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Heat engine

A heat engine is a physical or theoretical device that converts thermal energy to mechanical output. The mechanical output is called work, and the thermal energy input is called heat. Heat engines typically run on a specific thermodynamic cycle. Heat engines are often named after the thermodynamic cycle they are modeled by. They often pick up alternate names, such as gasoline/petrol, turbine, or steam engines. Heat engines can generate heat inside the engine itself or it can absorb heat from an external source. Heat engines can be open to the atmospheric air or sealed and closed off to the outside (Open or closed cycle).

In engineering and thermodynamics, a heat engine performs the conversion of heat energy to mechanical work by exploiting the temperature gradient between a hot "source" and a cold "sink". Heat is transferred from the source, through the "working body" of the engine, to the sink, and in this process some of the heat is converted into work by exploiting the properties of a working substance (usually a gas or liquid).

Overview

Heat engines are often confused with the cycles they attempt to mimic. Typically when describing the physical device the term 'engine' is used. When describing the model the term 'cycle' is used.

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 gaining a good understanding of the (possibly simplified or idealized) 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 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 obtains) is equal to the temperature difference between the hot and cold ends divided by the temperature at the hot end, all expressed in absolute temperature or kelvins.

The efficiency of various heat engines proposed or used today ranges from 3 percent (97 percent waste heat) for the OTEC ocean power proposal through 25 percent for most automotive engines, to 45 percent for a supercritical coal plant, to about 60 percent for a steam-cooled combined cycle gas turbine. All of these processes gain their efficiency (or lack thereof) due to the temperature drop across them.

OTEC uses the temperature difference of ocean water on the surface and ocean water from the depths, a small difference of perhaps 25 degrees Celsius, and so the efficiency must be low. The combined cycle gas turbines use natural-gas fired burners to heat air to near 1530 degrees Celsius, a difference of a large 1500 degrees Celsius, and so the efficiency can be large when the steam-cooling cycle is added in.

Everyday examples

Examples of everyday heat engines include: the steam engine, the diesel engine, and the gasoline (petrol) engine in an automobile. A common toy that is also a heat engine is a drinking bird. All of these familiar heat engines are powered by the expansion of heated gases. The general surroundings are the heat sink, providing relatively cool gases which, when heated, expand rapidly to drive the mechanical motion of the engine.

Examples of heat engines

It is important to note that although some cycles have a typical combustion location (internal external), they often can be implemented as the other combustion cycle. For example, John Ericsson developed an external heated engine running on a cycle very much like the earlier Diesel cycle. In addition, the externally heated engines can often be implemented in open or closed cycles.

What this boils down to is there are thermodynamic cycles and a large number of ways of implementing them with mechanical devices called engines.

Phase change cycles

In these cycles and engines, the working fluids are gases and liquids. The engine converts the working fluid from a gas to a liquid.

Rankine cycle (classical steam engine)
Regenerative cycle (steam engine more efficient than Rankine cycle)
Vapor to liquid cycle (Drinking bird, Injector)
Liquid to solid cycle (Frost heaving — water changing from ice to liquid and back again can lift rock up to 60 m.)
Solid to gas cycle (Dry ice cannon — Dry ice sublimes to gas.)

Gas only cycles

In these cycles and engines the working fluid is always a gas (ie, there is no phase change):

Carnot cycle (Carnot heat engine)
Ericsson Cycle (Caloric Ship John Ericsson)
Stirling cycle (Stirling engine, thermoacoustic devices)
Internal combustion engine (ICE):

Otto cycle (eg. Gasoline/Petrol engine, high-speed diesel engine)
Diesel cycle (eg. low-speed diesel engine)
Atkinson Cycle (Atkinson Engine)
Brayton cycle or Joule cycle originally Ericsson Cycle (gas turbine)
Lenoir cycle (e.g., pulse jet engine)
Miller cycle

Liquid only cycles

In these cycles and engines the working fluid are always like liquid:

Stirling Cycle (Malone engine)

Electron cycles

Magnetic cycles

Thermo-magnetic motor (Tesla)

Cycles used for refrigeration

A refrigerator is a heat pump: a heat engine in reverse. Work is used to create a heat differential. Many cycles can run in reverse to move heat from the cold side to the hot side, making the cold side cooler and the hot side hotter. Internal combustion engine versions of these cycles are, by their nature, not reversible.

Evaporative Heat Engines

The Barton Evaporation Engine is a heat engine based on a cycle producing power and cooled moist air from the evaporation of water into hot dry air.

Efficiency

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

From the laws of thermodynamics:

dW = dQ_c - (-dQ_h)

where

dW = -PdV

is the work extracted from the engine. (It is negative since work is done by the engine.)

dQ_h = T_hdS_h

is the heat energy taken from the high temperature system. (It is negative since heat is extracted from the source, hence

(-dQ_h)

is positive.)

dQ_c = T_cdS_c

is the heat energy delivered to the cold temperature system. (It is positive since heat is added 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 delivering the rest to the cold temperature heat sink.

In general, the efficiency of a given heat transfer process (whether it be a refrigerator, a heat pump or an engine) is defined informally by the ratio of "what you get out" to "what you put in."

