Carnot Engine

The Carnot cycle

Four reversible steps:

Step	Process	Heat	Work	Entropy
1 → 2	Isothermal expansion at T_h	Q_h absorbed (positive)	Work done by gas (positive)	ΔS_gas = +Q_h/T_h
2 → 3	Adiabatic expansion	Q = 0	Work done by gas (positive)	0
3 → 4	Isothermal compression at T_c	Q_c released (positive value)	Work done on gas (negative for gas)	ΔS_gas = -Q_c/T_c
4 → 1	Adiabatic compression	Q = 0	Work done on gas	0

Total cycle: net work W = Q_h - Q_c. Total ΔS_gas = 0 (returns to start). Reversible — no entropy generated.

Carnot efficiency

η_Carnot = W / Q_h = 1 − Q_c / Q_h = 1 − T_c / T_h

(Both T in Kelvin.) The form 1 - T_c/T_h follows from reversibility: Q_c/Q_h = T_c/T_h.

T_h	T_c	η_Carnot
500 K	300 K	40%
800 K	300 K	62.5%
1500 K	300 K	80%
2200 K (combustion)	300 K	86.4%
5800 K (Sun's surface)	300 K	94.8%

Higher T_h or lower T_c → higher efficiency. This is why power plants seek high boiler temperatures and cool condenser temperatures.

Real engines vs Carnot

System	T_h	T_c	η_Carnot	Real η	η/η_Carnot
Steam locomotive	500 K	373 K	25%	~10%	40%
Modern coal plant	900 K	310 K	66%	~40%	61%
Combined cycle gas	1700 K	310 K	82%	~60%	73%
Internal combustion (gasoline)	~2200 K	~600 K	73%	~25%	34%
Diesel	~2200 K	~600 K	73%	~35%	48%
Refrigerator (room T)	—	—	COP ~14	COP ~3-5	~25%

JavaScript — Carnot calculations

// Carnot efficiency
function carnotEfficiency(T_hot_K, T_cold_K) {
  return 1 - T_cold_K / T_hot_K;
}

// Carnot refrigerator COP
function carnotRefrigCOP(T_cold_K, T_hot_K) {
  return T_cold_K / (T_hot_K - T_cold_K);
}

// Carnot heat pump COP
function carnotHeatPumpCOP(T_cold_K, T_hot_K) {
  return T_hot_K / (T_hot_K - T_cold_K);
}

// Sample calculations
console.log(`Steam plant (550°C, 30°C): ${(carnotEfficiency(823, 303) * 100).toFixed(1)}%`); // ~63%
console.log(`Solar thermal (350°C, 30°C): ${(carnotEfficiency(623, 303) * 100).toFixed(1)}%`); // ~51%

console.log(`Fridge (4°C, 25°C): COP_ref = ${carnotRefrigCOP(277, 298).toFixed(1)}`);
console.log(`Heat pump (-5°C outside, 22°C inside): COP_hp = ${carnotHeatPumpCOP(268, 295).toFixed(1)}`);

// Real efficiency expressed as fraction of Carnot
function efficiencyFraction(real_eta, T_hot, T_cold) {
  return real_eta / carnotEfficiency(T_hot, T_cold);
}

console.log(`Modern diesel (35%): ${(efficiencyFraction(0.35, 2200, 600) * 100).toFixed(0)}% of Carnot`);

// Carnot work output for given Q_h
function carnotWork(Q_hot, T_hot, T_cold) {
  return Q_hot * carnotEfficiency(T_hot, T_cold);
}

// Heat 1000 J at 800 K, dump to 300 K
console.log(`Carnot work from 1000 J at 800K: ${carnotWork(1000, 800, 300).toFixed(0)} J`);  // 625

// Min cold reservoir for given efficiency
function minColdReservoir(efficiency, T_hot) {
  // η = 1 - T_c/T_h → T_c = T_h * (1 - η)
  return T_hot * (1 - efficiency);
}

console.log(`For 70% efficiency at 800K hot: T_cold < ${minColdReservoir(0.7, 800)} K`); // 240 K (-33°C)

Where Carnot matters

• Engine and power plant design. Sets theoretical max efficiency; engineers optimize toward this limit.
• Refrigeration and heat pumps. Carnot COP is the upper bound for cooling/heating efficiency.
• Solar thermal energy. High-temperature receivers approach Carnot limits at concentrated solar plants.
• Geothermal and OTEC (ocean thermal). Low ΔT means low Carnot efficiency; OTEC (~25°C ΔT) has fundamental limit ~7%.
• Cryogenics. Liquefying gases (helium, nitrogen) requires substantial energy; Carnot bounds efficiency of cryocoolers.
• Engineering education. Foundational concept — sets a "good vs perfect" benchmark for all heat engines.
• Theoretical thermodynamics. Used to derive other key relationships (Clausius inequality, entropy, free energy).

Common mistakes

• Using Celsius in η = 1 - T_c/T_h. MUST be Kelvin. Using Celsius gives wrong (and possibly negative) efficiency.
• Treating Carnot as practical efficiency goal. It's a theoretical limit. Aim for 70-80% of Carnot in real engines; 100% requires infinite time.
• Confusing Carnot for an engine type. Carnot is an IDEALIZATION — no real engine is Carnot. Real engines (Otto, Diesel, Brayton, Rankine cycles) have different cycles.
• Forgetting it's reversible. Carnot assumes infinite time. Faster operation (real engines) generates entropy — efficiency drops.
• Conflating engine efficiency with refrigerator COP. Engines have η ≤ 1 (always less than 1). Refrigerators have COP > 1 typically (they MOVE heat efficiently, not convert it). Different metrics.
• Treating Carnot as universal upper bound. It's the bound for engines with TWO heat reservoirs at fixed T_h, T_c. Some advanced systems (multi-stage, fuel cells using chemical energy directly) can exceed Carnot in specific configurations.