$5\mathrm{.}106$ In Section $5\mathrm{.}5,$ the one$-$term approximution th the

series solution for the temperature discribution wat

developed for a plane wall of thickncss $22.$ that in ini$-$

tially at a uniform temperarure and suddenly subjected

to convection heat transfer. If the waill cin

lumped capacitance $($Equation $5\mathrm{.}7).$ For the condition

shown schematically, we wish to compare predictions

based on the one$-$term approximation, the lutmped

Capacitance method, and a finite$-$difference solytion.

$Tx,f$

$p=7800ke{m}^3$

$c=449$MkeK

$t=15$ Win $h$

$($a$)$ Determine the midplane, $T0,1,$ and surfoce, $1(4,t).$

temperatures at $t=100,200,$ and $500$ susing the

one$-$term approximation to the senes solution, Equa$-$

tion $5\mathrm{.}43.$ What is the Biot number for the system?

$($b$)$ Treating the wall as a lumped capacalangen cal.

culate the temperatures at $t=50,100,200,$ and

$500$ s$.$ Did you expect these results to compare

favorably with those from part $($a$)?$ Why are the

temperarures considerably higher?

$($c$)$ Consider the $2-$ and $5-$node networks shown sche$-$

matically, Write the implicit form of the finite$-$

difference equations for each network, and determine

the temperature distributions for $r=50,100,200,$

and $500$ s using a time increment of $t=1$ s$.$ Pre$-$

pare a table summarizing the results of parts $(a),$

$($b$),$ and $($c$).$ Comment on the relative differences of