Thermodynamics

Carnot Efficiency Calculator — Thermal Efficiency Limit & Reservoir Temperature Analyzer

Calculate the maximum theoretical Carnot efficiency or solve for hot/cold reservoir temperature, with optional reversible work and rejected heat from a heat-input rate.

Unit-aware inputs Deterministic calculation Engineering interpretation
Calculation workspace

Enter the known values and review the calculated result

Deterministic calculation
01
Parameters

Input parameters

Use consistent values and select the intended engineering units.

Temperatures & Carnot efficiency

Optional heat input

02
Output

Results

Live
Ready to calculate Complete the required inputs and run the calculation.
—
Engineering Pro Save, document and continue this calculation

Turn this result into a reusable engineering record with saving, PDF export and reporting workflows.

Engineering reference

Method, application and limitations

Review the calculation method, intended application and engineering assumptions before using the result in a design decision.

01
Method

Formula and calculation method

Carnot efficiency formula:

ηC = 1 − Tc / Th

This relation gives the maximum theoretical thermal efficiency of a reversible heat engine operating between a hot reservoir and a cold reservoir. The temperature ratio must use absolute temperatures.

Rearranged forms:

  • Th = Tc / (1 − ηC)
  • Tc = Th · (1 − ηC)

Additional calculated quantities:

  • ηC,% = 100 · ηC
  • ΔT = Th − Tc
  • Tc / Th = 1 − ηC

Optional reversible heat-rate balance:

Wmax = ηC · Qh

Qc = (Tc / Th) · Qh

Qc = Qh − Wmax

In this calculator, Qh is entered as a heat-input rate. Therefore Wmax and Qc represent the corresponding maximum theoretical work-output rate and rejected-heat rate on the same power basis.

where:

  • ηC — Carnot efficiency, dimensionless (-)
  • ηC,% — Carnot efficiency expressed as a percentage (%)
  • Th — absolute hot-reservoir temperature (K)
  • Tc — absolute cold-reservoir temperature (K)
  • ΔT — reservoir temperature difference (K)
  • Qh — optional heat-input rate
  • Wmax — maximum theoretical work-output rate at the Carnot limit
  • Qc — heat-rejection rate at the Carnot limit

The calculator solves exactly one of ηC, Th, or Tc from the other two values.

02
Application

When to use this calculator

Use this calculator to determine the Carnot efficiency of an ideal heat engine or to solve the Carnot relation in reverse for one of the reservoir temperatures.

  • Calculate the maximum theoretical heat-engine efficiency from hot and cold reservoir temperatures.
  • Calculate the hot-reservoir temperature required for a specified Carnot efficiency and cold-reservoir temperature.
  • Calculate the cold-reservoir temperature corresponding to a specified Carnot efficiency and hot-reservoir temperature.
  • Evaluate the cold-to-hot absolute temperature ratio Tc/Th.
  • Calculate the reservoir temperature difference ΔT = Th − Tc.
  • Estimate the maximum theoretical work-output rate and rejected-heat rate when the optional heat-input rate is provided.
  • Use the Carnot limit as an ideal benchmark when comparing thermal power cycles or heat-engine concepts.

Do not use this calculator as a prediction of the actual efficiency of a steam turbine, gas turbine, internal-combustion engine, Rankine cycle, Brayton cycle, or other real thermal system. Actual performance depends on irreversibilities, component efficiencies, heat-transfer limitations, pressure losses, working-fluid properties, and operating conditions that are not included here.

This calculator does not calculate refrigerator COP, heat-pump COP, entropy generation, actual thermal efficiency from measured heat and work, or detailed thermodynamic cycle performance.

03
Decision support

How to interpret the result

Carnot efficiency ηC is the maximum theoretical fraction of heat input that a reversible heat engine could convert into work while operating between the selected hot and cold reservoir temperatures.

For example, ηC = 0.60 corresponds to a Carnot efficiency of 60%. It is an ideal upper limit, not the expected efficiency of a real engine or power plant.

