Physics

John D. Cutnell, Kenneth W. Johnson

Chapter 13

The Transfer of Heat - all with Video Answers

Educators

SB

Chapter Questions

Problem 1

The amount of heat per second conducted from the blood capillaries beneath the skin to the surface is 240 J/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.

SB

Sarah Brandsen

Numerade Educator

Problem 2

In an electrically heated home, the temperature of the ground in contact with a concrete basement wall is $12.8^{\circ} \mathrm{C}$ . The temperature at the inside surface of the wall is $20.0^{\circ} \mathrm{C}$ . The wall is 0.10 $\mathrm{m}$ thick and has an area of 9.0 $\mathrm{m}^{2}$ . Assume that one kilowatt hour of electrical energy costs $\$ 0.10 .$ How many hours are required for one dollar's worth of energy to be conducted through the wall?

Patrick Coffey

Numerade Educator

Problem 3

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 out- side 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?

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 4

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. Find the ratio $Q_{a} / Q_{b}$

Mayukh Banik

Numerade Educator

Problem 5

One end of an iron poker is placed in a fire where the temperature is $502^{\circ} \mathrm{C},$ and the other end is kept at a temperature of $26^{\circ} \mathrm{C}$ . The poker is 1.2 $\mathrm{m}$ long and has a radius of $5.0 \times 10^{-3} \mathrm{m} .$ Ignoring the heat lost along the length of the poker, find the amount of heat conducted from one end of the poker to the other in 5.0 $\mathrm{s} .$

Mayukh Banik

Numerade Educator

Problem 6

The block in the drawing has dimensions $L_{0} \times 2 L_{0} \times 3 L_{0}$
iwhere $L_{0}=0.30 \mathrm{m} .$ The block has a thermal conductivity of where $L_{0}=0.30 \mathrm{m} .$ The block has a thermal conductivity of 250 $\mathrm{J} /\left(\mathrm{s} \cdot \mathrm{c}^{\circ}\right) . \mathrm{In}$ drawings $\mathrm{A}, \mathrm{B},$ and $\mathrm{C},$ heat is conducted through the block in three different directions; in each case the temperature of
the warmer surface is $35^{\circ} \mathrm{C}$ and that of the cooler surface is $19^{\circ} \mathrm{C}$ . Determine the heat that flows in 5.0 $\mathrm{s}$ for each case.

Mayukh Banik

Numerade Educator

Problem 7

In the conduction equation $Q=(k A \Delta T) t / L,$ the combination of terms $k A / L$ is called the conductance. The human body has the ability to vary the conductance of the tissue beneath the skin by means of vasoconstriction and vasodilation, in which the flow of blood to the veins and capillaries is decreased and increased, respectively. The conductance can be adjusted over a range such that the tissue beneath the skin is equivalent to a thickness of 0.080 $\mathrm{mm}$ of Styrofoam or 3.5 $\mathrm{mm}$ of air. By what factor (high/low) can the body adjust the conductance?

Mayukh Banik

Numerade Educator

Problem 8

A closed box is filled with dry ice at a temperature of $-78.5^{\circ} \mathrm{C}$ while the outside temperature is $21.0^{\circ} \mathrm{C} .$ The box is cubical, measuring 0.350 $\mathrm{m}$ on a side, and the thickness of the walls is $3.00 \times 10^{-2} \mathrm{m} .$ In one day, $3.10 \times 10^{6} \mathrm{J}$ of heat is conducted through the six walls. Find the thermal conductivity of the material from which the box is made.

Mayukh Banik

Numerade Educator

Problem 9

One end of a brass bar is maintained at $306^{\circ} \mathrm{C},$ while the other
end is kept at a constant, but lower, temperature. The cross-sectional area of the bar is $2.6 \times 10^{-4} \mathrm{m}^{2} .$ Because of insulation, there is negligible heat loss through the sides of the bar. Heat flows through the bar, however, at the rate of 3.6 $\mathrm{J} / \mathrm{s}$ . What is the temperature of the bar at a point 0.15 $\mathrm{m}$ from the hot end?

