Chapter 13

at illustrates the concepts pertinent to this problem. A refrigerator has a surface area of \(5.3 \mathrm{~m}^{2}\). It is lined with \(0.075\) -m-thick insulation whose thermal conductivity is \(0.030 \mathrm{~J} /\left(\mathrm{s} \cdot \mathrm{m} \cdot \mathrm{C}^{\circ}\right) .\) The interior temperature is kept at \(5{ }^{\circ} \mathrm{C}\), while the temperature at the outside surface is \(25^{\circ} \mathrm{C}\). How much heat per second is being removed from the unit?

Concept Simulation 13.1 at illustrates the concepts pertinent to this problem. A refrigerator has a surface area of \(5.3 \mathrm{~m}^{2}\). It is lined with \(0.075-\mathrm{m}\) -thick insulation whose thermal conductivity is \(0.030 \mathrm{~J} /\left(\mathrm{s} \cdot \mathrm{m} \cdot \mathrm{C}^{\circ}\right) .\) The interior temperature is kept at \(5^{\circ} \mathrm{C}\) while the temperature at the outside surface is \(25^{\circ} \mathrm{C}\). How much heat per second is being removed from the unit?

The amount of heat per second conducted from the blood capillaries beneath the skin to the surface is \(240 \mathrm{~J} / \mathrm{s}\). The energy is transferred a distance of \(2.0 \times 10^{-3} \mathrm{~m}\) through a body whose surface area is \(1.6 \mathrm{~m}^{2}\). Assuming that the thermal conductivity is that of body fat, determine the temperature difference between the capillaries and the surface of the skin.

Two pots are identical except that the flat bottom of one is aluminum, whereas that of the other is copper. Water in these pots is boiling away at \(100.0^{\circ} \mathrm{C}\) at the same rate. The temperature of the heating element on which the aluminum bottom is sitting is \(155.0^{\circ} \mathrm{C}\). Assume that heat enters the water only through the bottoms of the pots and find the temperature of the heating element on which the copper bottom rests.

Multiple-Concept Example 3 discusses an approach to problems such as this. The ends of a thin bar are maintained at different temperatures. The temperature of the cooler end is \(11^{\circ} \mathrm{C},\) while the temperature at a point \(0.13 \mathrm{~m}\) from the cooler end is \(23{ }^{\circ} \mathrm{C}\) and the temperature of the warmer end is \(48^{\circ} \mathrm{C}\). Assuming that heat flows only along the length of the bar (the sides are insulated), find the length of the bar.

Refer to Interactive Solution \(\underline{13.13}\) at for help in solving this problem. In an aluminum pot, \(0.15 \mathrm{~kg}\) of water at \(100^{\circ} \mathrm{C}\) boils away in four minutes. The bottom of the pot is \(3.1 \times 10^{-3} \mathrm{~m}\) thick and has a surface area of \(0.015 \mathrm{~m}^{2}\). To prevent the water from boiling too rapidly, a stainless steel plate has been placed between the pot and the heating element. The plate is \(1.4 \times 10^{-3} \mathrm{~m}\) thick, and its area matches that of the pot. Assuming that heat is conducted into the water only through the bottom of the pot, find the temperature at (a) the aluminum-steel interface and (b) the steel surface in contact with the heating element.

Two cylindrical rods are identical, except that one has a thermal conductivity \(k_{1}\) and the other has a thermal conductivity \(k_{2}\). As the drawing shows, they are placed between two walls that are maintained at different temperatures \(T_{\mathrm{W}}\) (warmer) and \(T_{\mathrm{C}}\) (cooler). When the rods are arranged as in part \(a\) of the drawing, a total heat \(Q^{\prime}\) flows from the warmer to the cooler wall, but when the rods are arranged as in part \(b,\) the total heat flow is \(Q\). Assuming that the conductivity \(k_{2}\) is twice as great as \(k_{1}\) and that heat flows only along the lengths of the rods, determine the ratio \(Q^{\prime} / Q\).

A person is standing outdoors in the shade where the temperature is \(28{ }^{\circ} \mathrm{C}\). (a) What is the radiant energy absorbed per second by his head when it is covered with hair? The surface area of the hair (assumed to be flat) is \(160 \mathrm{~cm}^{2}\) and its emissivity is \(0.85 .\) (b) What would be the radiant energy absorbed per second by the same person if he were bald and the emissivity of his head were \(0.65 ?\)

How many days does it take for a perfect blackbody cube \((0.0100 \mathrm{~m}\) on a side, \(30.0^{\circ} \mathrm{C}\) ) to radiate the same amount of energy that a one-hundred-watt light bulb uses in one hour?