  • Higher Th: for a fixed Tc, increasing the hot-reservoir temperature increases Carnot efficiency.
  • Lower Tc: for a fixed Th, decreasing the cold-reservoir temperature increases Carnot efficiency.
  • Temperature ratio: ηC depends directly on Tc/Th. A smaller ratio produces a higher theoretical efficiency.
  • Temperature difference: ΔT is useful descriptive information, but ΔT alone does not determine Carnot efficiency because the absolute temperature levels also matter.

When the calculator solves for Th or Tc, the result is the reservoir temperature required by the ideal Carnot relation for the entered efficiency. It should not be interpreted as a guaranteed operating temperature for a real machine.

If the optional heat-input rate Qh is provided, Wmax represents the maximum theoretical work-output rate at the Carnot limit and Qc represents the corresponding rejected-heat rate. These quantities use the same rate basis as the heat-input value.

The calculator does not classify efficiency values as safe, unsafe, good, or poor. A high Carnot efficiency only indicates a higher theoretical thermodynamic limit; it does not establish component safety, feasibility, material suitability, or actual cycle performance.

04
Worked case

Calculation example

Example: calculate Carnot efficiency from reservoir temperatures.

Consider an ideal heat engine operating between a hot reservoir at 800 K and a cold reservoir at 300 K. Leave Carnot efficiency empty so that the calculator solves for ηC.

Input data:

  • Th = 800 K
  • Tc = 300 K
  • ηC = unknown

Step 1 — calculate the temperature ratio:

Tc / Th = 300 / 800 = 0.375

Step 2 — calculate Carnot efficiency:

ηC = 1 − 0.375 = 0.625

Step 3 — express the result as a percentage:

ηC,% = 0.625 · 100 = 62.5%

Step 4 — calculate the reservoir temperature difference:

ΔT = 800 − 300 = 500 K

Result:

  • Carnot efficiency ηC = 0.625
  • Carnot efficiency = 62.50%
  • Tc/Th = 0.375
  • ΔT = 500 K

The result means that 62.5% is the maximum theoretical heat-to-work conversion efficiency for a reversible engine operating between 800 K and 300 K. A real heat engine operating between comparable temperature levels will have a lower actual efficiency because real processes contain irreversibilities and other losses.

05
Model boundaries

Assumptions and limitations

  • The calculation represents an ideal reversible Carnot heat engine operating between two thermal reservoirs.
  • The reservoirs are represented by uniform hot and cold temperatures Th and Tc.
  • The Carnot temperature ratio requires absolute temperatures. Celsius or Fahrenheit values must not be substituted directly into Tc/Th without conversion to an absolute temperature scale.
  • Exactly one of ηC, Th, and Tc must be left unknown. The calculator solves that variable from the other two.
  • The implemented calculation requires Th > 0 K and Tc > 0 K.
  • The implemented heat-engine domain requires Tc < Th.
  • The implemented efficiency domain is 0 < ηC < 1. Boundary values of exactly 0 or 1 are not accepted by the calculator.
  • The optional Qh input must be greater than zero when it is provided.
  • The optional Qh field is configured as a heat-input rate. The calculated Wmax and Qc values should therefore be interpreted on the same power-rate basis.
  • The ideal Carnot result is independent of the working fluid because the calculator does not model fluid properties or a specific thermodynamic cycle.
  • The calculation neglects irreversibility, mechanical friction, finite-temperature heat transfer, pressure losses, leakage, auxiliary power, combustion losses, component inefficiencies, and other real-system effects.
  • The calculator does not determine actual engine efficiency, actual power output, equipment sizing, material temperature limits, economic feasibility, or compliance with a design standard.
  • Detailed design of a real thermal system requires an appropriate cycle model and component-level analysis in addition to the Carnot limit.
06
Questions

