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Thermodynamics

Carnot efficiency

Upper limit on heat-engine efficiency between two temperature reservoirs.

Ref: THE-07
Rev: v1
Field: Thermodynamics
Result: η_Carnot [%]

η = \frac{W_{net,out}}{Q_{H}}, η_{Carnot} = 1 - \frac{T_{C}}{T_{H}}

Use positive cycle energy magnitudes. Q_H and Q_C denote heat exchanged with the hot and cold reservoirs. Carnot efficiency applies to a reversible engine between two reservoirs; temperatures are absolute.

No heat engine can beat the Carnot limit, which depends only on how hot and how cold the reservoirs are.

Calculator

Solve for

η_CarnotCarnot efficiencyComputed

T_HHot reservoir

3502,500

T_CCold reservoir

200600

Maximum work per 100 kJ of heat62.5 kJ
Minimum heat rejected per 100 kJ37.5 kJ

What drives the result

Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.

Carnot efficiency (%) current value 62.5 %

T_HHot reservoirKey driver

14.388

T_CCold reservoir

2575

Raising the hot temperature from 800 K to 1200 K lifts the limit from 62.5% to 75%.
Real engines reach well below the Carnot value because of irreversibilities and heat loss.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing (%)	Local step	Change in result
T_H Hot reservoir	350 to 2,500 K	14.29	88	73.71	+10%	+5.45 %
T_C Cold reservoir	200 to 600 K	25	75	50	+10%	−6 %

Sweep of Hot reservoir

Hot reservoir (K)	Carnot efficiency (%)
350	14.29
618.8	51.52
887.5	66.2
1,156	74.05
1,425	78.95
1,694	82.29
1,963	84.71
2,231	86.55
2,500	88

Sweep one variable

Vary

Carnot efficiency (%)Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

η_Carnot

Carnot efficiency

Result · %

What the equation gives you. Everything else feeds it.

T_H

Hot reservoir

K · typical 350 to 2,500

Absolute temperature of the hot source. Raises the efficiency.

T_C

Cold reservoir

K · typical 200 to 600

Absolute temperature of the cold sink. Lowers the efficiency.

Worked example with the values above

T_C/T_H = 300 / 800 = 0.375.
1 − 0.375 = 0.625.
η_Carnot = 62.5 %.

Assumptions and limits

Reversible cycle between two constant-temperature reservoirs.
Absolute temperatures.

Common mistakes

Using °C.
Treating Carnot efficiency as achievable.

Statistics and uncertainty

Temperatures are measured to a few kelvin, a small error on the efficiency limit but important when T_H and T_C are close.

Where it is used

Setting the ceiling on engine and plant performance.
Comparing cycle designs.

Maths you need

Absolute temperature: kelvin
Second law: heat-engine limits

Related formulas

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