PHY1039 Properties of Matter Heat 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) Q1Q1 Cold Body (absorbs heat) Q2Q2 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. 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 Work per cycle = Area inside

Nuclear Power Plant: A Very Large Heat Engine

Efficiency of a Heat Engine Efficiency,  = Work out/Heat in: Apply First Law to the working substance:  U = Q 1 – Q 2 – W But in a cycle,  U = 0 Thus, W = Q 1 – Q 2. Substituting: Lesson:  is maximum when Q 2 is minimum. E Hot Body (source of heat) Q1Q1 Cold Body (absorbs heat) Q2Q2 W

The Stirling Engine Closed system Operates between two bodies with (small) different temperatures. Can use “stray” heat See:

isothermal = air temp =hot water Heat in Heat out T H >T C The Stirling Cycle (T H - T C ) 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 Cycle Hot Reservoir Fixed at T = T 1 Cold Reservoir Fixed at T = T 2 C Q1Q1 Q2Q2 W

Volume Pressure a b d T1T1 Q1Q1 Carnot Cycle Q2Q2 T2T2 c Q=0 Working substance = Ideal gas

Volume Pressure a b d T1T1 Q1Q1 Q2Q2 T2T2 c Q=0 W Carnot Cycle

From a to b: isothermal, so that  U = 0 and Q = - W Thus, Q 1 = +nRT 1 ln(V b /V a ) (+ve quantity) Carnot Cycle Similarly, from c to d: isothermal, so that  U = 0 and Q = - W Thus, Q 2 = +nRT 2 ln(V d /V c ) = -nRT 2 ln(V c /V d ) (-ve) From b to c: adiabatic, Q = 0, so that TV  -1 is constant. Thus, T 1 V b  -1 = T 2 V c  -1 or Similarly, d to a: adiabatic, Q = 0, so that TV  -1 is constant. Thus, T 2 V d  -1 = T 1 V a  -1 or

Carnot Cycle We see that: Which means 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. But as the volume ratios are equal:

What’s Special about a Carnot Cycle? (1) Heat is transferred to/from only two reservoirs at fixed temperatures, T 1 and T 2 - 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 T 1 and T 2 are the temperatures of the hot and cold reservoirs, respectively, in degrees Kelvin.  As T 2 > 0,  c is always <1. 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:

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) Q1Q1 Cold Body (absorbs heat) Q 2 = 0 W= -Q 1

Heat Engine E Hot Body (source of heat) Q 1 = 0 Cold Body (absorbs heat) Q 2 = W W POSSIBLE! Examples: friction creating heat; isothermal compression of ideal gas

Refrigerator: A heat engine operating in reverse E Hot Body Q1Q1 Cold Body Q2Q2 W Refrigerator Efficiency: Note that the cycle is going in the opposite direction to the engine.

Refrigerator Efficiency First Law tells us that Q 2 + W - Q 1 = 0. Thus, W = Q 1 – Q 2 For a Carnot refrigerator, the efficiency is: 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 = Q 1 -Q 2 So, substituting, we find:  hp is always > 1! For maximum , T 2 should be  T 1 (just slightly less). The 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, T 1 Cold Reservoir, T 2 C W Q1Q1 Q2Q2

Such that One could also consider the small amount of reversible heat flow  Q rev 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: For a Carnot cycle, some of the heat into the cycle is converted to work so that Q 1 > Q 2. We also know that From the definition of an integral, we find for the entire cycle that where the circle represent integration over the entire cycle. This relation can be shown to be true for any reversible cycle.

Clausius Inequality The Clausius statement tells us that for any reversible cycle: For non-reversible cycles, the Clausius Inequality States: where T o is the temp. of the reservoir (external heat source) and the circle represents integration over the entire cycle (contour integral).