(408g) A Diffuse Interface Model for Dissolution Processes with Variable Density
AIChE Annual Meeting
2010 Annual Meeting
Engineering Sciences and Fundamentals
Interfacial Transport Phenomena
Wednesday, November 10, 2010 - 10:35am to 10:55am
Dissolution of porous media or solids is widely concerned in many industrial fields, e.g., acid injection into petroleum reservoirs, ablation of composite layeres in rocket nozzles, dissolution of rocks caused by underground water, etc... In this latter example, rock dissolution may create underground cavities, which a potential risk of collapse. In most applications, modeling such dissolution problems is of paramount importance. There are mainly two ways for simulating liquid/solid dissolution processes with a moving interface: first, a direct treatment of the moving interface, for instance using an ALE technique, second, using a diffuse interface model (referred to as DIM in the paper sequel) to smooth the interface with continuous quantities, like the liquid phase volume fraction, species mass fractions, etc°(Collins and Levine, 1985; Anderson and Mcfadden, 1998; Leo et al., 1998) However, there are several difficulties associated to the use of ALE techniques for this problem: in particular, the need for fine meshes near the dissolving interface leads to severely deformed grid elements inducing numerical problems (instabilities, frequent need for remeshing, ...). On the contrary, it is easier to implement a DIM model because of the smoothing of the interface singularity, and resulting codes are more stable and efficient. For instance, in a previous study, Golfier et al. (2002) showed that a non-equilibrium Darcy-scale dissolution model was a good candidate for a DIM dissolution model. In particular, it was able to capture the instability (wormhole) formation during the dissolution of a porous medium, which is known to be a very difficult numerical problem. However, density variations with concentration were neglected in Golfier et al. (2002) model. For most problems, for instance saline cavity formations, we may have very high concentration variations around the solid/liquid interface, which may bring inaccuracy to the prediction of dissolution, especially under the case with considerable gravity effects. In this paper, a Darcy-scale diffuse-interface model (DIM) including density variation is deduced from the original liquid/solid dissolution model in the case of a binary system (following Golfier et al. suggestions). The conservation equations, which include the mass balance equations for the b and ¦" phases, and species A in the b-phase, are obtained in the form of two-equation models, with an exchange term between the b-phase and ¦"-phase. A Darcy-Brinkman model is used to describe the momentum equations. Several numerical examples with variable density effects are implemented in COMSOLTM, using both ALE and DIM, for different P¨¦clet numbers, Pe, and Rayleigh numbers, Ra. Fig. 1 shows the comparison between results obtained with ALE and DIM in the case Pe=100 without gravity. The thin black solid line represents the shape of the system at t=0. Both results show a very good agreement, thus qualifying the DIM model as an alternative to direct interface simulation. Fig. 2 shows the different phenomena observed for different Rayleigh numbers (Ra=10 and Ra=10000). For the larger Rayleigh number, the convective instability caused by the gravity effects is clearly displayed. Its impact on the position of the interface is also clearly put in evidence. Fig.1 Comparison of the mass fraction between ALE simulation and DIM simulation (Pe=100) Fig. 2 Comparison of the simulation results under Ra=10 and Ra=10000 As a conclusion, for the numerical simulation of solid/liquid dissolution processes, it is more convenient to use a diffuse interface model rather than a direct interface method, such as the ALE technique. References D. M. Anderson and G. B. McFadden, Diffuse-Interface Methods in Fluid Mechanics Annual Review of Fluid Mechanics, 30:139-165, 1998. J. B. Collins and H. Levine, Diffuse interface model of diffusion-limited crystal growth. Phys. Rev. B, 31: 6119-6122, 1985.  F. Golfier, C. Zarcone, B. Bazin, R. Lenormand, D. Lasseux, and M. Quintard. On the ability of a darcy-scale model to capture wormhole formation during the dissolution of a porous medium. Journal of Fluid Mechanics, 457:213¨C254, 2002. P. H. Leo, J. S. Lowengrub, and H. J. Jou, A diffuse interface model for microstructural evolution in elastically stressed solids. Acta Mater., 46:2113¨C2130, 1998.  M. Quintard and S. Whitaker. Transport in ordered and disordered porous media 1: The cellular average and the use of weighing functions. Transport in Porous Media, 14:163¨C177, 1994.  M. Quintard and S. Whitaker. Convection, dispersion, and interfacial transport of contaminant: Homogeneous porous media. Advances in Water Resources, 17:221¨C239, 1994.  M. Quintard and S. Whitaker. Dissolution of an immobile phase during flow in porous media. Ind. Eng. Chem. Res., 38:833¨C844, 1999.
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