KEVOS® MAKEOS · Engineering
Least Work to Chill a Product Below Ambient (Second-Law Minimum Power, Ideal COP, Chiller Claim Check)
What is the least power any machine could need to chill this product stream from its start temperature to below the surroundings, and is the chiller power quoted or measured for the duty possible?
Second-law minimum power to cool a steady stream with a constant specific heat from its start temperature to a final temperature below the surroundings, with the heat finally rejected to surroundings at a third temperature: the heat removed, the part of it above the surroundings temperature that needs no work, the least power from summing the Carnot work over the cooling, the ideal overall COP, the chiller power you enter as a COP and a second-law efficiency, a verdict on whether it is possible, and the heat, least and stated energy over a year of running.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Mass flow of the product or stream | kg/s | Mass flow of the product, water, glycol or air being chilled, from the process design or a flow meter. |
| Specific heat of the product at constant pressure | kJ/(kg·K) | Mean specific heat at constant pressure over the temperature range, from property data for the product or the supplier. |
| Product temperature at the start | °C | Temperature of the product entering the chilling step. |
| Final chilled temperature | °C | Temperature the product must leave at, below the surroundings temperature. |
| Temperature of the surroundings that receive the heat | °C | Temperature of the surroundings that finally receive the heat: ambient air for an air-cooled plant or the cooling water for a water-cooled one. |
| Chiller power drawn for this duty | kW | Electrical or shaft power the chiller plant draws for this duty, quoted or measured. |
| 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. |
| Running hours per year | h/year | Hours per year the chilling runs at this condition; used only for the yearly energy figures. |
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: Mass flow of the product or stream
- What it is
- The mass of product, water, glycol or air passing through the chilling step each second. With the specific heat it gives the heat removed per kelvin of cooling.
- Units and format
- Enter a number in kg/s: kg/s. The form accepts 0.000001 kg/s to 1000000 kg/s. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.5 kg/s as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The heat removed, least power and yearly energy scale in proportion; the COPs and second-law efficiency of a fixed chiller power change with it.
- Get it right
- For a batch, divide the batch mass by the chilling time to get an average flow.
What this means: Specific heat of the product at constant pressu…
- What it is
- The heat needed to warm one kilogram of the product by one kelvin at constant pressure, averaged over the chilling range.
- Units and format
- Enter a number in kJ/(kg·K): kilojoules per kilogram per kelvin. The form accepts 0.001 kJ/(kg·K) to 100 kJ/(kg·K). Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 3.9 kJ/(kg·K) as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The heat removed and least power scale in proportion.
- Get it right
- Mind the units: kJ/(kg·K), so water is about 4.18, not 4180.
What this means: Product temperature at the start
- What it is
- The temperature of the product as it enters the chilling step, converted to kelvin. Any cooling above the surroundings temperature can be done by rejecting heat directly and needs no work.
- 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 warmer start raises the heat removed; above the surroundings it adds no least work, only heat that precooling could remove.
- Get it right
- Measure it at the chiller inlet; a product that waits after cooking may arrive cooler than it left.
What this means: Final chilled temperature
- What it is
- The temperature the product must leave the chilling step at, which must be below both the start temperature and the surroundings.
- 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 2 °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 end raises the least power more steeply than the heat removed, so the ideal COP falls.
- Get it right
- Use the required product temperature, not the chilled-water or evaporating temperature inside the chiller.
What this means: Temperature of the surroundings that receive th…
- What it is
- The temperature of the surroundings that finally receive the heat: ambient air for an air-cooled plant or the cooling water for a water-cooled one. It is the dead state the least work is measured from.
- Units and format
- Enter a number in °C: degrees Celsius. The form accepts -100 °C to 100 °C. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 25 °C as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Warmer surroundings raise the least power and lower the ideal COP.
- Get it right
- Use the condition the chiller is rated or tested at; a hot day raises the least work.
What this means: Chiller power drawn for this duty
- What it is
- The electrical or shaft power the chiller plant draws while doing this duty, quoted by the supplier or measured on the plant.
- 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 20 kW as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Sets the stated COP and the second-law efficiency; below the least power the verdict is impossible.
