# Carnot efficiency

2009-06-02

For a bit of light entertainment, I have been reading Tom Rogers’ book Insultingly Stupid Movie Physics, which basically covers the same terrain as his entertaining website, though at greater length and with more detail. Of course, one can never entirely escape climate change related information, and the book includes a discussion of Carnot efficiency: the maximum theoretical efficiency with which heat engines can convert thermal energy into useful power.

The efficiency depends on two factors: the high temperature produced using combustion, solar energy, geothermal energy, etc, and the cold temperature where the heat is expended into the surrounding environment:

Efficiency = ( 1 – Cold temperature / Hot temperature ) * 100

This has implications for technologies like the co-generation of heat and power. If the heat source for a power plant is 375°C (648°K) and it is dumping waste heat into 10°C (283°K) outdoor weather, the Carnot efficiency is about 56.3% (the actual efficiency is lower, for various reasons). If, instead, it is dumping the heat into buildings at 25°C (198°K), the Carnot efficiency falls to 54.0%. In a case where the heat source is just 200°C (473°K), the difference between a 10°C cold area and a 25°C cold area cuts the Carnot efficiency from 40.2% to 37.0%. In many cases, cogeneration is still worthwhile, despite the loss of useful electrical or kinetic energy, but it should be appreciated that the redirection is not without cost.

Carnot efficiency also helps explain why waste heat is not always worth capturing. If the temperature difference between the source and an available destination for the thermal energy is not large, there isn’t much useful power that can be produced.

[Update: 4:47pm] Remember to express the temperatures in Degrees Kelvin, by adding 273.15 to the figure in Degrees Celsius.

Milan June 2, 2009 at 3:12 pm

This phenomenon is discussed in David MacKay’s book.

Also, I gather from Wikipedia that the formula provided by Rogers may be a simplification.

Matt June 2, 2009 at 4:28 pm

I don’t know about the validity of this equation. I will comment, though, that you absolutely have to use temperature in Kelvin for it to work:

Imagine your cold temperature is -40 °C and your hot temperature is 0 °C. Applying the equation: -40/0 = whoops! you can’t divide by zero.

Every good student of thermodynamics knows “Don’t hesitate, Kelvinate!”

Milan June 2, 2009 at 4:44 pm

Aha!

It was running a calculation with a negative cold temperature that made me doubt whether the equation told the whole story. I will fix the numbers in the post.

Milan June 2, 2009 at 4:59 pm

Here is a chart I made showing the Carnot efficiencies at different cool area temperatures for systems with heat sources at 200°C and 375°C.

. June 2, 2009 at 5:02 pm

The Third Law of Thermodynamics
Carnot Cycle, Efficiency, Heat Engines, and Absolute Zero

Carnot found that the efficiency of an engine operating on the Carnot cycle depends only on the temperatures of the hot and cold reservoirs. Carnot’s formula gives the efficiency as 1 – Tc/Th. Tc and Th are the temperatures of the cold and hot reservoirs in the Kelvin temperature scale. If Tc is zero, then the efficiency is 1 or 100%.

Matt June 2, 2009 at 5:03 pm

Previously, in the thread about Fuel Efficiency in 2050, I mentioned how engines that run hotter were more efficient. It is exactly the same phenomenon described in this thread.

Milan June 2, 2009 at 5:11 pm

It’s all coming together now. Hurray for science!

Tristan June 2, 2009 at 6:56 pm

Just to be specific, engines running hotter are more efficient because there is a lower, not higher, carnot efficiency in the heat transfer between the expanding gasoline and the cylinder walls, right? Gasoline engines are not heat extraction engines.

Milan June 2, 2009 at 7:11 pm

I don’t think the Carnot efficiency has anything to do with the design of the machine – it’s an upper bound that no machine design can exceed.

Carnot efficiency is: “I have one lump at X˚K and another at Y˚K. With the most perfect machine imaginable, what proportion of the thermal energy could I turn into useful power?”

