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Consider a homogenous spherical piece of radioactive material of radius r0=0.04 m that is generating heat at a constant rate of 5x10^7 W/m^3. The heat generated is dissipated to the environment steadily. The outer surface of the sphere is maintained at a uniform temperature of 110 C and the thermal conductivity of the sphere is k= 15 W/mK. Assuming steady one dimensional heat transfer, (a) express the differential equation and the boundary conditions for heat conduction through the sphere, (b) obtain a relation for the variation of temperature in the sphere by solving the differential equation, and (c) determine the temperature at the center of the sphere.
Electrically heated draw batch furnaces are commonly used in the heat treatment industry. Consider a draw batch furnace front made of a 20-mm thick steel plate with a thermal conductivity of 25 W/mK. The furnace is situated in a room with surrounding air temperature of 20 C and an average convection heat transfer coefficient of 10 W/m^2 K . If the inside surface of the furnace front is subjected to uniform heat flux of 5 kW/m^2 and the outer surface has an emissivity of 0.30, determine the inside surface temperature of the furnace front.
A pipe in a manufacturing plant is transporting superheated vapor at a mass flow rate of 0.3 kg/s. The pipe is 10 m long, has an inner diameter of 5 cm and pipe wall thickness of 6 mm. The pipe has a thermal conductivity of 17 W/mK, and the inner pipe surface is at a uniform temperature of 120 C. The temperature drop between the inlet and exit of the pipe is 7 C and the constant pressure specific heat of vapor is 2190 J/kgC. If the air temperature in the manufacturing plant is 25 C, determine the heat transfer coefficient as a result of convection between the outer pipe surface and the surrounding air.
Consider a 1.5 m high and 2.4 m wide double pane window consisting of two 3 mm thick layers of glass (k=0.78 W/mK) separated by a 12 mm wide stagnant air space (k=0.026 W/mK). Determine the steady rate of heat transfer through this double pane window and the temperature of its inner surface for a day during which the room is maintained at 21 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 h_1=10 W/m^2 K and h_2=25 W/m^2 K, and disregard any heat transfer by radiation.
To defrost ice accumulated on the outer surface of an automobile windshield, warm air is blown over the inner surface of the windshield. Consider an automobile windshield with thickness of 5 mm and thermal conductivity of 1.4 W/mK. The outside ambient temperature is -10 C and the convection heat transfer coefficient is 200 W/m^2 K, while the ambient temperature inside the automobile is 25 C. Determine the value of the convection heat transfer coefficient for the warm air blowing over the inner surface of the windshield necessary to cause the accumulated ice to begin melting.
Two 5 cm diameter, 15 cm long aluminum bars (k=176 W/mK) with ground surfaces are pressed against each other with a pressure of 20 atm. The bars are enclosed in an insulation sleeve and, thus, heat transfer from the lateral surfaces is negligible. If the top and bottom surfaces of the two bar system are maintained at temperatures of 150 C and 20 C, respectively, determine (a) the rate of heat transfer along the cylinders under steady conditions and (b) the temperature drop at the interface.
Superheated steam at an average temperature 200 C is transported through a steel pipe (k=50 W/mK, D_0=8.0 cm,D_i=6.0 cm,and L=20.0 m). The pipe is insulated with a 4-cm thick layer of gypsum plaster (k=0.5 W/mK). The insulated pipe is placed horizontally inside a warehouse where the average air temperature is 10 C. The steam and the air heat transfer coefficients are estimated to be 800 and 200 W/m^2 K, respectively. Calculate (a) the daily rate of heat transfer from the superheated steam, and (b) the temperature on the outside surface of the gypsum plaster insulation.
A 4-mm-diameter and 10-cm-long aluminum fin (k=237 W/mK) is attached to a surface. If the heat transfer coefficient is 12 W/m^2 K, determine the percent error in the rate of heat transfer from the fin when the infinitely long fin assumption is used instead of the adiabatic fin tip assumption.
Hot water at an average temperature of 53 C and an average velocity of 0.4 m/s is flowing through a 5-m section of a thin-walled hot-water pipe that has an outer diameter of 2.5 cm. The pipe passes through the center of a 14-cm-thick wall filled with fiberglass insulation (k=0.035 W/mK). If the surfaces of the wall are at 18 C, determine (a) the rate of heat transfer from the pipe to the air in the rooms and (b) the temperature drop of the hot water as it flows through this 5-m-long section of the wall.
A hot surface at 100 C is to be cooled by attaching 3-cm-long, 0.25-cm-diameter aluminum pin fins (k=237 W/mK) to it, with a center-to-center distance of 0.6 cm. The temperature of the surrounding medium is 30 C, and the heat transfer coefficient on the surfaces is 35 W/m^2 K. Determine the rate of heat transfer from the surface for a 1-m x 1-m section of the plate. Also determine the overall effectiveness of the fins.
Consider a 800-W iron whose base plate is made of 0.5-cm-thick aluminum alloy 2024-T6 (rho=2770 kg/m^3, c_p=875 J/kgK, alpha=7.3×10^(-5) m^2/s). The base plate has a surface area of 0.03 m^2. Initially, the iron is in thermal equilibrium with the ambient air at 22 C. Taking the heat transfer coefficient at the surface of the base plate to be 12 W/m^2 K and assuming 85 percent of the heat generated in the resistance wires is transferred to the plate, determine how long it will take for the plate temperature to reach 140 C. Is it realistic to assume the plate temperature to be uniform at all times?
