Q $11.$ A copper shell $($Area $A_c,$ volume $V_c,$ specific heat capacity $c_c,$ density ${}_c)$ is submerged at the center of another large container $($Outer surface area $A_m)$ filled with liquid mercury $($Volume $V_m,$ specific heat capacity $c_m,$ density ${}_m).$ The system is initially at a uniform temperature $T_i.$ The temperature of container wall is suddenly raised to $T_s.$ Assume negligible Biot number for either of the materials. Find variation of temperature of the copper shell with time. You can assume ${}_c={}_c{V}_c\frac{c_c}{h_c}{A}_c,{}_m={}_m{V}_m\frac{c_m}{h_m}{A}_m$

Ans $T_s+(T_i-T_s)[Aexp(-\frac{a}{2}+t\sqrt[\phantom{2}]{\frac{a^2}{4}-b})+$Bexp$(-\frac{a}{2}-t\sqrt[\phantom{2}]{\frac{a^2}{4}-b})],$ where $a=\frac{1}{{}_m}+\frac{1}{{}_c}+h_c\frac{A_c}{h_m}{A}_m{}_m,b=\frac{1}{{}_m}{}_c$ and $\frac{\text{:}\sqrt[\phantom{2}]{\frac{a^2}{4}-b}}{2}\sqrt[\phantom{2}]{\frac{a^2}{4}-b}$