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Heat and Mass Transfer
An aluminum pan whose thermal conductivity is 237 W/mK has a flat bottom with diameter 15 cm and thickness 0.4 cm. Heat is transferred steadily to boiling water in the pan through its bottom at a rate of 1400 W. If the inner surface of the bottom of the pan is at 105 C, determine the temperature of the outer surface of the bottom of the pan.
A transistor with a height of 0.4 cm and a diameter of 0.6 cm is mounted on a circuit board. The transistor is cooled by air flowing over it with an average heat transfer coefficient of 30 W/m^2K. If the air temperature is 55 C and the transistor case temperature is not to exceed 70 C, determine the amount of power this transistor can dissipate safely. Disregard any heat transfer from the transistor base.
A 0.3-cm-thick, 12-cm-high, and 18-cm-long circuit board houses 80 closely spaced logic chips on one side, each dissipating 0.06 W. The board is impregnated with copper fillings and has an effective thermal conductivity of 16 W/mK. All the heat generated in the chips is conducted across the circuit board and is dissipated from the back side of the board to the ambient air. Determine the temperature difference between the two sides of the circuit board.
A soldering iron has a cylindrical tip of 2.5 mm in diameter and 20 mm in length. With age and usage, the tip has oxidized and has an emissivity of 0.80. Assuming that the average convection heat transfer coefficient over the soldering iron tip is 25 W/m^2K, and the surrounding air temperature is 20 C, determine the power required to maintain the tip at 400 C.
An ice skating rink is located in a building where the air is at T_air=20 C and the walls are at T_w=25 C. The convection heat transfer coefficient between the ice and the surrounding air is h=10 W/m^2K. The emissivity of ice is 0.95. The latent heat of fusion of ice is 333.7 kJ/kg and its density is 920 kg/m^3. (a) Calculate the refrigeration load of the system necessary to maintain the ice at T_s=0 C for an ice rink of 12 m by 40 m. (b) How long would it take to melt 3 mm of ice from the surface of the rink if no cooling is supplied and the surface is considered insulated on the back side?
Consider a large 3-cm-thick stainless steel plate in which heat is generated uniformly at a rate of 5x10^6 W/m^3. Assuming the plate is losing heat from both sides, determine the heat flux on the surface of the plate during steady operation.
Consider a steel pan used to boil water on top of an electric range. The bottom section of the pan is L=0.3 cm thick and has a diameter of D=20cm. The electric heating unit on the range top consumes 1250 W of power during cooking, and 85 percent of the heat generated in the heating element is transferred uniformly to the pan. Heat transfer from the top surface of the bottom section to the water is by convection with a heat transfer coefficient of h. Assuming constant thermal conductivity and one-dimensional heat transfer, express the mathematical formulation (the differential equation and the boundary conditions) of this heat conduction problem during steady operation. Do not solve.
Consider a large plane wall of thickness L=0.3 m, thermal conductivity k=2.5 W/mK, and surface area A=12 m^2. The left side of the wall at x=0 is subjected to a net heat flux of 700 W/m^2 while the temperature at that surface is measured to be 80 C. Assuming constant thermal conductivity and no heat generation in the wall, (a) express the differential equation and the boundary conditions for steady one-dimensional heat conduction through the wall, (b) obtain a relation for the variation of temperature in the wall by solving the differential equation, and (c) evaluate the temperature of the right surface of the wall at x=L.
A transistor with a height of 0.4 cm and a diameter of 0.6 cm is mounted on a circuit board. The transistor is cooled by air flowing over it with an average heat transfer coefficient of 30 W/m^2K. If the air temperature is 55 C and the transistor case temperature is not to exceed 70 C, determine the amount of power this transistor can dissipate safely. Disregard any heat transfer from the transistor base.
A 0.3-cm-thick, 12-cm-high, and 18-cm-long circuit board houses 80 closely spaced logic chips on one side, each dissipating 0.06 W. The board is impregnated with copper fillings and has an effective thermal conductivity of 16 W/mK. All the heat generated in the chips is conducted across the circuit board and is dissipated from the back side of the board to the ambient air. Determine the temperature difference between the two sides of the circuit board.
A soldering iron has a cylindrical tip of 2.5 mm in diameter and 20 mm in length. With age and usage, the tip has oxidized and has an emissivity of 0.80. Assuming that the average convection heat transfer coefficient over the soldering iron tip is 25 W/m^2K, and the surrounding air temperature is 20 C, determine the power required to maintain the tip at 400 C.
