Heat Transfer - Yunus Cengel - 2ed - Chapter 4- Solutions

4–1C What is lumped system analysis? When is it applicable?
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4–2C Consider heat transfer between two identical hot solid bodies and the air surrounding them. The first solid is being cooled by a fan while the second one is allowed to cool naturally. For which solid is the lumped system analysis more likely to be applicable? Why?
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4–3C Consider heat transfer between two identical hot solid bodies and their environments. The first solid is dropped in a large container filled with water, while the second one is allowed to cool naturally in the air. For which solid is the lumped system analysis more likely to be applicable? Why?
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4–4C Consider a hot baked potato on a plate. The temperature of the potato is observed to drop by 4°C during the first minute. Will the temperature drop during the second minute be less than, equal to, or more than 4°C? Why?
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4–5C Consider a potato being baked in an oven that is maintained at a constant temperature. The temperature of the potato is observed to rise by 5°C during the first minute. Will the temperature rise during the second minute be less than, equal to, or more than 5°C? Why?
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4–6C What is the physical significance of the Biot number? Is the Biot number more likely to be larger for highly conducting solids or poorly conducting ones?
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4–7C Consider two identical 4kg pieces of roast bee
f. The first piece is baked as a whole, while the second is baked after being cut into two equal pieces in the same oven. Will there be any difference between the cooking times of the whole and cut roasts? Why?
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4–8C Consider a sphere and a cylinder of equal volume made of copper. Both the sphere and the cylinder are initially at the same temperature and are exposed to convection in the same environment. Which do you think will cool faster, the cylinder or the sphere? Why?
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4–9C In what medium is the lumped system analysis more likely to be applicable: in water or in air? Why?
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4–10C For which solid is the lumped system analysis more likely to be applicable: an actual apple or a golden apple of the same size? Why?
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4–11C For which kind of bodies made of the same material is the lumped system analysis more likely to be applicable: slender ones or well-rounded ones of the same volume? Why?
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4–12 Obtain relations for the characteristic lengths of a large plane wall of thickness 2L, a very long cylinder of radius ro, and a sphere of radius ro.
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4–13 Obtain a relation for the time required for a lumped system to reach the average temperature (Ti T ), where Ti is the initial temperature and T is the temperature of the environment.
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4–14 The temperature of a gas stream is to be measured by a thermocouple whose junction can be approximated as a 1.2-mm-diameter sphere. The properties of the junction are k 35 W/m · °C, 8500 kg/m3, and Cp 320 J/kg · °C, and the heat transfer coefficient between the junction and the gas is h 65 W/m2 · °C. Determine how long it will take for the thermocouple to read 99 percent of the initial temperature difference.
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4–15E In a manufacturing facility, 2-in.-diameter brass balls (k 64.1 Btu/h · ft · °F, 532 lbm/ft3, and Cp 0.092 Btu/lbm · °F) initially at 250°F are quenched in a water bath at 120°F for a period of 2 min at a rate of 120 balls per minute. If the convection heat transfer coefficient is 42 Btu/h · ft2 · °F, determine (a) the temperature of the balls after quenching and (b) the rate at which heat needs to be removed from the water in order to keep its temperature constant at 120°F.
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4–16E Repeat Problem 4–15E for aluminum balls.
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4–17 To warm up some milk for a baby, a mother pours milk into a thin-walled glass whose diameter is 6 cm. The height of the milk in the glass is 7 cm. She then places the glass into a large pan filled with hot water at 60°C. The milk is stirred constantly, so that its temperature is uniform at all times. If the heat transfer coefficient between the water and the glass is 120 W/m2 · °C, determine how long it will take for the milk to warm up from 3°C to 38°C. Take the properties of the milk to be the same as those of water. Can the milk in this case be treated as a lumped system? Why?
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4–18 Repeat Problem 4–17 for the case of water also being stirred, so that the heat transfer coefficient is doubled to 240 W/m2 · °C.
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4–19E During a picnic on a hot summer day, all the cold drinks disappeared quickly, and the only available drinks were those at the ambient temperature of 80°F. In an effort to cool a 12-fluid-oz drink in a can, which is 5 in. high and has a diameter of 2.5 in., a person grabs the can and starts shaking it in the iced water of the chest at 32°F. The temperature of the drink can be assumed to be uniform at all times, and the heat transfer coefficient between the iced water and the aluminum can is 30 Btu/h · ft2 · °F. Using the properties of water for the drink, estimate how long it will take for the canned drink to cool to 45°F.
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4–20 Consider a 1000-W iron whose base plate is made of 0.5-cm-thick aluminum alloy 2024-T6 ( 2770 kg/m3, Cp 875 J/kg · °C, 7.3 10 5 m2/s). The base plate has a surface area of 0.03 m2. 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/m2 · °C 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?
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4–21 Reconsider Problem 4–20. Using EES (or other) software, investigate the effects of the heat transfer coefficient and the final plate temperature on the time it will take for the plate to reach this temperature. Let the heat transfer coefficient vary from 5 W/m2 · °C to 25 W/m2 · °C and the temperature from 30°C to 200°C. Plot the time as functions of the heat transfer coefficient and the temperature, and discuss the results.
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4–22 Stainless steel ball bearings ( 8085 kg/m3, k 15.1 W/m · °C, Cp 0.480 kJ/kg · °C, and 3.91 10 6 m2/s) having a diameter of 1.2 cm are to be quenched in water. The balls leave the oven at a uniform temperature of 900°C and are exposed to air at 30°C for a while before they are dropped into the water. If the temperature of the balls is not to fall below 850°C prior to quenching and the heat transfer coefficient in the air is 125 W/m2 · °C, determine how long they can stand in the air before being dropped into the water.
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4–23 Carbon steel balls ( 7833 kg/m3, k 54 W/m · °C, Cp 0.465 kJ/kg · °C, and 1.474 10 6 m2/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/m2 · °C, 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.
