# Density of Fluids - with Changing Pressure and Temperature

## Calculating density of fluids when changing the pressure or temperature

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The density of a fluid change with temperature and pressure. The density of water versus temperature and pressure is indicated in the chart below:

### Density and change in Temperature

When temperature is changed the density of a fluid can be expressed as

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

where

ρ_{1}= final density (kg/m^{3})

ρ_{0}= initial density (kg/m^{3})

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

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

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

#### Volumetric Temperature Coefficients - *β *

- water :
*0.0002 (m*^{3}/m^{3 o}C) at 20^{o}C - ethyl alcohol :
*0.0011 (m*^{3}/m^{3 o}C) - volumetric expansion coefficient for some commonly used materials

**Note!** - volumetric temperature coefficients may vary strongly with temperature.

### Density and change in Pressure

When pressure is changed the density of a fluid can be expressed as

ρ_{1}= ρ_{0}/ (1 - (p_{1}- p_{0}) / E) (2)

where

E = bulk modulus fluid elasticity (N/m^{2})

ρ_{1}= final density (kg/m^{3})

ρ_{0}= initial density (kg/m^{3})

p_{1 }= final pressure (N/m^{2})

p_{0 }= initial pressure (N/m^{2})

#### Bulk Modulus Fluid Elasticity some common Fluids - *E*

- water :
*2.15 10*^{9 }(N/m^{2}) - ethyl alcohol :
*1.06 10*^{9}(N/m^{2}) - oil :
*1.5 10*^{9}(N/m^{2})

### Density of a fluid changing both Temperature and Pressure

The density of a fluid when changing both temperature and pressure can be expressed with (1) and (2) by substituting *ρ _{0}* in eq. 2 with the expression for

*ρ*in eq. 1.

_{1}

ρ_{1}=ρ_{1(from eq.1)}/ (1 - (p_{1}- p_{0}) / E)

=ρ_{0}/ (1 + β (t_{1}- t_{0}))/ (1 - (p_{1}- p_{0}) / E)(3)

### Example - Density of Water at 100 bar and 20^{o}C

- density of water 0
^{o}C:*999.8 (kg/m*^{3}) - expansion coefficient of water at 10
^{o}C:*0.000088 (**m*^{3}/m^{3}^{o}C) (average value between 0 and 20^{o}C) - bulk modulus of water:
*2.15 10*^{9 }(N/m^{2})

Density of water can be calculated with (3):

*ρ _{1} = (999.8 kg/m^{3}) / (1 + (0.000088 m^{3}/m^{3}^{o}C) ((20 ^{o}C) - (0 ^{o}C))) / (1 - ((100 10^{5} Pa) - (1 10^{5} Pa)) / (2.15 10^{9 }N/m^{2})) *

*= 998.0 / 0.995 *

*= 1002.7 (kg/m ^{3})*

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