Chapter 7: Thermal and Energy Systems
231
As part of a packing and distribution system, boxes are dropped onto a spring and
pushed onto a conveyor belt. The boxes are originally at a height h above the
uncompressed spring. Once dropped, the box of mass m compresses the spring a
distance ∆L. If all the potential energy of the box is converted into elastic energy in the
spring, find an expression for ∆L.
Approach:
Using Equation (7.1) for the gravitational potential energy and Equation (7.2) for the
Solution:
Balance the energy using Equations (7.1) and (7.2):
eg
UU =
k
Chapter 7: Thermal and Energy Systems
232
Discussion:
Chapter 7: Thermal and Energy Systems
Wind turbines convert the kinetic energy of wind to mechanical or electrical power.
The mass of air that hits a wind turbine each second is given by:
densityareavelocity
sec
mass ⋅⋅=
where the density of air is 1.23 kg/m
3
and the area is the area swept by the turbine rotor
blades. This mass flow rate can be used to calculate the amount of kinetic energy per
second that the air generates. One of the largest wind turbines in the world is in Norway
and is projected to generate 10 MW of power with winds of 35 mph. The diameter of
the rotor blades is 145 m. How much power is generated by the wind? Recall that
power is the amount of energy per unit time.
Approach:
Calculate mass flow rate of air using the velocity, turbine cross6sectional area, and density
Solution:
Calculate mass flow rate of air:
2
kg
m 145
h
m
mass .
s
Calculate kinetic energy of the air generated per second:
J
h
m
kg
1
2
s
Discussion:
The power generated by the wind is much larger than the projected power of the wind
turbine. This is because of mechanical and electrical inefficiencies in the wind turbine
system. Also, the Betz limit uses fundamental conservation of mass and energy principles
Chapter 7: Thermal and Energy Systems
234
Neglecting the presence of friction, air drag, and other inefficiencies, how much
gasoline is consumed when a 1300 kg automobile accelerates from rest to 80 km/h.
Express your answer in the units of mL. The density of gasoline is listed in Table 6.1.
Approach:
Find kinetic energy from Equation (7.3), and calculate the amount of heat supplied to the
Solution:
In consistent units, the automobile’s velocity is:
m
2222
hr
107782
m
0001
km
80
4
..v =
×
=
−
Discussion:
This seems like a small amount of gasoline, but the calculations are ideal in nature.
Chapter 7: Thermal and Energy Systems
235
In the summer of 2002, a group of miners in Quecreek, Pennsylvania, became
trapped 240 ft underground when a section of the coal seam they were working in
collapsed into an adjacent, but abandoned, mine that was not shown on their map. The
area became flooded with water, and the miners huddled in an air pocket at the end of a
passageway until they were safely rescued. As the first step in the rescue operation,
holes were drilled into the mine to provide the miners with warm fresh air and to pump
out the underground water. Neglecting friction in the pipes and the inefficiency of the
pumps themselves, what average power would be required to remove 20,000 gal of
water from the mine each minute? Express your answer in the units of horsepower. The
density of water is listed in Table 6.1.
Approach:
The power level required by the pumps balances the rate at which the gravitational
potential energy of the water is increased, following Equation (7.1). From Table 6.1, the
3
3
Solution:
In one minute, the mass of water removed by the pumps is:
slug
ft
3
3
Discussion:
To generate this amount of pumping power, it would require a huge specialized pumping
Chapter 7: Thermal and Energy Systems
236
Geothermal energy systems extract heat stored below the Earth’s crust. For every
300 ft below the surface, the temperature of groundwater increases by about 5 °F. Heat
can be brought to the surface by steam or hot water to warm homes and buildings, and
it also can be processed by a heat engine to produce electricity. Using the real
efficiency value of 8%, calculate the output of a geothermal power plant that processes
50 lb/s of groundwater at 180 °F and discharges it on the surface at 70 °F.
Approach:
The water transfers energy through its specific heat and temperature change following
F
lbm
Btu
Solution:
In one second, the mass of groundwater processed by the power plant is m = 50 lbm since a
one pound6mass object weighs one pound. The heat transferred is:
Btu
oo
s
Discussion:
This amount of geothermal power would easily provide enough power for multiple homes.
Chapter 7: Thermal and Energy Systems
237
A heat engine idealized as operating on the Carnot cycle is supplied with heat at the
boiling point of water (212 °F), and it rejects heat at the freezing point of water (32 °F).
If the engine produces 100 hp of mechanical work, calculate in units of Btu the amount
of heat that must be supplied to the engine each hour.
Approach:
Solution:
Efficiency:
R492
o
Discussion:
Since the performance of this engine is determined entirely by the temperatures of the two
Chapter 7: Thermal and Energy Systems
238
An inventor claims to have designed a heat engine that receives 120 Btu of heat and
generates 30 Btu of useful work when operating between a high6temperature energy
reservoir at 140 °F and a low6temperature energy reservoir at 20 °F. Is the claim valid?
Approach:
Solution:
Ideal Carnot efficiency:
(
)
( )
20%or 200
R67459140
R6745920
1
o
o
C
.
.
.=
+
+
−=
η
Real efficiency:
Btu 30Btu 120
−
Discussion:
Chapter 7: Thermal and Energy Systems
239
A person can blink an eye in approximately 7 ms. At what speed (in revolutions per
minute) would a four6stroke engine be operating if its power stroke took place literally
in the blink of an eye? Is that a reasonable speed for an automobile engine?
Approach:
Assume each stroke takes the same amount of time. Two piston strokes (one down, one up)
occur for each revolution of the engine’s crankshaft. Find the time one revolution requires.
Solution:
Two strokes occur during 14 ms.
Engine speed:
s
rev
471
s
.014
0
rev 1
s
m
14
rev 1 .==
min
rev
4286
min
s
06
s
rev
471 =
=.
4286 rpm
Discussion:
Chapter 7: Thermal and Energy Systems
A four6stroke gasoline engine produces an output of 35 kW. Using the density of
gasoline listed in Table 6.1, the heating value for gasoline in Table 7.3, and a typical
efficiency listed in Table 7.6, estimate the engine’s rate of fuel consumption. Express
your answer in the units of liters per hour.
Approach:
In Table 7.6, a typical internal combustion engine efficiency is 15–25%, so select an
average value of 20%. Find the fuel consumption using Q = mH (Equation (7.6)) with the
Solution:
In one second, the engine produces work W = (35 kW)(1 s) = 35 kJ with the required heat
input (Equation 7.12):
2
0
kJ 35 .
hr
Discussion:
Chapter 7: Thermal and Energy Systems
241
An automobile’s engine produces 30 hp while being driven at 50 mph on a level
highway. In those circumstances, the engine’s power is used to overcome air drag,
rolling resistance between the tires and the road, and friction in the drivetrain. Estimate
the vehicle‘s fuel economy rating in the units of miles per gallon. Use a typical engine
efficiency from Table 7.6 and the density of gasoline in Table 6.1.
Approach:
In Table 7.6, a typical internal combustion engine efficiency is 15–25%, so select an
average value of 20%. Find the fuel consumption using Q = mH (Equation (7.6)) with the
3
Solution:
In one second, the engine produces work
(
)
slbft
4
⋅
lbmBtu 10319
3
×
.
Convert to the consistent dimension of slug using the conversion factor in Table 3.5:
××=
−−
slug
23
Volume of fuel required:
4
−
s
mph, the vehicle’s fuel rating is:
gal
mi
3614
hrgal 483
hrmi 50 .
.=
242
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®
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gal
mi
414.
Discussion:
This fuel consumption value is a realistic value since it is estimated based on actual vehicle