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All Textbook Solutions for Principles of Heat Transfer (Activate Learning with these NEW titles from Engineering!)
1.1 On a cold winter day, the outer surface of a 0.2-m-thick concrete wall of a warehouse is exposed to temperature of –5°C, while the inner surface is kept at 20°C. The thermal conductivity of the concrete is 1.2 W/m K. Determine the heat loss through the wall, which is 10-m long and 3-m high.
Problem 1.1
1.2 The weight of the insulation in a spacecraft may be more important than the space required. Show analytically that the lightest insulation for a plane wall with a specified thermal resistance is the insulation that has the smallest product of density times thermal conductivity.
1.3 A furnace wall is to be constructed of brick having standard dimensions of Two kinds of material are available. One has a maximum usable temperature of 1040°C and a thermal conductivity of 1.7 W/(m K), and the other has a maximum temperature limit of 870°C and a thermal conductivity of 0.85 W/(m K). The bricks have the same cost and are laid in any manner, but we wish to design the most economical wall for a furnace with a temperature of 1040°C on the hot side and 200°C on the cold side. If the maximum amount of heat transfer permissible is 950 , determine the most economical arrangement using the available bricks.
1.4 To measure thermal conductivity, two similar 1-cm-thick specimens are placed in the apparatus shown in the accompanying sketch. Electric current is supplied to the guard heater, and a wattmeter shows that the power dissipation is 10 W. Thermocouples attached to the warmer and to the cooler surfaces show temperatures of 322 and 300 K, respectively. Calculate the thermal conductivity of the material at the mean temperature in W/m K.
Problem 1.4
To determine the thermal conductivity of a structural material, a large 15-cm-thick slab of the material is subjected to a uniform heat flux of 2500 W/m2 while thermocouples embedded in the wall at 2.5 cm. intervals are read over a period of time. After the system had reached equilibrium, an operator recorded the thermocouple readings shown below for two different environmental conditions: Distance from the Surface (cm) Temperature (C) Test 1 0 40 5 65 10 97 15 132 Test 2 0 95 5 130 10 168 15 208 From these data, determine an approximate expression for the thermal conductivity as a function of temperature between 40 and 208C.A square silicon chip 7mm7mm in size and 0.5-mm thick is mounted on a plastic substrate as shown in the sketch below. The top surface of the chip is cooled by a synthetic liquid flowing over it. Electronic circuits on the bottom of the chip generate heat at a rate of 5 W that must be transferred through the chip. Estimate the steady-state temperature difference between the front and back surfaces of the chip. The thermal conductivity of silicon is 150 W/m K. Problem 1.6A cooling system is to be designed for a food storage warehouse for keeping perishable foods cool prior to transportation to grocery stores. The warehouse has an effective surface area of 1860 m2 exposed to an ambient air temperature of 32C. The warehouse wall insulation (k=0.17W/(mK)) is 7.5 cm thick. Determine the rate at which heat must be removed (W) from the warehouse to maintain the food at 4C.1.80 Describe and compare the modes of heat loss through the single-pane and double-pane window assemblies shown in the sketch below.
Problem 1.80
Heat is transferred at a rate of 0.1 kW through glass wool insulation (density=100kg/m3) with a 5-cm thickness and 2-m2 area. If the hot surface is at 70C, determine the temperature of the cooler surface.1.10 A heat flux meter at the outer (cold) wall of a concrete building indicates that the heat loss through a wall of 10-cm thickness is . If a thermocouple at the inner surface of the wall indicates a temperature of 22°C while another at the outer surface shows 6°C, calculate the thermal conductivity of the concrete and compare your result with the value in Appendix 2, Table 11.
1.11 Calculate the heat loss through a glass window 7-mm thick if the inner surface temperature is 20°C and the outer surface temperature is 17°C. Comment on the possible effect of radiation on your answer.
1.12 A wall with a thickness is made of a material with a thermal conductivity that varies with its thickness according to the equation, , where and are constants. If the heat flux applied at the surface of one end of the wall is , derive an expression for the temperature gradient and the temperature distribution across the wall thickness (between and ). Use and define appropriate notations for the surface temperatures at each end of the wall.
1.13 If the outer air temperature in Problem is –2°C, calculate the convection heat transfer coefficient between the outer surface of the window and the air, assuming radiation is negligible.
Using Table 1.4 as a guide, prepare a similar table showing the orders of magnitude of the thermal resistances of a unit area for convection between a surface and various fluids.1.15 A thermocouple (0.8-mm-diameter wire) used to measure the temperature of the quiescent gas in a furnace gives a reading of . It is known, however, that the rate of radiant heat flow per meter length from the hotter furnace walls to the thermocouple wire is 1.1 W/m and the convection heat transfer coefficient between the wire and the gas is K. With this information, estimate the true gas temperature. State your assumptions and indicate the equations used.
Water at a temperature of 77C is to be evaporated slowly in a vessel. The water is in a low-pressure container surrounded by steam as shown in the sketch below. The steam is condensing at 107C. The overall heat transfer coefficient between the water and the steam is 1100W/m2K. Calculate the surface area of the container that is required to evaporate water at a rate of 0.01 kg/s. Problem 1.16The heat transfer rate from hot air by convection at 100C flowing over one side of a flat plate with dimensions 0.1m0.5m is determined to be 125 W when the surface of the plate is kept at 30C. What is the average convection heat transfer coefficient between the plate and the air?The heat transfer coefficient for a gas flowing over a thin float plate 3-m long and 0.3-m wide varies with distance from the leading edge according to hc(x)=10x1/4Wm2K If the plate temperature is 170C and the gas temperature is 30C, calculate (a) the average heat transfer coefficient, (b) the rate of heat transfer between the plate and the gas, and (c) the local heat flux 2 m from the leading edge. Problem 1.181.19 A cryogenic fluid is stored in a 0.3-m-diameter spherical container is still air. If the convection heat transfer coefficient between the outer surface of the container and the air is 6.8 , the temperature of the air is 27°C, and the temperature of the surface of the sphere is –183°C, determine the rate of heat transfer by convection.
