# Carnot heat engine

The Carnot heat engine uses a particular thermodynamic cycle studied by Nicolas Léonard Sadi Carnot in the 1820s and expanded upon by Benoit Paul Émile Clapeyron in the 1830s and 40s. A heat engine is an engine that uses heat to produce mechanical work by carrying a working substance through a cyclic process.

## The Carnot cycle

The Carnot cycle consists of the following steps:

1. Reversible isothermal expansion of the gas at the "hot" temperature, TH( Isothermal heat addition ). During this step, the expanding gas causes the piston to do work on the surroundings. The gas expansion is propelled by absorption of heat from the high temperature reservoir.
2. Reversible adiabatic expansion of the gas. For this step we assume the piston and cylinder are thermally insulated, so that no heat is gained or lost. The gas continues to expand, doing work on the surroundings. The gas expansion causes it to cool to the "cold" temperature, TC.
3. Reversible isothermal compression of the gas at the "cold" temperature, TC.( Isothermal heat rejection ) Now the surroundings do work on the gas, causing heat to flow out of the gas to the low temperature reservoir.
4. Reversible adiabatic compression of the gas. Once again we assume the piston and cylinder are thermally insulated. During this step, the surroundings do work on the gas, compressing it and causing the temperature to rise to TH. At this point the gas is in the same state as at the start of step 1.

## Properties and significance

The amount of work produced by the Carnot cycle, wcy, is the difference between the heat absorbed in step 1, qH and the heat rejected in step 3, qC. Or in equation form:

$w_{cy} = q_H - q_C \qquad \mbox{(1)}$

The efficiency η of a heat engine is defined as the ratio of the work done on the surroundings to the heat input at the higher temperature. Thus for the Carnot cycle:

$\eta = \frac{w_{cy}}{q_H} = \frac{q_H - q_C}{q_H} = 1 - \frac{q_C}{q_H} \qquad \mbox{(2)}$

It can also be shown that for the Carnot cycle qC/qH = TC/TH, so in terms of temperature, the efficiency is:

$\eta = 1 - \frac{T_C}{T_H} \qquad \mbox{(3)}$

From Equation 3 it is clear that in order to maximize efficiency one should maximize TH and minimize TC.

### Carnot's theorem

Carnot's theorem states that No engine operating between two heat reservoirs can be more efficient than a Carnot engine operating between the same reservoirs. Thus, Equation 3 gives the maximum efficiency possible for any engine using the corresponding temperatures. A corollary to Carnot's theorem states that: All reversible engines operating between the same heat reservoirs are equally efficient. So Equation 3 gives the efficiency of any reversible engine.

### Efficiency of real heat engines

Carnot realised that in reality it is not possible to build a thermodynamically reversible engine, so real heat engines are less efficient than indicated by Equation 3. Nevertheless, Equation 3 is extremely useful for determining the maximum efficiency that could ever be expected for a given set of thermal reservoirs.

Although Carnot's cycle is an idealisation, the expression of Carnot efficiency is still useful. Consider:

<TH>; and
<TC>.

-the average temperatures at which heat is, respectively, input and output. Replace TH and TC in Equation (3) by <TH> and <TC> respectively.

For the Carnot cycle, or its equivalent, <TH> is the highest temperature available and <TC> the lowest. For other less efficient cycles, <TH> will be lower than TH , and <TC> will be higher than TC. This can help illustrate, for example, why a reheater or a regenerator can improve thermal efficiency.

## References

• Kroemer, Herbert; Kittle, Charles (1980). Thermal Physics (2nd ed.), W. H. Freeman Company. ISBN 0716710889.
##### Toolbox

 Get A Wifi Network Switcher Widget for Android