U-v device

This will prove useful in connection with the u-v device, see below. 

The chronopotentiometry (controlled current) case has already been described for the EX method, where one simply finds a Co value that fits the known gradient G and the concentration points already established, as shown in (5.10). The situation is not quite so simple for implicit methods, and we introduce here both a preview of these, and the u-v device, which will be used extensively. 

Another, more convenient way is the u-v device. We establish a relation between the concentrations G[ and Cq, in order to obtain the extra information needed to solve for Cq. Taking the first equation in (6.3), we rewrite it explicitly for C[ ... 

Formally, the above process is equivalent to (6.4), extended for any n and solving that system. The u-v device is a more efficient way of solving it than any linear equation solver that might otherwise have been used, as n becomes larger. The u-v device will be extensively used in this book, even with implicit methods for coupled equation systems, where we must solve for a number of concentration profiles (see below). There are practitioners who believe that n = 2, that is the two-point G-approximation, is good enough. This is justified in cases where H is very small, as it often is, at least near the electrode, when unequal intervals are used (see Chap. 9). In that case, one can simply use (6.5). 

The u-v device can now be applied as before, the only complication being that there will be two sets of u s and u s the treatment is identical to the above one and results in the two equations... 

For those who prefer to keep the derivative approximation of G down to the two-point form, the above can perhaps be simplified a little the u-v device is not needed as such, as only the first substitution (6.6) is required. 

The process (which, as will be seen and already seen in the uncoupled case of the u-v device above) needs to be carried forward only to i = n — 1. It yields n — 1 equations... 

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