1. Chapter 15
Fluids

2. On Golden Pond
A boat carrying a large chunk of gold is floating on a lake. The chunk is then thrown overboard and sinks. What happens to the water level in the lake (with respect to the shore)?
a) rises
b) drops
c) remains the same
d) depends on the size of the gold
Answer: b

3. On Golden Pond
A boat carrying a large chunk of gold is floating on a lake. The chunk is then thrown overboard and sinks. What happens to the water level in the lake (with respect to the shore)?
a) rises
b) drops
c) remains the same
d) depends on the size of the gold
Initially the chunk of gold “floats” by sitting in the boat. The buoyant force is equal to the weight of the gold, and this will require a lot of displaced water to equal the weight of the gold. When thrown overboard, the gold sinks and only displaces its volume in water. This is not so much water—certainly less than before—and so the water level in the lake will drop.

4. Fluid Flow and Continuity
Continuity tells us that whatever the mass of fluid in a pipe passing a particular point per second, the same mass must pass every other point in a second. The fluid is not accumulating or vanishing along the way.
Volume per unit time
This means that where the pipe is narrower, the fluid is flowing faster

5. Continuity and Compressibility
Most gases are easily compressible; most liquids are not. Therefore, the density of a liquid may be treated as constant (not true for a gas).
mass flow is conserved
volume flow is conserved

6. Bernoulli’s Equation The kinetic energy of a fluid element is:
When a fluid moves from a wider area of a pipe to a narrower one, its speed increases; therefore, work has been done on it.
The kinetic energy of a fluid element is:
Equating the work done to the increase in kinetic energy gives:

7. Where fluid moves faster, pressure is lower
Bernoulli’s Equation
Where fluid moves faster, pressure is lower

8. Bernoulli’s Equation
If a fluid flows in a pipe of constant diameter, but changes its height, there is also work done on it against the force of gravity.
Equating the work done with the change in potential energy gives:

9. Bernoulli’s Equation
The general case, where both height and speed may change, is described by Bernoulli’s equation:
This equation is essentially a statement of conservation of energy in a fluid.

10. Dynamic lift
v high
P low
v low
P high
Aircraft wing

11. Applications of Bernoulli’s Equation
If a hole is punched in the side of an open container, the outside of the hole and the top of the fluid are both at atmospheric pressure.
Since the fluid inside the container at the level of the hole is at higher pressure, the fluid has a horizontal velocity as it exits.

12. Applications of Bernoulli’s Equation
If the fluid is directed upwards instead, it will reach the height of the surface level of the fluid in the container.
P1 = Patm
v1 = 0
P2 = Patm

13. The Falling Bucket
When a hole is made in the side of a cola can holding water, water flows out and follows a parabolic trajectory. If the container is dropped in free fall, the water flow will:
a) diminish
b) stop altogether
c) go out in a straight line
d) curve upward
Cola
Answer: b

14. The Falling Bucket
When a hole is made in the side of a cola can holding water, water flows out and follows a parabolic trajectory. If the container is dropped in free fall, the water flow will:
a) diminish
b) stop altogether
c) go out in a straight line
d) curve upward
Water flows out of the hole because the water pressure inside is larger than the air pressure outside. The water pressure is due to the weight of the water. When the can is in free fall, the water is weightless, so the water pressure is zero, and hence no water is pushed out of the hole!
Cola

15. Fluid Flow
Water flows through a 1-cm diameter pipe connected to a ½-cm diameter pipe. Compared to the speed of the water in the 1-cm pipe, the speed in the ½ -cm pipe is:
a) one-quarter
b) one-half
c) the same
d) double
e) four times
Answer: e

16. Fluid Flow
Water flows through a 1-cm diameter pipe connected to a ½-cm diameter pipe. Compared to the speed of the water in the 1- cm pipe, the speed in the ½ -cm pipe is:
a) one-quarter
b) one-half
c) the same
d) double
e) four times v1 v2
The area of the small pipe is less, so we know that the water will flow faster there. Because A  r2, when the radius is reduced by one-half, the area is reduced by one-quarter, so the speed must increase by four times to keep the flow rate (A × v) constant.

17. Surface Tension
A molecule in the center of a liquid drop experiences forces in all directions from other molecules. A molecule on the surface, however, experiences a net force toward the drop. This pulls the surface inward so that its area is a minimum.
Since there are forces tending to keep the surface area at a minimum, it tends to act somewhat like a spring – the surface acts as though it were elastic.

18. Surface Tension
This means that small, dense objects such as insects and needles can stay on top of water even though they are too dense to float.

