Finite differencing techniques

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... 

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

Oxygen, substrate and biomass are all transported by diffusion within the liquid phase contained in the aggregate. The modelling of this process is achieved via the use of a finite differencing technique. In this, the spherical aggregate is divided into a number of shells, as seen in Fig. 1. 

Prominent representatives of the first class are predictor-corrector schemes, the Runge-Kutta method, and the Bulir-sch-Stoer method. Among the more specific integrators we mention, apart from the simple Taylor-series expansion of the exponential in equation (57), the Cayley (or Crank-Nicholson) scheme, finite differencing techniques, especially those of second or fourth order (SOD and FOD, respectively) the split-operator, method and, in particular, the Chebychev and the shoit-time iterative Lanczos (SIL) integrators. Some of the latter integration schemes are norm-conserving (namely Cayley, split-operator, and SIL) and thus accumulate only... 

Many practical engineering problems involve not just a single variable and a single partial differential equation but a set of coupled partial differential equations. The extension of the finite differencing technique to coupled partial differential equations of the initial value, boundary value type is relatively straightforward and will be briefly discussed before developing code for solving such systems of equations. 

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. 

The mass-weighted Cartesian coordinates are denoted Using Eq. (3), the numerical value of the Blk matrix elements are determined for any value of the internal coordinates via a finite difference scheme. Once the B-matrix is constructed, expansions for Gtj and V, about the equilibrium configuration of the molecule, can be evaluated using higherordering differencing techniques. 

Although the application of quasilinearization requires a detailed and complicated derivation of equations, the basic technique is straightforward. First, one must consider a finite-differenced solution space and a marching-like solution. Here, the solution domain is of two dimensions dimensionless distance from 0 to 1 and open-ended time from the initial... 

The method of lines (14) is used as the numerical technique In this method, by "finite differencing" the space variable (here axial length of reactor), the reactor is divided into a number of cells. Then the partial differential equations are converted into ordinary differential equations where time is the only independent variable. Each cell corresponds to a continuous stirred tank reactor. 

For the centered-difference technique, the finite-difference analogs are written about the half point (z + 1/2, n -I-1/2). The reason for this is that this centered differencing develops second-order correct analogs for both the variable and its first derivatives. To get the first-order time partial, we develop Taylor series expansions for the functions u ti, Xn+112) and Xn+1/2) about the half point ( 1+1/2, a +i/2) then we have... 

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