Carnot Engine, Carnot Cycle & Efficiency
An ideal reversible heat engine, the four thermodynamic steps of the Carnot cycle, and efficiency derivation.
1. What is a Carnot Engine?
Proposed by the French engineer Nicolas Léonard Sadi Carnot in 1824, it serves as the theoretical benchmark for all real heat engines. Because it assumes ideal conditions (zero friction, non-conducting walls, and perfectly reversible processes), a Carnot engine cannot be built in practice.
Main Components:
- Cylinder & Piston: A cylinder fitted with a perfectly frictionless, non-conducting piston. The side walls are completely non-conducting, while the base is a perfect thermal conductor.
- Working Substance: An ideal gas enclosed inside the cylinder.
- Hot Reservoir (Source): A heat source kept at a constant high temperature T1 (in Kelvin).
- Cold Reservoir (Sink): A heat sink kept at a constant lower temperature T2 (in Kelvin).
- Insulating Stand: A perfectly non-conducting stand used to isolate the cylinder during adiabatic processes.
2. The Carnot Cycle
The Carnot engine operates on a closed reversible cycle consisting of four successive thermodynamic processes:
-
Isothermal Expansion:
The cylinder base is placed on the heat source at temperature T1. The gas is allowed to expand slowly. As the piston moves outward, the gas tends to cool down, but heat Q1 is continuously absorbed from the source to keep the temperature constant at T1. Pressure decreases while volume increases.
-
Adiabatic Expansion:
The cylinder is moved from the source onto the non-conducting insulating stand. The gas continues to expand slowly without exchanging any heat with the surroundings (Q = 0). The work done by the expanding gas causes its internal energy to fall, reducing its temperature from T1 down to T2.
-
Isothermal Compression:
The cylinder is placed on the heat sink at temperature T2. The piston is pushed inward to compress the gas slowly. Compression tends to heat the gas, but excess heat Q2 is rejected to the sink, maintaining a constant temperature T2. Pressure increases while volume decreases.
-
Adiabatic Compression:
The cylinder is placed back onto the insulating stand. The gas is further compressed adiabatically (no heat enters or leaves). Work done on the gas increases its internal energy, raising its temperature back from T2 to T1. The working substance returns to its initial state, completing one full cycle.
3. Efficiency of a Carnot Engine
The efficiency (η) of any heat engine is defined as the ratio of net mechanical work output to the total heat energy absorbed from the source:
η = Output Work / Input Heat = W / Q1 ——— (Equation 1)
From the First Law of Thermodynamics, for a complete cyclic process, the net work done equals the net heat absorbed:
W = Q1 - Q2
Substituting this value of W into Equation 1:
η = (Q1 - Q2) / Q1
η = 1 - (Q2 / Q1) ——— (Equation 2)
In Terms of Temperature:
For a reversible Carnot cycle, the amount of heat absorbed or rejected is directly proportional to the absolute temperature of the source and sink respectively:
Q2 / Q1 = T2 / T1
Replacing heat ratio with absolute temperature ratio in Equation 2 gives the final expression for Carnot efficiency:
Percentage Efficiency: η (%) = [1 - (T2 / T1)] × 100
Where:
- T1: Absolute temperature of Source (in Kelvin, K)
- T2: Absolute temperature of Sink (in Kelvin, K)
For efficiency η to reach 100% (1.0), the term (T2 / T1) must equal zero. This would require either:
- The sink temperature to be at absolute zero (T2 = 0 K), or
- The source temperature to be infinitely high (T1 = ∞).
Since absolute zero (0 K) is practically unattainable and an infinite temperature source does not exist, no heat engine—even an ideal Carnot engine—can ever achieve 100% efficiency. This is a direct consequence of the Second Law of Thermodynamics.
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