Chapter 3, Problem 3.104P

Consider the plane wall of thickness 2L, the infinite cylinder of radius $r_o$, and the sphere of radius $r_o$. Each configuration is subjected to a constant surface heat flux $q_s^{"}$. Using the approximate solutions of Table 5.2b for $Fo\ge 0.2$, derive expressions for each of the three geometries for the quantity $\left(T_{s,\text{act}}-T_i\right)/\left(T_{s,\text{lc}}-T_i\right)$. In this expression. $T_{s,\text{act}}$ is the actual surface temperature as determined by the relations of Table 5.2b, and $T_{s,\text{lc}}$ is the temperature associated with lumped capacitance behavior. Determine criteria associated with $\left(T_{s,\text{act}}-T_i\right)/\left(T_{s,\text{lc}}-T_i\right)\le 1.1$, that is, determine when the lumped capacitance approximation is accurate to within 10%.