Chapter 13 Part 3 The convergent-divergent nozzle

 

Introduction

In all probability before 1897 no one knew for certain that there are two regimes of flow for gases and vapours and only writers of science fiction thought in terms of travelling at speeds of thousands of miles an hour. The low-speed regime is common in nature but I think that steady high-speed flow of gases does not exist in nature. High-speed flow came about as a consequence of man’s ingenuity. It is likely that the first person to produce a steady supersonic flow was Gustaf de Laval (1845-1913) who developed, in 1897, a convergent-divergent nozzle for a small steam turbine for use in a creamery. His invention paved the way steam powered turbo-alternators, turbine powered flight and rocketry. It is important.

 

Text Box:  
Fig 13-4
We have looked at the engineering applications of the flow through pipes of uniform diameter and now we have to look at nozzles that are short (by comparison with a delivery pipe.) with changes in cross-sectional area. See figure 13-4.

 

This will lead to more applications of forms of the energy equation together with continuity and equations of state and, inevitably, lots of manipulation of mathematical equations. But before I do that I want to look at why a gas behaves as it does in a convergent-divergent nozzle.

 

 

The mode of operation of a convergent-divergent nozzle

I learnt about nozzles from the mathematical approach and it always seemed to me to be less than satisfactory. Whilst I knew how convergent-divergent nozzles behaved I did not know why they behaved that way. When I came to write this text I thought I ought really to make an attempt to put that right.

 

Those who first sought to make steam turbines were also the first to have a large steady supply of an elastic medium that is very like a gas, steam. They soon found themselves using nozzles to produce high-speed flow and they started by using convergent nozzles and they mostly still do. This was the intuitive design with its forerunner in use in hydraulic machinery. They soon found that whilst they could increase the speed of the jet formed by a given convergent nozzle by increasing the supply pressure, no comparable increase could be produced by reducing the back-pressure. They described the nozzles as “choked”. It must have been totally counter-intuitive to find that the fitting of a divergent cone to a convergent nozzle got rid of the problem.

 

So why is it that a convergent nozzle can be choked and a convergent-divergent nozzle is not choked? I want to explain without mathematics.

 

I had this to say about gases in chapter 6 :-

 

The kinetic theory of gases gives a very good picture of the structure of a gas. It postulates that a gas is composed of separate molecules, (which may be single atoms). The molecules of the gas have mass. They “fly” freely at high speed, colliding frequently with other molecules and with the walls of the container in which they are enclosed. The scale of the structure of a gas is indicated by the following figures. The common gases at room temperature and at a pressure of one atmosphere have about  molecules per cubic centimetre. Even with this concentration there is still space between the molecules for them to move freely at high-speed (about 350 m/s) through a distance of about 7 molecular "diameters" between collisions and the number of collisions made by each molecule each second is about . These are large numbers that I can accept but cannot imagine.

 

I think that an important observation to make to start is that when a gas flows steadily through a nozzle, whether it is convergent or convergent-divergent, a thermodynamic process takes place. By thermodynamic I mean that some of the random kinetic energy stored in the flowing gas is converted to mechanical energy in the form of the kinetic energy in a jet. It is mechanical energy because it has one direction. In de Laval’s turbine this kinetic energy was extracted from the flowing jet of steam emerging from the nozzle by a row of blades on a spinning rotor that reduced the speed of the jet in a controlled manner to a small fraction of its initial value. The mechanical energy was taken from the rotor through a shaft to drive machinery. What is important about this is that the process that takes place in the nozzle is the extraction of energy from the molecular structure of the gas.

 

We are quite familiar with the idea of extracting energy from the molecular structure by using a piston in a cylinder. There the gas exerts a pressure on the piston to create a force that moves and does work. The pressure is created by untold numbers of collisions between fast-moving molecules and the piston face. When the piston moves, the rebounding molecules lose energy in every collision and, as the piston continues to move, the stock of energy in the molecules, that is the internal energy in thermodynamic terms, is depleted and inevitably the temperature and the pressure falls.

 

Now we have a thermodynamic process going on in a nozzle and we have no piston. Instead the internal energy, that is the stock of random kinetic energy in the molecules, is transferred to the kinetic energy in one direction of the gas itself. It is this process that is of interest here.