A high-speed computer is located in a temperature-controlled room at 26C. When the machine is operating, its internal heat generation rate is estimated to be 800 W. The external surface temperature of the computer is to be maintained below 85C. The heat transfer coefficient for the surface of the computer is estimated to be 10W/m2K. What surface area would be necessary to assure safe operation of this machine? Comment on ways to reduce this area.1.21 In an experimental set up in a laboratory, a long cylinder with a 5-cm diameter, and an electrical resistance heater inside its entire length is cooled with water flowing crosswise over the cylinder at and a velocity of 0.8 m/s. For these flow conditions, 20 kW/m of power is required to maintain a uniform temperature of at the surface of the cylinder. When water is not available, air at is used with a velocity of 10 m/s to maintain the same surface temperature. However, in this case, the cylinder surface heat dissipation rate is reduced to 400 W/m. Calculate the convection heat transfer coefficients for both water and air, and comment on the reason for the differences in the values.
1.22 In order to prevent frostbite to skiers on chair lifts, the weather report at most ski areas gives both an air temperature and the wind-chill temperature. The air temperature is measured with a thermometer that is not affected by the wind. However, the rate of heat loss from the skier increases with wind velocity, and the wind-chill temperature is the temperature that would result in the same rate of heat loss in still air as occurs at the measured air temperature with the existing wind.
Suppose that the inner temperature of a 3-mm-thick layer of skin with a thermal conductivity of 0.35 W/m K is and the air temperature is . Under calm ambient conditions the heat transfer coefficient at the outer skin surface is about (see Table 1.4), but in a 40-mph wind it increases to .
(a) If frostbite occurs when the skin temperature drops to about , do you advise the skier to wear a face mask?
(b) What is the skin temperature drop due to the wind?
Using the information in Problem 1.22, estimate the ambient air temperature that could cause frostbite on a calm day on the ski slopes. 1.22 In order to prevent frostbite to skiers on chair lifts, the weather report at most ski areas gives both an air temperature and the wind-chill temperature. The air temperature is measured with a thermometer that is not affected by the wind. However, the rate of heat loss from the skier increases with wind velocity, and the wind-chill temperature is the temperature that would result in the same rate of heat loss in still air as occurs at the measured air temperature with the existing wind. Suppose that the inner temperature of a 3-mm-thick layer of skin with a thermal conductivity of 0.35W/mKis35C and the air temperature is 20C. Under calm ambient conditions the heat transfer coefficient at the outer skin surface is about 20W/m2K (see Table 1.4), but in a 40-mph wind it increases to 75W/m2K. If frostbite occurs when the skin temperature drops to about 10C, do you advise the skier to wear a face mask? What is the skin temperature drop due to the wind?Two large parallel plates with surface conditions approximating those of a blackbody are maintained at 816C and 260C, respectively. Determine the rate of heat transfer by radiation between the plates in W/m2 and the radiative heat transfer coefficient in W/m2K.1.25 A spherical vessel, 0.3 m in diameter, is located in a large room whose walls are at 27°C (see sketch). If the vessel is used to store liquid oxygen at –183°C and both the surface of the storage vessel and the walls of the room are black, calculate the rate of heat transfer by radiation to the liquid oxygen in watts and in Btu/h.
1.26 Repeat Problem 1.25 but assume that the surface of the storage vessel has an absorbance (equal to the emittance) of 0.1. Then determine the rate of evaporation of the liquid oxygen in kilograms per second and pounds per hour, assuming that convection can be neglected. The heat of vaporization of oxygen at –183°C is .
Determine the rate of radiant heat emission in watts per square meter from a blackbody at (a) 15C, (b) 600C, and (c) 5700C.1.28 The sun has a radius of and approximates a blackbody with a surface temperature of about 5800 K. Calculate the total rate of radiation from the sun and the emitted radiation flux per square meter of surface area.
1.29 A spherical interplanetary probe with a 30-cm diameter contains electronic equipment that dissipates 100 W. If the probe surface has an emissivity of 0.8, what is its surface temperature in outer space? State your assumptions in the calculations.
