Numerical Study of MHD Free Convection in a Packed Bed Square Enclosure Using Local Thermal Non-Equilibrium (LTNE) Model

In the present study, natural convection of fluid in a square packed bed enclosure is investigated numerically using a uniform magnetic field. The geometry model is heated from left-hand side vertical wall and cooled from opposite wall with adiabatic condition at both the top and bottom walls. Normally to this enclosure an electric coil was set to generate a uniform magnetic field. The Brinkman–Forchheimer extended Darcy model was used to solve the momentum equations, while the energy equations for fluid and solid phase were solved using the local thermal non-equilibrium (LTNE) model. Computations are performed for a range of the Darcy number from 10 to10, the porosity from 0.3 to 0.9, and Hartmann number from 0 to 75. The results showed that both the strength of applied magnetic field and the packed bed characteristics have significant effect on the flow field and heat transfer.


1-Introduction
Natural convection in MHD flows in packed bed is encountered in a number of problems with technological and scientific interest, which is covering a wide range of basic sciences such as metallurgy, crystal growth and nuclear engineering.Natural convection flows are characterized by a balance between pressure drop and buoyancy forces, in MHD natural convection flows in packed bed, the balance is achieved by inertial, viscous, electromagnetic and buoyancy forces, making the solution is more complicated.The unsteady free convection flow of an electrically conducting fluid between two heated vertical parallel plates with the presence of a magnetic field was investigated by Walker [1] and Singha and Deka [2].Makinde and Mhone [3] investigated the combined effect of a transverse magnetic field and a radiative heat transfer to unsteady free convective flow through a channel filled with porous medium.MHD free convective flow past a vertical porous plate was analyzed in the presence of a constant suction velocity which is normal to the porous wall by Pillai et.
al. [4] and Murthy et.al. [5].Singh [6]  porous medium and submitted to a strong magnetic field.However, the magnetic force has received more attention in the field of metallic materials, and less in the field of non-metallic materials.With the increase of magnetic field intensity, the magnetic force has more effects on the nonmetallic materials.The application of strong magnetic field may be found in the field of medical treatment such as magnetic resonance imaging, while the applications of natural convection in porous medium may be found in nuclear reactors, cooling of radioactive waste containers, heat exchangers, solar power collectors, grain storage, food processing, energy efficient drying process.So, we intend to study the effects of magnetic force on the natural convection in a square packed bed enclosure.The heat and flow characteristics will be studied on the effect of the packed bed parameters and the intensity of magnetic field.

2.1-Geometrical Shape of Studied Problem
The schematic view of the studied problem is shown in Fig. 1.The square enclosure has a side length (a) and it is filled with a saturated packed bed was made from steel.The left vertical wall of the square enclosure is isothermally heated and the opposite wall is isothermally cooled while the other walls are thermally insulated.A uniform magnetic field is applied on the enclosure which be normal to both (x & y) directions as shown in Fig. 1.

2.2-Governing Equations
In the model development, the following assumptions are adopted; the working fluid is mercury of Prandtl number Pr=0.025 and assumed to be incompressible and Newtonian fluid, no phase change occurs and the process is in a steady state, the Boussinesq approximation for buoyancy is adopted, the applied magnetic field is uniform throughout the enclosure, low magnetic Reynolds model is assumed where the induced magnetic field is neglected in comparison with the applied magnetic field, the effect of magnetic field on heating is negligible.The Brinkman-Forchheimer extended Darcy model is used to solve the momentum equations while the energy equations for fluid and solid phases are solved with the local thermal non-equilibrium (LTNE) model.Thus, the governing equations for the present study will take the following forms (Amiri & Vafai [11]); the terms (F Mx & F My ) in the momentum equations represent the magnetic body forces (Lorentz forces) in (x & y) directions respectively and they are defined as follow (Tillack and Morley [13]); where 0 B is the applied magnetic field vector, while ( ) represent the current densities due to the magnetic field and they are defined as follow (Tillack and Morley ) The geometric function F, specific surface area of the packed bed a sf and the fluidto-solid heat transfer coefficient in a packed bed h sf are determined as suggested by ) .9 ( ) 1 ( 6) .9 ( 150 75 .1 where the sphere particle diameter dp can be computed as follow (Amiri & Vafai [11]); while the effective thermal conductivity k feff and k seff in fluid and solid phase energy equations and the mean thermal diffusivity m can be computed as follow (Wang et.

al. [12]);
) . 10 ( Now we introduce the following non-dimensional quantities and parameters (Wang  2),( 3),( 4)&( 5)], we get the dimensionless forms of governing equations as follow; Using the (stream function-vorticity) formulation, the dependent variables will be reduced to only four variables, by differentiating equation ( 13) with respect to (Y) and differentiating equation ( 14) with respect to (X), after that, the first of the two resulted equations is subtracted from the second to eliminate the pressure terms from the momentum equations, thus, equations [(12),( 13)&( 14)] will be transformed to the following equations; where ( ) are the (stream function & vorticity) respectively, and they are defined as follow; After getting the final values of all dependent variables in the flow field, calculations will be made for local and mean Nusselt number, where the local Nusselt number at the hot wall can be found as follow (Wang et.al [12]);

2.3-Boundary Conditions
The hydrodynamic boundary conditions for the present problem at all enclosure walls will obey to the non-slip condition, while the thermal boundary conditions are (the left side wall was kept hot, the right side wall was kept cold and finally each one of the top and bottom walls were kept isothermally insulated), thus the boundary conditions will be as follow;

