Chapter 7: Thermal and Energy Systems
211
Chapter 7: Thermal and Energy Systems
212
Beginning from their definitions, determine the conversion factor between ft ⋅ lb and
kW ⋅ h in Table 7.1.
Approach:
Both ft ⋅ lb and kW ⋅ h are dimensions for work and energy. By definition of the derived
unit “watt,” 1 W = 1 (N⋅m)/s, and the “kilo)” prefix represents 1000 W. Convert force and
length using the factors in Table 3.6.
Solution:
Chapter 7: Thermal and Energy Systems
213
A remote6controlled toy car weighs 3 lb and moves at 15 ft/s. What is its kinetic
energy?
Approach:
The weight of the car is given, so the mass must first be calculated in consistent
dimensions. The kinetic energy is given by Equation (7.3). Conventional units for energy
in the USCS are ft ⋅ lb.
Solution:
The car’s mass is:
slb
103179
lb 3
2
2
2
⋅
×==
−
.m
Discussion:
It is critical to convert the mass to slugs before calculating the energy. If the car weighed
more, the kinetic energy would increase. The kinetic energy also increases proportional to
the square of the velocity.
Chapter 7: Thermal and Energy Systems
214
A lawn mower engine is started by pulling a cord wrapped around a hub of radius 6.0
cm. If a constant tension of 80 N is maintained in the cord and the hub makes three
revolutions before the motor starts, how much work is done?
Approach:
Calculate the distance the cord is pulled using the circumference of the hub. Calculate the
amount of work done using Equation (7.4).
Solution:
Calculate the circumference of the hub:
m
Discussion:
Chapter 7: Thermal and Energy Systems
215
In the movie Back to the Future, Doc Brown and the young Marty McFly need 1.21
GW of power for their time machine. (a) Convert that power requirement to
horsepower. (b) If a stock DeLorean sports car produces 145 hp, how many times more
power does the time machine need?
Approach:
Convert the power requirement using the SI prefix definition 1 GW = 10
9
W and the
conversion factor from Table 7.2 that 1 W = 1.341× 10
–3
hp.
Solution:
(a) Power in units of horsepower
hp
6399
××=×=
−
Chapter 7: Thermal and Energy Systems
216
A sprinter runs using a force of 200 N and a power output of 600 W. Calculate how
many minutes it takes for the runner to run 1 km.
Approach:
Substitute the equation for work (Equation (7.4)) into Equation (7.5) and then calculate the
amount of time required. Assume the force and power applied by the runner is constant.
Solution:
Chapter 7: Thermal and Energy Systems
217
A baseball catcher stops a 98 mph fastball over a distance of 0.1 m. What is the force
necessary to stop the 0.14 kg baseball?
Approach:
Convert the speed of the fastball to appropriate units using the factors in Table 3.6.
Calculate the kinetic energy of the fastball using Equation (7.3). The work done on the
fastball to stop it will be equivalent to its kinetic energy. Calculate the force required to
stop the fastball using Equation (7.4).
Solution:
Chapter 7: Thermal and Energy Systems
218
For the two automobiles of P6.31 in Chapter 6, how much power must the engines
produce just to overcome air drag at 60 mph?
Approach:
Solution:
Chapter 7: Thermal and Energy Systems
A light truck weighs 3100 lb and is rated at 30 miles per gallon for 60 mph highway
driving on level ground. Under those conditions, the engine must overcome air
resistance, rolling resistance, and other sources of friction. Give your answers in the
units shown. (a) The coefficient of drag is 0.6 at 60 mph, and the truck’s frontal area is
32 ft
2
. What is the drag force on the truck? (b) How much power must the engine
produce at 60 mph just to overcome air resistance? (c) In part (b), how much gasoline
would be consumed each hour (neglecting other frictional effects)?
Approach:
Find the drag force by using Equation (6.14), after converting velocity to the consistent
units of ft/s. The power to overcome air resistance is given by P = F
D
⋅v; convert to the
Solution:
(a) In consistent dimensions, the truck’s speed is:
ft
88
hr
1
ft
2805
mi
60 =
=v
(b) The required power is:
lbft
ft
4
⋅
(c) In one hour, the work performed to overcome air drag is (Equation 7.5):
lbft
4
⋅
Convert to consistent dimensions for heat and fuel calculation
−
Btu
37
Chapter 7: Thermal and Energy Systems
220
With ideal conversion of heat from the fuel’s combustion, the mass of fuel is (Equation
Using the density of gasoline from Table 6.1, the volume of fuel is:
Discussion:
Since the truck is rated at 30 mpg at 60 mph, the truck will use approximately 2 gal of
gasoline over the hour. Therefore almost 1.4 additional gallons are going into overcoming
Chapter 7: Thermal and Energy Systems
221
Suppose that the truck in P7.8 was going up a hill with a grade of 2%. How much
additional power must the engine produce to climb the hill, neglecting various
frictional effects?
Approach:
The tangent of the hill’s angle is 2% = 0.02. The angle is θ = tan
–1
(0.02) = 1.146°. In
consistent units, the truck‘s mass is:
ft
slb
sft 232
lb 1003
2
2
.
⋅
Solution:
Discussion:
Chapter 7: Thermal and Energy Systems
222
The heating value of agricultural residue biomass (e.g., crop residues, animal
manure and bedding, and organic material from food production) can range from 4300
to 7300 Btu/lbm. How much heat is released when 500 kg of biomass is burned?
