2019 Heat Transfer Final Exam  
Wednesday June 19th 2019
18:30 — 21:30


Question #1
You are working in the Hypersonic Branch at ADD (Korean Agency for Defense Development) in Daejeon and are the engineer in charge of installing the power generator on-board a ramjet flight vehicle. The power generator is needed to feed a megawatt-class energy weapon. Because of the high power requirements of the energy weapon and because of the necessity to minimize the weight of the power generating device, it is decided to use a MHD (magneto-hydro-dynamic) power generator rather than fuel cells or other alternatives. The MHD generator operates by converting some of the flow kinetic energy to electrical power through the Lorentz force. The Lorentz force appears when a magnetic field is present and when the airflow is sufficiently ionized to permit the flow of current. Your task is to design the cables linking the MHD generator located in the engine to the energy weapon located in the tail of the aircraft. When the energy weapon is activated, the power produced by the MHD generator is of 1 MW with a voltage difference of $300$ Volts. Noting that the MHD generator is located $2$ m away from the energy weapon, it is desired to find the optimal cable design that minimizes weight while keeping the temperature of the polyethylene insulator below melting point. The cable is located in an enclosed area in which there is stagnant air at a pressure of 0.05 atm and at a temperature of $-10^\circ$C. For safe operation the polyethylene layer is given a thickness of 0.5 cm. The electrical resistivity of copper at $20^\circ$C can be taken as $16.8~{\rm n\Omega\cdot m}$. The melting point and the thermal conductivity of polyethylene can be taken as $120^\circ$C and $0.5~{\rm W/m^\circ}$C, respectively. Design the cable with a safety margin: take into consideration that the convective heat transfer coefficient may have an error of 30% and do not let the maximum temperature within the polyethylene approach its melting point by less than $20^\circ$C.
Question #2
Consider the wing of an aircraft as a flat plate of 2.5 m length in the flow direction. The plane is moving at $100$ m/s in air that is at a pressure of 0.7 bar and a temperature of $-10^\circ$C. If the top surface of the wing absorbs solar radiation at a rate of $800$ $\rm W/m^2$, estimate its steady-state temperature with and without the effect of viscous dissipation. Assume the wing to be of solid construction and to have a single, uniform temperature. Ignore incident radiation on the bottom surface and take $\epsilon=0.4$ on the top and bottom surfaces of the wing.
Question #3
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 #4
A copper pipe with an internal diameter of 0.11 m, an external diameter of 0.12 m and a length of 10 m is located 3 m underground. The ground temperature at that depth is of $-20^\circ$C while the temperature of the soil on the surface is of $-30^\circ$C. It is known that there is a thin layer of ice that formed on the inner walls of the pipe. The ice layer has a thickness of 5 mm and is initially at the same temperature as the ground. To get rid of the ice layer within the pipe, it is proposed to pump some hot water at a temperature of 30$^\circ$C and a mass flow rate of 3 kg/s through the pipe. Specifically, knowing that the latent heat of melting for water is of 334 kJ/kg and that the liquid water, ice, copper, and ground properties correspond to:
PropertyCopperLiquid waterIceGround
$c_p$, kJ/kgK0.44.220.3
$k$, W/mK3860.62.31
$\mu$, kg/ms$10^{-3}$----
$\rho$, kg/m$^3$90001000920200
Estimate as accurately as possible the minimum time it will take in seconds for the hot water to melt all of the ice within the pipe. Specifically, do the following:
(a)  Find the bulk temperature of the water exiting the pipe.
(b)  Find the heat transfer from the hot water to the ice.
(c)  Find the minimum time in seconds that is required to melt the ice.
Hint: Take into consideration that the ice melts in a non-uniform manner leading to surface roughness.
Question #5
A piston with a length $L$ and a radius $R_{\rm p}$ moves in a casing at a speed of $u_{\rm p}$ with respect to the casing as follows:
Q5.png  ./download/file.php?id=5210&sid=807797b80c002ed611e40a83b845a465  ./download/file.php?id=5210&t=1&sid=807797b80c002ed611e40a83b845a465
Oil fills the gap between the piston and the casing. Knowing that the length of the piston is much greater than its radius, that the inner radius of the casing is of $R_{\rm c}$, that the temperature of the casing wall is of $T_{\rm c}$ and that no heat is transferred from the oil to the piston, do the following:
(a)  Derive an expression for the velocity of the oil located between the piston and the casing.
(b)  Derive an expression for the heat flux within the oil, $q^"_r$.
(c)  Derive an expression for the heat flux from the oil to the casing, $q^"_{r=R_{\rm c}}$.
Note: The derived expressions must be valid for any value of $R_{\rm p}$ and $R_{\rm c}$. Thus, you can not assume that the gap between the piston and the casing is much smaller than the radius of the piston.
Question #6
A spherical micro-satellite orbiting around the earth is composed of three layers with radii $R_{\rm A}=0.1$ m, $R_{\rm B}=0.3$ m, and $R_{\rm C}=0.4$ m as follows:
Q6.png  ./download/file.php?id=5212&sid=807797b80c002ed611e40a83b845a465  ./download/file.php?id=5212&t=1&sid=807797b80c002ed611e40a83b845a465
At one moment, the temperature within the 3 layers is uniform and equal to $-26^\circ$C. The power is suddenly turned off resulting in no more heat generation within the satellite. Further, the satellite is not subject to any incoming radiation from the sun. The emissivity of the outer layer (exposed to the space vacuum) is of 0.1 and the properties within each layer correspond to:
PropertyLayer ALayer BLayer C
$c$, J/kgK400800900
$k$, W/mK3862200
$\rho$, kg/m$^3$90001402700
Knowing the latter, calculate the amount of time needed for the temperature within the layers to drop to $-30^\circ$C. Hint: because of the slow heat transfer process by radiation, the temperature within all layers can be assumed uniform at one instant.
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