In an old house, the heating system uses radiators, which are hollow metal devices through which hot water or steam circulates. In one room the radiator has a dark color (emissivity \(=0.75\) ). it has a temperature of \(62^{\circ} \mathrm{C}\). The new owner of the house paints the radiator a lighter color (emissivity \(=0.50\) ). Assuming that it emits the same radiant power as it did before being painted, what is the temperature (in degrees Celsius) of the newly painted radiator?

The filament of a light bulb has a temperature of \(3.0 \times 10^{3}{\underline{\phantom{xx}}}^{\circ} \mathrm{C}\) and radiates sixty watts of power. The emissivity of the filament is \(0.36 .\) Find the surface area of the filament.

The amount of radiant power produced by the sun is approximately \(3.9 \times 10^{26} \mathrm{~W}\). Assuming the sun to be a perfect blackbody sphere with a radius of \(6.96 \times 10^{8} \mathrm{~m},\) find its surface temperature (in kelvins).

Liquid helium is stored at its boiling-point temperature of \(4.2 \mathrm{~K}\) in a spherical container \((r=0.30 \mathrm{~m})\). The container is a perfect blackbody radiator. The container is surrounded by a spherical shield whose temperature is \(77 \mathrm{~K}\). A vacuum exists in the space between the container and the shield. The latent heat of vaporization for helium is \(2.1 \times 10^{4} \mathrm{~J} / \mathrm{kg} .\) What mass of liquid helium boils away through a venting valve in one hour?

A solid cylinder is radiating power. It has a length that is ten times its radius. It is cut into a number of smaller cylinders, each of which has the same length. Each small cylinder has the same temperature as the original cylinder. The total radiant power emitted by the pieces is twice that emitted by the original cylinder. How many smaller cylinders are there?

Concept Simulation \(13.1\) at illustrates the concepts pertinent to this problem. A person's body is covered with \(1.6 \mathrm{~m}^{2}\) of wool clothing. The thickness of the wool is \(2.0 \times 10^{-3} \mathrm{~m}\). The temperature at the outside surface of the wool is \(11{ }^{\circ} \mathrm{C}\), and the skin temperature is \(36^{\circ} \mathrm{C}\). How much heat per second does the person lose due to conduction?

The temperature in an electric oven is \(160{ }^{\circ} \mathrm{C}\). The temperature at the outer surface in the kitchen is \(50^{\circ} \mathrm{C}\). The oven (surface area \(=1.6 \mathrm{~m}^{2}\) ) is insulated with material that has a thickness of \(0.020 \mathrm{~m}\) and a thermal conductivity of \(0.045 \mathrm{~J} /\left(\mathrm{s} \cdot \mathrm{m} \cdot \mathrm{C}^{\circ}\right) .\) (a) How much energy is used to operate the oven for six hours? (b) At a price of \(\$ 0.10\) per kilowatt. hour for electrical energy, what is the cost of operating the oven?

Multiple-Concept Example 8 discusses the ideas on which this problem depends. Interactive LearningWare \(13.2\) at reviews the concepts that are involved in this problem. Suppose the skin temperature of a naked person is \(34{ }^{\circ} \mathrm{C}\) when the person is standing inside a room whose temperature is \(25^{\circ} \mathrm{C}\). The skin area of the individual is \(1.5\) \(\mathrm{m}^{2}\). (a) Assuming the emissivity is \(0.80\), find the net loss of radiant power from the body. (b) Determine the number of food Calories of energy (1 food Calorie \(=4186 \mathrm{~J}\) ) that are lost in one hour due to the net loss rate obtained in part (a). Metabolic conversion of food into energy replaces this loss.

A car parked in the sun absorbs energy at a rate of 560 watts per square meter of surface area. The car reaches a temperature at which it radiates energy at this same rate. Treating the car as a perfect radiator \((e=1)\), find the temperature.

A solid sphere has a temperature of \(773 \mathrm{~K}\). The sphere is melted down and recast into a cube that has the same emissivity and emits the same radiant power as the sphere. What is the cube's temperature?

Two cylindrical rods have the same mass. One is made of silver (density \(=10\) \(\left.500 \mathrm{~kg} / \mathrm{m}^{3}\right)\), and one is made of iron (density \(\left.=7860 \mathrm{~kg} / \mathrm{m}^{3}\right)\). Both rods conduct the same amount of heat per second when the same temperature difference is maintained across their ends. What is the ratio (silver-to-iron) of (a) the lengths and (b) the radii of these rods?

Concept Questions A pot of water is boiling on a stove under one atmosphere of pressure. Assume that heat enters the pot only through its bottom, which is copper and rests on a heating element. In a certain time, a mass \(m\) of water boils away. (a) What is the temperature of the boiling water and does it change during this time? (b) What determines the amount of heat needed to boil the water? (c) Is the temperature of the heating element in contact with the pot greater than, smaller than, or equal to \(100^{\circ} \mathrm{C}\) ? Explain.

