2009 Heat Transfer Midterm Exam  
April 23rd 2009
19:00 — 21:20
 

NO NOTES OR BOOKS; USE HEAT TRANSFER TABLES THAT WERE DISTRIBUTED; ANSWER ALL 4 QUESTIONS; TOTAL POINTS: 100 PTS.
05.05.14
Question #1
25 pts. 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 3.0-cm-diameter aluminum tube. The fin length is 1.5 cm, and the contact conductance may be taken from the tables for a 100-$\mu$inch ground surface. The convection environment is at $30^\circ$C, and $h=140$ W/m$^2\cdot^\circ$C.
(a)  Calculate the heat transfer for each fin for a tube wall temperature of $200^\circ$C.
(b)  What percentage reduction in heat transfer is caused by the contact conductance?
Question #2
25 pts. A plate of stainless steel (18% Chrome, 8% Nickel) has a thickness of 4.0 cm and is initially uniform in temperature at $500^\circ$C. The plate is suddenly exposed to a convection environment on both sides at $80^\circ$C with $h=250~{\rm W/m}^2 \cdot ^\circ$C.
(a)  Calculate the time for the center temperature to reach $100^\circ$C.
(b)  Calculate the time for the face temperatures to reach $100^\circ$C.
Question #3
30 pts. A solid sphere of radius $r_0=1.0~{\rm m}$ is buried in earth. The center of the sphere is at a distance of 5.0 m below the earth's surface. The thermal conductivity of the earth is $k_{\rm earth}=1.2 ~{\rm W/m \cdot {^\circ}C}$, and its surface temperature is $25^\circ {\rm C}$. Inside the sphere, there is a volumetric rate of heat generation given by $S=S_0 [1- (r/r_0)^2 ]$ with $S_0=3750 ~{\rm W/m^3} = {\rm constant}$. The thermal conductivity of the sphere material is $k_{\rm sphere}=25 ~{\rm W/m\cdot {^\circ}C}$. At the interface between the earth and the solid sphere, there is a thermal contact conductance coefficient of $h_{\rm c}=500~{\rm W/m^2 \cdot {^\circ}C}$. Assuming that the temperature distribution inside the sphere is essentially 1-D radial, calculate the following for steady-state conditions:
(a)  the temperature of the earth at the interface between the earth and the sphere;
(b)  the surface temperature of the sphere;
(c)  the maximum temperature inside the sphere.
Question #4
20 pts. A solid cube is heated to a uniform temperature of $520^\circ$C. It is then exposed to a convective cooling environment: $h=1000~{\rm W/m^2\cdot{^\circ}C}$ and $T_\infty=20^\circ$C. The length of each side of the cube is 0.05 m. The properties of the cube material are the following: $\rho=8000~{\rm kg/m}^3$, $c=1000~{\rm J/kg \cdot {^\circ}C}$, and $k=25~{\rm W/m\cdot{^\circ}C}$.
(a)  How long does it take for the centre of each face of the cube to cool down to a temperature of $68.4^\circ$C?
(b)  At $t=200$ s into the cooling process, the cube is wrapped completely in excellent insulation. Calculate its equilibrium temperature.
Answers
1.  64.5 W, 26.4 %
2.  939 s, 879 s
3.  400 C, 401 C, 418.5 C
4.  200 s, 75.5 C
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