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4–24 Reconsider Problem 4–23. Using EES (or other) software, investigate the effect of the initial temperature of the balls on the annealing time and the total rate of heat transfer. Let the temperature vary from 500°C to 1000°C. Plot the time and the total rate of heat transfer as a function of the initial temperature, and discuss the results.
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4–25 An electronic device dissipating 30 W has a mass of 20 g, a specific heat of 850 J/kg · °C, and a surface area of 5 cm 2. The device is lightly used, and it is on for 5 min and then off for several hours, during which it cools to the ambient temperature of 25°C. Taking the heat transfer coefficient to be 12 W/m2 · °C, determine the temperature of the device at the end of the 5-min operating period. What would your answer be if the device were attached to an aluminum heat sink having a mass of 200 g and a surface area of 80 cm2? Assume the device and the heat sink to be nearly isothermal.
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4–26C What is an infinitely long cylinder? When is it proper to treat an actual cylinder as being infinitely long, and when is it not? For example, is it proper to use this model when finding the temperatures near the bottom or top surfaces of a cylinder? Explain.
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4–27C Can the transient temperature charts in Fig 4–13 for a plane wall exposed to convection on both sides be used for a plane wall with one side exposed to convection while the other side is insulated? Explain.
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4–28C Why are the transient temperature charts prepared using nondimensionalized quantities such as the Biot and Fourier numbers instead of the actual variables such as thermal conductivity and time?
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4–29C What is the physical significance of the Fourier number? Will the Fourier number for a specified heat transfer problem double when the time is doubled?
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4–30C How can we use the transient temperature charts when the surface temperature of the geometry is specified instead of the temperature of the surrounding medium and the convection heat transfer coefficient?
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4–31C Abody at an initial temperature of Ti is brought into a medium at a constant temperature of T . How can you determine the maximum possible amount of heat transfer between the body and the surrounding medium?
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4–32C The Biot number during a heat transfer process between a sphere and its surroundings is determined to be 0.02. Would you use lumped system analysis or the transient temperature charts when determining the midpoint temperature of the sphere? Why?
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4–33 A student calculates that the total heat transfer from a spherical copper ball of diameter 15 cm initially at 200°C and its environment at a constant temperature of 25°C during the first 20 min of cooling is 4520 kJ. Is this result reasonable? Why?
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4–34 An ordinary egg can be approximated as a 5.5-cmdiameter sphere whose properties are roughly k 0.6 W/m · °C and 0.14 10 6 m2/s. The egg is initially at a uniform temperature of 8°C and is dropped into boiling water at 97°C. Taking the convection heat transfer coefficient to be h 1400 W/m2 · °C, determine how long it will take for the center of the egg to reach 70°C.
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4–35 Reconsider Problem 4–34. Using EES (or other) software, investigate the effect of the final center temperature of the egg on the time it will take for the center to reach this temperature. Let the temperature vary from 50°C to 95°C. Plot the time versus the temperature, and discuss the results.
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4–36 In a production facility, 3-cm-thick large brass plates (k 110 W/m ·°C, 8530 kg/m3, Cp 380 J/kg · °C, and 33.9 10 6 m2/s) that are initially at a uniform temperature of 25°C are heated by passing them through an oven maintained at 700°C. The plates remain in the oven for a period of 10 min. Taking the convection heat transfer coefficient to be h 80 W/m2 · °C, determine the surface temperature of the plates when they come out of the oven.
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4–37 Reconsider Problem 4–36. Using EES (or other) software, investigate the effects of the temperature of the oven and the heating time on the final surface temperature of the plates. Let the oven temperature vary from 500°C to 900°C and the time from 2 min to 30 min. Plot the surface temperature as the functions of the oven temperature and the time, and discuss the results.
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4–38 Along 35-cm-diameter cylindrical shaft made of stainless steel 304 (k 14.9 W/m · °C, 7900 kg/m3, Cp 477 J/kg · °C, and 3.95 10 6 m2/s) comes out of an oven at a uniform temperature of 400°C. The shaft is then allowed to cool slowly in a chamber at 150°C with an average convection heat transfer coefficient of h 60 W/m2 · °C. Determine the temperature at the center of the shaft 20 min after the start of the cooling process. Also, determine the heat transfer per unit length of the shaft during this time period.
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4–39 Reconsider Problem 4–38. Using EES (or other) software, investigate the effect of the cooling time on the final center temperature of the shaft and theamount of heat transfer. Let the time vary from 5 min to 60 min. Plot the center temperature and the heat transfer as a function of the time, and discuss the results.
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4–40E Long cylindrical AISI stainless steel rods (k 7.74 Btu/h · ft · °F and 0.135 ft2/h) of 4-in. diameter are heattreated by drawing them at a velocity of 10 ft/min through a 30-ft-long oven maintained at 1700°F. The heat transfer coefficient in the oven is 20 Btu/h · ft2 · °F. If the rods enter the oven at 85°F, determine their centerline temperature when they leave.
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4–41 In a meat processing plant, 2-cm-thick steaks (k 0.45 W/m · °C and 0.91 10 7 m2/s) that are initially at 25°C are to be cooled by passing them through a refrigeration room at 11°C. The heat transfer coefficient on both sides of the steaks is 9 W/m2 · °C. If both surfaces of the steaks are to be cooled to 2°C, determine how long the steaks should be kept in the refrigeration room.
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4–42 A long cylindrical wood log (k 0.17 W/m · °C and 1.28 10 7 m2/s) is 10 cm in diameter and is initially at a uniform temperature of 10°C. It is exposed to hot gases at 500°C in a fireplace with a heat transfer coefficient of 13.6 W/m2 · °C on the surface. If the ignition temperature of the wood is 420°C, determine how long it will be before the log ignites.
