Heat Transfer Assignment 3 — Fins and Shapes  
$\xi$ 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.
Question #1
Fins are frequently installed on tubes by a press-fit process. Consider a circumferential aluminum fin having a thickness of 1.0 mm to be installed on a 2.5-cm-diameter aluminum tube. The fin length is 1.25 cm, and the contact conductance may be taken from the tables for a 100-$\mu$inch ground surface. The convection environment is at $20^\circ$C, and $h=125$ W/m$^2\cdot^\circ$C. Calculate the heat transfer for each fin for a tube wall temperature of $200^\circ$C. What percentage reduction in heat transfer is caused by the contact conductance?
Question #2
In certain locales, power transmission is made by means of underground cables. In one example an $8.0$-cm-diameter cable is buried at a depth of 1.3 m, and the resistance of the cable is $1.1\times 10^{-4}~\Omega/$m. The surface temperature of the ground is $25^\circ$C, and $k=1.2~$W/m$\cdot^\circ$C for earth. Calculate the maximum allowable current if the outside temperature of the cable cannot exceed $110^\circ$C. Hint: the heat generation in an electrical cable of length $L$ due to Joule heating is $LR_{\rm elect}I^2$ in Watts with $R_{\rm elect}$ the resistance in Ohms and $I$ the current in amperes and $L$ the length of the cable in meters.
Question #3
A thin rod of length $L$ and constant cross section area has its two ends connected to two walls which are maintained at temperatures $T_1$ and $T_2$, respectively. The rod loses heat to the environment at $T_\infty$ by convection. Derive an expression (i) for the temperature distribution in the rod and (ii) for the total heat lost by the rod through convection.
Question #4
Show that the fin efficiency of a fin with a rectangular cross-section and an insulated tip corresponds to: $$ \eta_{\rm f}= \frac{ {\rm tanh}\left(\sqrt{2}\cdot L^{1.5} \cdot\left(\frac{h}{k A_{\rm m}} \right)^{0.5} \right) } { \sqrt{2}\cdot L^{1.5} \cdot\left(\frac{h}{k A_{\rm m}} \right)^{0.5} } $$ with $A_{\rm m}\equiv L \cdot t$ with $L$ the length of the fin, $t$ the thickness of the fin, $k$ the thermal conductivity, and $h$ the convective heat transfer coefficient. Outline all assumptions.
1.  45.2 W.
2.  1181 A.
3.  $q=\left. {kAm({\rm cosh}(mL)-1)(T_2 + T_1 - 2T_\infty)}\right/ {{\rm sinh}(ml)}$.
Due on Thursday April 6th at 9:00. Do Questions #2, #3, and #4 only.
I added a hint at the end of Question #2.
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