Chapter 13 Part 4 Complete doc
Introduction
The plane shock wave occurs only in a convergent-divergent nozzle. Shock waves of all shapes occur around objects moving at supersonic speeds and in devices like engine intakes for supersonic flight. I do not think that the plane shock wave has any direct application in engineering in the sense that someone would design such a wave into a device. It may find some very special application but it is a process in which the velocity of a gas is suddenly reduced from supersonic to subsonic with a considerable loss of mechanical energy to internal energy. However we can make use of the outcome of analysing this shock wave in a nozzle in the more general field of supersonic flow. I think that graduates in engineering should be aware of the plane shock wave.
The plane shock wave
It
is a matter of observation that such a shock wave can occur in the divergent
cone of a nozzle and, in a very short distance, supersonic flow is reduced to
subsonic flow. Figure 13-12 shows the main features of the plane shock wave.
We have seen the parallel to this in the Venturi flume where, in a hydraulic jump, the flow changes from rapid flow to tranquil flow. The jump would have started as rolling wave (like a bore) that would have formed downstream and moved upstream against the flow until its speed is equal to that of the flow and that must occur in the divergence. The speed of the bore is greater than the critical speed for the flow.
In the nozzle we have a wave that has come to rest in a supersonic flow and its speed relative to the flow must also be supersonic.
I
want to look at the physics of a plane shock wave in the divergence of a
convergent-divergent nozzle.
In figure 13-13 I have drawn a plane shock wave at some arbitrary position in the divergence that I have been using throughout. The shock wave will be about 0.00005 mm thick. That seems to be very thin to us but in every square centimetre of the shock wave there will be about molecules. The 1.5 may not be accurate but the telling figure is the power 15 To all those molecules a shock wave is nothing special.
In the shock wave the properties of the gas change from to and the velocity changes from that is supersonic to that is subsonic. The pressure, the density and the temperature all increase but the velocity falls. The kinetic energy of the mass centre of the gas is depleted as it flows through the shock wave and some proportion of the energy is absorbed into the molecular structure of the gas. In the end we shall find that, for this model of a shock wave, . This result suggests an underlying order and is certainly very elegant.
The physics of the normal shock wave is so well established that the property relationships that come from it have been tabulated and published. This means that we need to follow the set procedure that leads to these tables to get an understanding of what has been done. So let us look at the physics of a shock wave in a nozzle by treating the flow up to the shock wave as reversible, adiabatic, one-dimensional flow, the flow through the shock wave as non-reversible, and the flow through the rest of the divergence as reversible, adiabatic, one-dimensional flow again. As the wave is very thin we regard the area of cross-section as the same on the two sides of the wave.
Now, as so often in these analyses, we start with momentum and equate the net force on the gas in the wave to the change in momentum/second that takes place. This gives :-
Now we must follow the set sequence to get the required outcome. First :-
(1)
Then we can use the energy equation in the form for one-dimensional flow and write :-.
where is the stagnation temperature at inlet to the nozzle.
However and and therefore or . It then follows that :- (3¢)
At this point we have a choice between proceeding in ordinary properties, changing to Mach number or to do both. I will do both in turn and start with properties.
So proceeding and hence . Then :-
(2¢)
(2)
Now we have to give some thought to the way in which we would like the result to turn out. As always we want the outcome to involve only physical constants and measurable quantities. We cannot measure velocity and certainly not just before and just after a shock wave nor can we measure temperature, pressure and density with any confidence. This means that we really want our answer to give ways to calculate these quantities from the stagnation conditions at entry to the nozzle. So let us first get rid of the velocities in favour of and to find using reversible adiabatic, one-dimensional flow to the shock wave.
Equation (1) gives :- and multiplying by gives:-
from (2)
Then :-
The next step is to multiply by to give :-
and then to divide by when :-
and
This is one form of the Rankine-Hugoniot expression that relates the pressures and densities before and after the shock wave. If the position of the shock wave is known the properties of the flow upstream of the shock wave can be found and the Rankine-Hugoniot expression will take us through the shock wave to give new start conditions.
It can be rearranged quite simply to give in terms of .
For air these become and .
Other forms of these equations can be derived but none of them eliminate the densities so we must use these equations and trial methods if necessary. In order to link the two sides of the shock wave we need another relationship. The one that we can have is between the velocities on either side of the shock wave.
I have already said that there is a rather elegant relationship between the velocities before and after the shock wave. Now I have to produce it.
Go back to equation (2¢) and it can be extended to include the throat.
