# Carnot heat engine

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A Carnot heat engine [2] is a theoretical engine that operates on the reversible Carnot cycle. The basic model for this engine was developed by Nicolas Léonard Sadi Carnot in 1824. The Carnot engine model was graphically expanded upon by Benoît Paul Émile Clapeyron in 1834 and mathematically explored by Rudolf Clausius in 1857 from which the concept of entropy emerged.

The Carnot cycle is a theoretical thermodynamic cycle proposed by French physicist Sadi Carnot in 1824 and expanded upon by others in the 1830s and 1840s. It provides an upper limit on the efficiency that any classical thermodynamic engine can achieve during the conversion of heat into work, or conversely, the efficiency of a refrigeration system in creating a temperature difference by the application of work to the system. It is not an actual thermodynamic cycle but is a theoretical construct.

Nicolas Léonard Sadi Carnot was a French military scientist and physicist, often described as the "father of thermodynamics". Like Copernicus, he published only one book, the Reflections on the Motive Power of Fire, in which he expressed, at the age of 27 years, the first successful theory of the maximum efficiency of heat engines. In this work he laid the foundations of an entirely new discipline, thermodynamics. Carnot's work attracted little attention during his lifetime, but it was later used by Rudolf Clausius and Lord Kelvin to formalize the second law of thermodynamics and define the concept of entropy.

Benoît Paul Émile Clapeyron was a French engineer and physicist, one of the founders of thermodynamics.

## Contents

Every thermodynamic system exists in a particular state. A thermodynamic cycle occurs when a system is taken through a series of different states, and finally returned to its initial state. In the process of going through this cycle, the system may perform work on its surroundings, thereby acting as a heat engine.

For thermodynamics, a thermodynamic state of a system is its condition at a specific time sheik, that is fully identified by values of a suitable set of parameters known as state variables, state parameters or thermodynamic variables. Once such a set of values of thermodynamic variables has been specified for a system, the values of all thermodynamic properties of the system are uniquely determined. Usually, by default, a thermodynamic state is taken to be one of thermodynamic equilibrium. This means that the state is not merely the condition of the system at a specific time, but that the condition is the same, unchanging, over an indefinitely long duration of time.

A thermodynamic cycle consists of a linked sequence of thermodynamic processes that involve transfer of heat and work into and out of the system, while varying pressure, temperature, and other state variables within the system, and that eventually returns the system to its initial state. In the process of passing through a cycle, the working fluid (system) may convert heat from a warm source into useful work, and dispose of the remaining heat to a cold sink, thereby acting as a heat engine. Conversely, the cycle may be reversed and use work to move heat from a cold source and transfer it to a warm sink thereby acting as a heat pump. At every point in the cycle, the system is in thermodynamic equilibrium, so the cycle is reversible.

A thermodynamic system is a group of material and/or radiative contents. Its properties may be described by thermodynamic state variables such as temperature, entropy, internal energy, and pressure.

A heat engine acts by transferring energy from a warm region to a cool region of space and, in the process, converting some of that energy to mechanical work. The cycle may also be reversed. The system may be worked upon by an external force, and in the process, it can transfer thermal energy from a cooler system to a warmer one, thereby acting as a refrigerator or heat pump rather than a heat engine.

A refrigerator is a popular household appliance that consists of a thermally insulated compartment and a heat pump that transfers heat from the inside of the fridge to its external environment so that the inside of the fridge is cooled to a temperature below the ambient temperature of the room. Refrigeration is an essential food storage technique in developed countries. The lower temperature lowers the reproduction rate of bacteria, so the refrigerator reduces the rate of spoilage. A refrigerator maintains a temperature a few degrees above the freezing point of water. Optimum temperature range for perishable food storage is 3 to 5 °C. A similar device that maintains a temperature below the freezing point of water is called a freezer. The refrigerator replaced the icebox, which had been a common household appliance for almost a century and a half.

A heat pump is a device that transfers heat energy from a source of heat to what is called a heat sink. Heat pumps move thermal energy in the opposite direction of spontaneous heat transfer, by absorbing heat from a cold space and releasing it to a warmer one. A heat pump uses a small amount of external power to accomplish the work of transferring energy from the heat source to the heat sink. The most common design of a heat pump involves four main components – a condenser, an expansion valve, an evaporator and a compressor. The heat transfer medium circulated through these components is called refrigerant.

