Heat engine

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Heat_engine

Figure 1: Heat engine diagram

Thermodynamics
The classical Carnot heat engine A heat engine is a system that converts heat to usable energy, particularly mechanical energy, which can then be used to do mechanical work. While originally conceived in the context of mechanical energy, the concept of the heat engine has been applied to various other kinds of energy, particularly electrical, since at least the late 19th century. The heat engine does this by bringing a working substance from a higher state temperature to a lower state temperature. A heat source generates thermal energy that brings the working substance to the higher temperature state. The working substance generates work in the working body of the engine while transferring heat to the colder sink until it reaches a lower temperature state. During this process some of the thermal energy is converted into work by exploiting the properties of the working substance. The working substance can be any system with a non-zero heat capacity, but it usually is a gas or liquid. During this process, some heat is normally lost to the surroundings and is not converted to work. Also, some energy is unusable because of friction and drag. In general, an engine is any machine that converts energy to mechanical work. Heat engines distinguish themselves from other types of engines by the fact that their efficiency is fundamentally limited by Carnot's theorem of thermodynamics. Although this efficiency limitation can be a drawback, an advantage of heat engines is that most forms of energy can be easily converted to heat by processes like exothermic reactions (such as combustion), nuclear fission, absorption of light or energetic particles, friction, dissipation and resistance. Since the heat source that supplies thermal energy to the engine can thus be powered by virtually any kind of energy, heat engines cover a wide range of applications.

Heat engines are often confused with the cycles they attempt to implement. Typically, the term "engine" is used for a physical device and "cycle" for the models.

Overview

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

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.

Examples

Although some cycles have a typical combustion location (internal or external), they can often be implemented with the other. For example, John Ericsson 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.

Everyday examples

Everyday examples of heat engines include the thermal power station, internal combustion engine, firearms, refrigerators and heat pumps. Power stations are examples of heat engines run in a forward direction in which heat flows from a hot reservoir and flows into a cool reservoir to produce work as the desired product. Refrigerators, air conditioners and heat pumps are examples of heat engines that are run in reverse, i.e. they use work to take heat energy at a low temperature and raise its temperature in a more efficient way than the simple conversion of work into heat (either through friction or electrical resistance). Refrigerators remove heat from within a thermally sealed chamber at low temperature and vent waste heat at a higher temperature to the environment and heat pumps take heat from the low temperature environment and 'vent' it into a thermally sealed chamber (a house) at higher temperature.

In general heat engines exploit the thermal properties associated with the expansion and compression of gases according to the gas laws or the properties associated with phase changes between gas and liquid states.

Earth's heat engine

Earth's atmosphere and hydrosphere—Earth's heat engine—are coupled processes that constantly even out solar heating imbalances through evaporation of surface water, convection, rainfall, winds and ocean circulation, when distributing heat around the globe.

A Hadley cell is an example of a heat engine. It involves the rising of warm and moist air in the earth's equatorial region and the descent of colder air in the subtropics creating a thermally driven direct circulation, with consequent net production of kinetic energy.

Phase-change cycles

In phase change cycles and engines, the working fluids are gases and liquids. The engine converts the working fluid from a gas to a liquid, from liquid to gas, or both, generating work from the fluid expansion or compression.

• Rankine cycle (classical steam engine)
• Regenerative cycle (steam engine more efficient than Rankine cycle)
• Organic Rankine cycle (Coolant changing phase in temperature ranges of ice and hot liquid water)
• Vapor to liquid cycle (drinking bird, injector, Minto wheel)
• Liquid to solid cycle (frost heaving – water changing from ice to liquid and back again can lift rock up to 60 cm.)
• Solid to gas cycle (firearms – solid propellants combust to hot gases.)

Gas-only cycles

In these cycles and engines the working fluid is always a gas (i.e., 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 (e.g. gasoline/petrol engine)
  • Diesel cycle (e.g. 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 (Miller engine)

Liquid-only cycles

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

• Stirling cycle (Malone engine)

Electron cycles

• Johnson thermoelectric energy converter
• Thermoelectric (Peltier–Seebeck effect)
• Thermogalvanic cell
• Thermionic emission
• Thermotunnel cooling

Magnetic cycles

• Thermo-magnetic motor (Tesla)

Cycles used for refrigeration

Main article: Refrigeration

A domestic refrigerator is an example of 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.