Tristan June 2, 2009 at 7:36 pm

Car engines don’t turn differences of heat into useful power. They turn the expanding force of a state change into useful power. The expanding force of a gas igniting is already mechanical (i.e. the movement of matter) So, it just turns mechanical force into useful power.

Matt June 2, 2009 at 7:38 pm

I’m not sure I completely understand the question, but maybe I can address it indirectly.

The Carnot cycle is an idealized situation which isn’t really something that is seen in practice because it operates on the principal of reversibility, which simplified means there are no energy losses. That being said, turbine engines, steam engines, gas and diesel engines all have similarities with the carnot engine. Specifically, compressing, heating and expanding gas (as in a gaseous substance, not gasoline) and deriving work from gas. Because of this, the carnot cycle is sort of a model of determining efficiency that does (partially) work for car engines. Therefore running an engine hotter, (or alternatively with a lower outside air temperature) is more efficient for the reasons Carnot says it is.

The part about the cylinder walls is separate from Carnot, because in a car for instance, heat out the cylinder walls winds up in the cooling system and out the radiator as a loss. It is true, though that the cylindrical shape of the engine cylinder is important, because the surface area (through which heat can be lost) to volume ratio (the volume of the gas being what you want to heat) is minimized. It is largely for this reason that rotary engines are inefficient. Their slab sided combustion chambers are hugely inefficient. If you look at the MPGs of a rotary equipped Mazda RX-8, you’ll find it gets around 15MPG (terrible) vs. any sort of similar car that would be in the 20s.

…between the expanding gasoline…
The gasoline is just there as fuel. It is mostly inert nitrogen (from the air) that is doing the actual work, from being heated, on the pistons in a car’s engine.

Tristan June 2, 2009 at 7:39 pm

For instance, a hydro electric dam is not in anyway straightforward way restricted by carnot efficiency. A gasoline engine is more like a hydro dam than a steam engine (depending on whether you include the fact you have to create the steam in your steam engine – if you ignore the energy you put into making the steam then its a mechanical engine – and the proof of this is you can run a steam engine just as easily on cold compressed air).

Matt June 2, 2009 at 7:41 pm

^
And by inert I mean ‘non-combustible.’ Nitrogen is of course reactive, and that’s why it forms NOx emissions.

Matt June 2, 2009 at 7:45 pm

Car engines don’t turn differences of heat into useful power. They turn the expanding force of a state change into useful power. The expanding force of a gas igniting is already mechanical (i.e. the movement of matter) So, it just turns mechanical force into useful power.

You’re mistaken. The air in a gasoline engine expands due to being heated (by burning fuel). If the temperatures inside and outside the combustion chambers were the same, the engine wouldn’t work.

. June 2, 2009 at 7:45 pm

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.

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.

Tristan June 2, 2009 at 7:49 pm

Here’s some good demonstrations of how a steam engine is mechanical. Notice how this: http://www.youtube.com/watch?v=UA8dpkFoq5A&feature=related
is essentially the same as this: http://www.youtube.com/watch?v=twvMO6TCKmo&feature=related

. June 2, 2009 at 7:53 pm

Thermal efficiency >> Heat engines

The second law of thermodynamics puts a fundamental limit on the thermal efficiency of heat engines. Surprisingly, even an ideal, frictionless engine can’t convert anywhere near 100% of its input heat into work. The limiting factors are the temperature at which the heat enters the engine, and the temperature of the environment into which the engine exhausts its waste heat, measured in the absolute Kelvin or Rankine scale.

This limiting value is called the Carnot cycle efficiency because it is the efficiency of an unattainable, ideal, lossless (reversible) engine cycle called the Carnot cycle. No heat engine, regardless of its construction, can exceed this efficiency.

For example, if an automobile engine burns gasoline at a temperature of 1089K and the ambient temperature is 294K then its maximum possible efficiency is: 73.0%.

