# diferenciranje-brzina zvuka

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• 8/9/2019 diferenciranje-brzina zvuka

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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

  ρ

ρ

( ) ( )   wqV e t

e 

 −=∇+

∂  ρ

ρ  .

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Conservation Laws Applied to  ! "tead# distur\$ance

( ) 0.   =∇+ ∂

∂ V

  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)%

( ) dx dp

U  dx

d  =

*  ρ

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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c-u   p&  ρ  ...

C P+dp&  ρ+ d  ρ  ...

ρ d

dp c  =

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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 epression!

•  The pressure for an ideal gas can be epressed as a

simple function of density and a function molecular structure or ratio of speci"c heats# γ  namely

γ    ρ ×= constant p

ρ   ρ

dp cdpd 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 by introducing the compressibility factor!

•  The compressibility factor represents the de\$iation from the ideal gas!

•  Thus# a real gas equation can be epressed in many 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

dp dh

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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

h dT

h dh

T  p   ∂

∂ +∂

∂ =

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The de"nition of pressure speci"c heat for a pure substanc

p

p T  hC

∂∂=

vdpdhTds   −=

5i\$\$s Function for constant pressure process %

p p   dhdsT    =

p p p   dT C dsT    =

p

p

s T C

∂ =

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pT    T

v T v

p

h

∂ −=

vdpdhTds   −=

dP   p

h dT

h dh

T  p   ∂

∂+ ∂ ∂=

p

p T

h C

∂ ∂

=

vdpdP   p

h dT

h Tds

T  p −∂

∂ +∂

∂ =

vdpdP

v T 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

z  T  z

p

dp

v

dv



  



  

 

  ∂ ∂

+

  

   ∂ ∂

+

=

v

p

z

T  z

z  T  z

n   γ

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p

dp n

v

dv −=

p

dp n

d  =  ρ

ρ

ρ  ρ

p nd

dp

=

nzRT

dp c   ==

ρ

*

"peed of sound in real gas nzRT c =

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Speed of Sound in Almost Incompressible Liquid

• E\$en %o&ing 'iquid normally is assumed to be incompressible in reality has a small and important compressible aspect!

•  The ratio of the change in the fractional \$olume to pressure or compression is referred to as the bul( modulus of the liquid!

• )or eample# 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 under this 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 as follo&ing5

ρ   ρ  d

dp  B =

ρ  ρ

B

dp c   ==*

Propert#3nertial

propert#6lastic

==  ρ   B

c

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"peed of "ound in "olids

•  The situation &ith solids is considerably more complicated# &ith di6erent speeds in di6erent directions# in di6erent (inds of geometries# and di6erences bet&een trans\$erse and longitudinal &a\$es!

• /e\$ertheless# the speed of sound in solids is larger than 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 other particles!

• In actuality# there could be more than one speed of sound for 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 a single \$elocity can be considered!

•  There can be se\$eral models that approached this pro

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