Eng6901 - Final Exam Sample Questions Eng6901 – Heat Transfer I 1 Final Exam Sample Questions The final exam in Eng6901 - Heat Transfer I consists of four questions: (1) energy balance (multimode heat transfer); (2) fins (analysis or design) with other stuff (e.g. sources, energy balances); (3) transient; and (4) radiation (maximum of three bodies). Permitted material in an exam are the text and one formula sheet. Below are some sample questions (with answers) from previous final exams. Solutions are not available for these questions. Enjoy. 1. The concrete slab at the entrance to a parking garage is heated to prevent the formation of ice. It is heated by a grid of wires embedded in the concrete such that the volumetric heat generation rate may be assumed constant within the slab. The slab is 20 cm thick, has thermal conductivity 1.2 W/m·K, and is well insulated on its underside. Unfortunately, over time the surface of the concrete has become pitted and it is being repaired. During the repairs the slab is covered temporarily by a 2 cm layer of wood (kw = 0.12 W/m·K). An air gap exists between the concrete and wood, and the convection coefficient in the air gap (between the surface temperatures of the concrete and wood) is 8 W/m2 ·K. On a particular day, the upper surface of the wood is exposed to 150 W/m2 of incident solar radiation, a convection environment of h = 30 W/m2 ·o C and T∞ = −10o C, and a clear sky with an effective temperature of Tsky = −35o C, i.e. surroundings temperature. The absorptivity of the wood to solar radiation is 0.5, and its emissivity is 0.8. (a) If the maximum power that can be supplied to the heating wires in the slab is 300 W/m2 of (exposed) slab surface area, is this sufficient to prevent water from freezing on the surface of the wood when it is exposed to the environment described above? State all assumptions. (No. Required q 00 = 331.4 W/m2 ) (b) What would be the maximum temperature of the slab if the maximum power was supplied to the slab and the environment was as described above? State all assumptions. (Tmax = 111.7o C) 2. Hot water flows inside a copper pipe (kc = 380 W/m·o C) with inner and outer diameters 24 mm and 26 mm, respectively. The convection coefficient in the water is 100 W/m2 ·o C and its temperature is 60o C. The pipe is located in an uninsulated crawlspace where the convection coefficient is 5 W/m2 ·o C and the air and surroundings temperatures are 5o C. What is the minimum thickness of foamed rubber insulation (kr = 0.03 W/m·o C, = 0.8) that must be placed around the pipe to limit the rate of heat loss from the hot water to less than 12.5 W/m length of pipe? State all assumptions. (tmin > 12.5 mm) 3. A molten aluminium alloy at 900 K is poured into a cylindrical container that is well insulated from large surroundings at 300 K. The inner diameter of the container is 250 mm, and the distance from the surface of the melt to the top of the container is 100 mm. Define the surface of the melt as surface 1, the inner surface of the cylinder above the melt as surface 2 and the surroundings as surface 3, and state all assumptions when answering the following questions. Eng6901 - Final Exam Sample Questions 2 (a) Determine the shape factors F12 , F13 and F23 . Note: If you cannot determine the shape factors assume F12 = 0.4, F13 = 0.6 and F23 = 0.25 to complete part (b). (F12 = 0.542, F13 = 0.458, F23 = 0.339) (b) If the oxidized surface of the melt has an emissivity of 0.55, what is the net rate of radiation heat transfer from the melt? (q1 = 823.6 W) 4. A finned aluminium (kAl = 200 W/m·o C) heat sink is attached to a very thin IC chip. The 00 ) chip and heat sink base dimensions are 40 mm × 40 mm. The contact conductance (Rt,c −4 2 o between the chip and the heat sink is 1 × 10 m · C/W. The base of the heat sink is 4 mm thick, and 12 uniformly spaced rectangular fins of thickness 0.8 mm are metallurgically attached to its upper surface. The heat sink is exposed to air, with ho = 50 W/m2 ·o C and T∞,o = 30o C. The bottom surface of the chip is joined to a circuit board of thickness 4 mm, and thermal conductivity 1 W/m·o C. The contact