The practical use of convergent and convergent-divergent nozzles.
I think that I have done enough to show that it is quite possible to understand how a convergent-divergent nozzle “works” and I have shown that useful calculations can be made to the point where an engineer can make a good start at designing a nozzle. However it makes sound sense to look at the nozzle in use to see what constraints the engineering imposes on the physics.
I can think of only three uses for convergent-divergent nozzles. There must be more but these three will serve my purpose. They are the nozzles in a steam turbine, the nozzles of a rocket engine and the intake duct of a gas turbine engine for supersonic flight. Let me look at these in turn.
Steam turbines used for electricity generation can use steam that is supplied at up to 170 bar (2,500 psi) and they all exhaust at something like 0.1 bar. Of course there are other smaller turbines working with steam at lower pressures. The field of application is very extensive and the study of these turbines is a vast undertaking. Nevertheless there are a few basic principles that can prove to be useful. A range of pressure from 170 bar to 0.1 bar is an extreme range and graph 13-15 shows more or less how the pressure varies with volume during the expansion in the steam turbine. The possible area under this curve is very small indeed for the ranges of pressure and volume involved. Only a steam turbine can make use of this range to extract work from the steam.
Steam turbines do not make one continuous expansion, they often make the expansion in three or four stages. This gives two engineering advantages, the first is to give opportunities for improvements in efficiency by heating the steam between stages and the second is that the thickness of the casings can be matched to the pressures that are exerted on them.
It is not unreasonable to think in terms of having each stage produce the same power. The power produced in a stage of a turbine is dependent on and in graph 13-16 I limited the ranges of pressure and volume so that areas can actually be seen on the graph. Then I have roughly divided the total area under an expansion into four equal areas. It immediately becomes evident that the first stage will involve a large pressure drop and small volumes of steam and for the last stage will be at low pressure and with very large volumes of steam.
We have seen that sonic speeds are associated with pressure ratios of 50+% and the pressure ratio on the first stage have about this value. As a result steam turbines often use a de Laval nozzle to extract work from the steam as kinetic energy and then transfer this kinetic energy to one or more rows of rotating blades and then the steam is expanded continuously in many stages each consisting of a fixed row of blades followed by a row of moving blades through the remaining three stages. In the first stage the blades operate at constant pressure and the best way to use the steam is to make a big drop in pressure in a ring of de Laval nozzles to start the expansion and have the blades operate in a drum at a much lower pressure than the boiler pressure. This is called an impulse stage and we now have the expressions that might let us design a nozzle.
Now we need to see what the mechanics of an impulse stage look like. I think that it is important to draw them reasonably accurately and not hide behind the words “schematic” or “ not to scale”. Figure 13-10 shows how two cambered, aerofoil-shaped blades can form a convergent-divergent nozzle or, indeed, just a convergent nozzle. These blades can be robust. The moving blades must also be robust just to stand the forces of very high-speed steam of high-density flowing over them and the centrifugal forces when rotating at 3,000 rpm as well. Fortunately they need to have this shape anyway to give the best flow pattern.
Look at the space between two fixed blades. It is a convergent-divergent nozzle but it is cropped off obliquely. This is done to ensure that the jet of steam has no unguided distance in which to start to mix and so lose kinetic energy to random motion and the gap between the exit from the fixed row and entry to the moving row is kept to a minimum. There is a price to be paid for cropping the nozzle in this way that is that the jet is diverted slightly as it flows with guidance on only one side but this is preferable to having a gap.
The nozzles are created in an annular ring and this ring has to withstand the very high pressure. It is not easy to ensure the surface finish of the nozzles. The engineering constraints of this design limit the value of attempts to refine the physics of the convergent-divergent nozzle as I have laid it out in this text. Nevertheless practical nozzles have efficiencies of 95% and higher.
