A numerical study of mixed and forced convection in a vertical packed tube and a packed channel

Abstract

A detailed numerical investigation has been performed for forced and mixed convection in a vertical channel and a cylindrical tube filled with a fluid-saturated porous medium, with particular emphasis on the developing region. The uniform wall temperature boundary condition has been assumed. The full momentum equations derived by Hsu and Cheng have been used, which accounts for variable porosity and permeability as well as viscous and inertia effects. A modification has been proposed to the dispersion conductivity model given by Hsu and Cheng, to take into account the ratio of the particle diameter to the characteristic length of the problem. An expression that accounts for the variation of porosity in the streamwise and cross-stream directions has been introduced to take into account the variation of porosity near the walls, entrance and the exit sections. The empirical constants N, C1 and φa, in the porosity function, and the Ergun constants A and B, in the permeability expression, have been determined by a comparison of the numerical and observed data for the pressure drop in a packed tube. The predicted hydrodynamic entrance length has been found to be 10 to 20 particle diameters long for 0.024 ≤ γ ≤ 0.097 and 1 ≤ ReD ≤ 10^3 (where γ is the ratio of the particle to the tube diameter and Red is the Reynolds number based on the particle diameter, dp) , with the shorter length corresponding to the smaller particle size. For all practical purpose the entrance length can be considered to be about the size of the diameter of the tube (or the plate separation distance). The empirical constants Cd and CJ)in the proposed dispersion model have been determined by comparing the predicted and observed heat flux data. The proposed porosity function with the present dispersion model have been found to predict the observed heat flux data of the air/glass sphere system to within 10% for 0.06 ≤ γ ≤ 0.12 and 10^3 ≤ Red≤2 x 10^4 (where ReD is the Reynolds number based on the tube diameter, D) for the packed tube. For the air/chrome steel sphere system (γ = 0.12 and 0.14, 10^3 ≤ ReD ≤ 2 x 104) the agreement was within 19%. The higher error in this case has been attributed to the large difference between the thermal conductivities of air and chrome steel, in which case the thermal equilibrium assumption invoked in the derivation of the energy equation may not be applicable. For the packed channel geometry (γ=0.06 and 0.12 for the Freon/glass sphere system and γ =0.125 for the 3 4 Freon/chrome steel sphere system, 2 x 10 ≤ ReD ≤ 2 x 10^4) the agreement between the observed and calculated heat flux was within 20%, the discrepancy being due to improper experimentation and variable property effects of the fluid next to the heated surface which was not taken into account in the numerical simulation. The volume averaged method that is used to derive the governing equations has been found to be applicable to problems in which γ≤0.15. The effect of using a fluid with a higher Prandlt number or a solid with a larger thermal conductivity has been observed to enhance heat transfer at high and low flow rates respectively. The exact values of the Reynolds number at which the above mentioned enhancement takes place was found to be dependent on the Prandlt number of the saturating fluid. Finally, it has been predicted in this study that the buoyancy force would play an important role in the heat transfer process for the air/glass sphere system with γ =0.06, if GrD/ReD > 9 x 10^5 for the packed tube and GrH/ReH > 9 x 10^4 for the packed channel configuration.