Carnot Engines and its Efficiency

A speculative Carnot engine runs on the Carnot cycle. Nicolas Léonard Sadi Carnot created the fundamental model for this engine in 1824. Benoît Paul Émile Clapeyron graphically enlarged the Carnot engine model in 1834, and Rudolf Clausius theoretically examined it in 1857, a work that led to the fundamental thermodynamic idea of entropy.

Every thermodynamic system has a specific state. When a system is brought through a sequence of stages before being returned to its initial condition, it is called a thermodynamic cycle. The Carnot engine may work on its surroundings while going through this cycle, thereby operating as a heat engine.

A heat engine moves energy from a warm part of space to an incredible region of space, transforming some of it into mechanical work in the process. The cycle can also be turned around. An external force can work on the system, allowing it to move thermal energy from a cooler to a warmer system, thereby working as a refrigerator or heat pump rather than a heat engine.

There are “two bodies A and B, kept each at a constant temperature, that of A being greater than that of B,” according to Carnot’s 1824 work, “Reflections on the Motive Power of Fire”. These two bodies serve as endless caloric reservoirs since we can provide or remove heat from them without causing their temperatures to alter. The one will be referred to as the furnace, while the second will be referred to as the refrigerator.” Carnot goes on to explain how we can get motive power, or “work,” by transporting a specific amount of heat from body A to body B. It can also be used as a refrigerator because it functions as a cooler.

Only cyclical machines, such as heat engines, are subject to Carnot principles, which state:

• The efficiency of an irreversible heat engine working between the same two reservoirs is always lower than that of a reversible one.
• The efficiency of all reversible heat engines that operate between two reservoirs is the same.

Carnot Theorem

This theorem states that no engine operating between two known temperatures can be more efficient than a reversible engine operating between the same two temperatures. All reversible engines operating between the same two temperatures, regardless of the working material, have the same efficiency. According to the Carnot theorem, a reversible engine will consistently outperform an irreversible one in productivity. The reversible heat engine is a heat pump that operates on a reverse cycle.

The change in entropy of the “working fluid” system is 0 since the “working fluid” returns to the same state after one cycle and the entropy of the system is a state function. As a result, for the process to be reversible and the engine’s efficiency to be optimum, the total entropy change of the furnace and sink must be zero. This derivation is completed in the following section.

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

Carnot Cycle

The Carnot cycle is a theoretical ideal thermodynamic cycle that determines the maximum efficiency of any classical thermodynamic engine when converting heat to work or the maximum efficiency of a refrigeration system when establishing a temperature differential by applying work to the system. It is a theoretical construct rather than a natural thermodynamic cycle. 

This theorem states that no engine operating between two known temperatures can be more efficient than a reversible engine operating between the same two temperatures. All reversible engines operating between the same two temperatures, regardless of the working material, have the same efficiency. According to the Carnot theorem, a reversible engine will consistently outperform an irreversible one in productivity. The reversible heat engine is a heat pump that operates on a reverse cycle.

The Carnot cycle is reversible, denoting an engine cycle’s maximum efficiency. Because practical engine cycles are irreversible, they have a substantially lower efficiency than Carnot while operating at the same temperatures. The addition and removal of the working fluid in the cycle are aspects that determine efficiency. Because all of the heat is directed to the working fluid at its highest temperature, the Carnot cycle achieves optimum efficiency.

There are four processes in  the Carnot cycle as follows:

Ist process: Isothermal gas expansion can be reversed. The ideal gas in the system receives qin quantity of heat from a heat source at a high-temperature Thigh, expands, and works on its surroundings in this process. In an isothermal process, no change in internal energy so that work done is equal to absorbed heat.

IInd process: A process of reversible adiabatic gas expansion. The system is thermally insulated throughout this process. Tlow explains that the gas conti expands and operates, causing the system to lower temperatures.

IIIrd process: Isothermal gas compression can be reversed. In this phase, the surroundings work on the gas at Tlow, causing a heat loss, qout. In this compression work, done will be equal to the rejected heat.

IVth process: The adiabatic gas compression process is reversible. The system is thermally insulated throughout this process. The surrounding environment continues to work on the gas, causing the temperature to increase back to the Thigh.

In the Carnot cycle, the total work done by the gas is given by

W = W1+ W2+W3+W4

Here W1 is work done in isothermal expansion

W2 is work done in adiabatic expansion

W3 is work done in isothermal compression

W4 is work done in adiabatic compression

The efficiency of the Carnot cycle is given by the following expression:

= 1- TcTh

Here Tc is the temperature of the  cold reservoir

Th is the temperature of the  hot reservoir

Conclusion:

Thus Carnot theorem states that no engine operating between two known temperatures can be more efficient than a reversible engine operating between the same two temperatures. All reversible engines operating between the same two temperatures, regardless of the working material, have the same efficiency.