Heat engine

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 to the sink from the source, 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).

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

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 (more efficient than Rankine cycle)
Drinking bird cycle
Frost heaving - water changing from ice to liquid and back again can lift rock up to 60m.

Gas only cycles

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

Carnot cycle (Carnot heat engine)
Brayton cycle or Joule cycle (Gas turbine)
Ericsson Cycle
Stirling cycle (Stirling engine)
Internal combustion engine (ICE):
- Otto cycle (eg. Gasoline/Petrol engine, high-speed diesel engine)
- Diesel cycle (eg. low-speed diesel engine)
- Atkinson Cycle
- Lenoir cycle (eg pulse jet engine)
- Miller cycle

Electron cycles

Photon cycles

Solar sail

Cycles used for refrigeration

A refrigerator is a heat pump: a heat engine in reverse. Work is used to create a heat differential.

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" 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, 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 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 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 processes


Cycle/Process	Compression	Heat Addition	Expansion	Heat Rejection
Carnot	adiabatic	isothermal	adiabatic	isothermal
Otto (Petrol)	adiabatic	isometric	adiabatic	isometric
Diesel	adiabatic	isobaric	adiabatic	isometric
Brayton (Jet)	adiabatic	isobaric	adiabatic	isobaric
Stirling	isothermal	isometric	isothermal	isometric
Ericsson	isothermal	isobaric	isothermal	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 working fluid)

References

External links

Heat Engine
Webarchive backup: Refrigeration Cycle Citat: "...The refrigeration cycle is basically the Rankine cycle run in reverse..."
Red Rock Energy Solar Heliostats: Heat Engine Projects Citat: "...Choosing a Heat Engine..."
Overview of heat engine types
The rotary piston array machine

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