Rockets are very simple engines. I have attempted to draw the essential features in figure 13-11. The design is dominated by the need to minimise its weight. Fuel and oxidant are pumped at very high rates into the combustion chamber where they combine to produce an enormous flow of gas. The creation of this gas and the resistance to its escape through the nozzle produces a high pressure in the combustion chamber. The gas flows out through the convergent-divergent nozzle under the pressure difference between the gas in the combustion chamber and the outside conditions. I have also drawn a series of arrows to indicate the way in which pressure is exerted on the inner surfaces much the same way as is commonly used to show pressure distribution over an aerofoil. All over the outer surface the pressure is whatever it may be around the engine. The pressure across the exit plane is not so easily known and I may have to deal with it separately.
I explained how these pressure distributions give rise to a net force in chapter 5 in the text associated with figures 5-14 and 5-15a and b. In essence there is a very large upward force on the top of the combustion chamber that is not balanced by a downward force on the convergent part of the nozzle so that there is a net upward force on the combustion chamber exerted by the gas. This force is exerted on the engine mounting and then on to whatever is being driven.
In the divergence the pressure falls steadily but, throughout, this pressure exerts a distributed force on the bell-shaped divergence that everywhere has a vertical component and there is a net upward force produced on the divergence. That force is exerted through the combustion chamber on to the engine mounting together with that from the combustion chamber.
This is simple in principle but not so easy to produce as an efficient mechanical device. I presume that it would be desirable to complete the combustion in the combustion chamber and, for this to be possible, the combustion chamber would have to be quite large. As it is a pressure vessel, if it is to be large it would also be heavy and there is a great incentive to keep its size to a minimum. If the combustion chamber were to be too small the combustion would take place partially in the nozzle. I have said that I cannot tell the difference between a hot luminous gas and a burning gas so just looking at pictures of rockets does not help. A compromise must be made between weight and performance and that compromise is affected by other factors.
Photographs of the inner surfaces of rocket nozzles suggest that they are lined with refractory material and it is not really smooth and, even if it were to be, it is not likely to survive a protracted burn unscathed.
The booster rocket works in just the same way as a liquid fuelled engine but now the upward force on the casing containing the burning fuel is exerted either on the fuel itself if it is not porous to gas or on the inner surface of the top of the casing if it is porous. So instead of exerting a force on the bottom of the rocket as the liquid-fuelled one does the force probably acts at least in part on the top. Fuel cases for rockets are columns and are designed as such.
So whilst we have modelled the convergent-divergent nozzle as an adiabatic frictionless flow the reality is very different. Nevertheless the model is an enormous help in getting a basic understanding of rocket nozzles and of possible ways to store experimental data.
The supersonic gas turbine engine is now used only in military applications. The gas turbine engine cannot operate with supersonic flow anywhere over its blading but it is used to power supersonic aeroplanes. The inlet conditions for the Olympus engines used on the Concorde require inlet conditions that are effectively those of ambient conditions at sea level. This may not be accidental because those engines, running on gas, also power electricity generators.
An aeroplane capable of supersonic speeds must also fly subsonically over a range of speeds up to 1,000 km/h. Then it must go on to fly at perhaps 3,000 km/h. There is only one way to cope with this and that is to fit intake ducts that can be used in conjunction with an engine management system to change the shape of the duct from one suitable for low speed to one suitable for high speed. At low speed it is essentially a straight through duct but at high speed the duct is a convergent-divergent nozzle with air entering it supersonic speed, possibly at low pressure and low temperature, and leaving the duct to enter the engine at about 1 bar absolute and about 300°K. The duct is rectangular in cross-section as the only possible design. In the supersonic configuration the air slows down in the convergence and slows down in the divergence with a steady increase in pressure. Surprisingly the net effect is that the ducts produce a very substantial forwards force that contributes about 40% of the total thrust. This sounds odd but we seldom ask the question “ how is the force produced by a jet engine and where is it exerted on the frame of the engine?” There are surprising answers.