Carnot Cycle

The Carnot cycle is a theoretical thermodynamic cycle that defines the maximum possible efficiency any heat engine can achieve when converting heat energy into mechanical work between two temperature reservoirs. It was proposed in 1824 by Nicolas Léonard Sadi Carnot, a French engineer often regarded as the father of thermodynamics. The Carnot cycle serves as a fundamental model for understanding the principles of heat transfer, entropy, and efficiency in thermal systems.

Concept and Significance

The Carnot cycle represents an idealised reversible process, meaning all changes occur infinitely slowly and without energy losses due to friction, turbulence, or non-equilibrium effects. No real engine can fully achieve these conditions, but the Carnot model establishes the upper limit of efficiency that all real-world heat engines can strive to approach.
The importance of the Carnot cycle lies in demonstrating that efficiency depends only on the temperatures of the heat source and sink, not on the specific working substance or construction of the engine. This principle underpins the Second Law of Thermodynamics, which states that no engine can be more efficient than a Carnot engine operating between the same two temperatures.

Components of the Carnot Engine

A Carnot engine operates between two thermal reservoirs:

• A hot reservoir at a higher absolute temperature THT_HTH​.
• A cold reservoir at a lower absolute temperature TCT_CTC​.

The working substance, often considered an ideal gas, absorbs heat from the hot reservoir, performs work on the surroundings, and rejects waste heat to the cold reservoir. The process is cyclic, returning the system to its initial state at the end of each cycle.
The Carnot cycle consists of four reversible processes, two isothermal (constant temperature) and two adiabatic (no heat exchange) transformations.

Four Stages of the Carnot Cycle

1. Isothermal Expansion (A → B)
   • The working substance is in contact with the hot reservoir at temperature THT_HTH​.
   • Heat QHQ_HQH​ is absorbed from the reservoir.
   • The gas expands slowly and performs work on the surroundings.
   • Since the process is isothermal, the internal energy remains constant, and the heat absorbed equals the work done by the gas:

     QH=WAB=nRTHln⁡VBVAQ_H = W_{AB} = nRT_H \ln\frac{V_B}{V_A}QH​=WAB​=nRTH​lnVA​VB​​

2. Adiabatic Expansion (B → C)
   • The system is now thermally insulated, preventing any heat exchange.
   • The gas continues to expand, performing work at the expense of its internal energy.
   • As a result, the temperature decreases from THT_HTH​ to TCT_CTC​.
   • For an ideal gas, the relation between temperature and volume during adiabatic expansion is:

     THVBγ−1=TCVCγ−1T_H V_B^{\gamma – 1} = T_C V_C^{\gamma – 1}TH​VBγ−1​=TC​VCγ−1​
     where γ=CP/CV\gamma = C_P / C_Vγ=CP​/CV​ (ratio of specific heats).

3. Isothermal Compression (C → D)
   • The gas comes into contact with the cold reservoir at temperature TCT_CTC​.
   • Heat QCQ_CQC​ is rejected to the cold reservoir.
   • The gas is compressed isothermally, and the surroundings do work on the gas.
   • Since internal energy remains constant:

     QC=WCD=nRTCln⁡VDVCQ_C = W_{CD} = nRT_C \ln\frac{V_D}{V_C}QC​=WCD​=nRTC​lnVC​VD​​
     Here, QCQ_CQC​ is a negative quantity (heat released).

4. Adiabatic Compression (D → A)
   • The system is again thermally insulated.
   • The gas is compressed without heat exchange, causing its temperature to rise from TCT_CTC​ back to THT_HTH​.
   • The relation during adiabatic compression is:

     TCVDγ−1=THVAγ−1T_C V_D^{\gamma – 1} = T_H V_A^{\gamma – 1}TC​VDγ−1​=TH​VAγ−1​

After this step, the gas returns to its initial pressure, volume, and temperature, completing the cycle.

