# Lecture 5: Heat Engines, Refrigerators, and the Carnot Engine

Don't Panic

Sections

A UV series of images of the Sun with an eruptive prominence.

An eruptive prominence is not the same as a coronal mass ejection: it is probably much less powerful. The blob can still make it to Earth. This is an extreme ultraviolet image from the SOHO probe and so the color is false.

Sol Invictus: The unconquered Sun.

The energy source for almost all of the biosphere.

The Sun's diameter is about 100 = 10**2 times the Earth's diameter.

The Sun's mass is about 3*10**5 times the Earth's mass.

Credit: NASA.

# Heat Engines

A heat engine is a machine that operates in a cycle.

Just a cute gym gif that illustrates a cycle.

Actually, a part of the cycle is off-screen.

Credit: Unknown to me, but I trust he/she is not aggrieved that I publicly display his/her wares.

Download site: None. The gym.gif was just sent me by someone and the details are long-forgotten.

Examples are steam engines, steam turbines (which are a kind of steam engine, but not so called usually), and internal combustion engines (ICEs)

A part of the machine is the working fluid (e.g., water in steam engine).

Caption: The aeolipile of Hero of Alexandria (c. 10--70 CE).

The high-pressure steam from the heated vessel flows out the pipes on the rotating sphere.

The reaction force of the steam on the outflow pipes drives the rotation which could be used for useful work of some kind.

The sphere rotates and in that sense there is a cycle. But the working fluid does not run in a cycle. But one could imagine recapturing it and feeding it back into the heated vessel to complete a cycle.

Really, there is a cycle. One feeds in water that's part of the whole water cycle and it returns to the water cycle as steam.

Credit: Knight's American Mechanical Dictionary, 1876. Uploaded by User: Quadell.

Linked source: Wikipedia image http://en.wikipedia.org/wiki/Image:Aeolipile_illustration.JPG.

Public domain.

Caption: "Work input on a working fluid by means of a cylinder-piston arrangement."

Credit: User Dmic0001 in 2008.

Linked source: Wikipedia image http://en.wikipedia.org/wiki/Image:Piston_cylinder.jpg.

Public domain.

The working fluid absorbs heat Q_H from a HOT BATH and changes some of it into macroscopic work W and rejects some heat Q_C to a COLD BATH.

The equation relating these quantities is dead simple:

# Q_H = Q_C + W

This is because the time averaged thermal energy of the working fluid is a constant: i.e.,

# 0 = (Delta E) = Q - W = Q_H - Q_C - W

where Q=C_H-Q_C is the net heat into the system and W is the work done by the system.

We are just obeying the law of the conservation of energy or the first law of thermodynamics.

Recall

# (Delta E) = Q - W

is the formula expression of this law for the cases we consider.

Ideally, the heat engine repeats the absorption-work-rejection process over and over again without change on average over time: i.e., it operates in a cycle.

To be even more precise, the exact state of the engine at any instant repeats one cycle time period later.

This is the ideal heat engine. In real heat engines periodic behavior can only be approximate.

For this to happen, of course, the thermal energy of the HOT BATH must be replenished to keep the HOT BATH on average unchanging.

And the excess thermal energy of the COLD BATH must be disposed of to keep the COLD BATH on average unchanging..

And the work W must go someplace: e.g., spinning car wheels or turning the turbine in an electric generator.

Here is a schematic heat engine.

A heat engine must reject enough heat per cycle to the COLD BATH so that the overall entropy of the whole heat engine system does NOT decrease on average over a cycle.

Note the replenishment of thermal energy to the HOT BATH and disposal of thermal energy from COLD BATH keeps the heat engine unchanging thermodynamically (i.e., which includes keeping the entropy constant) averaged over multiple cycles.

Entropy of the of the heat engine and the environment must increase or at least stay constant.

