$(50\%)1)$ Solve $3-91$;$2$ Also derive the corresponding equation for a slab $($plate$)$ with a thickness of $2$L$.$ Note: solving this requires you to review the concept of thermal circuit. A good place to start is to review Example $3\mathrm{.}12$ before trying this problem.

$3-91.$ A common situation in metallurgical practice is solidification from a melt in a cylindrical mold. If the Jakob number, Ja$,$ is small, we can assume quasi$-$steady conduction in the cylindrical shell of solidified melt, as well as in the mold wall. Show that the time required to solidify the complete melt is given by

$\frac{t}{r_2^2}=\frac{1}{4Ja}[1+\frac{2}{Bi}]$

Here the Biot number is a pseudo value that accounts for both the outside resistance and the mold wall,

$Bi=\frac{U_o{r}_2}{k}$

$\frac{1}{2r_{2}L{U}_o}=\frac{ln(\frac{r_3}{r_2})}{2k_{w}L}+\frac{1}{2r_{3}L{h}_o}$

where $k$ and $k_w$ are the conductivities of the solidified melt and mold wall, respectively.