Milan June 2, 2009 at 8:03 pm

The vacuum cleaner providing power for the beam engine works on a different principle from the steam engine.

The steam engine is turning heat into pressure. The work that can be done by that pressure is limited by Carnot efficiency.

I don’t think Carnot efficiency has relevance in relation to electrical motors, since they work using magnetic fields, not heat-induced pressure.

Given that automobile engines use gasoline to heat air to produce pressure, I think they are subject to Carnot efficiency, as specified in the linked Wikipedia articles.

Tristan June 2, 2009 at 8:12 pm

“If the temperatures inside and outside the combustion chambers were the same, the engine wouldn’t work.”

So, that means, if I place gasoline in a chamber that is already the temperature at which gasoline explodes, there will be no explosion?

What about an engine that ran on the expansive power of the reaction of francium and water? Would it still then be true that if the chamber were already the temperature of the reaction, that no mchanical force would result from combining the two materials?

It just seems wrong to say that every chemical reaction can result in useful energy only because of temperature differential between it and the ambient world.

Also, water expands as it freezes – and it should be in principle possible to extract energy from that. Because in some cases, its possible for the state to change without the temperature changing.

Oh – here’s one: Salt water freezes at a lower temperature than fresh water. I have a big tank of salt water at -5, liquid. The ambient temperature is -5. I remove the salt from the liquid, it freezes, and pushes up a lever. I’ve extracted mechanical work from the state change, and there is no temperature differential anywhere – my body is at -5, the salt is at -5, the water is at -5, all the desalination equipment is all at -5.

Milan June 2, 2009 at 8:20 pm

Desalination works in one of two ways, both very endothermic.

In one, you boil the water, leaving the salt behind, and then re-condense it.

In the other, you push water through a membrane that will not permit salt to pass through.

I sure either process would use more energy than you could get from the expansion of the freezing fresh water.

Milan June 2, 2009 at 8:21 pm

It just seems wrong to say that every chemical reaction can result in useful energy only because of temperature differential between it and the ambient world.

Chemical reactions can also do work through reduction and oxidation, as with batteries. Again, I think that is subject to different rules from those that apply to heat engines.

Milan June 2, 2009 at 8:23 pm

So, that means, if I place gasoline in a chamber that is already the temperature at which gasoline explodes, there will be no explosion?

I think the explosion happens because the combustion of gas produces heat that causes air pressure in a confined space to increase. If the air was already the temperature of burning gasoline, it would not expand further. The breakdown of gasoline itself into gasses might produce some pressure, but I don’t think it would be an explosion.

. June 2, 2009 at 8:24 pm

Redox (shorthand for reduction-oxidation reaction) describes all chemical reactions in which atoms have their oxidation number (oxidation state) changed. This can be either a simple redox process such as the oxidation of carbon to yield carbon dioxide or the reduction of carbon by hydrogen to yield methane (CH4), or it can be a complex process such as the oxidation of sugar in the human body through a series of very complex electron transfer processes.

. June 2, 2009 at 8:27 pm

Electrochemistry >> Fuel cells

Fossil fuels are used in power plants to supply electrical needs, however their conversion into electricity is an inefficient process. The most efficient electrical power plant may only convert about 40% of the original chemical energy into electricity when burned or processed.

To enhance electrical production, scientists have developed fuel cells where combustion is replaced by electrochemical methods, similar to a battery but requiring continuous replenishment of the reactants consumed.

Matt June 2, 2009 at 9:13 pm

So, that means, if I place gasoline in a chamber that is already the temperature at which gasoline explodes, there will be no explosion?

My answer was inelegant because it’s not really a full explanation. In a car engine, at the top of the power stroke the air is already compressed compared to the air on the other side of the piston. This alone would cause the piston to move, but not powerfully, and not enough to run the engine. Furthermore it was compressed from a previous power stroke elsewhere. So technically a piston with air compressed on one side, but at the same temperature with uncompressed air on the other side would move. But that’s not an engine.

