Eng6901 - Heat Transfer I 1 1 Thermal Resistance 1. A square silicon chip (k = 150 W/m K) is of width w = 5 mm on a side and thickness t = 1 mm. The chip is mounted in a substrate such that its sides and back surfaces are insulated, while the front surface is exposed to a coolant. If 4 W are being dissipated in circuits mounted on the back surface of the chip, what is the staedy-state temperature difference between the back and front surfaces? 2. An electric resistance heater is embedded in a long cylinder of diameter 30 mm. When water with a temperature of 25 C and velocity of 1 m/s flows crosswise over the cylinder, the power per unit length required to maintain the surface uniform at 90 C is 28 kw/m. When air, also at 25 C, but with a velocity of 10 m/s is flowing, the power per unit length required to maintain the same surface temperature is 400 W/m. Calculate and compare the convection coefficients for the flows of water and air. 3. The wall of an oven used to cure plastic parts is of thickness L = 0.05 m and is exposed to a large surroundings and air at its outer surface. The air and surroundings are at 300 K. If the temperature of the outer surface is 400 K and the convection coefficient and emissivity are h = 20 W/m 2 C and ɛ = 0.8, respectively, what is the temperature of the inner surface if the wall has a thermal conductivity k = 0.7 W/m C? 4. Plate glass at 600 C is cooled by passing air over its surface such that the convection heat transfer coefficient is h = 5 W/m 2 C. To prevent cracking the temperature gradient must not exceed 15 C/mm at any point in the cooling process. If the thermal conductivity of the glass is 1.4 W/m C and its surface emissivity is 0.8, what is the lowest temperature of the air that can be used for the cooling? Assume that the temperature of the air equals that of the surroundings. 5. The rear window of an automobile is defogged by pasing warm air over its inner surface. If the warm air is at T,i = 40 C and the convection environment is h i = 30 W/m 2 C, what are the inner and outer surface temperatures of the 4 mm thick window glass if the outside ambient temperature T,o = 10 C and the associated convection coefficient is h o = 65 W/m 2 C? 6. A house has a composite wall of wood, fiberglass insulation and plaster board of thicknesses 10 mm, 100 mm and 20 mm, respectively. On a cold day the convection environment on the inner wall surface is h i = 30 W/m 2 C and T,i = 20 C, while on the outer surface it is h o = 60 W/m 2 C and T,o = 15 C. The total wall surface area is 350 m 2. (a) Determine the total rate of heat loss through the wall. (b) If the wind is blowing violently, raising h o to 300 W/m 2 C, determine the percentage increase in heat loss. (c) What is the controlling resistance that determines the amount of heat flow through the wall? 7. The temperature of the steam at a certain location in the cycle of a steam powerplant is 100 C. The steam pipe is made of steel (k = 50 W/m C) and has inner and outer diameters of 5 cm and 5.5 cm, respectively. The pipe is wrapped with a 6 cm layer of insulation (k f = 0.2 W/m C), which is covered with a thin layer of oxidized foil with emissivity 0.9. The air temperature in the region of the pipe is T = 20 C, and the
Eng6901 - Heat Transfer I 2 convection coeficient is h o = 5 W/m 2 C. The temperature of the generating room walls is T s = 10 C. (a) Assuming the convection heat transfer rate in the steam is h i = 500 W/m 2 C, what is the rate of heat loss per meter length of pipe? (b) Suggest an economical means of reducing the heat loss and briefly explain how this method would work. 8. In an experiment, a stainless steel tube (k st = 13.4 W/m C), with 25 mm I.D. and wall thickness of 3 mm is covered with a 1 mm thick zirconia cloth (k z = 0.15 W/m C). A thin metallic electric heating element is placed around the cloth and is covered with 75 mm of fiberglass insulation (k f = 0.036 W/m C). Nitrogen at 40 C flows inside the tube with a convection heat transfer coefficient of 50 W/m 2 C. The surrounding air is at 20 C with a convection heat transfer coefficient of 10 W/m 2 C. The power input to the heating element is 515 W/m length of tube. What is the temperature of the heating element? State all assumptions. 