Chapter 7, Problem 7.113P

Packed beds of spherical panicles can be sintered at high temperature to form permeable, rigid foams. A foam sheet of thickness $t=10\text{mm}$ is comprised of sintered bronze spheres, each of diameter $D=0.6\text{mm}\text{.}$ The metal foam a porosity of $\epsilon =0.25,$ and the foam sheet fills the cross section of an $L=40\text{mm×W=40}\text{mm}$ wind tunnel. The upper and lower surfaces of the foam are at temperatures $T_s=80°\text{C,}$ and the two other foam edges (the front edge shown in the schematic and the corresponding back edge) are insulated. Air flows in the wind tunnel at an upstream temperature and velocity of $T_i=20°\text{C}$ and $V=10\text{m/s,}$ respectively.

1. Assuming the foam is at a uniform temperature $T_s,$ estimate the convection heat transfer rate to the air. Do you expect the actual heat transfer rate to be equal to, less than, or greater than your estimated value?
2. Assuming one-dimensional conduction in the x-direction, use an extended surface analysis to estimate the heat transfer rate to the air. To do so, show that the effective perimeter associated with Equation 3.70 is $P_{\text{eff}}=A_{p,t}/L.$ Determine the effective thermal conductivity of the foam $k_{\text{eff}}$ by using Equation 3.25. Do you expect the actual heat transfer rate to be equal to, less than, or greater than your estimated value?