Expanded grid

Finally, as mentioned earlier (Chap. 7, Sect. 7.2), Martinez-Ortiz [385] developed some rather simple formulae for derivative approximations for the special case of exponentially expanding grid spacings, and in the course of this work discovered that the four-point second-order derivative approximation u"(T), for the expansion factor 7 = /2 is third-order in accuracy, rather than second, as it is for other 7 values. This could be an easy and useful way to increase the accuracy, using the four-point formula. 

The problem of thin reaction layers are described sufficiently in Chap. 5. The solution is to use unequal intervals, that is, a few very small intervals near the electrode, so that there are sample points within the thin profile. This can be done up to a point by a fixed unequal grid such as the exponentially expanding grid described in Chap. 7. A more flexible approach is the moving adaptive grid also described in that chapter. This problem is thus solved and needs no further attention here. 

For simulation, a finite difference-based approach with expanding grid has been employed. 

The method can be easily extended to the case of breakage, and we refer the reader to the specialized literature for details. It is important to report also the many efforts to reduce and optimize the computational costs by using, for example, expanding grids, as illustrated in the work of Chiney et al. (2010). 

Grid pasting. Paste of the required density and consistency is spread on to the grids by means of a specially designed machine. When expanded grids are used, the paste is applied to the expanded strip and then both sides of the pasted strip are eovered with paper, after whieh the individual plates are separated. The pasted plates are partially dried in a tunnel oven and are then stacked on pallets. 

It has been established that on cycling of tubular positive plates with die-cut strap grids (SGTP) or of positive plates with expanded grids with flat ribs, a rapid capacity loss is observed (the PCL-1 effect, see Section 2.3, Chapter 2 and Chapter 9) [54]. The reason for this capacity loss is the formation of groups of PbS04 crystals in the layer of the PAM that contacts the current-collector (Fig. 3.39). These PbS04... 

Several approaches have been suggested for the more demanding problems, and they are described in detailed reviews (1,7, 8). One can abandon the model of boxes with equal widths and use space elements of variable dimensions (8, 28, 29). An exponentially expanding grid is frequently used, where the width of a box, Ax(y) depends on j ... 

Frequently, j8 is taken as 0.5. When jS 0, a uniform grid results. The effect of the expanding grid can be taken into account by writing (B.1.7) as (29)... 

Schematics of continuous process for manufacture of plates with expanded grids [99].

Fig. 4.1. Concentration profile of species A in the Cottrell experiment at T = Tmax-The solid line corresponds to the profile obtained with a uniform grid with h = 0.0001 the circle points indicate the position of the spatial nodes when an expanding grid is used with h = 0.0001 and ujx = 1-1. AT = 0.01. The value AT/VTinax is chosen as x-axis since this gives an estimation of the distance relative to the linear diffusion layer thickness X/ /Tmax = xly D t max ...

Exponentially expanding grids are widely employed in the simulation of electrochemical experiments such that the distance between consecutive points of the grid expands exponentially according to the following definition ... 

The distribution of the points in the grid can be adjusted by means of the parameters h = ho) and cox- For example, the distribution of nodes corresponding to ujx = 1.1 and h = 0.0001 in the Cottrell experiment are shown in Figure 4.1. The expanding grid enables us to have a very dense grid (Xi — Xo = 0.0001) next to the electrode surface and, at the same time, to cover all the simulation space from X = 0 to 6VTmax with only 92 points. 

Applying the philosophy of expanding grids to time discretisation, one can reason that the interval between timesteps can also be adjusted by placing more points at time values where changes in concentration are more rapid. This approach is particularly useful for the simulation of chronoamperometry and potential pulse techniques. 

Fig. 4.2. Variation of the current response with time in the Cottrell experiment. The circle points indicate the position of the timesteps when an expanding grid is used with AT = 0.01 and ujt = 105.

F. Martinez-Ortiz, A. Molina, and E. Laborda. Electrochemical digital simulation with highly expanding grid four point discretization Can Crank-Nicolson uncouple diffusion and homogeneous chemical reactions , Electrochim. Acta 56, 5707-5716 (2011). 

Finite Difference Methods Finite difference techniques have frequently been applied to the simulation of the SECM in the axisymmetric cylindrical geometry encountered with microdisk tips. Often an exponentially expanding grid is used in order to produce a fine mesh near... 

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