Mayukh Banik

Numerade Educator

Problem 10

A wall in a house contains a single window. The window consists of a single pane of glass whose area is 0.16 $\mathrm{m}^{2}$ and whose thickness is 2.0 $\mathrm{mm} .$ Treat the wall as a slab of the insulating material Styrofoam whose area and thickness are 18 $\mathrm{m}^{2}$ and 0.10 $\mathrm{m}$ , respectively. Heat is lost via conduction through the wall and the window. The temperature difference between the inside and outside is the same for the wall and the window.
Of the total heat lost by the wall and the window, what is the percentage lost by the window?

Mayukh Banik

Numerade Educator

Problem 11

A composite rod is made from stainless steel and iron and has a length of 0.50 $\mathrm{m}$ The
cross section of this composite rod is shown in the drawing and consists of a square within a circle.
The square cross section of the steel is 1.0 $\mathrm{cm}$ on a side. The temperature at one end of the rod is $78^{\circ} \mathrm{C}$ while it is $18^{\circ} \mathrm{C}$ at the other end. Assuming that no heat exits through the cylindrical outer surface, find the total amount of heat conducted through the rod in two minutes.

Mayukh Banik

Numerade Educator

Problem 12

Review Conceptual Example 5 before attempting this problem. To illustrate the effect of ice on the
aluminum cooling plate, consider the drawing shown here and the data that it contains. Ignore any limitations due to significant figures.
(a) Calculate the heat per second per square meter that is conducted through the ice–aluminum combination.
(b) Calculate the heat per second per square meter that would be conducted through the aluminum if the ice were not present. Notice how much larger the answer is in (b) as compared to (a).

Mayukh Banik

Numerade Educator

Problem 13

A cubical piece of heat-shield tile from the space shuttle measures 0.10 $\mathrm{m}$ on a side and has a thermal conductivity of 0.065 $\mathrm{J} /\left(\mathrm{s} \cdot \mathrm{m} \cdot \mathrm{C}^{\circ}\right) .$ The outer surface of the tile is heated to a temperature of $1150^{\circ} \mathrm{C},$ while the inner surface is maintained at a temperature of $20.0^{\circ} \mathrm{C}$ . (a) How much heat flows from the outer to the inner surface of the tile in five minutes? $(\mathrm{b})$ If this amount of heat were transferred to two liters $(2.0 \mathrm{kg})$ of liquid water, by how many Celsius degrees would the temperature of the water rise?

Mayukh Banik

Numerade Educator

Problem 14

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.

Mayukh Banik

Numerade Educator

Problem 15

A pot of water is boiling 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 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?

Mayukh Banik

Numerade Educator

Problem 16

In a house the temperature at the surface of a window is $25^{\circ} \mathrm{C}$ The temperature outside at the window surface is $5.0^{\circ} \mathrm{C} .$ Heat is lost through the window via conduction, and the heat lost per second has a certain value. The temperature outside begins to fall, while the conditions inside the house remain the same. As a result, the heat lost per second
increases. What is the temperature at the outside window surface when the heat lost per second doubles?

Mayukh Banik

Numerade Educator

Problem 17

In an aluminum pot, 0.15 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 $(\mathrm{a})$ the aluminum-steel interface and $(\mathrm{b})$ the steel surface in contact with the heating element.

Mayukh Banik

Numerade Educator

Problem 18

The drawing shows a solid cylindrical rod made from a center cylinder of lead and an outer concentric jacket of copper. Except for its ends, the rod is insulated (not shown), so that the loss of heat from the curved surface is negligible. When a temperature difference is maintained between its ends, this rod
conducts one-half the amount of heat that it would conduct if it were solid copper. Determine the ratio of the radii $r_{1} / r_{2}$

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 19

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 $Q^{\prime} / Q .$

Mayukh Banik

Numerade Educator

Problem 20

Light bulb 1 operates with a filament temperature of 2700 $\mathrm{K}$ whereas light bulb 2 has a filament temperature of 2100 $\mathrm{K}$ . Both filaments have the same emissivity, and both bulbs radiate the same power. Find the ratio $A_{1} / A_{2}$ of the filament areas of the bulbs.

Mayukh Banik

Numerade Educator

Problem 21

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).

