16. Unreasonable Results
A mountain stream is 10.0 m wide and averages 2.00 m in depth. During
the spring runoff, the flow in the stream reaches 100,000 m3 /s . (a)
What is the average velocity of the stream under these conditions? (b)
What is unreasonable about this velocity? (c) What is unreasonable or
inconsistent about the premises?
17. Verify that pressure has units of energy per unit volume.
18. Suppose you have a wind speed gauge like the pitot tube shown in
Figure 12.7(b). By what factor must wind speed increase to double the
Figure 12.29 The Huka Falls in Taupo, New Zealand, demonstrate flow rate. (credit:
RaviGogna, Flickr)
value of h in the manometer? Is this independent of the moving fluid and
the fluid in the manometer?
6. A major artery with a cross-sectional area of 1.00 cm2 branches into 19. If the pressure reading of your pitot tube is 15.0 mm Hg at a speed of
18 smaller arteries, each with an average cross-sectional area of
200 km/h, what will it be at 700 km/h at the same altitude?
0.400 cm2 . By what factor is the average velocity of the blood reduced 20. Calculate the maximum height to which water could be squirted with
when it passes into these branches?
the hose in Example 12.2 example if it: (a) Emerges from the nozzle. (b)
7. (a) As blood passes through the capillary bed in an organ, the
Emerges with the nozzle removed, assuming the same flow rate.
capillaries join to form venules (small veins). If the blood speed increases
21. Every few years, winds in Boulder, Colorado, attain sustained speeds
by a factor of 4.00 and the total cross-sectional area of the venules is
of 45.0 m/s (about 100 mi/h) when the jet stream descends during early
10.0 cm2 , what is the total cross-sectional area of the capillaries
spring. Approximately what is the force due to the Bernoulli effect on a
feeding these venules? (b) How many capillaries are involved if their
roof having an area of 220 m2 ? Typical air density in Boulder is
average diameter is 10.0 µ m ?
1.14 kg/m3 , and the corresponding atmospheric pressure is
8. The human circulation system has approximately 1×109 capillary
8.89×104 N/m2 . (Bernoulli’s principle as stated in the text assumes
vessels. Each vessel has a diameter of about 8 µ m . Assuming cardiac
laminar flow. Using the principle here produces only an approximate
result, because there is significant turbulence.)
output is 5 L/min, determine the average velocity of blood flow through
each capillary vessel.
22. (a) Calculate the approximate force on a square meter of sail, given
the horizontal velocity of the wind is 6.00 m/s parallel to its front surface
9. (a) Estimate the time it would take to fill a private swimming pool with a
and 3.50 m/s along its back surface. Take the density of air to be
capacity of 80,000 L using a garden hose delivering 60 L/min. (b) How
long would it take to fill if you could divert a moderate size river, flowing at
1.29 kg/m3 . (The calculation, based on Bernoulli’s principle, is
5000 m3 /s , into it?
approximate due to the effects of turbulence.) (b) Discuss whether this
force is great enough to be effective for propelling a sailboat.
10. The flow rate of blood through a 2.00×10–6-m -radius capillary is
23. (a) What is the pressure drop due to the Bernoulli effect as water
3.80×109 cm3 /s . (a) What is the speed of the blood flow? (This small goes into a 3.00-cm-diameter nozzle from a 9.00-cm-diameter fire hose
while carrying a flow of 40.0 L/s? (b) To what maximum height above the
speed allows time for diffusion of materials to and from the blood.) (b)
nozzle can this water rise? (The actual height will be significantly smaller
Assuming all the blood in the body passes through capillaries, how many
due to air resistance.)
of them must there be to carry a total flow of 90.0 cm3 /s ? (The large
number obtained is an overestimate, but it is still reasonable.)
