Editing Heat engine (section)

== Efficiency ==
The efficiency of a heat engine relates how much useful work is output for a given amount of heat energy input.

From the laws of [[thermodynamics]], after a completed cycle:<ref name="PlanckBook">{{cite book |last=Planck |first=M. |title=Treatise on Thermodynamics |page=§&nbsp;90 & §&nbsp;137 |quote=eqs.(39), (40), & (65) |publisher=Dover Publications |year=1945}}.</ref>
:<math> W + Q = \Delta_{cycle}U = 0 </math>
:and therefore
:<math> W = -Q = - (Q_c + Q_h) </math>
:where
:<math> W = -\oint PdV </math> is the net work extracted from the engine in one cycle. (It is negative, in the [[Work (thermodynamics)#Sign convention|IUPAC convention]], since work is ''done by'' the engine.)
:<math> Q_h > 0 </math> is the heat energy taken from the high temperature heat source in the surroundings in one cycle. (It is positive since heat energy is ''added'' to the engine.)
:<math> Q_c = -|Q_c|<0 </math> is the waste heat given off by the engine to the cold temperature heat sink. (It is negative<ref name="PlanckBook" /> since heat is ''lost'' by the engine to the sink.)

In other words, a heat engine absorbs heat energy from the high temperature heat source, converting part of it to useful work and giving off the rest as waste heat to the cold temperature heat sink.

In general, the efficiency of a given heat transfer process is defined by the ratio of "what is taken out" to "what is put in". (For a refrigerator or heat pump, which can be considered as a heat engine run in reverse, this is the [[coefficient of performance]] and it is ≥ 1.) In the case of an engine, one desires to extract work and has to put in heat <math> Q_h  </math>, for instance from [[combustion]] of a fuel, so the engine efficiency is reasonably defined as
:<math>\eta = \frac{|W|}{Q_h} = \frac{Q_h + Q_c}{Q_h} = 1 + \frac{Q_c}{Q_h} = 1 - \frac{|Q_c|}{Q_h}</math>

The efficiency is less than 100% because of the waste heat <math> Q_c<0  </math> unavoidably lost to the cold sink (and corresponding compression work put in) during the required recompression at the cold temperature before the [[Power stroke (engine)|power stroke]] of the engine can occur again.

The ''theoretical'' maximum efficiency of any heat engine depends only on the temperatures it operates between. This efficiency is usually derived using an ideal imaginary heat engine such as the [[Carnot heat engine]], although other engines using different cycles can also attain maximum efficiency. Mathematically, after a full cycle, the overall change of entropy is zero:

<math>\ \ \ \Delta S_h + \Delta S_c = \Delta_{cycle} S = 0</math>

Note that <math>\Delta S_h</math> is positive because isothermal expansion in the power stroke increases the [[Multiplicity (statistical mechanics)|multiplicity]] of the working fluid while <math>\Delta S_c</math> is negative since recompression decreases the multiplicity. If the engine is ideal and runs [[Reversible process (thermodynamics)|reversibly]], <math> Q_h = T_h\Delta S_h  </math> and <math> Q_c = T_c\Delta S_c  </math>, and thus<ref name="FermiBook">{{cite book |last=Fermi |first=E. |title=Thermodynamics |page=48 |quote= eq.(64) |publisher=Dover Publications (still in print) |year=1956}}.</ref><ref name="PlanckBook" />

<math> Q_h / T_h + Q_c / T_c = 0 </math>,

which gives <math> Q_c /Q_h = -T_c / T_h  </math> and thus the Carnot limit for heat-engine efficiency,
:<math>\eta_\text{max} = 1 - \frac{T_c}{T_h}</math>

where <math>T_h</math> is the [[absolute temperature]] of the hot source and <math>T_c</math> that of the cold sink, usually measured in [[kelvin]]s.

