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Analysis of Disturbance. P M V Subbarao Associate Professor Mechanical Engineering Department I I T Delhi. Modeling of A Quasi-static Process in A Medium …. Conservation Laws for a Blissful Fluid. Conservation Laws Applied to 1 D Steady disturbance. Conservation of Mass:.

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Analysis of disturbance

Analysis of Disturbance

P M V Subbarao

Associate Professor

Mechanical Engineering Department

I I T Delhi

Modeling of A Quasi-static Process in A Medium …..

Conservation laws applied to 1 d steady disturbance
Conservation Laws Applied to 1 D Steady disturbance

Conservation of Mass:

Conservation of Mass for 1DSF:


p, r ...


P+dp, r+dr ...

Change is final -initial

For infinitesimally small disturbance

Assume ideal gas conditions for Conservation of Momentum :

For steady flow momentum equation for CV:

For steady 1-D flow :

Nature of substance
Nature of Substance

  • The expressions for speed of sound can be used to prove that speed of sound is a property of a substance.

  • Using the momentum analysis :

  • If it is possible to obtain a relation between p and r, then c can be expressed as a state variable.

  • This is called as equation of state, which depends on nature of substance.

Steady disturbance in a medium
Steady disturbance in A Medium


p, r ...


P+dp, r+dr ...

Speed of sound in ideal and perfect gases
Speed of sound in ideal and perfect gases

  • The speed of sound can be obtained easily for the equation of state for an ideal gas because of a simple mathematical expression.

  • The pressure for an ideal gas can be expressed as a simple function of density and a function molecular structure or ratio of specific heats, gnamely

Speed of sound in a real gas
Speed of Sound in A Real Gas

  • The ideal gas model can be improved by introducing the compressibility factor.

  • The compressibility factor represents the deviation from the ideal gas.

  • Thus, a real gas equation can be expressed in many cases as

Isentropic relation for a real gas
Isentropic Relation for A Real Gas

Gibbs Equation for a general change of state of a substance:

Isentropic change of state:

Pfaffian analysis of enthalpy
Pfaffian Analysis of Enthalpy

For a pure substance :

For a change of state:

Enthalpy will be a property of a substance iff

The definition of pressure specific heat for a pure substance is

Gibbs Function for constant pressure process :

Gibbs Function for constant temperature process : substance is

Divide all terms by dp at constant temperature:

Speed of sound in almost incompressible liquid
Speed of Sound in Almost Incompressible Liquid substance is

  • Even flowing Liquid normally is assumed to be incompressible in reality has a small and important compressible aspect.

  • The ratio of the change in the fractional volume to pressure or compression is referred to as the bulk modulus of the liquid.

  • For example, the average bulk modulus for water is 2 X109N/m2.

  • At a depth of about 4,000 meters, the pressure is about 4 X 107N/m2.

  • The fractional volume change is only about 1.8% even under this pressure nevertheless it is a change.

  • The compressibility of the substance is the reciprocal of the bulk modulus.

  • The amount of compression of almost all liquids is seen to be very small.

Speed of sound in solids
Speed of Sound in Solids substance is

  • The situation with solids is considerably more complicated, with different speeds in different directions, in different kinds of geometries, and differences between transverse and longitudinal waves.

  • Nevertheless, the speed of sound in solids is larger than in liquids and definitely larger than in gases.

  • Sound speed for solid is:

Speed of sound in two phase medium
Speed of Sound in Two Phase Medium substance is

  • The gas flow in many industrial situations contains other particles.

  • In actuality, there could be more than one speed of sound for two phase flow.

  • Indeed there is double chocking phenomenon in two phase flow.

  • However, for homogeneous and under certain condition a single velocity can be considered.

  • There can be several models that approached this problem.

  • For simplicity, it assumed that two materials are homogeneously mixed.

  • The flow is mostly gas with drops of the other phase (liquid or solid), about equal parts of gas and the liquid phase, and liquid with some bubbles.