STEADY SOLUTE DISPERSION IN COMPOSITE POROUS MEDIUM BETWEEN TWO PARALLEL PLATES

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1 Journal of Porous Media, 6 : STEADY SOLUTE DISPERSION IN COMPOSITE POROUS MEDIUM BETWEEN TWO PARALLEL PLATES J. Prathap Kumar, J. C. Umavathi, & Ali J. Chamkha, Department of Mathematics, Gulbarga University, Gulbarga , Karnataka, India Manufacturing Engineering Department, Public Authority for Applied Education and Training, Shuweikh 70654, Kuwait Address all correspondence to Ali J. Chamkha achamkha@yahoo.com Original Manuscript Submitted: //0 Final Draft Received: 6/4/03 The Taylor dispersion of a solute for a composite porous medium between two parallel plates is studied analytically. The fluids in both the regions are incompressible and the transport properties are assumed to be constant. The closed-form solutions are obtained in both fluid regions of the channel. The results are presented graphically for various values of porosity, pressure gradient, height of the channel, viscosity, and diffusivity on the concentration. The volumetric flow rate in the channel for variation of viscosity, height, and porous parameter is also found. The validity of the results obtained for composite porous media is verified by comparison with the available porous medium results in the literature for one fluid model, and good agreement is found. KEY WORDS: Taylor dispersion, two-fluid model, horizontal channel, porous media. INTRODUCTION Studies on natural convection in porous enclosures have important applications in engineering and environmental research. Heat exchangers, underground spread of pollutants, environmental control, grain storage, food processing, material processing, geothermal systems, oil extraction, optimal design of furnaces, and packed-bed catalytic reactors are just a few examples of applications of this subject of study. Nield and Bejan 006 give an excellent summary of the subject. Dispersion in composite channels is a subject of intensive investigation. This is due to the rapid development of technology and numerous modern thermal applications relevant to this area, such as cooling of microelectronic devices. A new concept for a two-layer heat sink with countercurrent flow arrangement for cooling of the electronic components was studied by Vafai and Zhu 999. In realistic situations, however, the fluid system oftentimes consists of two and possibly more separate, immiscible liquids, a layer of one liquid overlying a layer of another liquid. The problem formulation now contains additional dynamical ingredients such as the interfacial stresses and deformation of the interface shape. Alzami and Vafai 00 analyzed fluid flow and heat transfer interfacial conditions between a porous medium and a fluid layer. Kuznetsov 000 gives detailed introduction to the applications and analytical studies of forced convection in partly filled porous configurations. The first exact solution for the fluid flow in the interface region was presented in Vafai and Kim 990. In this study, the shear stresses in the fluid and the porous medium were taken to be equal at the interface region. Vafai and Thiyagaraja 987 analytically studied the fluid flow and heat transfer for three types of interfaces, namely, the interface between two different porous media, the interface separating a porous medium from a fluid region, and the interface between a porous medium from a fluid region and the interface between a porous medium and an impermeable medium. Continuity of shear stress and heat flux were taken into account in their study while employing the Forchhiemer Darcy equation in their analysis X/3/$35.00 c 03 by Begell House, Inc. 087

