Carnot's Corollary
Introduction
Carnot’s theorem states that, “if we conduct a heat engine that is to operate between two temperature reservoirs, then that engine will have the maximum possible efficiency if it operates via a Carnot cycle. (1)”
Carnot’s corollary states that, “if there are several cyclic heat engines, some of which are reversible, operating around cycles between the same temperatures TH and TL, all the reversible ones have the same efficiency, while the nonreversible ones have efficiencies which can never exceed the efficiency of the reversible engines. (2)”
The statement of Carnot’s theorem is not confusing. The statement of Carnot’s corollary, however, is.
The inexperienced reader concludes that all reversible thermodynamic cycles whose highest temperature reached is TH and whose lowest temperature reached is TL have the same efficiency, although this is true only for Carnot cycles. Why is Carnot’s corollary not valid in such situations?
Logical Analysis
We must consider the corollary’s wording carefully. Particularly, “temperatures TH and TL” really refer to two–and only two–reservoirs at temperatures of TH and TL. Knowing this, we can show that the only reversible cycle that is governed by Carnot’s corollary is the Carnot cycle, and thus that Carnot’s corollary applies only to Carnot cycles.
The corollary requires that the reversible cycle be in contact with two thermal reservoirs, which translates to an isothermal expansion and an isothermal compression. Any non-isothermal contact with the reservoirs would indicate irreversibility.
The corollary further requires that the reversible cycle not be in contact with any other thermal reservoirs, which is only satisfied by an adiabatic expansion and an adiabatic compression. Any non-adiabatic process would require heat transfer, which requires contact with a thermal reservoir.
We thus have a cycle of two isotherms and two adiabats, which defines a Carnot cycle. This means that Carnot’s corollary is only valid for Carnot cycles. Thus we conclude that (a) all Carnot cycles with the same two operating temperatures are equally, and (b) all other cycles, reversible or irreversible, are less efficient.
Graphical Analysis
An alternative justification of Carnot’s corollary can be done graphically. While we often consider cycles using a P-V diagram, it is more instructive to use a T-S diagram in this case because a Carnot cycle is a rectangle when graphed in terms of T and S. (This can be proven mathematically via Jacobean transformations, or physically via the definition of a Carnot cycle as having two isotherms and two adiabats, or isentropes. Because adiabatic processes do not transfer heat, they also do not transfer entropy, and thus they are also isentropic processes.)
The efficiency of a cycle is work done on the surroundings divided by heat input, both of which have a graphical representation.
A thermodynamic cycle on a T-S diagram.
From the first law, the work done on the surroundings is equivalent to the heat input minus the heat output. The heat input is the integral of (the upper curve of) TdS between Smin and Smax, and is equal to the area of A + B. The heat output is the integral of (the lower curve of) TdS between Smin and Smax, and is equal to the area of B. The efficiency is then A/(A + B).
A thermodynamic cycle on a T-S diagram with boundary lines drawn.
We are only comparing cycles that work between temperatures of TH and TL, so we are restricted to cycles that lie between TH and TL. In order to maximize the efficiency, we must maximize A and minimize B, which is accomplished when A is a rectangle. While we have no entropic restrictions, B will increase at least by the same factor as A if we try to modify our bounds on entropy. Thus, we can create artificial bounds on entropy knowing that the most efficient cycle with bounds on entropy will be the same as that without bounds on entropy. We can see from the graph that the maximum efficiency results when A is the rectangle bounded by Tmin and Tmax, and by Smin and Smax. Then A is a Carnot cycle, and we have thus shown that the Carnot cycle is the most efficient cycle within two temperatures TH and TL.
(1) The efficiency of reversible heat engines. J. Chem. Educ., 1991, 68 (3), p 208 (2) Thermodynamics, E. Fermi (3) Engineering Thermodynamics, V. Kirillin, V. Sychev, A. Sheindlin
The explanation of Carnot’s corollary is adapted from 1, and the graphical analysis of heat engine efficiency is adapted from 3. “Carnot’s corollary” is coined by me; I did not wish to repeat “the corollary of Carnot’s theorem” multiple times.