KEVOS® MAKEOS · Hydraulics
Liquid Adiabatic Compression (Temperature Rise, cp − cv, Adiabatic Bulk Modulus, Wave Speed)
How much does a hydraulic oil or other liquid heat up when it is compressed quickly, and how much stiffer is it under rapid (adiabatic) compression than under slow (isothermal) compression?
Thermodynamics of a liquid compressed too quickly to exchange heat: the temperature rise from the expansion coefficient, density, specific heat and pressure rise, the difference and ratio of the specific heats from the isothermal bulk modulus, the adiabatic bulk modulus that governs fast pressure changes and oil-column stiffness, the speed of a pressure wave in the free liquid, and a check of the final temperature against your limit.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Liquid density | kg/m³ | Density of the liquid at the starting temperature and pressure, from the fluid supplier’s data sheet. |
| Specific heat at constant pressure | kJ/(kg·K) | Specific heat at constant pressure, from the fluid data sheet or property tables. |
| Volumetric thermal expansion coefficient | ×10⁻⁶ /K | Volumetric (cubical) thermal expansion coefficient, three times the linear coefficient, from the data sheet; water below 4 °C has a negative value. |
| Isothermal bulk modulus | MPa | Isothermal (slow-compression) bulk modulus of the liquid at this temperature and pressure, from the supplier or property tables; quote whether it is secant or tangent. |
| Liquid temperature before compression | °C | Temperature of the liquid just before the rapid pressure change. |
| Pressure rise (rapid, no heat exchange) | MPa | Pressure rise applied quickly enough that no heat leaves the liquid; a negative value is a rapid pressure drop. |
| Highest temperature allowed | °C | Your temperature limit for the liquid, seals or process after compression. |
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: Liquid density
- What it is
- The mass of one cubic metre of the liquid at the starting temperature and pressure. It converts the expansion coefficient and specific heat into a temperature change per unit pressure and sets the wave speed.
- Units and format
- Enter a number in kg/m³: kilograms per cubic metre. The form accepts 1 kg/m³ to 30000 kg/m³. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 870 kg/m³ as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A denser liquid warms less for the same pressure rise and has a lower wave speed for the same bulk modulus.
- Get it right
- Take the density at the working temperature; a 15 °C data-sheet value is slightly high for hot oil.
What this means: Specific heat at constant pressure
- What it is
- The heat needed to warm one kilogram of the liquid by one kelvin at constant pressure. The compression work that appears as heat is spread over this heat capacity.
- Units and format
- Enter a number in kJ/(kg·K): kilojoules per kilogram per kelvin. The form accepts 0.01 kJ/(kg·K) to 100 kJ/(kg·K). Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1.9 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
- A higher specific heat lowers the temperature rise and the difference between the two bulk moduli.
- Get it right
- Mind the units: kJ/(kg·K), so water is about 4.18, not 4180.
What this means: Volumetric thermal expansion coefficient
- What it is
- How much a unit volume of the liquid grows per kelvin of warming at constant pressure, in millionths per kelvin. It links compression to heating: a liquid that expands on heating warms on compression.
- Units and format
- Enter a number in ×10⁻⁶ /K: ×10⁻⁶ /K. The form accepts -2000 ×10⁻⁶ /K to 20000 ×10⁻⁶ /K. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 700 ×10⁻⁶ /K as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The temperature rise is proportional to it, and c_p − c_v rises with its square; a negative value makes the liquid cool on compression.
- Get it right
- Data sheets often give the linear coefficient for solids; for liquids the cubical coefficient is the one quoted, and three times any linear value.
What this means: Isothermal bulk modulus
- What it is
- The pressure change per unit fractional change in volume when the liquid is compressed slowly enough to stay at constant temperature. The tool turns it into the adiabatic modulus that applies to fast pressure changes.
- Units and format
- Enter a number in MPa: megapascals (1 MPa = 1 N/mm²). The form accepts 1 MPa to 1000000 MPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1600 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- c_p − c_v and the adiabatic modulus rise with it, and the wave speed rises with its square root.
- Get it right
- Ask whether a quoted modulus is isothermal or adiabatic: entering an adiabatic value here double-counts the correction.
What this means: Liquid temperature before compression
- What it is
- The temperature of the liquid just before the rapid pressure change, converted to kelvin because the temperature rise is proportional to the absolute temperature.
- Units and format
- Enter a number in °C: degrees Celsius. The form accepts -100 °C to 600 °C. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 40 °C as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The temperature rise and c_p − c_v both rise in proportion to the absolute temperature.
