Liquids - Densities vs. Pressure and Temperature Change
The density of a liquid will change with temperature and pressure. The density of water versus temperature and pressure is indicated below:
See also Water - Density, Specific Weight and Thermal Expantion Coefficient , for online calculator, figures and tables showing changes with temperature.
Density
The density of a liquid can be expressed as
ρ = m / V (1)
where
ρ = density of liquid (kg/m 3 )
m = mass of the liquid (kg)
V = volume of the liquid (m 3 )
The inverse of density is specific volume:
v = 1 / ρ
= V / m (2)
where
v = specific volume (m 3 /kg)
Volume and change in Temperature
When temperature increases - most liquids expands:
dV = V 1 - V 0
= V 0 β dt
= V 0 β (t 1 - t 0 ) (3)
where
dV = V 1 - V 0 = change in volume - difference between final and initial volume (m 3 )
β = volumetric temperature expansion coefficient (m 3 /m 3 o C)
dt = t 1 - t 0 = change in temperature - difference between final and initial temperature ( o C)
(3) can be modified to
V 1 = V 0 (1 + β (t 1 - t 0 )) (3b)
Density and change in Temperature
With (1) and (3b) the final density after a temperature change can be expressed as
ρ 1 = m / ( V 0 (1 + β (t 1 - t 0 ))) (4)
where
ρ 1 = final density (kg/m 3 )
- or combined with (2)
ρ 1 = ρ 0 / (1 + β (t 1 - t 0 )) (4b)
where
ρ 0 = initial density (kg/m 3 )
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
The influence of pressure on the volume of a liquid can be expressed with the three dimensional Hooke's law
E = - dp / (dV / V 0 )
= - (p 1 - p 0 ) / ((V 1 - V 0 ) / V 0 ) (5)
where
E = bulk modulus - liquid elasticity (N/m 2 )
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
V 1 = V 0 (1 - (p 1 - p 0 ) / E) (5b)
Combining (5b) with (1) - the final density can be expressed as:
ρ 1 = m / ( V 0 (1 - (p 1 - p 0 ) / E)) (6)
- or combined with (2) - the final density can be expressed as
ρ 1 = ρ 0 / (1 - (p 1 - p 0 ) / E) (6b)
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 )
Note! Bulk modulus for liquids varies with pressure and temperature.
Bulk modulus for water - Imperial Units
Bulk modulus for Water - SI units
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) :
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) (7)
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 ) )
= 1002.7 (kg/m 3 )
Related Topics
• Densities
Densities of solids, liquids and gases. Definitions and convertion calculators.
• Fluid Mechanics
The study of fluids - liquids and gases. Involving velocity, pressure, density and temperature as functions of space and time.
• Thermodynamics
Work, heat and energy systems.
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