Mayukh Banik

Numerade Educator

Problem 22

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$?$

Mayukh Banik

Numerade Educator

Problem 23

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 $0.85 (\mathbf {b})$ 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$?$

Mayukh Banik

Numerade Educator

Problem 24

The Kelvin temperature of an object is $T_{1},$ and the object radiates a certain amount of energy per second. The Kelvin temperature of the object is then increased to $T_{2},$ and the object radiates twice the energy per second that it radiated at the lower temperature. What is the ratio $T_{2} / T_{1} ?$

Mayukh Banik

Numerade Educator

Problem 25

Multiple-Concept Example 8 reviews the approach that is used in problems such as this. A person eats a dessert that contains 260 Calories. This "Calorie" unit, with a capital $C,$ is the one
used by nutritionists; 1 Calorie $=4186 \mathrm{J}$ . See Section $12.7 .$ The skin temperature of this individual is $36^{\circ} \mathrm{C}$ and that of her environment is $21^{\circ} \mathrm{C}$ . The emissivity of her skin is 0.75 and its surface area is 1.3 $\mathrm{m}^{2}$ How much time would it take for her to emit a net radiant energy from her body that is equal to the energy contained in this dessert?

Mayukh Banik

Numerade Educator

Problem 26

Consult Multiple-Concept Example 8 to see the concepts that are pertinent here. A person’s body is producing energy internally due to metabolic processes. If the body loses more energy than metabolic
processes are generating, its temperature will drop. If the drop is severe, it can be life-threatening. Suppose that a person is unclothed and energy is being lost via radiation from a body surface area of 1.40 $\mathrm{m}^{2}$ which has a temperature of $34^{\circ} \mathrm{C}$ and an emissivity of $0.700 .$ Also suppose that metabolic processes are producing energy at a rate of 115 $\mathrm{J} / \mathrm{s}$ . What is the temperature of the coldest room in which this person could stand and not
experience a drop in body temperature?

Mayukh Banik

Numerade Educator

Problem 27

A baking dish is removed from a hot oven and placed on a cooling rack. As the dish cools down to $35^{\circ} \mathrm{C}$ from $175^{\circ} \mathrm{C},$ its net radiant power decreases to 12.0 $\mathrm{W}$ . What was the net radiant power of the baking dish when it was first removed from the oven? Assume that the temperature in the kitchen remains at $22^{\circ} \mathrm{C}$ as the dish cools down.

Mayukh Banik

Numerade Educator

Problem 28

Sirius $B$ is a white star that has a surface temperature (in kelvins) that is four times that of our sun. Sirius $\mathrm{B}$ radiates only 0.040 times the power radiated by the sun. Our sun has a radius of $6.96 \times 10^{8} \mathrm{m} .$ Assuming that Sirius $\mathrm{B}$ has the same emissivity as the sun, find the radius of Sirius $\mathrm{B}$ .

Mayukh Banik

Numerade Educator

Problem 29

Multiple-Concept Example 8 discusses the ideas on which this problem depends. 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 $(1 \text { food }$ 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.

Mayukh Banik

Numerade Educator

Problem 30

A solar collector is placed in direct sunlight where it absorbs energy at the rate of 880 $\mathrm{J} / \mathrm{s}$ for each square meter of its surface. The emissivity of the solar collector is $e=0.75 .$ What equilibrium temperature does the collector reach? Assume that the only energy loss is due to the
emission of radiation.

Mayukh Banik

Numerade Educator

Problem 31

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?

Mayukh Banik

Numerade Educator

Problem 32

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

Mayukh Banik

Numerade Educator

Problem 33

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 thepieces is twice that emitted by the original cylinder. How many smaller cylinders are there?

Mayukh Banik

Numerade Educator

Problem 34

A small sphere (emissivity $=0.90,$ radius $=r_{1}$ ) is located at the center of a spherical asbestos shell (thickness $=1.0 \mathrm{cm},$ outer radius $=r_{2} )$ The thickness of the shell is small compared to the inner and outer radii of the shell. The temperature of the small sphere is $800.0 ~^{\circ} \mathrm{C},$ while the temperature of the inner surface of the shell is $600.0^{\circ} \mathrm{C},$ both temperatures remaining constant. Assuming that $r_{2} / r_{1}=10.0$ and
ignoring any air inside the shell, find the temperature of the outer surface of the shell.