Question: The ``must reject'' is because:

1. engine designers work hard to obey the laws.
2. is a lie. Heat engines don't have to reject any heat at all.
3. there is no physical way that heat engines can evade the second law of thermodynamics. They will always reject enough heat to cause the overall entropy of the whole heat engine system increase or at least stay constant.
4. All of the above.

Answer 3 is right.

The mathematical demonstration is beyond this course---and currently beyond me---but it can be shown that heat engines do as answer 3 says.

The unavoidable rejection of heat to the COLD BATH (which is a consequence of the second law of thermodynamics) has frustrated many.

Actual heat engines usually reject far more than the mimimum possible heat to the COLD BATH.

They have less than the ideal maximum engine efficiency.

The efficiency of heat engine is defined by the equation:

# F_eff = W / Q_H = ( Q_H - Q_C )/ Q_H = 1- Q_C/Q_H ,

where the second expression just follows from our earlier result from the first law of thermodynamics: on time average thermal energy in the working fluid is a constant: thus (Delta E)=0 on time average:

# 0 = (Delta E) = Q - W = Q_H - Q_C - W .

Since Q_H > = Q_C > 0 always, we must have:

# 0 =< F_eff < 1 .

We want, of course to have F_eff = W / Q_H as large as possible:

The most work for the fuel input that maintains the thermal energy of the HOT BATH.

In other words, the most bang for the buck.

It is perfectly possible to build at zero efficiency heat engine.

In fact, many actual heat engine are often run in zero-efficiency mode from time to time.

Except for helping to recharge the battery, your car's internal combustion engine is in zero-efficiency mode when idling.

Efficiencyizers are eager to store up the energy that is otherwise going into waste heat.

Actually the second law of thermodynamics is more stringent than F_eff < 1.

In order for the overall entropy of the whole heat engine system NOT to decrease, it turns out that

# F_eff = 1 - Q_C/Q_H =< F_eff_max = 1 - T_C/T_H ,

where T_C is the COLD BATH temperature and T_H is the HOT BATH temperature.

Those temperatures are Kelvin temperatures.

The F_eff_max = 1 - T_C/T_H is a bit of an idealization in that seldom in real heat engines are there fixed HOT BATH and COLD BATH temperatures.

But average temperatures can be used to find a characteristic F_eff_max = 1 - T_C/T_H.

Most heat engines work at far less than F_eff_max = 1 - T_C/T_H.

Typically, internal combustion engines of cars have an efficency of about 20 % although the hybrid cars (e.g., the Prius) may optimally reach 37 %.

Caption: "2004-2007 Toyota Prius photographed in USA."

The Prius: nothing special to look at, but it gets about 45 miles per gallon which makes it the most fuel-efficient car currently sold in the US.

Of course, in 1991, the standard GM Geo Metro got 60 miles per gallon at least according to the specifications.

Credit: IFCAR

Linked source: Wikipedia image http://en.wikipedia.org/wiki/Image:2nd-Toyota-Prius.jpg.

Public domain.

F_eff_max = 1 - T_C/T_H is often multipled by 100 % to get a percentage efficiency rather than a fraction.

This is often just left as understood in discussions.

I cannot seem to locate a F_eff_max = 1 - T_C/T_H for cars, but we can estimate it.

```
T_C = 300 K which is warmish air by human standards.

The COLD BATH is the ambient medium ultimately.

2000 K  which is a characteristic value adiabatic
value for liquid fuels
(combustion temperatures).

But that sounds way too high for engines.

Iron
melts at 1811 K under standard pressure of about 0.1 MPa (I'd guess).

Faute de mieux say 600 K.

Thus F_eff_max = 1 - T_C/T_H  = very roughly 1 - 300/6000 = 0.50 = 50 %.

Maybe this calculation is badly wrong.

```
Question: Is efficiency the only desideratum for a heat engine? Maybe one would also like:

1. Power (i.e., big energy per unit time: watts galore).
2. Environmental safety.
3. Operating safety
4. All of the above in some good ratio.

There's no absolute right or wrong, but I'd take 4 myself.