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4–43 In Betty Crocker’s Cookbook, it is stated that it takes 2h 45 min to roast a 3.2-kg rib initially at 4.5°C “rare” in an oven maintained at 163°C. It is recommended that a meat thermometer be used to monitor the cooking, and the rib is considered rare done when the thermometer inserted into the center of the thickest part of the meat registers 60°C. The rib can be treated as a homogeneous spherical object with the properties 1200 kg/m3, Cp 4.1 kJ/kg · °C, k 0.45 W/m · °C, and 0.91 10 7 m2/s. Determine (a) the heat transfer coefficient at the surface of the rib, (b) the temperature of the outer surface of the rib when it is done, and (c) the amount of heat transferred to the rib. (d) Using the values obtained, predict how long it will take to roast this rib to “medium” level, which occurs when the innermost temperature of the rib reaches 71°C. Compare your result to the listed value of 3 h 20 min. If the roast rib is to be set on the counter for about 15 min before it is sliced, it is recommended that the rib be taken out of the oven when the thermometer registers about 4°C below the indicated value because the rib will continue cooking evenafter it is taken out of the oven. Do you agree with this recommendation?
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4–44 Repeat Problem 4–43 for a roast rib that is to be “welldone” instead of “rare.” A rib is considered to be well-done when its center temperature reaches 77°C, and the roasting in this case takes about 4 h 15 min.
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4–45 For heat transfer purposes, an egg can be considered to be a 5.5-cm-diameter sphere having the properties of water. An egg that is initially at 8°C is dropped into the boiling water at 100°C. The heat transfer coefficient at the surface of the egg is estimated to be 800 W/m2 · °C. If the egg is considered cooked when its center temperature reaches 60°C, determine how long the egg should be kept in the boiling water.
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4–46 Repeat Problem 4–45 for a location at 1610-m elevation such as Denver, Colorado, where the boiling temperature of water is 94.4°C.
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4–47 The author and his 6-year-old son have conducted the following experiment to determine the thermal conductivity of a hot dog. They first boiled water in a large pan and measured the temperature of the boiling water to be 94°C, which is not surprising, since they live at an elevation of about 1650 m in Reno, Nevada. They then took a hot dog that is 12.5 cm long and 2.2 cm in diameter and inserted a thermocouple into the midpoint of the hot dog and another thermocouple just under the skin. They waited until both thermocouples read 20°C, which is the ambient temperature. They then dropped the hot dog into boiling water and observed the changes in both temperatures. Exactly 2 min after the hot dog was dropped into the boiling water, they recorded the center and the surface temperatures to be 59°C and 88°C, respectively. The density of the hot dog can be taken to be 980 kg/m3, which is slightly less than the density of water, since the hot dog was observed to be floating in water while being almost completely immersed. The specific heat of a hot dog can be taken to be 3900 J/kg · °C, which is slightly less than that of water, since a hot dog is mostly water. Using transient temperature charts, determine (a) the thermal diffusivity of the hot dog, (b) the thermal conductivity of the hot dog, and (c) the convection heat transfer coefficient.
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4–48 Using the data and the answers given in Problem 4–47, determine the center and the surface temperatures of the hot dog 4 min after the start of the cooking. Also determine the amount of heat transferred to the hot dog.
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4–49E In a chicken processing plant, whole chickens averaging 5 lb each and initially at 72°F are to be cooled in the racks of a large refrigerator that is maintained at 5°F. The entire chicken is to be cooled below 45°F, but the temperature of the chicken is not to drop below 35°F at any point during refrigeration. The convection heat transfer coefficient and thus the rate of heat transfer from the chicken can be controlled by varying the speed of a circulating fan inside. Determine the heat transfer coefficient that will enable us to meet both temperature constraints while keeping the refrigeration time to a minimum. The chicken can be treated as a homogeneous spherical object having the properties 74.9 lbm/ft3, Cp 0.98 Btu/lbm · °F, k 0.26 Btu/h · ft · °F, and 0.0035 ft2/h.
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4–50 Aperson puts a few apples into the freezer at 15°C to cool them quickly for guests who are about to arrive. Initially, the apples are at a uniform temperature of 20°C, and the heat transfer coefficient on the surfaces is 8W/m2 · °C. Treating the apples as 9-cm-diameter spheres and taking their properties to be 840 kg/m3, Cp 3.81 kJ/kg · °C, k 0.418 W/m · °C, and 1.3 10 7 m2/s, determine the center and surface temperatures of the apples in 1 h. Also, determine the amount of heat transfer from each apple.
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4–51 Reconsider Problem 4–50. Using EES (or other) software, investigate the effect of the initial temperature of the apples on the final center and surface temperatures and the amount of heat transfer. Let the initial temperature vary from 2°C to 30°C. Plot the center temperature, the surface temperature, and the amount of heat transfer as a function of the initial temperature, and discuss the results.
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4–52 Citrus fruits are very susceptible to cold weather, and extended exposure to subfreezing temperatures can destroy them. Consider an 8-cm-diameter orange that is initially at 15°C. A cold front moves in one night, and the ambient temperature suddenly drops to 6°C, with a heat transfer coefficient of 15 W/m2 · °C. Using the properties of water for the orange and assuming the ambient conditions to remain constant for 4 h before the cold front moves out, determine if any part of the orange will freeze that night.
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4–53 An 8-cm-diameter potato ( 1100 kg/m3, Cp 3900 J/kg · °C, k 0.6 W/m · °C, and 1.4 10 7 m2/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 25 W/m2 · °C, determine (a) how long the potato is baked in the oven and (b) the final equilibrium temperature of the potato after it is wrapped.