conductance between the chip and the circuit board is 1 × 10−3 m2 ·o C/W. The other surface of the board is exposed to ambient air for which hi = 10 W/m2 ·o C and T∞,i = 30o C. The dissipation rate of the chip is 20 W, and the maximum allowable temperature in the chip is 70o C. Determine the minimum length of the fins required to prevent the temperature of the chip from exceeding its maximum value. Neglect radiation. State all assumptions. (Lmin = 9 mm) 5. An electric heater consists of a 2 m long, 1 cm diameter rod with thermal conductivity of 2 W/m·K. Energy is generated uniformly within the 1 cm rod at the rate of 1000 W. The rod is covered by an insulating sleeve of thickness 2 mm and thermal conductivity 0.5 W/m·K. The insulating sleeve is covered by a 2 mm thick tube with thermal conductivity 50 W/m·K. The Eng6901 - Final Exam Sample Questions 3 00 = 1 × 10−4 m2 ·o C/W. contact conductance between the insulating sleeve and the tube is Rt,c Cylindrical pin fins of length 2 cm and diameter 4 mm are joined metallurgically to the tube. The fins have thermal conductivity 50 W/m·K. The exposed tube area and the fins are exposed to a convection environment with h = 10 W/m2 ·o C and T∞ = 20o C. How many fins are required to limit the maximum temperature in the heater assembly to 300o C? State all assumptions. (1536 fins) 6. A 20 cm diameter, 20 cm tall cylinder of ice (k = 2 W/m·K, ρ = 920 kg/m3 , cp = 2040 J/kg·K) initially at a uniform temperature of −20o C is placed on an insulated surface on its circular base in a room with air temperature 2.5o C. The heat transfer coefficient on the exposed surfaces of the ice is 10 W/m2 ·K. Determine the time elapsed before the ice cylinder begins to melt. State all assumptions. (4.04 hrs) 7. And now...not one but two, that’s right two transient questions: (a) To sense the temperature in a room, a transducer is made of a sphere of copper (k = 400 W/m·K, cp = 385 J/kg·K, ρ = 8900 kg/m3 ) with a thermocouple located at the center of the sphere. The transducer is used to turn on the air conditioning system when the temperature of the air rises to 21o C. When the temperature of the surrounding air (convection heat transfer coefficient 20 W/m2 ·o C) increases from 20o C to 21o C, the temperature at the center of the sphere should increase from 20o C to 20.1o C in 1 minute. What should be the diameter of the sphere? State all assumptions. (d = 2 cm) (b) Two slabs of aluminium (k = 200 W/m·K, cp = 900 J/kg·K, ρ = 2700 kg/m3 ), one at 100o C and the other at 0o C, are brought in to perfect thermal contact with each other. After two minutes, what is the temperature at a depth of 5 cm in each slab? State all assumptions. (Ta = 36.1o C, TB = 63.9o C) 8. A 0.02 m × 0.5 m strip heating element (surface 1) is centered in front of a 0.5 m diameter half-cylindrical reflector (surface 2) that has length 0.5 m. The reflector surface includes the two half circle end pieces. The heater element is located 0.05 m from the outer edge of the reflector. All of the energy generated in the heating element leaves the right face of surface 1 and the sides of the reflector not facing surface 1 are perfectly insulated. The heater is placed in a large room (surface 3) in which the surface temperature of the walls is maintained at 290 K. The emissivities of the heating element, reflector and room are 0.9, 0.1, and 0.5, respectively. The temperature of the heater surface is T1 = 1100 K. State all asumptions when answering the following questions. (a) Determine the shape factors F12 , F13 , and F23 . Note: If you are unable to determine the shape factors, assume F12 = 0.9, F13 = 0.1 and F23 = 0.5 to complete the remainder of the problem. (F12 = 0.98, F13 = 0.02, F23 = 0.385) (b) What is the rate of heat transfer from surface 1? (q1 = 716.26 W) (c) What is the temperature of the reflector? (T2 = 498.2 K) Eng6901 - Final Exam Sample Questions 9. Question 1, final 2006. (L = 17.5 mm) 10. Question 2, final 2006. (Ti = 47.4o C) 11. Question 3, final 2006. (a) (F12 = 0.1667, F13 = 0.8333, F23 = 0.9948) (b) (q 0 = 8541.2 W/m) (c) (T3 = 732.9 K) 12. Question 4, final 2006. (t = 211 s) 4