19. Surface Tension and Adhesion
Capillary action is the result of adhesion and surface tension. Adhesion of water to the walls of a vessel will cause an upward force on the liquid at the edges and result in a meniscus which turns upward. The surface tension acts to hold the surface intact, so instead of just the edges moving upward, the whole liquid surface is dragged upward.

20. Temperature and Heat

21. Objects are in thermal contact if heat can flow between them.
Definition of heat:
Heat is the energy transferred between objects because of a temperature difference.
Objects are in thermal contact if heat can flow between them.
When the transfer of heat between objects in thermal contact ceases, they are in thermal equilibrium.

22. The Zeroth Law of Thermodynamics
If object A is in thermal equilibrium with object B, and object C is also in thermal equilibrium with object B, then objects A and C will be in thermal equilibrium if brought into thermal contact.
Object B can then be a thermometer, providing a scale to compare objects
That is, temperature is the only factor that determines whether two objects in thermal contact are in thermal equilibrium or not.

23. Thermometric properties
Length of a metal rod
Volume of a gas held at constant pressure
Pressure of a gas held at constant volume
Electrical resistance of a metal conductor
Volume of a liquid
A good thermometric property should to be linear with temperature over a wide range of temperature.

24. Common Temperature Scales
The Celsius scale:
Water freezes at 0° Celsius.
Water boils at 100° Celsius.
The Fahrenheit scale:
Water freezes at 32° Fahrenheit.
Water boils at 212° Fahrenheit.
Andreas Celsius: , Swedish astronomer
Daniel Fahrenheit: , German physicist

25. A natural zero point for temperature
The pressure in a gas is proportional to its temperature.
The proportionality constant is different for different gases, but they all reach zero pressure at the same temperature, which we call absolute zero
Absolute Zero = oC
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26. Temperature Scales
The Kelvin scale is similar to the Celsius scale, except that the Kelvin scale has its zero at absolute zero.
Copyright © 2010 Pearson Education, Inc.

28. Thermal Expansion
Most substances expand when heated; the change in length or volume is typically proportional to the change in temperature.
The proportionality constant is called the coefficient of linear expansion.

29. Glasses
Two drinking glasses are stuck, one inside the other. How would you get them unstuck?
a) run hot water over them both
b) put hot water in the inner one
c) run hot water over the outer one
d) run cold water over them both
e) break the glasses
Answer: c

30. Glasses
Two drinking glasses are stuck, one inside the other. How would you get them unstuck?
a) run hot water over them both
b) put hot water in the inner one
c) run hot water over the outer one
d) run cold water over them both
e) break the glasses
Running hot water over only the outer glass will allow the outer one to expand, while the inner glass remains relatively unchanged. This should loosen the outer glass and free it.

31. Some typical coefficients of thermal expansion
a nickel steel alloy (65% Fe +35%) Ni noted for its very small expansion.
Charles Guillaume received the Nobel Prize in Physics in 1920 for the invention
Invar ~1.0 x 10-6

32. Thermal Expansion of a bi-metallic strip
A bimetallic strip consists of two metals of different coefficients of thermal expansion, A and B in the figure. It will bend when heated or cooled.
Thermostat

35. In northern Canada, where I grew up, the temperature ranges from around -40o F in the winter to around 100o F in the summer. If a train track rail is 10 m long when it is 68o F, how long is it on a cold winter night? On a hot summer day?

36. In northern Canada, where I grew up, the temperature ranges from around -40o F in the winter to around 100o F in the summer. If a train track rail is 10 m long when it is 68o F, how long is it on a cold winter night? On a hot summer day?

37. Thermal Expansion
The expansion of an area of a flat substance is derived from the linear expansion in both directions

38. Steel Expansion
Metals such as brass expand when heated. The thin brass plate in the movie has a circular hole in its center. When the plate is heated, what will happen to the hole?
a) gets larger
b) gets smaller
c) stays the same
d) vanishes
Answer: a

39. Steel Expansion
Metals such as brass expand when heated. The thin brass plate in the movie has a circular hole in its center. When the plate is heated, what will happen to the hole?
a) gets larger
b) gets smaller
c) stays the same
d) vanishes
Imagine drawing a circle on the plate. This circle will expand outward along with the rest of the plate. Now replace the circle with the hole, and you can see that the hole will expand outward as well. Note that the material does NOT “expand inward” to fill the hole!!
expansion

41. Thermal Volume Expansion
The change in volume of a solid is also derived from the linear expansion:
For liquids and gases, only the coefficient of volume expansion is defined:

42. Some typical coefficients of volume expansion
Pyrex Glass ~1.0 x 10-5

43. Thermal Expansion of Water
Water also expands when it is heated, except when it is close to freezing; it actually expands (density drops) when cooling from 4° C to 0° C. This is why frozen bottles burst.

