Chapter 13 Part 3 the Convergent-divergent Nozzle

10/4/13 Chapter 13 Part 3 The convergent-divergent nozzle

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Chapter 13 Part 3 The convergent-d iv ergent nozzl e IntroductionIn all probability before 1897 no one knew for certain that there are two regimes of flow for gases andvapours and only writers of science fiction thought in terms of travelling at speeds of thousands of milesan hour. The low-speed regime is common in nature but I think that steady high-speed flow of gasesdoes not exist in nature. High-speed flow came about as a consequence of man’s ingenuity. It is likelythat the first person to produce a steady supersonic flow was Gustaf de Laval (1845-1913) whodeveloped, in 1897, a convergent-divergent nozzle for a small steam turbine for use in a creamery. Hisinvention paved the way steam powered turbo-alternators, turbine powered flight and rocketry. It isimportant.

We have looked at the engineering applications of theflow through pipes of uniform diameter and now wehave to look at nozzles that are short (by comparisonwith a delivery pipe.) with changes in cross-sectionalarea. See figure 13-4. This will lead to more applications of forms of theenergy equation together with continuity andequations of state and, inevitably, lots ofmanipulation of mathematical equations. But before Ido that I want to look at why a gas behaves as it doesin a convergent-divergent nozzle.

The mode of operation of a convergent-divergent nozzleI learnt about nozzles from the mathematical approach and it always seemed to me to be less thansatisfactory. Whilst I knew how convergent-divergent nozzles behaved I did not know why theybehaved that way. When I came to write this text I thought I ought really to make an attempt to putthat right. Those who first sought to make steam turbines were also the first to have a large steady supply of anelastic medium that is very like a gas, steam. They soon found themselves using nozzles to producehigh-speed flow and they started by using convergent nozzles and they mostly still do. This was theintuitive design with its forerunner in use in hydraulic machinery. They soon found that whilst theycould increase the speed of the jet formed by a given convergent nozzle by increasing the supplypressure, no comparable increase could be produced by reducing the back-pressure. They described thenozzles as “choked”. It must have been totally counter-intuitive to find that the fitting of a divergentcone 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? Iwant 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 iscomposed 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 thecontainer in which they are enclosed. The scale of the structure of a gas is indicated by the followingfigures. 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 themolecules 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 eachsecond 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 anozzle, whether it is convergent or convergent-divergent, a thermodynamic process takes place. Bythermodynamic I mean that some of the random kinetic energy stored in the flowing gas is converted tomechanical energy in the form of the kinetic energy in a jet. It is mechanical energy because it has onedirection. In de Laval’s turbine this kinetic energy was extracted from the flowing jet of steam emergingfrom the nozzle by a row of blades on a spinning rotor that reduced the speed of the jet in a controlledmanner to a small fraction of its initial value. The mechanical energy was taken from the rotor through ashaft to drive machinery. What is important about this is that the process that takes place in the nozzleis 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 ina 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 pistonface. When the piston moves, the rebounding molecules lose energy in every collision and, as the pistoncontinues to move, the stock of energy in the molecules, that is the internal energy in thermodynamicterms, 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 theinternal energy, that is the stock of random kinetic energy in the molecules, is transferred to the kineticenergy 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 dropin pressure, a drop in temperature, an increase in volume and the formation of a jet that contains kineticenergy. It seems to me that it is impossible to avoid the observation that whatever goes on to producethese effects is a progressive process. The ways in which the properties change along the nozzle areultimately determined by the shape of the nozzle and the primary variable is the area of cross-sectionand, if the process taking place in the gas is to be continuous, that is, free from sudden changes, the areamust 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 tothe conclusion that I would have to think of the gas as flowing in lots and lots of thin layers. In figure13-5 I have drawn representations of these layers but only a very few compared with the number that Ireally imagine. These thin layers may be thin in the context of the nozzle but they are very thick whencompared 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, infact, create these properties. This must mean that, at the surface where the two layers abut, moleculesfrom upstream collide with molecules downstream and, as the upstream molecules have higher meanspeeds than those downstream, exert an accelerating force on the downstream layer to increase