2 088 Kumar, Umavathi, & Chamkha NOMENCLATURE C i concentration of the solute m ratio of the viscosities µ /µ D i molecular diffusion coefficient h i distance between the interface and plates Greek Symbols K i first-order reaction rate constant α i, β i dimensionless reaction rate parameters L i typical length along the flow direction µ i dynamic viscosity Q i volumetric rate κ permeability of the porous medium u i velocity σ porous parameter h i / κ i ū i average velocity ρ density of the fluid u ix fluid velocity ν kinematic viscosity dp i dx i pressure gradient h width ratio h /h Subscripts p pressure, quantities for region and region, respectively The linear encroachment in a two-immiscible-fluid system in a porous medium was studied by Vafai and Alzami 003. Malashetty and Umavathi 997, Malashetty et al. 00, 004, 005, and Umavathi et al. 005, 007 studied two fluid models in horizontal and inclined channels. Recently, Yang and Vafai 0 analyzed the phenomenon of heat flux bifurcation inside a porous medium by considering the convective heat transfer process within a channel partially filled with a porous medium under local thermal nonequilibrium conditions. The fundamental aspects of transport through biomaterials for a vast range of biorelated studies were discussed by Shafahi and Vafai 0. They considered the interfacial interactions in the biodegradation process and found that the porosity alteration as a result of biofilm formation within the carbon bed has a noticeable effect on the removable efficiency. Generally, the mixing of miscible fluids as they flow through porous media is referred to as hydrodynamical dispersion. With respect to the importance of dispersion processes, studies in water quality management, and pollution control, the dispersion has been referred to as a hydraulic mixing process by which the waste concentrations are attenuated while the waste pollutants are being transported downstream. There are large occasions when waste pollutant from industrial plants, urban areas, and other operations reach a natural water course. An ever-increasing pressure on the waste-assimilating capacity of our water resources does make it more difficult to ensure that concentrations of various contaminants will remain below the limits detected by a variety of competing uses. Interest in dispersion in porous media has also resulted from seawater intrusions into coastal aquifer seepage from canals and streams into and through acquifers and the deliberate release of herbicides into canals to kill weeds. The problem of dispersion in flow through packed beds is of central importance to chemical engineers, and it is treated in some detail in textbooks on chemical reaction engineering, at least since the pioneering text of Levenspial 96. Hydrologists and geophysicists have had a natural interest in the topic for nearly a hundred years now Slichter, 905, with good coverage on the subject offered in books devoted to the study of flow through porous media see, e.g., Bear, 97 Scheidegger, 974. This variety of fields of application has propelled the search for experimental data, with a marked peak of activity in the 950s and 960s. The vast majority of data on dispersion are either for air or water near ambient temperature, and some data also have been published for the flow of viscous liquids Blackwell, 96. At a macroscopic level, the quantitative treatment of dispersion is currently based on the use of an equation analogous to Fick s law, with a dispersion coefficient used instead of a molecular diffusion coefficient. The longitudinal or axial dispersion coefficient D L measures dispersion in the direction of flow, whereas dispersion perpendicular to the direction of flow is related to the transverse dispersion coefficient D T. Consideration was given only to the effect of fluid properties on D L. [The effect of fluid properties on D T was reported elsewhere by Delgado and Guedes de Carvalho 00.] Journal of Porous Media

3 Steady Solute Dispersion in Composite Porous Medium between Two Parallel Plates 089 In his pioneering work, Taylor 953 used an equation analogous to the equation φ c + φ v x c φd+φd m G c =0 t x x x to study the dispersion of a soluble substance in a slowmoving fluid in a small-diameter tube, and he primarily focused on modeling the molecular diffusion coefficient using concentration profiles along a tube for a large time. Following that work, Gill and Sankarasubramanian 970 developed an exact solution for the local concentration for the fully developed laminar flow in a tube for all time. Their work shows that the time-dependent dimensionless dispersion coefficient approaches an asymptotic value for a larger time, proving that Taylor s analysis is adequate for steady-state diffusion through tubes. Even though the preceding analyses are primarily concerned with the dispersive flow in small-diameter tubes, because a porous medium can be modeled as a pack of tubes, we could expect similar insights from the advection dispersion models derived for porous media flow. Verwoerd and Kulasiri 00 researched the effects of variable flow velocity on contaminant dispersion in porous flow. The effect of homogeneous and heterogeneous reactions on the dispersion of a solute in a porous medium in a magnetohydrodynamic laminar flow between two parallel plates was studied by Narasimha Murthy and Ali 975. The flow and heat transfers in biological tissues were analyzed by Khaled and Vafai 003. Pertinent works were reviewed to show how transport theories in porous media advance the progress in biology. The main concepts studied were transport in porous media using mass diffusion and different convective flow models such as the Darcy and the Brinkman models. The literature on hydrodynamic dispersion in a porous medium is very sparse despite its versatile applications in many branches of science, engineering, and technology. This dispersion in a porous medium is the macroscopic outcome of the actual movements of the individual solute particles through the pores and the various physical and chemical phenomena that take place within the pores. Keeping in view the various applications of composite porous medium, the objective of this work is to analyze dispersion of a solute for a composite porous medium using Taylor s 953 model. Consider the laminar flow of two immiscible fluids filled with different porous matrices with different permeabilities between two parallel plates distant h + h apart, taking the x-axis along the midsection of the channel and the y-axis perpendicular to the walls as shown in Fig.. Region h y 0 is filled with a fluid-saturated porous medium of density ρ, viscosity µ, and permeability κ under a uniform pressure gradient dp /dx, and region 0 y h is filled with another fluid saturated with a porous medium of density ρ, viscosity µ, and permeability κ under a uniform pressure gradient dp /dx. It is assumed that the fluids are incompressible and the flow is steady, laminar, fully developed, and that fluid properties are constant. The transport properties of both the fluids are assumed to be constant. The flow in both regions is assumed to be driven by a common constant pressure gradient. Under these assumptions, the governing equations of motion for incompressible fluids are as follows: Region : Region : d u dy d u dy κ u = µ dp dx κ u = µ dp dx where u i is the x-component of fluid velocity and p i is the pressure. The subscripts and denote the values for region and region, respectively. The boundary conditions on velocity are no-slip conditions requiring that the velocity vanish at the walls. In addition, continuity of velocity and shear stress at the interface is assumed. With these assumptions, the boundary and interface conditions on velocity become u = 0 at η =. MATHEMATICAL FORMULATION FIG. : Physical model and the coordinate system Volume 6, Number, 03

