Finite-difference formulation

Equations (4.12) and (4.13) now have to be integrated forward in time subject to the appropriate boundary conditions such as Eq. (4.16). Finite-diflerence methods are used. At t = 0, sigmoid-shaped starting profiles of species concentrations and enthalpy must be set up to represent the transfer from cold to hot boundary conditions. These are input as vectors O of values of each 

At the start of the integration, the specific enthalpy is assumed constant over the whole flame. For interconversion between enthalpy and temperature, the program must be provided with polynomial coefficients which allow the molar enthalpies of the pure components to be expressed as functions of temperature (Chapter 8). If the molar enthalpy of component i is expressed as 

The temperature at each grid point may be calculated by solution of the polynomial expression. The mixture density may then be found from the equation of state (2.25). 

The numerical integrations are performed by means of finite-difference techniques using a rectangular grid in the (co, t) plane having (nonuniform) intervals Sco and St, We denote by the vector of dependent variables at grid point n St)J S(o), The equations to be integrated [(4.12) and (4.13)] may then be written as the matrix equation 

The finite-difference integration at time n (St) involves computing the quantities d /dt = y by evaluating the remaining terms in 

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For many condensed media, the Mie-Gruneisen equation of state, based on a finite-difference formulation of the Gruneisen parameter (4.18), can be used to describe shock and postshock temperatures. The temperature along the isentrope (Walsh and Christian, 1955) is given by... 

The commercial CFD codes use the finite volume method, which was originally developed as a special finite difference formulation. The numerical algorithm consists of the following steps ... 

The difference between the two values results from the inaccuracy in determination of T4. Using the exact solution value of 207.017°C would give a heat loss of 6.2827 kW. For this problem the exact value of heat flow is 6.283 kW because the heat generation calculation is independent of the finite difference formulation. 

Derive a finite difference formulation for a steady-state reaction-diffusion system. 

Next, we develop the finite difference formulation of heat conduction problems by replacing the derivatives in the differential equations by differences. In the following section we do it using the energy balance method, which does not require any knowledge of differential equations. 

The heat conduction equation involves the second derivatives of temperature with respect to the space variables, such as d T/rfx , and the finite difference formulation is based on replacing the second derivatives by appropriate... 

Note that the boundary conditions have no effect on the finite difference formulation of interior nodes of the medium. This is not surprising since the control volume used in the development of the formulation does not involve any part of the boundary. You may recall that the boundary conditions had no effect on the differential equation of heat conduction in the medium either. 

The finite difference formulation above can easily be extended to two- nr threc-dimen.sinnal heat transfer problems by replacing each second derivative by a difference equation in that direction. For example, the finite difference fomiulalion for steady two-dimensional heat conduction in a region with... 

The finite difference formulation is given above to demonstrate how difference equations ate obtained from differential equations. However, we use the energy balance approach in the following section.s to obtain the numerical formulation because it is more intuitive and can handle boundary conditions more easily. Besides, the energy balance approach does not require having the differential equation before the analysis. 

In finite difference formulation, the temperature is assumed to vary linearly between the nodes. 

The assumed direction of heat transfer at surfaces of a volume element has no effect on the finite difference formulation. 

FIGURE 5-13 Finite difference formulation of specified temperature boundary conditions on both surfaces of a plane wall. 

FIGURE 5-14 Schematic for the finite difference formulation of the left boundary. node of a plane wall. 

One way of obtaining (he finite difference formulation of a node on an insulated boundary is to treat insulation as "zero heat flux and to write an energy balance, as done in Eq. 5 -23. Another and more practical way is to treat the node on an insulated boundary as an imerior node. Conceptually (his is done... 

Node 1 is an interior node, and the finite difference formulation at that node is obtained directly from Eq. 5-18 by setting m = 1 ... 

That is, the finite difference formulation of an interior node is obtained by adding the temperaiures of the four nearest neighbors of the node, subtructing four limes the temperature of the node itself and adding the heat generation lerm. It can also be expressed in this form, which is easy to remember ... 

The development of finite difference formulation of boundary nodes in two- (or three-) dimensional problems is similar to the development in the one-dimensional case discussed earlier. Again, the region is partitioned between the nodes by forming volume elements around the nodes, and an energy balance is written for each boundary node. Various boundary conditions can be handled as discussed for a plane wall, except that the volume elements ill the two-dimensional case involve heat transfer in the y-direction as well as the x-direction. Insulated surfaces can still be viewed as mirrors, and the... 

A) Node 8. This node is identical to node 7, and the finite difference formulation of this node can be obtained from that of node 7 by shifting the node numbers by 1 (i.e., replacing subscript m by rr + 1). It gives... 

This completes the development of finite difference formulation for this problem. Substituting the given quantities, the system of nine equations for the determination of nine unknown nodal temperatures becomes... 

Finite difference formulation of time-dependent problems involves discrete points in time as well as space. 

Note that the left side of this equation is simply the fiiiile difference formulation of the problem for the steady case. This is not surprising since the formulation must reduce to the steady case for = Tj,. Also, we are still not committed to explicit or implicit formulation since we did not indicate the time step on the left side of the equation. Wc now obtain the explicit finite difference formulation by expressing the left side at time step i as... 

Noie that in the case of no heal generation and t = 0.5, the explicit finite difference formulation for a general interior node reduces to T , = (T/,-1 +, )/2, which has the interesting interpretation that the temperature... 

To gain a better understanding of the stability criterion, consider the explicit finite difference formulation for an interior node of a plane wall (Eq. 5 47) for the case of no heat generation,... 

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