Work Done and Efficiency

The net work done (W) by the Carnot engine in one cycle is the difference between the heat absorbed from the hot reservoir and the heat rejected to the cold reservoir:
W=QH−QCW = Q_H – Q_CW=QH​−QC​
The efficiency (η) of the Carnot cycle, defined as the ratio of work output to heat input, is given by:
η=WQH=1−QCQH\eta = \frac{W}{Q_H} = 1 – \frac{Q_C}{Q_H}η=QH​W​=1−QH​QC​​
Since the cycle is reversible and depends only on the reservoir temperatures, the heat exchanged during the isothermal processes is proportional to the absolute temperatures:
QCQH=TCTH\frac{Q_C}{Q_H} = \frac{T_C}{T_H}QH​QC​​=TH​TC​​
Therefore, the Carnot efficiency becomes:
η=1−TCTH\eta = 1 – \frac{T_C}{T_H}η=1−TH​TC​​
This fundamental relation shows that efficiency depends solely on the two reservoir temperatures (measured in Kelvin). It also reveals that:

• Efficiency increases as THT_HTH​ rises or TCT_CTC​ decreases.
• No engine can achieve 100% efficiency unless TC=0 KT_C = 0 \, KTC​=0K, which is physically unattainable.

Representation on P–V and T–S Diagrams

The Carnot cycle can be visualised on thermodynamic diagrams:

• Pressure–Volume (P–V) Diagram: The isothermal expansion and compression appear as hyperbolic curves, while the adiabatic processes appear as steeper curves. The area enclosed by the cycle represents the net work done.
• Temperature–Entropy (T–S) Diagram: The isothermal processes are represented by horizontal lines (constant temperature, heat transfer occurs), and the adiabatic processes appear as vertical lines (no heat transfer). The area enclosed in the T–S diagram equals the net work done per cycle.

Reversed Carnot Cycle

When the Carnot cycle operates in reverse, it functions as a refrigerator or heat pump. In this mode, external work is applied to transfer heat from the colder reservoir to the hotter one. The reversed Carnot cycle thus represents the most efficient theoretical refrigeration system.
The coefficient of performance (COP) of the reversed cycle is given by:

• For a refrigerator:

  COPrefrigerator=TCTH−TC\text{COP}_{\text{refrigerator}} = \frac{T_C}{T_H – T_C}COPrefrigerator​=TH​−TC​TC​​

• For a heat pump:

  COPheat pump=THTH−TC\text{COP}_{\text{heat pump}} = \frac{T_H}{T_H – T_C}COPheat pump​=TH​−TC​TH​​

Importance in Thermodynamics

The Carnot cycle forms the cornerstone of modern thermodynamics for several reasons:

• It defines the maximum efficiency possible for any heat engine.
• It establishes the concept of reversible processes as ideal benchmarks.
• It provides a framework for understanding entropy and the direction of natural processes.
• It leads to the formulation of the Second Law of Thermodynamics, demonstrating that no heat engine operating between two reservoirs can exceed Carnot efficiency.

Furthermore, the cycle highlights that real engines—such as steam turbines, gas turbines, and internal combustion engines—operate with efficiencies lower than the Carnot limit due to irreversibilities like friction, turbulence, finite temperature differences, and material constraints.

Real-World Applications

Although no real engine perfectly follows the Carnot cycle, it provides a benchmark for evaluating practical systems. Engineers and scientists use Carnot efficiency to assess the performance of:

• Steam power plants, where water and steam act as working fluids.
• Gas turbines, comparing real efficiencies to the Carnot limit.
• Refrigeration and air-conditioning systems, designed to approach the ideal reversed Carnot efficiency.

In addition, the Carnot principle underlies the development of high-efficiency energy systems, such as cryogenic engines and low-temperature superconducting systems, where minimising heat loss is crucial.

Limitations

While the Carnot cycle represents the ideal standard, it is not achievable in practice for several reasons:

• Real processes are never perfectly reversible.
• Finite time and temperature gradients are required for heat transfer.
• Mechanical friction and thermal resistance always introduce energy losses.
• Materials cannot withstand the extreme temperature differences required for maximum efficiency.

Despite these limitations, the Carnot cycle remains a foundational theoretical construct, guiding the design and optimisation of real-world engines.