Kelvin Tam

Compressible Flow Project 3

Assuming isentropic flow in the nozzle, the mass flow rate leaving the nozzle can be written as

dmdt

=∀ ρo1

γ Po11γ

ρo11−γγ ( d Podt )

which is done using ideal gas and isentropic relations. For steady flow through an adiabatic nozzle, the energy equation can be used to solve for the exit velocity as a function of gamma, exit density and exit pressure:

V e=√ 2 γγ−1 ( Poρo )(1−( PePo )γ−1γ )

For a Stage I analysis, we assume choked flow in the nozzle hence M=1. From this, the following relation is realized:

P0Pe

=( γ+12 )γγ+1

Using this relation, the exit velocity of the nozzle can be written as

V e=√ 2 γγ+1 ( Poρo )=√ 2 γ Po11 1 /γ( γ+1 ) ρo1Poγ−12 γ

From the two equations above, solving for the exit density yields

ρe=ρo( PePo )1γ=( ρo1P011γ )( 2

γ+1 )1γ−1 Po

1γ

Next, the mass flow rate of the air leaving the nozzle is determined

m=( ρe AeV e)=( ρo1P011γ )( 2γ+1 )1γ−1 Ae √ 2 γ Po11 1 /γ(γ+1 ) ρo1

Poγ+12 γ

Setting the above equal to -dm/dt and introducing a constant C=(−γ A e∀ )( 2γ+1 )

1γ−1 √ 2 γ Po11 1 /γ( γ+1 ) ρo1

gives the simple relation

Po1−3 γ2 γ ( d Podt )=C

integrating from initial conditions to arbitrary time t gives:

Po=(Po1b +bCt )1b

where b is the constant (1-ϒ)/(2ϒ).

For a range of times from t=0 to t=5 seconds, a Matlab code is written to determine the ratio Po/P01. To begin, the initial conditions and properties of the nozzle are inputted which includes the volume V, stagnation pressure in the reservoir P01, initial temperature T0 and exit area Ae. Properties of the flow (air) is also inputted which include gamma (denoted by Y) equal to 1.4 and gas constant:

Using the ideal gas law, initial density denoted as rho1 is calculated:

Next, the constants C and b as defined in the derivation shown earlier is computed for the parameters of the given problem:

A vector t is defined which divides the interval from 0 to 5 seconds into 10 steps of size 0.5 seconds. Using the final result of Stage I analysis the pressure is computed and stored in the vector P0. This result is then divided by the initial reservoir pressure in order to obtain the desired ratios which are stored in the vector Ratio:

Results are then plotted using

The code returns the following output:

t Po/Po1

0 10.5 0.2622

1 0.08531.5 0.0324

2 0.01392.5 0.0065

3 0.00333.5 0.0018

4 0.0014.5 0.0006

5 0.0004