3-Numerical Solution
The governing equations for ( , , f & s ) can be written in a common form for the (convection-diffusion) problem as follow (Versteeg and Malalasekera [14]); ( ) where the general scalar stands for any one of the dependent variables under consideration, the diffusion coeffecient and the source term S in cartesian form are listed below for each governing equation; -Stream function equation The numerical solution of the governing equations will be made according to the finite volume method to transform the governing equations from partial differential form to discrete algebraic form, this method is based on principle of dividing the flow field to a number of volume elements, each one of them is called (control volume), after that a discretization process (Versteeg and Malalasekera [14]) was carried out by integrating the general conservation equation ( 22) over a control volume element, where this equation will be as follow; A computational program was written in Fortran-90 language to compute the values of the required variables, The discretized algebraic equations are solved by the tri-diagonal matrix algorithm (TDMA).Relaxation factors of about (0.5-0.7) are used for all dependent variables, Convergence was measured in terms of the maximum change in each variable during an iteration where the maximum change allowed for convergence check was (10 -6 ).

4-Results and Discussion
All results were carried out for mercury at (Ra = 10 4 ), where the results show that the flow and temperature field are very affected with the characteristics of both the applied magnetic field and the packed bed.Figs.(2, 3 & 4) show the effect of Darcy number on the stream function, fluid temperature and solid temperature contours respectively at low Hartmann number (Ha=25).Hartmann number and the porosity, at no applied magnetic field (Ha=0) or low Hartman number (Ha=25) it was noted that at porosity range of ( =0.4-0.7), the value of mean Nusselt number increases gradually with the increase of porosity in that range, but when the porosity increases above that range, the value of mean Nusselt number will decrease, and for explaining this phenomenon clearly we must return to Eq.( 20), where it is noted that the whole value of local Nusselt number is multiplicand by the porosity while the second term in this equation (solid phase term) was multiplicand by the inverse of dimensionless thermal conductivity -1 which decreases with the increase of porosity [ ] , so, when the porosity values equal to or less than (0.7), the value of local Nusselt number will increase, while it decreases after that value of porosity because the value of dimensionless thermal conductivity will be very small and it causes decreasing in Nusselt number more than the increasing in it because of porosity, After that with increasing in Hartmann number, we note that the levels of mean Nusselt number will decrease, also we note that at high Hartmann number (Ha=75), the value of Nusselt number decreases with increase of porosity because the magnetic forces will increase due to the increase of fluid volume with increase of porosity.To exhibit the reliability of the presented results, the variation of mean Nusselt number with Darcy number at (Ha=0 & =0.9) were compared with results of Wang et.al.
[12] at the same conditions as shown in Fig. (18), where it is clear the similarity in Nusselt number behavior with the mentionable study but there is a small difference between these values because of the difference in both working fluid and bed material of the compared studies.

5-Conclusions
This paper has presented a numerical investigation of MHD natural convective flow in a packed bed square enclosure by using local thermal non-equilibrium (LTNE) model, and from the resulted data we conclude the following points; 5.1-Generally, both the levels of flow and heat transfer decrease with the increase of Hartmann number as a result of increasing in magnetic forces.
studied the free convection and mass transfer flow of an electrically conducting fluid past a moving vertical porous plate in presence of large suction and under a uniform magnetic field, also in a close field, a study of MHD free convective flow past an infinite vertical oscillating plate through a porous medium was carried out by Chaudhary and Jain [7].Piazza and Ciofalo [8] investigated numerically the MHD buoyant flow in a cubic enclosure with differential heating at two of the enclosure walls, After that the same investigators studied the same field but with internal heating [9].Sarris et.al. [10] studied numerically the MHD free convection flows in circular annular cavities and square cavities.Amiri and Vafai [11] presented a numerical transient simulation of incompressible flow through a packed bed by using (LTNE) model.Wang et.al. [12] presented a numerical investigation of natural convection of fluid in an inclined square enclosure filled with by substituting equations [(7.a) & (7.b)] in equations [(6.a) & (6.b)] respectively we get the final forms of the magnetic body forces as follow; By substituting equations [(11.a)&(11.b)] in equations [(1),( terms of the discrete equation and their values for each governing equation are listed as follow; -For the stream function equation ) Figs.(5, 6 & 7)  show the effect of Darcy number on the stream function, fluid temperature and solid temperature contours respectively at high Hartmann number (Ha=75), where it is noted that the distributions of stream function and temperatures for both the fluid and solid phases at high Hartmann number will comport in a similar behavior to their distributions at low Hartmann number, but their levels will be less than them at low Hartmann number because of the strong effects of the magnetic force on the flow which will be opposite to the flow direction.Figs.(8, 9 & 10) show the effect of porosity on the stream function, fluid temperature and solid temperature contours respectively at low Hartmann number (Ha=25).Fig.(8.a) explains the stream function contour at low porosity ( =0.4), where the flow will take the direction of counterclockwise for the same mentionable reason previously, After that and with

5. 2 - 5 . 3 -
The value of mean Nusselt number increases with increase of Darcy Number at each Hartmann number values.At low Hartmann numbers, the value of mean Nusselt number increases with increase of porosity until the porosity reaches a certain value of about ( 0.6-0.7)where the value of mean Nusselt number will decrease after that value of porosity, while at high Hartmann numbers, the value of mean Nusselt number will decreases after a porosity value less than the mentionable value previously.