1.30PA spherical communications satellite, 2 m in diameter, is placed in orbit around the earth. The satellite generates 1000 W of internal power from a small nuclear generator. If the surface of the satellite has an emittance of 0.3, and is shaded from solar radiation by the earth, estimate its surface temperature.A long wire 0.7 mm in diameter with an emissivity of 0.9 is placed in a large quiescent air space at 270 K. If the wire is at 800 K, calculate the net rate of heat loss. Discuss your assumptions.Wearing layers of clothing in cold weather is often recommended because dead-air spaces between the layers keep the body warm. The explanation for this is that the heat loss from the body is less. Compare the rate of heat loss for a single 2-cm-thick layer of wool [k=0.04W/(mK)] with three 0.67-cm layers separated by 1.5 mm air gaps. The thermal conductivity of air is 0.024 W(mK).A section of a composite wall with the dimensions shown below has uniform temperatures of 200C and 50C over the left and right surfaces, respectively. If the thermal conductivities of the wall materials are: kA=70W/mK,kB=60W/mK, kC=40W/mK, and kD=20W/mK, determine the rate of heat transfer through this section of the wall and the temperatures at the interfaces.A section of a composite wall with the dimensions shown below has uniform temperatures of 200C and 50C over the left and right surfaces, respectively. If the thermal conductivities of the wall materials are: kA=70W/mK,kB=60W/mK, kC=40W/mK, and kP=20W/mK, determine the rate of heat transfer through this section of the wall and the temperatures at the interfaces. Repeat Problem 1.34, including a contact resistance of 0.1 K/W at each of the interfaces.Repeat Problem 1.35 but assume that instead of surface temperatures, the given temperatures are those of the air on the left and right sides of the wall and that the convection heat transfer coefficients on the left and right surfaces are 6 and 10W/m2K, respectively.1.37 Mild steel nails were driven through a solid wood wall consisting of two layers, each 2.5-cm thick, for reinforcement. If the total cross-sectional area of the nails is 0.5% of the wall area, determine the unit thermal conductance of the composite wall and the percent of the total heat flow that passes through the nails when the temperature difference across the wall is 25°C. Neglect contact resistance between the wood layers.
1.38P1.39 On a cold winter day, the outside wall of a home is exposed to an air temperature of when the inside temperature of the room is at . As a result of this temperature gradient, there is heat loss through the wall to the outside. Consider the convective heat transfer coefficients for the air inside the room and at the outside wall surface to be, respectively, 12.0 and . If the composite room wall is modeled as a plane wall with a thermal resistance per unit area of , determine the temperature at the outer surface of the wall as well as the rate of heat flow through the wall per unit area. If the homeowner were to consider using a fiberglass insulation layer on the inside wall surface for reducing this heat loss by 50%, what is the required thickness of this layer and the outside wall temperature for this case?
As a designer working for a major electric appliance manufacturer, you are required to estimate the amount of fiberglass insulation packing (k = 0.035 W/m K) that is needed for a kitchen oven shown in the figure below. The fiberglass layer is to be sandwiched between a 2-mm-thick aluminum cladding plate on the outside and a 5-mm-thick stainless steel plate on the inside that forms the core of the oven. The insulation thickness is such that the outside cladding temperature does not exceed 40C when the temperature at the inside surface of the oven is 300C. Also, the air temperature in the kitchen varies from 15Cto33C, and the average heat transfer coefficient between the outer surface of the oven and air is estimated to be 12.0W/m2K. Determine the thickness of the fiberglass insulation that is required for these conditions. What would be the outer surface temperature when the inside surface of the oven is at 475C?1.41 A heat exchanger wall consists of a copper plate 2 cm thick. The heat transfer coefficients on the two sides of the plate are 2700 and , corresponding to fluid temperatures of 92 and 32°C, respectively. Assuming that the thermal conductivity of the wall is , (a) compute the surface temperatures in °C and (b) calculate the heat flux in .
1.42P1.43 A simple solar heater consists of a flat plate of glass below which is located a shallow pan filled with water, so that the water is in contact with the glass plate above it. Solar radiation passes through the glass at the rate of . The water is at and the surrounding air is at . If the heat transfer coefficients between the water and the glass, and between the glass and the air are respectively, determine the time required to transfer of surface to the water in the pan. The lower surface of the pan is assumed to be insulated.
A composite refrigerator wall is composed of 5 cm of corkboard sandwiched between a 1.2-cm-thick layer of oak and a 0.8-mm-thick layer of aluminum lining on the inner surface. The average convection heat transfer coefficients at the interior and exterior wall are 11 and 8.5W/(m2K) respectively. (a) Draw the thermal circuit. (b) Calculate the individual resistances of the components of this composite wall and the resistances at the surfaces. (c) Calculate the overall heat transfer coefficient through the wall. (d) For an air temperature of 1C inside the refrigerator and 32C outside, calculate the rate of heat transfer per unit area through the wall.An electronic device that internally generates 600 mW of heat has a maximum permissible operating temperature of 70C. It is to be cooled in 25C air by attaching aluminum fins with a total surface area of 12cm2. The convection heat transfer coefficient between the fins and the air is 20W/m2K. Estimate the operating temperature when the fins are attached in such a way that (a) there is a contact resistance of approximately 50 K/W between the surface of the device and the fin array and (b) there is no contact resistance (in this case, the construction of the device is more expensive). Comment on the design options.1.46P1.47 A flat roof is modeled as a flat plate insulated on the bottom and placed in the sunlight. If the radiant heat that the roof receives from the sun is , the convection heat transfer coefficient between the roof and the air is 12 W/m2 K, and the air temperature is , determine the roof temperature for the following two cases: (a) Radiative heat loss to space is negligible, (b) The roof is black and radiates to space, which is assumed to be a black- body at 0 K.
A horizontal, 3-mm-thick flat-copper plate, 1-m long and 0.5-m wide, is exposed in air at 27C to radiation from the sun. If the total rate of solar radiation absorbed is 300 W and the combined radiation and convection heat transfer coefficients on the upper and lower surfaces are 20 and 15W/m2K, respectively, determine the equilibrium temperature of the plate.1.49 A small oven with a surface area of is located in a room in which the walls and the air are at a temperature of . The exterior surface of the oven is at , and the next heat transfer by radiation between the oven’s surface and the surroundings is 586 W. If the average convection heat transfer coefficient between the oven and the surrounding air is 11 calculate (a) the net heat transfer between the oven and the surroundings in W, (b) the thermal resistance at the surface for radiation and convection, respectively, in K/W, and (c) the combined heat transfer coefficient in .
