KEVOS® MAKEOS · Engineering
Heat Pump, Chiller or Heat Engine Claim Check (Carnot Limit, Second-Law Efficiency, Running Cost)
Is the COP or efficiency claimed for this heat pump, chiller or heat engine possible between these two temperatures, how close does it come to the Carnot limit, and what does it cost to run each year?
Second-law check of a cyclic machine between a hot and a cold temperature: the Carnot limit for a heat pump (T_h ÷ (T_h − T_c)), a chiller (T_c ÷ (T_h − T_c)) or a heat engine (1 − T_c ÷ T_h), the actual COP or efficiency from the useful output and the input, the second-law efficiency (actual ÷ limit) against your acceptance level, and the yearly input energy and its cost; for a heat pump also the saving against heating with direct electric resistance at the same price.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Machine | choice | What the machine is for: a heat pump delivers heat, a chiller removes heat, an engine turns heat into work. |
| Hot side temperature | °C | Temperature at which heat is delivered (heat pump, chiller) or taken in (engine): the water or air leaving the hot side at the rating point. |
| Cold side temperature | °C | Temperature at which heat is taken in (heat pump, chiller) or rejected (engine): the source water or outdoor air at the rating point. |
| Useful output (heat delivered, cooling or work) | kW | Rated or measured useful output at these temperatures: heating capacity, cooling capacity or shaft work. |
| Input (electrical or shaft power, or heat for an engine) | kW | Rated or measured input at the same conditions: electrical power for a heat pump or chiller, heat (fuel) input for an engine. |
| Lowest second-law efficiency you accept | % | The second-law efficiency below which you would question the machine or the claim; set it from your own comparisons. |
| Running cost | ||
| Running hours per year at this output | h/year | Equivalent full-load hours a year at this output. |
| Price of the input energy (including GST) | AUD/kWh | What one kilowatt-hour of the input energy costs you, including GST for a household (exclude it for a GST-registered business). |
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
- Which job the machine does: a heat pump whose product is heat, a chiller whose product is cooling, or an engine whose product is work.
- Units and format
- Choose one of 3 options from the list: Heat pump (heating is the useful output); Chiller or refrigeration (cooling is the useful output); Heat engine (work is the useful output).
- Example
- The form opens with “Heat pump (heating is the useful output)” selected as a worked example. Choose the option that matches your case.
- Effect on the result
- Selects which Carnot limit applies and how the output is read.
- Get it right
- A reversible air conditioner is a heat pump in winter and a chiller in summer; check each mode at its own temperatures.
What this means: Hot side temperature
- What it is
- The temperature at the warm end of the machine: where a heat pump or chiller rejects heat, or where an engine receives it.
- 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 55 °C as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A higher hot side temperature lowers the heat pump and chiller limits and raises the engine limit.
- Get it right
- Use the fluid temperature at the rating point, for example the hot water leaving a heat pump.
What this means: Cold side temperature
- What it is
- The temperature at the cold end: where a heat pump or chiller takes heat in, or where an engine rejects it.
- 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 7 °C as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A warmer cold side shrinks the lift, raising the heat pump and chiller limits and lowering the engine limit.
- Get it right
- For an air-source unit this is the outdoor air temperature at the rating point; for a chiller the chilled water temperature.
What this means: Useful output (heat delivered, cooling or work)
- What it is
- What the machine delivers that you want: heat to the building for a heat pump, heat removed for a chiller or shaft work for an engine.
- Units and format
- Enter a number in kW: kilowatts. The form accepts 0.000001 kW to 10000000 kW. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 12 kW as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The actual COP or efficiency, and so the second-law efficiency, rise in proportion.
- Get it right
- Take output and input from the same test point; data sheets often list several.
What this means: Input (electrical or shaft power, or heat for a…
- What it is
- What the machine consumes to deliver that output: electrical or shaft power for a heat pump or chiller, heat input for an engine.
- Units and format
- Enter a number in kW: kilowatts. The form accepts 0.000001 kW to 10000000 kW. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 3.5 kW as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The actual COP or efficiency is inversely proportional to the input; it also sets the yearly energy and cost.
- Get it right
- Include fans and pumps that the rating includes, so the COP matches the rating standard.
