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TThermodynamics-compressed flow in a nozzle equation derivation

  1. Aug 11, 2012 #1
    Hi everyone!

    1. The problem statement, all variables and given/known data

    Just one part of the lecture notes I couldn't work out how to derrive (apparently we need to know how).

    A/A*=1/M[2/(k+1)(1+(k-1)/2*M^2)]^(k+1/2(k-1))


    2. Relevant equations

    A/A*=1/M[2/(k+1)(1+(k-1)/2*M^2)]^(k+1/2(k-1))
    ρ0/ρ=[1+(k-1/2)*M^2]^(1/k-1)
    To/T=1+(k-2/2)*M^2
    Po/P=[1+(k-2/2)*M^2]^(k/k-1)
    ρ*A*v*=ρAv
    M=V/C (mach number)
    c=√(KRT)

    3. The attempt at a solution

    I tried fiddling around with quite a few things but got nothing.

    Thanks!
     
  2. jcsd
  3. Aug 12, 2012 #2
    Hint:

    rho*A*V = rhostar*Astar*Vstar (continuity)

    A/Astar = rhostar*Vstar/(rho*V)

    Relate Vstar/V to mach number and specific heat ratio by using supplemental equations below

    Relate rhostar/rho to mach number and specific ratio by using supplemental equations below



    Supplemental equations:

    T0/T =1 + (k-1)/2)M^2

    Tstar/T0 = 2/(k+1)

    0 subscript denotes stagnation condition
     
  4. Aug 14, 2012 #3
    okay so i think i've almost got it but i might be missing out on something:

    Vstar/V=Ap/Astar*pstar

    substituting in mac number,

    Mcstar/Mcr=RHS

    substituting C=sqrt(KRT)

    gives:

    M/Msqrt(KRTstar/KRT)

    =sqrt(T*/T)

    Tstar/T=Tstar/To*To/T=(1+k-1/2)M*2*2/k+1

    and pstar/p=po/p*pstar/po=(2/k+1)^1/(k-1)*(1+k-1/2*M^2)*1/k-1

    so back to original equation:

    A/Astar=sqrt(Tstar/T)*pstar/p

    sqrt((1+k-1/2)M*2*2/k+1)*(2/k+1)^1/(k-1)*(1+k-1/2*M^2)*1/k-1

    which doesn't quite give the right answer :(
     
  5. Aug 14, 2012 #4
    Derive an expression for Vstar/V that you'll eventually plug in to the continuity equation. After substituting Vstar=Cstar=sqrt(kRTstar) and also noting that V=C*M=M*sqrt(kRT) you should arrive at

    Vstar/V = (1/M) * sqrt(Tstar/T)

    Now write as

    Vstar/V = (1/M)*sqrt(Tstar/T0)*sqrt(T0/T)

    But Tstar/T0 = 2/(k + 1) and T0/T = 1 + (k-1)*M^2/2

    Substitute these in then use a similar tactic for rhostar/rho.

    Plug both into continuity equation and simplify. It'll work...
     
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