A steam pipe 200 mm in diameter passes through a large basement room. The temperature of the pipe wall is 500C, while that of the ambient air in the room is 20C. Determine the heat transfer rate by convection and radiation per unit length of steam pipe if the emissivity of the pipe surface is 0.8 and the natural convection heat transfer coefficient has been determined to be 10 W/m2K.1.51 The inner wall of a rocket motor combustion chamber receives by radiation from a gas at . The convection heat transfer coefficient between the gas and the wall is . If the inner wall of the combustion chamber is at
Problem 1.51
a temperature of , determine (a) the total thermal resistance of a unit area of the wall in and (b) the heat flux. Also draw the thermal circuit.
1.52 A flat roof of a house absorbs a solar radiation flux of . The backside of the roof is well insulated, while the outside loses heat by radiation and convection to ambient air at . If the emittance of the roof is 0.80 and the convection heat transfer coefficient between the roof and the air is K, calculate (a) the equilibrium surface temperature of the roof and (b) the ratio of convection to radiation heat loss. Can one or the other of these be neglected? Explain your answer.
Determine the power requirement of a soldering iron in which the tip is maintained at 400C. The tip is a cylinder 3 mm in diameter and 10 mm long. The surrounding air temperature is 20C, and the average convection heat transfer coefficient over the tip is 20W/m2K. The tip is highly polished initially, giving it a very low emittance.1.54 The soldering iron tip in Problem 1.53 becomes oxidized with age and its gray-body emittance increases to 0.8. Assuming that the surroundings are at 20°C, determine the power requirement for the soldering iron.
1.55PA pipe carrying superheated steam in a basement at 10C has a surface temperature of 150C. Heat loss from the pipe occurs by radiation (=0.6) and natural convection (hc=25W/m2K). Determine the percentage of the total heat loss by these two mechanisms.1.57PDraw the thermal circuit for heat transfer through a double-glazed or a double-paned window. Identify each of the circuit elements. Include solar radiation to the window and interior space.1.59P1.60 Two electric resistance heaters with a 20 cm length and a 2 cm diameter are inserted into a well-insulated 40-L tank of water that is initially at 300 K. If each heater dissipates 500 W, what is the time required for bringing the water temperature in the tank to 340 K? State your assumption for your analysis.
1.61P1.62P1.63 Liquid oxygen (LOX) for the space shuttle is stored at 90 K prior to launch in a spherical container 4 m in diameter. To reduce the loss of oxygen, the sphere is insulated with superinsulation developed at the U.S. National Institute of Standards and Technology's Cryogenic Division; the superinsulation has an effective thermal conductivity of 0.00012 W/m K. If the outside temperature is on the average and the LOX has a heat of vaporization of 213 J/g, calculate the thickness of insulation required to keep the LOX evaporation rate below 200 g/h.
The interior wall of a large, commercial walk-in type meat freezer is covered under normal operating conditions with a 2-cm thick layer of ice. One day, a power outage cuts electricity to the refrigeration system of the freezer. Estimate the time required to melt this layer of ice if it has a mass density of 700kg/m3 and a latent heat of fusion of 334 kJ/kg. Consider the air temperature inside the freezer to be 20C with a heat transfer coefficient of 2W/m2K for convection from the freezer surface to air, and clearly state the assumptions made in your calculations.1.65P1.66P1.67 In beauty salons and in homes, a ubiquitous device is the hairdryer. The front end of a typical hairdryer is idealized as a thin-walled cylindrical duct with a 6-cm diameter with a fan at the inlet that blows air over an electric heating coil as schematically shown in the figure. The design of this appliance requires two power settings, with which the air blown over the electric heating coil is heated from the ambient temperature of to an outlet temperature of and with exit air velocities of 1.0 m/s and 1.5 m/s. Estimate the electric power required for the heating coil to meet these conditions, assuming that heat loss from the
outside of the dryer duct is neglected.
1.68PThe heat transfer coefficient between a surface and a liquid is 57 W/(m2K). How many watts per square meter will be transferred in this system if the temperature difference is 10C?The thermal conductivity of fibreglass insulation at 67F is 0.02 Btu/h ft F. What is its value in SI units?1.71 The thermal conductivity of silver at 212°F is 238 Btu/h ft °F. What is the conductivity in SI units?
1.72 An ice chest (see sketch) is to constructed from styrofoam . If the wall of the chest is 5-cm thick, calculate its -value in .
Problem 1.72
Estimate the R-values for a 5-cm-thick fiberglass board and a 2.5-cm-thick polyurethane foam layer. Then, compare their respective conductivity-times-density products if the density for fiberglass is 50kg/m3 and the density of polyurethane is 30kg/m3. Use the units given in Figure 1.31.A manufacturer in the United States wants to sell a refrigeration system to a customer in Germany. The standard measure of refrigeration capacity used in the United States is the ton (T); a 1 T capacity means that the unit is capable of making about 1 T of ice per day or has a heat removal rate of 12,00 Btu/h. The capacity of the American system is to be guaranteed at 3 T. What is this guarantee in SI units?Referring to Problem 1.74, how many kilograms of ice can a 3-ton refrigeration unit produce in a 24-h period? The heat of fusion of water is 330 kJ/kg.1.76 Explain a fundamental characteristic that differentiates conduction from convection and radiation.
1.77 Explain each in your own words. (a) What is the mode of heat transfer through a large steel plate that has its surfaces at specified temperatures? (b) What are the modes when the temperature on one surface of the steel plate is not specified, but the surface is exposed to a fluid at a specified temperature?