4 090 Kumar, Umavathi, & Chamkha 3. SOLUTIONS u = 0 at η = u = u at η = η = 0 3 µ h du dη = µ h du dη at η = η = 0 Solutions of Eqs. and using boundary and interface conditions 3 become u = l cosh σ η + l sinh σ η P h σ 4 u = l 3 cosh σ η + l 4 sinh σ η P h σ 5 where σ = h / κ, σ = h / κ, P = /µ dp /dx, P = /µ dp /dx. The average velocity in region is defined by ū = 0 u dη Using the solution of u from Eq. 4, the average velocity becomes ū = l sinh σ σ + l cosh σ σ P h σ The average velocity in region is defined by ū = 0 u dη 6 Using the solution of u from Eq. 5, the preceding equation becomes ū = l 3 sinh σ σ + l 4 cosh σ σ P h σ We assume that a solute diffuses in the absence of a first-order irreversible chemical reaction in the liquid under isothermal conditions. The equation for the concentration C of the solute for the first region satisfies C t + u C x = D C x + C y 7 8 Similarly, the equation for the concentration C of the solute for the second region satisfies C C C + u = D t x x + C y 9 in which D and D are the molecular diffusion coefficients assumed constants for the first and second region, respectively. We now assume that the longitudinal diffusion is much less than the transverse diffusion, that is, C x C y and C x C y If we now consider convection across a plane moving with the mean speed of the flow, then relative to this plane, the fluid velocity is given by u x = u ū = l cosh σ η + l sinh σ η + l 5 0 for region and u x = u ū = l 3 cosh σ η + l 4 sinh σ η + l 6 for region. Introducing the dimensionless quantities θ = t t, t = L u, ξ = x ū t L, η = y h, θ = t t, t = L u, ξ = x ū t L, η = y h Equations 8 and 9 using Eqs. 0 and and using the nondimensional quantities as defined in Eq. become Region : Region : C + u x C = D C t θ L ξ h η C + u x C = D C t θ L ξ h η 3 4 where L and L are the typical lengths along the flow direction for the first and second regions, respectively. Following Taylor 953, we now assume that the partial equilibrium is established in any cross section of the channel so that the variations of C with η and C with η from Eqs. 3 and 4 become Region : Region : C η C η = h D L u x C ξ 5 = h D L u x C ξ 6 Journal of Porous Media

