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
[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.