# Volumetric (Cubic) Thermal Expansion

Specific volume of a unit can be expressed as

v = 1 / ρ = V / m (1)

where

v = specific volume(m^{3}/kg,ft^{3}/lb)

ρ = density(kg/m^{3},lb/ft^{3})

V = volume of unit (m^{3}, ft^{3})

m = mass of unit (kg, lb)

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

dV = V_{0}β (t_{1 }- t_{0}) (2)

where

dV = V_{1}- V_{0}= change in volume (m^{3}, ft^{3})

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

t_{1 }= final temperature (^{o}C,^{o}F)

t_{0 }= initial temperature(^{o}C,^{o}F)

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

ρ_{1}= m / V_{0}(1 + β (t_{1}- t_{0}))

= ρ_{0}/ (1 + β (t_{1}- t_{0})) (3)

where

ρ_{1}= final density (kg/m^{3},lb/ft^{3})

ρ_{0}= initial density(kg/m^{3},lb/ft^{3})

### 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.

**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 ^{o}C (39.2 ^{o}F)*. The volumetric coefficient for water is negative below 4

^{o}C and indicates that the a volume decreases when temperature moves from

*0*.

^{o}C (*32*) to 4^{o}F^{o}C### 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

### Volumetric Temperature Coefficients - *β *-* for some Fluids *

- water at
*0*:^{o}C*-0.00005**0 (1/*^{o}C) - water at
*4*:^{o}C*0 (1/*^{o}C) - water at
*10*:^{o}C*0.000088 (1/*^{o}C) - water at
*20*:^{o}C*0.000207 (1/*^{o}C) *water at**30*:^{o}C*0.000303 (1/*^{o}C)*water at**40*:^{o}C*0.000385 (1/*^{o}C)*water at**50*:^{o}C*0.000457 (1/*^{o}C)*water at**60*:^{o}C*0.000522 (1/*^{o}C)*water at**70*:^{o}C*0.000582 (1/*^{o}C)*water at**80*:^{o}C*0.000640 (1/*^{o}C)*water at**90*:^{o}C*0.000695 (1/*^{o}C)- ethyl alcohol:
*0.00109 (1/*^{ o}C), 0.00061 (1/^{o}F) - oil:
*0.00070 (1/*^{o}C), 0.00039 (1/^{o}F)

### Convert between Metric and Imperial Volumentric Temperature Coefficients

*1 (1/*^{o}C) = 0.56 (1/^{o}F)*1 (1/*^{o}F) = 1.8 (1/^{o}C)

### Example - Cubic Expansion of Oil

*100 liters* - *0.1 m ^{3} *- of oil with volumetric expansion coefficient

*0.00070 1/*is heated from

^{o}C*20*to

^{o}C*40*. The volumetric expansion can be calculated using equation

^{o}C*(2)*

*dV = (0.1 m ^{3})_{ }(0.00070 1/^{o}C) ((40 ^{o}C) - (20 ^{o}C)) *

* = 0.0014 m ^{3}*

* = 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 7*0 ^{o}F* to

*100*. The volumetric expansion can be calculated using equation

^{o}F*(2)*

*dV = (30 gallons) _{ }(0.00039 1/^{o}F) ((100 ^{o}F) - (70 ^{o}F)) *

* = 0.351 gallons *

The final volume is* *

*30 gallons + 0.351 gallon = 30.351 gallons*

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### • Thermodynamics

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