Eng Heat Transfer I: Sample Final Exam Questions 1

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1 Eng Heat Transfer I: Sample Final Exam Questions 1 The final exam in Eng Heat Transfer I consists of four questions involving steady-state heat transfer. The questions could involve: (a) an energy balance (multimode heat transfer); (b) fins (analysis or design) with other stuff (e.g. sources, energy balances); (c) convection alone or in the context of a larger heat transfer problem, including (a) and (b); and (d) radiation (maximum of three bodies). A formula sheet and properties will be provided. 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 (k w = 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/m 2 K. On a particular day, the upper surface of the wood is exposed to 150 W/m 2 of incident solar radiation, a convection environment of h = 30 W/m 2 oc and T = 10 C, and a clear sky with an effective temperature of T sky = 35 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/m 2 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? (No. Required q = W/m 2 ) (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? (T max = C) 2. Hot water flows inside a copper pipe (k c = 380 W/m C) withinner and outer diameters 24 mm and 26 mm, respectively. The convection coefficient in the water is 100 W/m 2 C and its temperature is 60 C. The pipe is located in an uninsulated crawlspace where the convection coefficient is 5 W/m 2 C and the air and surroundings temperatures are 5 C. What is the minimum thickness of foamed rubber insulation (k r = 0.03 W/m C, ɛ = 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? (t min > 12.5 mm) 3. 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 contact conductance between the insulating sleeve and the tube is R t,c = m 2 C/W. 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/m 2 C and T = 20 C. How many fins are required to limit the maximum temperature in the heater assembly to 300 C? (1536 fins)

2 Eng Heat Transfer I: Sample Final Exam Questions 2 4. 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. (a) Determine the shape factors F 12, F 13 and F 23. (F 12 = 0.542, F 13 = 0.458, F 23 = 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? (q 1 = W) 5. A finned aluminium (k Al = 200 W/m C) heat sink is attached to a very thin IC chip. The chip and heat sink base dimensions are 40 mm 40 mm. The contact conductance (R t,c) between the chip and the heat sink is m 2 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 h o = 50 W/m 2 C and T,o = 30 C. The bottom surface of the chip is joined to a circuit board of thickness 4 mm, and thermal conductivity 1 W/m C. The contact conductance between the chip and the circuit board is m 2 C/W. The other surface of the board is exposed to ambient air for which h i = 10 W/m 2 C and T,i = 30 C. The dissipation rate of the chip is 20 W, and the maximum allowable temperature in the chip is 70 C. Determine the minimum length of the fins required to prevent the temperature of the chip from exceeding its maximum value. Neglect radiation. (L min = 9 mm)

3 Eng Heat Transfer I: Sample Final Exam Questions 3 6. A long rod of 20 mm diameter and thermal conductivity k r = 1.5 W/m K has a uniform internal volumetric thermal energy generation rate of 10 6 W/m 3. The rod is covered with an electrically insulating sleeve of 2 mm thickness and thermal conductivity k i = 0.5 W/m K. A spider with 12 rectangular profile ribs of thickness 4 mm is used to support the rod and maintain concentricity with a thin tube. The base of the spider has thickness 5 mm, and the spider and ribs have a thermal conductivity k s = 175 W/m K. The outside of the thin tube is perfectly insulated. The contact conductance between the rod and sleeve, sleeve and the spider, and fins and spider base is m 2 K/W. Air at 25 C with convection coefficient 20 W/m 2 K passes over the spider surface. If the maximum temperature in the rod is limited to 100 C, determine the minimum required length, L = r 3 r 2, of the ribs. (L = 17.5 mm) 7. A long cylindrical heater element of diameter D = 10 mm, temperature T 1 = 1500 K, and emissivity ɛ 1 = 1 is used in a furnace. The bottom area A 2 has an emissivity ɛ 2 = 0.6 and is maintained at T 2 = 500 K. The side and top walls are fabricated from an insulating, refractory brick with ɛ 3 = 0.9. The length of the furnace normal to the page is very large compared to the width w and the height h. Neglect convection and treat the side and top walls of the furnace as isothermal. (a) Determine the shape factors F 12, F 13 and F 23. (F 12 = , F 13 = , F 23 = ) (b) Determine the power per unit length that must be supplied to the heating element to maintain steady state conditions. (q = W/m) (c) What is the temperature of the furnace wall? (T 3 = K)

4 Eng Heat Transfer I: Sample Final Exam Questions 4 8. Hot water flows inside a stainless steel pipe (k s = 15 W/m K) with inner and outer diameters of 28 mm and 30 mm, respectively. The convection coefficient in the water is 100 W/m 2 K. The pipe is wrapped in a 25 mm thick layer of foamed rubber insulation (k r = 0.03 W/m K, ɛ = 0.8). The insulated pipe is located in a region where the air and surroundings temperature are 5 C. The convection heat transfer coefficient on the outer surface of the rubber insulation is 10 W/m 2 K. If the local heat transfer rate from the water is limited to 7.5 W/m, determine the water temperature. (T i = 47.4 C) 9. An aluminium (k AL = 170 W/m C) heat sink is attached to the 5 cm 2 cm face of a chip. The heat sink is 2 mm thick, and has three evenly spaced rectangular profile fins of height 1.5 cm, thickness 1 mm, and depth 5 cm. Air at 20 C is drawn over the heat sink parallel to the 5 cm dimension. If the contact resistance between the chip and the heat sink is R t,c = m 2 C/W, and the chip dissipates 5 W by heat transfer from the heat sink, estimate the minimum required air velocity if the surface temperature of the chip is limited to 75 C. Use the air properties below. Air properties: ρ = kg/m 3, c p = 1008 J/kg K, k = W/m K, µ = kg/m s, P r =

5 Eng Heat Transfer I: Sample Final Exam Questions 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 T 1 = 1100 K. (a) Determine the shape factors F 12, F 13, and F 23. (F 12 = 0.98, F 13 = 0.02, F 23 = 0.385) (b) What is the rate of heat transfer from surface 1? (q 1 = W) (c) What is the temperature of the reflector? (T 2 = K)