 

Looking from the outside what we see is gas flowing steadily through the nozzle with a consequent drop in pressure, a drop in temperature, an increase in volume and the formation of a jet that contains kinetic energy. It seems to me that it is impossible to avoid the observation that whatever goes on to produce these effects is a progressive process. The ways in which the properties change along the nozzle are ultimately determined by the shape of the nozzle and the primary variable is the area of cross-section and, if the process taking place in the gas is to be continuous, that is, free from sudden changes, the area must change in a continuous (mathematically) way or as nearly continuous as practice permits.

 

I spent a long time trying out several ideas that might lead to an explanation but in the end I came to the conclusion that I would have to think of the gas as flowing in lots and lots of thin layers. In figure 13-5 I have drawn representations of these layers but only a very few compared with the number that I really imagine. These thin layers may be thin in the context of the nozzle but they are very thick when compared with the size of molecules. So I can think of adjacent layers having different pressures, different temperatures, different specific volumes and different mean speeds for their molecules that, in fact, create these properties. This must mean that, at the surface where the two layers abut, molecules from upstream collide with molecules downstream and, as the upstream molecules have higher mean speeds than those downstream, exert an accelerating force on the downstream layer to increase the speed of its mass centre. This is a thermodynamic process in which random kinetic energy in the upstream layer is used to increase the kinetic energy of the mass centre of the downstream layer.

 

This process goes on at every thin surface and the net effect is that there is a gradient in the mean speed of the molecules in the gas. As it is this mean speed that determines temperature and pressure there will be gradients in these as well. This is exactly what is observed.

 

However convergent nozzles “choke” and this can be explained by considering what happens at the thin layer as the speed of the mass centre of the gas in the layer approaches the mean speed of the molecules as it must do at some point along the nozzle. Then the layer downstream of it is moving away at the speed of the molecules in the upstream layer and even though collisions continue there is no net transfer of kinetic energy or any net force. The flow will have reached a maximum speed for a convergent nozzle and no change in conditions beyond the exit will affect the flow. The nozzle is choked.

 

That speed is given by  and as all three terms are properties of the gas this whole term is also a property.  is of course the local absolute temperature. It is no surprise to find that this for air at 20°C and 1 bar absolute is equal to the 350 m/s that I highlighted in bold above.

 

However we know that there can be higher speeds and so there must be a mechanism and it involves sideways expansion. In the convergence the gas has been expanding even though the duct has been reducing in area of cross-section. The expansion is in the direction of motion. If the duct now becomes divergent the gas can expand sideways and this will reduce the pressure, the mean speed of the molecules and the temperature. Then the layer of gas that is moving at the same speed as the mean speed of its molecules will be in contact with a layer of gas at lower pressure, lower temperature and with a lower mean speed of its molecules. Then the downstream layer will accelerate as before to give a layer moving faster than the mean speed of its molecules. As the gas advances into the divergence the speed of the mass centres of successive layers will increase but the mean speed of the molecules in successive layers will fall.

 

So, in the convergent-divergent nozzle, the speed of the gas can increase from end to end, the mean speed of the molecules in the molecular structure of the gas will fall from end to end and at some point the speed of flow will equal the mean speed of the molecules. We have seen that this is a special point where the duct must change from convergence to divergence. Then the flow in the convergence depends on expansion in the direction of flow and in the divergence on expansion across the flow. The speed at the junction, that is, in the throat, is  where  is of course the absolute temperature at the throat.

 

This immediately raises the issue of the use of the Mach number in connection with nozzles. In the early part of the 20th century there was great optimism that the use of non-dimensional groups of physical properties would lead to significant improvements in the ways in which we store experimentally gathered data.[1] One of the important groups is the Mach number denoted  and named in honour of Ernst Mach (1838-1916).

 

It is the ratio of two speeds and is most commonly used in connection with the flow round a body such as an aeroplane moving through stationary air. The speed of the aeroplane is measured relative to the stationary air and therefore relative to the ground and the quantity  is evaluated using either the measured value of  or is calculated from a notional value of a . Then a value for  is found by dividing the speed of the aeroplane by the . This gives a linear scale of numbers for  that is simply the speed of the aeroplane divided by a fixed speed. The speed at which  =1 has entered the collective conscience as Mach 1 and taken on some sort of charisma.

