Consider a $Sn-Pb$ alloy $Sn-Pb,$ directionally solidified at a rate of $R={10}^{-4}c\frac{m}{s}$ and use the phase diagram

below, answer questions: $D_L=6\mathrm{.}7{10}^{-6}c\frac{m^2}{s},{}_a=11\mathrm{.}5\frac{g}{c}{m}^3,{}_{}=7\mathrm{.}3\frac{g}{c}{m}^3{m}_L=-1\mathrm{.}2\frac{C}{a}t\%Pb(35$ point$)$

$($a$)$ Use the experimentally determined value of $$ to calculate $S_{}$ and $S_{}.$

$($b$)$ What is the liquid composition at the center of the $$ and $$ lamellar?

$($c$)$ At what distance in front of the eutectic interface does the deviation of the liquid composition from $C_E$

become negligible? Show that this is small compared with $\frac{D_L}{R}.$

$($d$)$ Consider a $Sn-15at.\%Pb$ system, what thermal gradient is required according to the constitutional

supercooling criterion for stability?

$($e$)$ From $($d$),$ draw composition profile when there is forced convection in liquid.

$($f$)$ From $($d$),$ draw composition profile if there is no convection in liquid.

$($g$)$ Calculate fraction of $2-$phase region in each $($e$)$ and $($f$)$ case.