Velocity

The nozzle exit velocity (at ) can be solved for from Eq. 3–2:

3–15a

As stated, this relation strictly applies when there are no heat losses. This equation also holds between any two locations within the nozzle, but hereafter subscripts 1 and 2 will only designate nozzle inlet and exit conditions. For constant k the above expression may be rewritten with the aid of Eqs. 3–6 and 3–7:

3–15b

When the chamber cross section is large compared the nozzle throat, the chamber velocity or nozzle entrance velocity is comparatively small and the term may be neglected above. The chamber temperature T₁, which is located at the nozzle inlet, under isentropic conditions differs little from the stagnation temperature or (for chemical rocket propulsion) from the combustion temperature T₀. Hence we arrive at important equivalent simplified expressions of the velocity , ones commonly used for analysis:

3–16

As can be seen above, the gas velocity exhausting from an ideal nozzle is a function of the prevailing pressure ratio p₁/p₂, the ratio of specific heats k, and the absolute temperature at the nozzle inlet T₁, as well as of the gas constant R. Because the gas constant for any particular gas is inversely proportional to the molecular mass , exhaust velocities or their corresponding specific impulses strongly depend on the ratio of the absolute nozzle entrance temperature (which is close to T₀) divided by the average molecular mass of the exhaust gases, as is shown in Fig. 3–2. The fraction T₀/ plays an important role in optimizing the mixture ratio (oxidizer to fuel flow) in chemical rocket propulsion defined in Eq. 6–1.

Equations 2–13 and 2–16 give relations between the thrust F and the velocities and c, and the corresponding specific impulse I_s which is plotted in Fig. 3–2 for two pressure ratios and three values of k. Equation 3–16 clearly shows that any increase in entrance gas temperatures (arising from increases in chamber energy releases) and/or any decrease of propellant molecular mass (commonly achieved from using low molecular mass gas mixtures rich in hydrogen) will enhance the ratio T₀/; that is, they will increase the specific impulse I_s through increases in the exhaust velocity and, thus, the performance of the rocket vehicle. Influences of pressure ratio across the nozzle p₁/p₂ and of specific heat ratio k are less pronounced. As can be seen from Fig. 3–2, performance does increase with an increase of the pressure ratio; this ratio increases when the value of the chamber pressure p₁ rises and/or when the exit pressure p₂ decreases, corresponding to high altitude designs. The small influence of k values is fortuitous because low molecular masses, found in many diatomic and monatomic gases, correspond to the higher values of k.

Values of the pressure ratio must be standardized when comparing specific impulses from one rocket propulsion system to another or for evaluating the influence of various design parameters. Presently, a chamber pressure of 1000 psia (6.894 MPa) and an exit pressure of 1 atm (0.1013 MPa) are used as standard reference for such comparisons, or .

For optimum expansion and the effective exhaust velocity c (Eq. 2–15) and the ideal rocket exhaust velocity become equal, namely,

3–17

Thus, here c can be substituted for in Eqs. 3.–15 and 3–16. For a fixed nozzle exit area ratio, and constant chamber pressure, this optimum condition occurs only at the particular altitude where the ambient pressure p₃ just equals the nozzle exhaust pressure p₂. At all other altitudes .

The maximum theoretical value of nozzle outlet velocity is reached with an infinite nozzle expansion (when exhausting into a vacuum):

3–18

This maximum theoretical exhaust velocity is finite, even though the pressure ratio is not, because it corresponds to finite thermal energy contents in the fluid. In reality, such an expansion cannot occur because, among other things, the temperature of most gas species will eventually fall below their liquefaction or their freezing points; thus, they cease to be a gas and can no longer contribute to the expansion or to any further velocity increases.

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 can be appreciable. In the example above the temperature changed 1115 °C in a relatively short distance. This should not be surprising, for the increase in the kinetic energy of the gases comes from a decrease of the enthalpy, which in turn is roughly proportional to the decrease in temperature. Because the exhaust gases are still relatively hot (1107 K) when leaving the nozzle, they contain considerable thermal energy which is not available for conversion because it is seldom realistic to match exit and ambient temperatures, (see Fig. 2–2).

Nozzle Flow and Throat Condition

Supersonic nozzles (often called De Laval nozzles after their inventor) always consist of a convergent section leading to a minimum area followed by a divergent section. From the continuity equation, this area is inversely proportional to the ratio /V. This quantity has also been shown in Fig. 3–3, where there is a maximum in the curve of /V because at first the velocity increases at a greater rate than the specific volume; however, in the divergent section, the specific volume increases faster.