A better answer is that the air in the cylinder is pressurized from being heated by burning fuel. The ideal gas law states PV=nRT. n and R are going to be constant for my example, so let’s simplfy this by saying PV=T. Pressure * Volume = Temperature. If you have a cylinder with a piston in it, and you examine the instant the fuel ignites, you can see that as the temperature increase (from burning fuel) the pressure or volume or both is going to also increase. Now say the volume is being held constant by the face of the piston in that split second. Because the volume can’t change, the pressure will. It goes up. This increase in pressure then pushes on the piston, moves it, and the engine runs.

The change of phase, has nothing to do with the mechanical forces in an engine, the volume of gasoline per stroke is so trivial compared to the volume of the cylinder it’s in that a phase change would have no effect. The heat energy it produces is not trivial, however. Like I said previously, most of the work on the piston face is done by nitrogen gas, found in the intake air. Nitrogen and gasoline don’t combust or otherwise react in any meaningful way.

Incidentally, gasoline in a piston engine doesn’t explode, it deflagrates. The distinction is meaningful because when explosions do occur, this is called detonation, aka knock, or ping. It’s detrimental to engines, and if unabated will lead to their rapid demise.

. June 2, 2009 at 9:26 pm

Typical thermodynamic system, showing input from a heat source (boiler) on the left and output to a heat sink (condenser) on the right. Work is extracted, in this case by a series of pistons.

Tristan June 3, 2009 at 1:51 am

Ok, so it turns out I was pretty mistaken. I’m still right about the water thing – the salt won’t work, but in certain situations a change of state can be caused by just mechanical shock. This happens when a liquid is super cooled, or super heated (meaning, raised above or below the temperature it would normally remain a liquid, but without boiling/freezing). At the moment of shock, there could be neutral ambient temperature, and work could still be extracted from the state change. This isn’t interesting, however, it just shows that state changes, like oxidization, are natural reactions that aren’t combustion.

Tristan June 3, 2009 at 1:53 am

“The vacuum cleaner providing power for the beam engine works on a different principle from the steam engine.

The steam engine is turning heat into pressure. The work that can be done by that pressure is limited by Carnot efficiency.”

I meant that I was ignoring about the fact steam is produced, and that there is nothing special about the steam being hot, just that it is pressurized. The fact you can run the steam engine on compressed air shows the steam engine doesn’t care what temperature gas is powering it.

Matt June 3, 2009 at 2:08 am

I’ve thought of a good practical, mechanical example of a phase change being useful in engineering: frangible bolts. If you need to have a bolted connection that you can break off on cue, you can make the bolt out of an alloy that changes crystal structure at relatively low temperature. Then, you wrap the bolt in a heating element and when you need it to break, you turn the heating element on. The change in volume associated with the change in crystal structure causes the bolt to fracture and the connection to break.

Matt June 3, 2009 at 3:25 am

I meant that I was ignoring about the fact steam is produced, and that there is nothing special about the steam being hot, just that it is pressurized. The fact you can run the steam engine on compressed air shows the steam engine doesn’t care what temperature gas is powering it.

Well, sort of. A steam engine is considered an external combustion engine, as opposed to an internal combustion engine. Like you say, the motor (piston and related assembly) will run on compressed air, or any gas under pressure, I suppose. But the motor is only part of the engine. The heat of combustion is still important for building the pressure of the steam in the first place. Additionally, the whole system still has a lot of elements of the carnot cycle in it. For instance, the steam will be of a lower temperature after the piston has extracted work from it.

