Bulk Modulus and Fluid Elasticity
Introduction to - and definition of - Bulk Modulus Elasticity commonly used to characterize the compressibility of fluids.
The Bulk Modulus Elasticity - or Volume Modulus - is a material property characterizing the compressibility of a fluid - how easy a unit volume of a fluid can be changed when changing the pressure working upon it.
The Bulk Modulus Elasticity can be calculated as
K = - dp / (dV / V0)
= - (p1 - p0) / ((V1 - V0) / V0) (1)
where
K = Bulk Modulus of Elasticity (Pa, N/m2)
dp = differential change in pressure on the object (Pa, N/m2)
dV = differential change in volume of the object (m3)
V0 = initial volume of the object (m3)
p0 = initial pressure (Pa, N/m2)
p1 = final pressure (Pa, N/m2)
V1 = final volume (m3)
The Bulk Modulus Elasticity can alternatively be expressed as
K = dp / (dρ / ρ0)
= (p1 - p0) / ((ρ1 - ρ0) / ρ0) (2)
where
dρ = differential change in density of the object (kg/m3)
ρ0 = initial density of the object (kg/m3)
ρ1 = final density of the object (kg/m3)
An increase in the pressure will decrease the volume (1). A decrease in the volume will increase the density (2).
- The SI unit of the bulk modulus elasticity is N/m2 (Pa)
- The imperial (BG) unit is lbf/in2 (psi)
- 1 lbf/in2 (psi) = 6.894 103 N/m2 (Pa)
A large Bulk Modulus indicates a relative incompressible fluid.
Bulk Modulus for some common fluids:
| Fluid | Bulk Modulus - K - | |
|---|---|---|
| Imperial Units - BG (105 psi, lbf/in2) | SI Units (109 Pa, N/m2) | |
| Acetone | 1.34 | 0.92 |
| Benzene | 1.5 | 1.05 |
| Carbon Tetrachloride | 1.91 | 1.32 |
| Ethyl Alcohol | 1.54 | 1.06 |
| Gasoline | 1.9 | 1.3 |
| Glycerin | 6.31 | 4.35 |
| ISO 32 mineral oil | 2.6 | 1.8 |
| Kerosene | 1.9 | 1.3 |
| Mercury | 41.4 | 28.5 |
| Paraffin Oil | 2.41 | 1.66 |
| Petrol | 1.55 - 2.16 | 1.07 - 1.49 |
| Phosphate ester | 4.4 | 3 |
| SAE 30 Oil | 2.2 | 1.5 |
| Seawater | 3.39 | 2.34 |
| Sulfuric Acid | 4.3 | 3.0 |
| Water (10 oC) | 3.12 | 2.09 |
| Water - glycol | 5 | 3.4 |
| Water in oil emulsion | 3.3 | 2.3 |
- 1 GPa = 109 Pa (N/m2)
Stainless steel with Bulk Modulus 163 109 Pa is aprox. 80 times harder to compress than water with Bulk Modulus 2.15 109 Pa.
Example - Density of Seawater in the Mariana Trench
- the deepest known point in the Earth's oceans - 10994 m.
The hydrostatic pressure in the Mariana Trench can be calculated as
p1 = (1022 kg/m3) (9.81 m/s2) (10994 m)
= 110 106 Pa (110 MPa)
The initial pressure at sea-level is 105 Pa and the density of seawater at sea level is 1022 kg/m3.
The density of seawater in the deep can be calculated by modifying (2) to
ρ1 = ((p1 - p0) ρ0 + K ρ0) / K
= (((110 106 Pa) - (1 105 Pa)) (1022 kg/m3) + (2.34 109 Pa) (1022 kg/m3)) / (2.34 109 Pa)
= 1070 kg/m3
Note! - since the density of the seawater varies with dept the pressure calculation could be done more accurate by calculating in dept intervals.
Bulk Modulus of Water vs. Temperature
| Temperature (oC) | Bulk Modulus (109 Pa) |
|---|---|
| 0.01 | 1.96 |
| 10 | 2.09 |
| 20 | 2.18 |
| 30 | 2.23 |
| 40 | 2.26 |
| 50 | 2.26 |
| 60 | 2.25 |
| 70 | 2.21 |
| 80 | 2.17 |
| 90 | 2.11 |
| 100 | 2.04 |