What this means: Lowest second-law efficiency you accept
- What it is
- The share of the Carnot limit below which you would treat the machine or claim as poor, your own acceptance level for the comparison.
- Units and format
- Enter a number in %: percent: the number is read as a percentage, so type 8 for 8 %. The form accepts 0% to 100%. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 30% as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A second-law efficiency below this value turns the verdict to a warning.
- Get it right
- Set it from machines you have measured or from other quotations at the same temperatures.
Running cost
What this means: Running hours per year at this output
- What it is
- The equivalent number of hours a year the machine runs at the output entered, used for the yearly energy and cost.
- Units and format
- Enter a number in h/year: h/year. The form accepts 0 h/year to 8784 h/year. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 2000 h/year as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Yearly energy, cost and saving scale in direct proportion.
- Get it right
- For part-load running, divide the yearly output energy by the rated output to get equivalent full-load hours.
What this means: Price of the input energy (including GST)
- What it is
- What one kilowatt-hour of the input energy costs you, used for the yearly running cost and the saving over resistance heating.
- Units and format
- Enter a number in AUD/kWh: Australian dollars per kilowatt-hour. The form accepts 0 AUD/kWh to 100 AUD/kWh. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.3 AUD/kWh as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Cost and saving scale in direct proportion.
- Get it right
- A household compares prices including GST; a GST-registered business compares them ex GST.
Method and formulas
- Carnot efficiency of a reversible engine between T_h and T_c: η = 1 − T_c ÷ T_h, in kelvin (Greiner eq. 2.56); reversed, the same cycle is the best possible heat pump, COP_h = T_h ÷ (T_h − T_c), or chiller, COP_c = T_c ÷ (T_h − T_c) (Greiner eqs 2.63–2.65).
- Actual COP or efficiency = useful output ÷ input; second-law efficiency = actual ÷ Carnot limit. Above 100 % the claim breaks the second law as stated.
- Yearly input energy = input × hours; cost = energy × price; heat pump saving = output × hours × price − input cost (direct resistance heating needs one kilowatt-hour per kilowatt-hour of heat).
- Example: actual 3.4286 against a Carnot limit of 6.8365, a second-law efficiency of 50.151 %; 7,000 kWh a year costs A$2,100.
Limitations
- The Carnot limit is the absolute ceiling for the temperatures entered; real machines also lose to the temperature differences across their heat exchangers, so use the source and sink temperatures at the rating point.
- One operating point: seasonal performance depends on how the temperatures and load vary through the year.
- Ratings to compare must be at the same conditions; heat pump and air-conditioner ratings in Australia follow the AS/NZS 3823 series of test conditions.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Test a supplier’s COP or efficiency claim against what physics allows
- Compare heat pumps or chillers on how close they come to the limit
- Estimate running cost and the saving of a heat pump over resistance heating
Who it is for
- Mechanical and building services engineers
- Energy and facilities managers
- Buyers comparing quotations
What you get
- Actual COP or efficiency and the Carnot limit
- Second-law efficiency against your acceptance level
- Yearly input energy and cost (and, for a heat pump, the saving over direct electric heating)
Interpreting the result
- Real vapour-compression machines fall well short of the Carnot limit because of heat-exchanger temperature differences and compressor losses; a claim close to the limit deserves its test conditions checked, and one above 100 % is impossible as stated.
- The limit falls as the temperature lift grows, so lowering the delivery temperature or raising the source temperature improves the COP more than most equipment choices.
References
- Greiner W., Neise L. and Stöcker H., Thermodynamics and Statistical Mechanics (Springer-Verlag, New York, 1995), pp. 52–58: thermodynamic engines, the Carnot efficiency (eq. 2.56) and heating with a heat pump against electricity (Exercise 2.7, eqs 2.63–2.65). Method reference; no text, tables or worked examples are reproduced.
- AS/NZS 3823 series, Performance of electrical appliances — Air conditioners and heat pumps: rating and test conditions; requirements are not reproduced here.
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.
Method explained here · KEVOS® original book
Cryocoolers and the heat budget
In Cryogenic Design by the Numbers
What it covers: Assemble CT-40’s heat budget, find where the cooler balances it, check the cooler against the Carnot limit, commission the budget by measurement, and a claim that broke the second law.
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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