KEVOS® MAKEOS · Engineering
Heat Engine and Heat Pump Second-Law Check (Carnot Limit, Entropy Generated, Lost Work)
Is the efficiency or COP claimed for this engine, chiller or heat pump possible between its two temperatures, how close is it to the Carnot limit, and how much work does its irreversibility cost?
Second-law check of a heat engine, refrigerator or chiller, or heat pump working between a hot and a cold temperature: the actual efficiency or coefficient of performance from the heat and work rates, the Carnot limit from the absolute temperatures, the share of that limit achieved, the heat at the other reservoir from the energy balance, the entropy generated from the Clausius inequality and the lost work against an ideal reversible machine, against a credible share of the Carnot limit you set.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Machine | choice | What the machine does: turn heat into work (engine), remove heat from a cold space (refrigerator or chiller) or deliver heat to a warm space (heat pump). |
| Heat rate: taken in (engine), cooling (chiller) or heating delivered (heat pump) | kW | The useful heat rate: heat taken in by an engine, cooling capacity of a chiller, or heating capacity of a heat pump, at the same operating point as the power. |
| Work or power: delivered (engine) or taken in (chiller, heat pump) | kW | Shaft work delivered by an engine, or electrical or shaft power taken in by a chiller or heat pump, at the same operating point. |
| Hot reservoir temperature | °C | Temperature of the hot side the machine exchanges heat with (the heat sink for a chiller or heat pump, the heat source for an engine). |
| Cold reservoir temperature | °C | Temperature of the cold side (the cooled space or fluid for a chiller, the outdoor air, ground or water for a heat pump, the sink for an engine). |
| Largest credible share of the Carnot limit | % | The highest share of the Carnot limit you would accept without measured evidence, from your own test data or experience of comparable machines; no value is built in. |
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: Machine
- What it is
- The kind of machine being checked: an engine turns heat into work, a refrigerator or chiller pumps heat out of a cold space, and a heat pump delivers heat into a warm space. It sets which Carnot limit applies.
- Units and format
- Choose one of 3 options from the list: Heat engine (heat in, work out); Refrigerator or chiller (cooling, work in); Heat pump (heating, work in).
- Example
- The form opens with “Heat pump (heating, work in)” selected as a worked example. Choose the option that matches your case.
- Effect on the result
- Changes the Carnot limit formula and the meaning of the heat rate and power fields.
- Get it right
- A reverse-cycle air conditioner is a refrigerator when cooling and a heat pump when heating; check each mode with its own figures.
What this means: Heat rate: taken in (engine), cooling (chiller)…
- What it is
- The useful heat flow of the machine at one operating point: the heat an engine takes in, the cooling a chiller provides, or the heating a heat pump delivers.
- Units and format
- Enter a number in kW: kilowatts. The form accepts 0.000001 kW to 1000000 kW. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 10 kW as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- For a chiller or heat pump the COP, and its share of the Carnot limit, rise in proportion to it.
- Get it right
- Take the heat rate and the power from the same test point; mixing a rated capacity with a different power reading gives a false COP.
What this means: Work or power: delivered (engine) or taken in (…
- What it is
- The work flow of the machine at the same operating point: shaft work an engine delivers, or the electrical or shaft power a chiller or heat pump takes in.
- Units and format
- Enter a number in kW: kilowatts. The form accepts 0.000001 kW to 1000000 kW. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 2.5 kW as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A larger power input lowers the COP of a chiller or heat pump; a larger output raises an engine’s efficiency.
- Get it right
- Include the compressor, fans and pumps that the claimed COP includes, and no more; ratings usually state which loads are counted.
What this means: Hot reservoir temperature
- What it is
- The temperature of the hot side the machine exchanges heat with: the heat source of an engine, or the space, water or air a chiller or heat pump rejects heat to or heats.
- Units and format
- Enter a number in °C: degrees Celsius. The form accepts -200 °C to 2000 °C. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 45 °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 hot side raises an engine’s Carnot limit but lowers the Carnot COP of a chiller or heat pump.