44. Steel Ring II a) heat the thing up b) cool the thing down
You want to take apart a couple of aluminum parts held together by steel screws, but the screws are stuck. What should you do?
a) heat the thing up
b) cool the thing down
c) blow the thing up
10−5
10−4
10−6
Coefficient of volume expansion β (1/°C )
Glass Hg
Quartz
Air
10−3 Al
Steel
Answer: a

45. Steel Ring II a) heat the thing up b) cool the thing down
You want to take apart a couple of aluminum parts held together by steel screws, but the screws are stuck. What should you do?
a) heat the thing up
b) cool the thing down
c) blow the thing up
10−5
10−4
10−6
Coefficient of volume expansion β (1/°C )
Glass Hg
Quartz
Air
10−3 Al
Steel
Because aluminum has a larger β value, that means aluminum expands more than steel. Thus, by heating the part, the aluminum holes will expand faster than the steel screws and the screws will come loose.

46. An aluminum saucepan with a diameter of 23 cm and a height of 6
An aluminum saucepan with a diameter of 23 cm and a height of 6.0 cm is filled to the brim with water. The initial temperature of the pan and water is 19 oC. The pan is now placed on a stove burner and heated to 88 oC.
Will water overflow from the pan, or will the water level in the pan decrease?
Calculate the volume of water the overflows , or the drop in water level in the pan, whichever is appropriate.
αAl = 24 x 10-6 / K
βWater = 0.21 x 10-3 / K

47. αAl = 24 x 10-6 / K βWater = 0.21 x 10-3 / K (a) (b)
An aluminum saucepan with a diameter of 23 cm and a height of 6.0 cm is filled to the brim with water. The initial temperature of the pan and water is 19 oC. The pan is now placed on a stove burner and heated to 88 oC.
Will water overflow from the pan, or will the water level in the pan decrease?
Calculate the volume of water the overflows , or the drop in water level in the pan, whichever is appropriate.
αAl = 24 x 10-6 / K
βWater = 0.21 x 10-3 / K
(a)
The coefficient of volumetric expansion for the Al pan is:
Since the coefficient of volumetric expansion is larger for water than for the pan, the water will overflow as the pan is heated.
(b)

48. Heat is another form of energy.
Heat and Mechanical Work
Heat is another form of energy.
James Joule used a device similar to this one to measure the mechanical equivalent of heat:
One kilocalorie (kcal) is defined as the amount of heat needed to raise the temperature of 1 kg of water from 14.5° C to 15.5° C.

49. Heat Capacity
The heat capacity of an object is the amount of heat added to it divided by its rise in temperature:
Q = C ΔT
Heat capacity tells you how much heat flow for a given ΔT
Q is positive if ΔT is positive; that is, if heat is added to a system.
Q is negative if ΔT is negative; that is, if heat is removed from a system.

50. Specific Heat
The heat capacity of an object depends on its mass and on a property of the material itself: the specific heat
“heat capacity per kilogram”

51. Specific heats of various materials
Heat capacity is
mass x specific heat
C = mc

52. A ceramic coffee cup, with c=1090 J/(kg K) and m =116 g, is initially at room temperature (24.0 °C). If 225 g of 80.3 °C coffee and 12.2 g of 5.00 °C cream are added to the cup, what is the equilibrium temperature of the system?
Assume that no heat is exchanged with the surroundings, and that the specific heat of coffee and cream are the same as the specific heat of water.
cwater = 4186 J / (kg K)

54. Heat Transfer Mechanisms
Thermal equilibrium is reached by means of thermal contact, which in turn can occur through three different mechanisms
conduction : it occurs when objects at different temperature are in physical contact (e.g. when holding a hot potato). Faster moving molecules in the hotter object transfer some of their energy to the colder one
convection : this occurs mainly in fluids. In a pot of water on a stove, the liquid at the bottom is heated by conduction. The hot water has lower density and rises to the top, cold water from the top falls to the bottom and gets heated, etc.
radiation : any object at non-zero temperature emits radiation (in the form of electromagnetic waves). The effect is more noticeable when standing next to a red-hot coal fire, or in the sun rays

55. Conduction is the flow of heat directly through a physical material
The amount of heat Q that flows through a rod:
increases proportionally to the cross-sectional area A
increases proportionally to ΔT from one end to the other
increases steadily with time
decreases inversely with the length of the rod
The constant k is called the thermal conductivity of the material

56. Some Typical Thermal Conductivities
Substances with high thermal conductivities are good conductors of heat; those with low thermal conductivities are good insulators.