The minimum nozzle cross‐sectional area is commonly referred to as the throat area A_t. The ratio of the nozzle exit area A₂ to the throat area A_t is called the nozzle expansion area ratio and is designated here by the Greek letter ε. It represents an important nozzle parameter:

3–19

The maximum gas flow rate per unit area occurs at the throat where a unique gas pressure ratio exists, which is only a function of the ratio of specific heats k. This pressure ratio is found by setting in Eq. 3–13:

3–20

The throat pressure p_t for which an isentropic mass flow rate attains its maximum is called the critical pressure. Typical values of the above critical pressure ratio range between 0.53 and 0.57 of the nozzle inlet pressure. 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 the value 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 usually 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 location of critical pressure, namely the throat, the Mach number is always one and the values of the specific volume and temperature can be obtained from Eqs. 3–7 and 3–12:

3–21

3–22

In Eq. 3–22 the nozzle inlet temperature T₁ is typically very close to the combustion chamber temperature and hence close to the nozzle flow stagnation temperature T₀. At the throat there can only be a small variation for these properties. Take, for example, a gas with ; the critical pressure ratio is about 0.56 (which means that p_t is a little more than half of the chamber pressure p₁; the temperature drops only slightly (), and the specific volume expands by over 60% (). Now, from Eqs. 3–15, 3–20, and 3–32, the critical or throat velocity is obtained:

3–23

The first version of this equation permits the critical velocity to be calculated directly from the nozzle inlet conditions and the second one gives when throat temperature is known. The throat is the only nozzle section where the flow velocity is also the local sonic velocity ( because ). The inlet flow from the chamber is subsonic and downstream of the nozzle throat it is supersonic. The divergent portion of the nozzle provides for 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 per unit area will remain its maximum. Sonic and supersonic flow conditions can only be attained when the critical pressure prevails at the throat, that is, when p₂/p₁ is equal to or less than the quantity defined by Eq. 3–20. There are, therefore, three basically different types of nozzles: subsonic, sonic, and supersonic, and these are described in Table 3–1.

	Subsonic	Sonic	Supersonic
Throat velocity	< at
Exit velocity	< a2
Mach number	M2 < 1		M2 > 1
Pressure ratio
Shape

The supersonic nozzle is the only one of interest for rocket propulsion systems. It achieves a high degree of conversion of enthalpy to kinetic energy. The ratio between the inlet and exit pressures in all rocket propulsion systems must be designed sufficiently large to induce supersonic flow. Only when the absolute chamber pressure drops below approximately 1.78 atm will there be subsonic flow in the divergent portion of any nozzle during sea‐level operation. This pressure condition actually occurs for very short times during start and stop transients.

The velocity of sound, a, is 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 location within a steady flow system, it is impossible for pressure disturbances to travel upstream past that location. Thus, any partial obstruction or disturbance of the flow downstream of a nozzle throat running critical 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 sonic velocity at the throat or the flow rate in a fixed configuration nozzle by further lowering the exit pressure or even evacuating the exhaust section. This important condition has been described as choking the flow, which is always established at the throat (and not at the nozzle exit or any other plane). Choked mass flow rates through the critical section of a supersonic nozzle may be derived from Eqs. 3–3, 3–21, and 3–23. From continuity, always equals to the mass flow at any other cross section within the nozzle for steady flow.

3–24

The (chocked) mass flow through a rocket nozzle is therefore proportional to the throat area A_t and the chamber (stagnation) pressure p₁; it is also inversely proportional to the square root of as well as a function of other gas properties. For supersonic nozzles the ratio between the throat and any downstream area at which a pressure p_y prevails can be expressed as a function of pressure ratios and the ratio of specific heats, by using Eqs. 3–4, 3–16, 3–21, and 3–23, as follows:

3–25

When , then and Eq. 3–25 shows the inverse nozzle exit area expansion ratio. For low‐altitude operation (sea level to about 10,000 m) nozzle area ratios are typically between 3 and 30, depending on chamber pressure, propellant combinations, and vehicle envelope constraints. For high altitudes (100 km or higher) area ratios are typically between 40 and 200, but there have been some as high as 400. Similarly, an expression for the ratio of the velocity at any point downstream of the throat with a pressure p_y to the throat velocity may be written from Eqs. 3–15 and 3–23 as:

3–26

Note that Eq. 3–26 is dimensionless but does not represent a Mach number. Equations 3–25 and 3–26 permit direct determination of the velocity ratio or the area ratio for any given pressure ratio, and vice versa, in ideal rocket nozzles. They both are plotted in Fig. 3–4. When , Eq. 3–26 describes the velocity ratio between the nozzle exit area and the throat section. When the exit pressure coincides with the atmospheric pressure (, see Fig. 2–1), these equations apply to optimum nozzle expansions. For rockets that operate at high altitudes, it can be shown that not much additional exhaust velocity may be gained by increasing the area ratio above 1000. In addition, design difficulties and a heavy inert nozzle mass have made applications above area ratios of about 350 marginal.