2 Steady-State Conduction with Source Terms 1. A resistance heating element is constructed of a 4 mm diameter wire (k w = 20 W/m C) covered with a 2 mm thick layer of ceramic (k c = 3 W/m C), and a 1 mm thick AISI stainless steel (k st = 15 W/m C, ɛ = 0.7) tube. The contact conductances between the wire and ceramic, and the ceramic and steel are 0.001 C m 2 /W. The resistivity of the wire is ρ = 0.03 Ω cm. The insulated wire is exposed to air with h = 10 W/m 2 C and T = 20 C. The wire is in a room where the walls are maintained at 15 C. The heater element is 2 m long and operates from a 240 V system. Determine the maximum and minimum temperatures in the heater. State all assumptions. (Note: R = ρ L A, where A is the cross-sectional area of the wire, and P = i 2 R.) 2. A planar resistance heater is constructed of a 4 mm thick heating element (k w = 5 W/m C) sandwiched between two 4 mm thicknesses of ceramic (k c = 3 W/m C). One side of the ceramic is insulated with a material having a thermal conductivity of k i = 0.1 W/m C. Both sides of the assembly are covered with 2 mm thick sheets of AISI stainless steel (k st = 15 W/m C). The contact conductances at each material interface are estimated to be 0.001 C m 2 /W. A current is passed through the heating element to uniformly generate heat. The insulated surface of the heater is exposed to air with h = 10 W/m 2 C and T = 20 C, and the other surface rests on a perfectly insulated surface. State all assumptions when answering the following questions. (a) Sketch the temperature distribution in the complete heater assembly. (b) What is the required thickness of insulation if the maximum temperature of the outer surface of the stainless steel exposed to the air is limited to 50 C and the maximum temperature in the heater cannot exceed 300 C? 3. 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 15cm thick, has thermal conductivity 1.2W/m C, and is well insulated on its underside. On a particular day, the slab is exposed to 200W/m 2 of incident solar radiation, a convection environment
Eng6901 - Heat Transfer I 3 of h = 30W/m 2 C and T = 10 C, and a clear sky with an effective temperature of T sky = 30 C. The absorptivity of the slab to solar radiation is 0.9, and its emissivity is 0.5. (97.1) (a) If the maximum power that can be supplied to the heating wires in the slab is 400W/m 2 of (exposed) slab surface area, is this sufficient to prevent water from freezing on the surface of the slab when it is exposed to the environment described above? (b) What would be the maximum temperature of the slab if it was exposed to the environment described above, and the maximum power was supplied to the slab? 4. A very thin IC chip with a heat dissipation rate of 30,000 W/m 2 is exposed to a dielectric liquid at its outer surface, with h o = 1000 W/m 2 C and T,o = 20 C, and is joined to a circuit board at its inner surface. The thermal contact conductance between the chip and board is 10 4 m 2 C/W, and the board thickness and thermal conductivity are L b = 5 mm and k b = 1 W/m C, respectively. The other surface of the board is exposed to ambient air for which h i = 40 W/m 2 C and T,i = 20 C. What is the chip temperature? State all assumptions. (98.1) 3 Fins 1. 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 50 mm 50 mm. The contact conductance (1/h c ) between the chip and the heat sink is 1 10 3 m 2 C/W. The base of the heat sink is 4 mm thick, and an n n array of uniformly spaced square cross-section pin fins of thickness 2 mm and length L are metallurgically attached to its upper surface. The heat sink is exposed to air, with h o = 60 W/m 2 C and T,o = 30 C. The bottom surface of the chip is joined to a circuit board of thickness 5 mm, and thermal conductivity 1 W/m C. The contact conductance between the chip and the circuit board is 1 10 3 m 2 C/W. The other surface of the board is exposed to ambient air for which h i = 8 W/m 2 C and T,i = 30 C. The maximum allowable temperature in the chip is 70 C. Neglect radiation and state all assumptions when answering the following questions. (a) If n = 8 and L = 20 mm, determine the maximum heat transfer rate from the chip. (b) If L = 20 mm and the rate of heat transfer from the chip is 22.9 W, determine n. (c) If n = 8 and the rate of heat transfer from the chip is 22.9 W, determine the required length, L, of the fins.
