Chapter 3, Problem 3.89P

A plane wall of thickness and thermal conductivity k experiences a uniform volumetric generation rate $\stackrel{.}{q}.$ As shown in the sketch for Case 1, the surface at $x=-L$ is perfectly insulated, while the other surface is maintained at a uniform, constant temperature $T_o.$ For Case 2, a very thin dielectric strip is inserted at the midpoint of the wall $\left(x=0\right)$ in order to electrically isolate the two sections, A and B. thermal resistance of the strip is $R_t^n=0.0005{\text{m}}^{\text{2}}\cdot \text{K/W}\text{.}$ The parameters associated with the wall are $k=50\text{W/m}\cdot \text{K,}L=20\text{mm,}\stackrel{.}{q}=5\times {10}^6{\text{W/m}}^{\text{3}},$ and $T_o=50°\text{C}\text{.}$

1. Sketch the temperature distribution for Case 1 on $T-x$coordinates. Describe the key features of this distribution. Identify the location of the maximum temperature in the wall and calculate thistemperature.
2. Sketch the temperature distribution for Case 2 on the same $T-x$ coordinates. Describe the key features of this distribution.
3. What is the temperature difference between the two walls at $x=0$ for Case 2?
4. What is the location of the maximum temperature in the composite wall of Case 2? Calculate this temperature.