A lot of devices go back to carnot, for instance air conditioning/refrigeration. The two things to know that are important in refrigeration are that when you compress a gas you heat it, and when you expand it you cool it. Say you compress a gas that is initially at 25˚C and thus heat it to 60˚C. Then, while it’s still compressed, you run it through a radiator, let’s call it an condenser, and cool it back down to 25˚C. Now, you expand it back to it’s original pressure in another sort of radiator, and let’s call this radiator an evaporator. Because the compressed gas has been cooled to 25˚C, when you expand it again, it will cool below that temperature (and its initial temperature), maybe to 2˚C. If you blow a fan over the evaporator, you get cold air.

Next time you pump up your bike tires, put your hand on the cylinder of the pump, I guarantee it will be hot. This is due to the air getting hotter from compressing it.

. June 26, 2009 at 10:23 am

Twist 1.5
Major’s Hill Park
Alex Wyse and Ken Guild 1978

. June 30, 2009 at 3:51 pm

Here are the formulae for the ideal efficiency of a heat pump, that is, the electrical energy required per unit of heat pumped. If we are pumping heat from an outside place at temperature T1 into a place at higher temperature T2, both temperatures being expressed relative to absolute zero (that is, T2, in kelvin, is given in terms of the Celsius temperature Tin, by 273.15 + Tin), the ideal efficiency is:

efficiency = T2 / ( T2 – T1)

If we are pumping heat out from a place at temperature T2 to a warmer exterior at temperature T1, the ideal efficiency is:

efficiency = T2 / (T1 – T2)

These theoretical limits could only be achieved by systems that pump heat infinitely slowly. Notice that the ideal efficiency is bigger, the closer the inside temperature T2 is to the outside temperature T1.

Stevmoon July 8, 2009 at 9:20 pm

I think you’ll find that if you consider that a heat pump is actually a sort of Stirling engine, you will see that efficiency for the heat pump goes down as efficiency of the Stirling goes up… kind of an inverse equation

Milan July 8, 2009 at 9:36 pm

It is logical.

A heat engine makes a hot place and a cold place more similar, and makes energy.

A heat pump makes a hot place and a cold place more different, and requires energy.

. September 11, 2009 at 12:11 am

“As more and more renewable energy enters the grid, it gets increasingly difficult to match supply and demand 24/7. The answer of German power company Lichtblick and Volkswagen is a swarm of 100,000 flexible base-load generators. These fridge-sized CHP (Combined Heat and Power) generators that will be installed in people’s basements in Hamburg starting early next year will feed electricity into the grid and the waste heat into their home’s water/heating. The “ZuhauseKraftwerk” (HomePowerPlant) features a vanilla VW Golf natural-gas engine that generates 20kW electrical and 34 kW heat with an efficiency of 92%. The units are remotely controlled via a mobile network or DSL; they can ramp up in a minute if needed. A water tank ensures that heat is continuously available, while electricity is produced on demand. The swarm will replace two nuclear plants, they say. And your old oil heating needed replacement anyway.”

Milan January 27, 2010 at 7:29 pm

“Don’t hesitate, Kelvinate!”

The surface of the sun is only about twenty times hotter than the
average surface temperature of the Earth: about 6,000°K compared with

People thinking in Celsius would likely get the ratio wildly wrong.

Dan Horn January 27, 2010 at 8:39 pm

Aha! I converted the figures I posted on facebook to Kelvin, but I was certain my result of the sun only being 22 times hotter had to have been wrong. I thought I had missed some unit conversion along the way and had to rethink my whole answer. This makes sense though, because something at 288K actually has a significant amount of kinetic energy.

Milan January 27, 2010 at 9:17 pm

Mark January 28, 2010 at 7:18 am

Surface of the sun (photosphere) is fairly cool at about 6,000 Kelvin. The corona,. on the other hand, which might also be considered as the outer layer of the sun, is anywhere from 1 million to 10 million Kelvin.
So your other Facebook guessers are not necessarily as far off as they might seem!

Why the corona is so much hotter is still not precisely known:
http://en.wikipedia.org/wiki/Corona#Coronal_heating_problem

Milan January 28, 2010 at 9:26 am

Good point.

