Chapter 14 : Heat and Heat Transfer Methods - all with Video Solutions
Problems & Exercises
Section 14.2: Temperature Change and Heat Capacity
Rubbing your hands together warms them by converting work into thermal energy. If a woman rubs her hands back and forth for a total of 20 rubs, at a distance of 7.50 cm per rub, and with an average frictional force of 40.0 N, what is the temperature increase? The mass of tissues warmed is only 0.100 kg, mostly in the palms and fingers.
(a) The number of kilocalories in food is determined by calorimetry techniques in which the food is burned and the amount of heat transfer is measured. How many kilocalories per gram are there in a 5.00-g peanut if the energy from burning it is transferred to 0.500 kg of water held in a 0.100-kg aluminum cup, causing a temperature increase? (b) Compare your answer to labeling information found on a package of peanuts and comment on whether the values are consistent.
Following vigorous exercise, the body temperature of an 80.0-kg person is . At what rate in watts must the person transfer thermal energy to reduce the the body temperature to in 30.0 min, assuming the body continues to produce energy at the rate of 150 W? (1 watt = 1 joule / second or 1W = 1 J/s).
Even when shut down after a period of normal use, a large commercial nuclear reactor transfers thermal energy at the rate of 150 MW by the radioactive decay of fission products. This heat transfer causes a rapid increase in temperature if the cooling system fails (1 watt = 1 joule/second or 1 W = 1 J/s and 1 MW = 1 megawatt) . (a) Calculate the rate of temperature increase in degrees Celsius per second () if the mass of the reactor core is kg and it has an average specific heat of 0.3349 kJ/kgo ⋅ C . (b) How long would it take to obtain a temperature increase of , which could cause some metals holding the radioactive materials to melt? (The initial rate of temperature increase would be greater than that calculated here because the heat transfer is concentrated in a smaller mass. Later, however, the temperature increase would slow down because the steel containment vessel would also begin to heat up.)
Section 14.3: Phase Change and Latent Heat
A bag containing ice is much more effective in absorbing energy than one containing the same amount of water. a. How much heat transfer is necessary to raise the temperature of 0.800 kg of water from to ? b. How much heat transfer is required to first melt 0.800 kg of ice and then raise its temperature? c. Explain how your answer supports the contention that the ice is more effective.
(a) How much heat transfer is required to raise the temperature of a 0.750-kg aluminum pot containing 2.50 kg of water from to the boiling point and then boil away 0.750 kg of water? (b) How long does this take if the rate of heat transfer is 500 W? 1 watt = 1 joule / second (1W = 1 J/s).
The formation of condensation on a glass of ice water causes the ice to melt faster than it would otherwise. If 8.00 g of condensation forms on a glass containing both water and 200 g of ice, how many grams of the ice will melt as a result? Assume no other heat transfer occurs.
(a) How much heat transfer is necessary to raise the temperature of a 0.200-kg piece of ice from to , including the energy needed for phase changes? (b) How much time is required for each stage, assuming a constant 20.0 kJ/s rate of heat transfer? (c) Make a graph of temperature versus time for this process.
In 1986, a gargantuan iceberg broke away from the Ross Ice Shelf in Antarctica. It was approximately a rectangle 160 km long, 40.0 km wide, and 250 m thick. (a) What is the mass of this iceberg, given that the density of ice is ? (b) How much heat transfer (in joules) is needed to melt it? (c) How many years would it take sunlight alone to melt ice this thick, if the ice absorbs an average of , 12.00 h per day?
How many grams of coffee must evaporate from 350 g of coffee in a 100-g glass cup to cool the coffee from to ? You may assume the coffee has the same thermal properties as water and that the average heat of vaporization is 2340 kJ/kg (560 cal/g). (You may neglect the change in mass of the coffee as it cools, which will give you an answer that is slightly larger than correct.)
(a) It is difficult to extinguish a fire on a crude oil tanker, because each liter of crude oil releases of energy when burned. To illustrate this difficulty, calculate the number of liters of water that must be expended to absorb the energy released by burning 1.00 L of crude oil, if the water has its temperature raised from to , it boils, and the resulting steam is raised to . (b) Discuss additional complications caused by the fact that crude oil has a smaller density than water.
