Publication de 12 numéros par an
ISSN Imprimer: 1091-028X
ISSN En ligne: 1934-0508
Indexed in
A CAHN-HILLIARD APPROACH TO THERMODIFFUSION IN POROUS MEDIA
RÉSUMÉ
We consider a fluid-saturated porous medium exposed to a nonuniform temperature field and describe it as a nonisothermal biphasic mixture comprising a solid and a two-constituent fluid. We model such a system by assuming that the fluid free energy density depends on the gradient of the solute mass fraction. This constitutive choice induces a coupling between the temperature gradient and the solute diffusive mass flux, which adds itself to the standard Soret effect. We present numerical simulations of a thermogravitational cell to show how the modified constitutive framework, which is mandatory in diffuse-interface problems (e.g., the Cahn-Hilliard model), could lead to some novel interpretations of thermodiffusion and enrich the phenomenological description of the considered benchmarks.
-
Anderson, D.M. andMcFadden, G.B., Diffuse-Interface Methods in Fluid Mechanics, Ann. Rev. FluidMech., vol. 30, pp. 139-165, 1998.
-
Bear, J. and Bachmat, Y., Introduction to Modeling of Transport Phenomena in Porous Media, Dordrecht, Boston, London: Kluwer, 1990.
-
Benano-Melly, L.B., Caltagirone, J.-P., Faissat, B., Montel, F., and Costeseques, P., Modeling Soret Coefficient Measurement Experiments in Porous Media Considering Thermal and Solutal Convection, Int. J. Heat Mass Transf., vol. 44, pp. 1285-1297, 2001.
-
Bennethum, L., Murad, M.A., and Cushman, J.H., Macroscale Thermodynamics and the Chemical Potential for Swelling Porous Media, Transp. Porous Media, vol. 39, pp. 187-225, 2000.
-
Brezis, H., Analisi Funzionale - Teoria e Applicazioni, Napoli (Italy): Liguori Editore, pp. 248, 1986 (in Italian).
-
Celestino, A., Leonardi, E., andMaciocco, L., A Computational Study of Salt Diffusion and Heat Extraction in Solar Pond Plants, Solar Energy, vol. 80, pp. 1498-1508, 2006.
-
Chandra Shekar, B., Kishan, N., and Chamka, A.J., Soret and Dufour Effects on MHD Natural Convective Heat and Solute Transfer in a Fluid-Saturated Porous Cavity, J. Porous Media, vol. 19, no. 8, pp. 669-686,2016.
-
Chavepeyer, G., Dutrieux, J.F., Van Vaerenbergh, S., and Legros, J.C., A Survey of the Thomaes Flow Cell Method for the Soret Coefficient, in Thermal Nonequilibrium Phenomena in Fluid Mixtures, W. Kohler and S. Wiegand, Eds.: Berlin, Heidelberg: Springer-Verlag, pp. 211-232,2002.
-
Chen, C.-Y. and Yan, P.-Y., A Diffuse Interface Approach to Injection Driven Flows of Different Miscibility in Heterogeneous Porous Media, Phys. Fluids, vol. 27, pp. 083101-1-083101-19, 2015.
-
Chen, C.-Y. and Yan, P.-Y., Radial Flows in Heterogeneous Porous Media with Linear Injection Scheme, Comput. Fluids, vol. 142, pp. 30-36,2017.
-
Choi, Y.J. and Anderson, P.D., Cahn-Hilliard Modeling of Particles Suspended in Two-Phase Flows, Int. J. Numer. Methods Fluids, vol. 69, no. 5, pp. 995-1015, 2012.
-
Collins, C., Shen, J., and Wise, S.M., An Efficient, Energy Stable Scheme for the Cahn-Hilliard-Brinkman System, Commun. Comput. Phys, vol. 13, pp. 929-957, 2013.