thespeed of its mass centre. This is a thermodynamic process in which random kinetic energy in theupstream 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 speedof the molecules in the gas. As it is this mean speed that determines temperature and pressure there willbe 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 thinlayer as the speed of the mass centre of the gas in the layer approaches the mean speed of the moleculesas it must do at some point along the nozzle. Then the layer downstream of it is moving away at thespeed of the molecules in the upstream layer and even though collisions continue there is no net transferof kinetic energy or any net force. The flow will have reached a maximum speed for a convergent nozzleand 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°Cand 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 involvessideways expansion. In the convergence the gas has been expanding even though the duct has beenreducing in area of cross-section. The expansion is in the direction of motion. If the duct now becomesdivergent the gas can expand sideways and this will reduce the pressure, the mean speed of themolecules and the temperature. Then the layer of gas that is moving at the same speed as the meanspeed of its molecules will be in contact with a layer of gas at lower pressure, lower temperature andwith a lower mean speed of its molecules. Then the downstream layer will accelerate as before to give alayer moving faster than the mean speed of its molecules. As the gas advances into the divergence thespeed of the mass centres of successive layers will increase but the mean speed of the molecules insuccessive layers will fall. So, in the convergent-divergent nozzle, the speed of the gas can increase from end to end, the meanspeed of the molecules in the molecular structure of the gas will fall from end to end and at some pointthe speed of flow will equal the mean speed of the molecules. We have seen that this is a special pointwhere the duct must change from convergence to divergence. Then the flow in the convergencedepends 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 atthe throat. This immediately raises the issue of the use of the Mach number in connection with nozzles. In theearly part of the 20th century there was great optimism that the use of non-dimensional groups ofphysical 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 ofErnst Mach (1838-1916). It is the ratio of two speeds and is most commonly used in connection with the flow round a body suchas 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 thecollective 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 thesteady 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 samepoint. 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 willbe 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 nozzleWhen one sets about the use of a convergent-divergent nozzle in engineering it is necessary to considerreal 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 notdepart much from streamlined flow and that any effect of friction is likely to be small. We also foundthat the flow in the divergence was likely to break down. However, in the divergence of a Venturi-meterwhether used with liquid or gas, the flow is slowing down and pressure energy is being recovered. It is adiffuser and there is a best angle for the divergence to give the best pay-off between increased loss tofriction when using a long divergence and increased loss in random internal motion caused by thebreakdown in the streamlined flow. In the high-speed convergent-divergent nozzle the flow in thedivergence increases in speed in the divergence and now both high friction loss and high loss to internalrandom motion cannot be set one against the other. Both will occur and engineers will be called upon todevelop shapes for the divergence that are as efficient as possible in performing its function ofextracting energy from the internal structure of the gas as orderly kinetic energy. This means thatengineers 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 andthen found expressions relating the pressure variation, the volume variation and the temperaturevariation 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 swingfor ever if there were to be no friction because we know in our bones that friction always involves anunrecoverable loss of energy. Yet, throughout that eternity, the bob would be continually exchangingpotential energy for kinetic energy . So it is not just about having no friction at the bearings or no airfriction, it is also about having no energy loss in this never-ending exchange of energy as well. When thisidea is applied to an expanding gas the most efficient process would involve no mechanical friction suchas that between the gas and the surfaces of the nozzles but there must be no viscous loss and any heatexchange that cannot be reversed. When we seek our ideal expansion we have no alternative but tosuppose that there is no heat exchange because a heat transfer that can take place in either direction asrequired 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 takeplace without heat exchange with the surroundings. In the terminology of thermodynamics it will be areversible, adiabatic expansion. If the flowing fluid were to be air we could then use all the usual gas laws and, most importantly, usethe 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 formwith some new index is over-whelming as we shall see.