Even though most heat engines do not operate at all near F_eff_max = 1 - T_C/T_H, there is usually a gain if you can make the HOT BATH hotter.

Practicality and safety limit us.

Fuels only burn so hot and engines can only take so much heat before they melt.

One can't usually do much about the COLD BATH. It's the ambient medium. Trying to get a colder COLD BATH would require refrigeration and that would cost energy.

The AMBIENT MEDIUM is what nature has given us.

# How Heat Engines Work In Outline

Well steam turbines are most easy to understand in bare outline.

Some heat source (e.g., burning coal, burning natural gas, nuclear reaction heat generation) heats water and boils it into high-pressure steam.

The high-pressure steam is forced through tubes and through a turbine.

The simplest turbine is just a rotor blade like a windmill.

But more complex turbines with stators to direct the steam flow on the rotar are more efficient depending on the case.

The hot steam loses some energy turning the turbine, but it cannot lose it all.

So the steam is collected and cooled in a condenser and it returns to liquid state.

Cooling the steam is usually done contact with cool water in cooling towers or in more primitive days from rivers and the like.

It's not environmentally sound to dump the hot cooling water back in the natural environment (i.e., rivers and the like).

So power stations with steam turbines usually have cooling towers as one of their most conspicuous and recognizable features.

Of course, my outline here does not cover all the variations and may not even be the best practices, but it will do for now.

Here is a diagram of thermal power station which illustrates my outline.

A diagram of thermal power station

```
Legend by image creator

1. Cooling tower           10. Steam governor valve        19. Superheater
2. Cooling water pump      11. High pressure turbine       20. Forced draught fan
3. Three-phase             12. Deaerator                   21. Reheater
transmission line
4. Unit transformer        13. Feed heater                 22. Air intake
5. Three-phase electrical  14. Coal conveyor               23. Economiser
generator
6. Low pressure turbine    15. Coal hopper                 24. Air preheater
7. Boiler feed pump        16. Pulverised fuel mill        25. Precipitator
8. Condenser               17. Boiler drum                 26. Induced draught fan
9. Intermediate pressure   18. Ash hopper                  27. Chimney stack
turbine

```

A detailed description can be found at the download source Wikipedia: Image:PowerStation2.svg.

Credit: Wikipedia contributor Magicflame. According to Wikipedia permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version.

The steam turbine is a kind of steam engine, of course.

But the traditional reciprocating steam engine is what is often first thought of as a steam engine.

It uses pistons and cylinders like an internal combustion engine.

But the combustion (as with the steam turbine) is external.

The cylinder and the turbine are really the only widespread practical heat engine designs I believe.

The cylinder and the turbine were both hit on long ago.

The cylinder appeared with the first practical steam engines circa 1700 or a bit earlier.

Modern turbines were invented in the 19th century if you don't count windmills. The modern steam turbine was invented in 1884, although precursor devices back to ancient times---i.e., the aeolipile of Hero of Alexandria (c. 10--70 CE).

At some point the steam turbine outperformed the traditional reciprocating steam engine and these are now little used.

Nowadays steam turbines and steam engines (if there are any left) are used primarily for the generation of electrical power.

This may be almost their only use.

# Refrigerators

The words are synonyms as I and many use them and I prefer refrigerator, but Wikipedia likes to use refrigerator for a food storage device only---which is a thing I call a FRIG.

An air conditioner is another kind of refrigerator.

A refrigerator is in many essentials much like a heat engine run in reverse.

But now the WORKING FLUID absorbs heat Q_C from a COLD BATH and by means of some macroscopic work W and rejects some heat Q_H to a HOT BATH.

The equation relating these quantities and expressing energy conservation is the same as before:

# Q_H = Q_C + W

The interpretation is a bit different.

The work W now goes into the WORKING FLUID and is transformed into heat that is rejected to the HOT BATH.

So running a refrigerator takes work (i.e., an ordered energy input).

The gain is that the COLD BATH can be made colder by taking thermal energy out of it or maintained at a cold temperature if heat flows from elsewhere are trying to heat it up.

