Bulk Modulus and Fluid Elasticities

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 / V ₀ )

= - ( p ₁ - p ₀ ) / ((V ₁ - V ₀ ) / V ₀ )                       (1)

where

K = Bulk Modulus of Elasticity (Pa, N/m ² )

dp = differential change in pressure on the object (Pa, N/m ² )

dV = differential change in volume of the object (m ³ )

V ₀ = initial volume of the object  (m ³ )

p ₀ = initial pressure ( Pa, N/m ² )

p ₁ = final pressure ( Pa, N/m ² )

V ₁ = final volume ( m ³ )

The Bulk Modulus Elasticity can alternatively be expressed as

K = dp / (dρ / ρ ₀ )

= ( p ₁ - p ₀ ) / (( ρ ₁ - ρ ₀ ) / ρ ₀ )                              (2)

where

dρ = differential change in density of the object   (kg/m ³ )

ρ ₀ = initial density of the object (kg/m ³ )

ρ ₁ = final density of the object ( kg/m ³ ) <

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/m ² (Pa)
• The imperial (BG) unit is lb _f /in ² (psi)
• 1 lb _f /in ² (psi) = 6.894 10 ³ N/m ² (Pa)

A large Bulk Modulus indicates a relative incompressible fluid.

Bulk Modulus for some common fluids:

• 1 GPa = 10 ⁹ Pa (N/m ² )

Stainless steel with Bulk Modulus 163 10 ⁹ Pa is aprox. 80 times harder to compress than water with Bulk Modulus 2.15 10 ⁹ 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

p ₁ = (1022 kg/m ³ ) (9.81 m/s ² ) (10994 m)

= 110 10 ⁶ Pa  (110 MPa)

The initial pressure at sea-level is 10 ⁵ Pa and the density of seawater at sea level is 1022 kg/m ³ .

The density of seawater in the deep can be calculated by modifying (2) to

ρ ₁ = ( ( p ₁ - p ₀ ) ρ ₀ + K ρ ₀ ) / K

= (((110 10 ⁶ Pa) - (1 10 ⁵ Pa)) (1022 kg/m ³ ) + (2.34 10 ⁹ Pa) (1022 kg/m ³ )) / ( 2.34 10 ⁹ Pa )

= 1070 kg/m ³

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