Refrigeration cycles include:

• Air cycle machine
• Gas-absorption refrigerator
• Magnetic refrigeration
• Stirling cryocooler
• Vapor-compression refrigeration
• Vuilleumier cycle

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.

Mesoscopic heat engines

Mesoscopic heat engines are nanoscale devices that may serve the goal of processing heat fluxes and perform useful work at small scales. Potential applications include e.g. electric cooling devices. In such mesoscopic heat engines, work per cycle of operation fluctuates due to thermal noise. There is exact equality that relates average of exponents of work performed by any heat engine and the heat transfer from the hotter heat bath. This relation transforms the Carnot's inequality into exact equality. This relation is also a Carnot cycle equality

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:

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

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 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:

${\displaystyle \mathrm{\Delta }{S}_h+\mathrm{\Delta }{S}_c={\mathrm{\Delta }}_{cycle}S=0}$

Note that ${\displaystyle \mathrm{\Delta }{S}_h}$ is positive because isothermal expansion in the power stroke increases the multiplicity of the working fluid while ${\displaystyle \mathrm{\Delta }{S}_c}$ is negative since recompression decreases the multiplicity. If the engine is ideal and runs reversibly, ${\displaystyle Q_h=T_h\mathrm{\Delta }{S}_h}$ and ${\displaystyle Q_c=T_c\mathrm{\Delta }{S}_c}$, and thus

${\displaystyle Q_h/T_h+Q_c/T_c=0}$,

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, 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: ${\displaystyle T_h^{\prime }}$ and ${\displaystyle T_c^{\prime }}$. The heat transfers between the reservoirs and the substance are considered as conductive (and irreversible) in the form ${\displaystyle d{Q}_{h,c}/dt=\alpha (T_{h,c}-T_{h,c}^{\prime })}$. 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, ${\displaystyle T\approx T^{\prime }}$ 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

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.

History

Main article: Timeline of heat engine technology

See also: History of the internal combustion engine and History of thermodynamics

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.

Enhancements

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.

Heat engine processes

Cycle	Compression, 1→2	Heat addition, 2→3	Expansion, 3→4	Heat rejection, 4→1	Notes
Power cycles normally with external combustion - or heat pump cycles:
Bell Coleman	adiabatic	isobaric	adiabatic	isobaric	A reversed Brayton cycle
Carnot	isentropic	isothermal	isentropic	isothermal	Carnot heat engine
Ericsson	isothermal	isobaric	isothermal	isobaric	The second Ericsson cycle from 1853
Rankine	adiabatic	isobaric	adiabatic	isobaric	Steam engines
Hygroscopic	adiabatic	isobaric	adiabatic	isobaric
Scuderi	adiabatic	variable pressure and volume	adiabatic	isochoric
Stirling	isothermal	isochoric	isothermal	isochoric	Stirling engines
Manson	isothermal	isochoric	isothermal	isochoric then adiabatic	Manson and Manson-Guise engines
Stoddard	adiabatic	isobaric	adiabatic	isobaric
Power cycles normally with internal combustion:
Atkinson	isentropic	isochoric	isentropic	isochoric	Differs from Otto cycle in that V1 < V4.
Brayton	adiabatic	isobaric	adiabatic	isobaric	Ramjets, turbojets, -props, and -shafts. Originally developed for use in reciprocating engines. The external combustion version of this cycle is known as the first Ericsson cycle from 1833.
Diesel	adiabatic	isobaric	adiabatic	isochoric	Diesel engine
Humphrey	isentropic	isochoric	isentropic	isobaric	Shcramjets, pulse- and continuous detonation engines
Lenoir		isochoric	adiabatic	isobaric	Pulse jets. 1→2 accomplishes both the heat rejection and the compression. Originally developed for use in reciprocating engines.
Otto	isentropic	isochoric	isentropic	isochoric	Gasoline / petrol engines

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)

Carnot heat engine

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Carnot_heat_engine 

Axial cross section of Carnot's heat engine. In this diagram, abcd is a cylindrical vessel, cd is a movable piston, and A and B are constant–temperature bodies. The vessel may be placed in contact with either body or removed from both (as it is here).