The reasoning behind this being the '''maximal''' efficiency goes as follows. It is first assumed that if a more efficient heat engine than a Carnot engine is possible, then it could be driven in reverse as a heat pump. Mathematical analysis can be used to show that this assumed combination would result in a net decrease in [[entropy]]. Since, by the [[second law of thermodynamics]], this is statistically improbable to the point of exclusion, the Carnot efficiency is a theoretical upper bound on the reliable efficiency of ''any'' thermodynamic cycle.

Empirically, no heat engine has ever been shown to run at a greater efficiency than a Carnot cycle heat engine.

Figure 2 and Figure 3 show variations on Carnot cycle efficiency with temperature. Figure 2 indicates how efficiency changes with an increase in the heat addition temperature for a constant compressor inlet temperature. Figure 3 indicates how the efficiency changes with an increase in the heat rejection temperature for a constant turbine inlet temperature.

{| cellpadding="2" style="border:1px solid darkgrey; margin:auto;" class=skin-invert-image
|-
|[[File:Carnot Efficiency.svg|none|thumb|385x385px|Figure 2: Carnot cycle efficiency with changing heat addition temperature.]]
|[[File:Carnot Efficiency2.svg|none|thumb|450x450px|Figure 3: Carnot cycle efficiency with changing heat rejection temperature.]]
|}

=== Endo-reversible heat-engines ===
By its nature, any maximally efficient Carnot cycle must operate at an [[infinitesimal]] temperature gradient; this is because any transfer of heat between two bodies of differing temperatures is irreversible, therefore the Carnot efficiency expression applies only to the infinitesimal limit. The major problem is that the objective of most heat-engines is to output power, and infinitesimal power is seldom desired.

A different measure of ideal heat-engine efficiency is given by considerations of [[endoreversible thermodynamics]], where the system is broken into reversible subsystems, but with non reversible interactions between them. A classical example is the Curzon–Ahlborn engine,<ref name=CurzonAhlborn1975>F. L. Curzon, B. Ahlborn (1975). "Efficiency of a Carnot Engine at Maximum Power Output". ''Am. J. Phys.'', Vol. 43, pp. 24.</ref> very similar to a Carnot engine, but where the thermal reservoirs at temperature <math>T_h</math> and <math>T_c</math> are allowed to be different from the temperatures of the substance going through the reversible Carnot cycle: <math>T'_h</math> and <math>T'_c</math>. The heat transfers between the reservoirs and the substance are considered as conductive (and irreversible) in the form <math>dQ_{h,c}/dt = \alpha (T_{h,c}-T'_{h,c})</math>. In this case, a tradeoff has to be made between power output and efficiency. If the engine is operated very slowly, the heat flux is low, <math>T\approx T'</math> and the classical Carnot result is found
:<math>\eta = 1 - \frac{T_c}{T_h}</math>,
but at the price of a vanishing power output. If instead one chooses to operate the engine at its maximum output power, the efficiency becomes
:<math>\eta = 1 - \sqrt{\frac{T_c}{T_h}}</math> (Note: ''T'' in units of [[kelvin|K]] or [[Rankine scale|°R]])

This model does a better job of predicting how well real-world heat-engines can do (Callen 1985, see also [[endoreversible thermodynamics]]):

{| class="wikitable"
|+'''Efficiencies of power stations'''<ref name=CurzonAhlborn1975 />
|-
! ''Power station'' !! <math>T_c</math> (°C) !! <math>T_h</math> (°C) !! <math>\eta</math> (Carnot) !! <math>\eta</math> (Endoreversible) !! <math>\eta</math> (Observed)
|-
! [[West Thurrock]] (UK) [[coal-fired power station]]
| 25 || 565 || 0.64 || 0.40 || 0.36
|-
! [[CANDU reactor|CANDU]] (Canada) [[nuclear power station]]
| 25 || 300 || 0.48 || 0.28 || 0.30
|-
! [[Larderello]] (Italy) [[geothermal power]] station
| 80 || 250 || 0.33 || 0.178 || 0.16
|}

As shown, the Curzon–Ahlborn efficiency much more closely models that observed.