5 Steady Solute Dispersion in Composite Porous Medium between Two Parallel Plates 09 To solve Eqs. 5 and 6, we use the following three types of boundary conditions: The first one connected with an insulated type of boundary conditions, namely, C η = 0 at η = and C η = 0 at η = 7 which expresses the fact that the walls of the channel are impermeable. However, in many biological problems, the condition at the upper wall is conducting and the lower is insulating. In other words, C η = 0 at η = and C = at η = 8 where the former condition represents the impermeable and the latter the permeable. Since the fluids in both the regions is filled with porous matrix, we can also consider the condition that the upper wall is insulating and the lower wall is conducting. That is, C = at η = and C η = 0 at η = 9 3. Case : Concentration Distribution with Impermeable Wall Conditions Equations 5 and 6 are solved exactly for C and C using boundary conditions as defined in Eq. 7, which are given by Region : C = h C l cosh σ η D L ξ + l 5η Region : σ + l sinh σ η σ + l 7 η 0 C = h C l3 cosh σ η D L ξ + l 6η σ + l 4 sinh σ η σ + l 9 η where integrating constants are determined using the entry condition. The volumetric rates at which the solute is transported across a section of the channel of unit breadth Q region and Q region using Eqs. 0, and 0,, respectively, are given by Q = 0 + sinh σ σ sinh σ { h C u x dη = h3 C l D L ξ σ cosh σ + l l σ σ cosh σ sinh σ σ σ cosh σ + sinh σ σ sinh σ σ σ σ + sinh σ σ 3 + l σ + l 5 l [ cosh σ ] } + l 5 6 σ 3 ] +h l 7 l cosh σ σ sinh σ σ + l 5 l +h l 7 l [ cosh σ σ h l 5 l 7 { Q = h C u x dη = h3 C l 3 D L ξ σ 0 + sinh σ + l 3l 4 cosh σ σ σ σ sinh σ + l 4l 6 cosh σ σ σ σ sinh σ cosh σ σ σ + sinh σ σ 3 + l 6 l 4 [ cosh σ + cosh σ sinh σ σ σ 3 + h l 9 l 3 cosh σ σ cosh σ σ sinh σ σ + sinh σ σ + h l 6 l 9 σ + l 4 σ + l 6 l 3 ] } + l h l 9 l 4 3. Case : Concentration Distribution with Upper Wall Conducting and Lower Wall Insulating Condition 3 To find the exact solutions of Eqs. 5 and 6, we require two more interface conditions along with boundary conditions defined as in Eq. 8, which are defined as C = C D C y = D C y at y = y = 0 4 The solutions of Eqs. 5 and 6 using boundary and interface conditions given in Eqs. 8 and 4 are Region : C =a cosh σ η +a sinh σ η +a 3 η +l 7 η +l 8 5 Volume 6, Number, 03

6 09 Kumar, Umavathi, & Chamkha Region : C =a 4 cosh σ η +a 5 sinh σ η +a 6 η +l 9 η +l 0 6 The volumetric rates at which the solute is transported across a section of the channel of unit breadth Q region and Q region are Q = 0 + h a l +a l h C u x dη = h a l sinh σ + h a l σ cosh σ σ + sinh σ σ 3 cosh σ + sinh σ σ +h a l 5 +h l 8 l σ sinh σ + h a l 5 σ sinh σ + h a 3 l cosh σ σ σ cosh σ +h l 7 l sinh σ σ σ +h a 3 l [ cosh σ + sinh σ σ σ cosh σ σ 3 + h a 3 l 5 3 h l 7 l 5 Q = 0 + h l 7 l [ sinh σ σ + cosh σ + h l 8 l cosh σ σ h C u x dη = h a 4 l 3 + h a 4 l 4 + a 5 l 3 cosh σ σ σ ] ] + h l 8 l sinh σ σ + h a 4 l 6 sinh σ σ sinh σ + h l 0 l 3 + h a 5 l 4 σ sinh σ cosh σ σ σ + sinh σ σ 3 + h l 0 l 4 cosh σ σ sinh σ σ + h a 6 l cosh σ σ 3 +h a 6 l 3 + h a 5 l 6 + h a 6 l 4 [ cosh σ σ ] + h l 9 l 6 + h l 0 l 6 ] + h l 9 l 3 [ sinh σ σ + cosh σ + h l 9 l 4 cosh σ σ sinh σ σ σ Case 3: Concentration Distribution with Upper Wall Insulating and Lower Wall Conducting The solutions of Eqs. 5 and 6 using boundary and interface conditions given in Eqs. 9 and 4 are Region : C =a cosh σ η +a sinh σ η +a 3 η +l 7 η +l 8 9 Region : C =a 4 cosh σ η +a 5 sinh σ η +a 6 η +l 9 η +l 0 30 The volumetric rates at which the solute is transported across a section of the channel of unit breadth Q region and Q region are Q = 0 h C u x dη = h a l + sinh σ σ + h a l + a l cosh σ + h a l 5 sinh σ 4σ σ + h l 8 l sinh σ + h a l sinh σ σ σ + h a l 5 cosh σ sinh σ +h a 3 l cosh σ σ σ + h a 3 l sinh σ σ 3 + h a 3 l [ sinh σ σ + h a 3 l h l 7 l cosh σ σ σ sinh σ σ cosh σ σ cosh σ cosh σ + h l 7 l sinh σ σ + h l 8 l cosh σ σ Q = 0 + h a 4 l 4 + a 5 l 3 sinh σ + h l 0 l 3 + h a 5 l 4 σ σ σ 3 ] h l 7 l 5 + h l 8 l 5 3 h C u x dη = h a 4 l 3 + sinh σ σ cosh σ + h a 4 l 6 σ sinh σ σ cosh σ + h a 5 l 6 + h l 0 l 4 σ sinh σ cosh σ σ σ + sinh σ σ 3 + h a 6 l 3 + h a 6 l 6 3 Journal of Porous Media