 

Now we must think about the flow through nozzles and we find that we have a different definition for . The change comes about because of the mathematics that follows the application of physics to the steady flow through a nozzle and to the flow when there is a shock wave in the divergence. It is then most convenient to redefine  using the same expression  but with the changes that  is now the speed of the gas at some point in the flow and  is now its absolute temperature at that same point. This means that there is no longer a linear relationship between  and .

 

This decision to redefine  means that we cannot just assume that the flow upstream of the throat will be subsonic and that downstream it will be supersonic but this is, in fact, the case. Consequently we can say that in the convergence the ratio  and in the divergence  where  is the velocity of the gas and  is its absolute temperature

 

Modelling the flow through a convergent-divergent nozzle

When one sets about the use of a convergent-divergent nozzle in engineering it is necessary to consider real flows. The first thing that is important is that nozzles may carry a variety of gases. In gas turbine engines the gas is essentially air, in steam turbines it is steam and in rockets it can be steam,[2] or the gases resulting from combustion.[3] These gases can be very hot and I really do not know whether they are still burning in rocket nozzles.[4]

 

We have already seen for the Venturi meter that the flow in a convergence is likely to be orderly and not depart much from streamlined flow and that any effect of friction is likely to be small. We also found that the flow in the divergence was likely to break down. However, in the divergence of a Venturi-meter whether used with liquid or gas, the flow is slowing down and pressure energy is being recovered. It is a diffuser and there is a best angle for the divergence to give the best pay-off between increased loss to friction when using a long divergence and increased loss in random internal motion caused by the breakdown in the streamlined flow. In the high-speed convergent-divergent nozzle the flow in the divergence increases in speed in the divergence and now both high friction loss and high loss to internal random motion cannot be set one against the other. Both will occur and engineers will be called upon to develop shapes for the divergence that are as efficient as possible in performing its function of extracting energy from the internal structure of the gas as orderly kinetic energy. This means that engineers need some yardstick against which to assess their success in developing their nozzles.

 

Others have used the ordinary ideas of thermodynamics to decide what is the best that can be done and then found expressions relating the pressure variation, the volume variation and the temperature variation of the gas with the variation of velocity as the area of cross-section changes along the nozzle. The most efficient expansion that we can conceive would one that is “reversible”. This is a profound idea that has its roots in mechanics[5] where it is taken for granted that, say, a pendulum would swing for ever if there were to be no friction because we know in our bones that friction always involves an unrecoverable loss of energy. Yet, throughout that eternity, the bob would be continually exchanging potential energy for kinetic energy . So it is not just about having no friction at the bearings or no air friction, it is also about having no energy loss in this never-ending exchange of energy as well. When this idea is applied to an expanding gas the most efficient process would involve no mechanical friction such as that between the gas and the surfaces of the nozzles but there must be no viscous loss and any heat exchange that cannot be reversed. When we seek our ideal expansion we have no alternative but to suppose that there is no heat exchange because a heat transfer that can take place in either direction as required is beyond our imagination and not found in nature.

 

So our most efficient expansion in a nozzle would be free from viscous effects of any sort and take place without heat exchange with the surroundings. In the terminology of thermodynamics it will be a reversible, adiabatic expansion.

 

If the flowing fluid were to be air we could then use all the usual gas laws and, most importantly, use the relationship  to relate pressure and specific volume. If the flowing fluid is not air but steam or the products of combustion the case for finding a  relationship[6] of this same form with some new index is over-whelming as we shall see.

 

Text Box:  
Fig 13-5
Now we have to think out how these ideas have been applied to a convergent-divergent nozzle. Whilst we do so remember the fundamental rule of science for engineers “ if you are creating a yard-stick make sure that you know what the yard-stick depends on so that you know when you depart from it”. This rule comes into operation when you look at figure 13-5 and see that the flow-pattern is not made up of parallel lines crossing plane sections but curved lines crossing convex and concave sections. A decision is needed and that decision is not in our hands, it was taken long ago, and it was to use one-dimensional flow. This decision makes a useful outcome possible and leads to a basis for a nozzle efficiency.

