Properties of a diffuse interface model based on a porous medium theory for solid–liquid dissolution problems

Haishan Luo1,2,3, Michel Quintard2,3, Gérald Debenest2, Farid Laouafa1
1Institut National de l’Environnement Industriel et des Risques, Parc techncologique ALATA BP2, Verneuil-en-Halatte, France
2Institut de Mécanique des Fluides de Toulouse, Université de Toulouse; INPT, UPS, Toulouse, France
3CNRS, IMFT, Toulouse, France

Tóm tắt

In this paper, a local non-equilibrium diffuse interface model is introduced for describing solid–liquid dissolution problems. The model is developed based on the analysis of Golfier et al. (J Fluid Mech 457:213–254, 2002) upon the dissolution of a porous domain, with the additional requirement that density variations with the mass fraction are taken into account. The control equations are generated by the upscaling of the balance equations for a solid–liquid dissolution using a volume averaging theory. This results into a diffuse interface model (DIM) that does not require an explicit treatment of the dissolving interface, e.g., the use of arbitrary Lagrangian–Eulerian (ALE) methods, for instance. Test cases were performed to study the features and influences of the effective coefficients inside the DIM. In particular, an optimum expression for the solid–liquid exchange coefficient is obtained from a comparison with the referenced solution by ALE simulations. Finally, a Ra–Pe diagram illustrates the interaction of natural convection and forced convection in the dissolution problem.

Từ khóa


Tài liệu tham khảo

Anderson, D.M., McFadden, G.B.: Diffuse-interface methods in fluid mechanics. Annu. Rev. Fluid Mech. 30, 139–165 (1998)

Anderson, R.Y., Kirkland, D.W.: Dissolution of salt deposits by brine density flow. Geology 8, 66–69 (1980)

Beckermann, C., Diepers, H.-J., Steinbach, I., Karma, A., Tong, X.: Modeling melt convection in phase-field simulations of solidification. J. Comput. Phys. 154, 468–496 (1999)

Bekri, S., Thovert, J.F., Adler, P.M.: Dissolution of porous media. Chem. Eng. Sci. 50, 2765–2791 (1995)

Boettinger, W.J., Warren, J.A., Beckermann, C., Karma, A.: Phase-field simulation of solidification. Annu. Rev. Mater. Res. 32, 163–194 (2002)

Bousquet-Melou, P., Neculae, N., Goyeau, B., Quintard, M.: Averaged solute transport during solidification of a binary mixture: active dispersion in dendritic structures. Metall. Mater. Trans. B 33, 365–376 (2002)

Cahn, J.W., Hilliard, J.: Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28, 258–267 (1958)

Collins, J.B., Levine, H.: Diffuse interface model of diffusion-limited crystal growth. Phys. Rev. B 31, 6119–6122 (1985)

Dell’Isola, F., Gouin, H., Rotoli, G.: Nucleation of spherical shell-like interfaces by second gradient theory: numerical simulations. Eur. J. Mech. B Fluid 15, 545–568 (1996)

Dijk, P.E., Berkowitz, B.: Buoyancy-driven dissolution enhancement in rock fractures. Geology 28, 1051–1054 (2000)

Dijk, P.E., Berkowitz, B., Yechieli, Y.: Measurement and analysis of dissolution patterns in rock fractures. Water Resour. Res. 38 (2002). doi:10.1029/2001WR000246

Donea, J., Giuliani, S., Halleux J.P.: An arbitrary Lagrangian–Eulerian finite element method for transient dynamic fluid–structure interactions. Comput. Methods Appl. Mech. Eng. 33, 689–723 (1982)

Glimm, J., Grove, J.W., Li, X.L., Shyue, K.M., Zeng, Y., Zhang, Q.: Three-dimensional front tracking. SIAM J. Sci. Comput. 3, 703–727 (1998)

Golfier, F., Quintard, M., Whitaker, S.: Heat and mass transfer in tubes: an analysis using the method of volume averaging. J. Porous Media 5(3), 169–185 (2002)

Golfier, F., Zarcone, C., Bazin, B., Lenormand, R., Lasseux, D., Quintard, M.: On the ability of a Darcy-scale model to capture wormhole formation during the dissolution of a porous medium. J. Fluid Mech. 457, 213–254 (2002)

Hirt, C.W., Nicolas, B.D.: Volume of fluid (VOF) method for the dynamics of free boundaries. J. Comput. Phys. 39, 201–225 (1981)

Kang, Q., Zhang, D., Chen, S.: Simulation of dissolution and precipitation in porous media. J. Geophys. Res. 108, 2505 (2003). doi:10.1029/2003JB002504

Kovacevic, I., Sarlera, B.: Solution of a phase-field model for dissolution of primary particles in binary aluminum alloys by an r-adaptive mesh-free method. Mater. Sci. Eng. A 413, 423–428 (2005)

Leo, P.H., Lowengrub, J.S., Jou, H.J.: A diffuse interface model for microstructural evolution in elastically stressed solids. Acta Mater. 46, 2113–2130 (1998)

Lopez, J., Hernandez, J., Gomez, P., Faura, F.: An improved PLIC-VOF method for tracking thin fluid structures in incompressible two-phase flows. J. Comput. Phys. 208(1), 51–74 (2005)

Mao, X., Prommer, H., Barry, D.A., Langevin, C.D., Panteleit, B., Li, L.: Three-dimensional model for multi-component reactive transport with variable density groundwater flow. Environ. Modell. Softw. 21, 615–628 (2006)