An ice skating rink is located in a building where the air is at T_air=20 C and the walls are at T_w=25 C. The convection heat transfer coefficient between the ice and the surrounding air is h=10 W/m^2K. The emissivity of ice is 0.95. The latent heat of fusion of ice is 333.7 kJ/kg and its density is 920 kg/m^3. (a) Calculate the refrigeration load of the system necessary to maintain the ice at T_s=0 C for an ice rink of 12 m by 40 m. (b) How long would it take to melt 3 mm of ice from the surface of the rink if no cooling is supplied and the surface is considered insulated on the back side?
Consider a large 3-cm-thick stainless steel plate in which heat is generated uniformly at a rate of 5x10^6 W/m^3. Assuming the plate is losing heat from both sides, determine the heat flux on the surface of the plate during steady operation.
Consider a steel pan used to boil water on top of an electric range. The bottom section of the pan is L=0.3 cm thick and has a diameter of D=20cm. The electric heating unit on the range top consumes 1250 W of power during cooking, and 85 percent of the heat generated in the heating element is transferred uniformly to the pan. Heat transfer from the top surface of the bottom section to the water is by convection with a heat transfer coefficient of h. Assuming constant thermal conductivity and one-dimensional heat transfer, express the mathematical formulation (the differential equation and the boundary conditions) of this heat conduction problem during steady operation. Do not solve.
Consider a large plane wall of thickness L=0.3 m, thermal conductivity k=2.5 W/mK, and surface area A=12 m^2. The left side of the wall at x=0 is subjected to a net heat flux of 700 W/m^2 while the temperature at that surface is measured to be 80 C. Assuming constant thermal conductivity and no heat generation in the wall, (a) express the differential equation and the boundary conditions for steady one-dimensional heat conduction through the wall, (b) obtain a relation for the variation of temperature in the wall by solving the differential equation, and (c) evaluate the temperature of the right surface of the wall at x=L.
In a food processing facility, a spherical container of inner radius r_1=40 cm, outer radius r_2=41 cm, and thermal conductivity k=1.5 W/mK is used to store hot water and to keep it at 100 C at all times. To accomplish this, the outer surface of the container is wrapped with a 800-W electric strip heater and then insulated. The temperature of the inner surface of the container is observed to be nearly 120 C at all times. Assuming 10 percent of the heat generated in the heater is lost through the insulation, (a) express the differential equation and the boundary conditions for steady one-dimensional heat conduction through the container, (b) obtain a relation for the variation of temperature in the container material by solving the differential equation, and (c) evaluate the outer surface temperature of the container. Also determine how much water at 100 C this tank can supply steadily if the cold water enters at 20 C.
Consider a large 5-cm-thick brass plate (k=111 W/mK) in which heat is generated uniformly at a rate of 2x10^5 W/m^3. One side of the plate is insulated while the other side is exposed to an environment at 25 C with a heat transfer coefficient of 44 W/m^2K. Explain where in the plate the highest and lowest temperatures will occur, and determine their values.
Heat is generated uniformly at a rate of 4.2x10^6 W/m^3 in a spherical ball (k=45 W/mK) of diameter 24 cm. The ball is exposed to iced-water at 0 C with a heat transfer coefficient of 1200 W/m^2K. Determine the temperatures at the center and the surface of the ball.
Consider a 1.5-m-high and 2.4-m-wide glass window whose thickness is 6 mm and thermal conductivity is k=0.78 W/mK. Determine the steady rate of heat transfer through this glass window and the temperature of its inner surface for a day during which the room is maintained at 24 C while the temperature of the outdoors is -5 C. Take the convection heat transfer coefficients on the inner and outer surfaces of the window to be 10 W/m^2K and 25 W/m^2K, and disregard any heat transfer by radiation.
The wall of a refrigerator is constructed of fiberglass insulation (k=0.035 W/mK) sandwiched between two layers of 1-mm-thick sheet metal (k=15.1 W/mK). The refrigerated space is maintained at 2 C, and the average heat transfer coefficients at the inner and outer surfaces of the wall are 4 W/m^2K and 9 W/m^2 K, respectively. The kitchen temperature averages 28 C. It is observed that condensation occurs on the outer surfaces of the refrigerator when the temperature of the outer surface drops to 20 C. Determine the minimum thickness of fiberglass insulation that needs to be used in the wall in order to avoid condensation on the outer surfaces.
