Carnot Thermodynamic Efficiency & Heat Pump COP Calculator

Thermodynamics: Sadi Carnot & The Second Law of Thermodynamics

Published in 1824 by French physicist Sadi Carnot in Reflections on the Motive Power of Fire, the Carnot Theorem established the foundation of modern thermodynamics:

• The Carnot Cycle: The theoretical ideal reversible thermodynamic cycle consisting of two reversible isothermal processes and two reversible isentropic (adiabatic) processes.

• The Absolute Upper Bound on Thermal Efficiency: The Second Law of Thermodynamics dictates that no heat engine can ever be 100% efficient. Even in a frictionless, ideal world, heat must be rejected to a colder reservoir; the maximum possible efficiency depends strictly on the absolute temperature ratio .

• Heat Pumps vs. Resistance Heaters: An electric baseboard heater has a maximum COP of 1.

• A Carnot heat pump operating across a small temperature lift (e.g. 21^ inside vs 0^ outside) has a theoretical delivering 14 kW of heat for every 1 kW of electrical work.

How to Use This Calculator

Enter your Thermodynamic Cycle Application — choose from Carnot Heat Engine — Power Generation (η_carnot = 1 - T_C / T_H), Carnot Heat Pump — Space Heating (COP_hp = T_H / (T_H - T_C)), Carnot Refrigerator — Cooling / Cryogenics (COP_ref = T_C / (T_H - T_C)), Hot Source Reservoir Temperature (T_H in °C) (e.g. 550 (550°C steam boiler / combustion)), Cold Sink Reservoir Temperature (T_C in °C) (e.g. 25 (25°C ambient air / cooling tower)), Thermal Heat Energy Input (Q_H in kW) (e.g. 100 (100 kW thermal power)), and Actual Real-World Engine Efficiency (% [Optional]) (e.g. 38 (38% real turbine / engine efficiency)). Results update instantly as you type — adjust any value to compare scenarios.

The Formula

Carnot Thermodynamic Theorem (Nicolas Léonard Sadi Carnot, 1824) equations:

Absolute Temperatures: T_H = T_H,^ + 273.15, T_C = T_C,^ + 273.15

Carnot Heat Engine Efficiency: η_Carnot = 1 - (T_C) / (T_H) = (T_H - T_C) / (T_H) × 100\%

Maximum Work Output: W_ = η_Carnot × Q_H

Minimum Waste Heat Rejection: Q_C = Q_H - W_ = Q_H ((T_C) / (T_H))

Second Law Exergy Efficiency: η_II = η_actualη_Carnot × 100\%

Carnot Heat Pump COP: COP_HP = (T_H) / (T_H - T_C)

Carnot Refrigerator COP: COP_Ref = (T_C) / (T_H - T_C) = COP_HP - 1

Worked Example

For example, evaluating a coal-fired or nuclear steam power plant operating between boiler temperature \ (823.15 K) and river cooling tower \ (298.15 K) with \ thermal heat input and 38\% actual electrical generator efficiency: The Carnot theoretical limit is . Maximum possible work is requiring at least 36.22 MW of waste heat rejection. Generating 38 MW in real life achieves a Second Law Exergy Efficiency of \.

Understanding Your Result

Automotive benchmarks for reference: engine displacement affects power density (target 100+ hp/liter for performance), power-to-weight ratio above 0.1 hp/lb is considered sporty. Tire sizing: the middle number (e.g., 60 in 225/60R17) is the aspect ratio — lower numbers indicate sportier, wider tires with less cushioning. Braking distance from 60 mph: average car 120–140 ft on dry pavement; wet roads add 40–60%. Fuel efficiency varies significantly by driving style — aggressive acceleration and braking can reduce MPG by 15–30%.