Analysis of a Model for Anomalous-Diffusion Behavior of CO2 in the Macromolecular-Network Structure of Coal

Summary This paper gives an analysis of the Thomas and Windle model (Thomas and Windle 1982) to determine its usefulness for describing anomalous diffusion of CO2 in coal and its relation to matrix swelling. In addition, a finite-element description for this model is derived. For reasons of easy reference, a shortened derivation of the Thomas and Windle model is presented, which was originally derived to describe diffusion in polymers. The derivation includes the surface saturation effects proposed by Hui et al. (1987a, 1987b). Because the cumulative sorption showed tα behavior with α > 0.5, the behavior was described as enhanced diffusion or even superdiffusion. Analysis of the model equation shows no evidence for superdiffusion even if non-Fickian behavior is observed [i.e., there is (1) an initial phase in which the coal surface gets saturated with a slope > 0.5 in a log-log plot of cumulative sorption vs. time, (2) an intermediate phase that shows the typical square-root-of-time behavior of an ordinary diffusion process, and (3) a final phase with a slope < 0.5 toward equilibrium]. The cumulative mass is for all times less than what would have been obtained for pure diffusion in a particle characterized by a rubber diffusion coefficient. The slow saturation at the surface masks a process where fast stress-induced diffusion dominates, which indeed can be faster than Fickian. The cumulative sorption rates give behavior similar to the Rückenstein model (Rückenstein et al. 1971), but the advantage of the Thomas and Windle model is that it can also calculate the resulting coal-swelling effects.