To help prevent frost damage, 4.00 kg of water is sprayed onto a fruit tree. (a) How much heat transfer occurs as the water freezes? (b) How much would the temperature of the 200-kg tree decrease if this amount of heat transferred from the tree? Take the specific heat to be , and assume that no phase change occurs.
A 0.250-kg aluminum bowl holding 0.800 kg of soup at is placed in a freezer. What is the final temperature if 377 kJ of energy is transferred from the bowl and soup, assuming the soup’s thermal properties are the same as that of water? Explicitly show how you follow the steps in Problem-Solving Strategies for the Effects of Heat Transfer.
Indigenous people sometimes cook in watertight baskets by placing hot rocks into water to bring it to a boil. What mass of rock must be placed in 4.00 kg of water to bring its temperature to , if 0.0250 kg of water escapes as vapor from the initial sizzle? You may neglect the effects of the surroundings and take the average specific heat of the rocks to be that of granite.
In some countries, liquid nitrogen is used on dairy trucks instead of mechanical refrigerators. A 3.00-hour delivery trip requires 200 L of liquid nitrogen, which has a density of . (a) Calculate the heat transfer necessary to evaporate this amount of liquid nitrogen and raise its temperature to . (Use and assume it is constant over the temperature range.) This value is the amount of cooling the liquid nitrogen supplies. (b) What is this heat transfer rate in kilowatt-hours? (c) Compare the amount of cooling obtained from melting an identical mass of ice with that from evaporating the liquid nitrogen.
Section 14.5: Conduction
(a) Calculate the rate of heat conduction through house walls that are 13.0 cm thick and that have an average thermal conductivity twice that of glass wool. Assume there are no windows or doors. The surface area of the walls is and their inside surface is at , while their outside surface is at . (b) How many 1-kW room heaters would be needed to balance the heat transfer due to conduction?
The rate of heat conduction out of a window on a winter day is rapid enough to chill the air next to it. To see just how rapidly the windows transfer heat by conduction, calculate the rate of conduction in watts through a window that is 0.635 cm thick (1/4 in) if the temperatures of the inner and outer surfaces are and , respectively. This rapid rate will not be maintained—the inner surface will cool, and even result in frost formation.
Suppose you stand with one foot on ceramic flooring and one foot on a wool carpet, making contact over an area of with each foot. Both the ceramic and the carpet are 2.00 cm thick and are on their bottom sides. At what rate must heat transfer occur from each foot to keep the top of the ceramic and carpet at ?
- A firewalker runs across a bed of hot coals without sustaining burns. Calculate the heat transferred by conduction into the sole of one foot of a firewalker given that the bottom of the foot is a 3.00-mm-thick callus with a conductivity at the low end of the range for wood and its density is . The area of contact is , the temperature of the coals is , and the time in contact is 1.00 s.
- What temperature increase is produced in the of tissue affected?
- What effect do you think this will have on the tissue, keeping in mind that a callus is made of dead cells?
(a) What is the rate of heat conduction through the 3.00-cm-thick fur of a large animal having a surface area? Assume that the animal’s skin temperature is , that the air temperature is , and that fur has the same thermal conductivity as air. (b) What food intake will the animal need in one day to replace this heat transfer?
A walrus transfers energy by conduction through its blubber at the rate of 150 W when immersed in water. The walrus’s internal core temperature is , and it has a surface area of . What is the average thickness of its blubber, which has the conductivity of fatty tissues without blood?
Compare the rate of heat conduction through a 13.0-cm- thick wall that has an area of and a thermal conductivity twice that of glass wool with the rate of heat conduction through a window that is 0.750 cm thick and that has an area of , assuming the same temperature difference across each.
Some stove tops are smooth ceramic for easy cleaning. If the ceramic is 0.600 cm thick and heat conduction occurs through the same area and at the same rate as computed in Example 14.6, what is the temperature difference across it? Ceramic has the same thermal conductivity as glass and brick. Note: From the example, the circular stove element has a diameter of 14.0 cm, and the rate of heat transfer is 2256 J/s.