Eng6901 - Heat Transfer I 4 2. Feedwater in a steam powerplant is being heated by exhaust gases in a heat exchanger. The feedwater flows inside a tube of inner and outer diameters 2.2 cm and 3 cm, respectively. Rectangular profile circumferential fins are press-fit on to the outer surface of the tube. The fins have length 3 cm, thickness 1 mm, are spaced 9 mm (center to center) and the contact conductance between the fin and tube wall is 1 10 4 m 2 K/W. The thermal conductivity of the fins and tube is 50 W/m K. The outside of the tube is exposed to exhaust gases at 850 K with a convection heat transfer coefficient of 100 W/m 2 /K. At a certain location the water inside the tube has a temperature of 300 K, and the convection coefficient in the water is 400 W/m 2 /K. Determine the rate of heat transfer to the water per meter length of finned tube. State all assumptions (neglect radiation). 3. An aluminium heat sink (k Al = 230 W/m oc) is attached to a thin 50 mm 50 mm IC chip. The heat sink has dimensions 50 mm 50 mm 2 mm, and has 12 uniformly spaced rectangular profile fins of length 5 mm, thickness 1 mm, and depth 50 mm. The contact conductance between the chip and the heat sink is 1 10 4 m 2 oc/w. If the heat sink is exposed to a dielectric fluid with h = 1000 W/m 2 oc, and T = 20 o C, determine the maximum heat dissipation rate of the chip if its temperature is limited to 50 o C. Neglect heat transfer from the underside of the chip. State all assumptions. (177.6 W)
Eng6901 - Heat Transfer I 5 4 Forced Convection 1. Copper tubing of inner diameter 10 mm and total length 8 m is soldered to the back of a solar collector plate, which is maintained at a uniform temperature of 70 C by solar radiation. If water enters the tubes at 25 C at a flow rate of 0.02 kg/s, what is the exit water temperature, and the total heat transfer rate for the tube? State all assumptions. 2. A thin walled, uninsulated square cross-section duct of dimension 0.3 m is used to route chilled air at 0.05 kg/s through the attic of a large commercial building. At one location the attic air at 37 C is in crossflow, parallel to a side, with the duct at a velocity of 4 m/s. If chilled air enters the 15 m long duct at 7 C, what is the exit temperature of the air, and the rate of heat gain? Properties of both air streams may be evaluated at 300K. State all assumptions. 3. Consider a thin-walled circular tube of diameter D = 0.025 m submerged in a container of paraffin, which is used to store thermal energy. The container has internal dimensions 0.25 m 0.25 m 3 m. As hot water flows through the tube, heat is transferred to the paraffin, converting it from the solid to the liquid state at the phase change temperature of T = 27.4 C. The latent heat of fusion and density of the paraffin are h if = 244 kj/kg and ρ = 770 kg/m 3, respectively. The thermophysical properties of the hot water may be taken as c p = 4.185 kj/kg C, k = 0.653 W/m C, and µ = 4.67 10 4 kg/m s. When the inlet water temperature is 60 C, the time required to completely melt the paraffin from an initial state of solid at 27.4 C is 2560 s. Determine the mass flow rate of the water. Assume the tube surface temperature to be uniform at the phase change value. State all assumptions. 4. A flat plate of width 1 m is maintained at a uniform surface temperature of T s = 150 C by using independently controlled, heat-generating rectangular modules of thickness a = 10 mm and length b = 50 mm. Each module is insulated from its neighbours, as well as on its back side. Atmospheric air at 25 C flows over the plate at a velocity of 30 m/s. The thermophysical properties of the module are: k = 5.2 W/m C, c p = 320 J/kg C, and ρ = 2300 kg/m 3. (a) Find the required power generation, q in W/m 3, in a module positioned at a distance 700 mm from the leading edge. (b) Find the maximum temperature T max in the heat-generating module. 5. A thin, flat plate of length L = 1 m separates two airstreams that are in parallel flow over opposite surfaces of the plate. One airstream has a temperature of T,1 = 200 C and a velocity of u,1 = 60 m/s, while the other airstream has temperature of T,2 = 25 C and a velocity u,2 = 10 m/s. What is the heat flux between the two streams at the midpoint of the plate? 6. The cover plate of a 1 m 2 m flat-plate solar collector is at 15 C, while ambient air at 10 C is flows over the plate parallel to the 1 m side, with u = 2 m/s. (a) What is the convective heat loss from the plate? (b) If the plate is installed 2 m from the leading edge of a roof and flush with the roof surface, what is the rate of convective heat loss?
