The finite difference equation

For the introduction and explanation of the method we will discuss the case of transient, geometric one-dimensional heat conduction with constant material properties. In the region x0 x xn the heat conduction equation 

A grid is established along the strip x0 x xnl t t0, with mesh size Ax in the x-direction and At in the f-direction, Fig. 2.43. A grid point (i,k) has the coordinates 

We refrain from indicating the approximation value of the temperature by a different symbol to the exact temperature, e.g. 0 instead of as is usually the case in mathematics literature. The time level tk is indicated by the superscript k without brackets because there is no danger of it being confused with the fc-th power of d. 

The derivatives which appear in (2.236) are replaced by difference quotients, whereby a discretisation error has to be taken into account 

The discretisation error goes to zero with a reduction in the mesh size Ax or At. 

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Since this equation introduces a new variable, Cq, another equation is needed and is obtained by writing the finite difference equation for = 1, too. 

If average diffusion coefficients are used, then the finite difference equation is as follows. 

The most general method of tackling the problem is the use of the finite-element technique 8 to determine the temperature distribution at any time by using the finite difference equation in the form of equation 9.40. 

This then provides a physical derivation of the finite-difference technique and shows how the solution to the differential equations can be propagated forward in time from a knowledge of the concentration profile at a series of mesh points. Algebraic derivations of the finite-difference equations can be found in most textbooks on numerical analysis. There are a variety of finite-difference approximations ranging from the fully explicit method (illustrated above) via Crank-Nicolson and other weighted implicit forward. schemes to the fully implicit backward method, which can be u.sed to solve the equations. The methods tend to increase in stability and accuracy in the order given. The difference scheme for the cylindrical geometry appropriate for a root is... 

Smith and Brinkley developed a method for determining the distribution of components in multicomponent separation processes. Their method is based on the solution of the finite-difference equations that can be written for multistage separation processes, and can be used for extraction and absorption processes, as well as distillation. Only the equations for distillation will be given here. The derivation of the equations is given by Smith and Brinkley (1960) and Smith (1963). For any component i (suffix i omitted in the equation for clarity)... 

The general solution of the finite differences equation (eq 29) is given by... 

We will discuss the solution of steady-state and unsteady-state heat conduction problems in this chapter, using the finite-difference method.. The finite-difference method comprises the replacement of the governing equations and corresponding boundary conditions by a set of algebraic equations. The discussion here is not meant to be exhaustive in its mathematical rigor. The basics are presented, and the solution of the finite-difference equations by numerical methods are discussed. The solution of convection problems using the finite-difference method is discussed in a later chapter. 

In this explicit scheme, the first-order forward difference approximation is used for the time derivative. The second-order central difference approximation is used for the spatial derivatives. Hence, the finite difference equation (FDE) of the partial differential equation (PDE) Eq. (10.2) is... 

It is observed that the above finite difference scheme is implicit if p> Vi. The finite difference equation (10.41) may be used as the continuity equation for both the fiilly implicit and the explicit methods. 

The conditions at nodal point lying at the intersection of the i and j grid lines are denoted by the subscript i, y. In order to derive the finite difference equations, attention is given to conditions at the four grid points shown in Fig. 3.20. 

Consider the boundary conditions on the end walls. When the temperature gradient is zero on the end walls, i.e., when the end walls are adiabatic, since to the same order of accuracy as used in deriving the finite-difference equations ... 

Chemical reactions couple the matrix equations for each species so they cannot be solved independently. The easy way around this is to approximate the kinetic terms explicitly (using concentrations at the old time), for example in an ECE mechanism species C is made from species B. The finite difference equation for species C could therefore use the concentration of species B from the previous time step as in (115). 

The Neumann boundaries (which involve derivatives) are converted into finite difference form and substituted into the finite difference equations for the nodes in the specified region (e.g. above the electrode surface). The Dirichlet conditions (which fix the concentration value) may be substituted directly. 

The finite difference equations will be formulated with five radial increments m = 0, 1, 2, 3, 4, 5, and for as many axial Increments as necessary to obtain 50 percent conversion. Accordingly,... 

The differential equation is valid at every point of a medium, whereas the finite difference equation is valid at discrete points (the nodes) only. 

Above we have developed a general relation for obtaining the finite difference equalion for each interior node of a plane wall. This relation is not applicable to the nodes on the boundaries, however, since it requires the presence of nodes on both sides of the node under consideration, and a boundary node does not have a neighbor ing node on at least one side, Therefore, we need to obtain the finite difference equations of boundary nodes separately. This is best done by applying an energy balance on the volume elements of boundary nodes. 

Note that thermodynamic temperatures must be used in radiation heat transfer calculations, and ail temperatures should be expressed in K or R when a boundary condition involves radiation to avoid mistakes. We usually try to avoid the radiation boundary condition even in numerical solutions since it causes the finite difference equations to be nonlinear, wlu ch are more difficult to solve. 

The finite difference equation for the boundary node 5 is obtained by writing an energy balance on the volume element of length Ax/2 at that boundary, again by assuming heat transfer to be into the medium at all sides (Fig. 5-21) ... 

When there is no heat generation in the medium, the finite difference equation for interior node further Simplifies to + Ti + T ghi +... 

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