PHY1039 Properties of Matter Heat Engines, Thermodynamic Efficiency, and Carnot Cycles

1. PHY1039 Properties of Matter Heat Engines, Thermodynamic Efficiency, and Carnot Cycles April 30 and May 3, 2012 Lectures 17 and 18

2. 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.

3. Hot Body (source of heat) Q1 W Q2 Cold Body (absorbs heat) Heat Engine E

4. 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. Combustion of wood, oil, gas and coal http://www.dailymail.co.uk/news/article-1043161/Anti-terror-patrols-secretly-stepped-power-stations.html http://www.nearfield.com/~dan/sports/bike/river/coyote/index.htm

5. Example of a Heat Engine Open system that is closed in part of the cycle

6. Internal Combustion Engine d a

7. Work per cycle = Area inside Q=0 Q=0 Comparison of Otto and Diesel Cycles combustion

8. Nuclear Power Plant: A Very Large Heat Engine http://science.howstuffworks.com/inside-nuclear-power-plant-pictures6.htm

9. Hot Body (source of heat) Q1 W Q2 Substituting: Cold Body (absorbs heat) Efficiency of a Heat Engine Efficiency,h = Work out/Heat in: Apply First Law to the working substance: DU = Q1 – Q2 – W But in a cycle,DU= 0 Thus, W = Q1 – Q2. E Lesson:his maximum when Q2 is minimum.

10. The Stirling Engine See: http://www.animatedengines.com/ltdstirling.shtml • Closed system • Operates between two bodies with (small) different temperatures. • Can use “stray” heat

11. isothermal Heat in Heat out isothermal The Stirling Cycle TH >TC (TH - TC ) is proportional to the amount of work that is done in a cycle. = air temp =hot water 2

12. 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.

13. Hot Reservoir Fixed at T = T1 Q1 W Q2 Cold Reservoir Fixed at T = T2 Carnot Cycle C

14. a • Q1 b • T1 Q=0 Q=0 • d • T2 Q2 c Carnot Cycle Pressure Working substance = Ideal gas Volume

15. a • Q1 b • T1 W Q=0 Q=0 • d • T2 Q2 c Carnot Cycle Pressure Volume

16. From b to c: adiabatic, Q = 0, so that TVg-1 is constant. Thus, T1Vbg-1 = T2Vcg-1 or Similarly, d to a: adiabatic, Q = 0, so that TVg-1 is constant. Thus, T2Vdg-1 = T1Vag-1 or Carnot Cycle From a to b: isothermal, so that DU = 0 and Q = - W Thus, Q1 = +nRT1ln(Vb/Va)(+ve quantity) Similarly, from c to d: isothermal, so thatDU= 0 and Q = - W Thus, Q2 = +nRT2ln(Vd/Vc) = -nRT2ln(Vc/Vd) (-ve)

17. Which means that But as the volume ratios are equal: Carnot Cycle We see that: Now also: This is an important result. Temperature can be defined (on the absolute (Kelvin) scale) in terms of the heat flows in a Carnot Cycle.

18. 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!

19. • Recall that for any cycle, the efficiency of a heat engine is given as: • For an engine using a Carnot cycle, the efficiency is also equal to: 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,hcis always <1.

20. 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, h < 1!

21. Hot Body (source of heat) Q1 Q2 = 0 Heat Engine E W= -Q1 Cold Body (absorbs heat)

22. Q2 = W Heat Engine Hot Body (source of heat) Q1= 0 Examples: friction creating heat; isothermal compression of ideal gas E W POSSIBLE! Cold Body (absorbs heat)

23. Refrigerator: A heat engine operating in reverse Hot Body Q1 Refrigerator Efficiency: E W Note that the cycle is going in the opposite direction to the engine. Q2 Cold Body

24. For a Carnot refrigerator, the efficiency is: Refrigerator Efficiency First Law tells us that Q2 + W - Q1 = 0. Thus, W = Q1 – Q2 Efficiency is usually >1! The smaller the T difference, the more efficient is the refrigerator.

25. 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.

26. The efficiency is defined as the amount of heat pumped in to the hot body per the amount of work done: The First Law tells us that W = Q1-Q2 So, substituting, we find: 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). hhpis always > 1! For maximumh, T2 should be T1 (just slightly less).

27. 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.

28. Heat Flows in a Carnot Cycle Hot Reservoir, T1 Q1 C W Q2 Cold Reservoir, T2

29. One could also consider the small amount of reversible heat flowdQrevthat flows at a temperature T at each point in the cycle. The net heat flow is equal to the sum of the differential flows: Such that From the definition of an integral, we find for the entire cycle that where the circle represent integration over the entire cycle. For a Carnot cycle, some of the heat into the cycle is converted to work so that Q1> Q2. We also know that This relation can be shown to be true for any reversible cycle.

30. Clausius Inequality 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).