Water Journal : Water Journal May 2012
refereed paper water MAY 2012 65 pipeline cleaning & maintenance were generated by taking the true uptake at times 0.1, 0.2, ..., 0.8, and rounding the uptake to two decimal places, in order to introduce a small experimental imprecision. To employ the short-time formula, the obvious choice for M is the value of M at the last observation, namely M ~ 0.89. Applying the formula to all eight observations results in an 11% overestimate of D. However, plotting this curve over the observations shows that it does not reflect the true initial uptake, as only the first few points closely approximate a relation. Hence a revised fit is applied to only the first two observations. Interestingly, this results in a worse estimate of D, being 25% larger than the true value. Evidently the error due to mis-specification of M was partially compensated by an error of opposite sign due to the inappropriate curve fit when all observations were included. (This is a fortuitous circumstance, and not a recommended 'correction'.) To use Equation 5 it is necessary to first estimate a value of M for which Taking M ~ 0.89 means that T50% would occur at M ~ 0.45. The experimental data points in this example are fairly sparse, and so linear interpolation is applied to obtain an estimate T50% ~ 0.161, and hence D ~ 1.22. (Interpolation on instead of t yields T50% ~ 0.157 and D ~ 1.26.) This estimate is as bad as the preceding one. These estimates can be compared with application of Equation 2. It might be considered easier to specify M ~ 0.89 again, so that only one parameter need be estimated. This expediency results in an unacceptable overestimation of D, by 39%. It is seen that the prediction clearly reaches an asymptote at M = 0.89, and the estimate of D is very sensitive to this. Finally, when both M and D are simultaneously estimated, by minimising the sum of squared errors, a good match is attained between the predicted curve and the real underlying response. (An exact match could not be expected, given the experimental errors introduced.) M has been almost perfectly estimated, and D is only underestimated by 3%. Mechanism of Degradation and Relevance of Diffusivities There are two important routes by which degradation might occur: physical and chemical. One might also include microbiological processes as a third route, but it is convenient to describe these as chemical processes, because ultimately it will be secretions of an enzyme, acid, Figure 1. Uptake of penetrating species into coating of unit thickness for various combinations of diffusivity and equilibrium uptake, in comparison with the early-time approximation (which is the same for these three combinations), plotted against time so that the initial response follows a square-root profile. Figure 2. Same as Figure 1, but plotted against square root of time, so that the initial response is linear. Figure 3. Comparison of uptake curves estimated by fitting 'observed' data with true underlying response. Unit thickness of coating.
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