__Theory of flow in isentropic nozzle:__

In a converging-diverging nozzle a large fraction of the thermal energy of the gases in the chamber is converted into kinetic energy. As will be explained, the gas pressure and temperature drop dramatically and the gas velocity can reach values in excess of two miles per second. This is a reversible, essentially isentropic flow process and its analysis is described here. If a nozzle inner wall has a flow obstruction or a wall protrusion (a piece of weld splatter or slag), then the kinetic gas energy is locally converted back into thermal energy essentially equal to the stagnation temperature and stagnation pressure in the chamber. Since this would lead quickly to a local overheating and failure of the wall, nozzle inner walls have to be smooth without any protrusion. Stagnation conditions can also occur at the leading edge of a jet vane or at the tip of a gas sampling tube inserted into the flow.

This equation also holds for any two points within the nozzle. When the chamber section is large compared to the nozzle throat section, the chamber velocity or nozzle approach velocity is comparatively small and the term ν_{1}^{2} can be neglected. The chamber temperature T1 is at the nozzle inlet and, under isentropic conditions, differs little from the stagnation temperature or (for a chemical rocket) from the combustion temperature. This leads to an important simplified expression of the exhaust velocity *v*_{2}*, *which is often used in the analysis.

It can be seen that the exhaust velocity of a nozzle is a function of the pressure ratio P_{1} /P_{2},* *the ratio of specific heats k, and the absolute temperature at the nozzle inlet T_{1}, as well as the gas constant R. Because the gas constant for any particular gas is inversely proportional to the molecular mass m, the exhaust velocity or the specific impulse is a function of the ratio of the absolute nozzle entrance temperature divided by the molecular mass, as is shown in Fig. 3-2. This ratio plays an important role in optimizing the mixture ratio in chemical rockets. Equations 2-14 and 2-15 give the relations between the velocity v_{2} , the thrust F, and the specific impulse I_{s}; it is plotted in Fig. 3-2 for two pressure ratios and three values of k. Equation 3-16 indicates that any increase in the gas temperature (usually caused by an increase in energy release) or any decrease of the molecular mass of the propellant (usually achieved by using light molecular mass gases rich in hydrogen content) will improve the performance of the rocket; that is, they will increase the specific impulse I_{s} or the exhaust velocity v_{2} or c and, thus, the performance of the vehicle. The influences of the pressure ratio across the nozzle P_{1} /P_{2} and of the specific heat ratio k are less pronounced. As can be seen from Fig. 3-2, performance increases with an increase of the pressure ratio; this ratio increases when the value of the chamber pressure Pl increases or when the exit pressure P_{2} decreases, corresponding to high altitude designs. The small influence of k-values is fortuitous because low molecular masses are found in diatomic or monatomic gases, which have the higher values of k.

For comparing specific impulse values from one rocket system to another or for evaluating the influence of various design parameters, the value of the pressure ratio must be standardized. A chamber pressure of 1000 psi (6.894 MPa) and an exit pressure of 1 atm (0.1013 MPa) are generally in use today.

For optimum expansion P2 = P3 and the effective exhaust velocity c (Eq. 2- 16) and the ideal rocket exhaust velocity are related, namely

and c can be substituted for v_{2} in Eqs. 3-15 and 3-16. For a fixed nozzle exit area ratio, and constant chamber pressure, this optimum condition occurs only at a particular altitude where the ambient pressure P_{3} happens to be equal to the nozzle exhaust pressure P_{2}. At all other altitudes c≠v_{2}. The maximum theoretical value of the nozzle outlet velocity is reached with an infinite expansion (exhausting into a vacuum).

This maximum theoretical exhaust velocity is finite, even though the pressure ratio is infinite, because it represents the finite thermal energy content of the fluid. Such an expansion does not happen, because, among other things, the temperature of many of the working medium species will fall below their liquefaction or the freezing points; thus they cease to be a gas and no longer contribute to the gas expansion.

