KEVOS® MAKEOS · Engineering
Heat Engine Carnot Limit Check (Ideal Efficiency, Second-Law Efficiency, Most Work, Least Heat Rejected)
Is the efficiency claimed or measured for this heat engine possible between its hot source and cold sink, how close is it to the Carnot limit, and what work and rejected heat does it imply?
The Carnot limit for a heat engine working between a hot source and a cold sink at fixed temperatures: the ideal efficiency 1 − T_C ÷ T_H from the two absolute temperatures, the stated efficiency as a share of that limit (the second-law efficiency), the most work and the least heat rejected for the heat supplied, the work, rejected heat, entropy generated and work lost at the stated efficiency, and a verdict on whether the figure breaks the limit or needs test evidence.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Hot source temperature | °C | Temperature at which the engine takes in heat: the steam, hot gas or hot fluid it is driven by, held fixed while the heat is supplied. |
| Cold sink temperature | °C | Temperature at which the engine rejects heat: the condenser, cooling water or ambient air it finally gives its waste heat to. |
| Stated engine efficiency | % | Thermal efficiency stated by the supplier or measured: net work out ÷ heat supplied, in per cent. |
| Heat supplied to the engine | kW | Heat supplied to the engine per second at the operating point, from a fuel or heat balance. |
| Highest second-law efficiency accepted without test evidence | % | Your own screening threshold: the second-law efficiency above which you ask for independent test results before accepting a claim. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Hot source temperature
- What it is
- The temperature at which the engine takes in its heat, such as the steam, hot gas or thermal oil driving it. It is converted to kelvin because the Carnot limit depends on the ratio of absolute temperatures.
- Units and format
- Enter a number in °C: degrees Celsius. The form accepts -270 °C to 3000 °C. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 450 °C as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A hotter source raises the Carnot efficiency and the most work, and lowers the second-law efficiency of a fixed stated figure.
- Get it right
- Use the temperature of the fluid the engine actually receives heat from, not a flame or furnace temperature further upstream.
What this means: Cold sink temperature
- What it is
- The temperature at which the engine rejects its waste heat: the condenser, cooling water or ambient air. With the source temperature it fixes the ideal efficiency 1 − T_C ÷ T_H.
- Units and format
- Enter a number in °C: degrees Celsius. The form accepts -270 °C to 3000 °C. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 30 °C as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A colder sink raises the Carnot efficiency and lowers the least heat rejected.
- Get it right
- Take the cooling medium temperature the engine sees in service, which in summer can be well above the design figure.
What this means: Stated engine efficiency
- What it is
- The thermal efficiency claimed by the supplier or measured on the plant: net work out divided by heat supplied, as a percentage. It is compared with the Carnot efficiency.
- Units and format
- Enter a number in %: percent: the number is read as a percentage, so type 8 for 8 %. The form accepts 0.001% to 100%. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 35% as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Sets the work, heat rejected, entropy generated and second-law efficiency; above the Carnot efficiency the verdict is impossible.
- Get it right
- Check the basis: gross or net, shaft or electrical, and on the higher or lower heating value of the fuel.
What this means: Heat supplied to the engine
- What it is
- The heat supplied to the engine each second at the operating point, from a fuel flow and heating value or a heat balance on the hot stream.
- Units and format
- Enter a number in kW: kilowatts. The form accepts 0.000001 kW or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1000 kW as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The most work, work, heat rejected and work lost scale in proportion; the efficiencies do not change.
- Get it right
- Enter heat input, not fuel energy before a boiler or heat exchanger loss, unless the stated efficiency is on that basis too.
What this means: Highest second-law efficiency accepted without…
- What it is
- Your own screening threshold for the second-law efficiency: a stated figure above it is still possible but is flagged for independent test evidence before you accept it.
- Units and format
- Enter a number in %: percent: the number is read as a percentage, so type 8 for 8 %. The form accepts 1% to 100%. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 75% as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Moves the line between the inside-the-limit and ask-for-evidence verdicts only; the calculated values do not change.
- Get it right
- Set it from what comparable engines you trust have achieved, not from the claim under review.
Method and formulas
- Absolute temperatures T_H = θ_H + 273.15 K and T_C = θ_C + 273.15 K; the source must be hotter than the sink.
