Water Journal : Water Journal May 2012
refereed paper technical features 64 MAY 2012 water pipeline cleaning & maintenance in which M represents the uptake of penetrant mass at time t, while M is the corresponding value at equilibrium; D is the diffusivity, which has units of cm/s2 or similar; l is half the thickness of the slab, both of whose faces are exposed to the fluid. A coating has only one face exposed to fluid, with the other bonded to the pipe; hence, l can also be taken as the thickness of a coating. Care must be taken lest the wrong interpretation of l be used (e.g. Valix and Bustamante, 2011). The ellipsis indicates that the series continues indefinitely; however, for all but the shortest times the summation is sufficiently accurate when only a few terms in the series are retained. A series of algebraic transformations to express Equation 2 in terms of the first integral of the complementary error function (see Crank, 1975) provides a more numerically tractable computation for the shorter times. In principle, that form of the equation can be used for all times, but at longer times it becomes impractical to evaluate, and at shorter times a simpler alternative can be employed (see Simplifications). A number of assumptions are made in the derivation of Equation 2. These include: 1. Diffusion occurs along one axis only. This means that the test specimen must be much thinner than it is wide or tall. 2. The material is uniform, chemically stable and of constant dimensions. 3. The diffusivity is constant. An average effective diffusivity is readily defined, but may be difficult to evaluate from absorption measurements. 4. The rate-limiting process is the internal diffusion within the solid. 5. The concentration of the penetrating species at the surface is constant. 6. For a coating bonded to a substrate, and thus exposed on one face only, the mass of penetrant propagating into the substrate is negligible. Simplifications At small times the behaviour described in Equation 2 always simplifies to a power law, in which M is proportional to (Hill, 1928). It is important to realise that "small" times is a relative concept. If the diffusion is a slow process (low diffusivity), then weeks might be considered small times. Conversely, if the diffusion proceeds quickly (high diffusivity), then even a minute might not be considered a small time. For the first phase of the diffusion, when , the full equation can be well approximated by (Barrer, 1951; Crank, 1975): Then M is conveniently plotted as a straight line against , the gradient being proportional to . Despite what is sometimes seen in the literature (e.g. Legghe et al., 2009; Valix and Bustamante, 2011), it is incorrect to apply this formula to the long-time response. When the diffusivity, D, is a function of concentration, the range of applicability of Equation 4 may be either extended or curtailed from the default situation (Crank and Park, 1968). The limiting behaviour for glassy polymers is found to be a direct proportionality between M and t, rather than (Alfrey et al., 1966). A combination of the two can also occur, as can powers intermediate to 0.1 and 1.0. Other 'anomalous' behaviours include power- law variation with indexes greater than 1.0 (Vieth, 1991), sigmoidal and two-stage uptake curves (Crank, 1975; Crank and Park, 1968; Ivanova et al., 2001), and Langmuir adsorption (Liu et al., 2008b). An even simpler formula has been derived to relate the diffusivity to the normalised time at which half of the absorption has occurred, namely in which T50% is the value of at which (cf. Crank, 1975). Again l is the half-width of a sheet exposed on both sides, or the full thickness of a lining exposed only on the outer face. While this last formula is simpler, its estimates are more sensitive to any dependence of the diffusivity upon concentration than those of the initial- gradient method (Crank, 1975). The claims of remarkable accuracy associated with the T50% formula (Crank, 1975) are not practically meaningful, as they ignore any kind of experimental error. Application of the Theory The formulae have been derived for coatings that are fulfilling their function, which means that they are applied to a substrate to protect it. One interface of the coating is bonded to the pipe, while the other is exposed to the fluids flowing through the pipe. This asymmetry is the reason for specifying l to be the thickness of a bonded coating, exposed at one face only. As mentioned, if the specimen is exposed on both faces, then its thickness is set equal to 2l. In the formulae, M is the mass uptake of the penetrant at equilibrium. Formally, that means the value at infinite time. Fortunately, sufficient accuracy is maintained if one merely waits for long times. How long is "long"? It means times that are long relative to the rate of diffusion. Hence, this can only really be determined by monitoring the mass uptake over time and ensuring that it has reached a practically constant value (assuming that no other transport mechanisms become important). In that case, any of the above formulae could be used to estimate D with reasonable accuracy. If the experiment has proceeded for a "short" time only, then M is still unknown. Even though a "maximum" value of acid uptake will have been recorded in the experiment (e.g. Valix and Bustamante, 2011), this should not be confused with the asymptotic value, which could only be identified by continuing the observations. In this case, there is no possibility of correctly evaluating D. Cursory inspection of Equation 4 might lead to the misapprehension that any arbitrary value of M , such as the maximum value observed, can be used. Rearranging the equation demonstrates more clearly that D cannot be disentangled from M , if M is unknown: This is demonstrated in Figure 1 and Figure 2. Even though the diffusivities vary by a factor of 16, the fourfold variation in solubility compensates, so that all three cases are described by the same curve at early times. Finally, if the experiment has proceeded for an 'intermediate' length of time, then M is still unknown. However, it can be predicted by using the complete formula, consistent with Equation 2. Figure 3 presents an example in which only a few observations are available of the true underlying response, for which D = 1 and M = 1. In the scenario, the coating has unit thickness; equivalently, the horizontal axis could have been plotted as or , to obtain a 'reduced sorption plot' (Crank and Park, 1968). The observations (2) =182e e9 9 e25 25 e49 49 , (3) D = 19 5 , (5) (6) 2 2D 2. 2D 4 2 . (4) 2D 2.
Water Journal July 2012
Water Journal April 2012