In the case of an engine, one desires to extract work and puts in a heat transfer.

eta = frac{-dW}{-dQ_h} = frac{-dQ_h - dQ_c}{-dQ_h} = 1 - frac{dQ_c}{-dQ_h}

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, this is because in reversible processes, the change in entropy of the cold reservoir is the negative of that of the hot reservoir (i.e., $dS_c = -dS_h$ ), keeping the overall change of entropy zero. Thus:

eta_text{max} = 1 - frac{T_cdS_c}{-T_hdS_h} = 1 - frac{T_c}{T_h}

where $T_h$ is the absolute temperature of the hot source and $T_c$ that of the cold sink, usually measured in kelvin. Note that $dS_c$ is positive while $dS_h$ is negative; in any reversible work-extracting process, entropy is overall not increased, but rather is moved from a hot (high-entropy) system to a cold (low-entropy one), decreasing the entropy of the heat source and increasing that of the heat sink.

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, the Carnot efficiency is a theoretical upper bound on the reliable efficiency of any process.

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

Here are two plots, Figure 2 and Figure 3, for the Carnot cycle efficiency. One plot indicates how the cycle efficiency changes with an increase in the heat addition temperature for a constant compressor inlet temperature, while the other indicates how the cycle efficiency changes with an increase in the heat rejection temperature for a constant turbine inlet temperature.

Other criteria of heat engine performance

One problem with the ideal Carnot efficiency as a criterion of heat engine performance is the fact that 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 at differing temperatures is irreversible, and therefore the Carnot efficiency expression only applies in the infinitesimal limit. The major problem with that is that the object of most heat engines is to output some sort of power, and infinitesimal power is usually not what is being sought.

A different measure of heat engine efficiency is given by the endoreversible process, which is identical to the Carnot cycle except in that the two processes of heat transfer are not reversible. As derived in Callen (1985), the efficiency for such a process is given by:

eta = 1 - sqrt{frac{T_c}{T_h}}

(Note: Units K or °R)

(Note: This equation is quite frequently traced to a paper by F.L. Curzon and B. Ahlborn, American Journal of Physics, vol. 43, pp. 22-24 (1975). The book by Herbert Callen probably copied from this paper. In a 1996 review paper by Adrian Bejan (J. Appl. Phys., vol. 79, pp. 1191-1218, 1 Feb. 1996), Adrian Bejan pointed out that this equation was also derived by P. Chambadal and I.I. Novikov earlier than Curzon and Ahlborn in the 1950s. Probably, this equation was just re-discovered by Curzon and Ahlborn in 1975. Therefore, some scientists call this efficiency the Chambadal-Novikov-Curzon-Ahlborn efficiency.)

This model does a better job of predicting how well real-world heat engines can do, as can be seen in the following table (Callen):

(Note: This table appeared in the paper by F.L. Curzon and B. Ahlborn, American Journal of Physics, vol. 43, pp. 22-24 (1975). The book by Herbert Callen probably copied from this paper.)

Efficiencies of Power Plants
Power Plant	$T_c$ (°C)	$T_h$ (°C)	$eta$ (Carnot)	$eta$ (Endoreversible)	$eta$ (Observed)
West Thurrock (UK) coal-fired power plant	25	565	0.64	0.40	0.36
CANDU (Canada) nuclear power plant	25	300	0.48	0.28	0.30
Larderello (Italy) geothermal power plant	80	250	0.33	0.178	0.16

As shown, the endoreversible efficiency much more closely models the observed data.

Heat engine enhancements

Engineers have studied the various heat engine cycles extensively in an effort 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 worked out at least two ways to possibly go around that limit, and one way to get better efficiency without bending any rules.

Increase the temperature difference in the heat engine. The simplest way to do this is to increase the hot side temperature, and is the approach used in modern combined-cycle gas turbines. Unfortunately, NO_x production and material limits (melting the turbine blades) place a hard limit to how hot you can make a workable heat engine. Modern gas turbines are about as hot as they can become and still maintain acceptable NO_x pollution levels. Another way of increasing efficiency is to lower the output temperature. One new method of doing so is to use mixed chemical working fluids, and 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.
Exploit the physical properties of the working fluid. The most common such exploitation is the use of water above the so-called critical point, or so-called supercritical steam. 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 CO₂. SO₂ and xenon have also been considered for such applications, although SO₂ is a little toxic for most.
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 causes it to recombine into N₂O₄. This is then fed back to 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.

Heat engine processes


Cycle/Process	Compression	Heat Addition	Expansion	Heat Rejection
Power cycles normally with external combustion
Carnot	adiabatic	isothermal	adiabatic	isothermal
Stirling	isothermal	isometric	isothermal	isometric
Ericsson	isothermal	isobaric	isothermal	isobaric
Rankine (Steam)	adiabatic	isobaric	adiabatic	isobaric
Stoddard	adiabatic	isobaric	adiabatic	isobaric
Power cycles normally with internal combustion
Otto (Petrol)	adiabatic	isometric	adiabatic	isometric
Diesel	adiabatic	isobaric	adiabatic	isometric
Brayton (Jet)	adiabatic	isobaric	adiabatic	isobaric

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)
adiabatic (no heat is added or removed from the system during adiabatic process which is equivalent to saying that the entropy remains constant)

References

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.