4. Worked Examples

Question 1

Air flows through a converging – diverging nozzle from a reservoir where the pressure is 500kPa and the temperature is 400K. The diameter at the throat is 9mm and the diameter at the exit is 20mm.

a) Find the two back pressures which cause the nozzle to choke yet still produce an isentropic flow.

b) Find the maximum velocity of the resulting flow.

Question 2

Argon flows isentropically through a converging - diverging nozzle with a stagnation temperature of 300K. The throat pressure is 800Kpa and the velocity of argon at the exit is 500ms-1 . For the following back pressures determine if the flow in the throat is choked.

a) Pb = 20Kpa

b) Pb = 100Kpa

Worked Answers

Question 1

a) We have both the throat area and exit area so we can find their ratio:

Using the table in The Little Book Of Thermofluids, 6th edition, we can find Mach numbers that match to this value. This gives us the two possible values of Ma.

Either Ma = 0.12 or Ma = 3.15

In the case Ma = 0.12:

and in the case Ma = 3.15:

This gives the two possible back pressures as Pb = 490.2Kpa and Pb = 10.89Kpa.

b) The maximum mass flow rate for the system can be determined using the equation:

Which gives us:

This maximum mass flow rate applies to the entire system. Mass flow rate can also be expressed as:

We need to find the maximum velocity, u. We know the area and value of R already, and just need the pressure and temperature at the exit. Logically the back pressure with the highest mach number is going to give the highest velocity, so Ma = 3.15 should be used to find the maximum possible velocity. From The Little Book Of Thermofluids, Ma = 3.15 gives the following properties:

We know stagnation pressure and stagnation temperature so we can calculate the values needed to determine the velocity like so:

and substituting into the velocity equation gives us the maximum velocity:

So the maximum velocity of the exit stream is 723m/s.

Question 2

For both part a and part b the ratio of stagnation pressure over back pressure can be determined first.

The values we have been given, as well as argon's properties are:

We can find the exit temperature with the following equation:

and now that we know the exit temperature and exit velocity it is possible to find the Mach speed of the exit stream:

And subsequently this can be used for calculating the ratio of stagnation pressure to back pressure:

Now we know the ratio we can calculate the stagnation pressure for each back pressure and since we know the throat pressure we can then calculate the mach number in the throat.

a) Pb = 20kPa

Using the ratio we can calculate the stagnation pressure and from that and known throat pressure find the mach number in the throat.

Since Ma<1 the flow in the throat is subsonic. Therefore there is no choking at the throat when the back pressure is 20kPa.

a) Pb = 100kPa

Again, using the ratio we can calculate the stagnation pressure and from that and known throat pressure find the mach number in the throat.

In this case Ma>1 in the throat. However knowing that part of the throat has a mach number above 1 means that there is also a point where it is equal to 1 and sonic. Therefore when the back pressure is 100kPa there is choking in the nozzle.