Chapter 1, Problem 1.33P

If $T_s\approx T_{sur}$ in Equation 1.9, the radiation heat transfer coefficient may be approximated as
   $h_{r,a}=4\in \sigma {\bar{T}}^3$
where $\bar{T}\equiv \left(T_s+T_{sur}\right)/2$ . We wish to assess the validity of this approximation by comparing values of $h_r$ and $h_{r,a}$ for the following conditions. In each case, represent your results graphically and comment on the validity of the approximation.
(a) Consider a surface of either polished aluminum $\left(\in =0.05\right)$ or black paint $\left(\in =0.9\right)$ , whose temperature may exceed that of the surroundings $\left(T_{sur}=25°\text{C}\right)$ by 10 to 100°C. Also compare your results with values of the coefficient associated with flee convection in air $\left(T_{\infty }=T_{sur}\right)$ , where $h\left({\text{W/m}}^{\text{2}}\cdot \text{K}\right)=0.98\Delta T^{1/3}$ .
(b) Consider initial conditions associated with placing a workpiece at $T_s=25°\text{C}$ in a large furnace whose wall temperature may be varied over the range $100\le T_{sur}\le 1000°\text{C}$ . According to the surface finish or coating, its emissivity may assume values of 0.05, 0.2, and 0.9. For each emissivity, plot the relative error, $\left(h_r-h_{r,a}\right)/h_r,$ as a function of the furnace temperature.