Problem $1.$ Consider the $1$D plan composite wall shown in the figure made of three regions.

Only in region $B,$ there is a uniform thermal energy generation $q_B^{}.$ The left side of wall

A is insulated, and the right side of wall B is exposed to convection. There is a thermal

resistance between region $B$ and region $C.$ The numerical values are given below.

$k_A=25\frac{W}{m}*K$

$k_B=15\frac{W}{m}*K$

$k_C=50\frac{W}{m}*K$

$q_B^{}=4{10}^6\frac{W}{m^3}$

$T_{}=25C$

$h=1000\frac{W}{m^2}*K$

$L_A=10mm$

$L_B=40mm$

$L_C=20mm$

${?}^{n}C=v.v01m*\frac{N}{v}$

$($a$)$ Write the general solutions of the temperature distribution $T(x)$ in each region at the

steady state.

$($b$)$ Determine the constants in the general solutions using boundary conditions.

$($c$)$ Find the temperatures on all boundaries and interfaces.

$($d$)$ Verify the above solutions with the thermal resistance method.

$($e$)$ Plot the temperature distributions in all three regions.

$($f$)$ Plot the heat flux distribution in all three regions.