A number of interesting deductions can be made from this example. Very high gas velocities (over 1 km/sec) can be obtained in rocket nozzles. The temperature drop of the combustion gases flowing through a rocket nozzle is appreciable. In the example given the temperature changed 1117°C in a relatively short distance. This should not be surprising, for the increase in the kinetic energy of the gases is derived from a decrease of the enthalpy, which in turn is proportional to the decrease in temperature. Because the exhaust gases are still very hot (1105 K) when leaving the nozzle, they contain considerable thermal energy not available for conversion into kinetic energy of the jet.

**Nozzles and choking:**

The usual configuration for a converging diverging (CD) nozzle is shown in the figure. Gas flows through the nozzle from a region of high pressure (usually referred to as the chamber) to one of low pressure(referred to as the ambient or tank). The chamber is usually big enough so that any flow velocities here are negligible. The pressure here is denoted by the symbol pc. Gas flows from the chamber into the converging portion of the nozzle, past the throat, through the diverging portion and then exhausts into the ambient as a jet. The pressure of the ambient is referred to as the ‘back pressure’ and given the symbol pb.

A simple example to get a basic feel for the behaviour of the nozzle imagines performing the simple experiment shown in figure 2. Here we use a converging diverging nozzle to connect two air cylinders. Cylinder A contains air at high pressure, and takes the place of the chamber. The CD nozzle exhausts this air into cylinder B, which takes the place of the tank.

Imagine you are controlling the pressure in cylinder B, and measuring the resulting mass flow rate through the nozzle. You may expect that the lower you make the pressure in B the more mass flow you’ll get through the nozzle. This is true, but only up to a point. If you lower the back pressure enough you come to a place where the flow rate suddenly stops increasing all together and it doesn’t matter how much lower you make the back pressure (even if you make it a vacuum) you can’t get any more mass flow out of the nozzle. We say that the nozzle has become ‘choked’. You could delay this behaviour by making the nozzle throat bigger (e.g. grey line) but eventually the same thing would happen. The nozzle will become choked even if you eliminated the throat altogether and just had a converging nozzle.

The reason for this behaviour has to do with the way the flows behave at Mach 1, i.e. when the flow speed reaches the speed of sound. In a steady internal flow (like a nozzle) the Mach number can only reach 1 at a minimum in the cross-sectional area. When the nozzle isn’t choked, the flow through it is entirely subsonic and, if you lower the back pressure a little, the flow goes faster and the flow rate increases. As you lower the back pressure further the flow speed at the throat eventually reaches the speed of sound (Mach 1). Any further lowering of the back pressure can’t accelerate the flow through the nozzle anymore, because that would entail moving the point where M=1 away from the throat where the area is a minimum, and so the flow gets stuck. The flow pattern downstream of the nozzle (in the diverging section and jet) can still change if you lower the back pressure further, but the mass flow rate is now fixed because the flow in the throat (and for that matter in the entire converging section) is now fixed too.

The changes in the flow pattern after the nozzle has become choked are not very important in our thought experiment because they don’t change the mass flow rate. They are, however, very important however if you were using this nozzle to accelerate the flow out of a jet engine or rocket and create propulsion, or if you just want to understand how high-speed flows work.

The flow pattern:

Figure 3a shows the flow through the nozzle when it is completely subsonic (i.e. the nozzle isn’t choked). The flow accelerates out of the chamber through the converging section, reaching its maximum (subsonic) speed at the throat. The flow then decelerates through the diverging section and exhausts into the ambient as a subsonic jet. Lowering the back pressure in this state increases the flow speed everywhere in the nozzle.

Lower it far enough and we eventually get to the situation shown in figure 3b. The flow pattern is exactly the same as in subsonic flow, except that the flow speed at the throat has just reached Mach 1. Flow through the nozzle is now choked since further reductions in the back pressure can’t move the point of M=1 away from the throat. However, the flow pattern in the diverging section does change as you lower the back pressure further.