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4–54 White potatoes (k 0.50 W/m · °C and 0.13 10 6 m2/s) that are initially at a uniform temperature of 25°C and have an average diameter of 6 cm are to be cooled by refrigerated air at 2°C flowing at a velocity of 4 m/s. The average heat transfer coefficient between the potatoes and the air is experimentally determined to be 19 W/m2 · °C. Determine how long it will take for the center temperature of the potatoes to drop to 6°C. Also, determine if any part of the potatoes will experience chilling injury during this process.
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4–55E Oranges of 2.5-in. diameter (k 0.26 Btu/h · ft · °F and 1.4 10 6 ft2/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/h · ft2 · °F. 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.
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4–56 A 65-kg beef carcass (k 0.47 W/m · °C and 0.13 10 6 m2/s) initially at a uniform temperature of 37°C is to be cooled by refrigerated air at 6°C flowing at a velocity of 1.8 m/s. The average heat transfer coefficient between the carcass and the air is 22 W/m2 · °C. Treating the carcass as a cylinder of diameter 24 cm and height 1.4 m and disregarding heat transfer from the base and top surfaces, determine how long it will take for the center temperature of the carcass to drop to 4°C. Also, determine if any part of the carcass will freeze during this process.
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4–57 Layers of 23-cm-thick meat slabs (k 0.47 W/m · °C and 0.13 10 6 m2/s) initially at a uniform temperature of 7°C are to be frozen by refrigerated air at 30°C flowing at a velocity of 1.4 m/s. The average heat transfer coefficient between the meat and the air is 20 W/m2 · °C. Assuming the size of the meat slabs to be large relative to their thickness, determine how long it will take for the center temperature of the slabs to drop to 18°C. Also, determine the surface temperature of the meat slab at that time.
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4–58E Layers of 6-in.-thick meat slabs (k 0.26 Btu/h · ft ·°F and 1.4 10 6 ft2/s) initially at a uniform temperature of 50°F are cooled by refrigerated air at 23°F to a temperature of 36°F at their center in 12 h. Estimate the average heat transfer coefficient during this cooling process.
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4–59 Chickens with an average mass of 1.7 kg (k 0.45 W/m · °C and 0.13 10 6 m2/s) initially at a uniform temperature of 15°C are to be chilled in agitated brine at 10°C. The average heat transfer coefficient between the chicken and the brine is determined experimentally to be 440W/m2 · °C. Taking the average density of the chicken to be 0.95 g/cm3 and treating the chicken as a spherical lump, determine the center and the surface temperatures of the chicken in 2 h and 30 min. Also, determine if any part of the chicken will freeze during this process.
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4–60C What is a semi-infinite medium? Give examples of solid bodies that can be treated as semi-infinite mediums for heat transfer purposes
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4–61C Under what conditions can a plane wall be treated as a semi-infinite medium
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4–62C Consider a hot semi-infinite solid at an initial temperature of Ti that is exposed to convection to a cooler medium at a constant temperature of T , with a heat transfer coefficient of h. Explain how you can determine the total amount of heat transfer from the solid up to a specified time to
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4–63 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/m · °C and 0.15 80 6 m2/s. Assuming an initial uniform temperature of 8°C for the ground, determine the minimum burial depth to prevent the water pipes from freezing
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4–64 The soil temperature in the upper layers of the earth varies with the variations in the atmospheric conditions. Before a cold front moves in, theearth at a location is initially at a uniform temperature of 10°C. Then the area is subjected to a temperature of 10°C and high winds that resulted in aconvection heat transfer coefficient of 40 W/m2 · °C on the earth’s surface for a period of 10 h. Taking the properties of the soil at that location to be k 0.9W/m · °C and 1.6 10 5 m2/s, determine the soil temperature at distances 0, 10, 20, and 50 cm from the earth’s surface at the end of this 10-h period.
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4–65 Reconsider Problem 4–64. Using EES (or other) software, plot the soil temperature as a function of the distance from the earth’s surface as the distance varies from 0 m to 1m, and discuss the results
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4–66E The walls of a furnace are made of 1.5-ft-thick concrete (k 0.64 Btu/h · ft · °F and 0.023 ft2/h). Initially, the furnace and the surrounding air are in thermal equilibrium at 70°F. The furnace is then fired, and the inner surfaces of the furnace are subjected to hot gases at 1800°F with a very large heat transfer coefficient. Determine how long it will take for the temperature of the outer surface of the furnace walls to rise to 70.1°F
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4–67 Athick wood slab (k 0.17 W/m · °C and 1.28 10 7 m2/s) that is initially at a uniform temperature of 25°C is exposed to hot gases at 550°C for a period of 5 minutes. The heat transfer coefficient between thegases and the wood slab is 35 W/m2 · °C. If the ignition temperature of the wood is 450°C, determine if the wood will ignite
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4–68 Alarge cast iron container (k 52 W/m · °C and 1.70 10 5 m2/s) with 5-cm-thick walls is initially at a uniform temperature of 0°C and is filled with ice at 0°C. Now the outer surfaces of the container are exposed to hot water at 60°C with a very large heat transfer coefficient. Determine how long it will be before the ice inside the container starts melting. Also, taking the heat transfer coefficient on the inner surface of the container to be 250 W/m2 · °C, determine the rate of heat transfer to the ice through a 1.2-m-wide and 2-m-high section of the wall when steady operating conditions are reached. Assume the ice starts melting when its inner surface temperature rises to 0.1°C.
