Lecture - 3 ( properties of fluids (part 2) - compressibility)

Compressibility (β) 

Compressibility is defined as the variation of volume(V) or densityWith respect to pressure with constant mass (m). If variation of volume with respect to pressure is high then compressibility is high.
Liquids are generally incompressible whereas gases are generally highly compressible. Two Piston cylinder arrangement is shown in below figure one arrangement is filled with liquid and other arrangement is filled with gas. In case of liquid with increase in pressure, the volume and density remains unchange. In case of gas as we increase the pressure, volume start decreasing and density start increasing. Therefore liquid is incompressible and gas is compressible. Experimentally it was found that air is 20000 times more compressible then Water.

An experiment was conducted to check the compressibility of water. In that experiment the density of water at 1atm pressure was 998 kilogram per metre cube, after that when the pressure was increased by hundred times, the density of water increase by 5 kilogram per metre cube. At 100atm pressure water has density 1003 kilogram per metre cube, which is very negligible rise of density as compared to pressure. There is increment of only 0.5% in density. From this experiment we can conclude that liquids are  almost incompressible. 

(Remember all those fluids having Mac number less than 0.3 are considered as incompressible fluid)

Mathematically compressibility (β) is defined as the reciprocal of bulk modulus (K) of elasticity.

 β= 1/K  

Bulk modulus (K) is defined as the ratio of direct stress to volumetric strain. In fluid mechanics our direct stress is hydrostatic stress.

K = hydrostatic stress/volumetric strain 

(Volumetric strain is defined as ratio of change in volume to the original volume)

Therefore, K = -dP/(dV/V) = -VdP/dV ; here 'P' is pressure and 'V' is volume.

In terms of density (ρ) value of bulk modulus is :  

ρ = m/V ;  we know mass is constant 

m = ρV ; differentiate this term

0 = ρdV + Vdρ

-V/dV = ρ/dρ ; by putting this value in equation of bulk modulus we get value of bulk modulus in terms of density and that is: K = ρdP/dρ

Conclusion: β = 1/K = -dV/VdP = dρ/ρdP 

ISOTHERMAL BULK MODULUS (KT ) 

Isothermal is a process in which temperature will remain constant. The process must be infinitely slow, if we want to make it a isothermal process. 

We know ideal gas follows the equation: 

P*V = m*R*T

here P is pressure, V is volume, m is mass, R is universal gas constant and T is temperature.

P = (m/V)*R*T

P = ρ*R*T ; differentiate this equation with respect to density (ρ) we get, 

dP/dρ = R*T 

We know dP/dρ = K/ρ

Bulk modulus for isothermal process is 'KT ' so replacing 'K' by 'KT ' we get, 

KT = ρ*R*T = P 

KT = P 

ADIABATIC BULK MODULUS (Ka )

Adiabatic processes or defined as the process in which no heat transfer is there. 

In adiabatic process, 

P*(V)^γ = C : here Gama (γ) is the index of air and its value is 1.4 and 'C' is any constant value

P*(m/ρ)^γ = C

P/ (ρ^γ) = C/ (m^γ)

P/ (ρ^γ) = C

P = C*(ρ^γ) ; differentiate this equation with respect to density (ρ) we get,

dP/dρ = C*γ*(ρ^(γ-1)) 

now we know Ka / ρ = dP/dρ ; therefore we get, 

C*γ*(ρ^(γ-1)) = Ka / ρ

Ka = ρ*P

Bulk modulus is directly proportional to pressure so with increase in pressure, bulk modulus increase and therefore compressibility will decrease because with increase in pressure molecules comes closer to each other and it becomes difficult to compress them further as they resist further compression and hence compressibility decrease.

From the above 2 derivations mathematically we can conclude that bulk modulus of adiabatic process is greater than bulk modulus of isothermal process

(Ka > KT ). Therefore we can say that compressing a gas adiabatically is more difficult than compressing a gas isothermally.

Overall conclusion is that, compressing a gas adiabatically is more difficult than compressing a gas isothermally because due to increase in temperature, in case of adiabatic process the randomness of gas molecule become very high and they will strike the Piston and create resistance to compression and hence compressibility is less in case of adiabatic process.

(We will study detailed explanation of isothermal and adiabatic process in our thermodynamics lectures)