- For a reversible (Carnot) cycle the heats exchanged with the two reservoirs satisfy Q_H ÷ T_H = Q_C ÷ T_C (Bryan §§63–65, §74), so the ideal efficiency is η_C = 1 − T_C ÷ T_H.
- Second-law efficiency = stated efficiency ÷ η_C; above 100 % the figure is impossible, above your threshold it needs test evidence.
- Most work = Q × η_C and least heat rejected = Q × T_C ÷ T_H; at the stated efficiency the work is Q × η and the heat rejected Q × (1 − η).
- Entropy generated per second = heat rejected ÷ T_C − Q ÷ T_H, and the work lost = T_C × entropy generated, equal to the most work less the stated work (Bryan §77 relates it to energy made unavailable).
- Example: an engine taking heat at 450 °C and rejecting it at 30 °C has a Carnot efficiency of 58.08 %; a stated 35 % is a second-law efficiency of 60.26 %, and 1000 kW of heat gives at most 580.8 kW of work against 350 kW at the stated efficiency.
Limitations
- Reservoirs at fixed temperatures: where the heat comes from a stream that cools as it gives up heat, the limit is lower (the waste heat recovery limit tool integrates over a changing temperature).
- The limit is only as good as the temperatures entered: use the temperatures of the source and sink the engine actually serves, not a flame or adiabatic temperature it never receives heat at.
- Part-load, auxiliary power, generator and gearbox losses and standard rating conditions are not modelled; enter efficiencies on the same basis (net or gross, shaft or electrical).
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Screen a heat engine, power cycle or heat-to-power proposal before it goes into a business case
- Find the most work any engine could make from a heat supply between two temperatures
- Compare engines on second-law efficiency rather than raw thermal efficiency
Who it is for
- Mechanical and energy engineers
- Power and process plant engineers
- Engineering managers reviewing supplier proposals
What you get
- Carnot efficiency between the two temperatures
- Second-law efficiency of the stated figure
- Most work and least heat rejected for the heat supplied
- Work, heat rejected, entropy generated and work lost at the stated efficiency
- Verdict: inside the limit, needs evidence, or impossible
Interpreting the result
- Real engines reach well under the Carnot efficiency, so a second-law efficiency close to 100 % is a sign of a mistaken temperature, a different efficiency basis or an unsupported claim.
- Raising the source temperature or lowering the sink temperature raises the limit; a low-temperature heat source gives a low limit however good the machine.
Before relying on the result
- Confirm the efficiency basis and the conditions behind any stated figure before comparing engines.
References
- Bryan G. H., Thermodynamics: An Introductory Treatise Dealing Mainly with First Principles and Their Direct Applications (B. G. Teubner, Leipzig, 1907), Chapter VI §§61–65 (Second Law, Carnot’s cycle, absolute temperature) and Chapter VII §§74, 77 (reversible cycles, entropy and unavailable energy), pp. 50–54, 62–67. Public-domain source; the method is restated in original words and no text, tables or figures are reproduced.
- Çengel Y. A. and Boles M. A., Thermodynamics: An Engineering Approach (McGraw-Hill): the second law and Carnot heat engines.
Learn the Concepts Behind This Tool
These KEVOS® Library readings explain the method this tool uses, with worked examples.
Method explained here · KEVOS® original book
Absolute temperature and the Carnot limit
What it covers: The kelvin, the reversible engine and the limit it sets: why no heat engine can beat 1 − Tc/Th, why temperatures must be absolute, and how to reject a heat-to-power claim in five minutes.
Method explained here · KEVOS® original book
The second law and its limits
In Heat Pumps and Refrigeration by the Numbers
What it covers: Understand the second law in practical terms: heat flows downhill, entropy and irreversibility, the Carnot limit on turning heat into work, energy quality, and checking claims against the limit.
Method explained here · KEVOS® original book
Heat into work: the first law and the Carnot ceiling
In On-Site Power and Heat by the Numbers
What it covers: How the first law accounts for every kilowatt of fuel, what efficiency and heat rate mean, and how the Carnot limit at real temperatures tests any claim to turn heat into power.
KEVOS® tools are screening and decision-support utilities. Verify important financial, engineering, safety, legal or operational decisions using appropriate professional standards and your organisation’s approved data.