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4–69C What is the product solution method? How is it used to determine the transient temperature distribution in a twodimensional system
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4–70C How is the product solution used to determine the variation of temperature with time and position in threedimensional systems
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4–71C Ashort cylinder initially at a uniform temperature Ti is subjected to convection from all of its surfaces to a medium at temperature T . Explain how you can determine the temperature of the midpoint of the cylinder at a specified time t
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4–72C Consider a short cylinder whose top and bottom surfaces are insulated. The cylinder is initially at a uniform temperature Ti and is subjected to convection from its side surface to a medium at temperature T with a heat transfer coefficient of h. Is the heat transfer in this short cylinder one- or twodimensional? Explain
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4–73 A short brass cylinder ( 8530 kg/m3, Cp 0.389 kJ/kg · °C, k 110 W/m · °C, and 3.39 10 5 m2/s) of diameter D 8 cm and height H 15 cm is initially at a uniform temperature of Ti 150°C. The cylinder is now placed in atmospheric air at 20°C, where heat transfer takes place by convection with a heat transfer coefficient of h 40 W/m2 ·°C. Calculate (a) the center temperature of the cylinder, (b) the center temperature of the top surface of the cylinder, and (c) the total heat transfer from the cylinder 15 min after the start of the cooling.
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4–74 Reconsider Problem 4–73. Using EES (or other) software, investigate the effect of the cooling time on the center temperature of the cylinder, the center temperature of the top surface of the cylinder, and the total heat transfer. Let the time vary from 5 min to 60 min. Plot the center temperature of the cylinder, the center temperature of the top surface, and the total heat transfer as a function of the time, and discuss the results
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4–75 Asemi-infinite aluminum cylinder (k 237 W/m · °C, 9.71 10 5 m2/s) of diameter D 15 cm is initially at a uniform temperature of Ti 150°C. The cylinder is now placed in water at 10°C, where heat transfer takes place by convection with a heat transfer coefficient of h 140 W/m2 ·°C. Determine the temperature at the center of the cylinder 5 cm from the end surface 8 min after the start of cooling
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4–76E A hot dog can be considered to be a cylinder 5 in. long and 0.8 in. in diameter whose properties are 61.2 lbm/ft3, Cp 0.93 Btu/lbm · °F, k 0.44 Btu/h · ft · °F, and 0.0077 ft2/h. A hot dog initially at 40°F is dropped into boiling water at 212°F. If the heat transfer coefficient at the surface of the hot dog is estimated to be 120 Btu/h · ft2 · °F, determine the center temperature of the hot dog after 5, 10, and 15 min by treating the hot dog as (a) a finite cylinder and (b) an infinitely long cylinder.
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4–77E Repeat Problem 4–76E for a location at 5300 ft elevation such as Denver, Colorado, where the boiling temperature of water is 202°F
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4–78 A5-cm-high rectangular ice block (k 2.22 W/m · °C and 0.124 10 7 m2/s) initially at 20°C is placed on a table on its square base 4 cm 4 cm in size in a room at 18°C. The heat transfer coefficient on the exposed surfaces of the ice block is 12 W/m2 · °C. Disregarding any heat transfer from the base to the table, determine how long it will be before the ice block starts melting. Where on the ice block will the first liquid droplets appear?
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4–79 Reconsider Problem 4–78. Using EES (or other) software, investigate the effect of the initial temperature of the ice block on the time period before the ice block starts melting. Let the initial temperature vary from 26°C to 4°C. Plot the time versus the initial temperature, and discuss the results
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4–80 A2-cm-high cylindrical ice block (k 2.22 W/m · °C and 0.124 10 7 m2/s) is placed on a table on its base of diameter 2 cm in a room at 20°C. The heat transfer coefficient on the exposed surfaces of the ice block is 13 W/m2 · °C, and heat transfer from the base of the ice block to the table is negligible. If the ice block is not to start melting at any point for at least 2 h, determine what the initial temperature of the ice block should be
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4–81 Consider a cubic block whose sides are 5 cm long and a cylindrical block whose height and diameter are also 5 cm. Both blocks are initially at 20°C and are made of granite (k 2.5 W/m · °C and 1.15 10 6 m2/s). Now both blocks are exposed to hot gases at 500°C in a furnace on all of their surfaces with a heat transfer coefficient of 40 W/m2 · °C. Determine the center temperature of each geometry after 10, 20, and 60 min
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4–82 Repeat Problem 4–81 with the heat transfer coefficient at the top and the bottom surfaces of each block being doubled to 80 W/m2 · °C.
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4–83 A 20-cm-long cylindrical aluminum block ( 2702 kg/m3, Cp 0.896 kJ/kg · °C, k 236 W/m · °C, and 9.75 10 5 m2/s), 15 cm in diameter, is initially at a uniform temperature of 20°C. The block is to be heated in a furnace at 1200°C until its center temperature rises to 300°C. If the heat transfer coefficient on all surfaces of the block is 80 W/m2 · °C, determine how long the block should be kept in the furnace. Also, determine the amount of heat transfer from the aluminum block if it is allowed to cool in the room until its temperature drops to 20°C throughout
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4–84 Repeat Problem 4–83 for the case where the aluminum block is inserted into the furnace on a low-conductivity material so that the heat transfer to or from the bottom surface of the block is negligible
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4–85 Reconsider Problem 4–83. Using EES (or other) software, investigate the effect of the final center temperature of the block on the heating time and the amount of heat transfer. Let the final center temperature vary from 50°C to 1000°C. Plot the time and the heat transfer as a function of the final center temperature, and discuss the results
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4–86C What are the common kinds of microorganisms? What undesirable changes do microorganisms cause in foods
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4–87C How does refrigeration prevent or delay the spoilage of foods? Why does freezing extend the storage life of foods for months
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4–88C What are the environmental factors that affect the growth rate of microorganisms in foods
