Compressibility

• Compressibility of any substance is the measure of its change in volume under the action of external forces.
  
  

• The normal compressive stress on any fluid element at rest is known as hydrostatic pressure p and arises as a result of innumerable molecular collisions in the entire fluid.
  
  

• The degree of compressibility of a substance is characterized by the bulk modulus of elasticity E defined as
  

  (2.3)

  
  Where Δ and Δp are the changes in the volume and pressure respectively, and is the initial volume. The negative sign (-sign) is included to make E positive, since increase in pressure would decrease the volume i.e for Δp>0 , Δ<0) in volume.

• For a given mass of a substance, the change in its volume and density satisfies the relation

  Dm = 0,    D( ρ ) = 0   

  (2.4)

  
  

  using

  we get

  (2.5)

       

• Values of E for liquids are very high as compared with those of gases (except at very high pressures). Therefore, liquids are usually termed as incompressible fluids though, in fact, no substance is theoretically incompressible with a value of E as .
  
  
• For example, the bulk modulus of elasticity for water and air at atmospheric pressure are approximately 2 x 10⁶ kN/m ² and 101 kN/m ² respectively. It indicates that air is about 20,000 times more compressible than water. Hence water can be treated as incompressible.
  
  
  
• For gases another characteristic parameter, known as compressibility K, is usually defined , it is the reciprocal of E
  
  

  (2.6)

  K is often expressed in terms of specific volume .

• For any gaseous substance, a change in pressure is generally associated with a change in volume and a change in temperature simultaneously. A functional relationship between the pressure, volume and temperature at any equilibrium state is known as thermodynamic equation of state for the gas.
  
  For an ideal gas, the thermodynamic equation of state is given by
  
  

  p = ρRT

  (2.7)

  
  
• where T is the temperature in absolute thermodynamic or gas temperature scale (which are, in fact, identical), and R is known as the characteristic gas constant, the value of which depends upon a particular gas. However, this equation is also valid for the real gases which are thermodynamically far from their liquid phase. For air, the value of R is 287 J/kg K.
  
  
• K and E generally depend on the nature of process