Finite differencing

The concentration gradient may have to be approximated in finite difference terms (finite differencing techniques are described in more detail in Secs. 4.2 to 4.4). Calculating the mass diffusion rate requires a knowledge of the area, through which the diffusive transfer occurs, since... 

The main process variables in differential contacting devices vary continuously with respect to distance. Dynamic simulations therefore involve variations with respect to both time and position. Thus two independent variables, time and position, are now involved. Although the basic principles remain the same, the mathematical formulation, for the dynamic system, now results in the form of partial differential equations. As most digital simulation languages permit the use of only one independent variable, the second independent variable, either time or distance is normally eliminated by the use of a finite-differencing procedure. In this chapter, the approach is based very largely on that of Franks (1967), and the distance coordinate is treated by finite differencing. 

Figure 4.1. Finite-differencing a tubular reactor with the stepwise approximation of the...

Figure 4.3. Finite-differenced equivalent of the depth of solid.

The concentration gradient terms, dC/dZ, both in and out of segment n, can be approximated by means of their finite-differenced equivalents. Substituting these into the component balance equation, gives... 

The unsteady model, originally formulated in terms of a partial differential equation, is thus transformed into N difference differential equations. As a result of the finite-differencing, a solution can be obtained for the variation with respect to time of the water concentration, for every segment, throughout the bed. 

Figure 4.11. Finite-differencing for a dynamic tubular reactor model.

A similar finite-differenced equivalent for the energy balance equation (including axial dispersion effects) may be derived. The simulation example DISRET involves the axial dispersion of both mass and energy and is based on the work of Ramirez (1976). A related model without reaction is used in the simulation example FILTWASH. 

Thus the system is defined by two coupled partial differential equations, which can be solved by finite-differencing. 

Figure 4.27. Finite-differencing of heat exchanger length.

Simplification gives the resulting finite-differenced model equations as... 

As shown in this chapter, in the simulation of systems described by partial differential equations, the differential terms involving variations with respect to length are replaeed by their finite-differenced equivalents. These finite-differenced forms of the model equations are shown to evolve as a natural eonsequence of the balance equations, according to the manner of Franks (1967). The approximation of the gradients involved may be improved, if necessary, by using higher order approximations. Forward and end sections can... 

Thus the problem involves the two independent variables, time t and length Z. The distance variable can be eliminated by finite-differencing the reactor length into N equal-sized segments of length AZ such that N AZ equals L, where L is the total reactor length. 

UNSTEADY-STATE FINITE DIFFERENCED MODEL REACTOR LENGTH DIVIDED INTO 8 EQUAL ELEMENTS... 

In these equations the designation for dimensionless concentration c, has been dropped. Note that in the above equation, the finite differencing of the convection term has been done over two neighbouring segments. Again special relationships apply to the end segments, owing to the absence of axial dispersion, exterior to the cake. 

Figure 5.236. Finite differencing the heat exchanger for dynamic modelling.

This example involves the same diffusion-reaction situation as in the previous example, ENZSPLIT, except that here a dynamic solution is obtained, using the method of finite differencing. The substrate concentration profile in the porous biocatalyst is shown in Fig. 5.252. 

Using finite differencing techniques for any given element n, these relations... 

Figure 5.255. The finite differencing of the spherical bead geometry.

Chapter 4 eoncerns differential applications, which take place with respect to both time and position and which are normally formulated as partial differential equations. Applications include diffusion and conduction, tubular chemical reactors, differential mass transfer and shell and tube heat exchange. It is shown that such problems can be solved with relative ease, by utilising a finite-differencing solution technique in the simulation approach. 

Central differences were used in Equation (5.8), but forward differences or any other difference scheme would suffice as long as the step size h is selected to match the difference formula and the computer (machine) precision with which the calculations are to be executed. The main disadvantage is the error introduced by the finite differencing. 

UNSTEADY-STATE FINITE-DIFFERENCED MODEL FOR AXIAL DISPERSION AND REACTION IN A TUBULAR REACTOR ... 

Finite-differencing the dimensionless length of the reactor (Z = l) into N equal-sized segments of length AZ, such that N AZ = L, gives for segment n... 

A consideration of axial dispersion is essential in any realistic description of extraction column behaviour. Here a dynamic method of solution is demonstrated, based on a finite differencing of the column height coordinate. Figure 1 below shows the extraction column approximated by N finite-difference elements. 

For solution by digital simulation the depth of the filter cake is finite differenced, each element having a dimensionless thickness Ax. For any element n the resulting difference differential equation is given by... 

Fig. 1 Chromatographic column showing finite differencing into column segments.

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