A 1-mm-thick copper plate (k=386 W/mK) is sandwiched between two 7-mm-thick epozy boards (k=0.26 W/mK) that are 15 cm x 20 cm in size. If the thermal contact conductance on both sides of the copper plate is estimated to be 6000 W/mK, determine the error involved in the total thermal resistance of the plate if the thermal contact conductances are ignored.
Consider a 5-m-high, 8-m-long, and 0.22-m-thick wall whose representative cross section is as given in the figure. The thermal conductivities of various materials used, in W/mK, are k_A=k_F=2, k_B=8, k_C=20, k_D=15, and k_E=35. The left and right surfaces of the wall are maintained at uniform temperatures of 300 C and 100 C, respectively. Assuming heat transfer through the wall to be one-dimensional, determine (a) the rate of heat transfer through the wall; (b) the temperature at the point where the sections B, D, and E meet; and (c) the temperature drop across the section F. Disregard any contact resistances at the interfaces.
Steam at 450 F is flowing through a steel pipe (k=8.7 Btu/h ft F) whose inner and outer diameters are 3.5 in and 4.0 in, respectively, in an environment at 55 F. The pipe is insulated with 2-in-thick fiberglass insulation (k=0.020 Btu/h ft F). If the heat transfer coefficients on the inside and the outside of the pipe are 30 and 5 Btu/ h ft^2 F, respectively, determine the rate of heat loss from the steam per foot length of the pipe. What is the error involved in neglecting the thermal resistance of the steel pipe in calculations?
Consider a large 5-cm-thick brass plate (k=111 W/mK) in which heat is generated uniformly at a rate of 2x10^5 W/m^3. One side of the plate is insulated while the other side is exposed to an environment at 25 C with a heat transfer coefficient of 44 W/m^2K. Explain where in the plate the highest and lowest temperatures will occur, and determine their values.
Heat is generated uniformly at a rate of 4.2x10^6 W/m^3 in a spherical ball (k=45 W/mK) of diameter 24 cm. The ball is exposed to iced-water at 0 C with a heat transfer coefficient of 1200 W/m^2K. Determine the temperatures at the center and the surface of the ball.
Consider a 1.5-m-high and 2.4-m-wide glass window whose thickness is 6 mm and thermal conductivity is k=0.78 W/mK. Determine the steady rate of heat transfer through this glass window and the temperature of its inner surface for a day during which the room is maintained at 24 C while the temperature of the outdoors is -5 C. Take the convection heat transfer coefficients on the inner and outer surfaces of the window to be 10 W/m^2K and 25 W/m^2K, and disregard any heat transfer by radiation.
The wall of a refrigerator is constructed of fiberglass insulation (k=0.035 W/mK) sandwiched between two layers of 1-mm-thick sheet metal (k=15.1 W/mK). The refrigerated space is maintained at 2 C, and the average heat transfer coefficients at the inner and outer surfaces of the wall are 4 W/m^2K and 9 W/m^2 K, respectively. The kitchen temperature averages 28 C. It is observed that condensation occurs on the outer surfaces of the refrigerator when the temperature of the outer surface drops to 20 C. Determine the minimum thickness of fiberglass insulation that needs to be used in the wall in order to avoid condensation on the outer surfaces.
A 1-mm-thick copper plate (k=386 W/mK) is sandwiched between two 7-mm-thick epozy boards (k=0.26 W/mK) that are 15 cm x 20 cm in size. If the thermal contact conductance on both sides of the copper plate is estimated to be 6000 W/mK, determine the error involved in the total thermal resistance of the plate if the thermal contact conductances are ignored.
Consider a 5-m-high, 8-m-long, and 0.22-m-thick wall whose representative cross section is as given in the figure. The thermal conductivities of various materials used, in W/mK, are k_A=k_F=2, k_B=8, k_C=20, k_D=15, and k_E=35. The left and right surfaces of the wall are maintained at uniform temperatures of 300 C and 100 C, respectively. Assuming heat transfer through the wall to be one-dimensional, determine (a) the rate of heat transfer through the wall; (b) the temperature at the point where the sections B, D, and E meet; and (c) the temperature drop across the section F. Disregard any contact resistances at the interfaces.
Steam at 450 F is flowing through a steel pipe (k=8.7 Btu/h ft F) whose inner and outer diameters are 3.5 in and 4.0 in, respectively, in an environment at 55 F. The pipe is insulated with 2-in-thick fiberglass insulation (k=0.020 Btu/h ft F). If the heat transfer coefficients on the inside and the outside of the pipe are 30 and 5 Btu/ h ft^2 F, respectively, determine the rate of heat loss from the steam per foot length of the pipe. What is the error involved in neglecting the thermal resistance of the steel pipe in calculations?