- Get it right
- Use the temperature of the fluid at the heat exchanger, not a room set point; a heat pump heating water to 55 °C works against 55 °C or more.
What this means: Cold reservoir temperature
- What it is
- The temperature of the cold side: the sink of an engine, the chilled space or water of a refrigerator, or the outdoor air, ground or water a heat pump draws heat from.
- Units and format
- Enter a number in °C: degrees Celsius. The form accepts -270 °C to 2000 °C. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 5 °C as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Moving it closer to the hot side raises the Carnot COP and lowers an engine’s Carnot efficiency.
- Get it right
- It must be lower than the hot side; for a heat pump on a cold night use that night’s outdoor temperature, not an annual average.
What this means: Largest credible share of the Carnot limit
- What it is
- 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 60% as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Only moves the line between a credible claim and one close to the limit; the calculated values do not change.
- Get it right
- Base it on measured data for comparable machines at similar temperatures; a single figure for all machine types is rarely right.
Method and formulas
- Absolute temperatures T = t + 273.15. Carnot limits: engine 1 − Tc ÷ Th; refrigerator Tc ÷ (Th − Tc); heat pump Th ÷ (Th − Tc).
- Energy balance for the heat at the other reservoir; entropy generated by the Clausius inequality, S_gen = Σ Q_out ÷ T − Σ Q_in ÷ T, which must not be negative.
- Lost work = the difference between the actual work and the work of a reversible machine with the same heat duty; it equals T × S_gen, with T the cold reservoir for an engine or heat pump and the hot reservoir for a refrigerator.
- Example: a heat pump giving 10 kW for 2.5 kW between 5 °C and 45 °C has a COP of 4 against a Carnot COP of 7.9537, 50.291 % of the limit; it generates 4.4678 W/K and loses 1.2427 kW of work.
Limitations
- Two reservoirs at constant temperature; machines working over a temperature glide need the mean temperatures of heat transfer, which lowers the true limit.
- Steady operation at one operating point; seasonal or part-load figures need the temperatures and rates that go with them.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Test a supplier’s COP or efficiency claim before buying
- Compare a chiller or heat pump with the best any machine could do
- Show how much power the irreversibility of a machine costs
Who it is for
- Mechanical and building services engineers
- Energy auditors
- Plant and procurement engineers
What you get
- Actual efficiency or COP and the Carnot limit
- Share of the Carnot limit and the verdict
- Heat at the other reservoir, entropy generated, reversible and lost work
Interpreting the result
- A claim above 100 % of the Carnot limit is impossible: the figures, the temperatures or the operating point do not belong together.
- Real vapour-compression machines reach well under the Carnot limit because their heat exchangers need temperature differences; the closer the temperatures, the larger the Carnot COP and the harder it is to approach.
Before relying on the result
- Ask for the test standard and conditions behind any claimed COP or efficiency.
- Measure inlet and outlet temperatures of both heat exchangers on site before using real operating temperatures.
References
- Linder, B., Thermodynamics and Introductory Statistical Mechanics (Wiley-Interscience, 2004), sections 4.1–4.4 and 4.11, printed pp. 32–41 and 48: Carnot efficiency 1 − T1 ÷ T2, the Clausius inequality and entropy change with surroundings. Method reference; no source text, tables or examples are reproduced.
- Rated COP for air conditioners and heat pumps in Australia comes from tests to the AS/NZS 3823 series; use the rating conditions stated with the figure.
Learn the Concepts Behind This Tool
These KEVOS® Library readings explain the method this tool uses, with worked examples.
Explains the concept · 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.
Explains the concept · KEVOS® original book
COP and the Carnot limit
In Heat Pumps and Refrigeration by the Numbers
What it covers: Define the coefficient of performance for cooling and heating, calculate its Carnot limit from two temperatures, estimate realistic COPs, and see why temperature lift drives the cost of moving heat.
Also in The second law and its limits
Method explained here · KEVOS® original book
The process chiller and its limit
In Process Chillers by the Numbers
What it covers: What a process chiller is and how its water circuit works, how it is rated, the second law's limit on CH-30, where the shortfall goes, COP, EER and tons, and a quoted figure that broke the second law.
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