. March 14, 2010 at 7:04 pm

Technology Quarterly

Heat scavenging
Stealing the heat
Energy: The idea of recycling paper, glass, metal and plastics has become commonplace. New technologies allow heat to be recycled, too

Mar 4th 2010 | From The Economist print edition

“By constructing a computer rack similar to that used in the office test, the researchers were able to provide the greenhouse with badly needed heat. A short while later, the rack was joined by three more racks that today provide the greenhouse with enough heat to cut its gas bills by \$15,600 a year—while simultaneously saving Notre Dame \$38,000 in cooling costs.

Another way to recycle heat that is being explored is to capture infrared with photovoltaic cells similar to those used in solar panels. Photovoltaic cells depend on packets of light (photons) knocking electrons free from atoms. They then employ the electrons so liberated to create a current. Photovoltaic cells are usually most responsive to photons in the visible and ultraviolet parts of the spectrum, but they can also respond to high-frequency infrared photons. Objects at a temperature of 1,000-1,500ºC produce plenty of such photons.

But only those that are travelling at a near-perfect right-angle to the surface of the hot material can escape and travel outwards. Photons travelling at any other angle within the material are reflected back inside when they reach the surface. As a result, photovoltaic cells placed near hot objects have only been able to generate around 0.02 watts per square centimetre. By contrast, photovoltaic cells absorbing sunlight can produce about 20 watts per square centimetre, provided the light is carefully concentrated using mirrors.

So Dr Hagelstein and his colleagues changed the design of the cell, adding tiny metal wires to the usual sandwich of semiconductor materials in order to pick up the liberated electrons and allow them to be carried off to create an electric current. Although the new device is still at an experimental stage, the team’s calculations, published in a paper in the Journal of Applied Physics in November, suggest that it could convert heat to electricity at a rate of 100 watts per square centimetre. Installed on a laptop, it could recycle heat from the microprocessor and extend running time by around 20%. One way or another, it seems likely that the abundant reservoirs of waste heat are about to be tapped.”

. May 13, 2010 at 5:00 pm

He saw very clearly, intuitively, that he could give very definite answers to the two questions set before the reader. The Carnot cycle is the most efficient possible engine, not only because of the (trivial) absence of friction and other incidental wasteful processes; the main reason is that it assumes no conduction of heat between parts of the engine at different temperatures. He knew that conduction of heat between bodies at different temperatures is a wasteful, irreversible process and must be eliminated if the heat engine is to have the maximum efficiency.”

. July 1, 2010 at 2:48 pm

Efficiency of a Carnot Engine
Definition of efficiency for a heat engine. Efficiency of a Carnot Engine.

Carnot Efficiency 2: Reversing the Cycle
Seeing how we can scale and or reverse a Carnot Engine (to make a refrigerator)

Carnot Efficiency 3: Proving that it is the most efficient
Proving that a Carnot Engine is the most efficient engine

bryant April 17, 2012 at 6:54 am

EXPLAIN THE REVERSIBLE AND IRREVERSIBLE PROCESSES?URGENT!!!!!

. April 17, 2012 at 8:06 am

Reversible process (thermodynamics)

In thermodynamics, a reversible process, or reversible cycle if the process is cyclic, is a process that can be “reversed” by means of infinitesimal changes in some property of the system without loss or dissipation of energy. Due to these infinitesimal changes, the system is in thermodynamic equilibrium throughout the entire process. Since it would take an infinite amount of time for the reversible process to finish, perfectly reversible processes are impossible. However, if the system undergoing the changes responds much faster than the applied change, the deviation from reversibility may be negligible. In a reversible cycle, the system and its surroundings will be exactly the same after each cycle.

An alternative definition of a reversible process is a process that, after it has taken place, can be reversed and causes no change in either the system or its surroundings. In thermodynamic terms, a process “taking place” would refer to its transition from its initial state to its final state.