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  • Identify a Carnot cycle.
  • Calculate maximum theoretical efficiency of a nuclear reactor.
  • Explain how dissipative processes affect the ideal Carnot engine.
Photograph of a novelty toy known as the drinking bird. It is made up of two glass bulbs connected to each other by a glass tube. The upper bulb is shaped like a bird’s head, and the tube looks like its neck. The lower bulb, which may be compared to the abdomen, contains methylene chloride that has been colored red. The bottom of the neck is attached to a pivot, and in front of the bird’s head is a glass of water.
This novelty toy, known as the drinking bird, is an example of Carnot’s engine. It contains methylene chloride (mixed with a dye) in the abdomen, which boils at a very low temperature—about 100ºF . To operate, one gets the bird’s head wet. As the water evaporates, fluid moves up into the head, causing the bird to become top-heavy and dip forward back into the water. This cools down the methylene chloride in the head, and it moves back into the abdomen, causing the bird to become bottom heavy and tip up. Except for a very small input of energy—the original head-wetting—the bird becomes a perpetual motion machine of sorts. (credit: Arabesk.nl, Wikimedia Commons)

We know from the second law of thermodynamics that a heat engine cannot be 100% efficient, since there must always be some heat transfer Q c size 12{Q rSub { size 8{c} } } {} to the environment, which is often called waste heat. How efficient, then, can a heat engine be? This question was answered at a theoretical level in 1824 by a young French engineer, Sadi Carnot (1796–1832), in his study of the then-emerging heat engine technology crucial to the Industrial Revolution. He devised a theoretical cycle, now called the Carnot cycle    , which is the most efficient cyclical process possible. The second law of thermodynamics can be restated in terms of the Carnot cycle, and so what Carnot actually discovered was this fundamental law. Any heat engine employing the Carnot cycle is called a Carnot engine    .

What is crucial to the Carnot cycle—and, in fact, defines it—is that only reversible processes are used. Irreversible processes involve dissipative factors, such as friction and turbulence. This increases heat transfer Q c size 12{Q rSub { size 8{c} } } {} to the environment and reduces the efficiency of the engine. Obviously, then, reversible processes are superior.

Carnot engine

Stated in terms of reversible processes, the second law of thermodynamics    has a third form:

A Carnot engine operating between two given temperatures has the greatest possible efficiency of any heat engine operating between these two temperatures. Furthermore, all engines employing only reversible processes have this same maximum efficiency when operating between the same given temperatures.

[link] shows the PV size 12{ ital "PV"} {} diagram for a Carnot cycle. The cycle comprises two isothermal and two adiabatic processes. Recall that both isothermal and adiabatic processes are, in principle, reversible.

Carnot also determined the efficiency of a perfect heat engine—that is, a Carnot engine. It is always true that the efficiency of a cyclical heat engine is given by:

Eff = Q h Q c Q h = 1 Q c Q h . size 12{ ital "Eff"= { {Q rSub { size 8{h} } - Q rSub { size 8{c} } } over {Q rSub { size 8{h} } } } =1 - { {Q rSub { size 8{c} } } over {Q rSub { size 8{h} } } } } {}

What Carnot found was that for a perfect heat engine, the ratio Q c / Q h size 12{Q rSub { size 8{c} } /Q rSub { size 8{h} } } {} equals the ratio of the absolute temperatures of the heat reservoirs. That is, Q c / Q h = T c / T h size 12{Q rSub { size 8{c} } /Q rSub { size 8{h} } =T rSub { size 8{c} } /T rSub { size 8{h} } } {} for a Carnot engine, so that the maximum or Carnot efficiency     Eff C size 12{ ital "Eff" rSub { size 8{c} } } {} is given by

Eff C = 1 T c T h , size 12{ ital "Eff" rSub { size 8{c} } =1 - { {T rSub { size 8{c} } } over {T rSub { size 8{h} } } } } {}

where T h size 12{T rSub { size 8{h} } } {} and T c size 12{T rSub { size 8{c} } } {} are in kelvins (or any other absolute temperature scale). No real heat engine can do as well as the Carnot efficiency—an actual efficiency of about 0.7 of this maximum is usually the best that can be accomplished. But the ideal Carnot engine, like the drinking bird above, while a fascinating novelty, has zero power. This makes it unrealistic for any applications.

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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