Heat Transfer Assignment 2 — 1D Steady Heat Transfer
 Instructions
$\xi$ is a parameter related to your student ID, with $\xi_1$ corresponding to the last digit, $\xi_2$ to the last two digits, $\xi_3$ to the last three digits, etc. For instance, if your ID is 199225962, then $\xi_1=2$, $\xi_2=62$, $\xi_3=962$, $\xi_4=5962$, etc. Keep a copy of the assignment — the assignment will not be handed back to you. You must be capable of remembering the solutions you hand in.
 05.05.14
 Question #1
Radioactive wastes are packed in a thin-walled spherical container. The wastes generate thermal energy nonuniformly according to the relation $S=S_0 [1+\xi_2 (r_0/r)^2]\times [1-(r/r_0)^2 ]$, where $S$ is the local rate of energy generation per unit volume, $S_0$ is a constant, and $r_0$ is the radius of the container. Steady-state conditions are maintained by submerging the container in a liquid which is at $T_\infty$ and provides a uniform convection coefficient $h$.
Obtain an expression for the total rate at which thermal energy is generated in the container. Use this result to obtain an expression for the temperature $T_{\rm w}$ of the container wall.
 Question #2
A plane wall is constructed of a material having a thermal conductivity that varies as the square of temperature according to the relation $k=k_0 [1+ \beta T^2]$. Derive an expression for the heat transfer in such a wall.
 Question #3
The temperature distribution in a certain plane wall is $$\frac{T-T_1}{T_2-T_1}=C_1+C_2 x^2 + C_3 x^3$$ where $T_1$ and $T_2$ are the temperatures on each side of the wall. If the thermal conductivity of the wall is constant and the wall thickness is $L$, derive an expression for the heat generation per unit volume as a function of $x$, the distance from the plane where $T=T_1$. Let the heat generation be $S_0$ at $x=0$.
 Question #4
Consider steady-state one-dimensional heat conduction through a composite wall:
with the lengths $L_{\rm a}=0.1$ m, $L_{\rm b}=0.1$ m, $L_{\rm c}=0.1$ m, $L_{\rm d}=0.1$ m, the thermal conductivities $k_{\rm a}=20$ W/m$\cdot^\circ$C, $k_{\rm b}=50$ W/m$\cdot^\circ$C, $k_{\rm c}=10$ W/m$\cdot^\circ$C, $k_{\rm d}=5$ W/m$\cdot^\circ$C, the heat generation per unit volume $S_{\rm a}=0$, $S_{\rm b}=10^4$ W/m$^3$, $S_{\rm c}=0$, $S_{\rm d}=0$, the convection heat transfer coefficient $h=50$ W/m$^2\cdot^\circ$C, and the temperature of the environment $T_\infty=20^\circ$C. Do the following tasks:
 (a) Sketch a qualitatively accurate temperature profile for this composite wall (b) Find the maximum temperature in the wall.
 Question #5
An electrical current of $700$ Amperes flows through a stainless steel cable having a diameter of 5 mm and an electrical resistance of $6 \times 10^{-4}~\Omega$/m (i.e. per meter of cable length). The cable is in an environment having a temperature of $30^\circ$C, and the total coefficient associated with convection and radiation between the cable and the environment is approximately 25 W/m$^2 \cdot$K.
 (a) If the cable is bare, what is its surface temperature? (b) If a very thin coating of electrical insulation is applied to the cable, with a contact resistance of 0.02 m$^2 \cdot$K/W, what are the insulation and cable surface temperatures? (c) There is some concern about the ability of the insulation to withstand elevated temperatures. What thickness of this insulation $(k=0.5~{\rm W/m \cdot K})$ will yield the lowest value of the maximum insulation temperature? What is the value of the maximum temperature when this insulation thickness is used?
 1. $T_\infty+ \frac{2}{15} S_0 r_0 h^{-1}$. 2. $-\frac{A k_0}{L} \left(T_2-T_1+\frac{\beta}{3}(T_2^3-T_1^3) \right)$. 3. $S_0 - 3\frac{x}{L} S_0- 6 k \frac{x}{L^3} (T_2-T_1)$. 4. $71^\circ$C. 5. $778.66^\circ$C, $778.66^\circ$C, 0.0175 m, $318.18^\circ$C.
 $\pi$