A pot of water is boiling on a stove under one atmosphere of pressure. Assume that heat enters the pot only through its bottom, which is copper and rests on a heating element. In a certain time, a mass \(m\) of water boils away. (a) What is the temperature of the boiling water and does it change during this time? (b) What determines the amount of heat needed to boil the water? (c) Is the temperature of the heating element in contact with the pot greater than, smaller than, or equal to \(100{ }^{\circ} \mathrm{C} ?\) Explain. Problem In two minutes, the mass of water boiled away is \(m=0.45 \mathrm{~kg} .\) The radius of the pot bottom is \(R=6.5 \mathrm{~cm}\) and the thickness is \(L=2.0 \mathrm{~mm}\). What is the temperature \(T_{\mathrm{E}}\) of the heating element in contact with the pot? Verify that your answer is consistent with your answers to the Concept Questions.

Two objects are maintained at constant temperatures, one hot and one cold. Two identical bars can be attached end to end, as in part \(a\) of the drawing, or one on top of the other, as in part \(b\). When either of these arrangements is placed between the hot and the cold objects for the same amount of time, heat \(Q\) flows from left to right. (a) Is the area through which the heat flows greater for arrangement \(a\) or arrangement \(b ?\) (b) Is the thickness of the material through which the heat flows greater for arrangement \(a\) or arrangement \(b ?(\mathrm{c})\) Is \(Q_{a}\) less than, greater than, or equal to \(Q_{b} ?\)

Light bulb 1 operates with a higher filament temperature than light bulb 2 , but both filaments have the same emissivity. (a) How is the power \(P\) expressed in terms of the energy \(Q\) radiated by a bulb and the time \(t\) during which the energy is radiated? (b) Does a higher filament temperature generate more radiated power or less radiated power? (c) Does a smaller area for radiation promote more radiated power or less radiated power? (d) Suppose that both bulbs radiate the same power. Is the filament area of bulb 1 greater than, less than, or the same as the filament area of bulb \(2 ?\) Give your reasoning.

Via radiation, an object emits more power than it absorbs from the room in which it is located. (a) The object has a temperature \(T\) (in kelvins). According to the Stefan-Boltzmann law, the power radiated by the object is \(Q / t=e \sigma T^{4} A\), where \(A\) is the area from which the radiation is emitted. What is the expression for the power absorbed from the room, which has a temperature \(T_{0}\) (in kelvins)? (b) Is the temperature of the object greater than, less than, or equal to the temperature of the room? Explain, assuming the temperatures are constant.

Concept Questions Sirius \(\mathrm{B}\) is a white star that has a much greater surface temperature than our sun does. Assume that both Sirius B and our sun are spherical and have the same emissivity. (a) Other things being equal, would the greater surface temperature imply that the power radiated by Sirius \(\mathrm{B}\) is greater than, less than, or equal to the power radiated by our sun? (b) The fact is that Sirius B radiates much less power than our sun does. Considering this fact, is the surface area of Sirius \(\mathrm{B}\) greater than, less than, or equal to the surface area of our sun? (c) Is the radius of Sirius B greater than, less than, or equal to the radius of our sun? Explain your answers.

Concept Questions The block shown in the drawing has dimensions \(L_{0} \times 2 L_{0} \times 3 L_{0}\). In drawings \(\mathrm{A}, \mathrm{B}\), and \(\mathrm{C}\), heat is conducted through the block in three different directions. In each case, the same temperature difference exists between the opposite surfaces through which the heat passes, and the time during which the heat flows is the same. (a) The cross-sectional area of the opposite surfaces in \(\mathrm{C}\) is greater than that in A. Does this fact alone mean that the heat conducted in \(\mathrm{C}\) is greater than that conducted in A? Provide a reason for your answer. (b) The length of material through which heat is conducted is greater in A than in B. Does this fact alone imply that the heat conducted in A is smaller than that conducted in B? Why? (c) Rank the heat conducted in each of the three cases, largest first.

Concept Questions Part (a) of the drawing shows a rectangular bar whose dimensions are \(L_{0} \times 2 L_{0} \times 3 L_{0}\). The bar is at the same temperature as the room (not shown) in which it is located. (a) Is the net radiant power emitted by the bar greater than zero, equal to zero, or less than zero? Provide a reason for your answer. (b) The bar is then cut, lengthwise, into two equal pieces, as shown in part \((b)\) of the drawing. The temperature of the bars does not change. Which situation, if either, emits more power into the room, the single bar in part \((a)\) or the two bars in part \((b)\) of the drawing? Why? (c) Which situation, if either, absorbs more power from the room, the single bar in part (a) or the two bars in part ( \(b\) ) of the drawing? Justify your reasoning.