 

So we need to treat the convergent-divergent nozzle as having one-dimensional, adiabatic, frictionless flow. There is then no doubt about this model of the flow.

 

Now we come to a difficult point. There are two versions of the treatment of the physics of the convergent-divergent nozzle. The first, chronologically, uses the physical properties of the gas, the second, uses Mach numbers. I am attempting to construct an empirical science for engineers and engineers want their analyses that are based on physics to end up relating quantities that can be measured or can be calculated from quantities that can be measured. They also want the analysis to be simple and transparent. Given the problem of the non-linearity of Mach numbers with speed some may find one treatment more convenient that the other. I will give both.

 

Modelling using properties

I will start with the treatment using properties.  The basic argument is that, in a nozzle, the work done by the gas as it expands goes to increase the kinetic energy of the gas in one direction. This means that a thermodynamic process takes place in which some of the random energy that is stored in the molecular structure is converted to mechanical work in the form of kinetic energy. This is done in a flow process and we must write :-

        .

Text Box:  
Fig 13-6
In figure 13-6 I have drawn a pressure-volume graph for a fully resisted adiabatic expansion of air. The terms in the equation above can be picked out as areas on the diagram.  is the area under A®1,  is the area under the curve 1®2 and  is the area under 2®B. Summing these as in the equation shows that the work done on the gas as it flows through the nozzle is the area A®1®2®B®A.

 

Then it is evident that:-

Text Box:  
Fig 13-7
           . This is a regular integral in thermodynamics and the equation becomes :-

            

In fact this is a steady flow of air from some region where the velocity  can be regarded as very small and then the temperature is the stagnation temperature. I have shown this in principle in figure 13-7. Then the measurable stagnation conditions are .

 

Now, if we choose to consider an expansion from stagnation conditions of  to some state where the conditions are  equation  can be rearranged to give velocity :-

                                             .

Now this must be manipulated to eliminate the volume terms.

                                            

But  and then:-

                                            

This expression for  can be made more convenient if  to give

                                          

 


(It is worth noting that there is a maximum velocity for the air obtained by putting . For a stagnation temperature0f 293°K it is 767 m/s or 2,760 km/h or 1716 mph.)

 

A graph of  could be plotted against  very easily and  we can find the diameter of the nozzle at the point where the pressure is  at the same time.

We have from continuity  or . This requires another manipulation

       :             .

Then :-                                  and .

Whilst I am plotting graphs I can add the graphs of  and . This generates the two graphs 13-7 and 13-8.

 

Graph 13-8 shows the profile of a nozzle that could, in principle, have a uniform pressure drop from inlet to outlet. It has a divergence that is hopelessly impractical.

 

I think that this topic of flow in a nozzle is complicated and it leads to a complicated result and it needs to be explored just to get a feel for the numbers. What is really needed is a graph showing to give the property distribution in a nozzle with a practical profile.I made a couple of attempts just to get practical shapes and start conditions. In the end I settled to look at a nozzle having a throat of 12 mm diameter with air supplied at 10 bar gauge and 100°C with a mass flow of 0.26 kg/s. What I thought that I would like to know is the way that the velocity and the pressure would change along the length of a nozzle having the sort of profile shown in figure 13-7. I let my nozzle have an inlet radius of 15 mm and a divergence of 16° included angle for no other reason than it is much the same as that for the divergence in a Venturi meter.

 

The result is graph 13-9.

 

It should be noted that in order to draw graph 13-7 and 13-8 I had to start with values for the stagnation conditions. At one time this would have meant that the calculations for drawing the graphs would have been very time consuming. That problem has disappeared with the emergence of mathematics packages on computers. We can draw as many graphs as we like almost instantly.

 

Text Box:  
Graph13-9
I think that this graph gives a good idea of what goes on in a convergent-divergent nozzle. It is surprising how short the divergence needs to be for the Mach number to reach 2 and then how much more of the divergence is required to reach Mach 4 but the increase in Mach number from 2 to 4 takes place at absolute pressures less than 1.5 bar. This reflects the shape of the  curve in graph 13-2 where small changes in low pressures produce large changes in volume.