A Carnot heat engine is a theoretical heat engine that operates on the Carnot cycle. The basic model for this engine was developed by Nicolas Léonard Sadi Carnot in 1824. The Carnot engine model was graphically expanded by Benoît Paul Émile Clapeyron in 1834 and mathematically explored by Rudolf Clausius in 1857, work that led to the fundamental thermodynamic concept of entropy. The Carnot engine is the most efficient heat engine which is theoretically possible. The efficiency depends only upon the absolute temperatures of the hot and cold heat reservoirs between which it operates.

A heat engine acts by transferring energy from a warm region to a cool region of space and, in the process, converting some of that energy to mechanical work. The cycle may also be reversed. The system may be worked upon by an external force, and in the process, it can transfer thermal energy from a cooler system to a warmer one, thereby acting as a refrigerator or heat pump rather than a heat engine.

Every thermodynamic system exists in a particular state. A thermodynamic cycle occurs when a system is taken through a series of different states, and finally returned to its initial state. In the process of going through this cycle, the system may perform work on its surroundings, thereby acting as a heat engine.

The Carnot engine is a theoretical construct, useful for exploring the efficiency limits of other heat engines. An actual Carnot engine, however, would be completely impractical to build.

Carnot's diagram

In the adjacent diagram, from Carnot's 1824 work, Reflections on the Motive Power of Fire, there are "two bodies A and B, kept each at a constant temperature, that of A being higher than that of B. These two bodies to which we can give, or from which we can remove the heat without causing their temperatures to vary, exercise the functions of two unlimited reservoirs of caloric. We will call the first the furnace and the second the refrigerator." Carnot then explains how we can obtain motive power, i.e., "work", by carrying a certain quantity of heat from body A to body B. It also acts as a cooler and hence can also act as a refrigerator.

Modern diagram

Carnot engine diagram (modern) - where an amount of heat Q_H flows from a high temperature T_H furnace through the fluid of the "working body" (working substance) and the remaining heat Q_C flows into the cold sink T_C, thus forcing the working substance to do mechanical work W on the surroundings, via cycles of contractions and expansions.

The previous image shows the original piston-and-cylinder diagram used by Carnot in discussing his ideal engine. The figure at right shows a block diagram of a generic heat engine, such as the Carnot engine. In the diagram, the "working body" (system), a term introduced by Clausius in 1850, can be any fluid or vapor body through which heat Q can be introduced or transmitted to produce work. Carnot had postulated that the fluid body could be any substance capable of expansion, such as vapor of water, vapor of alcohol, vapor of mercury, a permanent gas, air, etc. Although in those early years, engines came in a number of configurations, typically Q_H was supplied by a boiler, wherein water was boiled over a furnace; Q_C was typically removed by a stream of cold flowing water in the form of a condenser located on a separate part of the engine. The output work, W, is transmitted by the movement of the piston as it is used to turn a crank-arm, which in turn was typically used to power a pulley so as to lift water out of flooded salt mines. Carnot defined work as "weight lifted through a height".

Carnot cycle

Main article: Carnot cycle

Figure 1: A Carnot cycle illustrated on a PV diagram to illustrate the work done.

Figure 2: A Carnot cycle acting as a heat engine, illustrated on a temperature-entropy diagram. The cycle takes place between a hot reservoir at temperature T_H and a cold reservoir at temperature T_C. The vertical axis is temperature, the horizontal axis is entropy.

The Carnot cycle when acting as a heat engine consists of the following steps:

