2014 Heat Transfer Midterm Exam  
May 2nd 2014
19:00 — 21:00


NO NOTES OR BOOKS; USE HEAT TRANSFER TABLES THAT WERE DISTRIBUTED; ANSWER ALL 4 QUESTIONS; ALL QUESTIONS HAVE EQUAL VALUE.
06.02.14
Question #1
Two large black plates are separated by a vacuum. On the outside of one plate is a convection environment of $T=80^\circ$C and $h=100~{\rm W/m^2 \cdot ^\circ}$C, while the outside of the other plate is exposed to 20$^\circ$C and $h=15~{\rm W/m^2 \cdot ^\circ}$C. Make an energy balance on the system and determine the plate temperatures.
Question #2
You are working for KAI (Korea Aerospace Industries) and are in charge of the design of the cooling system of the altimeter installed in the cockpit of the A50 fighter jet. The altimeter requires 50 Watts of power to operate and has dimensions of 10 cm$~\times~$10 cm$~\times~$10 cm. The design of the cooling system should be such that it keeps the back surface of the altimeter below 60$^\circ$C while minimizing additional weight. Recalling the theory learned in your Heat Transfer course that you took several years ago at PNU, you decide to cool the altimeter by installing on its backside 10 aluminum fins with a thickness of 2 mm. The fins are rectangular, have a width equal to the one of the altimeter, and are long enough that the tips can be considered insulated. Knowing that the air behind the instrument panel is at a temperature of $20^\circ$C with an associated convective heat transfer coefficient of $h=12$ W/m$^2\cdot^\circ$C, find the value of the fin length that matches the design constraints. Take into consideration the fact that the convective heat transfer coefficient is not known accurately and may vary by as much as 30%.
Question #3
Some radioactive wastes generating 10 W per gram are located in a concrete spherical shell which is itself buried 3 meters in the earth (its center is 1.5 meter below the earth surface). Knowing that the inner and outer radius of the shell are of 15 cm and 25 cm, respectively, and that the concrete has properties of

$c_{\rm c}=900$ J/kg$^\circ$C, $\rho_{\rm c}=2000$ kg/m$^3$, $k_{\rm c}=0.3$ W/m$^\circ$C

that the earth has properties of:

$c_{\rm e}=1000$ J/kg$^\circ$C, $\rho_{\rm e}=700$ kg/m$^3$, $k_{\rm e}=2.0$ W/m$^\circ$C

and that the temperature of the earth surface is of 20$^\circ$, and that the contact resistance between the earth and the shell is of 0.1 m$^2$$^\circ$C/W determine the maximum amount of radioactive wastes (in grams) that can be inserted in the spherical shell such that the temperature anywhere within the concrete shell does not exceed 50$^\circ$C.
Question #4
Consider a block of concrete initially at a temperature of 100$^\circ$C cooled by some air flow as follows:
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Knowing that the dimensions and properties of the concrete can be taken as:

$L=0.4$ m,   $H=0.2$ m,  $D=0.2$ m

$c=900$ J/kg$^\circ$C,   $\rho=2000$ kg/m$^3$,   $k=1.4$ W/m$^\circ$C

and that $h$ can be taken as 14 W/m$^2$$^\circ$C and that $T_\infty$ corresponds to 20$^\circ$C, find the following temperatures at a time 3 hours after the concrete starts to be cooled by the air flow:
(a)  The temperature at the point ($x=0$, $y=H$, $z=L$)
(b)  The average temperature within the concrete
Answers
1.  350 K, 312 K.
2.  7.9 cm.
3.  3.06 gr.
4.  $\rm 24.1^\circ C$, $\rm39.2^\circ C$.
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