- Get it right
- Include the compressor and any condenser fans or pumps that the quoted duty depends on, and leave out loads that serve other duties.
What this means: Highest second-law efficiency accepted without…
- What it is
- Your own screening threshold for the second-law efficiency: a stated chiller power that is possible but gives a second-law efficiency above it is flagged for independent test evidence.
- 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
- 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 plants you trust have achieved, not from the claim under review.
What this means: Running hours per year
- What it is
- The hours per year the chilling duty runs at this condition. It turns the heat removed, least power and stated power into energy per year.
- 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 4000 h/year as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The yearly energy figures scale in proportion; the power results do not change.
- Get it right
- Use running hours at this duty only, not the hours the plant is switched on.
Method and formulas
- Absolute temperatures in kelvin; heat removed Q = ṁ c_p (T₁ − T₂). The final temperature must be below both the start temperature and the surroundings.
- Heat removed above the surroundings temperature can be rejected to the surroundings directly and needs no work; that part is ṁ c_p (T₁ − T₀) when the product starts above ambient.
- Below the surroundings each element of heat dQ taken from the product at temperature T needs at least dQ (T₀ ÷ T − 1) of work (Bryan §90), and summing over the cooling (Bryan §93) gives the least power W = ṁ c_p [T₀ ln(T_s ÷ T₂) − (T_s − T₂)], with T_s the lower of the start and surroundings temperatures.
- Ideal overall COP = Q ÷ W; stated COP = Q ÷ P; second-law efficiency = W ÷ P. A chiller drawing less than W is impossible; above your threshold the figure needs test evidence.
- Example: 0.5 kg/s of product at 3.9 kJ/(kg·K) chilled from 30 °C to 2 °C with surroundings at 25 °C gives up 54.6 kW, of which 9.75 kW is above ambient; the least power is 1.824 kW (ideal COP 29.9), so a chiller drawing 20 kW has a second-law efficiency of 9.12 %.
Limitations
- Constant specific heat, constant pressure and no phase change: freezing a product or condensing moisture from air needs its latent heat treated separately.
- The least power assumes reversible machines at every temperature of the product; a real chiller works from one evaporating temperature below the final product temperature, so even a perfect single-temperature machine needs more power than this limit.
- Pump, fan and heat-gain loads outside the product duty are not included; enter the chiller power on the same basis as the duty.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Find the least power a product chiller, process cooler or glycol plant could ever need for a duty
- Check a quoted chiller power or COP for a product cooling duty against the second law
- Rank chilling duties by the work they need rather than by heat alone
Who it is for
- Process and refrigeration engineers
- Food, beverage and plant engineers
- Energy engineers and managers reviewing supplier proposals
What you get
- Heat removed and the part above ambient that needs no work
- Least power to chill the product
- Ideal and stated COP
- Second-law efficiency of the stated power and a verdict
- Heat, least and stated energy per year
Interpreting the result
- Most of the least work is spent on the coldest part of the cooling, so each further degree below ambient costs more work per kilowatt of heat removed than the one before.
- Precooling with ambient air or cooling water before the chiller removes the heat above ambient without refrigeration work.
Before relying on the result
- Measure the product flow and temperatures and the chiller power on the running plant before relying on the comparison.
References
- Bryan G. H., Thermodynamics: An Introductory Treatise Dealing Mainly with First Principles and Their Direct Applications (B. G. Teubner, Leipzig, 1907), Chapter VIII §§90, 93 (available energy of a system in a medium of temperature T₀; available energy of a hot or cold body), pp. 82–87. Public-domain source; the method is restated in original words and no text, tables or figures are reproduced.
- Kotas T. J., The Exergy Method of Thermal Plant Analysis (Butterworths): physical exergy of a stream and exergetic efficiency.
- NIST Chemistry WebBook, https://webbook.nist.gov/chemistry/: thermophysical property data for specific heats.
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
Lost work and exergy
What it covers: Turn entropy into kilowatts: the lost-work theorem, the exergy of heat, of flowing streams and of cold, an exergy balance for each piece of plant, and why every audit must state its dead state.
Useful alongside · 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.
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.