Chapter 5, Problem 5.84P

A plane wall of thickness 0.6 m $\left(L=0.3\text{m}\right)$ is made of steel $\left(k=30\text{W/m}\cdot \text{K,}\rho =7900{\text{kg/m}}^{\text{3}},c=640\text{J/kg}\cdot \text{K}\right).$ It is initially at a uniform temperature and is then exposed to air on both surfaces. Consider two different convection conditions: natural convection, by $h=10{\text{W/m}}^{\text{2}}\cdot \text{K,}$ and forced convection. with $h=100{\text{W/m}}^{\text{2}}\cdot \text{K}\text{.}$ You are to calculate the surface temperature at three different times— $t=2.5\min ,$ 25 min, and 250 min—for a total of six different cases.

1. For each of these six cases. calculate the nondimensional surface temperature, ${\theta }_s^{*}=\left(T_s-T_{\infty }\right)/\left(T_i-T_{\infty }\right),$ using four different methods: exact solution, first- term-of-the-series solution. lumped capacitance, and semi-infinite solid. Present your results in a table.
2. Briefly explain the conditions for which (i) the first-term solution is a good approximation to the exact solution, (ii) the lumped capacitance solution is a good approximation. (iii) the semi-infinite solid solution is a good approximation.