1. Reversible isothermal expansion of the gas at the "hot" temperature, T_H (isothermal heat addition or absorption). During this step (A to B) the gas is allowed to expand and it does work on the surroundings. The temperature of the gas (the system) does not change during the process, and thus the expansion is isothermic. The gas expansion is propelled by absorption of heat energy Q_H and of entropy ΔS_H = Q_H / T_H from the high temperature reservoir.
2. Isentropic (reversible adiabatic) expansion of the gas (isentropic work output). For this step (B to C) the piston and cylinder are assumed to be thermally insulated, thus they neither gain nor lose heat. The gas continues to expand, doing work on the surroundings, and losing an equivalent amount of internal energy. The gas expansion causes it to cool to the "cold" temperature, T_C. The entropy remains unchanged.
3. Reversible isothermal compression of the gas at the "cold" temperature, T_C (isothermal heat rejection) (C to D). Now the gas is exposed to the cold temperature reservoir while the surroundings do work on the gas by compressing it (such as through the return compression of a piston), while causing an amount of waste heat Q_C < 0 (with the standard sign convention for heat) and of entropy ΔS_C = Q_C/T_C < 0 to flow out of the gas to the low temperature reservoir. (In magnitude, this is the same amount of entropy absorbed in step 1. The entropy decreases in isothermal compression since the multiplicity of the system decreases with the volume.) In terms of magnitude, the recompression work performed by the surroundings in this step is less than the work performed on the surroundings in step 1 because it occurs at a lower pressure due to the lower temperature (i.e. the resistance to compression is lower under step 3 than the force of expansion under step 1). We can refer to the first law of thermodynamics to explain this behavior: ΔU= W+Q .
4. Isentropic compression of the gas (isentropic work input) (D to A). Once again the piston and cylinder are assumed to be thermally insulated and the cold temperature reservoir is removed. During this step, the surroundings continue to do work to further compress the gas and both the temperature and pressure rise now that the heat sink has been removed. This additional work increases the internal energy of the gas, compressing it and causing the temperature to rise to T_H. The entropy remains unchanged. At this point the gas is in the same state as at the start of step 1.

Carnot's theorem

Main article: Carnot's theorem (thermodynamics)

Real ideal engines (left) compared to the Carnot cycle (right). The entropy of a real material changes with temperature. This change is indicated by the curve on a T–S diagram. For this figure, the curve indicates a vapor-liquid equilibrium (See Rankine cycle). Irreversible systems and losses of heat (for example, due to friction) prevent the ideal from taking place at every step.

Carnot's theorem is a formal statement of this fact: No engine operating between two heat reservoirs can be more efficient than a Carnot engine operating between the same reservoirs.

$${\displaystyle {\eta }_I=\frac{W}{Q_{\mathrm{H}}}=1-\frac{T_{\mathrm{C}}}{T_{\mathrm{H}}}}$$

Explanation

This maximum efficiency η_I is defined as above:

• W is the work done by the system (energy exiting the system as work),
• Q_H is the heat put into the system (heat energy entering the system),
• T_C is the absolute temperature of the cold reservoir, and
• T_H is the absolute temperature of the hot reservoir.

A corollary to Carnot's theorem states that: All reversible engines operating between the same heat reservoirs are equally efficient.

It is easily shown that the efficiency η is maximum when the entire cyclic process is a reversible process. This means the total entropy of system and surroundings (the entropies of the hot furnace, the "working fluid" of the heat engine, and the cold sink) remains constant when the "working fluid" completes one cycle and returns to its original state. (In the general and more realistic case of an irreversible process, the total entropy of this combined system would increase.)

Since the "working fluid" comes back to the same state after one cycle, and entropy of the system is a state function, the change in entropy of the "working fluid" system is 0. Thus, it implies that the total entropy change of the furnace and sink is zero, for the process to be reversible and the efficiency of the engine to be maximum. This derivation is carried out in the next section.

The coefficient of performance (COP) of the heat engine is the reciprocal of its efficiency.

Efficiency of real heat engines

For a real heat engine, the total thermodynamic process is generally irreversible. The working fluid is brought back to its initial state after one cycle, and thus the change of entropy of the fluid system is 0, but the sum of the entropy changes in the hot and cold reservoir in this one cyclical process is greater than 0.

The internal energy of the fluid is also a state variable, so its total change in one cycle is 0. So the total work done by the system W is equal to the net heat put into the system, the sum of ${\displaystyle Q_{\text{H}}}$ > 0 taken up and the waste heat ${\displaystyle Q_{\text{C}}}$ < 0 given off:

${\displaystyle W=Q=Q_{\text{H}}+Q_{\text{C}}}$

(2)

For real engines, stages 1 and 3 of the Carnot cycle, in which heat is absorbed by the "working fluid" from the hot reservoir, and released by it to the cold reservoir, respectively, no longer remain ideally reversible, and there is a temperature differential between the temperature of the reservoir and the temperature of the fluid while heat exchange takes place.

During heat transfer from the hot reservoir at ${\displaystyle T_{\text{H}}}$ to the fluid, the fluid would have a slightly lower temperature than ${\displaystyle T_{\text{H}}}$, and the process for the fluid may not necessarily remain isothermal. Let ${\displaystyle \mathrm{\Delta }{S}_{\text{H}}}$ be the total entropy change of the fluid in the process of intake of heat.

