The Carnot Cycle: The Theoretical Limit of Heat Engine Efficiency
Thermodynamics is the study of energy, heat, work, and the ways in which energy is transferred
between systems. One of the most important applications of thermodynamics is the heat
engine, a device that converts thermal energy into useful mechanical work. Heat engines are
found throughout modern engineering, including power plants, automobiles, aircraft engines,
and many industrial systems. However, no heat engine can convert all the heat supplied to it
into useful work. Some energy must ultimately be rejected to the surroundings. The Carnot
cycle, introduced by French engineer and physicist Sadi Carnot in 1824, provides an idealized
model for determining the maximum possible efficiency of a heat engine operating between two
temperatures. Although an actual Carnot engine cannot be constructed because its processes
are completely reversible and idealized, the Carnot cycle remains fundamental to
thermodynamics because it establishes the theoretical efficiency limit against which real heat
engines can be compared.
The Carnot cycle consists of four reversible thermodynamic processes: isothermal expansion,
adiabatic (isentropic) expansion, isothermal compression, and adiabatic (isentropic)
compression. During the first process, the working fluid is placed in contact with a
high-temperature reservoir at a constant temperature, (T_H). Heat, (Q_H), enters the system
while the working fluid expands and performs work on its surroundings. Because this process is
isothermal, the temperature of the working fluid remains constant even though energy is being
transferred into the system. For an ideal gas, the increase in energy caused by the heat transfer
is balanced by the work performed during expansion.
During the second process, the working fluid continues expanding, but it is thermally isolated
from its surroundings. Therefore, there is no heat transfer between the system and its
surroundings, making the process adiabatic. As the gas expands and performs work, its
temperature decreases from the high reservoir temperature, (T_H), to the lower reservoir
temperature, (T_L). Because the ideal Carnot cycle assumes that this expansion is both
adiabatic and reversible, it is also an isentropic process. Entropy remains constant during this
stage.
The third process is a reversible isothermal compression at the lower temperature, (T_L). The
working fluid is placed in contact with a cold reservoir while an external force compresses it.
Heat, (Q_L), leaves the working fluid and enters the low-temperature reservoir. The temperature
remains constant throughout this compression. Finally, the fourth process consists of reversible
adiabatic compression. No heat is transferred during this process. Work is performed on the
working fluid, causing its temperature to increase from (T_L) back to (T_H). At the end of this
process, the working fluid has returned to its initial thermodynamic state, completing the cycle.
One of the most significant results of the Carnot cycle is its relationship to thermal efficiency.
The thermal efficiency of any heat engine can be defined as the ratio of the net work produced
by the engine to the heat supplied from the high-temperature reservoir:
[
\eta = \frac{W_{net}}{Q_H}
]
Because energy must be conserved, the net work produced during a complete cycle is equal to
the heat entering the system minus the heat rejected:
[
W_{net}=Q_H-Q_L
]
Therefore,
[
\eta = 1-\frac{Q_L}{Q_H}
]
For a completely reversible Carnot cycle, the ratio of heat transfers is related directly to the
absolute temperatures of the two reservoirs. This produces the famous Carnot efficiency
equation:
[
\boxed{\eta_{Carnot}=1-\frac{T_L}{T_H}}
]
where (T_H) is the absolute temperature of the high-temperature reservoir and (T_L) is the
absolute temperature of the low-temperature reservoir. Temperatures must be expressed on an
absolute scale, such as Kelvin. This equation demonstrates that the maximum possible
efficiency of a heat engine depends only on the temperatures of the reservoirs rather than the
particular working fluid or mechanical design of the engine.
For example, consider an ideal Carnot engine operating between a high-temperature reservoir
at 600 K and a low-temperature reservoir at 300 K. Its maximum theoretical efficiency would be
[
\eta_{Carnot}=1-\frac{300}{600}=0.50
]
or 50 percent. This means that even a completely reversible engine operating between these
temperatures could convert no more than half of the heat supplied to it into useful work. The
remaining energy would have to be rejected to the low-temperature reservoir. Increasing the
high-temperature reservoir to 900 K while maintaining the cold reservoir at 300 K would
increase the theoretical efficiency to approximately 66.7 percent. Consequently, the Carnot
equation demonstrates why engineers generally seek higher operating temperatures and lower
heat-rejection temperatures when designing efficient thermal systems.
The Carnot cycle also illustrates the second law of thermodynamics. The first law establishes
conservation of energy, but conservation alone does not explain why all thermal energy cannot
simply be transformed into work. The second law places restrictions on the direction and quality
of energy transformations. One important consequence is that a heat engine operating in a
cycle cannot convert all of the heat it receives from a single reservoir into work. Some heat must
be rejected to a lower-temperature reservoir. If a Carnot engine were 100 percent efficient, the
efficiency equation would require (T_L=0) K. Reaching absolute zero is physically unattainable,
meaning that a heat engine operating between finite temperatures cannot achieve 100 percent
thermal efficiency.
The concept of entropy provides another way of understanding the importance of the Carnot
cycle. Because every process in the ideal Carnot cycle is reversible, the cycle does not
generate entropy. During the high-temperature isothermal process, entropy enters the working
fluid with heat. During the low-temperature isothermal process, the same amount of entropy
leaves the working fluid. The working fluid eventually returns to its original state, so its net
entropy change over the complete cycle is zero. Real processes, however, contain
irreversibilities such as friction, heat transfer across finite temperature differences, turbulence,
pressure losses, and other effects. These phenomena generate entropy and reduce the amount
of energy that can be converted into useful work. As a result, the efficiency of a real heat engine
must always be below the Carnot efficiency when operating between the same two reservoir
temperatures.
Despite its theoretical importance, a practical Carnot engine is impossible to construct. For the
heat-transfer processes to be perfectly reversible, heat would have to flow across an
infinitesimally small temperature difference. Such heat transfer would occur extremely slowly
and would require unrealistically large heat exchangers. Similarly, friction and other mechanical
losses would have to be completely eliminated. A real engine must produce useful power within
a reasonable amount of time, requiring finite temperature and pressure differences that
introduce irreversibilities. Consequently, actual engineering cycles such as the Rankine cycle
used in steam power plants, the Otto cycle associated with spark-ignition engines, the Diesel
cycle, and the Brayton cycle used in gas turbines provide more realistic representations of
practical energy-conversion systems.
The Carnot cycle nevertheless remains one of the most important theoretical models in
engineering thermodynamics. Its purpose is not to provide a blueprint for constructing an actual
engine but rather to establish a fundamental performance limit. It demonstrates that heat-engine
efficiency is restricted by the temperatures between which the engine operates and that no real
engine can exceed the efficiency of a reversible engine operating between the same reservoirs.
It also provides a clear connection among heat, work, entropy, reversibility, and the second law
of thermodynamics. Nearly two centuries after Sadi Carnot introduced these ideas, the Carnot
cycle continues to provide engineers with a benchmark for understanding and improving thermal
systems.
Works Cited
Çengel, Yunus A., Michael A. Boles, and Mehmet Kanoğlu. Thermodynamics: An Engineering
Approach. 10th ed., McGraw Hill, 2024.
Moran, Michael J., et al. Fundamentals of Engineering Thermodynamics. 9th ed., Wiley, 2018.
National Institute of Standards and Technology. “Thermodynamics.” NIST, U.S. Department of
Commerce.
Sonntag, Richard E., Claus Borgnakke, and Gordon J. Van Wylen. Fundamentals of
Thermodynamics. 8th ed., Wiley, 2013.