Frequently asked questions

What is Carnot efficiency?
Carnot efficiency is the maximum theoretical thermal efficiency that a reversible heat engine can achieve between a hot reservoir and a cold reservoir. It is defined by ηC = 1 − Tc/Th, where both reservoir temperatures are absolute temperatures.
How do you calculate Carnot efficiency?
Calculate Carnot efficiency using ηC = 1 − Tc/Th. For example, with Th = 800 K and Tc = 300 K, ηC = 1 − 300/800 = 0.625, or 62.5%.
Why must Kelvin or another absolute temperature scale be used in the Carnot efficiency formula?
Carnot efficiency depends on the ratio of absolute reservoir temperatures. Celsius and Fahrenheit have arbitrary zero points, so their numerical values cannot be used directly in Tc/Th. Convert the temperatures to kelvin or another absolute scale before evaluating the ratio.
How do you calculate the required hot-reservoir temperature from Carnot efficiency?
Rearrange the Carnot equation to Th = Tc / (1 − ηC). For example, if ηC = 0.60 and Tc = 300 K, the required ideal hot-reservoir temperature is 300 / (1 − 0.60) = 750 K.
How do you calculate the cold-reservoir temperature from Carnot efficiency?
Rearrange the Carnot equation to Tc = Th · (1 − ηC). If Th = 750 K and ηC = 0.60, the corresponding ideal cold-reservoir temperature is 750 · 0.40 = 300 K.
Can Carnot efficiency reach 100%?
The Carnot equation approaches 100% efficiency only as the cold-reservoir temperature approaches absolute zero for a finite hot-reservoir temperature. The calculator restricts the heat-engine solution to 0 < ηC < 1 and Tc > 0 K, so an efficiency of exactly 100% is not accepted.
Is Carnot efficiency the same as actual thermal efficiency?
No. Carnot efficiency is an ideal upper limit determined only by the hot and cold reservoir temperatures. Actual thermal efficiency is determined from the real heat input and useful work output of a system and is reduced by irreversibilities and component losses. This calculator evaluates the Carnot limit, not measured or actual machine efficiency.
Is this a general thermal efficiency calculator?
No. This calculator specifically evaluates the Carnot heat-engine limit from reservoir temperatures. A general thermal efficiency calculation based on measured heat input and work output uses η = W/Qin and represents a different calculation unless the process is assumed to operate reversibly at the Carnot limit.
Does a larger temperature difference always determine a higher Carnot efficiency?
Carnot efficiency is controlled by the absolute temperature ratio Tc/Th, not by temperature difference alone. Two systems can have the same ΔT but different Carnot efficiencies if their absolute reservoir temperatures are different.
What does the optional heat-input rate calculate?
When Qh is provided, the calculator applies the reversible Carnot heat balance to calculate Wmax = ηCQh and Qc = (Tc/Th)Qh. Because Qh is configured as a heat-input rate, these results represent a maximum theoretical work-output rate and rejected-heat rate on the same power basis.
What happens if the cold-reservoir temperature is equal to or higher than the hot-reservoir temperature?
The calculator rejects the input because its implemented model is a heat engine and requires Tc < Th. If the temperature ordering is reversed, the system is no longer operating in the heat-engine direction represented by this calculator.
Does this calculator calculate refrigerator COP, heat-pump COP, or entropy generation?
No. The current calculator solves Carnot heat-engine efficiency, hot or cold reservoir temperature, temperature difference, temperature ratio, and the optional reversible heat-rate balance. It does not calculate refrigerator COP, heat-pump COP, or entropy generation.
Add an engineering note
→
Continue analysis

Related engineering calculators

Engineering tool Ideal Gas Law Calculator — Pressure, Volume, Amount of Substance and Temperature Solver Calculate pressure, volume, amount of substance or temperature using the ideal gas equation and the three provided state variables. Open calculator Engineering tool Steam Turbine Wetness & Condensation Risk Calculator — Steam Quality, Outlet Enthalpy and Specific Work Calculate steam quality, outlet enthalpy and specific work using inlet pressure, outlet pressure, inlet enthalpy, inlet entropy and turbine efficiency. Open calculator Engineering tool Heat Exchanger Effectiveness NTU Calculator — Heat Transfer Rate & Outlet Temperature Validator Calculate heat transfer rate using inlet temperatures, heat capacity rates, UA, and flow configuration based on the ε-NTU method. Open calculator
Was this calculator useful?
Engineering notice. Verify input data, assumptions, applicable standards and required safety factors before using results for design release or safety-critical decisions.