Question $3)$: For the wall assembly shown below, the "wall $1"$ is generating volumetric energy of

$q_{G1}^{}[\frac{W}{m^3}]=q_w^{}(\frac{x}{L_1})$

Where $q_w^{}[\frac{w}{m^3}]$ is constant, positive, and known.

On the other hand, the "wall $3"$ is generating volumetric energy at constant rate of $q_{G3}^{}[\frac{W}{m^3}].$ You can ignore the contact resistances.

a$)$ Find the heat flux at $x=L_1($i$.$e$.$ the interface of "wall $1"$ and "wall $2").$

b$)$ Find the temperature at $x=L_1+L_2+L_3($i$.$e$.$ the interface of "wall $3"$ and "wall $4").$

c$)$ Assume the temperature profile in "wall $3"$ has the form of:

$T_3(x)=\frac{-q_{G3}^{}}{2k_3}{x}^2+ax+b,$ where a and $b$ are constant and unknow

i$.$ Find values of a and $b.$

ii$.$ Calculate the value of temperature at $x=L_1+L_2($i$.$e$.$ the interface of "wall $2"$ and "wall $3").$

d$)$ Sketch the temperature profile in this wall assembly qualitatively $($use the sketch below$).$

Sketch the temperature peofile here

Given data:

$q_w^{}=5{10}^5[\frac{W}{m^3}],q_{G3}^{}=1{10}^5[\frac{W}{m^3}]$

$k_1=100[\frac{W}{mK}],k_2=50[\frac{W}{mK}],k_3=500[\frac{W}{mK}],k_4=200[\frac{W}{mK}]$

$T_{}=300[K],h_{}=200[\frac{W}{m^{2}K}]$

$L_1=0\mathrm{.}1[m],L_2=0\mathrm{.}4[m],L_3=0\mathrm{.}1[m],L_4=0\mathrm{.}4[m]$