Carnot Efficiency Calculator

Carnot efficiency is the maximum theoretical thermal efficiency any heat engine can achieve when operating between two temperature reservoirs. It was derived by Sadi Carnot in 1824 and is expressed as eta = 1 - Tc/Th, where Tc is the absolute temperature of the cold reservoir and Th is the absolute temperature of the hot reservoir, both in kelvin. No real engine can exceed this limit; most practical engines achieve 30 to 60 percent of their Carnot efficiency due to friction, heat losses, and irreversible processes.

The second law of thermodynamics states that in any heat engine, some heat must be rejected to the cold reservoir. For 100% efficiency, the cold reservoir would need to be at absolute zero (0 K, -273.15°C), which is physically unattainable. Even approaching absolute zero requires infinite work. Therefore, all heat engines must reject some waste heat, and 100% conversion of heat to work is impossible. This is one of the most fundamental results in thermodynamics.

Coefficient of Performance (COP) applies to refrigerators and heat pumps, which move heat rather than convert it to work. A refrigerator's COP is the ratio of heat removed from the cold space to the work input: COP_ref = Tc / (Th - Tc). A heat pump's COP is the ratio of heat delivered to the hot space to the work input: COP_hp = Th / (Th - Tc). COP can be greater than 1, meaning you move more heat energy than the electrical energy you put in, which is why heat pumps are more efficient than electric resistance heaters.

Real engines fall well short of Carnot efficiency due to irreversibilities such as friction, heat conduction across finite temperature differences, turbulence, and component losses. A typical gasoline car engine achieves about 25 to 35% thermal efficiency, compared to a theoretical Carnot efficiency of around 55 to 65% for the same temperature range. Combined-cycle gas turbine power plants are among the most efficient, reaching about 60% thermal efficiency, still well below the Carnot limit for their operating temperatures.

The Carnot formula is used to set upper bounds on engine performance during design, to evaluate how well a real engine performs relative to the theoretical maximum, and to guide selection of working temperatures. In power generation, engineers push the hot reservoir temperature as high as materials allow (higher Th raises efficiency) and reject heat at the lowest feasible temperature. In refrigeration and air conditioning, the Carnot COP guides compressor sizing and operating pressure selection. The formula also appears in chemistry, where it underlies the thermodynamic analysis of chemical equilibrium.