TThermodynamics-compressed flow in a nozzle equation derivation

[luk3tm]

Hi everyone!

Homework Statement

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))

Homework 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)

The Attempt at a Solution

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

Thanks!

 

[LawrenceC]

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

 

[luk3tm]

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 :(

 

[LawrenceC]

Derive an expression for Vstar/V that you'll eventually plug into 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...