Now we have to think out how these ideas have beenapplied to a convergent-divergent nozzle. Whilst we doso remember the fundamental rule of science forengineers “ if you are creating a yard-stick make surethat you know what the yard-stick depends on so thatyou know when you depart from it”. This rule comesinto operation when you look at figure 13-5 and see thatthe flow-pattern is not made up of parallel lines crossingplane sections but curved lines crossing convex andconcave sections. A decision is needed and that decisionis 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 anozzle efficiency. So we need to treat the convergent-divergent nozzle as having one-dimensional, adiabatic, frictionlessflow. 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 theconvergent-divergent nozzle. The first, chronologically, uses the physical properties of the gas, thesecond, uses Mach numbers. I am attempting to construct an empirical science for engineers andengineers want their analyses that are based on physics to end up relating quantities that can bemeasured or can be calculated from quantities that can be measured. They also want the analysis to besimple and transparent. Given the problem of the non-linearity of Mach numbers with speed some mayfind one treatment more convenient that the other. I will give both. Modelling using propertiesI will start with the treatment using properties. The basic argument is that, in a nozzle, the work doneby the gas as it expands goes to increase the kinetic energy of the gas in one direction. This means that athermodynamic process takes place in which some of the random energy that is stored in the molecularstructure is converted to mechanical work in the form of kinetic energy. This is done in a flow processand we must write :-

.

In figure 13-6 I have drawn a pressure-volume graph fora fully resisted adiabatic expansion of air. The terms inthe 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 under2®B. Summing these as in the equation shows that thework done on the gas as it flows through the nozzle isthe area A®1®2®B®A. Then it is evident that:-



. This is a regular integral in

thermodynamics and the equation becomes :-

In fact this is a steady flow of air from some regionwhere the velocity can be regarded as very small andthen the temperature is the stagnation temperature. Ihave shown this in principle in figure 13-7. Then themeasurable stagnation conditions are . Now, if we choose to consider an expansion fromstagnation 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 astagnation 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 thepoint where the pressure is at the same time.

We have from continuity or . This requires another manipulation

: .



Then :- and .

Whilst Iamplottinggraphs Ican addthe graphsof

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 frominlet 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 needsto be explored just to get a feel for the numbers. What is really needed is a graph showing to give theproperty distribution in a nozzle with a practical profile.I made a couple of attempts just to get practicalshapes and start conditions. In the end I settled to look at a nozzle having a throat of 12 mm diameterwith air supplied at 10 bar gauge and 100°C with a mass flow of 0.26 kg/s. What I thought that I wouldlike to know is the way that the velocity and the pressure would change along the length of a nozzlehaving the sort of profile shown in figure 13-7. I let my nozzle have an inlet radius of 15 mm and adivergence of 16° included angle for no other reason than it is much the same as that for the divergencein 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 thestagnation conditions. At one time this would have meant that the calculations for drawing the graphswould have been very time consuming. That problem has disappeared with the emergence ofmathematics packages on computers. We can draw as many graphs as we like almost instantly.



I think that thisgraph gives a goodidea of what goeson in a convergent-divergent nozzle. Itis surprising howshort thedivergence needs tobe for the Machnumber to reach 2and then how muchmore of thedivergence isrequired to reachMach 4 but theincrease in Machnumber from 2 to 4takes place atabsolute pressuresless than 1.5 bar.This reflects the

shape of the curve in graph 13-2 where small changes in low pressures produce large changes involume. Modelling the flow using the Mach numberWe 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 sectionsand equate them. The forms of energy are internal energy, kinetic energy, pressure energy and, to bepedantic, potential energy. In this analysis it is better to work, not in head (energy per unit weight) aswe 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 thatthis equation can be applied to any two points in the flow. Then, if as before, we refer to the stagnationconditions 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 shallend 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 andmass 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 wayas yet to link the values of the properties with the nozzle. We really want some link with the area ofcross-section so that, if we knew the profile of the nozzle, we can plot properties against length. Wecannot continue with the stagnation conditions because the idea of having a cross-section at thestagnation conditions is meaningless. But we can use the throat where we not only have the minimumdiameter but also conditions that can be related to the stagnation conditions. This leads us to find aratio 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 convenientform that is known and each of the ratios is expressed in terms of and . In the lower part I havesubstituted 1.4 for to give much more manageable expressions for air. They can easily be plottedusing, say, Mathcad and the result is shown in graph 13-11 and in graph 13-12. It is also easy toevaluate 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 airthrough a convergent-divergent nozzle. There will be similar graphs for other gases and there must besome similar graph, probably based on an approximation, for steam although that is outside of the scopeof 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.

