Thermal Characteristics Of Ceramic Packed Bed

Convection heat transfer in horizontal channel filled with saturated packed bed has been studied numerically using finite difference technique. The channel wall is heated at constant heat flux and packed with a fluid – saturated spherical ceramic for diameter ratio (0.287). Air , helium and carbon dioxide are used as working fluid at Reynold number ranging (100 – 2500). The results show a significant effect of varying prandtle number on heat transfer rate and friction factor at different Reynold number. The radial temperature profile increase as the prandtle number decrease. The heat transfer rate increase as the prandtle number increase , Carbon dioxide is greater than for air and the last is greater than that for helium at the same flow conditions . Friction factor proportional inversely with prandtl number. New Correlations are obtained in this work:


Introduction
Porous media have been employed widely in thermal energy storage system, food processing , geothermal system, chemical reactors engineering , building insulation and infiltration [1,2] .The structure and porosity of the porous media affected the flow patterns and thermal transport phenomena in porous channel.For that various studies investiged convection heat transfer in a channel packed with spherical particle focused on particle diameter and thermal properties.Vafai etal [3] performed an experiment on the forced convection of water in a channel filled with glass spheres (5,8mm in diameter), the average Nusselt number were measured at selected Reynolds number and increased linearly with it.Renken and poulikatos [4] carried out a similar experiment with (3mm) glass spheres and reported the local Nusselt number.
Hwang etal [5] used another prandtl number fluid at similer experiment, forced convection of Freon -113 (Pr  : 8.06) in a channel packed with small glass spheres (3,5 and 6mm in diameter) and with chrome steel spheres (6.35mm in diameter).It was found that the Nusselt number increased as the particle diameter was decreased.
Most of heat transfer investigation in packed beds are presented in terms of correlations based on Reynolds number [4] , effective thermal conductivity [6] and particle size [7] .The effect of working fluid prandtle number on the pressure drop and heat transfer coefficient have not taken into account in this correlation.
The purpose of the present paper is to investigate numerically forced convection heat transfer in a packed bed of ceramic spherical particles.Air, heluim and carbon dioxide are used as working fluids at constant diameter ratio

Physical Problem Formulation
Consider the basic problem of a cylindrical horizontal channel of inner diameter (Di) heated symmetrically with a constant heat flux, packed with a fluid -saturated spherical ceramic of diameter (dc).The physical model being studied is shown in Fig( 1).Various kinds of thermal properties fluids at the same temperature (Ti) are used as working fluids like air, heluim and Carbon dioxide.
The ratio of diameter of the channel to the packing diameter should be a minimum of 8:1 to 10:1 for wall effects to be small [8] according the ratio (Di/dc) will be taken 7:1 and the core channel porosity is 0.399 in this study.
The governing equations (momentum and energy) are made on the following assumptions : 1-The fluid and porous medium are in 2-Fluid and ceramic physical properties are constant .3-Intraparticle conduction is neglected .
The momentum equation based on Darcy flow model which is popular in porous media convective heat transfer investigations because of its simplicity and good performance [4,6] It is essential to know the permeability (K) of the bed in order to relate the fluid flow rate to the pressure gradient with Darcy law : For a packed -sphere bed, the permeability of the bed is related to the porosity by [5] : For thermal analysis the following energy equation govering the convection phenomena in the channel [9] : There are various numerical techniques to discretize and solve the equations governing the present problem, among which the finite difference approach is the most straight forward method [10] .
In the finite difference approach the flow domain is discretized so that the dependent variables are considered only at discrete points.The typical two-dimensional grid system is shown in Fig( 2) , where (i , J) is the grid point in radial and axial coordinates.
The following central finite difference formules are used : Knowing the velocity distribustion from numerical solution of momentum energy then applied at the analysis of the energy equation .The temperature profiles and the local Nusselt number at any cross section can be calculated.The average values of Nusselt number (Nu x ) can be found: Also pressure drop across the ceramic packed bed are obtained from numerical solution and presented in terms of friction factor (F) [11] .

Results and Discussion
Results of the 2-D model ceramic packed bed analysis taken into account the effect of prandtl number on heat transfer rate and friction factor are presented for Reynold number ranging (100 -2500) and diameter ratio (0.287).The gases considered in this work are CO 2 , air and helium.These gases have different heat transfer properties (specfic heat, thermal conductivity and prandtl number) as shown in table (1) [12] .This difference in properties has a significant effect on the radial temperature distribution as shown in Fig ( 3).The dimensionless temperature profile across the channel radius at volumetric flow rate (0.5 lit/sec)at different position of X are illustrated .It can be seen that the temperature in the packed bed progressively decreases away from the heated channel wall.The temperature profile increase as the prandtle number decrease for that helium is greater than air and carbon dioxide.
The local Nusselt number along the flow direction for different volumetric flow rate are presented in Figs ( 4-6) .It begin with a high value at the inlet of the channel and decreases greatly with an increase in X.As the volumetric flow rate interms of Reynold number increase the local Nusselt number profile will be increased.
Fig. (7) shows the effect of prandtl number on the heat transfer rate at the same inlet and boundary condition .It is seen that the local Nusselt number profile increase as the prandtle number increase, CO 2 is greater than that for air and the last is greater than that for helium at the same flow conditions.Fig. ( 8) shows the correlation of average Nusselt number with Reynolds number.Increasing (Re) yields increasing (Nu av ) , but incremente at CO 2 is greater than that for air and helium.This variables are correlated by : Nu av = 24.548Re 0.238 Pr 0.0787 ---(11) The shap of curves in Fig( 8) is similer to those presented in ref( 5) and shown in Fig( 9).It is seen that average Nusselt number increase as Reynolds number increase at the different particle diameter.
From the numerical solution ( ∆ P/L) is obtained the value of friction factor (F) is evaluation by using Eq.( 10).Fig ( 9) shows the value of (F) versus [Red / (1-φ )] where Red is defined as particle Reynolds number based on ceramic particle diameter.It is noted that (F) proportional inversely with prandtl number.The following equation is obtained to prdict the friction factor across the ceramic packed bed .d ) in apacked channel [5] equivalent tridiagonal set of algebraic equations.

F
Fig (6) Local Nusselt number variation with the horizontal coordinate for Heluim