Appendix 2 is a table of several properties of the Earth's atmosphere with established standard values. It gives ambient pressure for different altitudes. These properties do vary somewhat from day to day (primarily because of solar activity) and between hemispheres. For example, the density of the atmosphere at altitudes between 200 and 3000 km can change by more than an order of magnitude, affecting satellite drag estimates.

Thrust and Thrust Coefficient

As already stated, the efflux of propellant gases (i.e., their momentum flowing out) causes thrust—a reaction force on the rocket structure. Because this flow is supersonic, the pressure at the exit plane of the nozzle may differ from the ambient pressure, and a pressure component in the thrust equation adds (or subtracts) to the momentum thrust as given by Eq. 2–13, which is repeated here:

Maximum thrust for any given nozzle operation is found in a vacuum where . Between sea level and the vacuum of space, Eq. 2–13 gives the variation of thrust with altitude, applying the properties of the atmosphere as listed in Appendix 2. A typical variation of thrust with altitude from a chocked nozzle would be represented by a smooth increase curving to an asymptotic value (see Example 2–2). To modify values calculated for the optimum operating condition (), for other conditions of p₁, k, and , the following expressions may be used. For the thrust,

3–27

For the specific impulse, using Eqs. 2–5, 2–17, and 2–13,

3–28

When, for example, the specific impulse at a new exit pressure p₂ corresponding to a new area ratio A₂/A_t is to be calculated, the above relations may be used.

Equation 2–13 can be modified by substituting , , and V_t from Eqs. 3–16, 3–21, and 3–23:

3–29

The first version of this equation is general and applies to the gas expansion of all rocket propulsion systems, the second form applies to an ideal rocket propulsion system with k being constant throughout the expansion process. This equation shows that thrust is proportional to the throat area A_t and the chamber pressure (or the nozzle inlet pressure) p₁ and is a function of the pressure ratio across the nozzle p₁/p₂, the specific heat ratio k, and of the pressure thrust. Equation 3–29 is called the ideal thrust equation. A thrust coefficient C_F may now defined as the thrust divided by the chamber pressure p₁ and the throat area A_t. Equation 3–30 then results:

3–30

The thrust coefficient C_F is dimensionless. It is a key parameter for analysis and depends on the gas property k, the nozzle area ratio ε, and the pressure ratio across the nozzle p₁/p₂, but is not directly dependent on chamber temperature. For any fixed ratio p₁/p₃, the thrust coefficient C_F and the thrust F have a peak when as area ratio varies. This peak value is known as the optimum thrust coefficient and is an important criterion in nozzle designs. The introduction of the thrust coefficient permits the following form of Eq. 3–29:

3–31

Equation 3–31 may be solved for C_F to experimentally determine thrust coefficient from measured values of chamber pressure, throat diameter, and thrust. Even though the thrust coefficient is a function of chamber pressure, it is not simply proportional to p₁, as can be seen from Eq. 3–30. However, it is directly proportional to throat area. Thrust coefficient has values ranging from just under 1.0 to roughly 2.0. Thrust coefficient may be thought of as representing the amplification of thrust due to the gas expansion in the supersonic nozzle as compared to the thrust that would be exerted if the chamber pressure acted over the throat area only. It is a convenient parameter for visualizing effects of chamber pressure and/or altitude variations in a given nozzle configuration, or for modifying sea‐level results to flight altitude conditions.

Figure 3–5 shows variations of the optimum expansion thrust coefficient () with pressure ratio p₁/p₂, for different values of k and area ratio ε. Complete thrust coefficient variations for and 1.30 are plotted in Figs. 3–6 and 3–7 as a function of pressure ratio p₁/p₃ and area ratio ε. These sets of curves are very useful in examining various nozzle problems for they permit the evaluation of under‐ and overexpanded nozzle operation, as explained below. The values given in these figures are ideal and do not consider such losses as divergence, friction, or internal expansion waves.