## Carnot's diagram

In the adjacent diagram, from Carnot's 1824 work, Reflections on the Motive Power of Fire , [3] there are "two bodies A and B, kept each at a constant temperature, that of A being higher than that of B. These two bodies to which we can give, or from which we can remove the heat without causing their temperatures to vary, exercise the functions of two unlimited reservoirs of caloric. We will call the first the furnace and the second the refrigerator.” [4] Carnot then explains how we can obtain motive power, i.e., “work”, by carrying a certain quantity of heat from body A to body B. It also acts as a cooler and hence can also act as a Refrigerator.

Reflections on the Motive Power of Fire and on Machines Fitted to Develop that Power is a book published in 1824 by French physicist Sadi Carnot. The 118-page book's French title was Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance. It is a significant publication in the history of thermodynamics about a generalized theory of heat engines.

The caloric theory is an obsolete scientific theory that heat consists of a self-repellent fluid called caloric that flows from hotter bodies to colder bodies. Caloric was also thought of as a weightless gas that could pass in and out of pores in solids and liquids. The "caloric theory" was superseded by the mid-19th century in favor of the mechanical theory of heat, but nevertheless persisted in some scientific literature—particularly in more popular treatments—until the end of the 19th century.

A furnace is a device used for high-temperature heating. The name derives from Latin word fornax, which means oven. The heat energy to fuel a furnace may be supplied directly by fuel combustion, by electricity such as the electric arc furnace, or through induction heating in induction furnaces.

## Modern diagram

The previous image shows the original piston-and-cylinder diagram used by Carnot in discussing his ideal engines. The figure at right shows a block diagram of a generic heat engine, such as the Carnot engine. In the diagram, the “working body” (system), a term introduced by Clausius in 1850, can be any fluid or vapor body through which heat Q can be introduced or transmitted to produce work. Carnot had postulated that the fluid body could be any substance capable of expansion, such as vapor of water, vapor of alcohol, vapor of mercury, a permanent gas, or air, etc. Although, in these early years, engines came in a number of configurations, typically QH was supplied by a boiler, wherein water was boiled over a furnace; QC was typically supplied by a stream of cold flowing water in the form of a condenser located on a separate part of the engine. The output work, W, represents the movement of the piston as it is used to turn a crank-arm, which in turn was typically used to power a pulley so as to lift water out of flooded salt mines. Carnot defined work as “weight lifted through a height”.

In systems involving heat transfer, a condenser is a device or unit used to condense a substance from its gaseous to its liquid state, by cooling it. In so doing, the latent heat is given up by the substance and transferred to the surrounding environment. Condensers can be made according to numerous designs, and come in many sizes ranging from rather small (hand-held) to very large. For example, a refrigerator uses a condenser to get rid of heat extracted from the interior of the unit to the outside air. Condensers are used in air conditioning, industrial chemical processes such as distillation, steam power plants and other heat-exchange systems. Use of cooling water or surrounding air as the coolant is common in many condensers.

## Carnot cycle

The Carnot cycle when acting as a heat engine consists of the following steps:

1. Reversible isothermal expansion of the gas at the "hot" temperature, TH (isothermal heat addition or absorption). During this step (1 to 2 on Figure 1, A to B in Figure 2) the gas is allowed to expand and it does work on the surroundings. The temperature of the gas does not change during the process, and thus the expansion is isothermic. The gas expansion is propelled by absorption of heat energy Q1 and of entropy ${\displaystyle \Delta S_{\text{H}}=Q_{\text{H}}/T_{\text{H}}}$ from the high temperature reservoir.
2. Isentropic (reversible adiabatic) expansion of the gas (isentropic work output). For this step (2 to 3 on Figure 1, B to C in Figure 2) the piston and cylinder are assumed to be thermally insulated, thus they neither gain nor lose heat. The gas continues to expand, doing work on the surroundings, and losing an equivalent amount of internal energy. The gas expansion causes it to cool to the "cold" temperature, TC. The entropy remains unchanged.
3. Reversible isothermal compression of the gas at the "cold" temperature, TC. (isothermal heat rejection) (3 to 4 on Figure 1, C to D on Figure 2) Now the gas is exposed to the cold temperature reservoir while the surroundings do work on the gas by compressing it (such as through the return compression of a piston), while causing an amount of heat energy Q2 and of entropy ${\displaystyle \Delta S_{\text{C}}=Q_{\text{C}}/T_{\text{C}}}$ to flow out of the gas to the low temperature reservoir. (This is the same amount of entropy absorbed in step 1.) This work is less than the work performed on the surroundings in step 1 because it occurs at a lower pressure given the removal of heat to the cold reservoir as the compression occurs (i.e. the resistance to compression is lower under step 3 than the force of expansion under step 1).
4. Isentropic compression of the gas (isentropic work input). (4 to 1 on Figure 1, D to A on Figure 2) Once again the piston and cylinder are assumed to be thermally insulated and the cold temperature reservoir is removed. During this step, the surroundings continue to do work to further compress the gas and both the temperature and pressure rise now that the heat sink has been removed. This additional work increases the internal energy of the gas, compressing it and causing the temperature to rise to TH. The entropy remains unchanged. At this point the gas is in the same state as at the start of step 1.