 

Modelling the flow using the Mach number

We must start by applying the energy equation to any two points 1 and 2 in the flow through the nozzle. There is no work done to the surroundings so we can sum all the forms of energy at the two sections and equate them. The forms of energy are internal energy, kinetic energy, pressure energy and, to be pedantic, potential energy. In this analysis it is better to work, not in head (energy per unit weight) as we did for liquids, but in energy per unit mass.

Then the equation becomes :-   where  is the internal energy per unit mass and the other symbols their usual meanings.

 

Nozzles are short so the potential energy terms are pointless and the equation becomes :-

                                           

We have  and  . From this  as is well known in thermodynamics and called the enthalpy[7]. Then the energy equation reduces to :-

                                            

I have argued for the flow through a convergent-divergent nozzle to be continuous and this means that this equation can be applied to any two points in the flow. Then, if as before, we refer to the stagnation conditions  we can proceed.

For any section in the nozzle we can write :-  where  is the absolute temperature at the section.

 

If we work with ratios between of the properties in the nozzle and the stagnation properties we shall end up with a set of ratios that are the same for every convergent-divergent nozzle. In order to use these ratios we shall have to find values of the reference properties for every set of stagnation properties and mass flow.

 Now  can be changed to Mach number form by dividing through by  to give :-

                 and then by using  to give .

Then substituting in the energy equation gives :-

                               .

 The first of these ratios is between absolute temperatures     .

This gives us  in terms of  and we may need  in terms of  and , and,  in terms of  and .

 

We have  and . One gives  and the other gives :-

             . Then

Therefore :-                                       

Finally            and then  

 

These expressions allow us to find out how the properties vary with Mach number but we have no way as yet to link the values of the properties with the nozzle. We really want some link with the area of cross-section so that, if we knew the profile of the nozzle, we can plot properties against length. We cannot continue with the stagnation conditions because the idea of having a cross-section at the stagnation conditions is meaningless. But we can use the throat where we not only have the minimum diameter but also conditions that can be related to the stagnation conditions. This leads us to find a ratio between the cross-sectional area at any point and that at the throat.

 

We have the continuity equation  and we can write  where the subscript  indicates throat conditions. From this  and we have

                     .

This does not match the other equations and we want it in terms of  so another manipulation is required. We can use  and we can write :-

                               which reduces to :-

                                and that deals with the density term. Now we need to eliminate the temperature term. We can use :-

                             

Now we can substitute to find the ratio of areas.

                          and this reduces to:-

                                            

                                       

In the top of figure 13-9 I have gathered the four resulting expressions. They are in the most convenient form that is known and each of the ratios is expressed in terms of  and . In the lower part I have substituted 1.4 for  to give much more manageable expressions for air. They can easily be plotted using, say, Mathcad and the result is shown in graph 13-11 and in graph 13-12. It is also easy to evaluate any of the ratios accurately using the calculating facility.

 

It should be noted that this is the graph for a reversible, adiabatic, one-dimensional, expansion of air through a convergent-divergent nozzle. There will be similar graphs for other gases and there must be some similar graph, probably based on an approximation, for steam although that is outside of the scope of this text.

 

Using these expressions

 

These are my calculations:-

I must find the mass flow and it must come from the throat conditions where ;  ; ; ; ;

Therefore  and this is ready for Mathcad.  and

 and then

This is enough for plotting a graph of  and of  versus diameter. Using Mathcad.

Text Box:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The plots are shown in graph 13-13 and 13-14. I was able to plot graph 13-9 again.

 

 

 

 

Text Box:  
Graphs 13-13 and 13-14
 

Assessment of these two methods

I think that it is right to put these into context. The properties treatment would have been time consuming to use as I have said but all the quantities involved are readily understood. The Mach number treatment that leads to one set of ratios of properties for all reversible, adiabatic one-dimensional expansions permits the construction of tables with Mach number as the independent variable. This table would have reduced the calculating time significantly. But the digital computer has reversed this position. It is now easier to use the earlier method of direct use of properties.

 

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.

 

Text Box:  Graph 13-15Steam turbines used for electricity generation can use steam that is supplied at up to 170 bar[8] (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.

Text Box:  
Graph 13-16

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.

 

Text Box:  
Fig 13-10
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.

