COMPRESSIBILITY AND ELASTICITY

 

All fluids may be compressed by the application of external force, and when the external force is removed the compressed volumes of fluids expand to their original volumes. Thus fluids also possess elastic characteristics like elastic solids. Compressibility of a fluid is quantitatively expressed as inverse of the bulk modulus of elasticity K of the fluid, which is defined as:

Thus bulk modulus of elasticity K is a measure of the incremental change in pressure dp which takes place when a volume V of the fluid is changed by an incremental amount dV. Since a rise in pressure always causes a decrease in volume, dV is always negative, and the minus sign is included in the equation to give a positive value of K.

For example, consider a cylinder containing a fluid of volume V, which is being compressed by a piston. Now if the piston is moved so that the volume V decreases by a small amount dV, then the pressure will increase by amount dp, the magnitude of which depends upon the bulk modulus of  elasticity of the fluid, as expressed in Eq. 1.

In SI units the bulk modulus of elasticity is expressed in N/m². In the metric gravitational system of units it is expressed in either kg(f)/cm² or kg(f)/m². In the English system of units it is expressed either in lb(f)/in² or lb(f)/ft². The bulk modulus of elasticity for water and air at normal temperature and pressure is approximately 2.06 × 109 N/m² [or 2.1 × 108 kg (f)/m²] and 1.03 × 105 N/m² [or 1.05 × 104 kg (f)/m²] respectively. Thus air is about 20,000 times more compressible than water. The bulk (volume) modulus of elasticity of mild steel is about 2.06 × 1011 N/m² [or 2.1 × 1010 kg(f)/m²] which shows that water is about 100 times more compressible than steel.

However, the bulk modulus of elasticity of a fluid is not constant, but it increases with increase in pressure. This is so because when a fluid mass is compressed, its molecules become close together and its resistance to further compression increases i.e., K increases. Thus for example, the bulk modulus of water roughly doubles as the pressure is raised from 1 atmosphere to 3500 atmospheres.

 The temperature of the fluid also affects the bulk modulus of elasticity of the fluid. In the case of liquids there is a decrease of K with increase of temperature. However, for gases since pressure and temperature are inter-related and as temperature increases, pressure also increases, an increase in temperature results in an increase in the value of K.

For liquids since the bulk modulus of elasticity is very high, the change of density with increase of pressure is very small even with the largest pressure change encountered. Accordingly in the case of liquids the effects of compressibility can be neglected in most of the problems involving the flow of liquids. However, in some special problems such as rapid closure of valve or water hammer, where the changes of pressure are either very large or very sudden, it is necessary to consider the effect of compressibility of liquids.

 On the other hand gases are easily compressible and with the change in pressure the mass density of gases changes considerably and hence the effects of compressibility cannot ordinarily be neglected in the problems involving the flow of gases. However, in a few cases where there is not much change in pressure and so gases undergo only very small changes of density, the effects of compressibility may be disregarded e.g., the flow of air in a ventilating system is a case where air may be treated as incompressible.