phy1039 properties of matter heat engines, thermodynamic efficiency, and carnot cycles april 30 and...
TRANSCRIPT
PHY1039
Properties of MatterHeat Engines, Thermodynamic
Efficiency, and Carnot Cycles
April 30 and May 3, 2012
Lectures 17 and 18
Heat Engine
• A heat engine is a device that absorbs heat (Q) and uses it to do useful work (W) on the surroundings when operating in a cycle.
• Sources of heat include the combustion of coal, petroleum or carbohydrates and nuclear reactions.
• Working substance: the matter inside the heat engine that undergoes addition and rejection of heat and that does work on the surroundings. Examples include air and water vapour (steam).
• In a cycle, the working substance must be in the same thermodynamic state at the end as at the start.
Heat Engine
E
Hot Body(source of heat)
Q1
Cold Body (absorbs heat)
Q2
W
Sources of Energy (Heat and Work)•Nuclear reactions are a source of heat (which can then be converted to work).
•Solar energy comes in the form of thermal radiation given off by the Sun. (Thermal radiation is a way to transfer heat from a hotter object to a colder object.) The origin of the heat of the Sun is a nuclear reaction.
•Chemical reactions are another source of heat (and hence work).
•Gravitational forces can likewise be a source of mechanical energy (work), which can be converted to electrical energy.
•Tidal energy originates from gravitational forces from the moon; can do work.
http://www.nearfield.com/~dan/sports/bike/river/coyote/index.htmhttp://www.dailymail.co.uk/news/article-1043161/Anti-terror-patrols-secretly-stepped-power-stations.html
Combustion of wood, oil, gas and coal
Open system that is closed in part of the cycle
Example of a Heat Engine
a
d
Internal Combustion Engine
Comparison of Otto and Diesel Cycles
combustion
Q=0
Q=0
Work per cycle = Area inside
Nuclear Power Plant: A Very Large Heat Engine
http://science.howstuffworks.com/inside-nuclear-power-plant-pictures6.htm
Efficiency of a Heat Engine
Efficiency, = Work out/Heat in:
Apply First Law to the working substance:
U = Q1 – Q2 – W
But in a cycle, U = 0
Thus, W = Q1 – Q2.
1
2
1
21
1
1Q
Q
Q
Q
W
Substituting:
1Q
W
Lesson: is maximum when Q2 is minimum.
E
Hot Body(source of heat)
Q1
Cold Body (absorbs heat)
Q2
W
The Stirling Engine
•Closed system
•Operates between two bodies with (small) different temperatures.
• Can use “stray” heat
See: http://www.animatedengines.com/ltdstirling.shtml
isothermal
isothermal
= air temp
=hot water
Heat in
Heat out
TH >TC
The Stirling Cycle
(TH - TC ) is proportional to the amount of work that is done in a cycle.
2
Nicolas Carnot
• Appreciated that to increase efficiency of an engine, as much heat as possible must be converted into work.
• Proposed an engine that operates on the reversible cycle named after him.
• Proved that reversible cycles are the most efficient possible.
Carnot CycleHot ReservoirFixed at T = T1
Cold Reservoir Fixed at T = T2
C
Q1
Q2
W
Volume
Pressure
•
•
a
b
•d
T1
Q1
Carnot Cycle
Q2
VnRT
P 1=
V
constP
.=
T2•c
Q=0Q=0
VnRT
P 2=
Working substance = Ideal gas
Volume
Pressure
•
•
a
b
•d
T1
Q1
Q2
VnRT
P 1=
V
constP
.=
T2•c
Q=0Q=0
VnRT
P 2=
W
Carnot Cycle
From a to b: isothermal, so that U = 0 and Q = - W
Thus, Q1 = +nRT1ln(Vb/Va) (+ve quantity)
Carnot Cycle
Similarly, from c to d: isothermal, so that U = 0 and Q = - W
Thus, Q2 = +nRT2ln(Vd/Vc) = -nRT2ln(Vc/Vd) (-ve)
From b to c: adiabatic, Q = 0, so that TV-1 is constant.Thus, T1Vb
-1 = T2Vc-1 or 1
2
1
b
c
V
V
T
T
Similarly, d to a: adiabatic, Q = 0, so that TV-1 is constant.Thus, T2Vd
-1 = T1Va-1 or 1
2
1
a
d
V
V
T
T
Carnot Cycle
We see that:11
2
1
a
d
b
c
V
V
V
V
T
T
a
b
d
c
V
V
V
V
Which means that
Now also:)/ln()/ln(
)/ln()/ln(
2
1
2
1
2
1
dc
ab
dc
ab
VVTVVT
VVnRTVVnRT
This is an important result. Temperature can be defined (on the absolute (Kelvin) scale) in terms of the heat flows in a Carnot Cycle.
