Question$-2(40$ points$)$: A solid rod of length $\backslash ($ L $\backslash )$ and radius $\backslash ($ R $\backslash )$ moves downward with a uniform speed $\backslash ($ U $\backslash )$ into a liquid metal bath at temperature $\backslash ($ T$\_\{0\}\backslash ).$ The temperature of the environment is $\backslash ($ T$\_\{\backslash$infty$\}\backslash ),$ and the convection heat transfer coefficient between the rod and the environment is h which is assumed to be uniform and constant all over the rod. It is desired to analyze the temperature distribution in the rod.

a$)$ Starting with the general energy balance equation, obtain a differential equation for the unsteadystate temperature profile in the axial and radial directions, $\backslash ($ T$($z$,$ r$,$ t$)\backslash ).$ State all your assumptions and indicate the reason clearly when you neglect a term. Do not solve the differential equation.

b$)$ Write the proper initial and boundary conditions to solve this equation.

c$)$ Assuming that the temperature gradient in the radial direction is negligible, simplify the differential equation of part $($a$)$ for obtaining the steady$-$state axial temperature profile.

d$)$ Solve the simplified differential equation of part $($c$)$ for constant physical properties $\backslash (\backslash$left$($C$\_\{$p$\},$ k$\backslash$right$)\backslash )$ to get the steady$-$state temperature profile in the axial direction, $\backslash (\backslash$mathrm$\{$T$\}(\backslash$mathrm$\{$z$\})\backslash ).$

e$)$ Calculate the heat transfer rate $\backslash ((\backslash$mathrm$\{$Q$\})\backslash )$ from the rod to environment under steady$-$state conditions.

Question$-3(25$ points$)$: Use scaling to determine under what condition one can neglect the radial temperature gradient in Question$-2.$