Volumetric (Cubic) Thermal Expansion

Specific volume of a unit can be expressed as

v = 1 / ρ = V / m                                                          (1)

where

v = specific volume (m³/kg, ft³/lb)

ρ = density (kg/m³, lb/ft³)

V = volume of unit (m³, ft³)

m = mass of unit (kg, lb)

The change in the units volume when temperature changes can be expressed as

dV = V₀ β (t₁ - t₀)                                                        (2)

where

dV = V₁ - V₀ = change in volume (m³, ft³)

β = volumetric temperature expansion coefficient (m³/m^{3 o}C, ft³/ft^{3 o}F)

t₁ = final temperature (^oC, ^oF)

t₀ = initial temperature (^oC, ^oF)

The density of a fluid when temperature changes can be expressed as

ρ₁ = m / V₀ (1 + β (t₁ - t₀))

    = ρ₀ / (1 + β (t₁ - t₀))                                                (3)

where

ρ₁ = final density (kg/m³, lb/ft³)

ρ₀ = initial density (kg/m³, lb/ft³)

Online Thermal Cubic Expansion Calculator - Expansion coefficient and Temperatures

Be aware that the expansion coefficient for some liquids - like water - may vary with temperature. The calculator below is generic and can be used for metric and imperial units as long as the use of units is consistent.

V₀ - initial volume (m³, ft³, gallons ..)

β - volumetric temperature expansion coefficient (1/^oC, 1/^oF)

t₀ - initial temperature (^oC, ^oF)

t₁ - final temperature (^oC, ^oF)

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Note that the volumetric expansion coefficient used in the calculator is constant. If you want to calculate volumetric change for a liquid over a temperature range where the volumetric expansion coefficient for the liquid changes a lot - interpolate the coefficient values, or split the calculation in the different temperature ranges. Example - water is a liquid where the volumetric expansion coefficient changes a lot with temperature. Water has its highest density and smallest volume at 4^oC (39.2 ^oF). The volumetric coefficient for water is negative below 4^oC and indicates that the a volume decreases when temperature moves from 0^oC (32 ^oF) to 4^oC.

Online Thermal Cubic Expansion Calculator - Densities

This calculator can be used to calculate expansion volume when initial volume and initial and final densities for the liquid are known

V₀ - initial volume (m³, ft³)

ρ₀ - initial density (kg/m³, lb/ft³)

ρ₁ - final density (kg/m³, lb/ft³)

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• Density water
• Density of Ethylene Glycol based Water Solution

Volumetric Temperature Coefficients - β - for some Fluids

• water at 0^oC: -0.000050 (1/^oC)
• water at 4^oC: 0 (1/^oC)
  
• water at 10^oC: 0.000088 (1/^oC)
  
• water at 20^oC: 0.000207 (1/^oC)
• water at 30^oC: 0.000303 (1/^oC)
• water at 40^oC: 0.000385 (1/^oC)
• water at 50^oC: 0.000457 (1/^oC)
• water at 60^oC: 0.000522 (1/^oC)
• water at 70^oC: 0.000582 (1/^oC)
• water at 80^oC: 0.000640 (1/^oC)
• water at 90^oC: 0.000695 (1/^oC)
• ethyl alcohol: 0.00109 (1/ ^oC), 0.00061 (1/^oF)
• oil: 0.00070 (1/^oC), 0.00039 (1/^oF)

• Volumetric expansion coefficients for commonly used liquids

Convert between Metric and Imperial Volumentric Temperature Coefficients

• 1 (1/^oC) =  0.56 (1/^oF)
• 1 (1/^oF) =  1.8 (1/^oC)

Example - Cubic Expansion of Oil

100 liters - 0.1 m³ - of oil with volumetric expansion coefficient 0.00070 1/^oC is heated from 20^oC to 40^oC. The volumetric expansion can be calculated using equation (2)

dV = (0.1 m³) (0.00070 1/^oC) ((40 ^oC) - (20 ^oC)) 

  = 0.0014 m³

  = 1.4 liter

The final volume is

100 liters + 1.4 liters = 101.4 liters

Example - Cubic Expansion of Oil

30 US gallons of oil is heated from 70^oF to 100^oF. The volumetric expansion can be calculated using equation (2)

dV = (30 gallons) (0.00039 1/^oF) ((100 ^oF) - (70 ^oF)) 

    = 0.351 gallons

The final volume is

30 gallons + 0.351 gallon = 30.351 gallons