7 Steady Solute Dispersion in Composite Porous Medium between Two Parallel Plates 093 cosh σ + h l 9 l 4 sinh σ σ σ + cosh σ sinh σ σ + cosh σ σ σ 3 [ cosh σ + h a 6 l 4 σ ] [ sinh σ + h l 9 l 3 σ ] + h l 9 l 6 + h l 0 l 6 3 The constants appearing in all the preceding equations are defined in the appendix. The total volumetric rate Q can be computed using Q = Q + Q for all the three cases. 4. RESULTS AND DISCUSSION The problem of longitudinal dispersion of a solute subject to molecular diffusion through a horizontal channel filled with two immiscible viscous fluids saturated with a porous matrix of different permeabilities following Taylor s dispersion model is studied. Three different types of boundary conditions are studied with the continuity of diffusivity and continuity of shear stress at the interface. The three different types of boundary conditions are both the walls insulating, the lower wall conducting and upper wall insulating, and the lower wall insulating and the upper wall conducting. For insulating-insulating boundary conditions, the effects of various parameters of the flow are shown graphically in Figs. 8. The effect of porous parameter σ σ = σ = σ of region and region on the velocity is shown in Fig.. It is observed that the velocity decreases in both the regions with the increase of porous parameter σ. The dragging effect in both the regions is the same as we are considering σ = σ = σ. This graph is drawn explicitly to understand the interface conditions. The effect of the pressure gradients P and P on the concentration is shown in Fig. 3. As the pressure gradient increases, the solute concentration increases in magnitude in both the regions. Since the boundary conditions are insulating-insulating, the concentration profiles are symmetric. Varying the height of the porous matrix h of region, the concentration decreases in both the regions but is more influential in region compared to region, as shown in Fig. 4a. For variations of h, results show the variation of concentration in both the regions due to the coupling effect. A similar effect is observed varying the height of permeable fluid in region, as seen in Fig. 4b. That is varying h, concentration decreases in both the regions, but its effect is more significant in region compared to region. Figure 5 displays the effect of viscosity on the FIG. : Velocity profiles for different values of porous parameter σ Volume 6, Number, 03

8 094 Kumar, Umavathi, & Chamkha FIG. 3: Concentration profiles for different pressure gradients P and P FIG. 4: Concentration profiles for different values of height ratios: a height ratio h and b height ratio h Journal of Porous Media

9 Steady Solute Dispersion in Composite Porous Medium between Two Parallel Plates 095 FIG. 5: Concentration profiles for different values of viscosity: a viscosity µ and b viscosity µ FIG. 6: Concentration profiles for different values of diffusivity: a diffusivity D and b diffusivity D Volume 6, Number, 03

10 096 Kumar, Umavathi, & Chamkha FIG. 7: Concentration profiles for different values of σ FIG. 8: Volumetric rate Q versus viscosity µ = µ Journal of Porous Media