-
Costeseques, P., Fargue, D., and Jamet, P., Thermodiffusion in Porous Media and Its Consequences, in Thermal Nonequilibrium Phenomena in Fluid Mixtures, W. Kohler and S. Wiegand, Eds., Berlin, Heidelberg, New York: Springer-Verlag, pp. 389-427, 2002.
-
Cueto-Felgueroso, L. and Juanes, R., A Phase Field Model of Unsaturated Flow, Water Resour. Res., vol. 45, pp. W10409-1- W10409-23,2009.
-
Davarzani, H., Marcoux, M., and Quintard, M., Theoretical Predictions of the Effective Thermodiffusion Coefficients in Porous Media, Int. J. Heat Mass Transf, vol. 53, pp. 1514-1528, 2010.
-
Davarzani, H. and Marcoux, M., Influence of Solid Phase Thermal Conductivity on Species Separation Rate in Packed Thermo- gravitational Columns: A Direct Numerical Simulation Model, C. R. Mec., vol. 339, no. 5, pp. 355-361, 2011.
-
Davis, H.T., A Theory of Tension at a Miscible Displacement Front, in Proc. of the Symposium on Numerical Simulation in Oil Recovery, M.F. Wheeler, Ed., New York, NY, USA: Springer-Verlag, pp. 105-110, 1988.
-
De Groot, S.R. and Mazur, P., Non-Equilibrium Thermodynamics, Mineola, NY: Dover Publications, Inc., 1984.
-
Dias, E.O. and Miranda, J., Control of Radial Fingering Patterns: A Weakly Nonlinear Approach, Phys. Rev. E, vol. 81, no. 1, pp. 016312-1-016312-7, 2010.
-
Diegel, A.E., Feng, X.H., and Wise, S.M., Analysis of a Mixed Finite Element Method for a Cahn-Hilliard-Darcy-Stokes System, SIAMJ. Numer. Anal., vol. 53, no. 1, pp. 127-152, 2015.
-
Emery, A.H. and Lorenz, M., Thermal Diffusion in a Packed Column, Chem. Eng. J, vol. 9, no. 5, pp. 661-663,1963.
-
Fargue, D., Jamet, P., and Costeseque, P., Dispersion Phenomena in Thermal Diffusion and Modelling of Thermogravitational Experiments in Porous Media, Transp. Porous Media, vol. 30, no. 3, pp. 323-344, 1998.
-
Fargue, D., Costeseque, P., Jamet, P., and Girard-Gaillard, S., Separation in Vertical Temperature Gradient Packed Thermodiffusion Cells: An Unexpected Physical Explanation to a Controversial Experimental Problem, Chem. Eng. Sci., vol. 59, no. 24, pp. 5847-5852, 2004.
-
Grillo, A., Lampe, M., and Wittum, G., Modelling and Simulations of Temperature-Density-Driven Flow and Thermodiffusion in Porous Media, J Porous Media, vol. 14, no. 8, pp. 671-690, 2011.
-
Gurtin, M.E., Fried, E., and Anand, L., The Mechanics and Thermodynamics of Continua, New York: Cambridge University Press, 2010.
-
Gurtin, M.E., Generalized Ginzburg-Landau and Cahn-Hilliard Equations based on a Microforce Balance, Physica D, vol. 92, pp. 178-192, 1996.
-
Harinath Reddy, S., Raju, M.C., and Keshava Reddy, E., Soret and Dufour Effects on Radiation Absorption Fluid in the Presence of Exponentially Varying Temperature and Concentration in a Conducting Field, Spec. Topics Rev. Porous Media: Int. J, vol. 7, no. 2, pp. 115-129,2016.
-
Hassanizadeh, S.M., Derivation of Basic Equations of Mass Transport in Porous Media, Part 2. Generalized Darcy's and Fick's Laws, Adv. Water Resour., vol. 9, pp. 207-222, 1986.