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



Assessment of these two methodsI think that it is right to put these into context. The properties treatment would have been timeconsuming 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 independentvariable. This table would have reduced the calculating time significantly. But the digital computer hasreversed 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 anengineer can make a good start at designing a nozzle. However it makes sound sense to look at thenozzle 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 willserve my purpose. They are the nozzles in a steam turbine, the nozzles of a rocket engine and the intakeduct 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[8]

(2,500 psi) and they all exhaust at something like0.1 bar. Of course there are other smallerturbines working with steam at lower pressures.The field of application is very extensive and thestudy of these turbines is a vast undertaking.Nevertheless there are a few basic principles thatcan prove to be useful. A range of pressure from170 bar to 0.1 bar is an extreme range and graph13-15 shows more or less how the pressurevaries with volume during the expansion in thesteam turbine. The possible area under thiscurve 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 fromthe steam. Steam turbines do not make one continuous expansion, they often make the expansion in three or fourstages. This gives two engineering advantages, the first is to give opportunities for improvements inefficiency by heating the steam between stages and the second is that the thickness of the casings canbe 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 powerproduced 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 thetotal area under an expansion into four equal areas. It immediately becomes evident that the first stagewill involve a large pressure drop and small volumes of steam and for the last stage will be at lowpressure and with very large volumes of steam.We have seen that sonic speeds are associated with pressure ratios of 50+% and the pressure ratio onthe first stage have about this value. As a result steam turbines often use a de Laval nozzle to extractwork from the steam as kinetic energy and then transfer this kinetic energy to one or more rows ofrotating blades and then the steam is expanded continuously in many stages each consisting of a fixedrow of blades followed by a row of moving blades through the remaining three stages. In the first stagethe blades operate at constant pressure and the best way to use the steam is to make a big drop inpressure in a ring of de Laval nozzles to start the expansion and have the blades operate in a drum at amuch lower pressure than the boiler pressure. This is called an impulse stage and we now have theexpressions that might let us design a nozzle.



Now weneed to seewhat themechanicsof animpulsestage looklike. Ithink thatit isimportantto drawthemreasonablyaccuratelyand nothide

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. Theseblades can be robust. The moving blades must also be robust just to stand the forces of very high-speedsteam 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 offobliquely. This is done to ensure that the jet of steam has no unguided distance in which to start to mixand so lose kinetic energy to random motion and the gap between the exit from the fixed row and entryto the moving row is kept to a minimum. There is a price to be paid for cropping the nozzle in this waythat is that the jet is diverted slightly as it flows with guidance on only one side but this is preferable tohaving a gap. The nozzles are created in an annular ring and this ring has to withstand the very high pressure. It is noteasy to ensure the surface finish of the nozzles. The engineering constraints of this design limit thevalue of attempts to refine the physics of the convergent-divergent nozzle as I have laid it out in thistext. Nevertheless practical nozzles have efficiencies of 95% and higher.

Rockets are very simple engines. I have attempted todraw the essential features in figure 13-11. The design isdominated by the need to minimise its weight. Fuel andoxidant are pumped at very high rates into thecombustion chamber where they combine to produce anenormous flow of gas. The creation of this gas and theresistance to its escape through the nozzle produces ahigh pressure in the combustion chamber. The gas flowsout through the convergent-divergent nozzle under thepressure difference between the gas in the combustionchamber and the outside conditions. I have also drawn aseries of arrows to indicate the way in which pressure isexerted on the inner surfaces much the same way as iscommonly used to show pressure distribution over anaerofoil. All over the outer surface the pressure iswhatever it may be around the engine. The pressureacross the exit plane is not so easily known and I mayhave to deal with it separately.



