Carnot Engine: Learn its Definition, Efficiency, Derivation, Theorem, Applications & Limitations

The Carnot engine is a theoretical cycle proposed by Nicholas Leonard Sadi Carnot. It is an engine that operates between two temperatures which gives the thermal efficiency of that engine and involves thermal and adiabatic processes.

In this article, you will learn briefly about the Carnot Engine, its theorem, different applications and related solved examples. 

What is Carnot Engine?

The Carnot engine is a thermodynamic cycle that gives the maximum efficiency of a heat engine while working between two temperatures in reversible thermal and adiabatic expansion and compression processes. It works between two temperatures T1 and T2 with a hot and a cold reservoir.

A process is irreversible if a process is quasi-static or non-dissipative and heat should be absorbed and released isothermally. There are two steps involved in the process:  isothermal process at a temperature T1 absorbing heat Q1 from the hot reservoir and isothermal process at temperature T2 releasing heat Q2 from the cold reservoir.

Carnot’s Theorem

Carnot’s theorem can be stated as

1. An engine working between two temperatures T1 and T2 of hot and cold reservoirs cannot have efficiency more than that of Carnot’s engine.
2. The efficiency of the Carnot engine is independent of the nature of the working substance.

Efficiency of Carnot Engine Formula

The efficiency of Carnot Engine is given as

Where is the thermal efficiency of the engine.

T1 and T2 are the two temperatures between which the engine works.

The efficiency of carnot engine depends on the absolute temperature range of operation.

Derivation of Carnot Engine

A reversible heat engine working between two temperatures T1 and T2 is known as Carnot Engine.

Consider the graph below – Carnot’s cycle for a heat engine with an ideal gas as the working substance.

• Suppose we have a hot reservoir at temperature T1 and a cold reservoir at temperature T2, such that the heat absorbed by temperature T1 is given by Q1 and the heat released from temperature T2 is given by Q2.
• The pressure and volume of the gas are given by P1 and P2 and the volume by V1 and V2. 
• Step 1 →2: The isothermal expansion of gas takes place from (P1, V1, T1) to (P2, V2, T2).
• The work done - equation (1)
• Step 2→3 : Adiabatic expansion of gas from (P2, V2, T1) to (P3, V3, T2).

Work done by gas,

• Step 3→4: Isothermal compression of the gas from (P3, V3, T2) to (P4, V4, T2)

The heat released Q2 by the gas to the reservoir in temperature T2.

The work done by the gas on the environment is given by,

• Step 4→1 : Work done on the environment,

• The total work done by the gas,

• The efficiency of a Carnot engine is given by,

• Step 2→3 is an adiabatic process,

– equation (1)

• Similarly, step 4→1 is an adiabatic process,

- equation (2)

Hence from equation (1) and (2) we get,

.

• – known as the efficiency of Carnot Engine.

Carnot’s Engine Cycle

• We have a hot reservoir working at temperature T1 and a cold reservoir at temperature T2.
• Heat(Q1) is absorbed from the hot reservoir and is released in the cold reservoir in the form of heat (Q2).
• We need to take the engine system from temperature T1 to temperature T2 and back to temperature T1.
• Each step of the Carnot cycle can be reversed.
• Heat taken (Q2) from a cold reservoir at temperature (T2), does work (W) on the system and transfers heat (Q1) in the hot reservoir working at temperature (T1). Theoretically, this can be a reversible refrigeration system.
• To prove Carnot’s theorem, imagine a reversible Carnot engine R and an irreversible engine I working between the sink and the reservoir.
• Engines I and R are arranged in such a way that I act like a heat engine and R acts like a refrigerator.
• I absorb heat Q1 from the source and give a work output as W’, releasing heat Q1-W’ into the sink.
• R returns the same heat Q1 to the source, takes heat Q2 from the sink and gives a work output W = Q1-Q2.
• The efficiencies of engines R and I are given as,

, i.e , if R were to work like an engine it would give less work output than I.

• Thus on a whole, for the I-R system, it extracts heat (Q1-W) – (Q1-W’) = (W’-W), from the cold reservoir and delivers the same amount of work in a single cycle – which is against the second law of thermodynamics. 
• Hence the assertion is wrong. No engine can have efficiency greater than Carnot’s engine.
• The maximum efficiency of a Carnot engine is independent of the nature of the system, eventually justifying the working of a Carnot engine in an ideal gas system.
• \(\frac {Q_1}{Q_2}=\frac {T_1}{T_2}\) is the universal relation of the system and can be used to define a universal thermodynamic temperature scale independent of properties used in the system.

Applications of Carnot Engine

Carnot engine has no practical application but is used as a theoretical reference.

1. Used for comparing efficiencies of thermal devices like, heat pumps, ac and refrigerators (to yield cooling).
2. Used to compare efficiencies of petrol and diesel engines.

Limitations of Carnot Cycle

There are a number of limitations to the Carnot cycle, including:

• Irreversibility: All real-world thermodynamic processes are irreversible, meaning that they cannot be perfectly reversed. This reduces the efficiency of all real-world heat engines.
• Heat transfer: Heat transfer between the working fluid and the reservoirs is not instantaneous, which also reduces the efficiency of real-world heat engines.
• Friction: Friction between the moving parts of a heat engine also reduces its efficiency.

Solved Examples on Carnot Engine

Example 1. A carnot engine works at the efficiency of 20%. By what must be the temperature of the reservoir, if the temperature of the sink is at 300K?

Solution 1. Given data,

= 20

T2 = 300K

To find temperature: T1

Hence temperature T1 at 20% efficiency = 375K.

The temperature of the reservoir is 375K.

Example 2. A carnot engine is working between the temperatures T1 = 300K and T2 = 500K.

What would be the working efficiency of the engine?

Solution 2: Given data.

Temperature of the reservoir (T1) = 300K.

Temperature of the sink (T2) = 500K

The efficiency of the engine,

= 0.4

The efficiency of the engine is 0.4

Example 3: An ideal Carnot's engine works between 227°C and 57°C. Find the efficiency of the engine.

Solution. =227°C=(227+273)

K=500K

=57°C=(57+273)

K=330K

Efficiency of carnot engine,

or η=34%

Example 4: A carnot engine, having an efficiency of η=1/10 is used as a refrigerator. If the work done on the system is 10 J, what is the amount of energy absorbed from the reservoir at lower temperature?

Solution.

By using the formula,

Example 5: Two Carnot engines A and B are operated in series. The first one, A, receives heat at = 600 K and rejects to a reservoir at temperature . The second engine B receives heat rejected by the first engine and, in turn, rejects to a heat reservoir at = 400 K. Calculate the temperature if the work outputs of the two engines are equal.

Solution.

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