Heat engine

In thermodynamics, heat engines are often modeled using a standard engineering model such as the Otto cycle. The theoretical model can be refined and augmented with actual data from an operating engine, using tools such as an indicator diagram. Since very few actual implementations of heat engines exactly match their underlying thermodynamic cycles, one could say that a thermodynamic cycle is an ideal case of a mechanical engine. In any case, fully understanding an engine and its efficiency requires a good understanding of the (possibly simplified or idealised) theoretical model, the practical nuances of an actual mechanical engine and the discrepancies between the two.

In general terms, the larger the difference in temperature between the hot source and the cold sink, the larger is the potential thermal efficiency of the cycle. On Earth, the cold side of any heat engine is limited to being close to the ambient temperature of the environment, or not much lower than 300 kelvin, so most efforts to improve the thermodynamic efficiencies of various heat engines focus on increasing the temperature of the source, within material limits. The maximum theoretical efficiency of a heat engine (which no engine ever attains) is equal to the temperature difference between the hot and cold ends divided by the temperature at the hot end, each expressed in absolute temperature.

The efficiency of various heat engines proposed or used today has a large range:

• 3%[6] (97 percent waste heat using low quality heat) for the ocean thermal energy conversion (OTEC) ocean power proposal
• 25% for most automotive gasoline engines[7]
• 49% for a supercritical coal-fired power station such as the Avedøre Power Station
• 60% for a combined cycle gas turbine[8]

The efficiency of these processes is roughly proportional to the temperature drop across them. Significant energy may be consumed by auxiliary equipment, such as pumps, which effectively reduces efficiency.

It is important to note that although some cycles have a typical combustion location (internal or external), they often can be implemented with the other. For example, John Ericsson[9] developed an external heated engine running on a cycle very much like the earlier Diesel cycle. In addition, externally heated engines can often be implemented in open or closed cycles. In a closed cycle the working fluid is retained within the engine at the completion of the cycle whereas is an open cycle the working fluid is either exchanged with the environment together with the products of combustion in the case of the internal combustion engine or simply vented to the environment in the case of external combustion engines like steam engines and turbines.

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:[15]

${\displaystyle W+Q=\Delta _{cycle}U=0}$

and therefore

${\displaystyle W=-Q=-(Q_{c}+Q_{h})}$

where

${\displaystyle W=-\oint PdV}$ is the net work extracted from the engine in one cycle. (It is negative, in the IUPAC convention, since work is done by the engine.)

${\displaystyle Q_{h}>0}$ 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.)

${\displaystyle Q_{c}=-|Q_{c}|<0}$ is the waste heat given off by the engine to the cold temperature heat sink. (It is negative[15] 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 ${\displaystyle Q_{h}}$, for instance from combustion of a fuel, so the engine efficiency is reasonably defined as

${\displaystyle \eta ={\frac {|W|}{Q_{h}}}={\frac {Q_{h}+Q_{c}}{Q_{h}}}=1+{\frac {Q_{c}}{Q_{h}}}=1-{\frac {|Q_{c}|}{Q_{h}}}}$

The efficiency is less than 100% because of the waste heat ${\displaystyle Q_{c}<0}$ unavoidably lost to the cold sink (and corresponding compression work put in) during the required recompression at the cold temperature before the 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:

${\displaystyle \ \ \ \Delta S_{h}+\Delta S_{c}=\Delta _{cycle}S=0}$

Note that ${\displaystyle \Delta S_{h}}$ is positive because isothermal expansion in the power stroke increases the multiplicity of the working fluid while ${\displaystyle \Delta S_{c}}$ is negative since recompression decreases the multiplicity. If the engine is ideal and runs reversibly, ${\displaystyle Q_{h}=T_{h}\Delta S_{h}}$ and ${\displaystyle Q_{c}=T_{c}\Delta S_{c}}$, and thus[16][15]

${\displaystyle Q_{h}/T_{h}+Q_{c}/T_{c}=0}$,

which gives ${\displaystyle Q_{c}/Q_{h}=-T_{c}/T_{h}}$ and thus the Carnot limit for heat-engine efficiency,

${\displaystyle \eta _{\text{max}}=1-{\frac {T_{c}}{T_{h}}}}$

where ${\displaystyle T_{h}}$ is the absolute temperature of the hot source and ${\displaystyle T_{c}}$ that of the cold sink, usually measured in kelvins.

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.

Figure 2: Carnot cycle efficiency with changing heat addition temperature.