11 Steady Solute Dispersion in Composite Porous Medium between Two Parallel Plates 097 solute concentration. As the viscosity of permeable fluid in region increases, concentration decreases in region and increases in region, considering equal permeabilities, as shown in Fig. 5a. A similar effect is observed for varying viscosity of the fluid in region, that is, the solute concentration decreases in region and increases in region for varying the viscosity of the fluid in region, considering the same porous matrix in both the regions, as seen in Fig. 5b. Keeping the diffusivity coefficient of region fixed and varying the diffusivity coefficient D of region, the solute concentration decreases in region and remains constant in region, as shown in Fig. 6a. Figure 6b displays that the solute concentration decreases in region and remains constant in region for variations of diffusivity coefficient D. Considering the equal permeabilities in both the regions, the effect of porous parameter σ is to decrease the solute concentration in both the regions, as shown in Fig. 7. It is interesting to note that there is no effect of σ at the interface on the concentration profiles. Here also the concentration profiles are symmetric. The volumetric flow rate for different values of viscosity µ = µ = µ is shown in Fig. 8. Volumetric flow rate increases in magnitude for values of µ approximately up to 0.4, and then there is no change in flow rate for values of µ = 0.4 and onward. Figures 9 8 display the effect of various parameters on the flow for conducting-insulating and insulatingconducting wall conditions. The effect of pressure gradient on solute concentration is to increase for P = P = P > 0 and decrease for P = P = P < 0 in both the regions, as seen in Fig. 9 for conductinginsulating wall boundaries. A similar effect is observed for insulating-conducting wall boundaries, as seen in Fig. 4. The effect of porous parameter σ σ = σ = σ is to decrease the solute concentration in both the regions for conducting-insulating wall boundaries, as shown in Fig. 0. The concentration profiles for variations of σ are symmetric. A similar result is observed for insulatingconducting wall boundaries, as seen in Fig. 5. Varying the height of the fluid region decreases the solute concentration in region and increases it in region, as shown in Fig. a for conducting-insulating wall boundaries, and its effect is very significant near the lower boundary. For insulating-conducting wall boundaries, the effect of h is to increase the solute concentration in both the regions, and its effect is more significant at the upper wall [Fig. 6a]. For conducting-insulating wall bound- FIG. 9: Concentration profiles for different pressure gradients P and P Volume 6, Number, 03

12 098 Kumar, Umavathi, & Chamkha FIG. 0: Concentration profiles for different values of porous parameter σ FIG. : Concentration profiles for different values of height: a height h and b height h Journal of Porous Media

13 Steady Solute Dispersion in Composite Porous Medium between Two Parallel Plates 099 FIG. : Concentration profiles for different values of viscosity: a viscosity µ and b viscosity µ FIG. 3: Concentration profiles for different values of diffusivity: a diffusivity D and b diffusivity D Volume 6, Number, 03

14 00 Kumar, Umavathi, & Chamkha FIG. 4: Concentration profiles for different values of P and P FIG. 5: Concentration profiles for different values of porous parameter σ Journal of Porous Media

15 Steady Solute Dispersion in Composite Porous Medium between Two Parallel Plates 0 FIG. 6: Concentration profiles for different values of height: a height h and b height h FIG. 7: Concentration profiles for different values of viscosity: a viscosity µ and b viscosity µ Volume 6, Number, 03

16 0 Kumar, Umavathi, & Chamkha FIG. 8: Concentration profiles for different values of diffusivity: a diffusivity D and b diffusivity D aries, the effect of height of the fluid h in region is to increase the solute concentration in both the regions, and its effect is significant at the lower plate [Fig. b]. Figure 6b shows that increasing h, the concentration increases in region and decreases in region, and its effect is more significant at the upper plate for insulatingconducting wall boundaries. Figure shows the effect of variations of viscosities of the fluids on concentration for conducting-insulating wall boundaries. Increasing viscosity of the fluid in region decreases the solute concentration in both the regions, as seen in Fig. a. A similar effect is observed for variations of viscosity of the fluid in region, as seen in Fig. b. Concentration profiles are symmetric in both the regions for variations of µ and µ. It is also observed that the concentration profiles for variations of µ and µ are similar for insulating-conducting wall boundaries, as seen in Fig. 7. Fixing the diffusivity coefficient D of region, the variations of diffusivity coefficient D are to increase the solute concentration in region without any variations in region. The effect of D is influential near the lower boundary, as seen in Fig. 3a conducting-insulating. For insulating-conducting wall boundaries, variation of diffusion coefficient D is to decrease the concentration in both the regions. Its effect is more significant at the upper plate, as seen in Fig. 8a. Keeping the diffusivity coefficient D of region fixed, the variations of diffusivity coefficient D of region decrease the solute concentration in both the regions. The variations of D are more effective at the lower plate, as seen in Fig. 3b conducting-insulating. Figure 8b depicts that the solute concentration increases for increasing values of diffusivity coefficient D in region and is invariant for region. Its effect is more significant near the boundary of the upper plate for insulating-conducting wall boundaries. The effect of variations of viscosity µ µ = µ = µ on flow rate for conducting-insulating and insulatingconducting wall boundaries is similar to insulatinginsulating wall boundaries and is not shown graphically. The effect of porous parameter σ on the flow rate is shown in Table. As porous parameter σ increases, flow rate increases in magnitude in both the regions. To validate the results obtained for the two-fluid model, results are compared with Rudraiah and Ng 007 for a onefluid model. An excellent agreement is observed for the values of volumetric flow rate obtained by Rudraiah and Ng 007 for the one-fluid model. Journal of Porous Media