${\displaystyle \mathrm{\Delta }{S}_{\text{H}}={\int }_{Q_{\text{in}}}\frac{\text{d}{Q}_{\text{H}}}{T}}$

(3)

where the temperature of the fluid T is always slightly lesser than ${\displaystyle T_{\text{H}}}$, in this process.

So, one would get:

${\displaystyle \frac{Q_{\text{H}}}{T_{\text{H}}}=\frac{\int \text{d}{Q}_{\text{H}}}{T_{\text{H}}}\le \mathrm{\Delta }{S}_{\text{H}}}$

(4)

Similarly, at the time of heat injection from the fluid to the cold reservoir one would have, for the magnitude of total entropy change ${\displaystyle \mathrm{\Delta }{S}_{\text{C}}}$< 0 of the fluid in the process of expelling heat:

${\displaystyle \mathrm{\Delta }{S}_{\text{C}}⩾\frac{Q_{\text{C}}}{T_{\text{C}}}<0}$

(5)

where, during this process of transfer of heat to the cold reservoir, the temperature of the fluid T is always slightly greater than ${\displaystyle T_{\text{C}}}$.

We have only considered the magnitude of the entropy change here. Since the total change of entropy of the fluid system for the cyclic process is 0, we must have

${\displaystyle \mathrm{\Delta }{S}_{\text{H}}+\mathrm{\Delta }{S}_{\text{C}}=\mathrm{\Delta }{S}_{\text{cycle}}=0}$

(6)

The previous three equations, namely (3), (4), (5), substituted into (6) to give:

${\displaystyle -\frac{Q_{\text{C}}}{T_{\text{C}}}⩾\frac{Q_{\text{H}}}{T_{\text{H}}}}$

(7)

For [ΔSh ≥ (Qh/Th)] +[ΔSc ≥ (Qc/Tc)] = 0

[ΔSh ≥ (Qh/Th)] = - [ΔSc ≥ (Qc/Tc)]

= [-ΔSc ≤ (-Qc/Tc)]

it is at least (Qh/Th) ≤ (-Qc/Tc)

Equations (2) and (7) combine to give

${\displaystyle \frac{W}{Q_{\text{H}}}\le 1-\frac{T_{\text{C}}}{T_{\text{H}}}}$

(8)

To derive this step needs two adiabatic processes involved to show an isentropic process property for the ratio of the changing volumes of two isothermal processes are equal.

Most importantly, since the two adiabatic processes are volume works without heat lost, and since the ratio of volume changes for this two processes are the same, so the works for these two adiabatic processes are the same with opposite direction to each other, namely, one direction is work done by the system and the other is work done on the system; therefore, heat efficiency only concerns the amount of work done by the heat absorbed comparing to the amount of heat absorbed by the system.

Therefore, (W/Qh) = (Qh - Qc) / Qh

= 1 - (Qc/Qh)

= 1 - (Tc/Th)

And, from (7)

(Qh/Th) ≤ (-Qc/Tc) here Qc it is less than 0 (release heat)

(Tc/Th) ≤ (-Qc/Qh)

-(Tc/Th) ≥ (Qc/Qh)

1+[-(Tc/Th)] ≥ 1+(Qc/Qh)

1 - (Tc/Th) ≥ (Qh + Qc)/Qh here Qc<0,

1 - (Tc/Th) ≥ (Qh - Qc)/Qh

1 - (Tc/Th) ≥ W/Qh

Hence,

${\displaystyle \eta \le {\eta }_{\text{I}}}$

(9)

where ${\displaystyle \eta =\frac{W}{Q_{\text{H}}}}$ is the efficiency of the real engine, and ${\displaystyle {\eta }_{\text{I}}}$ is the efficiency of the Carnot engine working between the same two reservoirs at the temperatures ${\displaystyle T_{\text{H}}}$ and ${\displaystyle T_{\text{C}}}$. For the Carnot engine, the entire process is 'reversible', and Equation (7) is an equality. Hence, the efficiency of the real engine is always less than the ideal Carnot engine.

Equation (7) signifies that the total entropy of system and surroundings (the fluid and the two reservoirs) increases for the real engine, because (in a surroundings-based analysis) the entropy gain of the cold reservoir as ${\displaystyle Q_{\text{C}}}$ flows into it at the fixed temperature ${\displaystyle T_{\text{C}}}$, is greater than the entropy loss of the hot reservoir as ${\displaystyle Q_{\text{H}}}$ leaves it at its fixed temperature ${\displaystyle T_{\text{H}}}$. The inequality in Equation (7) is essentially the statement of the Clausius theorem.

