomez, and T.A. Osswald. Modeling the non-newtonian calendering process using a coupled flow and heat transfer radial basis functions collocation method. Journal of Polymer Technology, 2005. 

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