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4–89C What is the effect of cooking on the microorganisms in foods? Why is it important that the internal temperature of a roast in an oven be raised above 70°C
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4–90C How can the contamination of foods with microorganisms be prevented or minimized? How can the growth of microorganisms in foods be retarded? How can the microorganisms in foods be destroyed
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4–91C How does (a) the air motion and (b) the relative humidity of the environment affect the growth of microorganisms in foods
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4–92C The cooling of a beef carcass from 37°C to 5°C with refrigerated air at 0°C in a chilling room takes about 48 h. To reduce the cooling time, it is proposed to cool the carcass with refrigerated air at –10°C. How would you evaluate this proposal
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4–93C Consider the freezing of packaged meat in boxes with refrigerated air. How do (a) the temperature of air, (b) the velocity of air, (c) the capacity of the refrigeration system, and (d) the size of the meat boxes affect the freezing time
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4–94C How does the rate of freezing affect the tenderness, color, and the drip of meat during thawing
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4–95C It is claimed that beef can be stored for up to two years at –23°C but no more than one year at –12°C. Is this claim reasonable? Explain
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4–96C What is a refrigerated shipping dock? How does it reduce the refrigeration load of the cold storage rooms
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4–97C How does immersion chilling of poultry compare to forced-air chilling with respect to (a) cooling time, (b) moisture loss of poultry, and (c) microbial growth
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4–98C What is the proper storage temperature of frozen poultry? What are the primary methods of freezing for poultry
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4–99C What are the factors that affect the quality of frozen fish
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4–100 The chilling room of a meat plant is 15 m 18 m 5.5 m in size and has a capacity of 350 beef carcasses. The power consumed by the fans and the lights in the chilling room are 22 and 2 kW, respectively, and the room gains heat through its envelope at a rate of 11 kW. The average mass of beef carcasses is 280 kg. The carcasses enter the chilling room at 35°C, after they are washed to facilitate evaporative cooling, and are cooled to 16°C in 12 h. The air enters the chilling room at 2.2°C and leaves at 0.5°C. Determine (a) the refrigeration load of the chilling room and (b) the volume flow rate of air. The average specific heats of beef carcasses and air are 3.14 and 1.0 kJ/kg · °C, respectively, and the density of air can be taken to be 1.28 kg/m3
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4–101 Turkeys with a water content of 64 percent that are initially at 1°C and have a mass of about 7 kg are to be frozen by submerging them into brine at 29°C. Using Figure 4–45, determine how long it will take to reduce the temperature of the turkey breast at a depth of 3.8 cm to 18°C. If the temperature at a depth of 3.8 cm in the breast represents the average temperature of the turkey, determine the amount of heat transfer per turkey assuming (a) the entire water content of the turkey is frozen and (b) only 90 percent of the water content of the turkey is frozen at 18°C. Take the specific heats of turkey to be 2.98 and 1.65 kJ/kg · °C above and below the freezing point of 2.8°C, respectively, and the latent heat of fusion of turkey to be 214 kJ/kg.
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4–102E Chickens with a water content of 74 percent, an initial temperature of 32°F, and a mass of about 6 lbm are to be frozen by refrigerated air at 40°F. Using Figure 4–44, determine how long it will take to reduce the inner surface temperature of chickens to 25°F. What would your answer be if the air temperature were 80°F
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4–103 Chickens with an average mass of 2.2 kg and average specific heat of 3.54 kJ/kg · °C are to be cooled by chilled water that enters a continuous-flow-type immersion chiller at 0.5°C. Chickens are dropped into the chiller at a uniform temperature of 15°C at a rate of 500 chickens per hour and are cooled to an average temperature of 3°C before they are taken out. The chiller gains heat from the surroundings at a rate of 210 kJ/min. Determine (a) the rate of heat removal from the chicken, in kW, and (b) the mass flow rate of water, in kg/s, if the temperature rise of water is not to exceed 2°C
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4–104 In a meat processing plant, 10-cm-thick beef slabs (ρ 1090 kg/m3, Cp 3.54 kJ/kg · °C, k 0.47 W/m · °C, and α 0.13 10–6 m2/s) initially at 15°C are to be cooled in the racks of a large freezer that is maintained at 12°C. The meat slabs are placed close to each other so that heat transfer from the 10-cm-thick edges is negligible. The entire slab is to be cooled below 5°C, but the temperature of the steak is not to drop below 1°C anywhere during refrigeration to avoid “frost bite.” The convection heat transfer coefficient and thus the rate of heat transfer from the steak can be controlled by varying the speed of a circulating fan inside. Determine the heat transfer coefficient h that will enable us to meet both temperature constraints while keeping the refrigeration time to a minimum
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4–105 Consider two 2-cm-thick large steel plates (k 43 W/m · °C and 1.17 10 5 m2/s) that were put on top of each other while wet and left outside during a cold winter night at 15°C. The next day, a worker needs one of the plates, but the plates are stuck together because the freezing of the water between the two plates has bonded them together. In an effort to melt the ice between the plates and separate them, the worker takes a large hairdryer and blows hot air at 50°C all over the exposed surface of the plate on the top. The convection heat transfer coefficient at the top surface is estimated to be 40 W/m2 · °C. Determine how long the worker must keep blowing hot air before the two plates separate
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4–106 Consider a curing kiln whose walls are made of 30-cm-thick concrete whose properties are k 0.9 W/m · °C and 0.23 10 5 m2/s. Initially, the kiln and its walls are in equilibrium with the surroundings at 2°C. Then all the doors are closed and the kiln is heated by steam so that the temperature of the inner surface of the walls is raised to 42°C and is maintained at that level for 3 h. The curing kiln is then opened and exposed to the atmospheric air after the stream flow is turned of
f. If the outer surfaces of the walls of the kiln were insulated, would it save any energy that day during the period the kiln was used for curing for 3 h only, or would it make no difference? Base your answer on calculations.