57. kPb = 34.3 W / (kg-m) kCu = 395 W / (kg-m)
Two metal rods—one lead, the other copper—are connected in series, as shown. Note that each rod is m in length and has a square cross section 1.50 cm on a side. The temperature at the lead end of the rods is 2.00°C; the temperature at the copper end is 106°C.
(a) The average temperature of the two ends is 54.0°C. Is the temperature in the middle, at the lead-copper interface, greater than, less than, or equal to 54.0°C? Explain.
(b) find the temperature at the lead-copper interface.
kPb = 34.3 W / (kg-m)
kCu = 395 W / (kg-m)

58. kPb = 34.3 W / (kg-m) kCu = 395 W / (kg-m) Assumptions:
Two metal rods—one lead, the other copper—are connected in series, as shown. Note that each rod is m in length and has a square cross section 1.50 cm on a side. The temperature at the lead end of the rods is 2.00°C; the temperature at the copper end is 106°C.
(a) The average temperature of the two ends is 54.0°C. Is the temperature in the middle, at the lead-copper interface, greater than, less than, or equal to 54.0°C? Explain.
(b) find the temperature at the lead-copper interface.
kPb = 34.3 W / (kg-m)
kCu = 395 W / (kg-m)
Assumptions:
The end points are infinite heat reservoirs... so their temperature doesn’t change for this exercise
The temperature is constant in time at every point. This is not true at moment of thermal connection. We are solving the “steady state” condition, when the temperature at each point doesn’t change.

59. - The lead has a smaller thermal conductivity than the copper
Two metal rods—one lead, the other copper—are connected in series, as shown. Note that each rod is m in length and has a square cross section 1.50 cm on a side. The temperature at the lead end of the rods is 2.00°C; the temperature at the copper end is 106°C.
(a) The average temperature of the two ends is 54.0°C. Is the temperature in the middle, at the lead-copper interface, greater than, less than, or equal to 54.0°C? Explain.
(b) find the temperature at the lead-copper interface.
kPb = 34.3 W / (kg-m)
kCu = 395 W / (kg-m)
(a)
- The heat (per unit time) through the lead must equal that through the copper
- The lead has a smaller thermal conductivity than the copper
The lead requires a larger temperature difference across it than the copper, to get the same heat flow. So TJ > 54o C

60. (b) kPb = 34.3 W / (kg-m) kCu = 395 W / (kg-m)
Two metal rods—one lead, the other copper—are connected in series, as shown. Note that each rod is m in length and has a square cross section 1.50 cm on a side. The temperature at the lead end of the rods is 2.00°C; the temperature at the copper end is 106°C.
(a) The average temperature of the two ends is 54.0°C. Is the temperature in the middle, at the lead-copper interface, greater than, less than, or equal to 54.0°C? Explain.
(b) find the temperature at the lead-copper interface.
(b)
kPb = 34.3 W / (kg-m)
kCu = 395 W / (kg-m)

61. Convection
Convection is the flow of fluid due to a difference in temperatures, such as warm air rising. The fluid “carries” the heat with it as it moves.

62. Radiation
All objects give off energy in the form of radiation, as electromagnetic waves (light) – infrared, visible light, ultraviolet – which, unlike conduction and convection, can transport heat through a vacuum.
Objects that are hot enough will glow – first red, then yellow, white, and blue.

64. The surface of the Sun has a temperature of 5500 oC.
(a) Treating the Sun as a perfect blackbody, with an emissivity of 1.0, find the power that it radiates into space. The radius of the sun is 7.0x108 m, and the temperature of space can be taken to be 3.0 K
(b) the solar constant is the number of watts of sunlight power falling on a square meter of the Earth’s upper atmosphere. Use your result from part (a) to calculate the solar constant, given that the distance from the Sun to the Earth is 1.5x1011 m.
(a)
emissivity
(b)

65. Heat Conduction
Given your experience of what feels colder when you walk on it, which of the surfaces would have the highest thermal conductivity?
a) a rug
b) a steel surface
c) a concrete floor
d) has nothing to do with thermal conductivity
Answer: b

66. Heat Conduction
Given your experience of what feels colder when you walk on it, which of the surfaces would have the highest thermal conductivity?
a) a rug
b) a steel surface
c) a concrete floor
d) has nothing to do with thermal conductivity
The heat flow rate is k A (T1 − T2)/l. All things being
equal, bigger k leads to bigger heat loss.
From the book: Steel = 40, Concrete = 0.84,
Human tissue = 0.2, Wool = 0.04, in units of J/(s.m.C°).