Packed beds of spherical particles can be sintered at high temperature to form permeable, rigid foams. A foam sheet of thickness t = 10mm is comprised of sintered bronze spheres, each of diameter D = 0.6 mm. The metal foam has a porosity of ε = 0.25, and the foam sheet fills the cross section of an L = 40mm X W = 40mm wind tunnel. The upper and lower surfaces of the foam are at temperatures T_s = 80°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°C and V = 10 m/s, respectively.

(a) 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?
(b) Assuming one-dimensional conduction in the xdirection, 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_eff = A_p,t /L. Determine the effective thermal conductivity of the foam k_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?

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V, T, L T, X T₂ Unsintered spheres Sintered spheres