Worked examples on nozzles
All pressures are absolute
1 The diagram shows the profile of a
convergent nozzle that is fixed between flanges in a pipe of large diameter.
Air is allowed to flow through the nozzle from an upstream reservoir where the
pressure is 10 bar at 400°K.
The pressure downstream of the nozzle can be increased from atmospheric
pressure at 1 bar up to the supply pressure of 10 bar.
2 Air at 7 bar 400°C is to be expanded through a convergent-divergent nozzle to 1 bar.
(i) Using the reversible, adiabatic model for steady, shock-free flow through the convergent part of the nozzle find a value for the throat area required to pass 50 kg/min.
(ii) Using the adiabatic, one-dimensional model for steady, shock-free flow through the divergent part of the nozzle and an overall nozzle efficiency of 90% find a value for the area at exit.
For air take g = 1.4 and =1.04 kJ/kg°K
3 A nozzle is to be designed to produce a supersonic jet of air at 1 bar when supplied with air at 10 bar and 50°C. The required flow is 2 kg/s.
(i) Use the reversible, adiabatic, one-dimensional model for steady flow of a compressible fluid to find “ideal” areas for the throat and exit.
A nozzle was made to these dimensions and tested. It was found that the nozzle gave the correct flow but the correct back pressure for shock-free supersonic flow was 1.25 bar.
(ii) Calculate the nozzle efficiency and,
(iii) Use this nozzle efficiency to find a better value for the exit area so that the nozzle expands air to 1 bar as required.
For air take R=287J/kg°K, and
4 Air flows through a convergent-divergent nozzle and a shock wave forms in the divergence at a point where the diameter is 30 mm. Measurements made on the nozzle indicate that the pressure rises in the shock wave from 2 bar to 3 bar. The temperature before the shock wave was calculated to be -10°C.
Calculate :-
(i) The Mach number just before the shock wave
(ii) The rate of flow through the nozzle in kg/sec, and,
(iii) The temperature just after the shock wave.
You are given :_
For air , and . Mach number .
For a plane shock wave :-
5 (a) Given that the pressure ratio across a normal shock wave in a perfect gas is :-
show that the velocity of propagation of a shock wave through still air is given by :-
where is the density of the gas, is the ratio of the principal specific heats and subscripts 1 and 2 refer to conditions before and after the shock wave.
(b) The initial calculations on the design of a convergent-divergent nozzle are to be made using the reversible, adiabatic, one-dimensional model. Air is to be supplied to the nozzle at 5 bar and 40°C, the throat diameter is to be 6 mm and the Mach number at exit is to be 1.8.
Calculate using Mathcad or tables
(i) the diameter of the nozzle at exit,
(ii) the mass flow of air,
(iii) the pressure at exit,
(iv) the back pressure at which a plane shock wave could occur at exit.
6 Air at 2.3 bar and 20°C flows steadily through a convergent-divergent nozzle. The throat diameter is 10 mm and the exit diameter is 15.5 mm.
Find:-
(i) a value for the maximum possible mass flow,
(ii) the highest exit pressure for which this flow is possible,
(iii) the pressure at the exit plane if no shock wave occurs in the nozzle,
(iv) the range of back-pressures for which a normal shock wave can occur in the nozzle, and,
(v) if the nozzle efficiency based on reversible, adiabatic flow to the throat and shock free flow in the divergence is 85% , the required diameter at exit if the back pressure is that for (ii).
Solutions
1 (i) We have no alternative to using the reversible, adiabatic, one-dimensional flow model.
For maximum mass flow critical conditions must exist at the throat that is at the exit. The ratio of the throat pressure to the stagnation pressure is given by :- . For maximum mass flow pressure at exit = bar absolute. Maximum mass flow is given
.
The highest backpressure for maximum mass flow is 5.28 bar absolute.
(ii) We have in general:-
Using Mathcad
The graphs are valid for the range between the critical exit pressure of 5.28 bar shown by dashed marker and 10 bar.
2(i)
;
;
Then Throat area =
(ii) For an expansion where . Here we have and then
If we used the reversible, adiabatic, one-dimensional model we would have the exit temperature = 386°K and the drop in enthalpy would be . However for the real nozzle the enthalpy drop is . The nozzle efficiency is defined as we can find from . .
We also have . As
Now, if we suppose that the loss in the nozzle occurs wholly in the divergence, the mass flow will remain unchanged. Then we can find he exit area using continuity.
and
Therefore
.
Exit area = .
3(i)
;
Then Ideal throat area =
;
Then Ideal throat area =
(ii) The practical nozzle made to these ideal dimensions is said to have a real flow that emerges at 1.25 bar and not 1 bar as is required. We have the definition of a nozzle efficiency where will be the stagnation temperature, will be the temperature for a reversible, adiabatic, one–dimensional flow from 10 bar to 1.25 bar and
will be the actual temperature at exit.
. Now we need a value for . If, as in question 2, we presume that the expansion in the convergence are mass flow is reversible we can find the temperature at exit using the continuity and the equations of state for the gas. Continuity gives :- and and from this
which gives
Use Mathcad to solve by trial or simplify and use calculator
Nozzle efficiency =
(iii) For a real exit temperature of 192°K at 1 bar and So or
exit area =
(4) (i) We have . and
Therefore
(ii) from which
and then
(iii)
5 (a) and re-arranging gives
A stationary
would be depicted as shown in the diagram. If ,
that is the gas is stationary, the shock wave would advance at into the gas. Now which can be rewritten 
(b) (i) Use Mathcad or tables to find property ratios. At =1.8 .
As ;
(b)(ii)
and
. From Mathcad putting .
. Then :-
(b) (iii) Using Mathcad .
Therefore
(b)(iv) For a shock wave with .
Pressure at exit
6 (i)
When
(ii) The nozzle can flow with subsonic conditions to the throat where the conditions are critical and also have subsonic flow to the exit. We have that . By trial using Mathcad the value of at ext is 0.25. Then and then for subsonic flow throughout and critical conditions at the throat:-
(iii) For the case of shock free flow
to exit and supersonic flow in the divergence the ratio .
This can be used with Mathcad again to find Mach number at exit. It is 2.4.
Then and
(iv) We have that the Mach number at exit
is 2.4 and this must be the Mach number upstream of the shock wave. Mathcad
gives the ratio of It follows that the range of back pressure
for which a shock wave can be present in the nozzle is from 1.029 bar to 2.2
bar.
(v) Using the T-s plane shown in the sketch the nozzle efficiency is defined as . For the same exit pressure we had for reversible adiabatic flow we have a new higher temperature at exit, a lower velocity and a new density but the same mass flow.
For reversible adiabatic flow . Then
. Therefore
But and then Then :-