Eng6901 - Heat Transfer I 6 7. An array of 10 silicon chips, each of length L = 10 mm on a side is insulated on one surface and cooled on the opposite surface by atmospheric air in parallel flow with T = 24 C and u = 40 m/s. When is use, the same power is dissipated in each chip, maintaining a uniform heat flux over the entire cooled surface. If the temperature of each chip may not exceed 80 C, what is the maximum allowable power per chip? What is the maximum allowable power if a turbulence promoter is used to trip the boundary layer at the leading edge? Would it be preferable to orient the array normal, instead of parallel, to the air flow? 5 Radiation 1. Assuming the sun is a blackbody at 5780K determine: (a) The total emissive pwer of the sun. (b) The maximum monochromatic emissive power. (c) The wavelength at which the maximum monochromtaic emissive power occurs. (d) The percentage of the total emitted radiation that lies in the visible light range 0.35µm λ 0.75µm. 2. How does a greenhouse work? Silica glass transmits 92% of the incident solar radiation in the wavelength range 0.35µm λ 2.7µm. and is essentially opaque to radiation at longer and shorter wavelengths. (a) Estimate the percentage of solar radiation that the glass will transmit. Assume the sun is a blackbody at 5780K. (b) If the garden in the greenhouse radiates like a blackbody at 38 C, what is the percentage of the radiation that will be transmitted through the glass? 3. Consider a long right angle isosoceles triangular duct. Determine all shape factors for the three sides. 4. Consider a model of a radiant heater composed of a circular disk heating element of radius r 1 facing a hemispherical reflector of radius r 2. Determine the shape factors for the disk to reflector, reflector to the room, and the reflector to itself. 5. A circular plate (surface 1) of diameter 0.5 m is maintained at T 1 = 325 C by an electrical heater and is postioned coaxial to a conical shape (surface 2). The back side of surface 2 is perfectly insulated. The spacing between the cone and the plate is 0.5 m. The plate and cone are placed in a large room (surface 3) in which the surface temperature of the walls is maitained at T 3 = 20 C. The emissivities of the surfaces are: ɛ 1 = ɛ 3 = 0.8, ɛ 2 = 0.5. (a) Determine the shape factors F 12, F 13 and F 23. (Helpful hint #1: surface area of a conical surface is πr r 2 + h 2, where r is the radius of the base and h is the vertical height of the cone). (b) To save me a few hairs assume F 12 = 0.2, F 13 = 0.8 and F 23 = 0.6 when answering parts (b) and (c). What is the rate at which power is supplied to the heated plate? (c) What is the temperature of the conical surface?
