# 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^{ o }F ) to 4^{ 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 *

## Related Topics

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### • Heating

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### • Thermodynamics

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Hot and cold water service systems - design properties, capacities, sizing and more.

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