Explicit finite difference

A numerical method to simulate the performance of the storage with the PCM module was implemented using an explicit finite-difference method. The discretization of the model can be seen in Figure 145. 

By adding the forward (explicit) finite-difference approximation to each side of this equation, we can identify both the explicit Euler algorithm and an expression for the local truncation error ... 

This revision does not attempt to take many of these recent advances into account, even though some of them are cited in this chapter. Rather, it continues to provide a rigorous foundation for writing programs that will perform explicit finite difference simulations. In learning how to do this, the reader develops an appreciation of the method and, more importantly, its limitations. 

Write a short explicit finite difference program to compute the fiber orientation function in a compression molding process with stretching only in the x-direction. Test the program with an example, where a 3 mm thick 50 x 50 cm plate is compression molded with an initial mold coverage of 50%. Assume an interaction coefficient C/ = 0.05. Assume initial fiber orientation distribution that is random. 

The effectiveness factors and n, defined as the ratios of the actual reaction rates at time 0 to the maximum reaction rates on a clean catalyst, are obtained nEmerically from equations [4] -[9]. An explicit finite difference method was used to solve the partial differential equations without further simplifications. Densities, porosities and clean catalyst pore diameters were measured experimentally. The maximum coke content is assumed to be that which fills the pore completely. The tortuosity is taken as 2.3, as discussed by Satterfield et al. (14). 

The finite difference approximations can either be applied to the derivatives on the line from which the solution is advancing or on the line to which it is advancing, the former giving an explicit finite difference scheme and the latter an implicit scheme. The type of solution procedure obtained with the two schemes is illustrated in Fig. 3.18. 

An explicit finite-difference procedure will be used here in dealing with the momentum and energy equations. Consider the nodal points shown in Fig. 4.27. It will be seen that a uniform grid spacing is used in the -direction. 

Because an explicit finite-difference procedure is being used to solve the momentum and energy equations, the solution can become unstable, i.e., as the solution proceeds it can diverge increasingly from the actual solution as indicated in Fig. 4.29. 

Approximate solutions for the two limiting cases discussed above can be obtained (see below). However, most real flows are not well described by either of these two limiting solutions. For this reason, a numerical solution of the governing equations must usually be obtained. To illustrate how such solutions can be obtained, a simple forward-marching, explicit finite-difference solution will be discussed here. 

As previously mentioned, the solution to the above set of equations will here be obtained using a forward-marching, explicit finite-difference procedure. The solution starts with the known conditions on the inlet plane and marches forward in the -direction from grid line to grid line as indicated in Fig. 8.19. Consider the nodal points shown in Fig. 8.20. 

In descriptions of this problem, the names of Randles [460] and Sevclk [505] are prominent. They both worked on the problem and reported their work in 1948. Randles was in fact the first to do electrochemical simulation, as he solved this system by explicit finite differences (and using a three-point current approximation), referring to Emmons [218]. Sevclk attempted to solve the system analytically, using two different methods. The second of these was by Laplace transformation, which today is the standard method. He arrived at (9.116) and then applied a series approximation for the current. Galus writes [257] that there was an error in a constant. Other analytical solutions were described (see Galus and Bard and Faulkner for references), all in the form of series, which themselves require quite some computation to evaluate. 

Finite difference — Finite difference is an iterative numerical procedure that has been used to quantify current-voltage-time relationships for numerous electrochemical systems whose analyses have resisted analytic solution [i]. There are two generic classes of finite difference analysis 1. explicit finite difference (EFD), where a new set of parameters at t + At is computed based on the known values of the relevant parameters at t and 2. implicit finite difference (IFD), where a new set of parameters at t + At is computed based on the known values of the relevant parameters at t and on the yet-to-be-determined values at t + At. EFD is simple to encode and adequate for the solution of many problems of interest. IFD is somewhat more complicated to encode but the resulting codes are dramatically more efficient and more accurate - IFD is particularly applicable to the solution of stiff problems which involve a wide dynamic range of space scales and/or time scales. 

Eqs. (H) through (L) will be solved by the explicit finite difference method. Substitute... 

Equation (7) can most readily be solved by an explicit finite difference scheme which steps forward in time across the spatial grid. The value of G is updated at each spatial grid point in turn. When all of the spatial grid has been updated a solution at that point in time has been calculated for the problem considered. 

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... 

Schematic for the explicit finite difference formulation of the convection condition at the left boundary of a plane wall.

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... 

I or example, in the case of transient one-dimensional heat conduction in a plane wall with specified surface temperatures, the explicit finite difference equations for all the nodes (which are interior nodes) are obtained from Eq. 5-47. The coefficient of TjJ, in the T expression is 1 - 2t, which is independent of the node number / , and thus the stability criterion for all nodes in this case is 1 — 2t s 0 or... 

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,... 

Substituting this value of t and other quantities, the explicit finite difference Equations (a) and (b) reduce to... 

We number the nodes as O, 1, 2, 3, 4, and 5, with node 0 on the interior surface of the Trombe wall and node 5 on the exterior surface, as shown in Figure 5-47, Nodes 1 through 4 are interior nodes, and the explicit finite difference lormulations of these nodes are obtained directly from Eq. 5 47 to be... 

The exterior surface of the Trombe v/ail is subjected to convection as well as to heat flux. The explicit finite difference formulation at that boundary is obtained hy writing an energy balance on the volume element represented by node 5,... 

C 1 he explicit finite difference formulation of a general interior node for transient two-dimensional heat conduction is given by... 

S-73 Consider transient heat conduction in a plane wall whose left surface (node 0) i.s maintained at. >0°C while the tiglil surface (node 6) is subjeeted to a solar heal flux of 600 W/m. The wall is initially at a uniform temperature of 50°C. Express the explicit finite difference fomiulalion of the boundary nodes 0 and 6 for the case of no heal generation. Also, obtain die finite difference formulaiioti for the total amount of heat transfer at the left boundary during the first three lime steps. 

Consider transient heat conduction in a plane wall with variable heal generation and constant thermal conductivity. The nodal network of (he medium consists of nodes 0, 1, 2, 3, and 4 with a uniform nodal spacing of A.r. The wall is initially at a specified temperaWre. The temperature at the right bound ary (node 4) is specified. Using the energy balance approach, obtain the explicit finite difference formulation of the boundary... 

Starling with an energy balance on a volume element, obtain the two-dimensional transient explicit finite difference equation for a general interior node in rectangular coordinates for T x, y, t) for the case of constant thermal conductivity and no heal generation. 

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