Eng6901 - Heat Transfer I 7 6. A long duct has an equilateral shape. Surface one is maintained at T 1 = 1100K and has ɛ 1 = 0.6. Surface 2 is maintained at T 2 = 2100K and has ɛ 2 = 0.8. Surface three has ɛ 3 = 0.7 and inputs flux q 3 = 1000 W/m2 to the duct. Determine the radiation heat fluxes on surfaces 1 and 2, q 1 and q 2, and the temperature, T 3, of surface 3. 7. In the arrangement shown below, surface 1, a disk of diameter D 1 = 0.2 m, is being heated by surface 2, a ring heater of inner and outer diameters D i = 0.1 m and D o = 0.2 m, respectively. The spacing between surfaces 1 and 2 is L = 0.2 m. The side of surface 1 facing away from surface 2 is perfectly insulated. All of the radiant energy leaving surface 2 leaves from the side facing surface 1. The two surfaces are in a large enclosure, surface 3. The emissivities of the three surfaces are ɛ 1 = 0.8, ɛ 2 = 0.9, and ɛ 3 = 0.5. If the temperature of surface 3 is T 3 = 300 K, and the heat input at surface 2 is 200 W, answer the following questions. (a) What are the shape factors F 12, F 13 and F 23? Note: If you cannot determine the shape factors use the following (incorrect) values to complete part (b): F 12 = 0.2, F 13 = 0.8 and F 23 = 0.7. (b) What is temperature of the ring heater, i.e. T 2? State all assumptions. (c) What is the temperature of surface 1, i.e. T 1? 6 Fun Final Exam Questions Theses are my three favourite questions ever! No I will not ask questions like these on your final exam. 1. While χmas shopping you buy some Hershey s Kisses and put them in the trunk of your car. Next morning you need a chocolate fix and remember you left the Kisses in your
Eng6901 - Heat Transfer I 8 car overnight, when the temperature was 10. You rush outside for some chocolate, but being a chocoholic you know chocolate only tastes good when it is near room temperature. You place some Kisses on the kitchen counter (on their flat base). Assume the Kiss is a cone with base radius, r = 1 cm, and vertical height, h = 2 cm, with thermophysical properties: ρ = 1000 kg/m 3, c = 3000 J/kg C, and k = 0.5 W/m C. The emissivity of the foil covering the Kiss is 0.1, and assume a contact conductance between the Kiss and foil of R t,c = 1 10 3 m 2 C/W. Assume the kitchen counter is a perfectly insulated surface. The exposed surfaces of the Kiss are in a natural convection environment, h = 5 W/m 2 C, and the room air and surroundings temperatures are 20. The Kisses are at a uniform initial temperature of 10 C. (a) Determine how long you must wait before the Kisses are edible, i.e. they are 15 C. (Helpful Hint: Surface area of a cone = πr r 2 + h 2 + πr 2, Volume of a cone = πr 2 h/3) (b) Briefly describe a means of reducing the time found in part (a), and discuss how the method would work. The Kiss must maintain its shape, and no electricity is allowed. 2. It s Sunday afternoon, and you have decided to bake a cake. The recipe requires butter and eggs. Since you are a good cook you know the eggs and butter should be at room temperature for the fluffiest cake. The butter and eggs are at a uniform temperature of 5 C when they are removed from the refrigerator. The block of butter is 4 cm 4 cm 8 cm, and one of the 4 cm 8 cm faces is placed on the counter (assume a perfectly insulated surface). The eggs may be modelled as 4 cm diameter spheres. Assume a natural convection environment of h = 7.5 W/m 2 C and T = 20 C. (a) Assuming the minimum temperature in the eggs and butter should be 15 C before they are used in the recipe, estimate how long you must wait after removing them from the refrigerator before you may use them in the recipe. Approximate the egg and butter properties with those for water given below. (b) Suggest a means of decreasing the time found in part (a), and briefly explain how this proposal would work. (Melting the butter is not permitted, and a microwave oven is unavailable.) Butter and Egg Properties: k = 0.6 W/m C, ρ = 1000 kg/m 3, c p = 4189 J/kg C 3. Its Sunday afternoon and you have decided to have some roasted garlic with your beef tenderloin. The trimmed head of garlic can be approximated as a cylinder of diameter 4 cm and height 2 cm. Since you are good cook you brush the head of garlic with olive oil (k o = 0.2 W/m K), and then tightly wrap it in aluminium foil (shiny side out) such that there is a 1mm layer of oil between the garlic and the foil. The garlic is initially at a uniform temperature of 20 C and is baked in a preheated oven at 175 C. The wrapped garlic is placed on an oven rack, so assume all surfaces are exposed to the air in the oven. Assume the emissivity of the foil is ɛ f = 0.1, and the convection heat transfer coefficient inside the oven is h = 5 W/m 2 K. Use the following thermophysical properties for the garlic: k = 0.5 W/m K, α = 1.5 0 7 m 2 /s. Assuming the garlic will be cooked when its minimum temperature is 100 C what is the required cooking time?