17 Steady Solute Dispersion in Composite Porous Medium between Two Parallel Plates 03 TABLE : Height h = h versus volumetric rate Q porous-porous case when varying porous parameters with respect to width Height Porous parameter Volumetric rate One-fluid model Two-fluid model h = h σ Q Q Q Q ACKNOWLEDGMENT J.P.K. thanks UGC New Delhi for financial support under a UGC Major Research Project. REFERENCES Alzami, B. and Vafai, K., Analysis of fluid flow and heat transfer interfacial conditions between a porous medium and a fluid layer, Int. J. Heat Mass Transfer, vol. 44, pp , 00. Bear, J., Dynamics of Fluids in Porous Media, Elsevier, New York, 97. Blackwell, R. J., Laboratory studies of microscopic dispersion phenomena, Soc. Petrol. Eng. J., vol. 5Pt., pp. 8, 96. Delgado, J. M. P. Q. and Guedes de Carvalho, J. R. F., Measurement of the coefficient of transverse dispersion in packed bed over a range of values of Schmidt number , Transport Porous Media, vol. 44, pp , 00. Gill, W. N. and Sankarasubramanian, R., Exact analysis of unsteady convective diffusion, Proc. R. Soc. London, Ser. A, vol. 36, pp , 970. Khaled, A.-R. A. and Vafai, K., Review: The role of porous media in modeling flow and heat transfer in biological tissues, Int. J. Heat Mass Transfer, vol. 46, pp , 003. Kuznetsov, A. V., Analytical studies of forced convection in partly porous configurations. In: Handbook of Porous Media, K. Vafai, ed., Marcel Dekker, New York, pp. 69 3, 000. Levenspial, O., Chemical Reaction Engineering, John Wiley, New York, 96. Malashetty, M. S. and Umavathi, J. C., Two phase magneto hydrodynamic flow and heat transfer in an inclined channel, Int. J. Multiphase Flow, vol. 3, pp , 997. Malashetty, M. S., Umavathi, J. C., and Prathap Kumar, J., Convective flow and heat transfer in an inclined composite porous medium, J. Porous Media, vol. 5, pp. 5, 00. Malashetty, M. S., Umavathi, J. C., and Prathap Kumar, J., Two fluid flow and heat transfer in an inclined channel containing porous and fluid layer, Heat Mass Transfer, vol. 40, pp , 004. Malashetty, M. S., Umavathi, J. C., and Prathap Kumar, J., Flow and heat transfer in an inclined channel containing a fluid layer sandwiched between two porous layers, J. Porous Media, vol. 8, pp , 005. Narasimha Murthy, S. and Ali, A. A., Homogeneous and heterogeneous reactions on the dispersion of a solute in a porous medium, Ind. J. Pure Appl. Math., vol. 43A, pp , 975. Nield, D. A. and Bejan, A., Convection in Porous Media, 3rd ed., Springer, New York, 006. Rudraiah, N. and Ng, C.-O., Dispersion in porous media with and without reaction: A review, J. Porous Media, vol. 0, pp. 9 48, 007. Scheidegger, A. E., The Physics of Flow through Porous Media, 3rd ed., University of Toronto Press, 974. Shafahi, M. and Vafai, K., Interfacial interactions of biomaterials in water decontamination applications, J. Mater. Sci., vol. 46, pp , 0. Slichter, C. S., Field measurement of the rate of movement of underground waters, U.S. Geol. Surv., Water Supply Paper 40, pp. 6 5, 905. Taylor, G. I., Dispersion of a solute in a solvent under laminar conditions, Proc. R. Soc. London, Ser. A, vol. 9, pp , 953. Umavathi, J. C., Chamkha, A. J., Mateen, A., and Al- Volume 6, Number, 03

