A device that converts thermal energy into work is a heat engine. We will study in detail the Carnot cycle, which gives the maximum theoretical efficiency.
Basic Knowledge
A heat engine absorbs heat QH from a hot reservoir, outputs work W, and releases heat QC to a cold reservoir in a cycle.
Thermal efficiency: η = W / QH = 1 − QC/QH (0 ≤ η < 1)
The Carnot cycle is a reversible cycle consisting of four quasi-static processes:
① Isothermal expansion (absorbs QH at high temperature TH)
② Adiabatic expansion (TH → TC, Q = 0)
③ Isothermal compression (releases QC at low temperature TC)
④ Adiabatic compression (TC → TH, Q = 0)
Carnot efficiency (maximum efficiency): ηC = 1 − TC/TH (T is absolute temperature [K])
The efficiency of any reversible engine operating between the same two temperatures equals the Carnot efficiency, while an irreversible engine is necessarily less efficient.
Thermal efficiency η (η = W/QH = 1 − QC/QH)
Carnot cycle (four processes: isothermal expansion → adiabatic expansion → isothermal compression → adiabatic compression)
Carnot efficiency ηC (ηC = 1 − TC/TH; maximum efficiency of any reversible engine)
Hot reservoir and cold reservoir (TH: side supplying heat, TC: side receiving waste heat)
Refrigerator and heat pump (reverse Carnot cycle: uses work to move heat from low temperature to high temperature)
Further Exploration (Background and Significance)
Carnot efficiency increases as the temperature difference becomes larger. This is why thermal power plants raise steam to high temperatures. Real engines have lower efficiencies than Carnot efficiency because of irreversible factors such as friction and heat conduction (gasoline engines ~25–35%, latest thermal power generation ~40–50%).
The efficiency of a refrigerator is expressed by the coefficient of performance (COP) ε = QC / W (cooling amount / work consumed). The COP of a heat pump is ε' = QH / W = ε + 1. Air conditioners can have a COP of about 3–6, warming rooms more “efficiently” than electric heaters.
The Carnot efficiency equation ηC = 1 − TC/TH is also used in the thermodynamic definition of absolute temperature (the definition of the kelvin).
- η = W/QH = 1 − QC/QH (η < 1 follows from the second law)
- Carnot efficiency ηC = 1 − TC/TH (calculate using absolute temperature)
- Four Carnot-cycle processes: isothermal expansion → adiabatic expansion → isothermal compression → adiabatic compression
- Efficiency of an irreversible engine < Carnot efficiency
- A refrigerator is a reverse Carnot cycle. COP = QC/W
Points to Note (Common Confusions)
① Use absolute temperature T [K] in the Carnot-efficiency formula. Substituting temperatures in ℃ is incorrect. ② Efficiency η = 1 is impossible under the second law. TC = 0 K cannot be achieved. ③ The directions of QC, QH, and W are reversed in heat engines and refrigerators. ④ From W = QH − QC (conservation of energy), we can derive η = 1 − QC/QH.
Practice
- Find the Carnot efficiency when the hot reservoir is TH = 600 K and the cold reservoir is TC = 300 K.
- A heat engine receives 1000 J of heat from a hot source and releases 600 J to a cold source. Find its thermal efficiency.
- State the four processes of the Carnot cycle in order and explain its shape on a pV diagram.