 

Text Box:  
Fig 13-11
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[9]

 

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.

 

The idea of having a correct back-pressure

In a Venturi-meter where the flow is always subsonic any change in either the pressure at entry or the pressure at exit from the meter, or both, will alter the flow through the meter and this is how the meter works.

 

For a convergent divergent nozzle with an overall pressure drop that is great enough for supersonic flow to occur throughout its divergence a change in the inlet pressure will alter the flow but the exit pressure can change quite significantly without affecting the flow. Providing that the flow in the divergence is wholly supersonic the flow can emerge from the nozzle either below the back-pressure or above the back-pressure. If this is the case there must be a pressure of the gas at exit that equals the back-pressure. At this pressure the jet will emerge with no net transverse pressure difference to make it either expand or contract. This is called the “correct” and sometimes it is convenient to use this concept.

 

The concept of an efficiency for a nozzle

The physics given in this chapter gives a very useful insight to the behaviour of gases in a convergent-divergent nozzle. The actual behaviour will not be so much different but the major difference will lie in the effect of friction on the expansion. That friction will be have two components, the friction between the gas and the solid boundaries of the nozzle and the internal friction stemming from irregularities in the flow. The combined effect will be to reduce the amount of the original internal energy that is extracted and converted to kinetic energy of the flow in the emerging jet. We must decide what effect this has on the flow.

 

If we choose stagnation conditions and a diameter for the throat, according to our model, the pressure at exit for supersonic flow in the divergence will be determined by the diameter at exit. Using our model we can determine a temperature at exit. We can use

The expression  or its derivative form  can be used to find the velocity at exit. So, using the model we can find the exit area, the pressure and the temperature.

 

Now we want to know what happens when a nozzle with these proportions operates with the inevitable friction. We can go back to the origins of . This came from  which is the application of the energy equation to a gas. The sum of the terms  becomes  the enthalpy. Now there is nothing in this that requires the process involved between 1 and 2 to be reversible, the only requirement is that the process is adiabatic. It follows that it can be used for our model of reversible adiabatic flow and for a flow involving friction.

 

I do not think that there can be any doubt that the effect of friction in the nozzle will be to reduce the amount of energy transferred from the internal motion of the molecules of the gas to the kinetic energy of the mass centre of the gas. So, if the kinetic energy is reduced from that produced in a reversible adiabatic flow, the result must be an increase in the absolute temperature at the end of expansion over that for reversible adiabatic flow.

 

For a nozzle with stagnation temperature of  we can denote the temperature after a real expansion as  and after a reversible adiabatic expansion as  and then if the corresponding velocities are  and  and .

 

If the function of a nozzle is to produce kinetic energy then an efficiency  for the nozzle can be  and this is the same as .

I have pointed out that the function of a nozzle in a rocket is to produce a force and the creation of a jet of fast moving gas is the means to produce that force. For rockets one can equate force to the momentum of the jet and then a suitable efficiency would be the ratio of forces. But the only way to improve the ratio of the forces is to improve the process by which the kinetic energy is produced so one nozzle efficiency will do for both.

 

We have said in several contexts that there will be very little loss in convergent flow but significant loss in a divergence. This will be true of the convergent-divergent nozzle. There is an advantage to be gained if the flow up to the throat is treated as reversible and all the loss presumed to take place in the divergence. Then we can relate the throat diameter to stagnation conditions and mass flow and these quantities will not change during calculations concerning the divergence.

 

 

 



[1] Eventually we found the limits of the applicability of these non-dimensional groups The optimism has not waned. There are scores of them now.

[2] The shuttle main engines run on hydrogen and oxygen.

[3] The shuttle booster rockets run on aluminium as the fuel and perchlorate as the oxidant.

[4] I find that I cannot tell the difference between the appearance of gases that are burning and hot gases that are luminous.

[5] I will look at this much more carefully when I deal with hydraulic machinery.

[6] It is common to use one index for expansions from initially superheated conditions and another for initially wet conditions.

[7] At one time this was called total heat but as it was not total and not heat there was a good case for changing it.

[8]  This unlikely looking figure is almost the maximum pressure that that is practical for steam plant.

[9] In the early days rockets failed as columns and folded.