One easy way to reduce heating (and cooling) costs is to add extra insulation in the attic of a house. Suppose the house already had 15 cm of fiberglass insulation in the attic and in all the exterior surfaces. If you added an extra 8.0 cm of fiberglass to the attic, then by what percentage would the heating cost of the house drop? Take the single story house to be of dimensions 10 m by 15 m by 3.0 m. Ignore air infiltration and heat loss through windows and doors.
(a) Calculate the rate of heat conduction through a double-paned window that has a area and is made of two panes of 0.800-cm-thick glass separated by a 1.00-cm air gap. The inside surface temperature is , while that on the outside is . (Hint: There are identical temperature drops across the two glass panes. First find these and then the temperature drop across the air gap. This problem ignores the increased heat transfer in the air gap due to convection.) (b) Calculate the rate of heat conduction through a 1.60-cm- thick window of the same area and with the same temperatures. Compare your answer with that for part (a).
Many decisions are made on the basis of the payback period: the time it will take through savings to equal the capital cost of an investment. Acceptable payback times depend upon the business or philosophy one has. (For some industries, a payback period is as small as two years.) Suppose you wish to install the extra insulation in Exercise 14.41. If energy cost $1.00 per million joules and the insulation was $4.00 per square meter, then calculate the simple payback time. Take the average for the 120 day heating season to be .
For the human body, what is the rate of heat transfer by conduction through the body’s tissue with the following conditions: the tissue thickness is 3.00 cm, the change in temperature is , and the skin area is . How does this compare with the average heat transfer rate to the body resulting from an energy intake of about 2400 kcal per day? (No exercise is included.)
Section 14.6: Convection
The “steam” above a freshly made cup of instant coffee is really water vapor droplets condensing after evaporating from the hot coffee. What is the final temperature of 250 g of hot coffee initially at if 2.00 g evaporates from it? The coffee is in a Styrofoam cup, so other methods of heat transfer can be neglected.
On a hot dry day, evaporation from a lake has just enough heat transfer to balance the of incoming heat from the Sun. What mass of water evaporates in 1.00 h from each square meter? Explicitly show how you follow the steps in the Problem-Solving Strategies for the Methods of Heat Transfer.
One winter day, the climate control system of a large university classroom building malfunctions. As a result, of excess cold air is brought in each minute. At what rate in kilowatts must heat transfer occur to warm this air by (that is, to bring the air to room temperature)?
The Kilauea volcano in Hawaii is the world’s most active, disgorging about of lava per day. What is the rate of heat transfer out of Earth by convection if this lava has a density of and eventually cools to ? Assume that the specific heat of lava is the same as that of granite.
A glass coffee pot has a circular bottom with a 9.00-cm diameter in contact with a heating element that keeps the coffee warm with a continuous heat transfer rate of 50.0 W (a) What is the temperature of the bottom of the pot, if it is 3.00 mm thick and the inside temperature is ? (b) If the temperature of the coffee remains constant and all of the heat transfer is removed by evaporation, how many grams per minute evaporate? Take the heat of vaporization to be 2340 kJ/kg.
Section 14.7: Radiation
(a) Cherry-red embers in a fireplace are at and have an exposed area of 0.200 m2 and an emissivity of 0.980. The surrounding room has a temperature of . If 50% of the radiant energy enters the room, what is the net rate of radiant heat transfer in kilowatts? (b) Does your answer support the contention that most of the heat transfer into a room by a fireplace comes from infrared radiation?
(a) Calculate the rate of heat transfer by radiation from a car radiator at into a environment, if the radiator has an emissivity of 0.750 and a surface area. (b) Is this a significant fraction of the heat transfer by an automobile engine? To answer this, assume a horsepower of 200 hp and the efficiency of automobile engines as 25%.
Find the net rate of heat transfer by radiation from a skier standing in the shade, given the following. She is completely clothed in white (head to foot, including a ski mask), the clothes have an emissivity of 0.200 and a surface temperature of , the surroundings are at , and her surface area is .
Suppose you walk into a sauna that has an ambient temperature of . (a) Calculate the rate of heat transfer to you by radiation given your skin temperature is , the emissivity of skin is 0.98, and the surface area of your body is . (b) If all other forms of heat transfer are balanced (the net heat transfer is zero), at what rate will your body temperature increase if your mass is 75.0 kg?