CHAPTER 12 | FLUID DYNAMICS AND ITS BIOLOGICAL AND MEDICAL APPLICATIONS 425
24. (a) Using Bernoulli’s equation, show that the measured fluid speed v
33. A small artery has a length of 1.1×10−3 m and a radius of
for a pitot tube, like the one in Figure 12.7(b), is given by
1 / 2
2.5×10−5 m . If the pressure drop across the artery is 1.3 kPa, what is
v = ⎛2 ρ′ gh⎞
⎝ ρ ⎠ ,
the flow rate through the artery? (Assume that the temperature is 37º C
.)
where h is the height of the manometer fluid, ρ′ is the density of the
34. Fluid originally flows through a tube at a rate of 100 cm3 /s . To
manometer fluid, ρ is the density of the moving fluid, and g is the
illustrate the sensitivity of flow rate to various factors, calculate the new
acceleration due to gravity. (Note that v is indeed proportional to the
flow rate for the following changes with all other factors remaining the
square root of h , as stated in the text.) (b) Calculate v for moving air if
same as in the original conditions. (a) Pressure difference increases by a
factor of 1.50. (b) A new fluid with 3.00 times greater viscosity is
a mercury manometer’s h is 0.200 m.
substituted. (c) The tube is replaced by one having 4.00 times the length.
(d) Another tube is used with a radius 0.100 times the original. (e) Yet
12.3 The Most General Applications of Bernoulli’s
another tube is substituted with a radius 0.100 times the original and half
the length, and the pressure difference is increased by a factor of 1.50.
35. The arterioles (small arteries) leading to an organ, constrict in order
25. Hoover Dam on the Colorado River is the highest dam in the United
to decrease flow to the organ. To shut down an organ, blood flow is
States at 221 m, with an output of 1300 MW. The dam generates
reduced naturally to 1.00% of its original value. By what factor did the
electricity with water taken from a depth of 150 m and an average flow
radii of the arterioles constrict? Penguins do this when they stand on ice
rate of 650 m3 /s . (a) Calculate the power in this flow. (b) What is the
to reduce the blood flow to their feet.
ratio of this power to the facility’s average of 680 MW?
36. Angioplasty is a technique in which arteries partially blocked with
26. A frequently quoted rule of thumb in aircraft design is that wings
plaque are dilated to increase blood flow. By what factor must the radius
should produce about 1000 N of lift per square meter of wing. (The fact
of an artery be increased in order to increase blood flow by a factor of
that a wing has a top and bottom surface does not double its area.) (a) At
10?
takeoff, an aircraft travels at 60.0 m/s, so that the air speed relative to the
37. (a) Suppose a blood vessel’s radius is decreased to 90.0% of its
bottom of the wing is 60.0 m/s. Given the sea level density of air to be
original value by plaque deposits and the body compensates by
1.29 kg/m3 , how fast must it move over the upper surface to create the increasing the pressure difference along the vessel to keep the flow rate
constant. By what factor must the pressure difference increase? (b) If
ideal lift? (b) How fast must air move over the upper surface at a cruising
turbulence is created by the obstruction, what additional effect would it
speed of 245 m/s and at an altitude where air density is one-fourth that at
have on the flow rate?
sea level? (Note that this is not all of the aircraft’s lift—some comes from
the body of the plane, some from engine thrust, and so on. Furthermore,
38. A spherical particle falling at a terminal speed in a liquid must have
Bernoulli’s principle gives an approximate answer because flow over the
the gravitational force balanced by the drag force and the buoyant force.
wing creates turbulence.)
The buoyant force is equal to the weight of the displaced fluid, while the
drag force is assumed to be given by Stokes Law, F
27. The left ventricle of a resting adult’s heart pumps blood at a flow rate
s = 6 πrηv . Show
of 83.0 cm3 /s , increasing its pressure by 110 mm Hg, its speed from
that the terminal speed is given by v = 2 R 2 g
zero to 30.0 cm/s, and its height by 5.00 cm. (All numbers are averaged
9 η ( ρ s − ρ 1),
over the entire heartbeat.) Calculate the total power output of the left
ventricle. Note that most of the power is used to increase blood pressure.
where R is the radius of the sphere, ρ s is its density, and ρ 1 is the
28. A sump pump (used to drain water from the basement of houses built
density of the fluid and η the coefficient of viscosity.
below the water table) is draining a flooded basement at the rate of 0.750
39. Using the equation of the previous problem, find the viscosity of
L/s, with an output pressure of 3.00×105 N/m2 . (a) The water enters a motor oil in which a steel ball of radius 0.8 mm falls with a terminal speed
hose with a 3.00-cm inside diameter and rises 2.50 m above the pump.
