Analysis of Disturbance P M V Subbarao Associate Professor Mechanical Engineering Department I I T...
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Transcript of Analysis of Disturbance P M V Subbarao Associate Professor Mechanical Engineering Department I I T...
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 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
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.
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
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
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
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
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.