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4–107 The water main in the cities must be placed at sufficient depth below the earth’s surface to avoid freezing during extended periods of subfreezing temperatures. Determine the minimum depth at which the water main must be placed at a location where the soil is initially at 15°C and the earth’s surface temperature under the worst conditions is expected to remain at 10°C for a period of 75 days. Take the properties of soil at that location to be k 0.7 W/m · °C and 1.4 10 5 m2/s
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4–108 Ahot dog can be considered to be a 12-cm-long cylinder whose diameter is 2 cm and whose properties are 980 kg/m3, Cp 3.9 kJ/kg · °C, k 0.76 W/m · °C, and 4–110E In Betty Crocker’s Cookbook, it is stated that it takes 5 h to roast a 14-lb stuffed turkey initially at 40°F in an oven maintained at 325°F. It is recommended that a meat thermometer be used to monitor the cooking, and the turkey is considered done when the thermometer inserted deep into the thickest part of the breast or thigh without touching the bone registers 185°F. The turkey can be treated as a homogeneous spherical object with the properties 75 lbm/ft3, Cp 0.98 Btu/lbm ·°F, k 0.26 Btu/h · ft · °F, and 0.0035 ft2/h. Assuming the tip of the thermometer is at one-third radial distance from the center of the turkey, determine (a) the average heat transfer coefficient at the surface of the turkey, (b) the temperature of the skin of the turkey when it is done, and (c) the total amount of heat transferred to the turkey in the oven. Will the reading of the thermometer be more or less than 185°F 5 min after the turkey is taken out of the oven? 2 10 7 m2/s. Ahot dog initially at 5°C is dropped into boiling water at 100°C. The heat transfer coefficient at the surface of the hot dog is estimated to be 600 W/m2 · °C. If the hot dog is considered cooked when its center temperature reaches 80°C, determine how long it will take to cook it in the boiling water.
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4–109 Along roll of 2-m-wide and 0.5-cm-thick 1-Mn manganese steel plate coming off a furnace at 820°C is to be quenched in an oil bath (Cp 2.0 kJ/kg · °C) at 45°C. The metal sheet is moving at a steady velocity of 10 m/min, and the oil bath is 5 m long. Taking the convection heat transfer coefficient on both sides of the plate to be 860 W/m2 · °C, determine the temperature of the sheet metal when it leaves the oil bath. Also, determine therequired rate of heat removal from the oil to keep its temperature constant at 45°C. During a fire, the trunks of some dry oak trees (k 0.17 W/m · °C and 1.28 10 7 m2/s) that are initially at a uniform temperature of 30°C are exposed to hot gases at 520°C for a period of 5 h, with a heat transfer coefficient of 65 W/m2 ·°C on the surface. The ignition temperature of the trees is 410°C. Treating the trunks of the trees as long cylindrical rods of diameter 20 cm, determine if these dry trees will ignite as the fire sweeps through them.
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4–110E In Betty Crocker’s Cookbook, it is stated that it takes 5 h to roast a 14-lb stuffed turkey initially at 40°F in an oven maintained at 325°F. It is recommended that a meat thermometer be used to monitor the cooking, and the turkey is considered done when the thermometer inserted deep into the thickest part of the breast or thigh without touching the bone registers 185°F. The turkey can be treated as a homogeneous spherical object with the properties 75 lbm/ft3, Cp 0.98 Btu/lbm ·°F, k 0.26 Btu/h · ft · °F, and 0.0035 ft2/h. Assuming the tip of the thermometer is at one-third radial distance from the center of the turkey, determine (a) the average heat transfer coefficient at the surface of the turkey, (b) the temperature of the skin of the turkey when it is done, and (c) the total amount of heat transferred to the turkey in the oven. Will the reading of the thermometer be more or less than 185°F 5 min after the turkey is taken out of the oven?
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4–111 During a fire, the trunks of some dry oak trees (k 0.17 W/m · °C and 1.28 10 7 m2/s) that are initially at a uniform temperature of 30°C are exposed to hot gases at 520°C for a period of 5 h, with a heat transfer coefficient of 65 W/m2 ·°C on the surface. The ignition temperature of the trees is 410°C. Treating the trunks of the trees as long cylindrical rods of diameter 20 cm, determine if these dry trees will ignite as the fire sweeps through them.
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4–112 We often cut a watermelon in half and put it into the freezer to cool it quickly. But usually we forget to check on it and end up having a watermelon with a frozen layer on the top. To avoid this potential problem a person wants to set the timer such that it will go off when the temperature of the exposed surface of the watermelon drops to 3°C. Consider a 30-cm-diameter spherical watermelon that is cut into two equal parts and put into a freezer at 12°C. Initially, the entire watermelon is at a uniform temperature of 25°C, and the heat transfer coefficient on the surfaces is 30 W/m2 · °C. Assuming the watermelon to have the properties of water, determine how long it will take for the center of the exposed cut surfaces of the watermelon to drop to 3°C.
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4–113 The thermal conductivity of a solid whose density and specific heat are known can be determined from the relation k / Cp after evaluating the thermal diffusivity . Consider a 2-cm-diameter cylindrical rod made of a sample material whose density and specific heat are 3700 kg/m3 and 920 J/kg · °C, respectively. The sample is initially at a uniform temperature of 25°C. In order to measure the temperatures of the sample at its surface and its center, a thermocouple is inserted to the center of the sample along the centerline, and another thermocouple is welded into a small hole drilled on thesurface. The sample is dropped into boiling water at 100°C. After 3 min, the surface and the center temperatures are recorded to be 93°C and 75°C, respectively. Determine the thermal diffusivity and the thermal conductivity of the material
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4–114 In desert climates, rainfall is not a common occurrence since the rain droplets formed in the upper layer of the atmosphere often evaporate before they reach the ground. Consider a raindrop that is initially at a temperature of 5°C and has a diameter of 5 mm. Determine how long it will take for the diameter of the raindrop to reduce to 3 mm as it falls through ambient air at 18°C with a heat transfer coefficient of 400 W/m2 · °C. The water temperature can be assumed to remain constant and uniform at 5°C at all times
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4–115E Consider a plate of thickness 1 in., a long cylinder of diameter 1 in., and a sphere of diameter 1 in., all initially at 400°F and all made of bronze (k 15.0 Btu/h · ft · °F and 0.333 ft2/h). Now all three of these geometries are exposed to cool air at 75°F on all of their surfaces, with a heat transfer coefficient of 7 Btu/h · ft2 · °F. Determine the center temperature of each geometry after 5, 10, and 30 min. Explain why the center temperature of the sphere is always the lowest.