When p₁/p₃ becomes very large (e.g., expansions into near vacuum), the thrust coefficient approaches an asymptotic maximum as shown for and 1.30 in Figs. 3–6 and 3–7. These figures also give values of C_F for some mismatched nozzle conditions (), provided the nozzle is flowing full at all times (the working fluid does not separate or break away from the nozzle walls). Flow separation is discussed later in this section.

Characteristic Velocity and Specific Impulse

The characteristic velocity was defined by Eq. 2–17. From Eqs. 3–24 and 3–31, it can be shown that

3–32

This velocity is only a function of propellant characteristics and combustion chamber properties, independent of nozzle characteristics. Thus, it can be used as a figure of merit when comparing different propellant combinations for combustion chamber performance. The first version of this equation applies in general and allows determination of from experimental values of , and A_t. The last version gives the ideal value of as a function of working gas properties, namely k, chamber temperature, and effective molecular mass , determined from theory as presented in Chapter 5; some values of are shown in Tables 5–5 and 5–6.

The term ‐efficiency can be used to express the degree of completion of chemical energy releases in the generation of high‐temperature, high‐pressure gases in combustion chambers. It is the ratio of the actual value of , as determined from measurements (the first part of Eq. 3–32), to the theoretical value (last part of Eq. 3–32) and has typical values between 92 and 99.5%.

Combining now Eqs. 3–31 and 3–32 allows to express the thrust itself as the mass flow rate () times a function of combustion chamber parameters () times a function of nozzle expansion parameters (C_F), namely,

3–33

Some authors use a term called the discharge coefficient C_D, which is merely the reciprocal of . Both C_D and the characteristic exhaust velocity can be used only with chemical rocket propulsion systems.

The influence of variations in the specific heat ratio k on various quantities (such as c, A₂/A_t, /, or I_s) is not as large as those from changes in chamber temperature, pressure ratio, or molecular mass. Nevertheless, they are a noticeable factor, as can be seen by examining Figs. 3–2 and 3–4 to 3–7. The value of k is 1.67 for monatomic gases such as helium and argon, 1.4 for cold diatomic gases such as hydrogen, oxygen, and nitrogen, and for triatomic and beyond it varies between 1.1 and 1.3 (methane is 1.11 and ammonia and carbon dioxide 1.33), see Table 7–3. In general, the more complex the molecule the lower the value of k; this is also true for molecules at high temperatures when their vibrational modes have been activated. Average values of k and for typical rocket exhaust gases with several constituents depend strongly on the composition of the products of combustion (chemical constituents and concentrations), as explained in Chapter 5. Values of k and are given in Tables 5–4, 5–5, and 5–6.

Under‐ and Overexpanded Nozzles

An underexpanded nozzle discharges the gases at an exit pressure greater than the external pressure because its exit area is too small for an optimum expansion. Gas expansion is therefore incomplete within the nozzle, and further expansion will take place outside of the nozzle exit because the nozzle exit pressure is higher than the local atmospheric pressure.

In an overexpanded nozzle the gas exits at lower pressure than the atmosphere as it has a discharge area too large for optimum. The phenomenon of overexpansion inside a supersonic nozzle is indicated schematically in Fig. 3–8, from typical early pressure measurements of superheated steam along the nozzle axis and different back pressures or pressure ratios. Curve AB shows the variation of pressure with optimum back pressure fully corresponding to the nozzle area ratio. Curves AC to AH show pressure variations along the axis for increasingly higher external pressures. The expansion within the nozzle proceeds normally along its initial portion but, for example, at any location after I on curve AD the pressure is lower than the exit pressure and a sudden rise in pressure takes place that is accompanied by separation of the flow from the walls (separation is described later in this chapter).

Separation behavior in nozzles is deeply influenced by the presence of compression waves or shock waves inside the diverging nozzle section, which are strong discontinuities and exist only in supersonic flow. The sudden pressure rise in curve ID represents such a compression wave. Expansion waves (below point B), also strictly supersonic phenomena, act to match flows from the nozzle exit to lower ambient pressures. Compression and expansion waves are further described in Chapter 20.

