# 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:

Fluid | Bulk Modulus - K - | |
---|---|---|

Imperial Units - BG ( 10 ^{ 5 } psi, lb _{ f } /in ^{ 2 } ) | SI Units ( 10 ^{ 9 } Pa, N/m ^{ 2 } ) | |

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 ^{ o } C) | 3.12 | 2.09 |

Water - glycol | 5 | 3.4 |

Water in oil emulsion | 3.3 | 2.3 |

*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

Temperature ( ^{ o } C) | Bulk Modulus (10 ^{ 9 } 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 |

## Related Topics

### • Fluid Mechanics

The study of fluids - liquids and gases. Involving velocity, pressure, density and temperature as functions of space and time.

### • Material Properties

Material properties of gases, fluids and solids - densities, specific heats, viscosities and more.

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