Consider a very long rectangular fin attached to a flat surface such that the temperature at the end of the fin is essentially that of the surrounding air, i.e. 20 C. Its width is 5.0 cm; thickness is 1.0 mm; thermal conductivity is 200 W/mK; and base temperature is 40 C. The heat transfer coefficient is 20 W/m^2 K. Estimate the fin temperature at a distance of 5.0 cm from the base and the rate of heat loss from the entire fin.
A 0.2 –cm-thick, 10-cm-high, and 15-cm-long circuit board houses electronic components on one side that dissipate a total of 15 W of heat uniformly. The board is impregnated with conducting metal fillings and has an effective thermal conductivity of 12 W/m K. All the heat generated in the components is conducted across the circuit board and is dissipated from the back side of the board to a medium at 37 C, with a heat transfer coefficient of 45 W/m^2 K. (a) Determine the surface temperatures on the two sides of the circuit board. (b) Now a 0.1-cm-thick, 10-cm-high, and 15-cm-long aluminum plate (k=237 W/mK) with 20 0.2-cm-thick, 2-cm-long, and 15-cm-wide aluminum fins of rectangular profile are attached to the back side of the circuit board with a 0.03-cm-thick epoxy adhesive (k=1.8 W/mK). Determine the new temperatures on the two sides of the circuit board.
Circular fins of uniform cross section, with diameter of 10 mm and length of 50 mm, are attached to a wall with surface temperature of 350 C. The fins are made of material with thermal conductivity of 240 W/mK, and they are exposed to an ambient air condition of 25 C and the convection heat transfer coefficient is 250 W/m^2 K. Determine the heat transfer rate and plot the temperature variation of a single fin for the following boundary conditions: (a) infinitely long fin (b) Adiabatic fin tip (c) fin with tip temperature of 250 C (d) convection from the fin tip.
Consider a sphere of diameter 5 cm, a cube of side length 5 cm, and a rectangular prism of dimension 4 cm x 5 cm x 6 cm, all initially at 0 C and all made of silver (k=429 W/mK, rho=10500 kg/m^3, c_p=0.235 kJ/kgK). Now all three of these geometries are exposed to ambient air at 33 C on all of their surfaces with a heat transfer coefficient of 12 W/m^2 K. Determine how long it will take for the temperature of each geometry to rise to 25 C.
Carbon steel balls (rho=7833 kg/m^3, k=54 W/mK, c_p=0.465 kJ/kg C, and alpha=1.474x10^-6 m^2/s) 8 mm in diameter are annealed by heating them first to 900 C in a furnace and then allowing them to cool slowly to 100 C in ambient air at 35 C. If the average heat transfer coefficient is 75 W/m^2 K, determine how long the annealing process will take. If 2500 balls are to be annealed per hour, determine the total rate of heat transfer from the balls to the ambient air.
Long cylindrical AISI stainless steel rods (k=7.74 Btu/h ft F and alpha=0.135 ft^2/h) of 4-in-diameter are heat treated by drawing them at a velocity of 7 ft/min through a 21-ft-long oven maintained at 1700 F. The heat transfer coefficient in the oven is 20 Btu/h ft^2 F. If the rods enter the oven at 70 F, determine their centerline temperature when they leave.
A 9-cm-diameter potato (rho=1100 kg/m^3, c_p=3900 J/kgK, k=0.6 W/mK, and alpha=1.4x10^-7 m^2/s) that is initially at a uniform temperature of 25 C is baked in an oven at 170 C until a temperature sensor inserted to the center of the potato indicates a reading of 70 C. The potato is then taken out of the oven and wrapped in thick towels so that almost no heat is lost from the baked potato. Assuming the heat transfer coefficient in the oven to be 40 W/m^2 K, determine (a) how long the potato is baked in the oven and (b) the final equilibrium temperature of the potato after it is wrapped.
A 10-cm thick aluminum plate (rho=2702 kg/m^3, c_p=903 J/kg K, k=273 W/mK, and alpha=97.1x10^-6 m^2/s) is being heated in liquid with temperature of 500 C. The aluminum plate has a uniform initial temperature of 25 C. If the surface temperature of the aluminum plate is approximately the liquid temperature, determine the temperature at the center plane of the aluminum plate after 15 seconds of heating.