-
Ingle, S.E. and Horne, F.H., The Dufour Effect, J. Chem. Phys, vol. 59, no. 11, pp. 5882-5894,1973.
-
Jamet, D., Diffuse Interface Models in Fluid Mechanics, Adv. Water Resour., vol. 25, no. 3, pp. 335-348,2001.
-
Jamet, P., Fargue, D., Costeseque, P., de Marsily, G., and Cernes, A., The Thermogravitational Effect in Porous Media: A Modelling Approach, Transp. Porous Media, vol. 9, pp. 223-240, 1992.
-
Jasnow, D. and Vinals, J., Coarse-Grained Description of Thermo-Capillary Flow, Phys. Fluids A, vol. 8, pp. 660-669, 1996.
-
Joseph, D.D., Huang, A., and Hu, H., Non-Solenoidal Velocity Effects and Korteweg Stresses in Simple Mixtures of Incompressible Fluids, Phys. D, vol. 97, pp. 104-125, 1996.
-
Kita, R., Wiegand, S., and Luettmer-Strathmann, J., Sign Change of the Soret Coefficient of Poly (Ethylene Oxyde) in Water/Ethanol Mixtures Observed by Thermal Diffusion Forced Rayleigh Scattering, J. Chem. Phys, vol. 121, no. 8, pp. 3874.
-
Landau, L.D. and Liftschitz, E.M., Fluid Mechanics, 2nd Ed., Course of Theoretical Physics, vol. 6, London: Pergamon, 1984.
-
Lorenz, M. and Emery, A.H., The Packed Thermal Diffusion Column, Chem. Eng. Sci, vol. 11, pp. 16-23, 1959.
-
Lowengrub, J. and Truskinovsky, L., Quasi-Incompressible Cahn-Hilliard Fluids and Topological Transitions, Proc. R. Soc. Lond. A, vol. 454, pp. 2617-2654, 1998.
-
Madariaga, J.A., Santamaria, C., Barrutia, H., Mounir Bou-Ali, M., Ecenarro, O., and Valencia, J.J., Validity Limits of the FJO Thermogravitational Column Theory: Experimental and Numerical Analysis, C. R. Mec., vol. 339, no. 5, pp. 292-296, 2011.
-
Mallikarjuna, B., Chamkha, A.J., and Bhuvana Vijaya, R., Soret and Dufour Effects on Double Diffusive Convective Flow through a Non-Darcy Porous Medium in a Cylindrical Annular Region in the Presence of Heat Sources, J. Porous Media, vol. 17, no. 7, pp. 623-636, 2014.
-
Micunovic, M.V., Thermodynamics of Viscoplasticity - Fundamentals and Applications, New York: Springer, 2009.
-
Nasrabadi, H., Hoteit, H., and Firoozabadi, A., An Analysis of Species Separation in a Thermogravitational Column Filled with a Porous Medium, Transp. Porous Media, vol. 67, pp. 437-486, 2007.
-
Oldenburg, C.M. and Pruess, K., Layered Thermohaline Convection in Hypersaline Geothermal Systems, Transp. Porous Media, vol. 33, pp. 29-63, 1998.
-
Oldenburg, C.M. and Pruess, K., Layered Plume Separation by Transient Thermohaline Convection in Porous Media, Geophys. Res. Lett., vol. 26, no. 19, pp. 2997-3000,1999.
-
Platten, J.K., The Soret Effect: A Review of Recent Experimental Results, J. Appl. Mech., vol. 73, pp. 5-15,2006.
-
Quintard, M., Kaviany, M., and Whitaker, S., Two-Medium Treatment of Heat Transfer in Porous Media: Numerical Results for Effective Properties, Adv. Water Resour., vol. 20, nos. 2-3, pp. 77-94, 1997.