of 4.32 cm/s. The densities of the ball and the oil are 7.86 and 0.88 g/mL,
What is its pressure at this point? (b) The hose goes over the foundation
respectively.
wall, losing 0.500 m in height, and widens to 4.00 cm in diameter. What is 40. A skydiver will reach a terminal velocity when the air drag equals their
the pressure now? You may neglect frictional losses in both parts of the
weight. For a skydiver with high speed and a large body, turbulence is a
problem.
factor. The drag force then is approximately proportional to the square of
12.4 Viscosity and Laminar Flow; Poiseuille’s Law
the velocity. Taking the drag force to be F D = 12 ρAv 2 and setting this
29. (a) Calculate the retarding force due to the viscosity of the air layer
equal to the person’s weight, find the terminal speed for a person falling
between a cart and a level air track given the following information—air
“spread eagle.” Find both a formula and a number for v t , with
temperature is 20º C , the cart is moving at 0.400 m/s, its surface area is assumptions as to size.
2.50×10−2 m2 , and the thickness of the air layer is 6.00×10−5 m .
41. A layer of oil 1.50 mm thick is placed between two microscope slides.
(b) What is the ratio of this force to the weight of the 0.300-kg cart?
Researchers find that a force of 5.50×10−4 N is required to glide one
30. What force is needed to pull one microscope slide over another at a
over the other at a speed of 1.00 cm/s when their contact area is
speed of 1.00 cm/s, if there is a 0.500-mm-thick layer of 20º C water
6.00 cm2 . What is the oil’s viscosity? What type of oil might it be?
between them and the contact area is 8.00 cm2 ?
42. (a) Verify that a 19.0% decrease in laminar flow through a tube is
31. A glucose solution being administered with an IV has a flow rate of
caused by a 5.00% decrease in radius, assuming that all other factors
4.00 cm3 /min . What will the new flow rate be if the glucose is replaced remain constant, as stated in the text. (b) What increase in flow is
obtained from a 5.00% increase in radius, again assuming all other
by whole blood having the same density but a viscosity 2.50 times that of
factors remain constant?
the glucose? All other factors remain constant.
43. Example 12.8 dealt with the flow of saline solution in an IV system.
32. The pressure drop along a length of artery is 100 Pa, the radius is 10
mm, and the flow is laminar. The average speed of the blood is 15 mm/s.
(a) Verify that a pressure of 1.62×104 N/m2 is created at a depth of
(a) What is the net force on the blood in this section of artery? (b) What is
1.61 m in a saline solution, assuming its density to be that of sea water.
the power expended maintaining the flow?
(b) Calculate the new flow rate if the height of the saline solution is
426 CHAPTER 12 | FLUID DYNAMICS AND ITS BIOLOGICAL AND MEDICAL APPLICATIONS
decreased to 1.50 m. (c) At what height would the direction of flow be
54. A fire hose has an inside diameter of 6.40 cm. Suppose such a hose
reversed? (This reversal can be a problem when patients stand up.)
carries a flow of 40.0 L/s starting at a gauge pressure of
44. When physicians diagnose arterial blockages, they quote the
1.62×106 N/m2 . The hose goes 10.0 m up a ladder to a nozzle having
reduction in flow rate. If the flow rate in an artery has been reduced to
an inside diameter of 3.00 cm. Calculate the Reynolds numbers for flow
10.0% of its normal value by a blood clot and the average pressure
in the fire hose and nozzle to show that the flow in each must be
difference has increased by 20.0%, by what factor has the clot reduced
turbulent.
the radius of the artery?
