Slip and no-slip temperature boundary conditions at interface of porous, plain media'conduction

-The phase distribution nonuniformities near bounding surfaces result in anisotropy and nonuniformity of the effective thermal conductivity tensor. For a two-dimensional porous medium made of cylindrical particles, we evaluate the properties of this tensor for cases where the medium is bounded by the fluid saturating it or by a solid surface. The use of a uniform effective conductivity, such as the bulk (far from the surface) value, along with the near surface temperature distribution results in an error in the calculated heat flux. We examine this error and the errors resulting from the use of other approximations of the effective conductivity near the surface. We also point out a slip in the surface temperature occurring when the bulk effective conductivity and the temperature distribution away from the surface are used to extrapolate the temperature at the interface. A slip coefficient is used to account for this slip in temperature. 1. I N T R O D U C T I O N UNDER the assumption of local thermal equilibrium, a single energy equation can be written for sol idfluid heterogeneous systems such as saturated porous media. This single energy equation includes a local effective thermal conductivity tensor Kc which represents a local volume-averaged molecular conduction through both phases and a local dispersion tensor D ~ which represents the local volume-averaged hydrodynamic dispersion. This dispersion results from the simultaneous presence of a temperature and a velocity gradient within the pore. Because of the anisotropy and nonuniformity of the solid matrix structure, both Ko and D d are in general anisotropic and nonuniform. In this paper we examine the variation of K, near the bounding surface and for a two-dimensional structure. In a later paper we will combine the hydrodynamic analysis, already reported by Sahraoui and Kaviany [1], with a heat transfer analysis in order to obtain the variation of D ~_ Figure 1 (a) is a rendering of the phase distributions near a solid bounding surface where the conductivity of this solid k, b can be different to the solid particle and the fluid conductivities k~ and kr. The heat flows from the bound medium (here a solid) to the fluid phase through A,,r and to the solid phase through A~,. We expect each of the three conductivities k,~, k,, and kr to influence the magnitude of K, at the interface. Also, the distribution of the local porosity significantly influences the magnitude of K~ causing a nonuniformity. This nonuniformity near the bounding surfaces has been recognized by many investigators such as Yagi and Kunii [2], Ofuchi and Kunii [3] and Matsuura et al. [4]. Ofuchi and Kunii attempted to model the nonuniformity of K0 by including a modification which allows for a larger porosity near the bounding solid surface of a packed bed of spherical particles_ This treatment of the interface is called the layered model. In this model the average porosity for the distance of one half of a particle diameter from the boundary is used to evaluate the effective thermal conductivity at the bounding surface. This corresponds to an averaging volume which, although smaller than the representative elementary volume, is too large to represent the pointwise effective conductivity needed for the evaluation of the surface heat flux using the pointwise temperature gradient. The bulk (away from the interface) effective thermal conductivity is also used near the bounding surface, along with the extrapolation of the temperature field away from the boundary. This, in general (and when kr < k,), has resulted in a larger surface heat flux or a surface temperature slip. Then, attempts have been made to model this temperature slip by using a film heat transfer coefficient [4]. The existing rigorous analytical-numerical treatments of the bulk effective conductivity, which are mostly for packed beds of spherical particles, are in general not capable of handling anisotropy and nonuniformity of the particle arrangements near the boundary. The nonuniformity could be modeled using a variable k0± and imposing the continuity of the temperature, i.e. no temperature slip is allowed at the boundary. This is given by the steady state, onedimensional equation with negligible radiation effects