Approach:
Convert the mass of the biomass into lbm using Table (3.6). Calculate the range for the
Solution:
Convert the mass of the biomass:
lbm
Discussion:
While this is not as much heat as would be generated from an equivalent amount of wood,
Chapter 7: Thermal and Energy Systems
223
During processing in a steel mill, a 750 lb steel casting at 800 °F is quenched by
plunging it into a 500 gal oil bath, which is initially at a temperature of 100 °F. After
the casting cools and the oil bath warms, what is the final temperature of the two? The
weight per unit volume of the oil is 7.5 lb/gal.
Approach:
Calculate the mass of the oil using the volume and weight per unit volume. Then, recognize
that the steel casting and the oil will be the same temperature if the casting is allowed to sit
F
lbm
o
steel
⋅
F
lbm
o
oil
⋅
The conventional dimension for mass in thermal and energy systems calculations is the
Solution:
Mass of steel casting, in the USCS conventional unit of pound–mass:
Discussion:
This assumes the heat flows perfectly from the steel casting into the oil. Realistically, heat
Chapter 7: Thermal and Energy Systems
224
The interior contents and materials of a small building weigh 25 tons, and together
they have an average specific heat of 0.25 Btu/(lbm ⋅ °F). Neglecting any inefficiency
in the furnace, what amount of natural gas must be burned to raise the building’s
temperature from freezing to 70 °F?
Approach:
The weight of the building and its contents is w = (25 tons)(2000 lb/ton) = 50,000 lb, with
“ton” defined in Table 3.5. Find the quantity of heat needed to raise the temperature using
Solution:
Discussion:
This is equivalent to almost 500 ft
3
of natural gas. The amount of natural gas needed will
Chapter 7: Thermal and Energy Systems
225
A 5.0 kg steel gear is heated to 150
o
C and then placed into a 0.5 gal container of
water at 10
o
C. What is the final temperature of the metal and water?
Approach:
Convert the water volume to consistent units using Table 3.6. Calculate the mass of the
Solution:
Convert the water volume:
3
.
final
Discussion:
This assumes the heat flows perfect from the gear into the water. Realistically, heat will
Chapter 7: Thermal and Energy Systems
226
A hollow square box is made from 1 ft
2
sheets of a prototype insulating material
that is 1 in. thick. Engineers are performing a test to measure the new material’s thermal
conductivity. A 100 W electrical heater is placed inside the box. Over time,
thermocouples attached to the box show that the interior and exterior surfaces of one
face have reached the constant temperatures of 150 °F and 90 °F. What is the thermal
conductivity? Express your result in both the SI and USCS.
Approach:
In the steady state, 100 W of power is added to the interior of the box by the heater, and the
Solution:
J) (1 s) = 100 J. In the USCS, the
Chapter 7: Thermal and Energy Systems
227
A welding rod with κ = 30 (Btu/hr)/(ft ⋅ °F) is 20 cm long and has a diameter of 4
mm. The two ends of the rod are held at 500 °C and 50 °C. (a) In the units of Btu and J,
how much heat flows along the rod each second? (b) What is the temperature of the
welding rod at its midpoint?
Approach:
Find the heat flow using Equation (7.9). Convert thermal conductivity and heat units using
Solution:
(a) The rod‘s cross–sectional area is:
(
)
25
2
m 102571
4
m 0040
−
×== .
.
A
π
In the SI, the thermal conductivity is:
(
)
( )
( )
Cm
W
9351
FfthrBtu
CmW
7311
Fft
hrBtu
30
oo
o
o
⋅
=
⋅
⋅
⋅
=..
κ
(
)
(
)
(
)
(
)
( )
C450
m
2
0
s 1m 102571CmW 9351
>
o
25o
.
..
Q
−
×⋅
=
= 1.469 J
In the USCS,
( )
×=
−
J
Btu
104789J 4691>
4
..Q
= 1.39× 10
–3
Btu
(b) Midpoint temperature
(
)
C50C500
2
1
T
oo
+=
= 275°C
Discussion:
An increased diameter in the rod will increase the heat flow, but an increase the length of
Chapter 7: Thermal and Energy Systems
A brick wall 3 m high, 7.5 m wide, and 200 mm thick has a thermal conductivity of
0.7 W/(m⋅°C). The temperature on the inner face is 25 °C, and the temperature on the
outer face is 0 °C. How much heat is lost per day through the wall?
Chapter 7: Thermal and Energy Systems
229
A 2500 lb automobile comes to a complete stop from 65 mph. If 60% of the braking
capacity is provided by the front disk brake rotors, determine their temperature rise.
Each of the two cast iron rotors weighs 15 lb and has a specific heat of 0.14 Btu/(lbm ⋅
°F).
Approach:
Convert vehicle mass into consistent units. The kinetic energy (Equation (7.3)) is
converted to heat and stored as a temperature rise following Equation (7.7). Use the
conversion factor from Table 7.1: 1 ft ⋅ lb = 1.285 × 10
3
Btu.
Solution:
Vehicle mass, in consistent units of slugs for the kinetic energy calculation:
lb 2500
Discussion:
This is a significant increase in temperature over a short amount of time. However, iron has
Chapter 7: Thermal and Energy Systems
230
A small hydroelectric power plant operates with 500 gal of water passing through
the system each second. The water falls through a vertical distance of 150 ft from a
reservoir to the turbines. Calculate the power output, and express it in the units of both
hp and kW. The density of water is listed in Table 6.1.
Approach:
Calculate the change in the reservoir‘s gravitational potential energy from Equation (7.1)
Solution:
Discussion:
Not only do hydroelectric power plants utilize water for power production, but a number of
other renewable forms of power production leverage the kinetic and potential energy of