Consider one$-$dimensional steady$-$state heat conduction for the three cases shown below. The

conductivity of each wall is $k,$ and its thickness is $L.$ For the first two cases $($a$)$ and $($b$)$ write

down the governing equations and boundary conditions and sketch accurately the

temperature profile $T(x).$ In case $($c$)$ sketch the temperature distribution only. Do not solve

the problems.

$($a$)$ A wall insulated on one side and a specified temperature on the other side.

$($b$)$ A symmetric wall with volumetric heat generation and the same specified temperature on

both sides.

$($c$)$ A two$-$layer wall of equal conductivity and equal thickness $L.$ The outer left$-$hand$-$side

surface is maintained at temperature $T_0,$ and the outer right$-$hand$-$side is maintained at

temperature $T_L$ so that $T_0>T_L.$ Sketch the temperature distribution in both layers if the

heat flux $q_0^{\text{'}\text{'}}$ is known and the contact resistance is $R_{ct}.$