Top Banner

of 10

diferenciranje-brzina zvuka

Jun 01, 2018

Download

Documents

Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
  • 8/9/2019 diferenciranje-brzina zvuka

    1/22

    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 …..

  • 8/9/2019 diferenciranje-brzina zvuka

    2/22

    Conservation Laws for a Blissful Fluid

    ( )   pV V t 

    V −∇=∇+

    ∂  

     ρ  ρ 

    .

    ( ) 0.   =∇+∂

    ∂V 

     ρ 

     ρ 

    ( ) ( )   wqV et 

    e

    −=∇+

    ∂ ρ 

     ρ .

  • 8/9/2019 diferenciranje-brzina zvuka

    3/22

    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

  • 8/9/2019 diferenciranje-brzina zvuka

    4/22

     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 

  • 8/9/2019 diferenciranje-brzina zvuka

    5/22

    ( )   dpU d    =* ρ 

    ( ) ( )   pdp pcucd    −+=−−+ +,**  ρ  ρ  ρ 

    ( ) ( )   dpccuucd    =−−++*** *   ρ  ρ  ρ 

    For infinitesimall# small distur$ance

    *** * -  ccucu  

  • 8/9/2019 diferenciranje-brzina zvuka

    6/22

     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.

  • 8/9/2019 diferenciranje-brzina zvuka

    7/22

     "tead# distur$ance in A Medium

    c-u   p&  ρ  ...

    C P+dp&  ρ+ d  ρ  ...

     ρ d 

    dpc  =

  • 8/9/2019 diferenciranje-brzina zvuka

    8/22

    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 

    dpcdpd c   =⇒=*

  • 8/9/2019 diferenciranje-brzina zvuka

    9/22

    constant   −××=   γ   ρ γ  c

     ρ γ  

     ρ 

     ρ γ  

    γ   p

    c   ×⇒×

    ×=constant

     RT c   γ  =

  • 8/9/2019 diferenciranje-brzina zvuka

    10/22

    "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 =

  • 8/9/2019 diferenciranje-brzina zvuka

    11/22

    Compressi$ilit# C'art

  • 8/9/2019 diferenciranje-brzina zvuka

    12/22

    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

  • 8/9/2019 diferenciranje-brzina zvuka

    13/22

    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 

    hdh

    T  p  ∂

    ∂+∂

    ∂=

  • 8/9/2019 diferenciranje-brzina zvuka

    14/22

    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

     sT C 

    ∂=

  • 8/9/2019 diferenciranje-brzina zvuka

    15/22

  • 8/9/2019 diferenciranje-brzina zvuka

    16/22

     pT   T 

    vT v

     p

    h

    ∂−=

    vdpdhTds   −=

    dP  p

    hdT 

    hdh

    T  p  ∂

    ∂+∂∂=

     p

     pT 

    hC 

    ∂∂

    =

    vdpdP  p

    hdT 

    hTds

    T  p−∂

    ∂+∂

    ∂=

    vdpdP 

    vT vdT C Tds

     p

     p   −

    ∂−+=

  • 8/9/2019 diferenciranje-brzina zvuka

    17/22

    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  

  • 8/9/2019 diferenciranje-brzina zvuka

    18/22

     p

    dpn

    v

    dv−=

     p

    dpn

    d  = ρ 

     ρ 

     ρ  ρ 

     pnd 

    dp

    =

    nzRT 

    dpc   ==

     ρ 

    *

    "peed of sound in real gas nzRT c =

  • 8/9/2019 diferenciranje-brzina zvuka

    19/22

    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!

  • 8/9/2019 diferenciranje-brzina zvuka

    20/22

    • The mathematical de"nition of bul( modulus asfollo&ing5

     ρ  ρ d 

    dp B =

     ρ  ρ 

     B

    dpc   ==*

    Propert#3nertial

     propert#6lastic

    ==  ρ  B

    c

  • 8/9/2019 diferenciranje-brzina zvuka

    21/22

    "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

  • 8/9/2019 diferenciranje-brzina zvuka

    22/22

    "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!