-
RamReddy, Ch., Murthy, P.V.S.N., Rashad, A.M., and Chamkha, A.J., Soret Effect on Stagnation-Point Flow past a Stretch-ing/Shrinking Sheet in a Nanofluid-Saturated Non-Darcy Porous Medium, Spec. Topics Rev. Porous Media: Int. J, vol. 7, no. 3, pp. 229-243,2016.
-
Rauch, J., Diffusion and Thermal Diffusion in Polymer Solutions, PhD, Universitat Bayreuth, Germany, 2006 (in German).
-
Rauch, J. and Kohler, W., Diffusion and Thermal Diffusion of Semidilute to Concentrated Solutions of Polystyrene in Toluene in the Vicinity of the Glass Transition, Phys. Rev. Lett., vol. 88, no. 18, pp. 185901-1-185901-4, 2002.
-
Rauch, J. and Kohler, W., Collective and Thermal Diffusion in Dilute, Semidilute, and Concentrated Solutions of Polystyrene in Toluene, J. Chem. Phys, vol. 119, no. 22, pp. 11977-11988,2003.
-
Rosanne, R., Paszkuta, M., Tevissen, E., and Adler, P.M., Thermodiffusion in Compact Clays, J. ColloidInterf. Sci., vol. 267, no. 1,pp. 194-203,2003.
-
Rowley, R.L. and Horne, F.H., The Dufour Effect, III. Direct Experimental Determination of the Heat of Transport of Carbon Tetrachloride-Cyclohexane Liquid Mixtures, J. Chem. Phys., vol. 72, no. 1, pp. 131-139, 1980.
-
Salsa, S., Partial Differential Equations in Action-From Modelling to Theory, Milan, Berlin, Heidelberg, New York: Springer, 2008.
-
Srinivasacharya, D. and Kaladhar, K., Soret and Dufour Effects on Mixed Convection Flow of Couple Stress Fluid in a Non-Darcy Porous Medium with Heat and Mass Fluxes, J. Porous Media, vol. 17, no. 2, pp. 93-101, 2014.
-
Srinivasan, S. and Saghir, M.Z., Thermodiffusion in Multicomponent Mixtures-Thermodynamic, Algebraic, and Neuro-Computing Models, Springer Briefs in Thermal Engineering and Applied Science, New York: Springer, vol. 106, no. 9, 2013. DOI: 10.1007/978-1-4614-5599-8.
-
Swernsath, S., Malengier, B., and Pushpavanam, S., Effect of Korteweg Stress on Viscous Fingering of Solute Plugs in a Porous Medium, Chem. Eng. Sci., vol. 65, pp. 2284-2291, 2010.
-
Tyrrell, H.J.V., The Calculation of Diffusion Coefficients and Soret Coefficients from Optical Measurements on Pure Soret Effect Cells, Trans. Faraday Society, vol. 52, pp. 940-948, 1956.
-
Veeresh, C., Varma, S.V.K., Raju, M.C., and Rushi Kumar, B., Thermal Diffusion Effects on Unsteady Magnetohydrodynamic Boundary Layer Slip Flow past a Vertical Permeable Plate, Spec. Topics Rev. Porous Media: Int. J, vol. 7, no. 1, pp. 43-55, 2016.
-
Yadav, D. and Kim, M.C., The Onset of Transient Soret-Driven Buoyancy Convection in Nanoparticle Suspensions with Particle-Concentration-Dependent Viscosity in a Porous Medium, J. Porous Media, vol. 18, no. 4, pp. 369-378, 2015.
-
Yue, P., Feng, J.J., Liu, C., and Shen, J., A Diffuse-Interface Method for Simulating Two-Phase Flows of Complex Fluids, J. Fluid Mech., vol. 515, pp. 293-317, 2004.
-
Zhang, K.J., Briggs, M.E., Gammon, R.W., Sengers, J.V., and Douglas, J.F., Thermal and Mass Diffusion in a Semidilute Good Solvent-Polymer Solution, J. Chem. Phys, vol. 111, no. 5, pp. 2270-2282, 1999.