As P_{b} is lowered below that needed to just choke the flow a region of supersonic flow forms just downstream of the throat. Unlike a subsonic flow, the supersonic flow accelerates as the area gets bigger. This region of supersonic acceleration is terminated by a normal shock wave. The shock wave produces a near-instantaneous deceleration of the flow to subsonic speed. This subsonic flow then decelerates through the remainder of the diverging section and exhausts as a subsonic jet. In this regime if you lower or raise the back pressure you increase or decrease the length of supersonic flow in the diverging section before the shock wave.

If you lower p_{b} enough you can extend the supersonic region all the way down the nozzle until the shock is sitting at the nozzle exit (figure 3d). Because you have a very long region of acceleration (the entire nozzle length) in this case the flow speed just before the shock will be very large in this case. However, after the shock the flow in the jet will still be subsonic.

Lowering the back pressure further causes the shock to bend out into the jet (figure 3e), and a complex pattern of shocks and reflections is set up in the jet which will now involve a mixture of subsonic and supersonic flow, or (if the back pressure is low enough) just supersonic flow. Because the shock is no longer perpendicular to the flow near the nozzle walls, it deflects it inward as it leaves the exit producing an initially contracting jet. We refer to this as over-expanded flow because in this case the pressure at the nozzle exit is lowers than that in the ambient (the back pressure) – i.e. the flow has been expanded by the nozzle too much.

A further lowering of the back pressure changes and weakens the wave pattern in the jet. Eventually we will have lowered the back pressure enough so that it is now equal to the pressure at the nozzle exit. In this case, the waves in the jet disappear altogether (figure 3f), and the jet will be uniformly supersonic. This situation, since it is often desirable, is referred to as the ‘design condition’.

Finally, if we lower the back pressure even further we will create a new imbalance between the exit and back pressures (exit pressure greater than back pressure), figure 3g. In this situation (called ‘under expanded’) what we call expansion waves (that produce gradual turning and acceleration in the jet) form at the nozzle exit, initially turning the flow at the jet edges outward in a plume and setting up a different type of complex wave pattern.

The pressure distribution in the nozzle

A plot of the pressure distribution along the nozzle (figure 4) provides a good way of summarizing its behaviour. To understand how the pressure behaves you have to remember only a few basic rules

When the flow accelerates (sub or supersonically) the pressure drops

The pressure rises instantaneously across a shock

The pressure throughout the jet is always the same as the ambient (i.e. the back pressure) unless the jet is supersonic and there are shocks or expansion waves in the jet to produce pressure differences.

The pressure falls across an expansion wave.

The labels on figure 4 indicate the back pressure and pressure distribution for each of the flow regimes illustrated in figure 3. Notice how, once the flow is choked, the pressure distribution in the converging section doesn’t change with the back pressure at all.

**Nozzle Flow and Throat Condition**

The required nozzle area decreases to a minimum (at 1.130 MPa or 164 psi pressure in the previous example) and then increases again. Nozzles of this type (often called De Laval nozzles after their inventor) consist of a convergent section followed by a divergent section. From the continuity equation, the area is inversely proportional to the ratio *v/V. *This quantity has also been plotted in Fig. 3-3. There is a maximum in the curve of v/V because at first the velocity increases at a greater rate than the specific volume; however, in the divergent section, the specific volume increases at a greater rate. The minimum nozzle area is called the *throat area. *The ratio of the nozzle exit area A_{2} to the throat area A_{t} is called the nozzle area expansion ratio and is designated by the Greek letter e. It is an important nozzle design parameter.

The throat pressure Pt for which the isentropic mass flow rate is a maximum is called the *critical pressure. *Typical values of this critical pressure ratio range between 0.53 and 0.57. The flow through a specified rocket nozzle with a given inlet condition is less than the maximum if the pressure ratio is larger than that given by Eq. 3-20. However, note that this ratio is not that across the entire nozzle and that the maximum flow or choking condition (explained below) is always established internally at the throat and not at the exit plane. The nozzle inlet pressure is very close to the chamber stagnation pressure, except in narrow combustion chambers where there is an appreciable drop in pressure from the injector region to the nozzle entrance region. This is discussed in Section 3.5. At the point of critical pressure, namely the throat, the Mach number is one and the values of the specific volume and temperature can be obtained from Eqs. 3-7 and 3-12.

