Analysis of Disturbance

P M V SubbaraoAssociate Professor

Mechanical Engineering DepartmentI I T Delhi

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

Conservation Laws for a Blissful Fluid

pVVt

V

.

0.

Vt

wqVet

e

.

Conservation Laws Applied to 1 D Steady disturbance

0.

Vt

0Udx

d

Conservation of Mass:

0Ud

0 uducd

c-u p,

CP+dp, d

0 cucd

Conservation of Mass for 1DSF:

Change is final -initial

d

cuucducd 0

Assume ideal gas conditions for Conservation of Momentum :

pVV

.

For steady flow momentum equation for CV:

dx

dpU

dx

d2

For steady 1-D flow :

For infinitesimally small disturbance 0ud

dpUd 2

pdppcucd )(22

dpccuucd 222 2

For infinitesimally small disturbance

222 2 & ccucu

d

dpcdpdc 2

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 :

),( pfc

• If it is possible to obtain a relation between p and , 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

c-u p,

CP+dp, d

d

dpc

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, namely

constantp

d

dpcdpdc 2

1constant c

pc

constant

RTc

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

RTzp

Compressibility Chart

Isentropic Relation for A Real Gas

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

pdvduTds

vdpdhTds

Isentropic change of state:

0 vdpdh

0dp

dh

Pfaffian Analysis of Enthalpy

),( pTfh

For a pure substance :

NdPMdTdh For a change of state:

Enthalpy will be a property of a substance iff

dPp

hdT

T

hdh

Tp

The definition of pressure specific heat for a pure substance is

pp T

hC

vdpdhTds

Gibbs Function for constant pressure process :

ppdhdsT

pppdTCdsT

pp T

sTC

Gibbs Function for constant temperature process :

vdpdhTds

TTTdpvdhdsT

vp

sT

p

h

TT

TTTdpvdsTdh

Divide all terms by dp at constant temperature:

Tp p

s

T

v

pTT

vTv

p

h

vdpdhTds

dPp

hdT

T

hdh

Tp

pp T

hC

vdpdPp

hdT

T

hTds

Tp

vdpdPT

vTvdTCTds

pp

Isentropic Relation for A Real Gas

0

vdpdPT

vTvdTC

pp

zRTpv

v

p

v

p

Tz

Tz

Tz

Tz

C

C

p

dp

v

dv

v

p

Tz

Tz

Tz

Tz

n

p

dpn

v

dv

p

dpn

d

p

nd

dp

nzRTd

dpc

2

Speed of sound in real gas nzRTc

Speed of Sound in Almost Incompressible Liquid

• 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 X109 N/m2.

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

•The mathematical definition of bulk modulus as following:

d

dpB

B

d

dpc 2

Property Inertial

property Elastic

B

c

Speed of Sound in Solids

• 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:

Property Inertial

property Elastic

E

c

Speed of Sound in Two Phase Medium

• 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.