Numerical modelling on super-diffusion of carbon-dioxide in coal

Abstract

Case II (super)-diffusion is characterized by a diffusion like process that goes faster than the square root of time. Superdiffusion can occur in polymeric structures if a stress gradient enhances the diffusion process. In such a case the part of the polymer invaded by the penetrant, e.g. CO2, undergoes a transition in structure from the glass state to a rubber like state. In ideal cases the transition zone can propagate proportional to time. The Thomas & Windle model, a model that describes anomalous diffusion behavior in polymers, was also proposed to describe diffusion of CO2 in coal. The proposal was due to the similarities in structure between coal and glassy polymers. Coal has a molecular structure of carbon-carbon bonds with ‘open holes’ between the chains. If these holes are smaller in volume than the volume of a CO2-molecule, the uptake of CO2 leads to a geometric problem. The CO2 molecules will squeeze themselves into the structure, resulting into a swollen rubbery coal. This phenomenon is happening due to the fact that a carbon-carbon dioxide bond is thermodynamically favored above the carbon-carbon bond and is more stable than the other. Analysis of the Thomas & Windle model shows there is a time where diffusion is much faster than conventional diffusion. The results also show that super-diffusion is only a transitional effect, i.e., there is an initial phase where relaxation dominates, a transitional phase that exhibits anomalous diffusion and a long time phase that shows the typical square root of time behavior of an ordinary diffusion process. This study gives details of a numerical model used to solve the stress induced diffusion equation using the Finite Element Method implemented in Matlab. The numerical results show that for some favorable set of parameters diffusion can indeed be faster than Fickean diffusion during an intermediate stage. For other parameters the diffusion can be slower. A comparison to a laboratory result shows an example where diffusion appears to be slower.