In Eq. 3-22 the nozzle inlet temperature T_{1} is very close to the combustion temperature and hence closes to the nozzle flow stagnation temperature T_{0}. At the critical point there is only a mild change of these properties. Take for example a gas with k = 1.2; the critical pressure ratio is about 0.56 (which means that P_{t} equals almost half of the chamber pressure P_{1}); the temperature drops only slightly (T_{t} = 0.91T_{1}), and the specific volume expands by over 60% (V_{t} = 1.61V_{1}). From Eqs. 3-15, 3-20, and 3-22, the critical or throat velocity V_{t} is obtained:

The first version of this equation permits the throat velocity to be calculated directly from the nozzle inlet conditions without any of the throat conditions being known. At the nozzle throat the critical velocity is clearly also the sonic velocity. The divergent portion of the nozzle permits further decreases in pressure and increases in velocity under supersonic conditions. If the nozzle is cut off at the throat section, the exit gas velocity is sonic and the flow rate remains a maximum. The sonic and supersonic flow condition can be attained only if the critical pressure prevails at the throat, that is, if P2/Pl is equal to or less than the quantity defined by Eq. 3-20. There are, therefore, three different types of nozzles: subsonic, sonic, and supersonic, and these are described in Table 3-1.

The supersonic nozzle is the one used for rockets. It achieves a high degree of conversion of enthalpy to kinetic energy. The ratio between the inlet and exit pressures in all rockets is sufficiently large to induce supersonic flow. Only if the absolute chamber pressure drops below approximately 1.78 atm will there be subsonic flow in the divergent portion of the nozzle during sea-level operation. This condition occurs for a very short time during the start and stop transients. The velocity of sound is equal to the propagation speed of an elastic pressure wave within the medium, sound being an infinitesimal pressure wave. If, therefore, sonic velocity is reached at any point within a steady flow system, it is impossible for a pressure disturbance to travel past the location of sonic or supersonic flow. Thus, any partial obstruction or disturbance of the flow downstream of the nozzle throat with sonic flow has no influence on the throat or upstream of it, provided that the disturbance does not raise the downstream pressure above its critical value. It is not possible to increase the throat velocity or the flow rate in the nozzle by further lowering the exit pressure or even evacuating the exhaust section. This important condition is often described as choking the flow. It is always established at the throat and not the nozzle exit plane. Choked flow through the critical section of a supersonic nozzle may be derived from Eqs. 3-3, 3-21, and 3-23. It is equal to the mass flow at any section within the nozzle.

The mass flow through a rocket nozzle is therefore proportional to the throat area A t and the chamber (stagnation) pressure P_{1}; it is also inversely proportional to the square root of T/m and a function of the gas properties. For supersonic nozzle the ratio between the throat and any downstream area at which a pressure P_{x} prevails can be expressed as a function of the pressure ratio and the ratio of specific heats, by using Eqs. 3-4, 3-16, 3-21, and 3-23, as follows:

**Under-and Over-Expanded Nozzles**

An *under-expanded nozzle *discharges the fluid at an exit pressure greater than the external pressure because the exit area is too small for an optimum area ratio. The expansion of the fluid is therefore incomplete within the nozzle, and must take place outside. The nozzle exit pressure is higher than the local atmospheric pressure.

In an *over-expanded nozzle *the fluid attains a lower exit pressure than the atmosphere as it has an exit area too large for optimum. The phenomenon of over-expansion for a supersonic nozzle is shown in Fig. 3-9, with typical pressure measurements of superheated steam along the nozzle axis and different back pressures or pressure ratios. Curve *AB *shows the variation of pressure with the optimum back pressure corresponding to the area ratio. Curves *AC* and *AD *show the variation of pressure along the axis for increasingly higher external pressures. The expansion within the nozzle proceeds normally for the initial portion of the nozzle. At point I on curve AD, for example, the pressure is lower than the exit pressure and a sudden rise in pressure takes place which is accompanied by the separation of the flow from the walls (separation is described later).