- Get it right
- Use the liquid temperature at the point of compression, which in a running circuit is often well above the tank temperature.
What this means: Pressure rise (rapid, no heat exchange)
- What it is
- The rise in pressure applied so quickly that no heat leaves the liquid during it, as in a press stroke, intensifier or accumulator charge. A negative value models a rapid release of pressure, which cools the liquid.
- Units and format
- Enter a number in MPa: megapascals (1 MPa = 1 N/mm²). The form accepts -1000 MPa to 2000 MPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 35 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The temperature rise grows almost in proportion; the bulk moduli and wave speeds do not depend on it.
- Get it right
- Use the change in pressure, not the final gauge pressure, if the liquid started under pressure.
What this means: Highest temperature allowed
- What it is
- The highest liquid temperature you accept after compression, from the fluid’s rating, the seals or the process. The final temperature is checked against it.
- Units and format
- Enter a number in °C: degrees Celsius. The form accepts -100 °C to 1000 °C. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 60 °C as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Moves the pass and fail line only; the calculated values do not change.
- Get it right
- Remember that repeated rapid cycles add up if the heat cannot escape between them.
Method and formulas
- A Maxwell relation gives the temperature change on adiabatic compression: (∂T ÷ ∂p)_s = T (∂v ÷ ∂T)_p ÷ c_p = T α ÷ (ρ c_p) (Bryan §114, relation (131)).
- With α, ρ and c_p held constant this integrates to T₂ = T₁ exp[α Δp ÷ (ρ c_p)]; for small rises it equals T α Δp ÷ (ρ c_p).
- The specific heats differ by c_p − c_v = T α² K_T ÷ ρ (Bryan §116), so γ = c_p ÷ c_v.
- The adiabatic and isothermal moduli are in the ratio of the specific heats, K_s = γ K_T (Bryan §33, equation (35)), and the wave speed is √(K_s ÷ ρ) (Bryan §121).
- Example: an oil at 870 kg/m³, 1.9 kJ/(kg·K), 700 × 10⁻⁶ /K and 1600 MPa compressed rapidly by 35 MPa from 40 °C warms by 4.68 K; γ = 1.174 and K_s = 1879 MPa.
Limitations
- Properties are taken as constant over the pressure and temperature change; over large pressure rises use property data at the mean condition or step the calculation.
- The liquid alone: entrained air, hose and pipe wall flexibility lower the effective bulk modulus and wave speed far below these values.
- Rapid compression with no heat exchange; in practice the liquid then cools towards the surroundings.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Estimate oil heating in a press, intensifier or accumulator charged quickly
- Use the right bulk modulus for hydraulic natural frequency and stiffness estimates
- Estimate the pressure wave speed in a liquid for water hammer screening
Who it is for
- Hydraulic and fluid power engineers
- Machine designers
- High-pressure equipment engineers
What you get
- Temperature rise and final temperature
- c_p − c_v, c_v and the ratio γ = c_p ÷ c_v
- Adiabatic bulk modulus
- Wave speed in the free liquid, adiabatic and isothermal
- Verdict on the temperature limit
Interpreting the result
- The adiabatic modulus is never below the isothermal one (17 % higher in the example), so using the isothermal value understates the stiffness of a fast-acting circuit.
- Water has a small γ near room temperature and γ = 1 at its density maximum, where rapid compression causes no temperature change.
Before relying on the result
- Confirm whether the supplier quotes an isothermal or adiabatic, secant or tangent bulk modulus.
References
- Bryan G. H., Thermodynamics: An Introductory Treatise Dealing Mainly with First Principles and Their Direct Applications (B. G. Teubner, Leipzig, 1907), Chapter III §33 (adiabatic and isothermal moduli in the ratio of the specific heats, equation (35)), Chapter XI §§113–116 (Maxwell’s relations; specific heats and moduli in terms of the potentials) and Chapter XII §121 (wave speed), pp. 22, 108–113, 119. 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): thermodynamic property relations, c_p − c_v and the isothermal compressibility.
Learn the Concepts Behind This Tool
These KEVOS® Library readings explain the method this tool uses, with worked examples.
Useful alongside · KEVOS® original book
Wave speed in a pipe: water, wall and air
In Water Hammer by the Numbers
What it covers: Calculate pressure-wave speed from water's bulk modulus, the pipe wall's stiffness and its restraint, and see how a little entrained air or a plastic pipe changes it.
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