Explicit finite difference method

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. 

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). 

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

Considering both radiation and convection heal transfers and using the explicit finite difference method with a mesh size of iiv - 2.5 cm and a time step of dr = 5 min, determine the temperatures of tlie inner and outer surfaces of the roof at 6 am. Also, determine the average rale of heat transfer through the roof during that night. 

A.r = 2 cm and a time step to be Ar = 0.5 s, determine the nodal temperatures after 5 rain by using (he explicit finite difference method. Also, determine how long it will take for steady conditions to be reached. 

The explicit, finite difference method (9,10) was used to generate all the simulated results. In this method, the concurrent processes of diffusion and homogeneous kinetics can be separated and determined independently. A wide variety of mechanisms can be considered because the kinetic flux and the diffusional flux in a discrete solution "layer" can be calculated separately and then summed to obtain the total flux. In the simulator, time and distance increments are chosen for convenience in the calculations. Dimensionless parameters are used to relate simulated data to real world data. 

Initially (time zero) the values of c are known and therefore the values of cj (concentrations at t = At) can be calculated by direct application of (25.93). This approach can be repeated and the values of c"+1 can be calculated from the previously calculated values of c . This is an example of an explicit finite difference method, where, if approximate solution values are known at time t = nAt, then approximate values at time tn+1 = (n+ 1) At may be explicitly and immediately calculated using (25.93). Typically, explicit techniques require that constraints be placed on the size of Af that may be used to avoid significant numerical errors and for stable operation. In a stable method, unavoidable small errors in the solution are suppressed with time in an unstable method a small initial error may increase significandy, leading to erroneous results or a complete failure of the method. Equation (25.93) is stable only if At < (Ax)2/2, and therefore one is obliged to use small integration timesteps. 

In Fig. 9.4 we plot the corresponding bifurcation diagram. A forward or supercritical bifurcation occurs at T = L. We depict with symbols the values obtained by integrating (9.1) numerically using an explicit finite difference method with a... 

Output from Program 2 for the Explicit Finite-Difference Method... 

There are many numerical approaches one can use to approximate the solution to the initial and boundary value problem presented by a parabolic partial differential equation. However, our discussion will focus on three approaches an explicit finite difference method, an implicit finite difference method, and the so-called numerical method of lines. These approaches, as well as other numerical methods for aU types of partial differential equations, can be found in the literature [5,9,18,22,25,28-33]. 

It was stated earlier that one of the principles of the explicit finite difference method was that all the various parts of the electrochemical process can be treated sequentially. Thus for systems involving homogeneous reactions each iteration involves first a diffusional part as described above and then a kinetic part. To see what form this kinetic part takes let us consider the ece process described below ... 

This is how the explicit finite difference method works in solving numerically partial fferential equations of parabolic type, as those expressing the Pick s second law. 

Feldberg, Stephen W. (1990) A fast quasi-explicit finite difference method for simulating... 

Seeber, R. and Stefani, S. (1981) Explicit finite difference method in simulating electrode processes, Anal.Chem. 53, 1011-1016. 

Feldberg, S.W. (1990) A fast quasi-explicit finite difference method for simulating electrochemical phenomena. Part 1. Application to cyclic voltammetric problems, J. Electroanal. Chem. 290,49-65. 

In the past, Matsue et al. have discussed this issue in a study related to the characterization of diaphorase-pattemed surfaces by SECM. Using a digital simulation based on the explicit finite difference method that considered the heterogenous enzyme reaction at the substrate, they generated steady-state current vs. distance profiles that depended on the surface concentration of the enzyme. Using these working curves, they quantified the surface concentration of the active immobilized diaphorase (134). Using, the electrochemical... 

Although diffusion and chemical reaction are concurrent processes in the explicit finite difference method, they are calculated separately. This procedure. 

Today, there is hardly a paper written in electrochemistry, that does not casually mention the use, in some way, of a computer (or several) or hardly an electrochemist not routinely using one (or more). This was not the case in 1964, when Feldberg and Auerbach published their first paper on digital simulation. At that time, computers were much slower, more expensive to use and the electrochemists using them were a minority. This partly explains the anomaly of the box method the explicit finite difference method had been known since at least 1928 (Courant et al) the Crank-Nicolson improvement since 1947 (and it was immediately used by an electrochemist (Randles, 1948)) but Feldberg developed his particular style independently. 

Marques da SUva B, Avaca LA, Gonzalez ER (1989) New explicit finite difference methods in the digital simulation of electrochemical problems. J Electroanal Chem 269 1-14... 

In this first phase of the work the explicit finite difference method of Binder-Schmidt was applied W other methods being examined are the method of Dufort and Frankel (9) and orthogonal collocation (10). 

Another approach to TG/SC experiments does not rely on the mediator feedback [56]. The reactant galvanostatically electrogenerated at the tip diffuses to the substrate and undergoes the reaction of interest at its surface. The substrate current is recorded as a function of either time or the tip/ substrate separation distance (approach cnrves). The theory for transient responses, steady-state TG/SC approach curves, and polarization cnrves (i.e., 4 vs. E ) was generated solving the diffnsion problem numerically (an explicit finite difference method was used). The substrate process was treated as a first-order irreversible reaction, and the effects of its rate constant and the experimental parameters were illnstrated by families of the dimensionless working curves (Figure 5.11). 

For numerical analysis, the governing equations were discretized using the explicit finite difference method [2]. FIGURE 3 illustrates all the governing equations and boundary conditions used with respect to their location in the one dimensional analysis. These equations were then used iteratively to simulate the thermal cycle in the injection molding process. The final thermal profile for each iteration was used as the initial condition for the next cycle. 

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