The box method

This method, pioneered (in electrochemistry) by Feldberg (Feldberg and Auerbach 1964, Feldberg 1969) has been called the model approach by Abraham and Tiller (1972) the worker thinks in terms of the physical model itself, and uses, in the case of diffusion, only the more or less self-evident first equation of Fick (Eq. 2.2). This has the advantage that we do not lose sight of what we are simulating it can in theory easily adapt itself to all sorts of geometric complications and we never need to write down a partial differential equation. 

It is developed as follows imagine, as shown in Fig. 3.2, a thin, one-dimensional piece of solution of cross-sectional area A, divided into segments of equal length h, and consider three adjacent box elements numbered i-1, i and i+1, with corresponding concentrations as marked. These concentrations can be considered uniform within each element. What happens to c during the small time interval St Element i adjoins element i-1 and i+1 and there may be flow of substance into it as well as out of it. These fluxes can be computed using Eq. 2.2 (flux is 

Now the amount of substance, n, added to element i during the time interval 6t is simply 

Element 1 is bounded at the left by the electrode, where the (boundary) concentration is Cq. There are various physical models proposed for handling this, such as a fictitious element behind the electrode or an infinitely thin one right at the electrode these are really not needed. If we just assume the value Cq (obtained from some other, non-diffusional, calculation), we can postulate for the concentration gradient between electrode and element 1 (always at the centre of the element)  

We are now in a position to write a small computer program fragment for the above process. Only the innermost loop is shown - that is, just one time step. The above symbols c become C in the program, assumed to be 

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Often the inlet device (air supply) in a ventilated room is geometrically complicated. To resolve the flow around such a device would require a very fine grid. Instead of trying to resolve the complex flow near the inlet device, one can choose to use the box method or the prescribed velocity method.Both methods are based on the observation that downstream of the inlet, the flow behaves like a wall jet. Thus it is important that the bound-... 

Nielsen, P. V. The box method—A practical procedure for introduction of an air terminal device in CFD calculation. Report R9744, Dept, of Building Technology and Structural Engineering, Aalborg University, Aalborg, 1997. 

The low value of 1.0 in Table VII for the rigid Polyurethane foam in the box-methods may be explained by low ventilation rate. The experiment is over in less than 1 minute and most of the material is combusted during that period. The ventilation rate becomes to low to generate a "normal" ratio of C02/C0 for fast burning materials. In this particular case the oxygen concentration become very close to 0 oxygen. 

An adaptation of the Box method, however, seems to offer the advantage of improved efficiency while still being susceptible to automatic computation. Box s approach may be divided into two stages. The first, to which he has applied the name method of steepest ascents, is primarily for the purpose of approximately locating the optimum response. The second is a more intensive investigation in the local region of the optimum. This will permit a precise determination of the optimum and also indicate the behavior of the response in its neighborhood. 

FDM was applied to electrochemical problems very early [4], but it was in the 1960s when Feldberg developed the basis of digital simulation of electrochemical processes by means of the box method, which at present is considered as a FEM-like method (see [5]). 

Equation (2.2) is the only equation needed when using the box method and this is sometimes cited as an advantage. It brings one close to the microscopic system, as we shall see, and has - in theory - great flexibility in cases where the diffusion volume has an awkward geometry. In practice, however, most geometries encountered will be - or can be simplified to - one of but a few... 

A stretched stack of boxes was used by Feldberg [231] for the box-method, to be described in Chap. 9. Pao and Dougherty [433] developed the same idea... 

Some families combine traditional measurements with less conventional thinking out of the box methods. 

The box method will get brief treatment here but the accent is now on points. They make advanced methods (see Sect. 1.3.9), such as the implicit methods that have made possible great advances in efficiency, easier to implement. 

In this section, a very brief description of the box method is given. This is the method as originally devised by Feldberg [1]. It is, in the opinion of the present author, awkward and outdated, although many prefer it because it appears... 

To reduce memory, the box method is abandoned. Similar to that of FE, only the solid elements are recorded, ignoring the vacuum elements. The topology relationship is stored in the variable of each element. This algorithm is not used only in temperature calculation it will be even more helpful in microstructure calculation in further works. 

Several different methods have been used to solve the partial differential equations describing the diffusion processes diat take place in a cyclic voltammetric experiment [1, 4 - 6]. The use of methods that discretize the space coordinates separately from the time coordinate and then solve the corresponding ordinaiy differential equations, like the box method, have the distinct advantage, over the finite difference methods, of requiring much less conq)uter memory, and thus allowing the convenient use of a personal computer. 

We have used the box method and focused our attention in the choice of a suitable numerical integration scheme taking into account its stability, computation time and error. The number of boxes used was twenty with a fixed length, except for tte first one that had half the length of the other ones to increase accuracy in the computation of the current. 

Burning the base of the trunks was the first method used in North America to promote gum exudation by trees, followed by the boxing method, whereupon a cavity called box was cut at the base of trees and above it streaks were made, both highly destructive methods for the trees [2, 78]. Nevertheless, techniques of resin stimulation used worldwide currently (French, Chinese, Mazek, and American) [79] are variations of the boxing method briefly described above. 

Physical approaches not reqniring the numerical solution of the differential equations have also been developed. For example, an atomistic model considers the cell as a domain filled with a popnlation of particles and diffusion is simulated by the random walk of the particles within the domain (53, 54). The current is computed by counting the number of particles that reach the electrode per unit time. Convection and migration can even be included. Another model, the box method nsed in the early days of electrochemical simulation (55), divides the solntion in thin slabs (boxes wherein the concentration is assumed to be uniform) and calculates the movement of species between slabs nsing Fick s first law of diffusion. Althongh more intuitive, these approaches are in fact eqnivalent to solving the transport eqnation. 

The three points thus give us a new point at the next time. Compare this with the box-method expression, Eq. 3.5 they are identical. In fact, one might say that the box-method derives Fick s second diffusion equation in discrete form. Although Eqs. 3.5 and 3.26 are identical, it is clear that the point method derivation is much faster with only a little practice, the discretisation formulae 3.10 to 3.13 are easily memorised and expressions like Eq. 3.26 can be written down straight from the diffusion equation. Furthermore, it will be seen that if the... 

This is identical with the equation for box 1, using the box method, Eq. 3.9 note also that the points in Fig. 3.8 are spaced the same as the box centres in Fig. 3.3. Why should one do this It will be seen in the chapter on accuracy, that some believe this to produce more accurate results. It will be argued in that chapter that the expression is, in fact, wrong, despite the good results from it. It is for those who consider the results as evidence that it is correct, that we present it here. It is, of course, possible to use the correct expression (to be developed in the later chapter) with the h grid shift - there is an argument that a point close to the electrode means a more accurate current approximation and thus better simulation results. All this is to be discussed. 

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. 

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