These different possible flow conditions in supersonic nozzles may be stated as follows:

1. When the external pressure p₃ is below the nozzle exit pressure p₂, the nozzle will flow full but there will form external expansion waves at its exit (i.e., underexpansion). The expansion of the gas inside the nozzle is incomplete and the values of C_F and I_s will be less than at optimum expansion.
2. For external pressures p₃ somewhat higher than the nozzle exit pressure p₂, the nozzle will continue to flow full. This will occur until p₂ drops to a value between about 10 and 40% of p₃. The expansion is somewhat inefficient and C_F and I_s will have lower values than an optimum nozzle would have. Weak oblique shock waves will develop outside the nozzle exit section.
3. At higher external pressures, flow separation will begin to take place inside the divergent portion of the nozzle. The diameter of the exiting supersonic jet will be smaller than the nozzle exit diameter (with steady flow, separation remains typically axially symmetric). Figures 3–9 and 3–10 show diagrams for separated flows. The axial location of the separation plane depends on both the local pressure and the wall contour. The separation point travels upstream with increasing external pressure. At the nozzle exit, separated flow remains supersonic in the center portion 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 an optimum expansion nozzle at the available area ratio. Shock waves may exist outside the nozzle in the plume.

   

   

4. For nozzles in which the exit pressure is just below the value of the inlet pressure (i.e., curve AH in Fig. 3–8), the pressure ratio is below the critical pressure ratio (as defined by Eq. 3–20) and subsonic flow would prevail throughout the entire nozzle. This condition normally occurs in rocket nozzles for a short time during the start and stop transients.

Methods for estimating pressure at the location of the separation plane inside the diverging section of a supersonic nozzle have traditionally been empirical (see Ref. 3–4). Reference 3–5 describes a variety of nozzles, their behavior, and methods used to estimate the location and pressure at separation. Actual values of pressure for the overexpanded and underexpanded 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 steady flows tend to separate uniformly over the divergent portion of a nozzle, but during start and stop transients this separation may not be axially symmetric and may momentarily produce large side forces at the nozzle walls. During normal sea‐level transients in large rocket nozzles (before the chamber pressure reaches its full value), some momentary flow oscillations and nonsymmetric separation of the jet may occur while in overexpanded flow operation. The magnitude and direction of these transient side forces may change rapidly and erratically. Because these resulting side forces can be large, they have caused failures of nozzle exit cone structures and thrust vector control gimbal actuators. References , and 14 discuss techniques for estimating these side forces.

When flow separates, as it does in a highly overexpanded nozzle, the thrust coefficient C_F can be estimated only when the location of separation in the nozzle is known. The C_F thus determined relates to an equivalent smaller nozzle with an exit area equal to that at the separation plane. The effect of separation is move the effective value of ε in Figs. 3–6 and 3–7 to the left of the physical or design value. Thus, with separated gas flows, a nozzle designed for high altitude (large value of ε) could have a larger thrust at sea level than expected; in this case, separation may actually be desirable. But with continuous separated flow operation a large and usually heavy portion of the nozzle is not utilized making it bulkier and longer than necessary. Such added engine mass and size decrease flight performance. Designers therefore usually select an area ratio that will not cause separation.

Because of potentially uneven flow separation and its destructive side loads, sea‐level static tests of an upper stage or of a space propulsion system with high area ratios are usually avoided because they result in overexpanded nozzle operation; instead, tests are done with a sea‐level substitute nozzle (of a much smaller area ratio) often called a “stub nozzle.” However, actual or simulated altitude testing (in an altitude test facility, see Chapter 21) is done with nozzles having the correct area ratio. One solution that avoids separation at low altitudes and has high values of C_F at high altitudes would consist of a nozzle that changes area ratio in flight. This is discussed at the end of this section.

For most atmospheric flights, rocket systems have to operate over a range of altitudes; for a fixed chamber pressure this implies a range of nozzle pressure ratios. The most desirable nozzle for such an application is not necessarily one that gives optimum nozzle gas expansion, but one that gives the best vehicle overall flight performance (say, total impulse, or range, or payload); this can often be related to a time average over the powered flight trajectory.

Figure 3–10 shows behavior comparisons of altitude and sea‐level ground‐tests of three nozzles and their plumes at different area ratios for a typical three‐stage satellite launch vehicle. When fired at sea‐level conditions, the nozzle of the third stage with the highest area ratio will experience flow separation and suffer a major performance loss; the second stage will flow full but the external plume will contract; since , there is a loss in I_s and F. There is no effect on the first stage nozzle. When the second and third stages actually operate at their proper altitude, they would experience no separation.

Figures 3–9 and 3–10 suggest that a continuous optimum performance for an ascending (e.g., launch) rocket vehicle would require a “rubber‐like” diverging section that would lengthen and enlarge so that the nozzle exit area would increase as the ambient pressure is reduced. Such design would then allow the rocket vehicle to attain its best performance at all altitudes as it ascends. As yet we have not achieved such flexible, stretchable mechanical nozzle designs with full altitude compensation similar to “stretching rubber.” However, there are a number of practical nozzle configurations that can be used to alter nozzle shape with altitude and obtain peak performance at different altitudes. These are discussed in the next section.

