Interactive demonstration of ideal heat engine cycle
Q₁ = nRT₁ln(V₂/V₁)
TV^(γ-1) = 常数
Q₂ = nRT₂ln(V₄/V₃)
TV^(γ-1) = 常数
η = 1 - T₂/T₁
The Carnot cycle is a theoretical thermodynamic cycle proposed by French physicist Sadi Carnot in 1824. It is the most efficient heat engine cycle theoretically possible. The Carnot cycle consists of two isothermal processes and two adiabatic processes, forming a closed curve on a P-V diagram. Carnot's theorem states: Among all heat engines operating between two constant temperature reservoirs, the reversible engine has the highest efficiency, and the efficiency of a reversible engine depends only on the temperatures of the two reservoirs, not on the working substance.
1. Isothermal Expansion (Heat Absorption): The system contacts the high-temperature reservoir T₁, absorbs heat Q₁ from it, and expands from volume V₁ to V₂ while maintaining constant temperature. During this process, the system does work on the surroundings, and all absorbed heat is converted to work.
2. Adiabatic Expansion: Isolated from heat reservoirs, the system continues to expand from V₂ to V₃, with temperature dropping from T₁ to T₂. During this process, the system consumes internal energy to do work, with no heat exchange.
3. Isothermal Compression (Heat Rejection): The system contacts the low-temperature reservoir T₂, is compressed from V₃ to V₄ at constant temperature. During this process, work is done on the system, and the system rejects heat Q₂ to the low-temperature reservoir.
4. Adiabatic Compression: Isolated from heat reservoirs, the system continues to be compressed from V₄ to V₁, with temperature rising from T₂ to T₁. During this process, work done on the system is entirely converted to internal energy, raising the temperature.
The Carnot efficiency formula is η = 1 - T₂/T₁, where T₁ is the absolute temperature of the high-temperature reservoir and T₂ is the absolute temperature of the low-temperature reservoir. This formula tells us: heat engine efficiency depends only on the temperature difference between the two reservoirs—the larger the difference, the higher the efficiency. Carnot efficiency is the theoretical upper limit for all heat engines operating between the same temperatures; actual heat engines always have lower efficiency. For example, when T₁=500K and T₂=300K, Carnot efficiency is 40%, meaning even the most ideal heat engine can only convert 40% of absorbed heat into useful work, while the remaining 60% must be rejected to the low-temperature reservoir.
An important insight from the Carnot cycle is the concept of entropy. In a complete Carnot cycle, the system absorbs heat Q₁ from the high-temperature reservoir and rejects heat Q₂ to the low-temperature reservoir. Since Q₁/T₁ = Q₂/T₂ (for reversible processes), the system's entropy change is zero. However, the universe's total entropy increases: the high-temperature reservoir loses entropy Q₁/T₁, the low-temperature reservoir gains entropy Q₂/T₂, and since T₂ < T₁, we have Q₂/T₂ > Q₁/T₁, resulting in a positive net entropy change. This demonstrates the second law of thermodynamics: the entropy of an isolated system never decreases, and natural processes always proceed in the direction of increasing entropy.
Carnot's theorem has two important conclusions: (1) All reversible heat engines operating between two constant temperature reservoirs have equal efficiency, independent of the working substance; (2) Among all heat engines operating between the same reservoirs, reversible engines have the highest efficiency, and irreversible engines always have lower efficiency than reversible engines. This theorem laid the foundation for the second law of thermodynamics and also pointed out the fundamental way to improve heat engine efficiency: increase the high-temperature reservoir's temperature or decrease the low-temperature reservoir's temperature.
Although actual heat engines cannot reach Carnot efficiency, the Carnot cycle provides theoretical guidance for heat engine design and optimization. In internal combustion engines, efficiency is improved by increasing combustion temperature (raising T₁) and lowering exhaust temperature (reducing T₂); in steam turbines, superheated steam and multi-stage reheat are used to raise the average heat absorption temperature; in refrigerators, the Carnot cycle gives the theoretical upper limit for the coefficient of performance. The concept of the Carnot cycle extends to other fields, such as reversibility of chemical reactions and energy consumption in information processing (Landauer's principle), making it an important cornerstone of thermodynamic theory.