Heat Engine Efficiency Calculator

Calculate actual efficiency, Carnot efficiency, and compare real vs ideal heat engine performance.

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How Heat Engine Efficiency Connects Heat and Work

Heat-engine efficiency measures what fraction of the energy absorbed from a hot reservoir becomes useful work. A heat engine operates in a cycle: it receives heat QH, produces work W, and rejects the remaining heat QC to a colder reservoir. Energy conservation therefore gives QH = W + QC. Efficiency is a ratio, not an amount of energy, so a larger engine can produce more work while having the same efficiency as a smaller engine.

The second law of thermodynamics adds a stricter limit. Even an ideal reversible engine cannot convert all incoming heat into work while operating between two finite temperatures. The Carnot efficiency depends only on the absolute reservoir temperatures. It is an upper bound for any engine operating between the same TH and TC, not a prediction that a real engine will actually reach that value.

η = W / QH     ηCarnot = 1 − TC/TH
SymbolMeaningWhy it appears / units
QHHeat absorbed from hot reservoirEnergy input, usually joules
WNet work produced per cycleUseful energy output, joules
QCHeat rejected to cold reservoirQH − W by energy conservation
TH, TCHot and cold absolute temperaturesMust be kelvins in the Carnot expression

An actual efficiency of 35% means 35 J of work is obtained for every 100 J of heat absorbed, while 65 J must leave as rejected heat. If the Carnot limit for those reservoir temperatures is 50%, the real engine is below the thermodynamic ceiling, as it must be. The gap reflects irreversibilities such as friction, finite-temperature heat transfer, turbulence, and combustion losses.

Worked Examples

Example 1: Steam plant: Q_H=1000J, W=350J
η=350/1000
Result: 35% actual efficiency
Typical coal power plant
Example 2: Carnot: T_H=600K, T_C=300K
η_C=1-300/600
Result: 50% Carnot limit
Real plants reach 30-45% of Carnot
Example 3: Actual engine with rejected heat
QH = 2400 J, W = 720 J → η = 720/2400; QC = 2400 − 720
Result: η = 30%, QC = 1680 J
The efficiency and the rejected heat tell the same energy-balance story: only part of QH becomes work.
Example 4: Celsius temperatures must be converted
TH = 400°C = 673.15 K, TC = 30°C = 303.15 K → ηC = 1 − 303.15/673.15
Result: ηC ≈ 55.0%
Using 400 and 30 directly would give a physically wrong Carnot efficiency because temperature ratios require an absolute scale.

Common Mistakes

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Using Celsius in the Carnot formula

Carnot efficiency contains a temperature ratio, so both temperatures must be measured from absolute zero. Convert °C to K by adding 273.15 before dividing.

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Allowing work output to exceed heat input

For a cyclic heat engine, W cannot be greater than QH. If it is, the stated data violate the first-law energy balance.

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Treating Carnot efficiency as expected real performance

The Carnot value is the reversible upper limit. Real engines operate below it because irreversible processes generate entropy.

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Confusing efficiency with coefficient of performance

Heat-engine efficiency compares work output with heat input. Refrigerator and heat-pump COP compare transferred heat with work input and can therefore be greater than one.

Frequently Asked Questions

Why can't efficiency reach 100%?
The 2nd law of thermodynamics: some heat must always be rejected to the cold reservoir. Only at T_C=0K (absolute zero) could Carnot efficiency reach 100%, which is unattainable.
COP vs efficiency?
Heat-engine efficiency is W/QH and cannot exceed 1. A heat-pump COP is QH/W and a refrigerator COP is QC/W; either COP can exceed 1 because it compares heat moved with work input rather than representing conversion efficiency.
Why does lowering the cold-reservoir temperature improve the Carnot limit?
For fixed TH, lowering TC decreases the ratio TC/TH, so a larger fraction of the heat input can in principle be converted to work. Real systems still face material, cooling, friction, and heat-transfer constraints, so improving the theoretical limit does not guarantee the same improvement in actual efficiency.
What does a 40% heat-engine efficiency mean?
It means 40% of the heat absorbed from the hot source is converted into net work over the cycle. The remaining 60% is rejected as heat to the colder surroundings or cooling system. It does not mean that 40% of the fuel mass or 40% of the engine components are being used.
Can two engines have the same efficiency but different power?
Yes. Efficiency is an energy ratio, whereas power measures energy transferred per unit time. A large engine processing much more heat each second can produce far more power than a small engine even when both convert the same percentage of incoming heat into work.
Why must TH be greater than TC?
A conventional heat engine requires a temperature difference that drives heat from the hot reservoir toward the cold reservoir while part of that energy is converted to work. If the reservoirs are at the same temperature, the Carnot limit is zero and no cyclic heat engine can extract net work from that single equilibrium temperature.

Formula Explorer connections

Interpretation: This relationship connects motion, force, momentum, work or energy in a mechanical system. Assumption: Choose a consistent reference direction and unit system. The model may assume constant acceleration, rigid bodies, negligible losses or an isolated system.

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