Thermography is a technique for measuring radiant heat and detecting variations in surface temperatures that may be medically, environmentally, or militarily meaningful.(a) What is the percent increase in the rate of heat transfer by radiation from a given area at a temperature of compared with that at , such as on a person’s skin? (b) What is the percent increase in the rate of heat transfer by radiation from a given area at a temperature of compared with that at , such as for warm and cool automobile hoods?
The Sun radiates like a perfect black body with an emissivity of exactly 1. (a) Calculate the surface temperature of the Sun, given that it is a sphere with a radius that radiates into 3-K space. (b) How much power does the Sun radiate per square meter of its surface? (c) How much power in watts per square meter is that value at the distance of Earth, away? (This number is called the solar constant.)
A large body of lava from a volcano has stopped flowing and is slowly cooling. The interior of the lava is at , its surface is at , and the surroundings are at .
- Calculate the rate at which energy is transferred by radiation from of surface lava into the surroundings, assuming the emissivity is 1.00.
- Suppose heat conduction to the surface occurs at the same rate. What is the thickness of the lava between the surface and the interior, assuming that the lava’s conductivity is the same as that of brick?
Calculate the temperature the entire sky would have to be in order to transfer energy by radiation at — about the rate at which the Sun radiates when it is directly overhead on a clear day. This value is the effective temperature of the sky, a kind of average that takes account of the fact that the Sun occupies only a small part of the sky but is much hotter than the rest. Assume that the body receiving the energy has a temperature of .
- A shirtless rider under a circus tent feels the heat radiating from the sunlit portion of the tent. Calculate the temperature of the tent canvas based on the following information: The shirtless rider’s skin temperature is and has an emissivity of 0.970. The exposed area of skin is . He receives radiation at the rate of 20.0 W—half what you would calculate if the entire region behind him was hot. The rest of the surroundings are at .
- Discuss how this situation would change if the sunlit side of the tent was nearly pure white and if the rider was covered by a white tunic.
One day the relative humidity is 75.0% , and that evening the temperature drops to , well below the dew point. (a) How many grams of water condense from each cubic meter of air? (b) How much heat transfer occurs by this condensation? (c) What temperature increase could this cause in dry air?
Large meteors sometimes strike the Earth, converting most of their kinetic energy into thermal energy.
- What is the kinetic energy of a meteor moving at 25.0 km/s?
- If this meteor lands in a deep ocean and 80% of its kinetic energy goes into heating water, how many kilograms of water could it raise by
- Discuss how the energy of the meteor is more likely to be deposited in the ocean and the likely effects of that energy.
Frozen waste from airplane toilets has sometimes been accidentally ejected at high altitude. Ordinarily it breaks up and disperses over a large area, but sometimes it holds together and strikes the ground. Calculate the mass of ice that can be melted by the conversion of kinetic and gravitational potential energy when a 20.0 kg piece of frozen waste is released at 12.0 km altitude while moving at 250 m/s and strikes the ground at 100 m/s (since less than 20.0 kg melts, a significant mess results).
- A large electrical power facility produces 1600 MW of “waste heat,” which is dissipated to the environment in cooling towers by warming air flowing through the towers by . What is the necessary flow rate of air in ?
- Is your result consistent with the large cooling towers used by many large electrical power plants?
(a) Suppose you start a workout on a Stairmaster, producing power at the same rate as climbing 116 stairs per minute. Assuming your mass is 76.0 kg and your efficiency is 20.0% , how long will it take for your body temperature to rise if all other forms of heat transfer in and out of your body are balanced? (b) Is this consistent with your experience in getting warm while exercising?
In certain large geographic regions, the underlying rock is hot. Wells can be drilled and water circulated through the rock for heat transfer for the generation of electricity. (a) Calculate the heat transfer that can be extracted by cooling of granite by . (b) How long will this take if heat is transferred at a rate of 300 MW, assuming no heat transfers back into the 1.00 km of rock by its surroundings?
