4.5 The Carnot Cycle

Heat engines are fascinating devices that convert thermal energy into mechanical work. The Carnot cycle, a theoretical model, represents the most efficient possible heat engine. It consists of four key processes: isothermal expansion, adiabatic expansion, isothermal compression, and adiabatic compression.

The efficiency of a Carnot engine depends solely on the temperatures of its hot and cold reservoirs. This concept is crucial for understanding the limits of energy conversion in thermodynamics. Real-world heat engines, while less efficient, strive to approach the Carnot cycle's theoretical maximum efficiency.

The Carnot Cycle and Heat Engine Efficiency

Processes of Carnot cycle

• Isothermal expansion
  • Gas expands at constant temperature $T_h$ while in thermal contact with a hot reservoir (heat source)
  • Gas performs work on surroundings and absorbs heat from hot reservoir, maintaining constant temperature
• Adiabatic expansion
  • Gas continues expanding but is now thermally insulated from surroundings, preventing heat exchange
  • Temperature of gas decreases from $T_h$ to $T_c$ as it performs work on surroundings, converting internal energy to mechanical energy
  • This is an example of an isentropic process
• Isothermal compression
  • Gas is compressed at constant temperature $T_c$ while in thermal contact with a cold reservoir (heat sink)
  • Surroundings perform work on gas, and gas releases heat to cold reservoir, maintaining constant temperature
• Adiabatic compression
  • Gas continues being compressed but is again thermally insulated from surroundings
  • Temperature of gas increases from $T_c$ back to $T_h$ as work is done on gas, converting mechanical energy to internal energy
• Carnot cycle is a reversible process that can be run in reverse as a refrigerator (removes heat from cold reservoir) or heat pump (transfers heat to hot reservoir)
• Carnot cycle represents most efficient possible heat engine operating between two temperatures, serving as a theoretical limit for real engines

Evaluation of heat engine efficiency

• Efficiency of a Carnot heat engine is given by: $\eta = 1 - \frac{T_c}{T_h}$
  • $T_c$ is absolute temperature of cold reservoir (in Kelvin)
  • $T_h$ is absolute temperature of hot reservoir (in Kelvin)
• Efficiency depends only on temperatures of hot and cold reservoirs, not on working substance (gas) or engine design
• Larger temperature difference between hot and cold reservoirs results in higher efficiency (steam engines, internal combustion engines)
• No real heat engine can exceed efficiency of a Carnot engine operating between same temperatures due to irreversibilities (friction, heat loss)
• The thermal efficiency of a heat engine is a measure of how well it converts heat into useful work

Carnot principle vs second law

• Second law of thermodynamics states it is impossible to construct a heat engine that converts all heat it receives into work
  • Some heat must always be released to a cold reservoir, limiting efficiency
• Carnot principle is a consequence of second law, setting an upper limit on efficiency of any heat engine based on reservoir temperatures
• Carnot efficiency represents maximum theoretical efficiency for a heat engine, requiring a perfectly reversible engine (not possible in practice)
• Carnot principle and second law imply no heat engine can be 100% efficient, as some energy will always be lost as heat to environment (waste heat)

Thermodynamic Analysis

• The Carnot cycle is an ideal thermodynamic cycle that represents the most efficient heat engine possible
• A p-V diagram can be used to visualize the Carnot cycle, showing the relationship between pressure and volume during each process
• The concept of entropy is closely related to the Carnot cycle, as it helps explain why the cycle is the most efficient possible