The point method

The expressions just derived in Sects. 3.1.2 and 3.1.3 are now used. Let us take the second diffusion equation of Fick in one (space) dimension. 

The unknown, variable c, is a function of the two variables t and x and we can represent it graphically as in Fig. 3.6 as values in the (x,t) plane. We now sample this c function at a grid of points in the plane, spaced at distances h apart in the x-direction and 6t in the t-direction. We give the index k to time (t = kSt) and the index i to x (x ih). Assume that c values for all x up to time k6t are known, and the next row, for t+St = (k+l)St, is desired. We focus on the marked trio of points i+1 ich will generate the value cf. We must 

The right-hand side, second derivative, using Eq. 3.13, becomes 

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 

With the point method, what happens at the electrode At x = 0, we have the non-diffusionally derived Cq value. The first point affected by diffusion is c, one h-interval from the electrode. Here, the trio of points Cq, and C2 produce the new point c exactly as in Eq. 3.26, 

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The discussion will be restricted to the point method and to the onedimensional case. We will now work in normalised variables, see Sect. 2.3. We then have concentration points Cq, C, ... C/y, Cqv+i, at the locations... 

Consideration of (9.9) reveals that it is of the same form as that shown for the point method using arbitrarily spaced points, (8.8) in Chap. 8. We can proceed from here in the same way as in that chapter. That is, all methods described there (or even the explicit method) can be applied. By dividing (9.9) by St, we can even go into a MOlMype method (see below). There is thus no need to describe the procedure further from here. 

Box method Chap. 9, Sect. 9.1. The original electrochemical simulation method. With boxes, most of the above techniques can be applied. There is an unresolved issue of whether this method is inherently better than the point method, or not. 

As noted in the following sections, for both the ranking method and the factor comparison method, external and internal factors are incorporated throughout the job-evaluation process. In the classification method and the point method, internal factors and external factors are considered separately at first and are later reconciled with each other. In the point method, for example, point totals denoting relative internal worth can be reconciled with market data through statistical procedures such as regression analysis. 

With the point method, as with all job-evaluation plans, the first step is job analysis. The next steps are to choose the factors, scale them, establish the factor weights, and then evaluate jobs. 

Compensable factors play a pivotal role in the point method. In choosing factors, an organization must decide What factors are valued in our jobs What factors will be paid for in the work we do Compensable factors should possess the following characteristics ... 

In the point method, each job s relative value, and hence its location in the pay structure, is determined by the total points assigned to it. A job s total point value is the sum of the numerical values for each degree of compensable factor that the job possesses. In Table 5, the point plan has four factors skills required, effort required, responsibility, and working conditions. There are five degrees for each factor. 

TABLE 5 The Point Method of Job Evaluation Factors, Weights, and Degrees... 

M. Rudolph. Digital simulations on unequally spaced grids. Part 1. Critical remarks on using the point method by discretisation on a transformed grid, J. Electroanal. Chem. 529, 97-108 (2002). 

One last example should suffice to illustrate the way the point method can be used. If we take the diffusion equation in cylindrical coordinates and add a homogeneous first-order chemical reaction. 

The reader should now have a good idea of how the various forms of the diffusion equation are discretized, using the point method. 

This is again seen to be almost identical with the box result (Eq. 3.35) which has, in effect, a backward difference for 3c/3r. As suggested previously (Britz 1980), the above comparison provides rather strong argument in favour of the point method - it is faster to use and leaves no doubts. The argument will be further strengthened when we come to convection tetms (Chapt. 8). 

There are two ways at arriving at these approximations Taylor expansions and polynomials. They give the same results but the former method also gives an idea of errors. The following treatment has been given previously (Britz 1987). We start with the point-method distribution. 

The method can also be used in the box-mode, expressing it in terms of the point method with a i h-shift towards the electrode. The conversion is obvious. 

For a X value of 0.4, commonly used, we then get T 0.25T. A similar calculation for the point method gives 6T/nX or 0.86T. In effect, we are starting the simulation not at T 0 but at these (calculable) times. The actual value depends on the simulation technique and the current approximation used. In neither the box- or point case, though, is the starting time equal to i ST. 

Firstly, the question of box vs. point the end-element difficulty seen in the box method is not shared by the standard point method, and discretisation is always straightforward with points. Clearly, the point method with the concentration sample distribution 0, H, 2H,. .. is to be preferred. 

Using the point method (unlike Feldberg 1980), we define the usual set of concentrations Cq, C, . .., with Cq at the (moving) drop surface (R = Rq). The point spacing SR = H is conveniently kept the same at all T, so we have a point grid tied at one end to the moving dme surface. Eq. 8.4 or 8.8 is then discretised with no special problems (in contrast with the box method, where shell-elements, shrinking with time, are used). 

For theoretical references on the rde, see Levich (1962) and Albery and Hitchman (1971). Digital simulation studies include the classical work of Prater and Bard (1970), Clarenbach et al (1973) and, more recently, Feldberg (1980), to name but a few. These all used the box method, while Hoyer and Kryger (1985) used the point method. 

Britz D (1980) The point method for electrochemical digital simulation. Anal Chim Acta 122 331. 

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