What are the important modes of heat transfer for a person sitting quietly in a room? What if the person is sitting near a roaring fireplace?1.79 Consider the cooling of (a) a personal computer with a separate CPU and (b) a laptop computer. The reliable functioning of these machines depends on their effective cooling. Identify and briefly explain all modes of heat transfer involved in the cooling process.
Describe and compare the modes of heat loss through the single-pane and double-pane window assemblies shown in the sketch below.A person wearing a heavy parka is standing in a cold wind. Describe the modes of heat transfer determining heart loss from the person's body.Discuss the modes of heat transfer that determine the equilibrium temperature of the space shuttle Endeavour when it is in orbit. What happens when it reenters the earths atmosphere?1.1DP1.2DP1.3DPA plane wall, 7.5 cm thick, generates heat internally at the rate of 105 W/m3. One side of the wall is insulated, and the other side is exposed to an environment at 90C. The convection heat transfer coefficient between the wall and the environment is 500 W/m2 K. If the thermal conductivity of the wall is 12 W/m K, calculate the maximum temperature in the wall.
2.2 A small dam, which is idealized by a large slab 1.2 m thick, is to be completely poured in a short Period of time. The hydration of the concrete results in the equivalent of a distributed source of constant strength of 100 W/m3. If both dam surfaces are at 16°C, determine the maximum temperature to which the concrete will be subjected, assuming steady-state conditions. The thermal conductivity of the wet concrete can be taken as 0.84 W/m K.
2.3 The shield of a nuclear reactor is idealized by a large 25-cm-thick flat plate having a thermal conductivity of . Radiation from the interior of the reactor penetrates the shield and there produces heat generation that decreases exponentially from a value of at the inner surface to a value of at a distance of 12.5 cm from the interior surface. If the exterior surface is kept at 38°C by forced convection, determine the temperature at the inner surface of the field. Hint: First set up the differential equation for a system in which the heat generation rate varies according to .
A plane wall 15 cm thick has a thermal conductivity given by the relation k=2.0+0.0005T[W/mK] where T is in kelvin. If one surface of this wall is maintained at 150C and the other at 50C, determine the rate of heat transfer per square meter. Sketch the temperature distribution through the wall.2.5 Derive an expression for the temperature distribution in a plane wall in which there are uniformly distributed heat sources that vary according to the linear relation
where is a constant equal to the heat generation per unit volume at the wall temperature . Both sides of the plate are maintained at and the plate thickness is 2L.
A plane wall of thickness 2L has internal heat sources whose strength varies according to qG=qocos(ax) Where qo is the heat generated per unit volume at the center of the wall (x=0) and a is a constant. If both sides of the wall are maintained at a constant temperature of Tw, derive an expression for the total heat loss from the wall per unit surface area.2.7 A very thin silicon chip is bonded to a 6-mm thick aluminum substrate by a 0.02-mm thick epoxy glue. Both surfaces of this chip-aluminum system are cooled by air at , where the convective heat transfer coefficient of air flow is . If the heat dissipation per unit area from the chip is under steady-state conditions, draw the thermal circuit for the system and determine the operating temperature of the chip.
2.8P2.9 In a large chemical factory, hot gases at 2273 K are cooled by a liquid at 373 K with gas-side and liquid-side convection heat transfer coefficients of 50 and , respectively. The wall that separates the gas and liquid streams is composed of a 2-cm thick oxide layer on the gas side and a 4-cm thick slab of stainless steel on the liquid side. There is a contact resistance between the oxide layer and the steel of . Determine the rate of heat loss from hot gases through the composite wall to the liquid.
2.10P2.11P2.12P2.13P2.14 Calculate the rate of heat loss per foot and the thermal resistance for a 15-cm schedule 40 steel pipe covered with a 7.5-cm-thick layer of magnesia. Superheated steam at flows inside the pipe , and still air at is on the outside .
2.15 Suppose that a pipe carrying a hot fluid with an external temperature of and outer radius is to be insulated with an insulation material of thermal conductivity k and outer radius . Show that if the convection heat transfer coefficient on the outside of the insulation is and the environmental temperature is , the addition of insulation actually increases the rate of heat loss if , and the maximum heat loss occurs when . This radius, is often called the critical radius.