The non-ideal behaviour of nozzles is strongly influenced by the presence of compression waves or shock waves inside the diverging nozzle section, which are strong compression discontinuities and exist only in supersonic flow. The sudden pressure rise in the curve ID is such a compression wave. Expansion waves, also strictly supersonic phenomena, match the flow from a nozzle exit to lower ambient pressures.

The different possible flow conditions in a supersonic nozzle are as follows:

1. When the external pressure P_{3} is below the nozzle exit pressure P_{2}, the nozzle will flow full but will have external expansion waves at its exit (i.e., under-expansion). The expansion of the gas inside the nozzle is incomplete and the value of *C*_{F}* *and I_{s}, will be less than at optimum expansion.

2. For external pressures P_{3} slightly higher than the nozzle exit pressure P_{2}, the nozzle will continue to flow full. This occurs until P_{2} reaches a value between about 25 and 40% of P_{3}. The expansion is somewhat inefficient and C_{F} and I_{s} will have lower values than an optimum nozzle would have. Shock waves will exist outside the nozzle exit section.

3. For higher external pressures, separation of the flow will take place inside the divergent portion of the nozzle. The diameter of the supersonic jet will be smaller than the nozzle exit diameter. With steady flow, separation is typically axially symmetric. Figs. 3-10 and 3-11 show diagrams of separated flows. The axial location of the separation plane depends on the local pressure and the wall contour. The point of separation travels downstream with decreasing external pressure. At the nozzle exit the flow in the center portion remains supersonic, but is surrounded by an annular shaped section of subsonic flow. There is a discontinuity at the separation location and the thrust is reduced, compared to a nozzle that would have been cut off at the separation plane. Shock waves exist outside the nozzle in the external plume.

4. For nozzles in which the exit pressure is just below the value of the inlet pressure, the pressure ratio is below the critical pressure ratio (as defined by Eq. 3-20) and subsonic flow prevails throughout the entire nozzle. This condition occurs normally in rocket nozzles for a short time during the start and stop transients.

The method for estimating pressure at the location of the separation plane inside the diverging section of a supersonic nozzle has usually been empirical. Reference 3-4 shows separation regions based on collected data for several dozen actual conical and bell-shaped nozzles during separation. Reference 3-5 describes a variety of nozzles, their behaviour, and methods used to estimate the location and the pressure at separation. Actual values of pressure for the over-expanded and under-expanded regimes described above are functions of the specific heat ratio and the area ratio (see Ref. 3-1).

The axial thrust direction is not usually altered by separation, because a steady flow usually separates uniformly over a cross-section in a divergent nozzle cone of conventional rocket design. During transients, such as start and stop, the separation may not be axially symmetric and may cause momentary but large side forces on the nozzle. During a normal sea-level transient of a large rocket nozzle (before the chamber pressure reaches its full value) some momentary flow oscillations and non-symmetric separation of the jet can occur during over-expanded flow operation. Reference 3-4 shows that the magnitude and direction of transient side forces can change rapidly and erratically. The resulting side forces can be large and have caused failures of nozzle exit cone structures and thrust vector control gimbal actuators. References 3-5 and 3-6 discuss techniques for estimating these side forces.

When the flow separates, as it does in a highly over-expanded nozzle, the thrust coefficient C_{F} can be estimated if the point of separation in the nozzle is known. Thus, C_{F} can be determined for an equivalent smaller nozzle with an exit area equal to that at the point of separation. The effect of separation is to increase the thrust and the thrust coefficient over the value that they would have if separation had not occurred. Thus, with separated gas flow, a nozzle designed for high altitude (large value of e) would have a larger thrust at sea level than expected, but not as good as an optimum nozzle; in this case separation may actually be desirable. With separated flow a large and usually heavy portion of the nozzle is not utilized and the nozzle is bulkier and longer than necessary. The added engine weight and size decrease flight performance.

