Difference formulae for partial differential equations

Note that the outlet approximations must be consistent with a final steady-state heat balance. Note also that is easy to allow in the simulation for variations in the heat transfer coefficient, density and specific heats as a function of temperature. The modelling methods demonstrated in this section are applied in the simulation example HEATEX. 

As shown in this chapter for the simulation of systems described by partial differential equations, the differential terms involving variations with respect to length are replaced by their finite-differenced equivalents. These finite-differenced forms of the model equations are shown to evolve as a natural consequence of the balance equations, according to Franks (1967), and as derived for the various examples in this book. The approximation of the gradients involved may be improved, if necessary, by using higher-order approximations. Forward and end-sections can be better approximated by the forward and backward differences as derived in the previous examples. The various forms of approximation based on the use of central, forward and backward differences have been listed by Chu (1969). 

Central difference as extensively used in this chapter 

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Deactivating catalyst 319 Dead zones 159, 162, 163 Degree of segregation 471 Density influences 492 Desorption of solute 578, 579 Difference differential equation 579 Difference formulae for partial differential equations 268 Differential column 167... 

The partial differential equations describing the catalyst particle are discretized with central finite difference formulae with respect to the spatial coordinate [50]. Typically, around 10-20 discretization points are enough for the particle. The ordinary differential equations (ODEs) created are solved with respect to time together with the ODEs of the bulk phase. Since the system is stiff, the computer code of Hindmarsh [51] is used as the ODE solver. In general, the simulations progressed without numerical problems. The final values of the rate constants, along with their temperature dependencies, can be obtained with nonlinear regression analysis. The differential equations were solved in situ with the backward... 

Method of Lines. The method of lines is used to solve partial differential equations (12) and was already used by Cooper (I3.) and Tsuruoka (l4) in the derivation of state space models for the dynamics of particulate processes. In the method, the size-axis is discretized and the partial differential a[G(L,t)n(L,t)]/3L is approximated by a finite difference. Several choices are possible for the accuracy of the finite difference. The method will be demonstrated for a fourth-order central difference and an equidistant grid. For non-equidistant grids, the Lagrange interpolation formulaes as described in (15 ) are to be used. 

The original system of partial differential equations is transformed into a system of ordinary differential equations by replacing the time differential terms with time finite difference formulas. The number of equations in the new system is the same as the original number of equations. However, it is necessary to store intermediate results at spatial nodes for both current and previous time increments. 

The mathematical model forms a system of coupled hyperbolic partial differential equations (PDEs) and ordinary differential equations (ODEs). The model could be converted to a system of ordinary differential equations by discretizing the spatial derivatives (dx/dz) with backward difference formulae. Third order differential formulae could be used in the spatial discretization. The system of ODEs is solved with the backward difference method suitable for stiff differential equations. The ODE-solver is then connected to the parameter estimation software used in the estimation of the kinetic parameters. More details are given in Chapter 10. The comparison between experimental data and model simulations for N20/Ar step responses over RI1/AI2O3 (Figure 8.8) demonstrates how adequate the mechanistic model is. 

Most textbook discussions of relaxation methods for solving partial differential equations use the familiar second-order form for the Laplacian. Finite difference Laplacian representations result from Taylor series (i.e., polynomial) expansions of a function centered on the grid point x. We give the prescription for the second-order formula and then it is apparent how to proceed to higher orders. Expand the function in the positive and negative directions to fourth order ... 

Fiiiite Differences. Many of the equations used in numerical analysis contain ordinary or partial derivatives. One of the most important techniques used in numerical analysis is to replace the derivatives in an equation, or system of equations, with equivalent finite differences of the same order, and then develop an iterative formula from the equation. For example, in the case of a first-order differential equation with an initial value condition, such as/(x) = F[x,/( )],... 

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