Convective Heat Transfer Assignment 2 — Convective Heat Transfer on Solids  
Question #1
Consider a fin with rectangular cross-section attached to a wall maintained at a temperature $T_0$. The fin is cooled by a fluid with a convective heat transfer coefficient $h$ and a temperature $T_\infty$ (fluid temperature far from the fin). The fin has a length $L$, a depth $D$ and a thickness $t$. The cross-sectional area of the fin corresponds to $A=D\,t$:
question3.png  ./download/file.php?id=3379&sid=e9ddf4c0c005c5adb4d01bc224b8437b  ./download/file.php?id=3379&t=1&sid=e9ddf4c0c005c5adb4d01bc224b8437b
Given the thermal conductivity of the fin, $k$, and assuming that the convective heat transfer coefficient $h$ is constant over all the fin exposed surfaces, derive an expression for the conduction heat transfer at the base of the fin (i.e., where the fin is attached to the wall). Note: the fin tip is not insulated.
Question #2
Consider a combustor of a turbojet engine made of a 1 m long hollow steel cylinder, with the cylinder outer radius being of $r_{\rm o}=0.3$ m and the cylinder inner radius being of $r_{\rm i}=0.25$ m. Gases flow within the combustor at a temperature of $2000^\circ$C with a convective heat transfer coefficient between the cylinder and the gases of $h_{\rm in}=4$ W/m$^2$$^\circ$C (including radiation). On the outside of the combustor, some cool air is flowing at a temperature of 10$^\circ$C and a convective heat transfer of $h_{\rm out}=5$ W/m$^2$$^\circ$C (including radiation). You weld 8 steel fins on the outside of the combustor to cool it, as depicted below.
combustor-and-fins.png  ./download/file.php?id=3376&sid=e9ddf4c0c005c5adb4d01bc224b8437b  ./download/file.php?id=3376&t=1&sid=e9ddf4c0c005c5adb4d01bc224b8437b
Each fin is 3 mm thick with a width of 1 m (spanning the length of the cylinder). You wish to minimize the length of the fins $L_{\rm fin}$ as much as possible to keep the weight of the combustor down while resulting in sufficient cooling so that the combustor temperature anywhere (i.e., anywhere within the steel) doesn't exceed 800$^\circ$C. What would be the optimal fin length that would accomplish this? Specifically, given the following air, gases, and steel properties:
Matter$\rho,~{\rm kg/m}^3$$c,~{\rm J/kg^\circ C}$$k,~{\rm W/m^\circ C}$
Do the following:
1.  Find the heat transfer at the base of one fin $q_{\rm fin}$ that will result in the combustor steel temperature not exceeding $800^\circ$C anywhere.
2.  Find the fin length $L_{\rm fin}$ that yields the heat transfer found in part 1.
Hint: You can assume that the outer surface of the combustor is insulated except for the fins.
1.  $$ q=kA(T_0-T_\infty)m\left( \frac{(h/km){\rm cosh}(mL)+{\rm sinh}(mL)} {{\rm cosh}(mL)+(h/km){\rm sinh}(mL)} \right)$$ $${\rm with~~~} m=\sqrt{\frac{2h(D+t)}{kA}} $$
2.  942.5 W, $796^\circ$C, 0.274 m.
Due on Monday April 10th at 16:30. Do both problems.
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