Designers therefore select an area ratio that will not cause separation. Because of uneven flow separation and potentially destructive side loads, sea-level static tests of an upper stage or a space propulsion system with a high area ratio over-expanded nozzle are usually avoided; instead, a sea-level test nozzle with a much smaller area ratio is substituted. However, actual and simulated altitude testing (in an altitude test facility similar to the one described in Chapter 20) would be done with a nozzle having the correct large area ratio. The ideal solution that avoids separation at low altitudes and has high values of C_{F} at high altitudes is a nozzle that changes area ratio in flight. This is discussed at the end of this section.

For most applications, the rocket system has to operate over a range of altitudes; for a fixed chamber pressure this implies a range of nozzle pressure ratios. The condition of optimum expansion (P_{2} = P_{3}) occurs only at one altitude, and a nozzle with a fixed area ratio is therefore operating much of the time at either over-expanded or under-expanded conditions. The best nozzle for such an application is not necessarily one that gives optimum nozzle gas expansion, but one that gives the largest vehicle flight performance (say, total impulse, or specific impulse, or range, or payload); it can often be related to a time average over the powered flight trajectory.

**Nozzle efficiency**

Obviously nozzles are not perfectly efficient and there are several ways to define the nozzle efficiency. One of the effective ways is to define the efficiency as the ratio of the energy converted to kinetic energy and the total potential energy could be converted to kinetic energy. The total energy that can be converted is during isentropic process is

where is the enthalpy if the flow was isentropic. The actual energy that was used is

The efficiency can be defined as

The typical efficiency of nozzle is ranged between 0.9 to 0.99. In the literature some define also velocity coefficient as the ratio of the actual velocity to the ideal velocity,

There is another less used definition which referred as the coefficient of discharge as the ratio of the actual mass rate to the ideal mass flow rate.

##### Ejector nozzles

The simpler of the two is the ejector nozzle, which creates an effective nozzle through a secondary airflow and spring-loaded petals. At subsonic speeds, the airflow constricts the exhaust to a convergent shape. As the aircraft speeds up, the two nozzles dilate, which allows the exhaust to form a convergent-divergent shape, speeding the exhaust gasses past Mach 1. More complex engines can actually use a tertiary airflow to reduce exit area at very low speeds. Advantages of the ejector nozzle are relative simplicity and reliability. Disadvantages are average performance (compared to the other nozzle type) and relatively high drag due to the secondary airflow.

**Thrust reversal**

Thrust reversal, also called reverse thrust, is the temporary diversion of an aircraft engine‘s exhaust so that the exhaust produced is directed forward, rather than aft. This acts against the forward travel of the aircraft, providing deceleration. Thrust reverser systems are featured on many jet aircraft to help slow down just after touch-down, reducing wear on the brakes and enabling shorter landing distances. Such devices affect the aircraft significantly and are considered important for safe operation by airlines. There have been accidents involving thrust reversal systems.

Reverse thrust is also available on many propeller-driven aircraft through reversing the controllable pitch propellers to a negative angle. The equivalent concept for a ship is called astern propulsion.

**Nozzle efficiency**

In a large number of turbo-machinery components the flow process can be regarded as a purely nozzle flow in which the fluid receives an acceleration as a result of a drop in pressure. Such a nozzle flow occurs at entry to all turbo-machines and in the stationary blade rows in turbines. In axial machines the expansion at entry is assisted by a row of stationary blades (called guide vanes in compressors and nozzles in turbines) which direct the fluid on to the rotor with a large swirl angle. Centrifugal compressors and pumps, on the other hand, often have no such provision for flow guidance but there is still a velocity increase obtained from a contraction in entry flow area.

Figure 2.12a shows the process on a Mollier diagram, the expansion proceeding from state 1 to state 2. It is assumed that the process is steady and adiabatic such that h_{01} = h_{02}

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