---
title: "Carnot engine explanation"
chunk: 6/8
source: "https://en.wikipedia.org/wiki/Carnot_engine_explanation"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T04:19:16.023719+00:00"
instance: "kb-cron"
---

Another way to think about entropy is as a measurement of the availability of useful energy in a system. While energy cannot be created or destroyed, as the  approaches equilibrium the energy of that system becomes less available for use.
The concept entropy, though important in thermodynamics, is not necessary for an intuitive understanding of Carnot's theory.  There are many formulations of the Second Law that do not mention entropy at all, including the original Clausius and Thomson versions.

== Efficiency ==
It is sometimes stated that Carnot gave the formula for the efficiency of his engine. He could not have done, since his theory did not embrace the First Law of Thermodynamics, not then known.  Carnot himself was able to state that it depended on the temperature difference between the hot source and cold sink, and the temperature of the cold sink.. But he did not give the explicit formula.
The efficiency even of the ideal or Carnot engine turns out to be surprisingly poor, and therefore, that of real engines is even worse.  It has been said that the Second Law of Thermodynamics imposes an "energy tax", payable to Nature, every time heat is converted to work.

=== Of the Carnot engine ===
The Carnot engine's efficiency depends on only two temperatures and its calculation is simple.  It can be considered in terms of the fraction of heat that goes down the cold sink instead of being converted to work  — the "energy tax" that must be paid to nature.
This fraction is simply the temperature of the cold sink divided by the temperature of the hot sink; they must be measured in degrees kelvin.  (On this scale 0 °K is absolute zero.  Fahrenheit or Celsius temperatures would give erroneous results since these scales were arbitrarily defined.)
For example if the hot temperature is 373 °K (water boils) and the cold temperature is 273 °K (ice melts), then 73% of the heat must go down the cold sink, an escapable fact of nature.  The engine's efficiency working between those temperatures is thus only 27%.

=== In real time ===
In fact, the Carnot engine cannot deliver even that performance within a realistic timescale.
Of the four phases of the Carnot cycle, the two isothermals must be performed extremely slowly.  (If not, there would be an appreciable temperature gradient, implying heat loss and irreversibility, see above.)  But this means that the engine takes infinite time to perform a cycle, or put crudely, it never does.
If the engine is to operate in real time, it becomes necessary to sacrifice some of its reversibility.  It then develops real power, but it is no longer a true Carnot engine, and its efficiency is less.
It has been calculated that the fraction of waste heat down the cold sink then is, not the ratio of the two temperatures (as above), but the square root of that number.
This result was derived by Curzon and Ahlborn  —  though they were not the first to do so  — who claimed that it more closely predicts the performance of real thermal generators.
For example, if working between given temperatures a Carnot engine loses ⁠1/4⁠ of its heat down the cool sink, it will lose ⁠1/2⁠ in real time operation.

=== All practical heat engines are worse ===

The Carnot engine is supposed to be frictionless and have perfect insulation or conduction where required.  Real engines can never match these criteria and their efficiency is poorer.  Further, the hot temperature cannot be made extremely high, for practical materials reasons, and the cold temperature can rarely be made very low.

==== Materials limitations ====
For example, in the first commercial nuclear power stations the fuel rods could not operate above 450 °C for fear of melting the Magnox cladding. The thermal efficiency was 23%.  Later alloys allowed the temperature to be raised to 640 °C, which could deliver a thermal efficiency of 41%.

==== The steam locomotive ====
A good cold sink is needed for efficiency.  In the traditional steam railway locomotive such was lacking, since it had no condenser, and simply vented waste steam into the atmosphere.  It turned only 4% of its heat into mechanical work.  The rest went "straight to heat up the countryside".

==== Cars and trucks ====
Car engines can have efficiencies of 20% or less, compared to their Carnot Limit of 37%.  The highest efficiency for a commercial vehicle diesel engine (2021) was claimed to be 50%.

==== Power stations ====
According to Mitsubishi Heavy Industries, in 2022 the world's highest thermal efficiency was achieved at the Joetsu Thermal Power Station No 1, Japan, being certified by Guinness World Records.  It was 63.62%.

==== Solar cells ====
Solar cells are heat engines, and they start off with the advantage that the hot reservoir  —  the Sun — is at 6,000 °K.  Assuming a good cold sink this would give a Carnot efficiency of 95%.  However a solar cell is not a Carnot engine.  A 2016 review found that after allowing for various losses they achieved 7-8% efficiency, though it was hoped to raise this.

== Public recognition ==
Carnot has been compared to thinkers of the calibre of Euclid, Isaac Newton and Francis Bacon ("Only now and then, in the centuries, does such a genius come into view").
But he is little known to the general public, even in his native country.  In France the better known Carnots are his father, his nephew and his younger brother.

== Explanatory notes ==

== References and referenced notes ==