Chapter 4, Problem 4.8P

Consider Problem 4.5 for the case where the plate is of square cross section, $W=L.$
(a)Derive an expression for the shape factor, $S_{\max },$ associated with the maximum top surface temperature, such that $q=S_{\max }k\left(T_{2,\max }-T_1\right)$ where $T_{2,\max }$ is the maximum temperature along $y=W.$
(b)Derive an expression for the shape factor, $S_{avg},$ associated with the average top surface temperature, $q=S_{avg}k\left({\bar{T}}_2-T_1\right)$ where ${\bar{T}}_2$ is the average temperature along $y=W.$
(c)Evaluate the shape factors that can be used to determine the maximum and average temperatures along $y=W.$ Evaluate the maximum and average temperatures for $T_1=0°\text{C,}L=W=10\text{mm,}k=20\text{W/m}\cdot \text{K,}$ and $q_s^n=1000{\text{W/m}}^{\text{2}}.$