## Carnot's theorem

Carnot's theorem is a formal statement of this fact: No engine operating between two heat reservoirs can be more efficient than a Carnot engine operating between the same reservoirs.

${\displaystyle \eta _{\text{I}}={\frac {W}{Q_{\text{H}}}}=1-{\frac {T_{\text{C}}}{T_{\text{H}}}}}$

(1)

Explanation
This maximum efficiency ${\displaystyle \eta _{\text{I}}}$ is defined as above:

W is the work done by the system (energy exiting the system as work),
${\displaystyle Q_{\text{H}}}$ is the heat put into the system (heat energy entering the system),
${\displaystyle T_{\text{C}}}$ is the absolute temperature of the cold reservoir, and
${\displaystyle T_{\text{H}}}$ is the absolute temperature of the hot reservoir.

A corollary to Carnot's theorem states that: All reversible engines operating between the same heat reservoirs are equally efficient.

It is easily shown that the efficiency η is maximum when the entire cyclic process is a reversible process. This means the total entropy of the net system (the entropies of the hot furnace, the "working fluid" of the Heat engine, and the cold sink) remains constant when the "working fluid" completes one cycle and returns to its original state. (In the general case, the total entropy of this combined system would increase in a general irreversible process).

In thermodynamics, a reversible process is a process whose direction can be "reversed" by inducing infinitesimal changes to some property of the system via its surroundings. Throughout the entire reversible process, the system is in thermodynamic equilibrium with its surroundings. Having been reversed, it leaves no change in either the system or the surroundings. Since it would take an infinite amount of time for the reversible process to finish, perfectly reversible processes are impossible. However, if the system undergoing the changes responds much faster than the applied change, the deviation from reversibility may be negligible. In a reversible cycle, a cyclical reversible process, the system and its surroundings will be returned to their original states if one half cycle is followed by the other half cycle.

In statistical mechanics, entropy is an extensive property of a thermodynamic system. It is closely related to the number Ω of microscopic configurations that are consistent with the macroscopic quantities that characterize the system. Under the assumption that each microstate is equally probable, the entropy is the natural logarithm of the number of microstates, multiplied by the Boltzmann constant kB. Formally,

Since the "working fluid" comes back to the same state after one cycle, and entropy of the system is a state function; the change in entropy of the "working fluid" system is 0. Thus, it implies that the total entropy change of the furnace and sink is zero, for the process to be reversible and the efficiency of the engine to be maximum. This derivation is carried out in the next section.

The coefficient of performance (COP) of the heat engine is the reciprocal of its efficiency.

## Efficiency of real heat engines

For a real heat engine, the total thermodynamic process is generally irreversible. The working fluid is brought back to its initial state after one cycle, and thus the change of entropy of the fluid system is 0, but the sum of the entropy changes in the hot and cold reservoir in this one cyclical process is greater than 0.

The internal energy of the fluid is also a state variable, so its total change in one cycle is 0. So the total work done by the system W, is equal to the heat put into the system ${\displaystyle Q_{\text{H}}}$ minus the heat taken out ${\displaystyle Q_{\text{C}}}$.

${\displaystyle W=Q_{\text{H}}-Q_{\text{C}}}$

(2)

For real engines, sections 1 and 3 of the Carnot Cycle; in which heat is absorbed by the "working fluid" from the hot reservoir, and released by it to the cold reservoir, respectively; no longer remain ideally reversible, and there is a temperature differential between the temperature of the reservoir and the temperature of the fluid while heat exchange takes place.