I explained how these pressure distributions give rise to a net force in chapter 5 in the text associatedwith figures 5-14 and 5-15a and b. In essence there is a very large upward force on the top of thecombustion chamber that is not balanced by a downward force on the convergent part of the nozzle sothat there is a net upward force on the combustion chamber exerted by the gas. This force is exerted onthe 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 onthe bell-shaped divergence that everywhere has a vertical component and there is a net upward forceproduced on the divergence. That force is exerted through the combustion chamber on to the enginemounting 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 thatit would be desirable to complete the combustion in the combustion chamber and, for this to bepossible, the combustion chamber would have to be quite large. As it is a pressure vessel, if it is to belarge it would also be heavy and there is a great incentive to keep its size to a minimum. If thecombustion chamber were to be too small the combustion would take place partially in the nozzle. Ihave said that I cannot tell the difference between a hot luminous gas and a burning gas so just lookingat pictures of rockets does not help. A compromise must be made between weight and performance andthat compromise is affected by other factors. Photographs of the inner surfaces of rocket nozzles suggest that they are lined with refractory materialand it is not really smooth and, even if it were to be, it is not likely to survive a protracted burnunscathed. The booster rocket works in just the same way as a liquid fuelled engine but now the upward force onthe casing containing the burning fuel is exerted either on the fuel itself if it is not porous to gas or onthe inner surface of the top of the casing if it is porous. So instead of exerting a force on the bottom ofthe 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 realityis very different. Nevertheless the model is an enormous help in getting a basic understanding of rocketnozzles and of possible ways to store experimental data. The supersonic gas turbine engine is now used only in military applications. The gas turbine enginecannot operate with supersonic flow anywhere over its blading but it is used to power supersonicaeroplanes. The inlet conditions for the Olympus engines used on the Concorde require inlet conditionsthat are effectively those of ambient conditions at sea level. This may not be accidental because thoseengines, running on gas, also power electricity generators. An aeroplane capable of supersonic speeds must also fly subsonically over a range of speeds up to 1,000km/h. Then it must go on to fly at perhaps 3,000 km/h. There is only one way to cope with this andthat is to fit intake ducts that can be used in conjunction with an engine management system to changethe shape of the duct from one suitable for low speed to one suitable for high speed. At low speed it isessentially a straight through duct but at high speed the duct is a convergent-divergent nozzle with airentering it supersonic speed, possibly at low pressure and low temperature, and leaving the duct to enterthe engine at about 1 bar absolute and about 300°K. The duct is rectangular in cross-section as the onlypossible design. In the supersonic configuration the air slows down in the convergence and slows downin the divergence with a steady increase in pressure. Surprisingly the net effect is that the ducts producea very substantial forwards force that contributes about 40% of the total thrust. This sounds odd but weseldom ask the question “ how is the force produced by a jet engine and where is it exerted on the frameof 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 thepressure at exit from the meter, or both, will alter the flow through the meter and this is how the meterworks. For a convergent divergent nozzle with an overall pressure drop that is great enough for supersonic flowto occur throughout its divergence a change in the inlet pressure will alter the flow but the exit pressurecan change quite significantly without affecting the flow. Providing that the flow in the divergence iswholly supersonic the flow can emerge from the nozzle either below the back-pressure or above theback-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 eitherexpand or contract. This is called the “correct” and sometimes it is convenient to use this concept. The concept of an efficiency for a nozzleThe 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 inthe effect of friction on the expansion. That friction will be have two components, the friction betweenthe gas and the solid boundaries of the nozzle and the internal friction stemming from irregularities inthe flow. The combined effect will be to reduce the amount of the original internal energy that isextracted and converted to kinetic energy of the flow in the emerging jet. We must decide what effectthis has on the flow. If we choose stagnation conditions and a diameter for the throat, according to our model, the pressureat exit for supersonic flow in the divergence will be determined by the diameter at exit. Using our modelwe 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 itcan 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 theamount of energy transferred from the internal motion of the molecules of the gas to the kinetic energyof the mass centre of the gas. So, if the kinetic energy is reduced from that produced in a reversibleadiabatic flow, the result must be an increase in the absolute temperature at the end of expansion overthat 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 ajet of fast moving gas is the means to produce that force. For rockets one can equate force to themomentum of the jet and then a suitable efficiency would be the ratio of forces. But the only way toimprove the ratio of the forces is to improve the process by which the kinetic energy is produced so onenozzle efficiency will do for both. We have said in several contexts that there will be very little loss in convergent flow but significant lossin a divergence. This will be true of the convergent-divergent nozzle. There is an advantage to be gainedif the flow up to the throat is treated as reversible and all the loss presumed to take place in thedivergence. Then we can relate the throat diameter to stagnation conditions and mass flow and thesequantities 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 arescores 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.

Documents

Chapter 13 Part 3 the Convergent-divergent Nozzle