A schematic diagram of a heat engine.

With the arrows reversed, it is the schematic diagram of a refrigerator.

Legend

1. The box marked T_H is the hot bath: e.g., burning fuel in the cylinder of an internal combustion engine (ICE).
2. The box marked T_C is the cold bath: e.g., usually the ambient medium. Sometime the ambient medium is present indirectly. Nuclear reactors use water as a cold bath, but the water is cooled in cooling towers to something like the ambient medium.
3. The circle is the working fluid: e.g., the air and burning fuel in the cylinder of an ICE.
4. Q_H is heat absorbed from the hot bath into the working fluid.
5. Q_C is heat rejected from the working fluid to the cold bath.
6. W is the work done by the working fluid.

Now for a less schematic schematic diagram of a refrigerator.

A schematic diagram of a refrigerator.

Legend (by image creator: see below)

1. Condenser coil (hot side heat exchanger). The left, red side.
2. Expansion valve (gas expands, cools and liquifies).
3. Evaporator coil (cold side heat exchanger). The right, blue side
4. Compressor.
5. Red = Gas at high pressure and temperature.
6. Pink = Gas at high pressure and reduced temperature.
7. Blue = Liquid at low pressure and greatly reduced temperature.
8. Light Blue = Gas at low pressure and warmer temperature.

Credit: Wikipedia contributor Ilmari Karonen. The creator has put the image in the public domain.

Credit: Wikipedia contributor Ilmari Karonen. The creator has put the image in the public domain.

Let us go through the details of the cycle of a refrigerator. refrigerator

1. In the EVAPORATOR, the working fluid is a mixed phase of low density vapor and liquid.

The pressure is low and this causes evaporative cooling in the EVAPORATOR.

The cooling makes the EVAPORATOR lower in temperature than the surroundings which are, in fact, the COLD BATH (which could be the INSIDE of your FRIG or your air-conditioned house).

Because the EVAPORATOR is colder than the COLD BATH heat spontaneously flows from the COLD BATH to the EVAPORATOR. This cools the COLD BATH. It can take considerable heat flow FROM the COLD BATH to make up the latent heat of vaporization.

The EVAPORATOR is often a coil with fins to facilitate heat inflows from the COLD BATH. Fans may be used to blow the air in the COLD BATH by the fins to accelerate the heat flow.

The EVAPORATOR and COLD BATH are shielded by insulation in the inner region of the refrigerator from the outer region of the refrigerator.

2. The working fluid is being driven by a periodic pressure force and this forces the EVAPORATED WORKING FLUID through a one-way valve in the COMPRESSOR into the outer region of the refrigerator.

There a COMPRESSOR (e.g., a piston and cylinder) compresses the EVAPORATED WORKING FLUID.

The compression actually heats the EVAPORATED WORKING FLUID, but the higher pressure nonetheless leads to condensation when the working fluid is forced through another one-way valve in the CONDENSER.

The COMPRESSOR is also providing---I think---the pressure force to the keep the working fluid flowing throughout the refrigerator.

3. In the CONDENSER, the working fluid loses heat to the cooler surroundings which are in fact the HOT BATH.

The CONDENSER is heating the HOT BATH though this may not be noticed.

The heat flow occurs spontaneously because the working fluid is hotter than the HOT BATH initially.

The heat it loses came both from the COLD BATH and the work done it in the COMPRESSOR.

Because it is high density, the working fluid condenses even at the temperature of the HOT BATH or higher.

The working fluid may reach about the temperature of the HOT BATH before leaving the CONDENSER.

It can't get lower.

The CONDENSER is often a coil with fins to facilitate heat outflows to the HOT BATH. Fans may be used to blow the air in the HOT BATH by the fins to accelerate the heat flow.

4. The LIQUID WORKING FLUID is then forced through a EXPANSION VALVE or CONSTRICTION.

The EXPANSION VALVE has the effect of changing disordered pressure-causing thermal energy into partially ordered kinetic energy of a faster flowing LIQUID WORKING FLUID.