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4–116E Repeat Problem 4–115E for cast iron geometries (k 29 Btu/h · ft · °F and 0.61 ft2/h)
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4–117E Reconsider Problem 4–115E. Using EES (or other) software, plot the center temperature of each geometry as a function of the cooling time as the time varies fom 5 min to 60 min, and discuss the results.
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4–118 Engine valves (k 48 W/m · °C, Cp 440 J/kg · °C, and 7840 kg/m3) are heated to 800°C in the heat treatment section of a valve manufacturing facility. The valves are then quenched in a large oil bath at an average temperature of 45°C. The heat transfer coefficient in the oil bath is 650 W/m2 · °C. The valves have a cylindrical stem with a diameter of 8 mm and a length of 10 cm. The valve head and the stem may be assumed to be of equal surface area, and the volume of the valve head can be taken to be 80 percent of the volume of steam. Determine how long will it take for the valve temperature to drop to (a) 400°C, (b) 200°C, and (c) 46°C and (d) the maximum heat transfer from a single valve
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4–119 A watermelon initially at 35°C is to be cooled by dropping it into a lake at 15°C. After 4 h and 40 min of cooling, the center temperature of watermelon is measured to be 20°C. Treating the watermelon as a 20-cm-diameter sphere and using the properties k 0.618 W/m · °C, 0.15 10 6 m2/s, 995 kg/m3, and Cp 4.18 kJ/kg ·°C, determine the average heat transfer coefficient and the surface temperature of watermelon at the end of the cooling period
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4–120 10-cm-thick large food slabs tightly wrapped by thin paper are to be cooled in a refrigeration room maintained at 0°C. The heat transfer coefficient on the box surfaces is 25 W/m2 · °C and the boxes are to be kept in the refrigeration room for a period of 6 h. If the initial temperature of the boxes is 30°C determine the center temperature of the boxes if the boxes contain (a) margarine (k 0.233 W/m · °C and 0.11 10 6 m2/s), (b) white cake (k 0.082 W/m · °C and 0.10 10 6 m2/s), and (c) chocolate cake (k 0.106 W/m · °C and 0.12 10 6 m2/s)
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4–121 A30-cm-diameter, 3.5-m-high cylindrical column of a house made of concrete (k 0.79 W/m · °C, 5.94 10 7 m2/s, 1600 kg/m3, and Cp 0.84 kJ/kg · °C) cooled to 16°C during a cold night is heated again during the day by being exposed to ambient air at an average temperature of 28°C with an average heat transfer coefficient of 14 W/m2 · °C. Determine (a) how long it will take for the column surface temperature to rise to 27°C, (b) the amount of heat transfer until the center temperature reaches to 28°C, and (c) the amount of heat transfer until the surface temperature reaches to 27°C
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4–122 Long aluminum wires of diameter 3 mm ( 2702 kg/m3, Cp 0.896 kJ/kg · °C, k 236 W/m · °C, and 9.75 10 5 m2/s) are extruded at a temperature of 350°C and exposed to atmospheric air at 30°C with a heat transfer coefficient of 35 W/m2 · °C. (a) Determine how long it will take for the wire temperature to drop to 50°C. (b) If the wire is extruded at a velocity of 10 m/min, determine how far the wire travels after extrusion by the time its temperature drops to 50°C. What change in the cooling process would you propose to shorten this distance? (c) Assuming the aluminum wire leaves the extrusion room at 50°C, determine the rate of heat transfer from the wire to the extrusion room.
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4–123 Repeat Problem 4–122 for a copper wire ( 8950 kg/m3, Cp 0.383 kJ/kg · °C, k 386 W/m · °C, and 1.13 10 4 m2/s)
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4–124 Consider a brick house (k 0.72 W/m · °C and 0.45 10 6 m2/s) whose walls are 10 m long, 3 m high, and 0.3 m thick. The heater of the house broke down one night, and the entire house, including its walls, was observed to be 5°C throughout in the morning. The outdoors warmed up as the day progressed, but no change was felt in the house, which was tightly sealed. Assuming the outer surface temperature of the house to remain constant at 15°C, determine how long it would take for the temperature of the inner surfaces of the walls to rise to 5.1°C.
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4–125 A40-cm-thick brick wall (k 0.72 W/m · °C, and 1.6 10 7 m2/s) is heated to an average temperature of 18°C by the heating system and the solar radiation incident on it during the day. During the night, the outer surface of the wall is exposed to cold air at 2°C with an average heat transfer coefficient of 20 W/m2 · °C, determine the wall temperatures at distances 15, 30, and 40 cm from the outer surface for a period of 2 hours
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4–126 Consider the engine block of a car made of cast iron (k 52 W/m · °C and 1.7 10 5 m2/s). The engine can be considered to be a rectangular block whose sides are 80 cm, 40 cm, and 40 cm. The engine is at a temperature of 150°C when it is turned of
f. The engine is then exposed to atmospheric air at 17°C with a heat transfer coefficient of 6 W/m2 · °C. Determine (a) the center temperature of the top surface whose sides are 80 cm and 40 cm and (b) the corner temperature after 45 min of cooling
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4–127 A man is found dead in a room at 16°C. The surface temperature on his waist is measured to be 23°C and the heat transfer coefficient is estimated to be 9 W/m2 · °C. Modeling the body as 28-cm diameter, 1.80-m-long cylinder, estimate how long it has been since he died. Take the properties of the body to be k 0.62 W/m · °C and 0.15 10 6 m2/s, and assume the initial temperature of the body to be 36°C.
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