According to the second theorem, "The efficiency of the Carnot engine is independent of the nature of the working substance".

The Carnot engine and Rudolf Diesel

In 1892 Rudolf Diesel patented an internal combustion engine inspired by the Carnot engine. Diesel knew a Carnot engine is an ideal that cannot be built, but he thought he had invented a working approximation. His principle was unsound, but in his struggle to implement it he developed a practical Diesel engine.

The conceptual problem was how to achieve isothermal expansion in an internal combustion engine, since burning fuel at the highest temperature of the cycle would only raise the temperature further. Diesel's patented solution was: having achieved the highest temperature just by compressing the air, to add a small amount of fuel at a controlled rate, such that heating caused by burning the fuel would be counteracted by cooling caused by air expansion as the piston moved. Hence all the heat from the fuel would be transformed into work during the isothermal expansion, as required by Carnot's theorem.

For the idea to work a small mass of fuel would have to be burnt in a huge mass of air. Diesel first proposed a working engine that would compress air to 250 atmospheres at 800 °C (1,450 °F), then cycle to one atmosphere at 20 °C (50 °F). However, this was well beyond the technological capabilities of the day, since it implied a compression ratio of 60:1. Such an engine, if it could have been built, would have had an efficiency of 73%. (In contrast, the best steam engines of his day achieved 7%.)

Accordingly, Diesel sought to compromise. He calculated that, were he to reduce the peak pressure to a less ambitious 90 atmospheres, he would sacrifice only 5% of the thermal efficiency. Seeking financial support, he published the "Theory and Construction of a Rational Heat Engine to Take the Place of the Steam Engine and All Presently Known Combustion Engines" (1893). Endorsed by scientific opinion, including Lord Kelvin, he won the backing of Krupp and Maschinenfabrik Augsburg. He clung to the Carnot cycle as a symbol. But years of practical work failed to achieve an isothermal combustion engine, nor could have done, since it requires such an enormous quantity of air that it cannot develop enough power to compress it. Furthermore, controlled fuel injection turned out to be no easy matter.

Even so, the Diesel engine slowly evolved over 25 years to become a practical high-compression air engine, its fuel injected near the end of the compression stroke and ignited by the heat of compression, capable by 1969 of 40% efficiency.

As a macroscopic construct

The Carnot heat engine is, ultimately, a theoretical construct based on an idealized thermodynamic system. On a practical human-scale level the Carnot cycle has proven a valuable model, as in advancing the development of the diesel engine. However, on a macroscopic scale limitations placed by the model's assumptions prove it impractical, and, ultimately, incapable of doing any work. As such, per Carnot's theorem, the Carnot engine may be thought as the theoretical limit of macroscopic scale heat engines rather than any practical device that could ever be built.

For example, for the isothermal expansion part of the Carnot cycle, the following infinitesimal conditions must be satisfied simultaneously at every step in the expansion:

• The hot reservoir temperature T_H is infinitesimally higher than the system gas temperature T so heat flow (energy transfer) from the hot reservoir to the gas is made without increasing T (via infinitesimal work on the surroundings by the gas as another energy transfer); if T_H is significantly higher than T, then T may be not uniform through the gas so the system would deviate from thermal equilibrium as well as not being a reversible process (i.e. not a Carnot cycle) or T might increase noticeably so it would not be an isothermal process.
• The force externally applied on the piston (opposite to the internal force on the piston by the gas) needs to be infinitesimally reduced externally. Without this assistance, it would not be possible to follow a gas PV (Pressure-Volume) curve downward at a constant T since following this curve means that the gas-to-piston force decreases (P decreases) as the volume expands (the piston moves outward). If this assistance is so strong that the volume expansion is significant, the system may deviate from thermal equilibrium, and the process fail to be reversible (and thus not a Carnot cycle).

Such "infinitesimal" requirements as these (and others) cause the Carnot cycle to take an infinite amount of time, rendering the production of work impossible.

Other practical requirements that make the Carnot cycle impractical to realize include fine control of the gas, and perfect thermal contact with the surroundings (including high and low temperature reservoirs).