This lowers the pressure significantly.

The low pressure LIQUID WORKING FLUID is now back in the EVAPORATOR and is unstable toward evaporation because of its low pressure.

The LIQUID WORKING FLUID will now begin the cycle again.

I don't know of any refrigerators that work in with continuous constant flow, but they may exist.

But in many designs the COMPRESSOR has a piston that has be pushed in and pulled out in cycle.

So the working fluid flow has to be stopped and started.

This may be the cause of some FRIG and air conditioner noises. However, fans may cause much of the noise.

Also so much cooling can be achieved in a few cycles that continuous cycling is often not needed, and so FRIGS and air conditioners are often in a rest mode.

To be practical for a moment, what is the working fluid in a practical FRIG or air conditioner?

It has to be substance that can be condensed and evaporated over a large range of temperatures.

This enables the FRIG or air conditioner to operate under many conditions.

The substance shouldn't be toxic nor chemically active: we don't want leaks to poison us nor reactions to clog to foul up the inner workings.

For many years, chlorofluorcarbons (CFCs) were used: Freon is a DuPont trade name for CFCs.

But leakage of CFCs from frigs, air conditioners, and other sources turned out to be destructive of the ozone layer.

The ozone layer (at an altitude of approximately 15--35 km) protects the biosphere from dangerous solar ultraviolet light.

Caption: "Levels of ozone at various altitudes, and related blocking of several types of ultraviolet radiation."

A DU is a Dobson unit is a bizarre unit used in ozone science.

A DU/km is a ozone number density: i.e., ozone molecules per unit volume.

The volume unit, however, is very strange---and people think astronomers use wacko units.

Credit: NASA. Posted in 2005

Linked source: Wikipedia image http://en.wikipedia.org/wiki/Image:Ozone_altitude_UV_graph.jpg.

Public domain at least in USA.

Thus, CFCs are being/have been phased out in favor of hydrofluorocarbons which are much less destructive of the ozone layer.

Hydrofluorocarbons are NOT as good as CFCs in other respects.

# Refrigerator Performance

The permformance of refrigerators is usually measured using the coefficient of performance (COP) which is just 1/F_eff = Q_H/W with the meanings of the quantities changed, mutatis mutandis.

I just prefer to the use

# G_eff = W / Q_H = ( Q_H - Q_C )/ Q_H = 1- Q_C/Q_H

where everything is the same as for F_eff, except the interpretation.

The smaller G_eff, the better.

As a neologism, let us call G_eff the reverse efficiency.

The impossible would be to have W=0 and Q_H=Q_C.

This would be a spontaneous flow of heat from COLD to HOT

Spontaneous, because it happens without work done---and this never happens.

Actually, the minimum possible G_eff satisfies

# G_eff = 1 - Q_C/Q_H >= G_eff_min = 1 - T_C/T_H ,

where T_C and T_H are on the Kelvin temperature scale.

So you see a heat engine efficiency is bounded above by

# F_eff_max = 1 - T_C/T_H

and a refrigerator reverse efficiency is bounded below by

# G_eff_min = 1 - T_C/T_H .

We are now at the point where we can discuss the Carnot engine.

# The Carnot Engine

Carnot engine or Carnot cycle was discovered by Sadi Carnot at least in some form if not quite what we know today.

The actual history is beyond my scope.

Nicolas Leonard Sadi Carnot (1796--1832): a pioneer of thermodynamics.

Sadi Carnot in the dress uniform of a student of the Ecole Polytechnique.

Carnot is noted for his discovery of the Carnot engine or Carnot cycle.

The Carnot engine has the most efficient cycle possible for a heat engine or refrigerator.

Credit: Unknown artist to the web. Wikipedia judges its copyright status as uncertain. But it must have been painted no later than the 1820s and surely the artist's copyright is now long expired if it ever exited.

He also had a famous father,

Caption: "Lazare Carnot" with braids and epaulets.