18 04 Kumar, Umavathi, & Chamkha Mudhaf, A., Unsteady two fluid flow and heat transfer in a horizontal channel, Heat Mass Transfer, vol. 4, pp. 8 90, 005. Umavathi, J. C., Mateen, A., Chamkha, A. J., and Al- Mudhaf, A., Oscillatory flow and heat transfer in a horizontal composite porous medium, Int. J. Heat Technol., vol. 4, pp , 007. Vafai, K. and Alzami, B., On the linear encroachment in twoimmiscible fluid systems in a porous medium, J. Fluids Eng., vol. 5, pp , 003. Vafai, K. and Kim, S. J., Fluid mechanics of interface region between a porous medium and a fluid layer an exact solution, Int. J. Heat Fluid Flow, vol., pp , 990. Vafai, K. and Thiyagaraja, R., Analysis of flow and heat transfer at the interface region of a porous medium, Int. J. Heat Mass Transfer, vol. 30, pp , 987. Vafai, K. and Zhu, L., Analysis of two-layered micro-channel heat sink concept in electronic cooling, Int. J. Heat Mass Transfer, vol. 4, pp , 999. Verwoerd, W. S. and Kulasiri, G. D., The effects of variable flow velocity on contaminant dispersion in porous flow, MOD- SIM0 Proc., pp , ANU, Canberra, 00. Yang, K. and Vafai, K., Analysis of heat flux bifurcation inside porous media incorporating inertial and dispersion effects an exact solution, Int. J. Heat Mass Transfer, vol. 54, pp , 0. APPENDIX A. Case : Insulating-Insulating Conditions P = µ dp dx P = µ dp dx m = µ µ h = h σ = σ h σ P h σ mhσ sinh σ + P h σ. sinh σ P h σ P h σ sinh σ cosh σ l = mhσ. cosh σ sinh σ + sinh σ cosh σ l 3 = l + P h σ P h σ l = l coth σ P h σ sinh σ l 5 = l sinh σ σ l cosh σ σ l 6 = l 3 sinh σ σ l 4 cosh σ σ l 7 = h C l sinh σ D L ξ l 9 = h D L C ξ l cosh σ + l 5 σ σ l3 sinh σ + l 4 cosh σ + l 6 σ σ A. Case : Conducting-Insulating Conditions P = dp P = dp µ dx µ dx P h σ mhσ sinh σ + P h σ. sinh σ P h σ P h σ sinh σ cosh σ l = mhσ. cosh σ sinh σ + sinh σ cosh σ l =l coth σ P h σ l 3 =l + P h sinh σ σ P h σ l 4 = mhσl l 5 = l sinh σ σ l cosh σ σ l 6 = l 3 sinh σ σ l 4 cosh σ σ a = h C l D L ξ σ a = h C l D L ξ σ a 3 = h C l 5 D L ξ a 4 = h C l 3 D L ξ σ a 5 = h C l 4 D L ξ σ a 6 = h C l 6 D L ξ l 7 = a σ sinh σ a σ cosh σ + a 3 l 8 = a 4 + l 0 a l 9 = D h D [a σ + l 7 ] a 5 σ l 0 = a 4 cosh σ a 5 sinh σ a 6 l 9 A3. Case 3: Insulating-Conducting Conditions P = dp P = dp µ dx µ dx P h σ mhσ sinh σ + P h σ. sinh σ P h σ P h σ sinh σ cosh σ l = mhσ. cosh σ sinh σ + sinh σ cosh σ Journal of Porous Media

19 Steady Solute Dispersion in Composite Porous Medium between Two Parallel Plates 05 l = l coth σ P h σ l 3 = l + P h σ P h σ sinh σ l 4 = mhσl l 5 = l sinh σ σ l cosh σ σ l 6 = l 3 sinh σ σ l 4 cosh σ σ a = h C l D L ξ σ a = h C l D L ξ σ a 3 = h C l 5 D L ξ a 4 = h C l 3 D L ξ σ a 5 = h C l 4 D L ξ σ a 6 = h C l 6 D L ξ l 9 = a 4 σ sinh σ a 5 σ cosh σ a 6 l 0 = a a 4 + l 8 l 7 = D D h [a 5σ + l 9 ] a σ l 8 = a cosh σ + a sinh σ a 3 + l 7. Volume 6, Number, 03