2.16P2.17PEstimate the rate of heat loss per unit length from a 5-cm ID, 6-cm OD steel pipe covered with high-temperature insulation having a thermal conductivity of 0.11 W/(m K) and a thickness of 1.2 cm. Steam flows in the pipe. It has a quality of 99% and is at 150C. The unit thermal resistance at the inner wall is 0.0026(m2K)/W the heat transfer coefficient at the outer surface is 17W/(m2K) and the ambient temperature is 16C.The rate of heat flow per unit length q/L through a hollow cylinder of inside radius ri and outside radius ro is q/L=(AkT)/(rori) where A=2/(rori)/ln(ro/ri). Determine the percent error in the rate of heat flow if the arithmetic mean area (ro+ri) is used instead of the logarithmic mean area A for ratios of outside-to-inside diameters (Do/Dj) of 1.5, 2.0, and 3.0. Plot the results.A 2.5-cm-OD, 2-cm-ID copper pipe carries liquid oxygen to the storage site of a space shuttle at -183Cand0.04m3/min. The ambient air is at 21C and has a dew point of 10C. How much insulation with a thermal conductivity of 0.02 W/m K is needed to prevent condensation on the exterior of the insulation if hc+h=17W/m2 K on the outside?2.21PA cylindrical liquid oxygen (LOX) tank has a diameter of 1.22 m, a length of 6.1 m, and hemispherical ends. The boiling point of LOX is -179.4C. An insulation is sought that will reduce the boil-off rate in the steady state to no more than 11.3 kg/h. The heat of vaporization of LOX is 214 kJ/kg. If the thickness of this insulation is to be no more than 7.5 cm, what would the value of its thermal conductivity have to be?2.23P2.24PShow that the rate of heat conduction per unit length through a long, hollow cylinder of inner radius ri and outer radius ro, made of a material whose thermal conductivity varies linearly with temperature, is given by qkL=TiTo(rori)/kmA where Ti = temperature at the inner surface To = temperature at the outer surface A=2(rori)/ln(ro/ri)km=ko[1+k(Ti+To)/2]L=lenthofcyclinder2.26PDerive an expression for the temperature distribution in an infinitely long rod of uniform cross section within which there is uniform heat generation at the rate of 1 W/m. Assume that the rod is attached to a surface at Ts and is exposed through a convection heat transfer coefficient h to a fluid at Tf.Heat is generated uniformly in the fuel rod of a nuclear reactor. The rod has a long, hollow cylindrical shape with its inner and outer surfaces at temperatures of TiandTo, respectively. Derive an expression for the temperature distribution.2.29 In a cylindrical fuel rod of a nuclear reactor, heat is generated internally according to the equation
where = local rate of heat generation per unit volume at r
= outside radius
= rate of heat generation per unit volume at the centerline
Calculate the temperature drop from the centerline to the surface for a 2.5-cm-diameter rod having a thermal conductivity of if the rate of heat removal from its surface is 1.6 .
2.30 An electrical heater capable of generating 10,000 W is to be designed. The heating element is to be a stainless steel wire having an electrical resistivity of ohm-centimeter. The operating temperature of the stainless steel is to be no more than 1260°C. The heat transfer coefficient at the outer surface is expected to be no less than in a medium whose maximum temperature is 93°C. A transformer capable of delivering current at 9 and 12 V is available. Determine a suitable size for the wire, the current required, and discuss what effect a reduction in the heat transfer coefficient would have. (Hint: Demonstrate first that the temperature drop between the center and the surface of the wire is independent of the wire diameter, and determine its value.)
A hollow sphere with inner and outer radii of R1 and R2, respectively, is covered with a layer of insulation having an outer radius of R3. Derive an expression for the rate of heat transfer through the insulated sphere in terms of the radii, the thermal conductivities, the heat transfer coefficients, and the temperatures of the interior and the surrounding medium of the sphere.2.32P2.33P2.34 Show that the temperature distribution in a sphere of radius . made of a homogeneous material in which energy is released at a uniform rate per unit volume , is
2.35P2.36P2.37P
2.38 The addition of aluminum fins has been suggested to increase the rate of heat dissipation from one side of an electronic device 1 m wide and 1 m tall. The fins are to be rectangular in cross section, 2.5 cm long and 0.25 cm thick, as shown in the figure. There are to be 100 fins per meter. The convection heat transfer coefficient, both for the wall and the fins, is estimated to be K. With this information determine the percent increase in the rate of heat transfer of the finned wall compared to the bare wall.
The tip of a soldering iron consists of a 0.6-cm- diameter copper rod, 7.6 cm long. If the tip must be 204C, what are the required minimum temperature of the base and the heat flow, in watts, into the base? Assume that h=22.7W/m2KandTair=21C.One end of a 0.3-m-long steel rod is connected to a wall at 204C. The other end is connected to a wall that is maintained at 93C. Air is blown across the rod so that a heat transfer coefficient of 17W/m2 K is maintained over the entire surface. If the diameter of the rod is 5 cm and the temperature of the air is 38C, what is the net rate of heat loss to the air?Both ends of a 0.6-cm copper U-shaped rod are rigidly affixed to a vertical wall as shown in the accompanying sketch. The temperature of the wall is maintained at 93C. The developed length of the rod is 0.6 m, and it is exposed to air at 38C. The combined radiation and convection heat transfer coefficient for this system is 34W/m2K. (a) Calculate the temperature of the midpoint of the rod. (b) What will the rate of heat transfer from the rod be?2.42 A circumferential fin of rectangular cross section, 3.7-cm OD and 0.3 cm thick, surrounds a 2.5-cm- diameter tube as shown below. The fin is constructed of mild steel. Air blowing over the fin produces a heat transfer coefficient of K. If the temperatures of the base of the fin and the air are and , respectively, calculate the heat transfer rate from the fin.
2.43 A turbine blade 6.3 cm long, with cross-sectional area and perimeter , is made of stainless steel . The temperature of the root, , is . The blade is exposed to a hot gas at , and the heat transfer coefficient is K. Determine the temperature of the blade tip and the rate of heat flow at the root of the blade. Assume that the tip is insulated.
2.44 To determine the thermal conductivity of a long, solid 2.5-cm-diameter rod, one half of the rod was inserted into a furnace while the other half was projecting into air at . After steady state had been reached, the temperatures at two points 7.6 cm apart were measured and found to be and , respectively. The heat transfer coefficient over the surface of the rod exposed to the air was estimated to be 22.7 . What is thermal conductivity of the rod?
2.45 Heat is transferred from water to air through a brass wall . The addition of rectangular brass fins, 0.08 cm thick and 2.5 cm long, spaced 1.25 cm apart, is contemplated. Assuming a water-side heat transfer coefficient of and an airside heat transfer coefficient of , compare the gain in heat transfer rate achieved by adding fins to (a) the water side, (b) the air side, and (c) both sides. (Neglect temperature drop through the wall.)
