Quasi-1D flow in a converging-diverging nozzle

In this notebook we will explore the flow in a converging-diverging nozzle. The objective of this notebook is to explore the effect of back pressure relative to the upstream stagnation pressure.

Set up the module

using Gasdynamics1D

using Plots

Set up a nozzle

First, we will set up a nozzle. For this, we will create a representative nozzle shape: a bell-shaped converging inlet and a slowly increasing diverging section. We set the inlet area, the throat area, and the exit area:

Ai = Area(100u"cm^2")
At = Area(30u"cm^2")
Ae = Area(60u"cm^2")
noz = Nozzle(Ai,At,Ae)

Converging-diverging nozzle
   Inlet area (sq cm)= 100.0 cm^2
   Throat area (sq cm)= 30.0 cm^2
   Exit area (sq cm)= 60.0 cm^2

Let's plot this nozzle to see its shape

nozzleplot(noz)

Create a flow through the nozzle

Let's set the conditions to create a flow through this nozzle. We need to set the stagnation conditions upstream of the nozzle. We will use stagnation pressure and temperature.

p0 = StagnationPressure(700u"kPa")
T0 = StagnationTemperature(30u"°C")

StagnationTemperature = 303.15 K

We also need a back pressure, outside of the nozzle exit. Let's try setting it to 660 kPa, a little bit below the stagnation pressure

pb = Pressure(660u"kPa")

Pressure = 660000.0 Pa

Now we have enough information to solve this flow. Let's plot it. We need to specify which quantities we wish to show. We will show pressure and Mach number. We could also show temperature and density.

The plot below shows that the pressure reaches a minimum in the throat and the Mach number reaches a maximum there. However, it does not quite get to sonic conditions.

nozzleplot(noz,pb,p0,T0,fields=("pressure","machnumber"))

One note on the last plots: We could have specified the gas, but it defaults to air.

Lowering the back pressure

What happens if we lower the back pressure further? Let's try this. For several back pressures, we get a shock in the diverging section. As back pressure gets smaller, this shock appears closer to the exit. Finally, the shock leaves the nozzle entirely.

nozzleplot(noz,Pressure(660u"kPa"),p0,T0,fields=("pressure","mach"))
nozzleplot!(noz,Pressure(600u"kPa"),p0,T0,fields=("pressure","mach"))
nozzleplot!(noz,Pressure(500u"kPa"),p0,T0,fields=("pressure","mach"))
nozzleplot!(noz,Pressure(400u"kPa"),p0,T0,fields=("pressure","mach"))
nozzleplot!(noz,Pressure(360u"kPa"),p0,T0,fields=("pressure","mach"))
nozzleplot!(noz,Pressure(300u"kPa"),p0,T0,fields=("pressure","mach"))

Mass flow rate

What is the mass flow rate through the nozzle? Let's inspect this value for a few back pressures. First, the case with 660 kPa:

nozproc = NozzleProcess(noz,Pressure(660u"kPa"),p0,T0)
massflowrate(nozproc)

MassFlowRate = 4.663646959166167 kg s^-1

Now with 600 kPa

nozproc = NozzleProcess(noz,Pressure(600u"kPa"),p0,T0)
massflowrate(nozproc)

MassFlowRate = 4.874946341422668 kg s^-1

And now with 500 kPa

nozproc = NozzleProcess(noz,Pressure(500u"kPa"),p0,T0)
massflowrate(nozproc)

MassFlowRate = 4.874946341422668 kg s^-1

Notice that the mass flow rate is stuck. No matter how much we lower the back pressure, we cannot increase the mass flow rate. The flow is choked.

We can classify the type of flow with the flow_quality function:

nozproc = NozzleProcess(noz,Pressure(600u"kPa"),p0,T0)
flow_quality(nozproc)

"Supersonic with normal shock"

There is a normal shock in the diverging section

nozproc = NozzleProcess(noz,Pressure(100u"kPa"),p0,T0)
flow_quality(nozproc)

"Overexpanded supersonic"

This is over-expanded: the exit pressure is too small and the flow needs to pass through shocks outside the nozzle to get up to the back pressure.

nozproc = NozzleProcess(noz,Pressure(60u"kPa"),p0,T0)
flow_quality(nozproc)

"Underexpanded supersonic"

This is under-expanded: the exit pressure is not low enough and the flow needs to pass through an expansion fan outside the nozzle to lower its pressure further to get down to the back pressure.

When we seek to generate a supersonic exit flow, as is often the case, then our goal is to design it so that the flow is perfectly expanded. We can find the required back pressure easily, using isentropic relations:

Ae = nozzleexit(noz)
Astar = throat(noz)
pb = Pressure(SupersonicPOverP0(Ae,Astar,Isentropic)*p0)

Pressure = 65752.85201299141 Pa

This page was generated using Literate.jl.