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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 …..
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Conservation Laws for a Blissful Fluid
( ) pV V t
V −∇=∇+
∂
∂
ρ ρ
.
( ) 0. =∇+∂
∂V
t
ρ
ρ
( ) ( ) wqV et
e
−=∇+
∂
∂ ρ
ρ .
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Conservation Laws Applied to ! "tead# distur$ance
( ) 0. =∇+∂
∂V
t
ρ
ρ
( ) 0=U dx
d ρ
Conservation of Mass%
( ) 0=U d ρ
0=−− ρ ρ ρ ud ucd
c-u p& ρ ...
C P+dp& ρ+ d ρ ...
( )( ) 0=−−+ cucd ρ ρ ρ
Conservation of Mass for !"F%
C'ange is final -initial
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ρ
ρ ρ ρ ρ ρ
d cuucd ucd =⇒=⇒=− 0
Assume ideal gas conditions for Conservation of Momentum %
( ) pV V ∇=∇ ρ .For stead# flow momentum e(uation for C)%
( ) dxdp
U dx
d =
* ρ
For stead# -! flow %
For infinitesimall# small distur$ance 0≈ ρ ud
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( ) dpU d =* ρ
( ) ( ) pdp pcucd −+=−−+ +,** ρ ρ ρ
( ) ( ) dpccuucd =−−++*** * ρ ρ ρ
For infinitesimall# small distur$ance
*** * - ccucu
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ature of "u$stance
/ 'e e1pressions for speed of sound can $e used to prove
t'at speed of sound is a propert# of a su$stance.
/ 2sing t'e momentum anal#sis %
+&, ρ p f c =
/ 3f it is possi$le to o$tain a relation $etween p and ρ & t'en c
can $e e1pressed as a state varia$le.
/ 'is is called as e(uation of state& w'ic' depends on nature
of su$stance.
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"tead# distur$ance in A Medium
c-u p& ρ ...
C P+dp& ρ+ d ρ ...
ρ d
dpc =
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Speed of sound in ideal and perfect gases
• The speed of sound can be obtained easily for theequation of state for an ideal gas because of asimple mathematical epression!
• The pressure for an ideal gas can be epressed as a
simple function of density and a function molecularstructure or ratio of speci"c heats# γ namely
γ ρ ×= constant p
ρ ρ
d
dpcdpd c =⇒=*
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constant −××= γ ρ γ c
ρ γ
ρ
ρ γ
γ p
c ×⇒×
×=constant
RT c γ =
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"peed of "ound in A 4eal 5as
• The ideal gas model can be impro$ed byintroducing the compressibility factor!
• The compressibility factor represents thede$iation from the ideal gas!
• Thus# a real gas equation can be epressed inmany cases as
RT z p ρ =
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Compressi$ilit# C'art
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3sentropic 4elation for A 4eal 5as
5i$$s 6(uation for a general c'ange of state of a su$stance%
pdvduTds
vdpdhTds
+=
−=
3sentropic c'ange of state%
0=− vdpdh
0=− ρ
dpdh
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Pfaffian Anal#sis of 6nt'alp#
+&, pT f h =
For a pure su$stance %
NdP MdT dh +=For a c'ange of state%
6nt'alp# will $e a propert# of a su$stance iff
dP p
hdT
T
hdh
T p ∂
∂+∂
∂=
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The de"nition of pressure speci"c heat for a pure substanc
p
pT hC
∂∂=
vdpdhTds −=
5i$$s Function for constant pressure process %
p p dhdsT =
p p p dT C dsT =
p
p
T
sT C
∂
∂=
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pT T
vT v
p
h
∂
∂−=
∂
∂
vdpdhTds −=
dP p
hdT
T
hdh
T p ∂
∂+∂∂=
p
pT
hC
∂∂
=
vdpdP p
hdT
T
hTds
T p−∂
∂+∂
∂=
vdpdP
T
vT vdT C Tds
p
p −
∂
∂−+=
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3sentropic 4elation for A 4eal 5as
0=−
∂
∂
−+ vdpdP T
v
T vdT C p
p
zRT pv =
∂∂+
∂∂
+
−=
v
p
v
p
T z T z
T
z T z
C
C
p
dp
v
dv
∂∂
+
∂∂
+
=
v
p
T
z
T z
T
z T z
n γ
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p
dpn
v
dv−=
p
dpn
d = ρ
ρ
ρ ρ
pnd
dp
=
nzRT
d
dpc ==
ρ
*
"peed of sound in real gas nzRT c =
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Speed of Sound in Almost IncompressibleLiquid
• E$en %o&ing 'iquid normally is assumed to be incompressiblein reality has a small and important compressible aspect!
• The ratio of the change in the fractional $olume to pressure orcompression is referred to as the bul( modulus of the liquid!
• )or eample# the a$erage bul( modulus for &ater is * +,-. /0m*!
• At a depth of about 1#--- meters# the pressure is about 1 + ,-2 /0m*!
• The fractional $olume change is only about ,!34 e$en underthis pressure ne$ertheless it is a change!
• The compressibility of the substance is the reciprocal of the
bul( modulus!• The amount of compression of almost all liquids is seen to be
$ery small!
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• The mathematical de"nition of bul( modulus asfollo&ing5
ρ ρ d
dp B =
ρ ρ
B
d
dpc ==*
Propert#3nertial
propert#6lastic
== ρ B
c
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"peed of "ound in "olids
• The situation &ith solids is considerably morecomplicated# &ith di6erent speeds in di6erentdirections# in di6erent (inds of geometries# anddi6erences bet&een trans$erse and longitudinal&a$es!
• /e$ertheless# the speed of sound in solids is largerthan in liquids and de"nitely larger than in gases!
• Sound speed for solid is5
Propert#3nertial
propert#6lastic==
ρ
E c
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"peed of "ound in wo P'ase Medium
• The gas %o& in many industrial situations contains otherparticles!
• In actuality# there could be more than one speed of soundfor t&o phase %o&!
• Indeed there is double choc(ing phenomenon in t&o phase%o&!
• 7o&e$er# for homogeneous and under certain condition asingle $elocity can be considered!
• There can be se$eral models that approached this problem!
• )or simplicity# it assumed that t&o materials arehomogeneously mied!
• The %o& is mostly gas &ith drops of the other phase 8liquidor solid9# about equal parts of gas and the liquid phase#and liquid &ith some bubbles!