2.46 The wall of a liquid-to-gas heat exchanger has a surface area on the liquid side of with a heat transfer coefficient of . On the other side of the heat exchanger wall flows a gas, and the wall has 96 thin rectangular steel fins 0.5 cm thick and 1.25 cm high as shown in the accompanying sketch. The fins are 3 m long and the heat transfer coefficient on the gas side is . Assuming that the thermal resistance of the wall is negligible, determine the rate of heat transfer if the overall temperature difference is .
2.47PThe handle of a ladle used for pouring molten lead is 30 cm long. Originally the handle was made of 1.9cm1.25cm mild steel bar stock. To reduce the grip temperature, it is proposed to form the handle of tubing 0.15 cm thick to the same rectangular shape. If the average heat transfer coefficient over the handle surface is 14 W/m K, estimate the reduction of the temperature at the grip in air at 21C.2.49P2.50 Compare the rate of heat flow from the bottom to the top of the aluminum structure shown in the sketch below with the rate of heat flow through a solid slab. The top is at v, the bottom at . The holes are filled with insulation that does not conduct heat appreciably.
2.51 Determine by means of a flux plot the temperatures and heat flow per unit depth in the ribbed insulation shown in the accompanying sketch.
2.52PDetermine the rate of heat transfer per meter length from a 5-cm-OD pipe at 150C placed eccentrically within a larger cylinder of 85 magnesia wool as shown in the sketch. The outside diameter of the larger cylinder is 15 cm and the surface temperature is 50C.2.54P2.55 A long, 1-cm-diameter electric copper cable is embedded in the center of a 25-cm-square concrete block. If the outside temperature of the concrete is 25oC and the rate of electrical energy dissipation in the cable is 150 W per meter length, determine temperatures at the outer surface and at the center of the cable.
2.56P2.57P2.58P2.59P2.60P2.61P2.62P2.63P2.64P2.1DP2.2DP2.3DP2.4DPConsider a flat plate or a plane wall with a thickness L and a long cylinder of radius r0. Both of these are made of materials such that they can be treated as lumped capacitances (Bi0.1). Show that in each case, the characteristic length lc, defined lc=(V/As), can be approximated as (L/2) and (ro/2), respectively.3.2 High-strength steel is required for use in building structures and equipment (e.g., cranes). It is produced by heat treating quench-hardened steel in a process called tempering that reduces brittleness and imparts toughness. In a production facility, alloy steel plates (k = 50 W/m K, c = 460 J/kg K, and ρ = 7865 kg/m3) of thickness 3.0 cm have to be tempered in a convective oven by heating them to 550°C. If the plates are initially at 40°C and the air inside the heat treating oven is at 700°C with a convective heat transfer coefficient of 45 W/m2 K, determine how long the plate has to remain in the oven.
3.3P3.4P3.5 In a ball-bearing production facility, steel balls that are each of 15 mm in diameter are annealed by first heating them to 870°C and then slowly cooling in air to 125°C. If the cooling air stream temperature is 60°C, and it has a convective heat transfer coefficient of , determine the time required for the cooling.
A 0.6-cm diameter mild steel rod at 38C is suddenly immersed in a liquid at 93C with hc=110W/m2K. Determine the time required for the rod to warm to 88C.3.7P3.8P3.9 The heat transfer coefficients for the flow of 26.6°C air over a sphere of 1.25 cm in diameter are measured by observing the temperature-time history of a copper ball the same dimension. The temperature of the copper ball was measured by two thermocouples, one located in the center and the other near the surface. The two thermocouples registered, within the accuracy of the recording instruments, the same temperature at any given instant. In one test run, the initial temperature of the ball was 66°C, and the temperature decreased by 7°C in 1.15 min. Calculate the heat transfer coefficient for this case.
3.10 A spherical shell satellite (3-m-OD, 1.25-cm-thick stainless steel walls) re-enters the atmosphere from outer space. If its original temperature is 38°C, the effective average temperature of the atmosphere is 1093°C, and the effective heat transfer coefficient is , estimate the temperature of the shell after reentry, assuming the time of reentry is 10 min and the interior of the shell is evacuated.
3.11P3.12P3.13P3.14 A thin-wall cylindrical vessel (1 m in diameter) is filled to a depth of 1.2 m with water at an initial temperature of 15°C. The water is well stirred by a mechanical agitator. Estimate the time required to heat the water to 50°C if the tank is suddenly immersed in oil at 105°C. The overall heat transfer coefficient between the oil and the water is , and the effective heat transfer surface is .
A thin-wall jacketed tank heated by condensing steam at one atmosphere contains 91 kg of agitated water. The heat transfer area of the jacket is 0.9m2 and the overall heat transfer coefficient U=227W/m2K based on that area. Determine the heating time required for an increase in temperature from 16C to 60C.3.16 A large, 2.54-cm.-thick copper plate is placed between two air streams. The heat transfer coefficient on one side is and on the other side is . If the temperature of both streams is suddenly changed from 38°C to 93°C, determine how long it takes for the copper plate to reach a temperature of 82°C.
3.17 A 1.4-kg aluminum household iron has a 500-W heating element. The surface area is . The ambient temperature is 21°C, and the surface heat transfer coefficient is . How long after the iron is plugged in does its temperature reach 104°C?
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3.28 A long wooden rod at with a 2.5-cm-OD is placed into an airstream at 600°C. The heat transfer coefficient between the rod and air is . If the ignition temperature of the wood is , , and , determine the time between initial exposure and ignition of the wood.
