Effective Surface and Boundary Condition for Heterogeneous Salt Media with Insoluble Materia

Guo, Jianwei

Quintard, Michel

Veran-Tissoires, Stéphanie

English

Open Archive Toulouse Archive Ouverte

Any correspondence concerning this service should be sent to the repository administrator: staff-oatao@inp-toulouse.fr                    Official URL: http://www.interpore.org/       OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible.  This is an author-deposited version published in: http://oatao.univ-toulouse.fr/  Eprints ID: 9091 To cite this version:  Guo, Jianwei and Veran-Tissoires, Stéphanie and Quintard, Michel Effective Surface and Boundary Condition for Heterogeneous Salt Media with Insoluble Material. (2013) In: 5th International Conference on porous media & Annual meeting of the International Society for Porous Media (InterPore), 22-24 May 2013, Prague, Czech Republic. Open Archive Toulouse Archive Ouverte (OATAO)   Effective Surface and Boundary Condition for Heterogeneous Salt Media with Insoluble Material  J. Guoa, S. Veran-Tissoiresa,b, M. Quintarda,b a Université de Toulouse; INPT, UPS; IMFT (Institut de Mécanique des Fluides de Toulouse); Allée Camille Soula, F-31400 Toulouse, France b CNRS; IMFT; F-31400 Toulouse, France Key words: heterogeneous surface, multi-scale decomposition, effective surface Several mechanisms, such as dissolution of  salt material or drying of  porous media, involve transport from a heterogeneous, rough surface characterized by areas in thermodynamic equilibrium with the surrounding fluid and areas of  no flux (insoluble material, solid grains in the porous medium) [1], [2]. In general, the characteristic length-scale of  the surface is much smaller than the scale of  the process under investigation. To deal with this multi-scale aspect, the engineering practice is to replace the heterogeneous, rough surface by a homogeneous, smooth surface, the “effective” surface, over which is applied the Dirichlet condition corresponding to thermodynamic equilibrium. This idea has been checked partially against direct numerical simulation of  the small-scale problem, in particular in the case of  drying problems.  In this study, we develop a more complete theory of  the concept of  effective surface applied to such transport problems. The typical example under investigation corresponds to the laminar flow over a soluble salt medium which contains insoluble materials. To develop the concept of  effective surface we apply a multi-domain decomposition approach [3], [4], [5]. We first decompose the system, by introduction of  an arbitrary interface Σ0, into a macro-scale subdomain Ω0 where the variables are smooth and a series of  micro-scale Ωi subdomains which contain all the roughnesses (cf. Figure 1). We then make estimates of  velocity and concentration at the micro-scale with an asymptotic expansion of  deviation terms with respect to macro-scale velocity and concentration fields. Closure problems for the deviations are obtained and used to study the position of  the effective surface and the related boundary conditions.   After resolution of  the closure problems, it is shown that the boundary condition for Navier-Stokes equations is of  the Navier type (already known from the literature), and of  a mixed reactive type, for the mass balances. The effective surface can be positioned in different places, for instance in order to recover a no-slip condition. The associated surface is denoted Σeffv, and the effective boundary conditions are 00 u  at Σeffv                                                    ⑴     )-(- 0eff0 eqcckcDn   at Σeffv                                        ⑵ where u0 and c0 denote respectively the velocity and concentration in the first order effective model, while ceq is the thermodynamic equilibrium concentration, n is the normal vector pointing from the fluid towards the solid, D is the diffusion coefficient and keff is defined as an effective mass exchange coefficient. If  one wants to recover the original equilibrium condition, the position of  the related effective surface Σeffc is different from Σeffv. In that case we have eqcc 0  at Σeffc                                                      (3)                                   00 )-( uwwnuvxcx   at Σeffc                         (4) where wxc and wxv denote respectively the distance from Σ0 to Σeffc and Σeffv. Based on numerical resolution of  the closure problems, we investigate the evolution of  keff and the impact of  the effective surface position, for various geometries, Reynolds and Schmidt numbers. Finally, we make a comparison between the numerical results obtained with this effective model and those from direct numerical simulations. This allows us to discuss the practical implementation of  the effective surface model, in particular the optimal position of  the surface. An example of  the obtained concentration fields is shown in Figure 2. Concentration contours from direct numerical simulation and the first order effective model are well superposed, apart from a small entrance region where conditions for upscaling break down, which creates a small but acceptable discrepancy. This demonstrates the interest of  using an effective surface model, characterized by a homogeneous, smooth surface, to reproduce in a bulk sense flows over heterogeneous, rough surfaces.  Ω0∑0∑effvwxv wxc∑∑effcΩi Figure 1: Definition of  an effective surface.   Figure 2: Concentration field contours for the initial and the effective domain, for a flow entrance velocity of  1m∙s-1 and Schmidt number of  1.  References [1] M. Quintard and S. Whitaker. Dissolution of  an immobile phase during flow in porous media. Industrial & Engineering Chemistry Research, 38(3), 833-844(1999). [2] M. Suzuki and S. Maeda, On the mechanism of  drying of  granular bed - Mass transfer from discontinuous source. Journal of  Chemical Engineering of  Japan, 1(1), 26-31(1968). [3] Y. Achdou, O. Pironneau and F. Valentin. Effective boundary conditions for laminar flows over periodic rough boundaries. Journal of  computational physics, 147, 187-218(1998). [4] S. Veran, Y. Aspa and M. Quintard. Effective boundary conditions for rough reactive walls in laminar boundary layers. International Journal of  Heat and Mass Transfer, 52, 3712-3725(2009). [5] C. Introïni, M. Quintard and F. Duval. Effective surface modelling for momentum and heat transfer over rough surfaces: Application to a natural convection problem. International Journal of  Heat and Mass Transfer, 54, 3622-3641(2011). 

Dissolution of an immobile phase during flow in porous media.

Effective boundary conditions for laminar flows over periodic rough boundaries.

Effective boundary conditions for rough reactive walls in laminar boundary layers.

Effective surface modelling for momentum and heat transfer over rough surfaces: Application to a natural convection problem.

On the mechanism of drying of granular bed - Mass transfer from discontinuous source.

Effective Surface and Boundary Condition for Heterogeneous Salt Media with Insoluble Material

http://oatao.univ-toulouse.fr/9091/1/Guo_9091.pdf

Effective Surface and Boundary Condition for Heterogeneous Salt Media with Insoluble Material

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