(4)
We have from (1) :-
Therefore
or :- (5)
but we have from (4) :- and :-
.
From these two equations :-
and finally or
This also means that :-
Velocity ratio and pressure ratio across a shock wave in terms of Mach number
We already have as equation (3¢):-
.
It can be applied to the conditions at the throat and to any other section but it is the condition just before the shock wave that we need and is section 1 :-
When this is divided by it gives:-
which gives :-
and :-
Now we can use the relationship and note that
Therefore
And finally
This is quite an important equation because it can now be used with Rankine-Hugoniot to relate pressure ratio over a shock wave with
First use continuity applied between the two sides of the shock wave.
and as is unchanged :- . This can be used to eliminate the inconvenient density terms in favour of .
becomes . This can actually be simplified!
the last term cancels out and if we divide through by we get :-
This is a remarkable outcome from such an unlikely looking expression.
We need a relationship between the temperatures before and after the shock wave. It is inevitable that a the energy exchanges in a shock wave are adiabatic and then the expression that can be applied through the shock wave is the energy equation in the form:- where and are the absolute temperatures just before and just after the shock and is the stagnation temperature. It is interesting that the stagnation temperature is the same before and after the shock wave but this follows from the fact that, even though there is depletion of kinetic energy in the shock wave, it reappears in the internal energy of the flowing gas..
We can write and then :-
and then
Substituting we get which is a considerable simplification.
This ratio can also be expressed in terms of
But, using
These are all the ratios that are required to calculate the properties before and after a shock wave. It is the value of that determines all the other properties before and after the shock wave. The value of is of course dependent on and and, for a nozzle, these can be determined from reversible, adiabatic flow up to the shock wave.
At this point we need a worked example to show how one might use all these ratios. A suitable example follows in blue.
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.
Faced with this example and all the convoluted expressions that require lots of calculation one must think of ways to reduce the labour involved and reduce the opportunity for calculating error at the same time. There are tables in which all these calculations have been done and published. They were created around 1970 and are copyright. Since that time pc’s have evolved and these can reduce the work.
I am sure that there are people who can find a more elegant way to use Mathcad but all that engineers want is an answer. Basically this calculating programme finds the ratios between properties for any value of the Mach number in a nozzle for any value of . Mathcad will not accept a / in the designation of a variable so I have used z instead to mean “divided by”. The upper group of ratios is for a reversible, adiabatic, one-dimensional model of an expansion. The lower group is for a plane shock wave. I have put and let to give the values of the ratios at the throat but when is given some new value all the ratios are recalculated instantly..
Now we can attempt the worked example.
(i) Change to 1.8 in Mathcad and this gives the area ration between throat and exit as 1.439. As areas are proportional to diameter squared .
(ii) The programme above gives the ratios for the throat.
Therefore and
Therefore .
(iii) At exit =1.8 and Therefore
(iv) This question is pointing out that for the flow in the divergence to be free from shock waves the back pressure must be lower that that at which a shock wave forms at the exit. If a shock wave were to be at the exit the value of before the shock wave would be 1.8 and the ratio of pressures through the shock wave will be 3.613. So :-
The back-pressure
The specimen example shows that Mathcad can reduce the burden of calculation but, in examinations, it may be desirable to use tables. For real engineering the Mathcad programme or something like it will speed the design process.
The behaviour of the flow in a nozzle as the back-pressure is increased
So far I have introduced the physics of the plane shock wave in the divergence without saying anything more than the simple statement that it can exist. We need to know more than that.
The physics for reversible, adiabatic, one-dimensional flow shows us that when there is no shock wave in the nozzle the pressure at which the fluid leaves the nozzle depends only on the inlet conditions and the dimensions of the nozzle. However the back-pressure is clearly a quantity that can change and it could be below or above the pressure at exit from the nozzle or be greater than the exit pressure. We have already seen from the worked example that a shock wave can form at the exit and that the pressure at which the fluid leaves the nozzle and could be 3.14 bar with the flow in the divergence still wholly supersonic and falling to 0.87 bar just before the shock wave. We need an explanation for this behaviour.
Graph13-8
is not really suitable for the next stage of this text so I have redrawn it as
graph 13-16 with the divergence truncated so that the pressure at exit would be
1 bar absolute. This is just an arbitrary decision taken to give me increments
in pressure that can be seen and can be labelled. Notice how short the
divergence is for an exit Mach number of more than 2.