55. Concrete is pumped from a cement mixer to the place it is being laid,
45. During a marathon race, a runner’s blood flow increases to 10.0 times instead of being carried in wheelbarrows. The flow rate is 200 L/min
her resting rate. Her blood’s viscosity has dropped to 95.0% of its normal
through a 50.0-m-long, 8.00-cm-diameter hose, and the pressure at the
value, and the blood pressure difference across the circulatory system
has increased by 50.0%. By what factor has the average radii of her
pump is 8.00×106 N/m2 . Verify that the flow of concrete is laminar
blood vessels increased?
taking concrete’s viscosity to be 48.0 ⎛⎝N/m2⎞⎠ · s , and given its density is
46. Water supplied to a house by a water main has a pressure of
3.00×105 N/m2
2300 kg/m3 .
early on a summer day when neighborhood use is
low. This pressure produces a flow of 20.0 L/min through a garden hose.
56. At what flow rate might turbulence begin to develop in a water main
Later in the day, pressure at the exit of the water main and entrance to
with a 0.200-m diameter? Assume a 20º C temperature.
the house drops, and a flow of only 8.00 L/min is obtained through the
same hose. (a) What pressure is now being supplied to the house,
57. What is the greatest average speed of blood flow at 37º C in an
assuming resistance is constant? (b) By what factor did the flow rate in
artery of radius 2.00 mm if the flow is to remain laminar? What is the
the water main increase in order to cause this decrease in delivered
corresponding flow rate? Take the density of blood to be 1025 kg / m3 .
pressure? The pressure at the entrance of the water main is
5.00×105 N/m2 , and the original flow rate was 200 L/min. (c) How
58. In Take-Home Experiment: Inhalation, we measured the average
many more users are there, assuming each would consume 20.0 L/min in flow rate Q of air traveling through the trachea during each inhalation.
the morning?
Now calculate the average air speed in meters per second through your
47. An oil gusher shoots crude oil 25.0 m into the air through a pipe with
trachea during each inhalation. The radius of the trachea in adult humans
a 0.100-m diameter. Neglecting air resistance but not the resistance of
is approximately 10−2 m . From the data above, calculate the Reynolds
the pipe, and assuming laminar flow, calculate the pressure at the
number for the air flow in the trachea during inhalation. Do you expect the
entrance of the 50.0-m-long vertical pipe. Take the density of the oil to be
air flow to be laminar or turbulent?
900 kg/m3 and its viscosity to be 1.00 (N/m2 ) ⋅ s (or 1.00 Pa ⋅ s ). 59. Gasoline is piped underground from refineries to major users. The
Note that you must take into account the pressure due to the 50.0-m
flow rate is 3.00×10–2 m3 /s (about 500 gal/min), the viscosity of
column of oil in the pipe.
48. Concrete is pumped from a cement mixer to the place it is being laid,
gasoline is 1.00×10–3 (N/m2 ) ⋅ s , and its density is 680 kg/m3 . (a)
instead of being carried in wheelbarrows. The flow rate is 200 L/min
What minimum diameter must the pipe have if the Reynolds number is to
through a 50.0-m-long, 8.00-cm-diameter hose, and the pressure at the
be less than 2000? (b) What pressure difference must be maintained
pump is 8.00×106 N/m2 . (a) Calculate the resistance of the hose. (b)
along each kilometer of the pipe to maintain this flow rate?
What is the viscosity of the concrete, assuming the flow is laminar? (c)
60. Assuming that blood is an ideal fluid, calculate the critical flow rate at
How much power is being supplied, assuming the point of use is at the
which turbulence is a certainty in the aorta. Take the diameter of the aorta
same level as the pump? You may neglect the power supplied to
to be 2.50 cm. (Turbulence will actually occur at lower average flow rates,
increase the concrete’s velocity.
because blood is not an ideal fluid. Furthermore, since blood flow pulses,
turbulence may occur during only the high-velocity part of each
49. Construct Your Own Problem
heartbeat.)
Consider a coronary artery constricted by arteriosclerosis. Construct a
61. Unreasonable Results
problem in which you calculate the amount by which the diameter of the
artery is decreased, based on an assessment of the decrease in flow
A fairly large garden hose has an internal radius of 0.600 cm and a length
rate.
of 23.0 m. The nozzleless horizontal hose is attached to a faucet, and it
delivers 50.0 L/s. (a) What water pressure is supplied by the faucet? (b)
50. Consider a river that spreads out in a delta region on its way to the
What is unreasonable about this pressure? (c) What is unreasonable
sea. Construct a problem in which you calculate the average speed at
about the premise? (d) What is the Reynolds number for the given flow?
which water moves in the delta region, based on the speed at which it
was moving up river. Among the things to consider are the size and flow
(Take the viscosity of water as 1.005×10–3 ⎛⎝N / m2⎞⎠ ⋅ s .)
rate of the river before it spreads out and its size once it has spread out.