Lazare Carnot (1753--1823), one of the French Revolution leaders. Carnot was called the ``Organizer of Victory''.

Credit: Unknown artist. Uploaded by User: Nk in 2005.

Linked source: Wikipedia image http://en.wikipedia.org/wiki/Image:Lazare_carnot.jpg.

Public domain.

What Carnot imagined (at least the ideal Carnot if not the Carnot of history) was a reversible THERMODYNAMIC ENGINE.

Caption: "Sadi Carnot's pison-and-cylinder diagram from 1824".

A and B may Carnot's HOT BATH and COLD.

Credit: Sadi Carnot (1796--1832). Uploaded by User: Sadi Carnot who is not Sadi Carnot.

Linked source: Wikipedia image http://en.wikipedia.org/wiki/Image:Carnot-engine-1824.png.

Public domain at least in USA.

This machine now is called the Carnot engine.

Run forward, it is a heat engine.

Run in reverse, it is a refrigerator.

The Q_H, Q_C, and W quantities are the same if the Carnot engine is run forward as heat engine or in reverse as refrigerator.

Thus, one has

# F_eff = G_eff = W/Q_H = 1- Q_C/Q_H

for the Carnot engine.

Now for some logic.

Caption: "El libro mas antiguo que se conserva en la Biblioteca Publica del Estado - Biblioteca Provincial de Huelva. Una edicion de la Logica de Aristoteles impresa en Lyon en 1570".

Credit: Biblioteca Huelva and Aristotle (384--322 BCE).

Say you had two identical Carnot engines both operating between the same HOT BATH and COLD BATH.

One operates as a heat engine and the other as a refrigerator.

``Say a diagram would really help here.''

``Give me a break. I'm working on it.''

The work W from the heat engine is used to drive the refrigerator.

Since the Q_H, Q_C, and W quantities are the same for both Carnot engines, the overall effect of the two Carnot engines viewed a one system is NOTHING.

No net heat is removed from the HOT BATH and no net heat is reject to the COLD BATH and no net work is done and the entropy does NOT change.

Now imagine one had a heat engine more efficient than the Carnot heat engine.

It rejects the same heat to the COLD BATH, but takes more heat from the HOT BATH and thus does more work than the Carnot heat engine.

We replace Carnot heat engine with the more efficient heat engine and use its work to drive the Carnot refrigerator engine.

Now what is the overall system doing?

Net heat is extracted from the HOT BATH and net work is done, but no net heat is rejected to the COLD BATH.

In effect, an amount of thermal energy has been transformed entirely into thermodynamic work with no heat rejected to a COLD BATH by a heat engine operating in a cycle.

This violates 2nd law of thermodynamics by causing entropy to increase---as one can show by a calculation we will NOT do.

Now Sadi Carnot came before the word thermodynamics was coined and knew nothing about the formal 2nd law of thermodynamics or entropy as a well defined thermodynamic variable.

But he did know that no one had seen in nature or technology any cycle which turned thermal energy entirely into thermodynamic work with no heat rejected to a COLD BATH.

So Sadi Carnot (at least the ideal Carnot if not the Carnot of history) concluded that a reversible engine (which we call a Carnot engine) had to be the most efficient possible heat engine.

A similar argument shows that the Carnot engine has to be the most efficient refrigerator.

The refrigerator in gory mathematical form---which we omit from any classroom presentation. Now imagine that you had a HYPOTHETICAL FRIG with a lower reverse efficiency, than the Carnot heat engine.

We distinguish the HYPOTHETICAL FRIG quantities by subscript ``hyp''.

You could scale this HYPOTHETICAL FRIG to use the work W of the Carnot heat engine: i.e., set W_hyp=W.