A highway made of asphalt is initially at a uniform temperature of 55 C. Suddenly the highway surface temperature is reduced to 25 C by rain. Determine the temperature at the depth of 3 cm from the highway surface and the heat flux transferred from the highway after 60 minutes. Assume the highway surface temperature is maintained at 25 C.
A 0.2 –cm-thick, 10-cm-high, and 15-cm-long circuit board houses electronic components on one side that dissipate a total of 15 W of heat uniformly. The board is impregnated with conducting metal fillings and has an effective thermal conductivity of 12 W/m K. All the heat generated in the components is conducted across the circuit board and is dissipated from the back side of the board to a medium at 37 C, with a heat transfer coefficient of 45 W/m^2 K. (a) Determine the surface temperatures on the two sides of the circuit board. (b) Now a 0.1-cm-thick, 10-cm-high, and 15-cm-long aluminum plate (k=237 W/mK) with 20 0.2-cm-thick, 2-cm-long, and 15-cm-wide aluminum fins of rectangular profile are attached to the back side of the circuit board with a 0.03-cm-thick epoxy adhesive (k=1.8 W/mK). Determine the new temperatures on the two sides of the circuit board.
Circular fins of uniform cross section, with diameter of 10 mm and length of 50 mm, are attached to a wall with surface temperature of 350 C. The fins are made of material with thermal conductivity of 240 W/mK, and they are exposed to an ambient air condition of 25 C and the convection heat transfer coefficient is 250 W/m^2 K. Determine the heat transfer rate and plot the temperature variation of a single fin for the following boundary conditions: (a) infinitely long fin (b) Adiabatic fin tip (c) fin with tip temperature of 250 C (d) convection from the fin tip.
Consider a sphere of diameter 5 cm, a cube of side length 5 cm, and a rectangular prism of dimension 4 cm x 5 cm x 6 cm, all initially at 0 C and all made of silver (k=429 W/mK, rho=10500 kg/m^3, c_p=0.235 kJ/kgK). Now all three of these geometries are exposed to ambient air at 33 C on all of their surfaces with a heat transfer coefficient of 12 W/m^2 K. Determine how long it will take for the temperature of each geometry to rise to 25 C.
Carbon steel balls (rho=7833 kg/m^3, k=54 W/mK, c_p=0.465 kJ/kg C, and alpha=1.474x10^-6 m^2/s) 8 mm in diameter are annealed by heating them first to 900 C in a furnace and then allowing them to cool slowly to 100 C in ambient air at 35 C. If the average heat transfer coefficient is 75 W/m^2 K, determine how long the annealing process will take. If 2500 balls are to be annealed per hour, determine the total rate of heat transfer from the balls to the ambient air.
Long cylindrical AISI stainless steel rods (k=7.74 Btu/h ft F and alpha=0.135 ft^2/h) of 4-in-diameter are heat treated by drawing them at a velocity of 7 ft/min through a 21-ft-long oven maintained at 1700 F. The heat transfer coefficient in the oven is 20 Btu/h ft^2 F. If the rods enter the oven at 70 F, determine their centerline temperature when they leave.
A 9-cm-diameter potato (rho=1100 kg/m^3, c_p=3900 J/kgK, k=0.6 W/mK, and alpha=1.4x10^-7 m^2/s) that is initially at a uniform temperature of 25 C is baked in an oven at 170 C until a temperature sensor inserted to the center of the potato indicates a reading of 70 C. The potato is then taken out of the oven and wrapped in thick towels so that almost no heat is lost from the baked potato. Assuming the heat transfer coefficient in the oven to be 40 W/m^2 K, determine (a) how long the potato is baked in the oven and (b) the final equilibrium temperature of the potato after it is wrapped.
A 10-cm thick aluminum plate (rho=2702 kg/m^3, c_p=903 J/kg K, k=273 W/mK, and alpha=97.1x10^-6 m^2/s) is being heated in liquid with temperature of 500 C. The aluminum plate has a uniform initial temperature of 25 C. If the surface temperature of the aluminum plate is approximately the liquid temperature, determine the temperature at the center plane of the aluminum plate after 15 seconds of heating.
A highway made of asphalt is initially at a uniform temperature of 55 C. Suddenly the highway surface temperature is reduced to 25 C by rain. Determine the temperature at the depth of 3 cm from the highway surface and the heat flux transferred from the highway after 60 minutes. Assume the highway surface temperature is maintained at 25 C.
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