Chapter 5, Problem 5.104P

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 $q_s^n.$ Using the approximate solutions of Table 5.2b for $Fo\ge 0.2,$ derive expressions for each of the three geometries for the quantity $\left(T_{\text{s,act}}-T_i\right)/\left(T_{\text{s,lc}}-T_i\right).$ In this expression, $T_{\text{s,act}}$ is the actual surface temperature as determined by the relations of Table 5.2b. and $T_{s,\text{3c}}$ is the temperature associated with lumped capacitance behavior. Determine criteria associated with $\left(T_{\text{s,act}}-T_i\right)/\left(T_{s,1\text{c}}-T_i\right)\le 1.1,$ that is, determine when the lumped capacitance approximation is accurate to within $10\%.$