But as the volume ratios are equal: 2
1
2
1
TT
What’s Special about a Carnot Cycle?
(1) Heat is transferred to/from only two reservoirs at fixed temperatures, T1 and T2 - not at a variety of temperatures.
(2) Heat transfer is the efficient because the temperature of the working substance equals the temperature of the reservoirs. No heat is wasted in flowing from hot to cold. The heat transfer is reversible.
(3) The cycle uses an adiabatic process to raise and lower the temperature of the working substance. No heat is wasted in heating up the working substance.
(4) Carnot cycles are reversible. Not all cycles are!
What’s Special about a Carnot Cycle?
(5) The Carnot theorem states that the Carnot cycle (or any reversible cycle) is the most efficient cycle possible. Hence, the Carnot cycle defines an upper limit to the efficiency of a cycle.
• Where T1 and T2 are the temperatures of the hot and cold reservoirs, respectively, in degrees Kelvin.
As T2 > 0, c is always <1.
• Recall that for any cycle, the efficiency of a heat engine is given as:
1
2
11==QQ
QW
E
• For an engine using a Carnot cycle, the efficiency is also equal to:
1
21=TT
C
Kelvin-Planck Statement of the Second Law of Thermodynamics
“It is impossible to construct a device that - operating in a cycle - will produce no other effect than the extraction of heat from a single body and the performance of an equivalent amount of work”
Or…A cyclical engine cannot convert heat from a single body completely into work. Some heat must be rejected at a lower temperature. Thus, efficiency, < 1!
Heat Engine
E
Hot Body(source of heat)
Q1
Cold Body (absorbs heat)
Q2 = 0
W= -Q1
Heat Engine
E
Hot Body(source of heat)
Q1= 0
Cold Body (absorbs heat)
Q2 = W
WPOSSIBLE!
Examples: friction creating heat; isothermal compression of ideal gas
Refrigerator: A heat engine operating in reverse
E
Hot Body
Q1
Cold Body
Q2
WWQ
workheat
in
outR
2
Refrigerator Efficiency:
Note that the cycle is going in the opposite direction to the engine.
Refrigerator Efficiency
WQ
workheat
in
outR
2
21
2
QR
First Law tells us that Q2 + W - Q1 = 0.
Thus, W = Q1 – Q2
2
21
2
1
2
1
2
21 111
T
TT
T
T
Q
Q
Q
QQcR
For a Carnot refrigerator, the efficiency is:
21
2
TT
TcR
Efficiency is usually >1!
The smaller the T difference, the more efficient is the refrigerator.
Clausius Statement of the Second Law of Thermodynamics
(applies to refrigerators)
“It is impossible to construct a device that - operating in a cycle - will produce no other effect than heat transfer from a colder body to hotter body.”
“Or…Heat cannot flow from a cold body to a hotter body by itself. Work has to be done in the process.”
The Kelvin-Planck and Clausius statements are equivalent. See the proof in Chapter 4 of Finn’s book, Thermal Physics.
Efficiency of a Heat Pump
The purpose of a heat pump is to extract heat from a cold body (such as the River Thames) and “pump” it to a hot body (such as an office building).
The First Law tells us that W = Q1-Q2 So, substituting, we find:
1221
1
21
1
/1
1
TTTT
T
QChp
hp is always > 1! For maximum , T2 should be T1 (just slightly less).
W
QChp
1The efficiency is defined as the amount of heat pumped in to the hot body per the amount of work done:
The Clausius Inequality
• Expressions of inequality/equality relating to heat flow in a cycle.
• The expression is required for the derivation of an equation for entropy – which is our next main topic.
• Derived from a “thought experiment” using Carnot engines acting in a series. See Finn’s Thermal Physics, Chapter 5.
Heat Flows in a Carnot Cycle
Hot Reservoir, T1
Cold Reservoir, T2
CW
Q1
Q2
2
1
2
1 =QQ
TT
2
121=TT
Such that 02
2
1
1 T
Q
T
Q
One could also consider the small amount of reversible heat flow Qrev that flows at a temperature T at each point in the cycle. The net heat flow is equal to the sum of the differential flows:
02
2
1
1
T
Q
T
Q
T
Q
cycle
rev
For a Carnot cycle, some of the heat into the cycle is converted to work so that Q1 > Q2. We also know that
2
2
1
1
T
Q
T
Q
From the definition of an integral, we find for the entire cycle that
where the circle represent integration over the entire cycle.
0
T
Qrev
This relation can be shown to be true for any reversible cycle.
Clausius Inequality
0
T
Qrev
The Clausius statement tells us that for any reversible cycle:
For non-reversible cycles, the Clausius Inequality States:
where To is the temp. of the reservoir (external heat source) and the circle represents integration over the entire cycle (contour integral).