During heat transfer from the hot reservoir at ${\displaystyle T_{\text{H}}}$ to the fluid, the fluid would have a slightly lower temperature than ${\displaystyle T_{\text{H}}}$, and the process for the fluid may not necessarily remain isothermal. Let ${\displaystyle \Delta S_{\text{H}}}$ be the total entropy change of the fluid in the process of intake of heat.

${\displaystyle \Delta S_{\text{H}}=\int _{Q_{\text{in}}}{\frac {{\text{d}}Q_{\text{H}}}{T}}}$

(3)

where the temperature of the fluid T is always slightly lesser than ${\displaystyle T_{\text{H}}}$, in this process.

So, one would get

${\displaystyle {\frac {Q_{\text{H}}}{T_{\text{H}}}}={\frac {\int {\text{d}}Q_{\text{H}}}{T_{\text{H}}}}\leq \Delta S_{\text{H}}}$

(4)

Similarly, at the time of heat injection from the fluid to the cold reservoir one would have, for the magnitude of total entropy change ${\displaystyle \Delta S_{\text{C}}}$ of the fluid in the process of expelling heat:

${\displaystyle \Delta S_{\text{C}}=\int _{Q_{\text{out}}}{\frac {{\text{d}}Q_{\text{C}}}{T}}\leq {\frac {\int {\text{d}}Q_{\text{C}}}{T_{\text{C}}}}={\frac {Q_{\text{C}}}{T_{\text{C}}}}}$,

(5)

where, during this process of transfer of heat to the cold reservoir, the temperature of the fluid T is always slightly greater than ${\displaystyle T_{\text{C}}}$.

We have only considered the magnitude of the entropy change here. Since the total change of entropy of the fluid system for the cyclic process is 0, we must have

${\displaystyle \Delta S_{\text{H}}=\Delta S_{\text{C}}}$

(6)

The previous three equations combine to give:

${\displaystyle {\frac {Q_{\text{C}}}{T_{\text{C}}}}\geq {\frac {Q_{\text{H}}}{T_{\text{H}}}}}$

(7)

Equations ( 2 ) and ( 7 ) combine to give

${\displaystyle {\frac {W}{Q_{\text{H}}}}\leq 1-{\frac {T_{\text{C}}}{T_{\text{H}}}}}$

(8)

Hence,

${\displaystyle \eta \leq \eta _{\text{I}}}$

(9)

where ${\displaystyle \eta ={\frac {W}{Q_{\text{H}}}}}$ is the efficiency of the real engine, and ${\displaystyle \eta _{\text{I}}}$ is the efficiency of the Carnot engine working between the same two reservoirs at the temperatures ${\displaystyle T_{\text{H}}}$ and ${\displaystyle T_{\text{C}}}$. For the Carnot engine, the entire process is 'reversible', and Equation ( 7 ) is an equality.

Hence, the efficiency of the real engine is always less than the ideal Carnot engine.

Equation (7) signifies that the total entropy of the total system(the two reservoirs + fluid) increases for the real engine, because the entropy gain of the cold reservoir as ${\displaystyle Q_{\text{C}}}$ flows into it at the fixed temperature ${\displaystyle T_{\text{C}}}$, is greater than the entropy loss of the hot reservoir as ${\displaystyle Q_{\text{H}}}$ leaves it at its fixed temperature ${\displaystyle T_{\text{H}}}$. The inequality in Equation ( 7 ) is essentially the statement of the Clausius theorem.

According to the second theorem, "The efficiency of the Carnot engine is independent of the nature of the working substance".

## Notes

1. Figure 1 in Carnot (1824, p. 17) and Carnot (1890, p. 63). In the diagram, the diameter of the vessel is large enough to bridge the space between the two bodies, but in the model, the vessel is never in contact with both bodies simultaneously. Also, the diagram shows an unlabeled axial rod attached to the outside of the piston.
2. In French, Carnot uses machine à feu, which Thurston translates as heat-engine or steam-engine. In a footnote, Carnot distinguishes the steam-engine (machine à vapeur) from the heat-engine in general. (Carnot, 1824, p. 5 and Carnot, 1890, p. 43)
3. English translation by Thurston (Carnot, 1890, p. 51-52).

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## References

• Carnot, Sadi (1824). Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance (in French). Paris: Bachelier. (First Edition 1824) and (Reissue of 1878)
• Carnot, Sadi (1890). Thurston, Robert Henry, ed. Reflections on the Motive Power of Heat and on Machines Fitted to Develop That Power. New York: J. Wiley & Sons.