Carnot Efficiency Calculator

Results

How to Use the Carnot Efficiency Calculator

1. 1. Pick the mode. Maximum efficiency solves η = 1 − T_c/T_h for the efficiency or for either temperature. Refrigerator or heat pump gives both coefficients of performance. Check an engine's heat flows compares a real engine's numbers with the Carnot limit.
2. 2. Enter the temperatures. Each box has a unit menu for kelvin, °C, °F and °R, and the calculator converts to absolute temperature itself. The hot reservoir must be at least as hot as the cold one, and neither can be at or below absolute zero.
3. 3. Add the energy, if you want it. For a refrigerator or heat pump you can give the work input to see how much heat it moves. For the flow check, give the heat absorbed and either the heat rejected or the work done.
4. 4. Read the result. A schematic shows the heat flows with arrow thickness scaled to their size, and the working lists each equation with your numbers.

The Carnot Formulas

η_C = 1 − T_c / T_h

COP_cool = T_c / (T_h − T_c)

COP_heat = T_h / (T_h − T_c) = COP_cool + 1

ΔS = Q_c / T_c − Q_h / T_h ≥ 0

In 1824 Sadi Carnot showed that no engine working between two heat reservoirs can be more efficient than a reversible one, and that every reversible engine between the same two reservoirs has the same efficiency, which depends only on their temperatures. The temperatures must be absolute: 25 °C is 298.15 K. Efficiency rises as the cold side gets colder or the hot side hotter, and it reaches 100% only if the cold reservoir sits at absolute zero, which is impossible. This is the second law of thermodynamics expressed as a number. The derivation is in OpenStax University Physics Volume 2, section 4.5.

A refrigerator or heat pump runs the cycle backwards, spending work W to move heat from cold to hot. Its performance is measured by the coefficient of performance, heat moved per unit of work, rather than by an efficiency, because it can exceed 1. Heating gets a bonus: the heat delivered to the hot side is the heat taken from the cold side plus the work, so COP_heat = COP_cool + 1.

Worked Examples

1. A steam power plant. Steam at 540 °C (813.15 K) and cooling water at 25 °C (298.15 K) give η_C = 1 − 298.15/813.15 = 63.33%. Real coal plants reach roughly a third; the best combined-cycle gas plants about 60%.
2. Simple temperatures. Reservoirs at 600 K and 300 K give exactly 50%. To reach 40% with a 300 K cold sink, the hot side must be at least 500 K.
3. A steam engine at atmospheric pressure. Between boiling and freezing water, 100 °C and 0 °C, the limit is only 26.80%.
4. A refrigerator. Holding 4 °C in a 25 °C kitchen, the ideal COP is 277.15 / 21 = 13.20. A freezer at −18 °C has a lower ideal COP of 5.934. Real appliances reach a fraction of these.
5. A heat pump. Keeping a house at 20 °C when it is 0 °C outside has an ideal COP_heat of 14.66: one kWh of electricity could deliver 14.66 kWh of heat. Real heat pumps manage a COP around 3 to 5.
6. Checking an engine. An engine absorbs 1000 J at 600 K and rejects 700 J at 300 K. It does 300 J of work (η = 30%), which is 60% of the Carnot limit of 50%, and it generates 0.6667 J/K of entropy. A claimed 600 J of work from the same heat would be 60%, above the limit, and is impossible.

Assumptions and Limits

• Constant-temperature reservoirs. The reservoirs are so large that taking or giving heat does not change their temperature.
• Absolute temperatures. Always use kelvin in the formulas. The calculator converts °C and °F, but a hand calculation with °C gives wrong answers.
• Reversible cycles. The results are upper limits; they say nothing about how fast an engine runs or how much power it delivers.
• Cyclic operation. The first-law check W = Q_h − Q_c applies to an engine that returns to its starting state each cycle.

Frequently Asked Questions

What is the Carnot efficiency formula?

η = 1 − T_c / T_h, with both temperatures in kelvin. It is the greatest fraction of the heat drawn from the hot reservoir that any engine can turn into work. At 600 K and 300 K it is 1 − 300/600 = 50%.

Why must temperatures be in kelvin?

The ratio T_c/T_h only means something on an absolute scale where zero is the lowest possible temperature. In °C, 25 °C against 100 °C would look like a 75% efficiency; in kelvin it is 1 − 298.15/373.15 = 20.1%.

Can an engine be 100% efficient?

Not in practice. A Carnot engine reaches 100% only if its cold reservoir is at absolute zero, which can never be attained. Any real engine rejects some heat, which the second law requires.

What is the coefficient of performance?

The COP of a refrigerator is the heat removed from the cold space divided by the work used; for a heat pump it is the heat delivered divided by the work. Unlike an efficiency it can be larger than 1. The Carnot values T_c/(T_h − T_c) and T_h/(T_h − T_c) are the best possible.

How does an engine's efficiency relate to its Carnot limit?

Divide the two: a 30% engine working between 600 K and 300 K achieves 30/50 = 60% of its Carnot limit. That ratio is called the second-law efficiency, and it shows how much room for improvement is left.

Is a heat pump with a COP of 4 breaking the conservation of energy?

No. It does not create heat from electricity; it moves heat from outside to inside. With 1 kWh of work and 3 kWh taken from the outdoors, 4 kWh arrive indoors, so energy is conserved and the COP is the ratio of heat delivered to work paid.