The products of combustion from a burner are routed to an industrial application through a thin-walled metallic duct of diameter D; = I m and length L = 100 In. The gas enters the duct at atmospheric pressure and a mean temperature and velocity of Tm.i = 1600 K and um,i; = 10m/s, respectively. It must exit the duct at a temperature that is no less than Tm,o = 1400 K. What is the minimum thickness of an alumina-silica insulation (k ins = 0.125 W/m ∙ K) needed to meet the outlet requirement under worst case conditions for which the duct is exposed to ambient air al T∞ = 250 K and a cross-flow velocity of V = 15 m/s? The properties of the gas may be approximated as those of air and as a first estimate, the effect of the insulation thickness on the convection coefficient and thermal resistance associated with the cross flow may be neglected.