Carnot Engine: Introduction, Theorem, Cycle and Steps

Introduction

The Carnot engine is a theoretical thermodynamic cycle that is often used as a benchmark for the efficiency of real-world engines. It was first proposed by Nicolas Léonard Sadi Carnot in 1824 in his book “Reflections on the Motive Power of Fire”.

The Carnot engine consists of four processes: two isothermal processes, during which the temperature remains constant, and two adiabatic processes, during which there is no heat exchange between the system and its surroundings. The engine operates between two heat reservoirs, a hot reservoir at temperature TH and a cold reservoir at temperature TC, with TH > TC. The working fluid is usually a gas, such as hydrogen, helium or air.

The Carnot engine works as follows. In the first process, the gas is compressed adiabatically and its temperature increases. In the second process, the gas is brought into thermal contact with the hot reservoir and expands isothermally, doing work on the surroundings. In the third process, the gas expands adiabatically and its temperature decreases. Finally, in the fourth process, the gas is brought into thermal contact with the cold reservoir and is compressed isothermally, expelling heat to the surroundings.

The efficiency of the Carnot engine is given by the ratio of the work done by the engine to the heat absorbed from the hot reservoir:

η = (TH – TC) / TH

This efficiency is the highest possible for an engine operating between two heat reservoirs at temperatures TH and TC.

The Carnot engine is a theoretical idealization and cannot be constructed in practice, but it is a useful benchmark for real-world engines. Real engines always have some losses due to friction, heat transfer, and other factors, and their efficiencies are always lower than that of the Carnot engine. The Carnot cycle is also used as the basis for the development of the thermodynamic concept of entropy, which plays a fundamental role in modern physics and engineering.

Carnot Theorem

Carnot’s theorem, also known as Carnot’s principle or Carnot’s law, is a fundamental concept in thermodynamics. It states that the maximum efficiency of a heat engine operating between two fixed temperature reservoirs is dependent only on the temperatures of the reservoirs and not on the working substance or the details of the engine’s design.

The theorem was first proposed by French physicist Nicolas Léonard Sadi Carnot in his 1824 book “Réflexions sur la puissance motrice du feu” (“Reflections on the Motive Power of Fire”). Carnot’s work was based on the idea that heat is a form of energy that can be transformed into mechanical work, and that this transformation can be optimized by designing an engine that operates in a reversible cycle.

Carnot’s theorem states that the efficiency of a heat engine is given by the difference in temperature between the hot and cold reservoirs divided by the temperature of the hot reservoir. This is known as the Carnot efficiency and is expressed mathematically as:

η = 1 – (Tc/Th)

where η is the efficiency of the engine, Tc is the temperature of the cold reservoir, and Th is the temperature of the hot reservoir.

The significance of Carnot’s theorem lies in the fact that it sets an upper limit on the efficiency of any heat engine, regardless of its design or the working substance it uses. This limit is known as the Carnot limit and can only be approached but not exceeded. This means that any practical engine operating between two given temperatures will have an efficiency that is less than the Carnot efficiency.

Carnot’s theorem has important implications for the design and optimization of heat engines, and has played a central role in the development of thermodynamics as a field of study. It is also the basis for the development of the second law of thermodynamics, which states that the total entropy of a closed system always increases over time.

Carnot Cycle

The Carnot Cycle is a theoretical thermodynamic cycle that was developed by Nicolas Léonard Sadi Carnot in the early 19th century. It is considered to be the most efficient cycle possible for converting heat into work, and it forms the basis for the modern study of thermodynamics.

The Carnot Cycle is efficient because it operates between two heat reservoirs with different temperatures, which allows for the maximum possible work to be extracted from the system. The efficiency of the cycle is given by the ratio of the work output to the heat input, which is equal to 1 – (Tlow/Thigh), where Tlow is the temperature of the low-temperature reservoir and Thigh is the temperature of the high-temperature reservoir.

The Carnot Cycle is an idealized cycle, and no real cycle can be perfectly reversible.

Steps involved in a Carnot Cycle

The Carnot cycle is a theoretical thermodynamic cycle that consists of four reversible processes, namely, isothermal expansion, adiabatic expansion, isothermal compression, and adiabatic compression. It is an idealized cycle that represents the maximum possible efficiency that a heat engine can achieve, operating between two heat reservoirs at different temperatures. Here are the steps involved in a Carnot cycle:

1. Isothermal Expansion: The cycle begins with the system being in contact with a high-temperature reservoir, and the working substance (usually a gas) expands isothermally, i.e., at constant temperature, by absorbing heat from the reservoir. The heat absorbed by the system is given by QH = nRTln(V2/V1), where QH is the heat absorbed, n is the number of moles of the gas, R is the gas constant, T is the temperature of the reservoir, and V1 and V2 are the initial and final volumes of the gas.
2. Adiabatic Expansion: The system is then insulated, and the gas expands adiabatically, i.e., without exchanging heat with its surroundings, while doing work on its surroundings. The temperature and pressure of the gas decrease during this process, and the work done by the gas is given by W1 = (γ/(γ-1))nR(T1-T2), where γ is the specific heat ratio of the gas, T1 is the initial temperature of the gas, and T2 is the temperature at the end of the expansion.
3. Isothermal Compression: The gas is brought into contact with a low-temperature reservoir, and it is compressed isothermally, i.e., at constant temperature, while releasing heat to the reservoir. The heat released by the system is given by QC = nRTln(V3/V4), where QC is the heat released, V3 and V4 are the initial and final volumes of the gas, and T is the temperature of the reservoir.
4. Adiabatic Compression: The system is insulated again, and the gas is compressed adiabatically, i.e., without exchanging heat with its surroundings, while work is done on it by its surroundings. The temperature and pressure of the gas increase during this process, and the work done on the gas is given by W2 = (γ/(γ-1))nR(T4-T3), where T3 is the temperature at the end of the isothermal compression, and T4 is the final temperature of the gas.

The net work done by the system during a Carnot cycle is given by the difference between the work done during the expansion and compression, i.e., Wnet = W1 – W2, and the efficiency of the cycle is given by η = Wnet/QH, where QH is the heat absorbed by the system during the isothermal expansion. The efficiency of a Carnot cycle is independent of the working substance used and depends only on the temperatures of the two reservoirs, making it an idealized cycle for maximum efficiency heat engines.

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