Supratim Pal

Numerade Educator

Problem 35

Due to a temperature difference $\Delta T$ , heat is conducted through an aluminum plate that is 0.035 $\mathrm{m}$ thick. The plate is then replaced by a stainless steel plate that has the same temperature difference and cross sectional area. How thick should the steel plate be so that the same
amount of heat per second is conducted through it?

Mayukh Banik

Numerade Educator

Problem 36

A copper pipe with an outer radius of 0.013 $\mathrm{m}$ runs from an outdoor wall faucet into the interior of a house. The temperature of the faucet is $4.0^{\circ} \mathrm{C}$ , and the temperature of the pipe, at 3.0 $\mathrm{m}$ from the faucet,
is $25^{\circ} \mathrm{C}$ In fifteen minutes, the pipe conducts a total of 270 $\mathrm{J}$ of heat to
the outdoor faucet from the house interior. Find the inner radius of the
pipe. Ignore any water inside the pipe.

Mayukh Banik

Numerade Educator

Problem 37

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?

Mayukh Banik

Numerade Educator

Problem 38

An object is inside a room that has a constant temperature of 293 $\mathrm{K}$ . Via radiation, the object emits three times as much power as it absorbs from the room. What is the temperature (in kelvins) of the object? Assume that the temperature of the object remains constant.

Mayukh Banik

Numerade Educator

Problem 39

The concrete wall of a building is 0.10 $\mathrm{m}$ thick. The temperature inside
the building is $20.0^{\circ} \mathrm{C},$ while the temperature outside is $0.0^{\circ} \mathrm{C}$ . Heat is conducted through the wall. When the building is unheated, the inside temperature falls to $0.0^{\circ} \mathrm{C},$ and heat conduction ceases. However, the wall does emit radiant energy when its temperature is $0.0^{\circ} \mathrm{C}$ . The radiant energy emitted per second per square meter at $0.0^{\circ} \mathrm{C}$ is the same as the heat lost per second per square meter due to conduction when the temperature inside the building is $20.0^{\circ} \mathrm{C}$ . What is the emissivity of the wall?

Mayukh Banik

Numerade Educator

Problem 40

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 constant temperature as the room (not shown) in which it is located. The bar is then cut, length wise, into two identical picces, as shown in part (b) of the drawing. The temperature of each piece is the same as that of the original bar. (a) What is the ratio of the power absorbed by the two bars in part $(b)$ of the drawing to the single bar in part (a)? (b) Suppose that the temperature of the single bar in part (a) is 450.0 $\mathrm{K}$ . What would the temperature (in kelvins) of the room and the two bars in part $(b)$ have to be so that the two bars absorb the same power as the single bar in part ( $(a) ?$

Mayukh Banik

Numerade Educator

Problem 41

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.

Mayukh Banik

Numerade Educator

Problem 42

A copper rod has a length of 1.5 $\mathrm{m}$ and a cross-sectional area of $4.0 \times 10^{-4} \mathrm{m}^{2}$ One end of the rod is in contact with boiling water and the other with a mixture of ice and water. What is the mass of ice per second that melts? Assume that no heat is lost through the side surface of the rod.

Mayukh Banik

Numerade Educator

Problem 43

Two cylindrical rods have the same mass. One is made of silver (density $=10500 \mathrm{kg} / \mathrm{m}^{3} )$ , and one is made of iron (density $=7860 \mathrm{kg} / \mathrm{m}^{3} )$
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 $(\mathrm{a})$ the lengths and $(\mathrm{b})$ the radii of these rods?

Mayukh Banik

Numerade Educator

Problem 44

One end of a $0.25-\mathrm{m}$ copper rod with a cross-sectional area of $1.2 \times 10^{-4} \mathrm{m}^{2}$ is driven into center of a sphere of ice at $0^{\circ} \mathrm{C}$
(radius $=0.15 \mathrm{m} ) .$ The portion of the rod that is embedded in the ice is also at $0^{\circ} \mathrm{C}$ . The rod is horizontal and its other end is fastened to a wall in a room. The rod and the room are kept at a constant temperature of $24^{\circ} \mathrm{C}$ . The emissivity of the ice is 0.90 . What is the ratio of the heat per second gained by the sphere through conduction to the net heat per second gained by the ice due to radiation? Neglect any heat gained through the sides of the rod.

Manish Jain

Numerade Educator