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 $h$.
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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:
question1.png  ./download/file.php?id=1558&sid=58baf02b5b201e35620ce6f0089081b8  ./download/file.php?id=1558&t=1&sid=58baf02b5b201e35620ce6f0089081b8
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 the thickness is used?
Answers
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.
Due on Thursday March 30th at 9:00. Do questions #1, #3, and #4 only.
03.24.17
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