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.
Question #5
Consider a micro satellite in the shape of a hollow sphere orbiting around the earth in space as follows:
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Electrical circuits located within the satellite generate power with the amount $q_{\rm gen}$ (in Watts). The temperature within either matter A or matter B can not exceed 600 K for safety reasons. The incoming radiation heat flux from the sun varies between being 0 and being $q^"_{\rm sun}=1200~$W/m$^2$. The radiation heat flux from the sun may reflect on adjacent solar panels and may thus englobe the micro satellite from all directions. The thermal conductivities are of $k_{\rm A}=0.5~$W/mK and of $k_{\rm B}=0.2~$W/mK, while the contact conductance between matter A and matter B is of $h_{\rm c}=24.68~$W/m$^2$K. Knowing that the outer surface of the micro-satellite is a black body, and that the dimensions are of $r_1=8~$cm, $r_2=9~$cm, $r_3=10~$cm, do the following:
(a)  Indicate where the maximum temperature will occur (i.e. the precise location within either matter A or matter B).
(b)  Find the maximum allowable $q_{\rm gen}$ that maintains the temperature within both matter A and matter B to less than 600 K.
(c)  Find the temperature on the outer surface of the satellite when the maximum temperature within either matter A or B is of 600 K.
Question #6
Consider a rectangular fin resting on a table as follows:
question5.png  ./download/file.php?id=5035&sid=04ed226fdf62846cb52bc4b3337f07ac  ./download/file.php?id=5035&t=1&sid=04ed226fdf62846cb52bc4b3337f07ac
Knowing that the fin tip is not insulated, that $W \gg L$, and that there is no heat transfer between the fin and the table, do the following:
(a)  Find the temperature at the fin tip (at $x=L$) as a function of $T_0$ and $T_\infty$.
(b)  Find the heat transfer at the fin base (at $x=0$)
Note: you can not assume that the thickness $t$ is much smaller than the length $L$. Outline all assumptions.
1.  45.2 W.
2.  1181 A.
3.  $q=\left. {kAm({\rm cosh}(mL)-1)(T_2 + T_1 - 2T_\infty)}\right/ {{\rm sinh}(ml)}$.
5.  141.4 W.
6.  $T_{\rm L}=T_\infty+\frac{T_0-T_\infty}{\cosh(mL)+\frac{h}{km}\sinh(mL)}$
Due on Wednesday April 3rd at 9:00. Do Questions #1, #5, and #6 only.
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