Current approximation 3- point

Remark 2 The right-hand sides of the first three sets of constraints are the support functions that are represented as outer approximations (or linearizations) at the current solution point xk of the primal problem. If condition Cl is satisfied then these supports are valid underestimators and as a result the relaxed master problem provides a valid lower bound on the global solution of the MINLP problem. 

In this way, the coefficients for any y((n) can be calculated. Table A.l in Appendix A shows them all, as whole numbers m/3j, where m is the multiplier mentioned above. For each n, the Table shows forward differences (at index 1), backward derivatives (at index n) and derivatives applying at points between the two ends. For n up to 6, all possible forms are included, as they will be needed later, while for n = 7, only the forward and backward formulae are shown, as only these are needed. In case the reader wonders why all this is of interest the forms y[(n) will be used to approximate the current in simulations (see the next section) the backward forms y n(n) will be used in the section on the BDF method in Chaps. 4 and 9, and the intermediate forms shown in the Table will be used for the Kimble White (high-order) start of the BDF method, also described in these chapters. The coefficients have a long history. Collatz [169] derived some of them in 1935 and presents more of them in [170]. Bickley tabulated a number of them in 1941 [88], The three-point current approximation, essentially y((3) in the present notation, was first used in electrochemistry by Randles [460] (preempted by two years by Eyres et al. [225] for heat flow simulations), then by Heinze et al. [301], and schemes of up to seven-point were provided in [133]. 

One further new notation is useful here, used by Bieniasz. A given current approximation is denoted as n(m), where n is the number of points used to approximate it, and m is the order with respect to FI, the intervals in X. Thus, the formulae used so far make, for example, G2 a 2(1) form and G3 a 3(2) form. It will be seen that we can easily obtain, for example, 2(3) and 3(4), etc. 

For a small number n of points, it may be worthwhile using the algebraic solutions for the coefficients. The procedure is as described above, but instead of inserting actual hk values into the matrix in (3.48), that matrix is inverted algebraically and the coefficients expressed as a genera] formula. These are given, for a few approximations, both for first and second derivatives (restricted to those that are deemed to be of practical interest) in Appendix A. All the current approximations up to n = 4 are provided there, as well as... 

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. 

Here, the algorithm described in Chap. 3, Sect. 3.8, implemented in the very general subroutine U DERIV referred to in Appendix C can provide both the derivatives on an arbitrarily spaced set of points (.r, u). However, the reader may wish to restrict the expressions to those involving only up to four points (for which there are some good arguments, see Chap. 8, Sect. 8.4). This can be coupled with current approximations using up to four points. For this number of points, the expressions are not unreasonably long, and a few useful ones are therefore presented here. 

U DERIV can be used to compute Co, given the current (as in chronopoten-tiometry) and the concentration profile. As for equal intervals, the current approximation formula on n points is adapted, by the function CU. 

Figure 23.2 The general polarization curve of the catalyst layer with ideal feed transport. Solid lines, analytical solutions for the low- and high-current regimes points, the exact numerical solution to the system [Eqs.(23.6) and (23.7)] dashed line, the approximate analytical curve, Eq. (23.11),...

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. 

If the simple two-point current approximation is used, there is no problem we know the interval H to the first concentration point C away from the electrode, and the usual formula is used. If, however, we wish to use a higher-order expression, it is not so straight-forward. We have two choices to develop an n-point scheme in X-space, taking into account the actual X values corresponding to Cj, C2, / -1 compute (3C/3Y)q in Y-space, where we have equal intervals and transform this to (3C/3X)q. The latter is, of course, the easier method we have... 

As mentioned above in Sect. 6.2, use of the 5- or 6-point current approximation will eliminate the error from this source for all practical purposes that is, use Eq. 4.86 to compute G and Eq. 4.93 to compute Cq from a known concentration profile. The expressions for n = 5 or 6 are not too unwieldy and, in any case, if they are used routinely, it makes sense to write function procedures to evaluate them. In our laboratory we use library functions GOFUNC and COFUNC, with variable n (see Chapt. 9 for the FORTRAN code). 

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. 

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