Calculation
ΔV = V0 · γ · ΔT
Unlike for solids, liquids are usually characterised directly by the volumetric expansion coefficient γ, not three times the linear coefficient.
ΔV = V0·γ·ΔT
Liquid volumetric expansion in a closed vessel
Unlike for solids, liquids are usually characterised directly by the volumetric expansion coefficient γ, not three times the linear coefficient.
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Result
Select a target and calculate.
Understand the inputs
- Volume increase ΔV — Additional space the liquid needs at the stated temperature rise; must remain free as expansion reserve.
- Liquid volume at reference temperature V0 — Filled liquid volume at the colder reference temperature, e.g. at fill time.
- Volumetric expansion coefficient γ — Fluid property; water at 20 °C about 0.0002/K, mineral hydraulic oil per DIN 51757 about 0.00065/K, diesel fuel about 0.00095/K to 0.001/K.
- Temperature change ΔT — Difference between the highest expected operating temperature and the fill reference temperature.
Example
V0=200 l (0.2 m³) of hydraulic oil with γ≈0.00065/K and ΔT=40 K give ΔV=0.0052 m³, about 5.2 litres of additional expansion space.
Assumptions and limits
Constant γ over the considered range, a rigid vessel with no expansion of its own, and no evaporation or phase change; real γ values are also mildly temperature-dependent and should be checked against manufacturer data.
Understand Liquid volumetric expansion in a closed vessel
This calculator determines how much extra space must be available when a liquid enclosed in a vessel expands on heating. It complements the existing linear thermal expansion of solid components with the direct volumetric expansion coefficient customary for liquids.
What does this quantity describe?
For liquids, volume increase is usually given directly via a volumetric expansion coefficient γ, rather than the linear coefficient α used for solids (there, γ≈3α approximately). ΔV=V0·γ·ΔT applies, where V0 is the liquid volume at the colder reference temperature.
Formula and variables
ΔV = V0 · γ · ΔT
| Symbol / input | Meaning |
|---|---|
| Volume increase ΔV | Additional space the liquid needs at the stated temperature rise; must remain free as expansion reserve. |
| Liquid volume at reference temperature V0 | Filled liquid volume at the colder reference temperature, e.g. at fill time. |
| Volumetric expansion coefficient γ | Fluid property; water at 20 °C about 0.0002/K, mineral hydraulic oil per DIN 51757 about 0.00065/K, diesel fuel about 0.00095/K to 0.001/K. |
| Temperature change ΔT | Difference between the highest expected operating temperature and the fill reference temperature. |
Choose the inputs correctly
V0 is the filled liquid volume at the reference temperature, usually the fill temperature. γ is the volumetric expansion coefficient of the liquid, e.g. water at 20 °C about 0.0002/K, mineral hydraulic oil per DIN 51757 about 0.00065/K, and diesel fuel about 0.00095 to 0.001/K. ΔT is the difference between the highest expected operating temperature and the reference temperature.
How to use the calculator
Take V0 from the actual fill quantity. Take γ from a material datasheet for the liquid used; for hydraulic oils it is usually stated in the technical datasheet or per DIN 51757. Form ΔT from the difference between minimum and maximum expected operating temperature.
Worked example
V0=200 l (0.2 m³) of hydraulic oil with γ≈0.00065/K and ΔT=40 K give ΔV=0.2·0.00065·40=0.0052 m³, about 5.2 litres of additional expansion space.
Understand the result and units
ΔV indicates how much air or expansion space must remain free in the vessel so the liquid does not fill it completely at the warmest operating temperature. Too little expansion space in a closed system can lead to unacceptably high pressure build-up.
V0 and ΔV are volumes, γ a volumetric expansion coefficient per kelvin and ΔT a temperature difference in kelvin.
Where the coefficients come from
The formula ΔV=V0·γ·ΔT is the standard thermodynamic relation for liquids. The reference values quoted for water, hydraulic oil (DIN 51757) and diesel fuel are publicly documented material properties; γ remains a visible input for the user to verify rather than a fixed table built into the calculator.
Typical applications
Sizing the air cushion in hydraulic and gear-oil reservoirs, fuel tanks and other closed or partially open liquid vessels operated over a wide temperature range.
Assumptions, limits and common mistakes
Constant γ over the considered range, a rigid vessel with no thermal expansion of its own, and no evaporation, degassing or phase change. Real γ values are also mildly temperature- and pressure-dependent and should be checked against manufacturer data for safety-relevant designs.
Common mistake: Do not substitute the linear expansion coefficient α of solid materials for the liquid-specific γ; γ is significantly larger than 3α of a comparable solid for most liquids. Do not forget that the vessel itself also expands slightly, marginally reducing the required free space.
Frequently asked questions
Why is γ used for liquids instead of α?
Liquids have no fixed shape; their expansion is given directly as a volume change. For solids, γ can be approximated as 3α, but for liquids γ is a distinct, usually much larger material property.
Where do I get the right γ value for my liquid?
From the manufacturer's technical datasheet or material property tables; for mineral oils, DIN 51757 is the relevant reference.
Does the calculator account for the vessel's own expansion?
No. The vessel is assumed rigid; its own, usually much smaller thermal expansion slightly reduces the actually required free space.