Collocation axial

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... 

Since more than one axial collocation point is generally necessary. 

Although only 6 axial collocation points are used in most simulations, 12 points were necessary here and even then the results are less than optimum. 

Figure 26 shows the predicted axial gas temperature profiles during reactor start-up for standard type I conditions with varying numbers of axial collocation points. Eight or more axial collocation points provide similar results, and even simulations with six collocation points show minimal inaccuracy. However, reducing the number of collocation points below this leads to major discrepancies in the axial profiles. 

Figure 5. Comparison of measured profiles interpolated at the collocation points (left) and calculated profiles (right). Ranges of variables are the same as in Figure 3. Key a, time profiles for temperature and concentration at axial collocation points and b, time profiles for radial collocation points.

The results of the steady-state model for the reactor under the same operating conditions are displayed as the solid lines in Figure 2. The predicted catalyst and gas temperatures are shown at each of the axial collocation points. As discussed earlier, a priori values of kinetic parameters were used ( 1, 2) similarly, heat and mass transfer parameters (which are listed in Table II) were taken from standard correlations (15, 16, 17) or from experimental temperature measurements in the reactor under non-reactive conditions. The agreement with experimental data is encouraging, considering the uncertainty which exists in the catalyst activity and in the heat transfer parameters for beds with such large particles. 

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. 

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. 

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. 

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. 

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. 

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... 

The algebraic equations for the orthogonal collocation model consist of the axial boundary conditions along with the continuity equation solved at the interior collocation points and at the end of the bed. This latter equation is algebraic since the time derivative for the gas temperature can be replaced with the algebraic expression obtained from the energy balance for the gas. Of these, the boundary conditions for the mass balances and for the energy equation for the thermal well can be solved explicitly for the concentrations and thermal well temperatures at the axial boundary points as linear expressions of the conditions at the interior collocation points. The set of four boundary conditions for the gas and catalyst temperatures are coupled and are nonlinear due to the convective term in the inlet boundary condition for the gas phase. After a Taylor series expansion of this term around the steady-state inlet gas temperature, gas velocity, and inlet concentrations, the system of four equations is solved for the gas and catalyst temperatures at the boundary points. 

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. 

For the nonlinear case, the nonlinear two-point boundary value differential equation(s) for the catalyst pellet can be solved using the same method as used for the axial dispersion model in Section 5.1, i.e., by the orthogonal collocation technique of MATLAB s bvp4c. m boundary value solver. 

Discretization of the partial differential equation system in axial (z) and radial (r) direction by means of the orthogonal collocation method (7) leads to the following system of ordinary differential equations. 

Comparison of Measured and Calculated Profiles. In order to compare the measured time profiles shown in Figure 3 with the calculated time profiles, the former were axially and radially interpolated to obtain the corresponding profiles at the collocation points. Figures 5.a) and b) show the measured (I) and calculated (II) time profiles for the axial and radial collocation points, respectively. 

A model for transient simulation of radial and axial composition and temperature profiles In pressurized dry ash and slagging moving bed gasifiers Is described. The model Is based on mass and energy balances, thermodynamics, and kinetic and transport rate processes. Particle and gas temperatures are taken to be equal. Computation Is done using orthogonal collocation In the radial variable and exponential collocation In time, with numerical Integration In the axial direction. 

When radial dispersion is included, even the steady state equations are partial differential equations — in the axial and radial space variables. The dispersion model equations can be numerically solved by finite-difference schemes, or more efficiently, by orthogonal collocation methods (14, 15). 

The orthogonal collocation method using piecewise cubic Her-mite polynomials has been shown to give reasonably accurate solutions at low computing cost to the elliptic partial differential equations resulting from the inclusion of axial conduction in models of heat transfer in packed beds. The method promises to be effective in solving the nonlinear equations arising when chemical reactions are considered, because it allows collocation points to be concentrated where they are most effective. 

In the case of a breakthrough curve for a binary mixture (step injection), Liapis and Rippin [32,33] used an orthogonal collocation method to calculate numerical solutions of a kinetic model including axial dispersion, intraparticle diffusion, and surface film diffusion, and assuming constant coefficients of diffusion and... 

Beste et al. [104] compared the results obtained with the SMB and the TMB models, using numerical solutions. All the models used assumed axially dispersed plug flow, the linear driving force model for the mass transfer kinetics, and non-linear competitive isotherms. The coupled partial differential equations of the SMB model were transformed with the method of lines [105] into a set of ordinary differential equations. This system of equations was solved with a conventional set of initial and boundary conditions, using the commercially available solver SPEEDUP. Eor the TMB model, the method of orthogonal collocation was used to transfer the differential equations and the boimdary conditions into a set of non-linear algebraic equations which were solved numerically with the Newton-Raphson algorithm. 

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