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,[17] very similar to a Carnot engine, but where the thermal reservoirs at temperature ${\displaystyle T_{h}}$ and ${\displaystyle T_{c}}$ are allowed to be different from the temperatures of the substance going through the reversible Carnot cycle: ${\displaystyle T'_{h}}$ and ${\displaystyle T'_{c}}$. The heat transfers between the reservoirs and the substance are considered as conductive (and irreversible) in the form ${\displaystyle dQ_{h,c}/dt=\alpha (T_{h,c}-T'_{h,c})}$. 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, ${\displaystyle T\approx T'}$ and the classical Carnot result is found

${\displaystyle \eta =1-{\frac {T_{c}}{T_{h}}}}$,

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

${\displaystyle \eta =1-{\sqrt {\frac {T_{c}}{T_{h}}}}}$ (Note: T in units of K or °R)

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

Efficiencies of power stations[17]

Power station	${\displaystyle T_{c}}$ (°C)	${\displaystyle T_{h}}$ (°C)	${\displaystyle \eta }$ (Carnot)	${\displaystyle \eta }$ (Endoreversible)	${\displaystyle \eta }$ (Observed)
West Thurrock (UK) coal-fired power station	25	565	0.64	0.40	0.36
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.

Heat engines have been known since antiquity but were only made into useful devices at the time of the industrial revolution in the 18th century. They continue to be developed today.

Engineers have studied the various heat-engine cycles to improve the amount of usable work they could extract from a given power source. The Carnot cycle limit cannot be reached with any gas-based cycle, but engineers have found at least two ways to bypass that limit and one way to get better efficiency without bending any rules:

1. Increase the temperature difference in the heat engine. The simplest way to do this is to increase the hot side temperature, which is the approach used in modern combined-cycle gas turbines. Unfortunately, physical limits (such as the melting point of the materials used to build the engine) and environmental concerns regarding NO_x production (if the heat source is combustion with ambient air) restrict the maximum temperature on workable heat-engines. Modern gas turbines run at temperatures as high as possible within the range of temperatures necessary to maintain acceptable NO_x output . Another way of increasing efficiency is to lower the output temperature. One new method of doing so is to use mixed chemical working fluids, then exploit the changing behavior of the mixtures. One of the most famous is the so-called Kalina cycle, which uses a 70/30 mix of ammonia and water as its working fluid. This mixture allows the cycle to generate useful power at considerably lower temperatures than most other processes.
2. Exploit the physical properties of the working fluid. The most common such exploitation is the use of water above the critical point (supercritical water). The behavior of fluids above their critical point changes radically, and with materials such as water and carbon dioxide it is possible to exploit those changes in behavior to extract greater thermodynamic efficiency from the heat engine, even if it is using a fairly conventional Brayton or Rankine cycle. A newer and very promising material for such applications is supercritical CO₂. SO₂ and xenon have also been considered for such applications. Downsides include issues of corrosion and erosion, the different chemical behavior above and below the critical point, the needed high pressures and – in the case of sulfur dioxide and to a lesser extent carbon dioxide – toxicity. Among the mentioned compounds xenon is least suitable for use in a nuclear reactor due to the high neutron absorption cross section of almost all isotopes of xenon, whereas carbon dioxide and water can also double as a neutron moderator for a thermal spectrum reactor.
3. Exploit the chemical properties of the working fluid. A fairly new and novel exploit is to use exotic working fluids with advantageous chemical properties. One such is nitrogen dioxide (NO₂), a toxic component of smog, which has a natural dimer as di-nitrogen tetraoxide (N₂O₄). At low temperature, the N₂O₄ is compressed and then heated. The increasing temperature causes each N₂O₄ to break apart into two NO₂ molecules. This lowers the molecular weight of the working fluid, which drastically increases the efficiency of the cycle. Once the NO₂ has expanded through the turbine, it is cooled by the heat sink, which makes it recombine into N₂O₄. This is then fed back by the compressor for another cycle. Such species as aluminium bromide (Al₂Br₆), NOCl, and Ga₂I₆ have all been investigated for such uses. To date, their drawbacks have not warranted their use, despite the efficiency gains that can be realized.[18]

Each process is one of the following:

• isothermal (at constant temperature, maintained with heat added or removed from a heat source or sink)
• isobaric (at constant pressure)
• isometric/isochoric (at constant volume), also referred to as iso-volumetric
• adiabatic (no heat is added or removed from the system during adiabatic process)
• isentropic (reversible adiabatic process, no heat is added or removed during isentropic process)

• Einstein refrigerator
• Heat pump
• Reciprocating engine for a general description of the mechanics of piston engines
• Thermosynthesis
• Timeline of heat engine technology

Wikimedia Commons has media related to Heat engines.

• Kroemer, Herbert; Kittel, Charles (1980). Thermal Physics (2nd ed.). W. H. Freeman Company. ISBN 0-7167-1088-9.
• Callen, Herbert B. (1985). Thermodynamics and an Introduction to Thermostatistics (2nd ed.). John Wiley & Sons, Inc. ISBN 0-471-86256-8.
• Robinson, Clark (1943). The Thermodynamics of Firearms. MaGraw-Hill Book Company Inc.