# Heat engine

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Heat engine in engineering and thermodynamics[[1]] , 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 to the sink from the source, and in this process some of the heat is converted into work.

## Everyday examplesEdit

Examples of everyday heat engines include: the steam engine, the diesel engine, and the gasoline (petrol) engine in an automobile. 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.

## Engineering and physical conceptsEdit

Examples of heat engines:

## EfficiencyEdit

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[[14]] :

$dW \ = \ dQ_c \ - \ (-dQ_h)$
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" 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 due to the fact that 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:

$\eta_{max} = 1 - \frac{T_cdS_c}{-T_hdS_h} \equiv 1 - \frac{T_c}{T_h}$

where $T_h$ is the absolute temperature of the hot source and $T_c$ that of the cold sink, usually measured in kelvins. 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[[15]] , this is forbidden, the Carnot efficiency is a theoretical upper bound on the efficiency of any process.

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

## Other criteria of heat engine performanceEdit

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 due to the fact that 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 much more accurate 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 treated as reversible. As derived in Callen (1985), the efficiency for such a process is given by:

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

The accuracy of this model can be seen in the following table (Callen):

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 0.32 0.175 0.16

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

## Heat engine processesEdit

Cycle/Process Compression Heat Addition Expansion Heat Rejection
Stirling isothermal isometric isothermal isometric
Ericsson isothermal isobaric isothermal isobaric

Each process is one of the following:

• isothermal pphttp://en.wikipedia.org/wiki/Isothermal]] (at constant temperature, maintained with heat added or removed from a heat source or sink)
• isobaric [[16]] (at constant pressure)
• isochoric process [[17]] (at constant volume)
• adiabatic [[18]] (no heat is added or removed from the working fluid)