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...

 

[rezeew]

Hello there! I would be happy to provide a response to your question.

Firstly, it's important to note that the equation you have provided is known as the "compressible flow in a nozzle equation". This equation is derived from the principles of thermodynamics and fluid mechanics, specifically the conservation of mass, momentum, and energy.

To begin, we can start with the continuity equation, which states that the mass flow rate through a nozzle is constant. This equation is represented by the term ρ*A*v*, where ρ is the density, A is the area, and v is the velocity. We can also use the ideal gas law, which relates the density, temperature, and pressure of a gas.

Next, we can use the isentropic flow relations, which describe the changes in temperature, pressure, and density of a gas as it flows through a nozzle. These relations are represented by the equations you have provided: ρ0/ρ=[1+(k-1/2)*M^2]^(1/k-1), To/T=1+(k-2/2)*M^2, and Po/P=[1+(k-2/2)*M^2]^(k/k-1).

Finally, we can combine these equations and use the definition of the Mach number (M=V/C) to derive the compressible flow in a nozzle equation: A/A*=1/M[2/(k+1)(1+(k-1)/2*M^2)]^(k+1/2(k-1)). This equation represents the change in area (A/A*) with respect to the Mach number, which is a measure of the gas velocity.

I hope this helps to clarify the derivation of the compressible flow in a nozzle equation. Please let me know if you have any further questions. Keep up the good work in your studies!