Heat transfers from your lungs and breathing passages by evaporating water. (a) Calculate the maximum number of grams of water that can be evaporated when you inhale 1.50 L of air with an original relative humidity of 40.0%. (Assume that body temperature is also .) (b) How many joules of energy are required to evaporate this amount? (c) What is the rate of heat transfer in watts from this method, if you breathe at a normal resting rate of 10.0 breaths per minute?
Hot air rises because it has expanded. It then displaces a greater volume of cold air, which increases the buoyant force on it.
- Calculate the ratio of the buoyant force to the weight of air surrounded by air.
- What energy is needed to cause of air to go from to ?
- What gravitational potential energy is gained by this volume of air if it rises 1.00 m? Will this cause a significant cooling of the air?
A slightly deranged Arctic inventor surrounded by ice thinks it would be much less mechanically complex to cool a car engine by melting ice on it than by having a water-cooled system with a radiator, water pump, antifreeze, and so on.
- If 80.0% of the energy in 1.00 gal of gasoline is converted into “waste heat” in a car engine, how many kilograms of ice could it melt?
- Is this a reasonable amount of ice to carry around to cool the engine for 1.00 gal of gasoline consumption?
- What premises or assumptions are unreasonable?
A meteorite 1.20 cm in diameter is so hot immediately after penetrating the atmosphere that it radiates 20.0 kW of power. (a) What is its temperature, if the surroundings are at and it has an emissivity of 0.800? (b) What is unreasonable about this result? (c) Which premise or assumption is responsible?
Test Prep for AP® Courses
Section 14.1: Heat
Problem 1 (AP)
An ice cube is placed in a cup of hot water. Which of the following statements correctly describes energy transfer at the molecular level?
- Kinetic energy is transferred only from the hot water molecules to the water molecules in the ice.
- Kinetic energy is transferred only from the water molecules in the ice to the hot water molecules.
- Kinetic energy is transferred mostly from the hot water molecules to the water molecules in the ice.
- Kinetic energy is transferred mostly from the water molecules in the ice to the hot water molecules.
Problem 2 (AP)
The molecular description of heat transfer from higher to lower temperatures applies to ‘spontaneous’ processes, that is, processes in which no energy is added to or removed or from the systems by work or heat. Refrigeration is an example of work being done to remove energy from air within a given space, and thus lower the temperature of the air. Assume a typical kitchen refrigerator, where the air inside the unit forms the system with a temperature of 25°C, and the walls are kept at a constant temperature of 10°C. In terms of molecules and average kinetic energy, describe how the air is made colder.
Section 14.5: Conduction
Problem 5 (AP)
For the experiment that you devised Example 14.1, which variables can be changed, and how should they be changed, so as to shorten the time in which a measurement is made?
- Use a smaller quantity of ice (smaller m).
- Use containers with greater thickness (larger d).
- Use containers with smaller surface areas (smaller A).
- Use a lower ambient temperature outside the container (smaller T2).
Problem 6 (AP)
You wish to design a saucepan that has the same rate of thermal conduction as a pan made of silver. You need to use a less costly material, and limit the size of the pan so that the surface area in contact with a range heating element is no more than 50% greater than that of the hypothetical silver pan. Explain what other factor(s) can be adjusted, and by how much, so that your design will be successful. Use Table 14.3 to obtain thermal conductivity values for different substances.
Section 14.6: Convection
Problem 7 (AP)
Under which two conditions would convection in a fluid be greatest?
- The gravitational acceleration is large, and the fluid density varies greatly for a given temperature change.
- The gravitational acceleration is small, and the fluid density varies greatly for a given temperature change.
- The gravitational acceleration is large, and the fluid density varies slightly for a given temperature change.
- The gravitational acceleration is small, and the fluid density varies slightly for a given temperature change.
Problem 8 (AP)
Sea breezes occur along coastlines, and consist of cool air moving toward the shore from the ocean. However, this only occurs during the day, and is a stronger effect when the air temperature on the land is greatest and the air temperature above the water is coldest. At night, the breezes are reversed, moving from the land toward the ocean. Taking into consideration the specific heat capacities of water and sand (which is about the same as that of concrete), explain how sea breezes form during the day and change direction at night.