I have to start with the green line A-B-C in the diagram. It is the pressure distribution for typical subsonic flow throughout the nozzle. A represents the inlet pressure, C is the back-pressure at exit and B is the lowest pressure, that will always occur in the throat. We have seen this pressure distribution before for the Venturi meter. If the back-pressure is raised, the pressure at B will increase but, if the backpressure is lowered, the pressure at B will fall until the conditions at the throat are such that the velocity at the throat is equal to where is the absolute temperature in the throat and the pressure distribution will be the black line A-T-D. So D represents the lowest value that the backpressure can have and still be consistent with subsonic flow through the nozzle. It is called the upper critical back-pressure.
In fact it is much more likely that when the velocity at the throat goes to the expansion through the divergence will be supersonic and the pressure distribution will follow the line T-E and, because the nozzle has been truncated to produce these conditions, the exit pressure will be 1 bar and equal to the backpressure. The pressure at E is called the lower critical backpressure.
It
is what happens in between E and D that is now of interest because it is
obvious that the back-pressure can have any value between 0 and that at D and
still have a speed of in the throat. We have already seen
that there can be a plane shock wave in the divergence of a
convergent-divergent nozzle. Now I can add lines to scale on graph 13-14 to
give graph 13-17 and to show the pressure distribution when a shock wave is
present.
A typical line is A-T-J-K-L. A-T-J is the normal expansion. J-K is the rise in pressure in a plane shock wave and K-L is a subsonic expansion to the exit where the pressure just reaches the back pressure.
I have drawn a second line A-T-F-G-H and shown the special case of a shock wave at exit as A-T-E-P.
There is a line T-K-G-P, shown dotted, that is the locus of the points representing the pressure just after the shock wave. Points K-G and P can be calculated using the model for a plane shock wave and the lines K-L and G-H using the reversible, adiabatic, one-dimensional model.
The mechanics of a shock wave
It seems to me that not much attention is paid to what actually happens in a plane shock wave. We can see the result of a compressible fluid flowing through a shock wave in that the velocity falls, the pressure rises, the temperature rises and the density rises but which of these changes is the one that “drives” the process? I think that it is necessary to think how the steady flow comes into existence. As the back-pressure rises to some pressure above that at P a shock wave moves towards the throat until it comes to rest relative to the nozzle when the speed of the wave relative to the flow and towards the throat equals the speed of the flow. On one side of the wave the speed is supersonic and the density low, on the other the speed is subsonic and the density high. The high-speed flow of untold numbers of molecules suddenly changes its density by a factor of 2 or more. The molecules must be packed closer together to increase the number of collisions made by each molecule every second and with this change an increase of the mean kinetic energy of the molecules. This can only come from the mechanical energy possessed by the gas in the speed of its mass centre. Mechanical energy is lost back into the random energy of the molecules. As I have already pointed out this wave may seem thin to us but on a molecular scale it is quite thick.
Such a process could not operate in the reverse direction and take mechanical energy “instantly” from the random energy of the motion of the molecules. We had to use a carefully designed nozzle to make this change.
Practical consequences of the presence of a plane shock wave
I have said that I do not think that anyone would design a shock wave into an engineering device.
In section 3 I discussed the use of convergent-divergent nozzles in steam turbines. The nozzle is part of an engineering device comprising the ring of nozzles and a ring of moving blades. All the velocities involved are high and we have seen that the steam speed is likely to be around Mach 1. It is not easy to find the best proportions for the nozzle and the moving blades to get all these velocities sorted out to give a high efficiency for the extraction of energy from the steam. It may be that it would be desirable to have steam entering the moving blades at supersonic speed but this must be balanced against the consequences of having the inevitable shock waves form in the blading. Energy is lost in these oblique shock waves. So there is an incentive to avoid supersonic flow altogether. There is little likelihood that conditions leading to plane shock waves in the nozzles are necessary.
The rocket nozzle is rather different because the creation of a jet having a high Mach number also creates forces on the nozzle to give thrust. This being so the question of the existence and consequences of having a shock wave in the divergent cone must be relevant.
In figure 13-11 of the rocket engine I just drew a notional pressure distribution through the nozzle. Now I have to look at the discharge from the nozzle to see how it might affect the thrust
A rocket engine used to lift a space craft into orbit will start off discharging to atmospheric pressure and end up discharging to a pressure that is close to a vacuum. In graph 13-7 the pressure at exit is well below atmospheric pressure but atmospheric pressure is well below the back-pressure needed to cause a shock wave to form in the nozzle. What does change as the pressure surrounding the nozzle decreases is the net force on the rocket increases simply because the external pressure on outside of the rocket motor decreases.
Worked examples
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 :-
