This Carnot efficiency calculator computes η = 1 − T꜀/Tₕ — the maximum efficiency any heat engine operating between a hot reservoir at Tₕ and a cold reservoir at T꜀ can ever achieve, with absolute (Kelvin) temperatures. It also returns the reversed cycle’s limits: the Carnot refrigerator and heat-pump COPs, and the work/heat split for a given heat input.

–

The Ceiling Nobody Negotiates With

Carnot’s theorem (1824) is thermodynamics’ most quoted insult to engineering ambition: between two reservoirs, NO engine — steam, Stirling, photon-driven, yet-uninvented — can beat η = 1 − T꜀/Tₕ, and every reversible engine ties at exactly that figure. The defaults run a respectable industrial case: 600 K (327 °C) steam against a 300 K environment caps at 50.0% — half the fuel’s heat MUST be rejected to the cold side even by a perfect engine, and real plants at these temperatures deliver 35–40%. Push Tₕ to a combined-cycle’s ~1,900 K and the ceiling lifts to 84%; drop a geothermal source to 450 K against 300 K and it collapses to 33% before a single turbine blade is drawn. The formula is why power engineering is a materials war over Tₕ (superalloys, ceramic coatings, cooling passages) fought for single percentage points.

Running the cycle backwards splits the same ratio into COPs: a refrigerator between 300/270 K enjoys a Carnot COP of 9 (real units: 2–4), a heat pump 10 — the numbers behind heat pumps’ “300–400% efficiency” headlines (they MOVE heat at COP 3–4; resistance heaters convert at exactly 1). And the rejected-heat line matters at civilizational scale: Q꜀ is the cooling tower, the river-warming, the reason thermal plants site on water.

Worked Examples (Verified)

  • Defaults: η = 1 − 300/600 = 50.0%; from Qₕ = 1,000 J: W = 500 J, Q꜀ = 500 J. COPs: fridge 300/300 = 1.00… note this is a poor fridge (ΔT huge) — at 270/300 K the fridge COP is 9.0, heat pump 10.0.
  • Car engine reality check: peak gas temperature ~2,500 K, exhaust environment ~350 K → ceiling 86%; delivered tank-to-wheel ~25–30% — combustion irreversibility, friction and part-load operation consume the gap (the ceiling is necessary, never sufficient).
  • The Kelvin trap: boiler 300 °C, condenser 100 °C: in Kelvin 573/373 → η = 34.9%. The naive Celsius ratio suggests 66.7% — a factor-of-two fantasy from one skipped conversion; the calculator refuses non-Kelvin physics by labeling inputs K.

Carnot Discipline

  • Kelvin, always — the theorem is about absolute temperature ratios; even Rankine works (also absolute), Celsius and Fahrenheit do not.
  • The bound assumes reservoirs that STAY at Tₕ/T꜀ (infinite heat capacity idealization); finite streams that cool as they give heat have lower (exergy-based) limits.
  • Endoreversible reality: maximum-POWER efficiency (Curzon–Ahlborn, 1 − √(T꜀/Tₕ)) sits below Carnot — engines optimized for output rather than efficiency land near it; know which game your engine is playing.
  • Carnot efficiency is about heat engines; fuel cells and photovoltaics aren’t bound by it (they’re not heat engines) — comparing their efficiencies to Carnot ceilings is a category error examiners love.
  • Everyday consequence: the rejected fraction Q꜀/Qₕ = T꜀/Tₕ is a floor on waste heat — data centers and plants budget cooling for it before anything is built.

Related Tools

Frequently Asked Questions

How do you calculate Carnot efficiency?

η = 1 − T꜀/Tₕ with both temperatures in Kelvin. Defaults: 1 − 300/600 = 50.0%. Multiply by heat input for maximum work (500 J of each 1,000 J here); the rest is necessarily rejected heat. This calculator adds the reversed-cycle COP limits and the full work/heat split.

Why can’t any engine beat the Carnot efficiency?

Because beating it would break the second law: pair a super-Carnot engine with a reversed Carnot refrigerator and the combination would pump heat from cold to hot with no net work input — a perpetual-motion machine of the second kind. Carnot’s theorem is proved by that contradiction, so the limit binds every working fluid and every mechanism: it’s a property of temperature itself, not of steam or pistons.

Why must temperatures be in Kelvin?

The derivation uses ratios of absolute temperatures (through the ideal-gas and entropy relations), and only an absolute scale has a true zero. At 573 K / 373 K the ceiling is 34.9%; the same state described in Celsius (300/100) plugged raw into the formula hallucinates 66.7%. Fahrenheit users can use Rankine (also absolute); everyone else converts: K = °C + 273.15.

How close do real engines get?

Best-in-class: large combined-cycle gas plants ~60–64% against Carnot ceilings around 84% (≈ 75% of the limit); modern diesels ~45–50% peak; gasoline cars 25–35% at the wheels across a drive cycle; coal steam plants 33–47%. The remaining gap is friction, finite-rate heat transfer, combustion irreversibility and auxiliary loads — engineering’s permanent homework, graded against a ceiling that never moves.

What is COP and how does it relate?

Coefficient of performance — the reversed cycle’s scorecard: COP_fridge = Q꜀/W ≤ T꜀/(Tₕ − T꜀), COP_heat pump = Qₕ/W ≤ Tₕ/(Tₕ − T꜀) = COP_fridge + 1 (exactly one more, since Qₕ = Q꜀ + W). COPs above 1 are not “efficiency over 100%” — the device moves heat rather than converting work into it. Our 270 K fridge / 300 K room example: Carnot COP 9; real appliances achieve 2–4.

How can efficiency rise — hotter, or colder?

Read the derivatives: η depends on the RATIO T꜀/Tₕ. Cutting T꜀ from 300 to 270 at Tₕ 600 gains 5 points (50 → 55%); raising Tₕ 600 → 700 at T꜀ 300 gains 7.1 points. Both levers fight physics’ logistics: T꜀ is pinned near ambient (the environment is the cold reservoir you get), while Tₕ is pinned by materials — hence the turbine-blade materials war, and why a winter day measurably improves a power plant’s heat rate.

Does Carnot apply to electric vehicles or fuel cells?

No — they don’t run heat cycles. Fuel-cell limits are set by ΔG/ΔH of the reaction (often 60–80% theoretical), battery drivetrains by electrochemical and power-electronics losses. It’s still common to see EVs’ ~85–90% drivetrain efficiency marveled at “versus Carnot” — the comparison is meaningless except as a reminder that skipping the heat step entirely is the winning move when available.

Is this Carnot efficiency calculator free?

Yes — free, instant, no sign-up. Enter the two reservoir temperatures in Kelvin and a heat input for the efficiency ceiling, work, rejected heat and both COP bounds.