Since it has a lower reverse efficiency:

```
Q_H_hyp > Q_H

and

recall

W = Q_H - Q_C    where W is output

and

where Q_H is absorbed from the HOT BATH and Q_C is rejected to the COLD BATH

and

W_hyp = W = Q_H_hyp - Q_C_hyp  where W is work input

and

where Q_H_hyp is rejected to the HOT BATH and Q_C is absorbed from the COLD BATH.

Subtracting the former from the latter, we find

0 = ( Q_H_hyp - Q_H ) - ( Q_C_hyp - Q_C )

or

( Q_H_hyp - Q_H ) = ( Q_C_hyp - Q_C ) > 0

```
The upshot in words is that no net work is done and yet a finite amount of heat
```
( Q_H_hyp - Q_H ) > 0

```
has been moved from the COLD BATH to the HOT BATH.

From an overall perspective, there is a spontaneous flow of heat from HOT to COLD.

A similar argument shows that if you have a heat engine more efficient than a Carnot engine, then you can turn heat entirely into work without rejecting any net heat to a COLD BATH.

Carnot did not know of the second law of thermodynamics: it was formulated after his day.

But he did know that no one had every seen a spontaneous flow of heat from HOT to COLD nor a heat engine without rejection to a COLD BATH.

The ideal Carnot (if not the Carnot of history) argued that Carnot heat engine must be the most efficient heat engine and the most efficient refrigerator possible.

In the above, arguments our hypothetical heat engine and refrigerator that are more efficient than the Carnot engine DO violate the second law of thermodynamics and cannot exist.

Question: Now I know what you are thinking, can a Carnot engine exist.

1. No. It's a complete myth.
2. Yes. They used everywhere. They are wonders. You just don't hear much about them just like you never hear much about spontaneous human combustion though it happens to tens of people every year (according to David St. Hubbins).
3. Almost. Not quite.

Answer 3 is right.

You can almost build a Carnot engine.

The main trick is to only let heat flows occur when the working fluid is in thermal contact with the HOT BATH and the COLD BATH and only let those flows occur between vanishingly small temperature differences.

Caption: "Pressure-volume (p-V) diagram for the Carnot cycle."

Credit: User Keta in 2006.

Linked source: Wikipedia image http://en.wikipedia.org/wiki/Image:Carnot_cycle_p-V_diagram.svg.

So the Carnot engine is the reversible engine Sadi Carnot imagined.

Thus, the Carnot engine is the most efficient heat engine and refrigerator.

You can't quite build an ideal Carnot engine.

But you can get very close.

Question: If a nearly ideal Carnot heat engine can be built, why are they not widely used?

1. They are dreadfully dangerous.
2. They CANNOT built: the instructor has been lying.
3. They are pretty powerless.
4. All of the above.

A nearly ideal Carnot heat engine must operate very slowly since all the segments must be quasistatic.

In particular, the heat transfers must happen over NEARLY ZERO temperature gradients.

The lower the temperature gradients, the slower the heat transfer---we are just asserting this, but it's true.

Carnot engines do have special experimental uses.

But actually the only one I know of is to measure temperature in some special cases.

You can for example build a Carnot engine using a gas as a working fluid.

They may be mostly small desktop affairs with tubes and cylinders and pistons---just guessing.

The last point leads us to the ideal maximum efficiency and mimimum reverse efficiency we have discussed above.

The Carnot engine as it is reversible has

# F_eff = G_eff= 1 - Q_C/Q_H

and F_eff is the biggest that can be obtained for a heat engine and G_eff is the lowest that can be obtained for a refrigerator.

It can be shown---but we won't do it---that for the Carnot engine the ratio

```
Q_C/Q_H  =  T_C/T_H ,

```
where the temperatures are Kelvin temperatures.

This leads to our results

and

# G_eff_min = 1 - T_C/T_H .

These ideal results can never be quite obtained.

And, of course, people don't even try to get extremely close usually since they don't want the nearly ``powerless'' Carnot engine.

Nevertheless the ideal efficiency results set absolute limits on what is obtainable and guide designers in getting the best they can out heat engines and refrigerators.

For actual heat engines there is often something to be gained by making T_C/T_H as small as one can subject to other desiderata: i.e., safety and high power.