A mild-steel cylindrical billet 25 cm in diameter is to be raised to a minimum temperature of 760C by passing it through a 6-m long strip-type furnace. If the furnace gases are at 1538C and the overall heat transfer coefficient on the outside of the billet is 68W/m2K, determine the maximum speed at which a continuous billet entering at 204C can travel through the furnace.3.30P3.31P3.32P3.33P3.34P3.35P3.36P3.37P
3.38 An egg, which for the purposes of this problem is assumed to be a 5-cm-diameter sphere having the thermal properties of water, is initially at a temperature of . It is immersed in boiling water at for 15 min. The heat transfer coefficient from the water to the egg is assumed to be . What is the temperature of the egg center at the end of the cooking period?
3.39P3.40P3.41P3.42P3.43P3.1DP3.2DP3.3DP3.4DP4.1P4.2P4.3P4.4P4.5P4.6P4.7P4.8P4.9P4.10P4.11P4.12P4.13P4.14P4.15P4.16P4.17P4.18P4.19P4.20P4.21P4.22P4.23P4.24P4.25P4.26P4.27P4.28P4.29P4.30P4.31P4.32P4.33P4.34P4.35P4.36P4.37P4.38P4.39P4.40P4.41P4.42P4.43P4.44P4.45P4.46P4.47P4.48P4.49P4.50P4.51P4.52P4.53P4.54P4.55P4.56P4.57P4.58P4.1DP4.2DP4.3DPEvaluate the Reynolds number for flow over a tube from the following data: D=6cm,U=1.0m/s, =300kg/m3,=0.04Ns/m2.5.2 Evaluate the Prandtl number from the following data: , .
Evaluate the Nusselt number for flow over a sphere for the following conditions: D=0.15m,k=0.2W/mK, hc=102W/m2K.5.4 Evaluate the Stanton number for flow over a tube from the following data:
, , , , .
Evaluate the dimensionless groups hcD/k,UD/, and cp/k for water, n-butyl alcohol, mercury, hydrogen, air, and saturated steam at a temperature of 100C. Let D=1m,U=1m/sec, and hc=1W/m2K.5.6 A fluid flows at 5 over a wide, flat plate 15 cm long. For each from the following list, calculate the Reynolds number at the downstream end of the plate. Indicate whether the flow at that point is laminar, transition, or turbulent. Assume all fluids are at 40°C. (a) air, (b) , (c) water, (d) engine oil.
5.7 The average Reynolds number for air passing in turbulent flow over a 2-m-long, flat plate is . Under these conditions, the average Nusselt number was found to be equal to 4150. Determine the average heat transfer coefficient for an oil having thermal properties similar to those in Appendix 2, Table 18, at at the same Reynolds number and flowing over the same plate.
5.8PWhen a sphere falls freely through a homogeneous fluid, it reaches a terminal velocity at which the weight of the sphere is balanced by the buoyant force and the frictional resistance of the fluid. Make a dimensional analysis of this problem and indicate how experimental data for this problem could be correlated. Neglect compressibility effects and the influence of surface roughness.5.10 Experiments have been performed on the temperature distribution in a homogeneous long cylinder (0.1 m diameter, thermal conductivity of 0.2 W/m K) with uniform internal heat generation. By dimensional analysis, determine the relation between the steady-state temperature at the center of the cylinder , the diameter, the thermal conductivity, and the rate of heat generation. Take the temperature at the surface as your datum. What is the equation for the center temperature if the difference between center and surface temperature is when the heat generation is ?
5.11P5.12P5.13 The torque due to the frictional resistance of the oil film between a rotating shaft and its bearing is found to be dependent on the force F normal to the shaft, the speed of rotation N of the shaft, the dynamic viscosity of the oil, and the shaft diameter D. Establish a correlation among these variables by using dimensional analysis.
5.14P5.15P5.16P5.17PThe drag on an airplane wing in flight is known to be a function of the density of air (), the viscosity of air(), the free-stream velocity (U), a characteristic dimension of the wing (s), and the shear stress on the surface of the wing (s). Show that the dimensionless drag, sU2, can be expressed as a function of the Reynolds number, Us.5.19 Suppose that the graph below shows measured values of for air in forced convection over a cylinder of diameter D. plotted on a logarithmic graph of as a function of ReD Pr.
Write an appropriate dimensionless correlation for the average Nusselt number for these data and state any limitations to your equation.
5.20P5.21P5.22P5.23PEngine oil at 100C flows over and parallel to a flat surface at a velocity of 3 m/s. Calculate the thickness of the hydrodynamic boundary layer at a distance 0.3 m from the leading edge of the surface.5.25P5.26P5.27PFor flow over a slightly curved isothermal surface, the temperature distribution inside the boundary layer t can be approximated by the polynomial T(y)=a+by+cy2+d3(yt), where y is the distance normal to the surface. (a) By applying appropriate boundary conditions, evaluate the constants a, b, c, and d. Fluid (b) Then obtain a dimensionless relation for the temperature distribution in the boundary layer.Air at 20C flows at 1 m/s between two parallel flat plates spaced 5 cm apart. Estimate the distance from the entrance to the point at which the hydrodynamic boundary layers meet.Air at 1000C flows at an inlet velocity of 2 m/s between two parallel flat plates spaced 1 cm apart. Estimate the distance from the entrance to the point where the boundary layers meet.5.31P5.32P5.33P5.34P5.35P5.36P5.37P5.38P5.39P