B$3.(50$ pts$)$

A large industrial furnace is supported on a long column of fireclay brick, which is $1$ x $1$ m on a side. During steady$-$state operation, installation is such that three surfaces of the column are maintained at $600$ K while the remaining surface is exposed to an airstream for which $\backslash ($ T$\_\{\backslash$infty$\}=300\backslash$mathrm$\{$~K$\}\backslash )$ and $\backslash (\backslash$mathrm$\{$h$\}=12\backslash$mathrm$\{$~W$\}/\backslash$left$(\backslash$mathrm$\{$m$\}^\{2\}\backslash$mathrm$\{$~K$\}\backslash$right$)\backslash ).$ Determine the temperature at the midpoint of the exposed side of the column. Take k$=1\mathrm{.}4$ W$/($m K$).$

Solve

$($a$)$ by hand calculations $($with three finite elements as recommended by the dashed lines in the picture below$).$ Do you see any problems with your hand calculation results? If there is a problem, what should be done to mitigate it$?$

$($b$)$ using MARC with a reasonable number of elements.

Compare the results.