Density of a Liquid - when Changing Pressure and Temperature

Density and specific volume of a liquid when changing pressure or temperature

The density of liquids changes with temperature and pressure. The density of water versus temperature and pressure is indicated below:

water density temperature pressure

Density

The density of a liquid can be expressed as

ρ = m / V     (1)

where

ρ = density of liquid (kg/m3)

m = mass of the liquid (kg)

V = volume of the liquid (m3)

The inverse of density is specific volume:

v = 1 / ρ

   = V / m      (2)

where

v = specific volume (m3/kg)

Volume and change in Temperature

When temperature increases -  most liquids expands:

dV = V1 - V0

     = V0 β dt 

     = V0 β (t1 - t0)      (3)

where

dV = V1 - V0 = change in volume - difference between final and initial volume (m3)

β = volumetric temperature expansion coefficient (m3/m3 oC)

dt = t1 - t0 = change in temperature - difference between final and initial temperature (oC)

(3) can be modified to

V1 = V0 (1 + β (t1 - t0))       (3b)

Density and change in Temperature

With (1) and (3b) the final density after a temperature change can be expressed as

ρ1 = m / (V0 (1 + β (t1 - t0)))    (4)

where

ρ1 = final density (kg/m3)

- or combined with (2)

ρ1ρ0 / (1 + β (t1 - t0))     (4b)

where

ρ0 = initial density (kg/m3)

Volumetric Temperature Coefficients - β

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

Density and change in Pressure

The influence of pressure on the volume of a liquid can be expressed with the three dimensional Hooke's law

E = - dp / (dV / V0

   = - (p1 - p0) / ((V1 - V0) / V0)      (5)

where

E = bulk modulus - liquid elasticity (N/m2)

The minus sign corresponds to the fact that an increase in the pressure leads to a decrease in volume.

With (5) - the final volume after pressure change can be expressed as

V1 = V0 (1 - (p1 - p0) / E)      (5b)

Combining (5b) with (1) - the final density can be expressed as:

ρ1 = m / (V0 (1 - (p1 - p0) / E))    (6)

- or combined with (2) - the final density can be expressed as

ρ1 = ρ0 / (1 - (p1 - p0) / E)    (6b)

Bulk Modulus Fluid Elasticity some common Fluids - E

  • water : 2.15 109 (N/m2)
  • ethyl alcohol : 1.06 109 (N/m2)
  • oil : 1.5 109 (N/m2)

Note! Bulk modulus for liquids varies with pressure and temperature.

Bulk modulus for water - Imperial Units

bulk modulus (psi) water

Bulk modulus for Water - SI units

bulk modulus (GPa) water

Density of a fluid changing both Temperature and Pressure

The density of a fluid when changing both temperature and pressure can be expressed combining (4b) and (6b):

 ρ1ρ1(from eq.1) / (1 - (p1 - p0) / E)

      = ρ0 / (1 + β (t1 - t0)) / (1 - (p1 - p0) / E)   (7)

Example - Density of Water at 100 bar and 20oC

  • density of water 0oC: 999.8 (kg/m3)
  • expansion coefficient of water at 10oC: 0.000088 (m3/m3oC) (average value between 0 and 20oC)
  • bulk modulus of water: 2.15 10(N/m2)

Density of water can be calculated with (3):

 ρ1 = (999.8 kg/m3) / (1 + (0.000088 m3/m3oC) ((20 oC) - (0 oC))) / (1 - ((100 105 Pa) - (1 105 Pa)) / (2.15 109 N/m2))  

     = 998.0 / 0.995    

     = 1002.7 (kg/m3)

Related Topics

  • Fluid Mechanics - The study of fluids - liquids and gases. Involves velocity, pressure, density and temperature as functions of space and time
  • Thermodynamics - Effects of work, heat and energy on systems

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