You can construct the problem for the river spreading out into one large
river or into multiple smaller rivers.
12.7 Molecular Transport Phenomena: Diffusion,
Osmosis, and Related Processes
62. You can smell perfume very shortly after opening the bottle. To show
51. Verify that the flow of oil is laminar (barely) for an oil gusher that
that it is not reaching your nose by diffusion, calculate the average
shoots crude oil 25.0 m into the air through a pipe with a 0.100-m
distance a perfume molecule moves in one second in air, given its
diameter. The vertical pipe is 50 m long. Take the density of the oil to be
diffusion constant D to be 1.00×10–6 m2 /s .
900 kg/m3 and its viscosity to be 1.00 (N/m2 ) ⋅ s (or 1.00 Pa ⋅ s ). 63. What is the ratio of the average distances that oxygen will diffuse in a
52. Show that the Reynolds number N R is unitless by substituting units given time in air and water? Why is this distance less in water
(equivalently, why is D less in water)?
for all the quantities in its definition and cancelling.
53. Calculate the Reynolds numbers for the flow of water through (a) a
64. Oxygen reaches the veinless cornea of the eye by diffusing through
nozzle with a radius of 0.250 cm and (b) a garden hose with a radius of
its tear layer, which is 0.500-mm thick. How long does it take the average
0.900 cm, when the nozzle is attached to the hose. The flow rate through
oxygen molecule to do this?
hose and nozzle is 0.500 L/s. Can the flow in either possibly be laminar?
65. (a) Find the average time required for an oxygen molecule to diffuse
through a 0.200-mm-thick tear layer on the cornea. (b) How much time is
CHAPTER 12 | FLUID DYNAMICS AND ITS BIOLOGICAL AND MEDICAL APPLICATIONS 427
required to diffuse 0.500 cm3 of oxygen to the cornea if its surface area
is 1.00 cm2 ?
66. Suppose hydrogen and oxygen are diffusing through air. A small
amount of each is released simultaneously. How much time passes
before the hydrogen is 1.00 s ahead of the oxygen? Such differences in
arrival times are used as an analytical tool in gas chromatography.
428 CHAPTER 12 | FLUID DYNAMICS AND ITS BIOLOGICAL AND MEDICAL APPLICATIONS
CHAPTER 13 | TEMPERATURE, KINETIC THEORY, AND THE GAS LAWS 429
13
TEMPERATURE, KINETIC THEORY, AND THE
GAS LAWS
Figure 13.1 The welder’s gloves and helmet protect him from the electric arc that transfers enough thermal energy to melt the rod, spray sparks, and burn the retina of an
unprotected eye. The thermal energy can be felt on exposed skin a few meters away, and its light can be seen for kilometers. (credit: Kevin S. O’Brien/U.S. Navy)
Learning Objectives
• Define temperature.
• Convert temperatures between the Celsius, Fahrenheit, and Kelvin scales.
• Define thermal equilibrium.
• State the zeroth law of thermodynamics.
13.2. Thermal Expansion of Solids and Liquids
• Define and describe thermal expansion.
• Calculate the linear expansion of an object given its initial length, change in temperature, and coefficient of linear expansion.
• Calculate the volume expansion of an object given its initial volume, change in temperature, and coefficient of volume expansion.
• Calculate thermal stress on an object given its original volume, temperature change, volume change, and bulk modulus.
• State the ideal gas law in terms of molecules and in terms of moles.
• Use the ideal gas law to calculate pressure change, temperature change, volume change, or the number of molecules or moles in a given
volume.
• Use Avogadro’s number to convert between number of molecules and number of moles.
13.4. Kinetic Theory: Atomic and Molecular Explanation of Pressure and Temperature
• Express the ideal gas law in terms of molecular mass and velocity.
•