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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 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 

    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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