Carnot Engine: Carnot Cycle, Theorem, Carnot Engine Formula

Carnot engine is a reversible engine that operates between two temperatures T₁ (source) and T₂ (sink). It gives an estimate of the maximum possible efficiency that a heat engine can have in successful heat conversion between the two temperatures. Carnot engines can be found in refrigerators and air conditioners. Carnot equation can be used in understanding the working process of heat and internal energy change in adiabatic as well as isothermal processes.

Also read: Difference Between Work and Energy 

Key Terms: Carnot cycle, Carnot Theorem, Carnot Engine Formula, carnot vapour cycle, carnot heat engine

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Carnot Engine Definition

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Carnot engine, as proposed by Nicolas Léonard Sadi Carnot in 1824, is the one that operates on reversible thermodynamics. It has been named after its invention by Carnot. It is an analytical engine that gives the maximum efficiency that a heat engine can have while between two working reservoirs, it is in the process of conversion of heat into work and conversely.

Carnot Engine Video Explanation

Thermodynamic Carnot Engine was developed to describe conversion of heat to work. A Carnot engine in its reverse operation can also work as a model for heat pumps (refrigerators).
 

Carnot Engine Activity

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Carnot Theorem 

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The Carnot theorem states that ‘The systems working between the two temperatures T₁ (hot reservoir) and T₂ (cold reservoir), cannot have more efficiency than the Carnot engine that is working between the same two reservoirs.’

The efficiency of this engine depends on the temperature of the hot and cold reservoirs and is independent of the nature of the working substance.

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Carnot Cycle 

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The Carnot cycle is an ideal reversible closed thermodynamic cycle, which involves four stages of operations. 

The stages are:

• isothermal expansion, 
• adiabatic expansion, 
• isothermal compression, and 
• adiabatic compression. 

Through these operations, the expansion and compression of substances can be done to the desired point and back to the initial state.

Steps Involved in Carnot Cycle

For an ideal gas operating inside a Carnot cycle, the following are the steps involved:

Step 1:

Isothermal expansion: The ideal gas is taken from P₁, V₁, T₁ to P₂, V₂, T₂. Q₁ amount of heat is absorbed from the reservoir at temperature T₁. The gas then expands and isothermal. Thus the total internal energy change is zero, and the amount of heat absorbed by the gas is equal to the work done by it on the environment.
This is given by the equation: W₁→2=Q₁=μ*R*T₁ *ln(v₂/v₁)

Step 2:

Adiabatic expansion: The gas incurs an expansion of adiabatically from P₂, V₂, T₁ to P₃, V₃, T₂. 

The work done by the gas is given by: W₂→3=μR/(γ−1) *(T₁−T₂)

Step 3:

Isothermal compression: The gas is then compressed isothermally from(P₃, V₃, T₂) to (P₄, V₄, T₂).

Here, the work done by the environment on the gas is given by: W₃→4=μRT₂ln(v₃/v₄)

Step 4:

Adiabatic compression: In this step, the gas is compressed adiabatically from(P₄, V₄, T₂) to (P₁, V₁, T₁).

Hence, the work done on the gas by the environment is given by: W₄→1=μR/(γ−1) * (T₁−T₂)

Hence, the total work done is given by:

W= W₁→2+W₂→3+W₃→4+W₄→1

W = [μRT₁ln(v₂/v₁)] − [μRT₂ln(v₃/v₄)]

Net efficiency=Net work done by the gas / heat absorbed by the gas

Net efficiency=W/Q₁=(Q₁−Q₂)/Q₁ = 1−(Q₂/Q₁) = 1−(T₂/T₁) (lnv₃/v₄) / (lnv₂/v₁)

Since step 2→3 is an adiabatic process, we can write T₁V₂^Y-1-1 = T₂V₃^Y-1-1

Or,

v₂/v₃=(T₂/T₁)^1/γ−1

Similarly, for process 4 →1, we can write

v₁/v₂= (T₂/T₁)^1/γ−1 ⇒ v₂/v₃ = v₁/v₂

Hence, net efficiency can be given as Net efficiency=1−(T₂/T₁)

Steps Involved in Carnot Cycle

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Applications of Carnot Cycle 

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Carnot Cycle is used in the various fields like 

• thermal devices
• heat pumps
• refrigerators
• steam turbines
• Combustion engine
• Reaction turbines of airplane