Maury, B.: Characteristics ale method for the unsteady 3d Navier–Stokes equations with a free surface. Int. J. Comput. Fluid Dyn. 6, 175–188 (1996)

Mittal, R., Iaccarino, G.: Immersed boundary methods. Annu. Rev. Fluid Mech. 37, 239–261 (2005)

Olsson, E., Kreiss, G.: A conservative level set method for two phase flow. J. Comput. Phys. 210, 225–246 (2005)

Onuki, A., Kawasaki, K.: Nonequilibrium steady state of critical fluids under shear flow: a renormalization group approach. Ann. Phys. 121, 456–528 (1979)

Osher, S., Fedkiw, R.: Level set methods: an overview and some recent results. J. Comput. Phys. 169, 463–502 (2001)

Quintard, M., Whitaker, S.: Convection, dispersion, and interfacial transport of contaminant: homogeneous porous media. Adv. Water Resour. 17, 221–239 (1994)

Quintard, M., Whitaker, S.: Transport in ordered and disordered porous media 1: the cellular average and the use of weighting functions. Transp. Porous Media 14, 163–177 (1994)

Quintard, M., Whitaker, S.: Dissolution of an immobile phase during flow in porous media. Ind. Eng. Chem. Res. 38, 833–844 (1999)

Rider, W.J., Kothe, D.B.: Reconstructing volume tracking. J. Comput. Phys. 141(2), 112–152 (1998)

Roux, P., Goyeau, B., Gobin, D., Fichot, F., Quintard, M.: Chemical non-equilibrium modelling of columnar solidification. Int. J. Heat Mass Transfer 49(23–24), 4496–4510 (2006)

Sekerka, R.F.: Morphology: from sharp interface to phase field models. J. Cryst. Growth 264, 530–540 (2004)

Seppecher, P.: Moving contact lines in the Cahn–Hilliard theory. Int. J. Eng. Sci. 34, 977–992 (1996)

Sethian, J.A., Smereka, P.: Level set methods for fluid interfaces. Annu. Rev. Fluid Mech. 35, 341–372 (2003)

Stone, H.A., Leal, L.G.: The effects of surfactants on drop deformation and breakup. J. Fluid Mech. 220, 161–186 (1990)

Taylor, R., Krishna, R.: Multicomponent Mass Transfer. Wiley-Interscience, New York (1993)

Teigen, K.E., Li, X., Lowengrub, J., Wang, F., Voigt, A.: A diffuse-interface approach for modelling transport, diffusion and adsorption/desorption of material quantities on a deformable interface. Commun. Math. Sci. 7, 1009–1037 (2009)

Tiadena, J., Nestlerb, B., Diepersa, H.J., Steinbacha, I.: The multiphase-field model with an integrated concept for modelling solute diffusion. Physica D Nonlinear Phenom. 115, 73–86 (1998)

Tryggvason, G., Bunner, B., Esmaeeli, A., Juric, D., Al-Rawahi, N., Tauber, W., Han, J., Nas, S., Jan, Y.-J.: A front-tracking method for the computations of multiphase flow. J. Comput. Phys. 169, 708–759 (2001)

Turski, L.A., Langer, J.S.: Dynamics of a diffuse liquid–vapor interface. Phys. Rev. A 22, 2189–2195 (1980)

van Noorden, T.L.: Crystal precipitation and dissolution in a porous medium: effective equations and numerical experiments. Multiscale Model. Simul. 7, 1220–1236 (2009)

van Noorden, T.L., Eck, C.: Phase field approximation of a kinetic moving-boundary problem modelling dissolution and precipitation. Interfaces Free Bound. 13, 29–55 (2011)

vanderWaals, J.D.: The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density (translated by J.S. Rowlinson, 1893). J. Stat. Phys. 20, 197–244 (1979)

Vignoles, G.L., Aspa, Y., Quintard, M.: Modelling of carbon–carbon composite ablation in rocket nozzles. Compos. Sci. Technol. 70, 1303–1311 (2010)

Warren, J.A., Boettinger, W.J.: Prediction of dendritic growth and microsegregation patterns in a binary alloy using the phase-field method. Acta Metall. Mater. 43, 689–703 (1995)

Whitaker, S.: The Method of Volume Averaging. Kluwer Academic, Dordrecht (1999)

Wood, J.R., Hewett, T.A.: Fluid convection and mass transfer in porous sandstones—a theoretical model. Geochim. Cosmochim. Acta 46, 1907–1913 (1982)

Worster, M.G., Wettlaufer, J.S.: Natural convection, solute trapping, and channel formation during solidification of saltwater. J. Phys. Chem. B 101, 6132–6136 (1997)

Xu, Z., Meakin, P.: Phase-field modeling of solute precipitation and dissolution. J. Chem. Phys. 129, 014705 (2008)

Xu, Z., Meakin, P., Tartakovsky, A.M.: Diffuse-interface model for smoothed particle hydrodynamics. Phys. Rev. E 79, 036702 (2009)

Yu, D., Ladd, A.J.C.: A numerical simulation method for dissolution in porous and fractured media. J. Comput. Phys. 229, 6450–6465 (2010)

Zhao, C., Hobbs, B.E., Ord, A.: Fundamentals of Computational Geoscience: Numerical Methods and Algorithms. Springer, Berlin (2009)

Zhao, C., Hobbs, B.E., Ord, A.: Theoretical analyses of nonaqueous phase liquid dissolution-induced instability in two-dimensional fluid-saturated porous media. Int. J. Numer. Anal. Methods Geomech. 34, 1767–1796 (2010)