Carnot Cycle

How the Carnot cycle works

Take a piston cylinder containing an ideal gas. Run it through four reversible processes in sequence and return to the start. Two of those processes exchange heat at constant temperature; the other two are perfectly insulated. That's the Carnot cycle.

1. Isothermal expansion at T_h. The cylinder sits on a hot reservoir. Gas expands slowly, absorbing heat Q_h while pushing the piston. Temperature stays at T_h because heat flows in as fast as work flows out.
2. Adiabatic expansion. The cylinder is now insulated. Gas keeps expanding, but no heat enters. Internal energy drops, so temperature falls from T_h to T_c.
3. Isothermal compression at T_c. The cylinder contacts a cold reservoir. The piston compresses the gas, doing work on it. Heat Q_c flows out into the cold reservoir, holding temperature at T_c.
4. Adiabatic compression. Insulation back on. Compression continues until temperature climbs back to T_h. The cycle closes.

Net work output equals Q_h − Q_c (energy conservation). Efficiency is what you got out divided by what you paid for: η = W/Q_h = (Q_h − Q_c)/Q_h = 1 − Q_c/Q_h.

Carnot's insight was that for a reversible cycle, Q_c/Q_h = T_c/T_h exactly — a result that essentially defines absolute temperature. So the maximum possible efficiency between two reservoirs is:

           T_c
   η = 1 − ───
           T_h     (temperatures in kelvin)

Worked example: 600 K vs 300 K

Consider a heat engine with a hot reservoir at T_h = 600 K (327°C, roughly the steam temperature in an older thermal plant) and a cold reservoir at T_c = 300 K (27°C, ambient cooling water).

   η = 1 − T_c/T_h
     = 1 − 300/600
     = 1 − 0.5
     = 0.50  →  50%

That's the absolute ceiling. The best modern combined-cycle power plants — which cleverly stack a Brayton (gas turbine) cycle on top of a Rankine (steam) cycle to widen the temperature span — reach about 64% efficiency, against a Carnot bound near 75% for their actual T_h ≈ 1700 K and T_c ≈ 300 K. They lose the gap to friction, finite-rate heat transfer, exhaust losses, and combustion irreversibility.

The cycle on a P–V diagram

      P
      │
      │   1●─────● 2     isothermal at T_h (top arc)
      │   ╱       ╲
      │  ╱  Q_h    ╲     adiabatic ↓
      │ ╱            ╲
      │●              ●  4
      │ ╲            ╱
      │  ╲   Q_c    ╱
      │   ╲       ╱      adiabatic ↑
      │   3●─────●       isothermal at T_c (bottom arc)
      │
      └─────────────── V
       enclosed area = net work W

The same cycle on a T–S (temperature–entropy) diagram is a clean rectangle: two horizontal isotherms at T_h and T_c bridged by two vertical adiabats. The rectangle's area equals W; this is the cleanest way to see why no other cycle between the same two temperatures can do better.

Why real engines fall short

Carnot is reversible. Reversible means every step proceeds through a continuous sequence of equilibrium states — temperature differences across heat exchangers shrink to zero, friction vanishes, and the piston moves infinitely slowly. That ideal yields zero power. Useful engines accept irreversibility in exchange for finite power output:

• Finite ΔT in heat exchangers. Heat moves at a rate proportional to ΔT, so usable heat-transfer rates require non-zero temperature gaps, which generate entropy.
• Friction. Pistons, bearings, fluid viscosity — every moving part dissipates work as heat.
• Combustion losses. In any chemical-energy engine, mixing fuel and air at finite rate is a one-way irreversible process.
• Exhaust losses. Open-cycle engines (jet, gasoline) discard hot exhaust gas instead of cooling it to ambient — a deliberate trade for power density.

The endoreversible Curzon–Ahlborn analysis (1975) gives a more practical upper bound for power-producing engines: η_CA = 1 − √(T_c/T_h). For 600 K/300 K that's 1 − √0.5 ≈ 29%, which matches real fossil-plant performance far better than the 50% Carnot figure.

Carnot vs other thermodynamic cycles

Cycle	Strokes	Working fluid	Real-world use	Typical max efficiency	Carnot ratio
Carnot	2 isothermal, 2 adiabatic	Any	Theoretical reference only	50–75% (theoretical)	1.00
Otto (gasoline)	2 isochoric, 2 adiabatic	Air–fuel	Car engines	25–35%	~0.45
Diesel	1 isobaric, 1 isochoric, 2 adiabatic	Air–fuel	Trucks, ships, generators	40–50%	~0.65
Brayton (gas turbine)	2 isobaric, 2 adiabatic	Air	Jet engines, power gas turbines	35–42% (simple)	~0.55
Rankine (steam)	2 isobaric, 1 adiabatic, 1 phase-change	Water/steam	Coal, nuclear, geothermal plants	33–48%	~0.55
Stirling	2 isothermal, 2 isochoric	Gas (He, H₂)	Cryocoolers, niche generators	30–40%	~0.65
Combined cycle	Brayton + Rankine	Air + steam	Modern gas-fired plants	60–64%	~0.85

Carnot's 1824 result is more useful as a yardstick than as a blueprint. Every entry above is graded by how close it gets — combined-cycle plants are the current champion at roughly 85% of their Carnot ceiling.

The reversed cycle: heat pumps and refrigerators

Run a Carnot cycle backwards — pump heat from cold to hot — and you get the ideal refrigerator. Its coefficient of performance is COP = T_c/(T_h − T_c). For a freezer at 250 K dumping into a 300 K kitchen, COP = 250/50 = 5: each joule of compressor work moves 5 joules of heat. Real fridges hit COP 2–4; air-conditioner-style heat pumps in heating mode reach COP 3–5 against modest outdoor temperatures.

Common failure modes in real Carnot-equivalent equipment

• Heat exchanger fouling. Mineral scale, biofilm, or particulate buildup on tube walls increases the temperature gap needed to transfer the same heat. A 1 mm scale layer on a condenser can cost 5–10% in cycle efficiency. Power plants schedule chemical cleanings every 12–24 months.
• Air infiltration into condensers. A few percent air in steam reduces heat-transfer coefficient sharply, raises condenser pressure, and shrinks the effective T_h − T_c span.
• Cooling-water temperature creep. A 5°C rise in river or sea cooling water during a heat wave directly raises T_c; a plant rated for 35% efficiency may drop to 33%, costing megawatts.
• Working-fluid leaks in heat pumps. Refrigerant loss reduces mass flow, depresses evaporator pressure, and pulls COP toward 1 (zero net heat moved per joule of work).

Historical context: 1824 to today

Sadi Carnot, a 28-year-old French military engineer, published Réflexions sur la puissance motrice du feu ("Reflections on the motive power of fire") in 1824. He framed the central question — what is the upper limit on a heat engine's efficiency? — and derived an answer using a then-fashionable but soon-discredited theory of heat as a fluid called caloric. Despite the wrong microphysics, the bookkeeping was right.

The work was largely ignored until Lord Kelvin and Rudolf Clausius reformulated it in the 1850s using energy and entropy. Kelvin used Carnot's reversible cycle to define the absolute temperature scale that bears his name: two reservoirs are in the ratio T_1/T_2 if a reversible engine running between them rejects heat in that same ratio. The Carnot cycle thus underpins the entire definition of thermodynamic temperature.