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1、Summer University, Vrnjacka Banja, October 2007,1,Mathematical models for mass and heat transport in porous media II.,Agneta M.Balint and Stefan Balint West University of Timisoara, Romania Faculty of Mathematics- Computer Science Faculty of Physics balintbalint.uvt.ro; balintphysics.uvt.ro,Summer U
2、niversity, Vrnjacka Banja, October 2007,2,TOPICS:,MASS TRANSPORT IN POROUS MEDIA COMPUTATIONAL RESULTS TESTED AGAINST EXPERIMENTAL RESULTS HEAT TRANSPORT IN POROUS MEDIA COMPUTED CONDUCTIVITY FOR THE HEAT TRANSPORT IN POROUS MEDIA TESTED AGAINST EXPERIMENTAL RESULTS,Summer University, Vrnjacka Banja
3、, October 2007,3,5. MASS TRANSPORT IN POROUS MEDIA,We present the mass transport in porous media as it is described byAuriault I.L. and Lewandowska J. in “Diffusion, adsorption, advection, macrotransport in soils”, Eur.J.Mech. A/Solids 15,4, 681-704, 1996.The pollutant transport in soils can be stud
4、ied by means of a model in which the real heterogeneous medium is replaced by the macroscopic equivalent (effective continuum) like in the case of the fluid flow. The advantage of this approach is the “elimination” of the microscopic scale (the pore scale), over which the variables such as velocity
5、or the concentration are measured. In order to develop the macroscopic model the homogenization technique of periodic media may be employed. Although the assumption of the periodic structure of the soil is not realistic in many practical applications, it was found reasonably model to real situations
6、. It can be stated that this assumption is equivalent to the existence of an elementary representative volume in a non periodic medium, containing a large number of heterogeneities. Both cases lead to identical macroscopic models as presented in:Auriault I.L., “Heterogeneous medium, Is an equivalent
7、 macroscopic description possible?” Int.J.Engn,Sci.,29,7,785-795, 1995.,Summer University, Vrnjacka Banja, October 2007,4,The physical processes of molecular diffusion with advection in pore space and adsorption of the pollutant on the fixed solid particles surface can be described by the following
8、mass balance equation:(5.1)(5.2)where c is the concentration (mass of pollutant per unit volume of fluid), Dij is the molecular diffusion tensor, t is the time variable is the flow field and is the unit vector normal to . The coefficient denotes the adsorption parameter ( 0). For simplicity it is as
9、sumed that the adsorption is instantaneous, reversible and linear. The advective motion (the flow) is independent of the diffusion and adsorption. Therefore the flow model (Darcys law and the incompressibility condition)(5.3)(5.4)which has been already presented in the earlier sequence, will be dire
10、ctly used.,Summer University, Vrnjacka Banja, October 2007,5,The derivation of the macroscopic model is accomplished by the application of homogenization method using the double scale asymptotic developments. In the process of homogenization all the variables are normalized with respect to the chara
11、cteristic length l of the periodic cell. The representation of all the dimensional variables, appearing in eqs. (5.1) and (5.2) versus the non-dimensional variables iswhere the subscript “c” means the characteristic quantity (constant) and the superscript “*” denotes the non-dimensional variable. In
12、troducing the above set of variables into eqs. (5.1)-(5.2) we get the following dimensionless equations:,Summer University, Vrnjacka Banja, October 2007,6,In this way three dimensionless numbers appear:the Pclet numberthe Damkhler numberThe Pclet number measures the convection/diffusion ratio in the
13、 pores. The Damkhler number is the adsorption/diffusion ratio at the pore surface. Pl represents the time gradient of concentration in relation to diffusion in the pores. In practice, Pel and Ql are commonly used to characterize the regime of a particular problem under consideration. In the homogeni
14、zation process their order of magnitude must be evaluated with respect to the powers of the small parameter . Each combination of the orders of magnitude of the parameters Ql , Pl , and Pel corresponds to a phenomenon dominating the processes that take place at micro scale and different regime gover
15、ning migration at the macroscopic scale.,Summer University, Vrnjacka Banja, October 2007,7,i). Moderate diffusion, advection and adsorption,the case of: The process of homogenization leads to the traditional phenomenological dispersion equation for an adsorptive solute:(5.8)where: -the effective dif
16、fusion tensor Dij* is defined as:(5.9)and the vector field is the solution of the standard (representative) cell problem: is periodic (5.10)(5.11)(5.12)(5.13),Summer University, Vrnjacka Banja, October 2007,8,-the coefficient Rd, called the retardation factor, is defined as(5.14)with = the total vol
17、ume of the periodic cell= the volume of the fluid in the cellSp = the surface of the solid in the cellIn terms of soil mechanics(5.15)with = the porosityas = the specific surface of the porous medium defined as the global surface of grains in a unit volume of soil; . -the effective velocity is given by the Darcys law.,
