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Compressibility

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

(ii) 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.

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

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

using

we get,

(i) 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 ∞ .

(ii) For example, the bulk modulus of elasticity for water and air at atmospheric pressure are approximately 2 x 10⁶ kN/m² and 101 kN/m² respectively. It indicates that air is about 20,000 times more compressible than water. Hence water can be treated as incompressible.

(iii) For gases another characteristic parameter, known as compressibility K, is usually defined , it is the reciprocal of E

K is often expressed in terms of specific volume ∀.

(iv) 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 (5)

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.

(v) K and E generally depend on the nature of process

Distinction between an Incompressible and a Compressible Flow

(vi) In order to know, if it is necessary to take into account the compressibility of gases in fluid flow problems, we need to consider whether the change in pressure brought about by the fluid motion causes large change in volume or density.

Using Bernoulli's equation

p + (1/2)ρV²= constant (V being the velocity of flow), change in pressure, Δp, in a flow field, is of the order of (1/2)ρV² (dynamic head).

Invoking this relationship into

we get ,

So if Δρ/ρ is very small, the flow of gases can be treated as incompressible with a good degree of approximation.

(vii) According to Laplace's equation, the velocity of sound is given by

Where, Ma is the ratio of the velocity of flow to the acoustic velocity in the flowing medium at the condition and is known as Mach number. So we can conclude that the compressibility of gas in a flow can be neglected if Δρ/ρ is considerably smaller than unity, i.e. (1/2)Ma²<<1.

(vii) In other words, if the flow velocity is small as compared to the local acoustic velocity, compressibility of gases can be neglected. Considering a maximum relative change in density of 5 per cent as the criterion of an incompressible flow, the upper limit of Mach number becomes approximately 0.33. In the case of air at standard pressure and temperature, the acoustic velocity is about 335.28 m/s. Hence a Mach number of 0.33 corresponds to a velocity of about 110 m/s. Therefore flow of air up to a velocity of 110 m/s under standard condition can be considered as incompressible flow.