Consider a spherical shell satellite with outer diameter of 4 m and shell thickness of 10 mm is reentering the atmosphere. The shell satellite is made of stainless steel with properties of =8238 kg/m^3 , c_p=468 J/kgK , and k=13.4 W/mK. During the reentry, the effective atmosphere temperature surrounding the satellite is 1250 C with convection heat transfer coefficient of 130W/m^2 K. If the initial temperature of the shell is 10 C, determine the shell temperature after 5 minutes of reentry. Assume heat transfer occurs only on the satellite shell.
Oranges of 2.5-in-diameter (k=0.26 Btu/hftF and alpha=1.4×10^(-6) ft^2/s) initially at a uniform temperature of 78 F are to be cooled by refrigerated air at 25 F flowing at a velocity of 1 ft/s. The average heat transfer coefficient between the oranges and the air is experimentally determined to be 4.6 Btu/hft^2F. Determine how long it will take for the center temperature of the oranges to drop to 40 F. Also, determine if any part of the oranges will freeze during this process.
In areas where the air temperature remains below 0 C for prolonged periods of time, the freezing of water in underground pipes is a major concern. Fortunately, the soil remains relatively warm during those periods, and it takes weeks for the subfreezing temperatures to reach the water mains in the ground. Thus, the soil effectively serves as an insulation to protect the water from the freezing atmospheric temperatures in winter. The ground at a particular location is covered with snow pack at -8 C for a continuous period of 60 days, and the average soil properties at that location are k=0.35 W/mK and alpha=.15×10^(-6) m^2/s. Assuming an initial uniform temperature of 8 C for the ground, determine the minimum burial depth to prevent the water pipes from freezing.
The upper surface of a 50-cm-thick solid plate (k=237 W/mK) is being cooled by water with temperature of 20 C. The upper and lower surfaces of the solid plate maintained at constant temperatures of 60 C and 120 C, respectively. Determine the water convection heat transfer coefficient and the water temperature gradient at the upper plate surface.
Consider an airplane cruising at an altitude of 10 km where standard atmospheric conditions are -50 C and 26.5 kPa at a speed of 800 km/h. Each wing of the airplane can be modeled as a 25-m x 3-m flat plate, and the friction coefficient of the wings is 0.0016. Using the momentum-heat transfer analogy, determine the heat transfer coefficient for the wings at cruising conditions.
In an effort to prevent the formation of ice on the surface of a wing, electrical heaters are embedded inside the wing. With a characteristic length of 2.5 m, the wing has a friction coefficient of 0.001. If the wing is moving at a speed of 200 m/s through air at 1 atm and -20 C, determine the heat flux necessary to keep the wing surface above 0 C. Evaluate fluid properties at -10 C.
Water at 43.3 C flows over a large plate at a velocity of 30.0 cm/s. The plate is 1.0 m long (in the flow direction), and its surface is maintained at a uniform temperature of 10.0 C. Calculate the steady rate of heat transfer per unit width of the plate.
The top surface of the passenger car of a train moving at a velocity of 95 km/h is 2.8 m wide and 8 m long. The top surface is absorbing solar radiation at a rate of 380 W/m^2, and the temperature of the ambient air is 30 C. Assuming the roof of the car to be perfectly insulated and the radiation heat exchange with the surroundings to be small relative to convection, determine the equilibrium temperature of the top surface of the car.
Consider a fluid with a Prandtl number of 7 flowing through a smooth circular tube. Using the Colburn, Petukhov, and Gnielinski equations, determine the Nusselt numbers for Reynolds numbers at 3500, 10^4, and 5×10^5. Compare and discuss the results.
Oil at 15 C is to be heated by saturated steam at 1 atm in a double-pipe heat exchanger to a temperature of 25 C. The inner and outer diameters of the annular space are 3 cm and 5 cm respectively, and oil enters with a mean velocity of 1.2 m/s. The inner tube may be assumed to be isothermal at 100 C, and the outer tube is well insulated. Assuming fully developed flow for oil, determine the tube length required to heat the oil to the indicated temperature. In reality, will you need a shorter or longer tube?
A double-pipe parallel-flow heat exchanger is used to heat cold tap water with hot water. Hot water (c_p=4.25 kJ/kgK) enters the tube at 85 C at a rate of 1.4 kg/s and leaves at 50 C. The heat exchanger is not well insulated, and it is estimated that 3 percent of the heat given up by the hot fluid is lost from the heat exchanger. If the overall heat transfer coefficient and the surface area of the heat exchanger are 1150 W/m^2 K and 4 m^2, respectively, determine the rate of heat transfer to the cold water and the log mean temperature difference for this heat exchanger.
Engine oil (c_p=2100 J/kgK) is to be heated from 20 C to 60 C at a rate of 0.3 kg/s in a 2-cm-diameter thin-walled copper tube by condensing steam outside at a temperature of 130C (h_fg=2174 kJ/kg). For an overall heat transfer coefficient of 650 W/m^2 K, determine the rate of heat transfer and the length of the tube required to achieve it.
Glycerin (c_p=2400 J/kgK) at 20 C and 0.5 kg/s is to be heated by ethylene glycol (c_p=2500 J/kgK) at 60 C and the same mass flow rate in a thin-walled double-pipe parallel-flow heat exchanger. If the overall heat transfer coefficient is 380 W/m^2 K and the heat transfer surface area is 6.5 m^2, determine (a) the rate of heat transfer and (b) the outlet temperatures of the glycerin and the glycol.
A 15-cm-diameter aluminum ball is to be heated from 80 C to an average temperature of 200 C. Taking the average density and specific heat of aluminum in this temperature range to be rho=2700 kg/m^3 and c_p=0.90 kJ/kgK, respectively, determine the amount of energy that needs to be transferred to the aluminum ball.
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