Carnot Efficiency

What Is Efficiency? Everyday Usage and Physicists' Lingo

In daily life, we sometimes use the word "efficient" to mean "powerful". For example, someone who said "that air conditioner is very efficient," might be saying that the air conditioner is very effective and can cool a room quickly. Physicists, however, use a different definition of efficiency.

When physicists use the word "efficiency", they mean "how much useful work do you get out of a process compared to how much energy you put in." Since physicists measure work and energy in the same units, they calculate efficiency as the useful work out divided by the energy in. Since power is just energy per unit time, you can also find the efficiency by dividing the useful power output by the power input.

For example, if an electric motor that's supplying 25 watts of work is using 100 watts of electric power, then its efficiency is 25%.

Is There a Limit to Efficiency?

Unfortunately, there are limits to how efficient an engine can be. There's a simple formula for the maximum possible of efficiency of steam engines and other heat engines. A heat engine is an engine that, like a steam engine, by taking in a hot substance, using the substance to do work, and then releasing the substance as low temperature exhaust. A steam engine, for example, takes in hot steam, and uses the expansion of the steam to push a piston. As it expands, the steam cools. The steam engine then lets the cooled steam escape as exhaust.

The formula is:
Efficiency = 1 - Tc/Th, where Tc is the temperature of the cold end of the cycle, and Th is the temperature of the hot end .
That is, effeciency equals one minus the cold temperature divided by the hot temperature.
For a steam engine, the hot temperature would be the temperature of the incoming steam, and the cold temperature would be the temperature of "cold" steam exhaust.

What Temperature Scales Can Be Used in This Formula?

For this formula to work, you can't use the familiar Fahrenheit or Celsius (Centigrade) temperature scales. You must use a scale whose zero is absolute zero, the lowest temperature that can exist. For a discussion of absolute temperature, see the temperature scales page in our Introduction to Cryogenics section.

Why Is This Called "Carnot Efficiency"

This efficiency limit is named after the French engineer Sadi Carnot (1796-1832). In the early 19th century, when Carnot worked, the steam engine was the leading edge of technology. Despite the importance of the steam engine, however, the basic physics describing its operation had not been worked out. It was Carnot who developed many of the ideas which we use today to study the behavior of engines.

For example, Carnot realized that:

The fact that the steam comes in at a hotter temperature than it leaves the engine is an important part of how the engine works. As the steam cools, it gives up heat energy, which is converted by the engine to work.
The more the steam cools, the more work it can do. That seems almost self-evident to us today, because we realize that heat transfer is a form of energy flow.
Although steam is a convenient substance to use to run an engine, you can use other substances, instead. I personally have seen a tiny engine that works on air (which, like steam, expands when heated and contracts when cooled.) Its energy source is the heat from a computer monitor. It sits on top of the monitor, turning a little wheel. Not very practical as a major contributor to the economy, perhaps, but it does show that Carnot was right about other substances besides steam being usable in engines.
No matter what the substance used in the engine - steam, air, whatever - the same efficiency limits apply.

In the years since Carnot worked, his conclusions have been expanded and generalized, eventually producing such concepts as entropy and the Second Law of Thermodynamics. Unfortunately, I don't have time to discuss all that, even if I did have the knowledge at my fingertips.

Links:

Curator: Mark O. Kimball
NASA Official: Eric A. Silk
Last Updated: 09/11/2014