Influence of Chamber Geometry

When the chamber has a cross section that is about four times larger than the throat area , the chamber velocity , may be neglected, as was mentioned in introducing Eqs. 3–15 and 3–16. However, some vehicle space‐ or weight‐constraints for liquid propellant engines often require smaller thrust chamber areas, and for solid propellant motors grain design considerations may lead to small void volumes or small perforations or port areas. In these cases, 's contribution to performance can no longer be neglected. Gases in the chamber expand as heat is being added. The energy expended in accelerating these expanding gases within the chamber will also cause a pressure drop, and processes in the chamber can be adiabatic (no heat transfer) but not isentropic. Losses are a maximum when the chamber diameter equals the nozzle throat diameter, which means that there is no converging nozzle section. This condition is called a throatless rocket motor and is found in a few tactical missile booster applications, where there is a premium on minimum inert mass and length. Here, flight performance improvements due to inert mass savings have supposedly outweighed the nozzle performance loss of a throatless motor.

Because of the significant pressure drops within narrow chambers, chamber pressures are lower at the nozzle entrance than they would be with a larger A₁/A_t. This causes a small loss in thrust and specific impulse. The theory of such losses is given in the Second and Third Editions of this book, and some results are listed in Table 3–2.

Cone‐ and Bell‐Shaped Nozzles

The conical nozzle is the oldest and perhaps the simplest configuration. It is relatively easy to fabricate and is still used today in many small nozzles. A theoretical correction factor λ must be applied to the nozzle exit momentum in any ideal rocket propulsion system using a conical nozzle. This factor is the ratio between the momentum of the gases exhausting with a finite nozzle angle 2α and the momentum of an ideal nozzle with all gases flowing in the axial direction:

3–34

For ideal rockets . For a rocket nozzle with a divergence cone angle of 30° (half angle α = 15°), the exit momentum and therefore the axial exhaust velocity will be 98.3% of the velocity calculated by Eq. 3–15b. Note that the correction factor λ only applies to the first term (momentum thrust) in Eqs. 2–13, 3–29, and 3–30 and not to the second term (pressure thrust).

A half angle value of 15° has become the unofficial standard of reference for comparing correction factors or energy losses or lengths from different diverging conical nozzle contours.

Small nozzle divergence angles may allow most of the momentum to remain axial and thus produce high specific impulses, but they result long nozzles introducing performance penalties in the rocket propulsion system and the vehicle mass, amounting to small losses. Larger divergence angles give shorter, lightweight designs, but their performance may become unacceptably low. Depending on the specific application and flight path, there is an optimum conical nozzle divergence (typically between 12 and 18° half angle).

Bell‐shaped or contour nozzles (see Figs. 3–11 and 3–13) are possibly the most common nozzle shapes used today. They have a high angle expansion section (20 to 50°) immediately downstream of the nozzle throat followed by a gradual reversal of nozzle contour slope so that at the nozzle exit the divergence angle is small, usually less than a 10° half angle. It is possible to have large divergence angles immediately behind the throat (of 20 to 50°) because high relative local pressures, large pressure gradients, and rapid expansions of the working fluid do not allow separation in this region (unless there are discontinuities in the nozzle contour). Gas expansion in supersonic bell nozzles is more efficient than in simple straight cones of similar area ratio and length because wall contours can be designed to minimize losses, as explained later in this section.

Changing the direction of supersonic flows with an expanding wall geometry requires expansion waves. These occur at thin regions within the flow, where the flow velocity increases as it slightly changes its direction, and where the pressure and temperature drop. These wave surfaces are at oblique angles to the flow. Beyond the throat, the gas crosses a series of expansion waves with essentially no loss of energy. In the bell‐shaped configuration shown in Fig. 3–13 these expansions occur internally in the flow between the throat and the inflection location I; here the area is steadily increasing (like a flare on a trumpet). The contour angle θ_i is a maximum at this inflection location. Between the inflection point I and the nozzle exit E the flow area increases at a diminishing rate. Here, the nozzle wall contour is different and the rate of change in cross‐sectional area per unit length is decreasing. One purpose of this last nozzle segment is to reduce the nozzle exit angle θ_e and thus to reduce flow divergence losses as the gas leaves the nozzle exit plane. The angle at the exit θ_e is small, usually less than 10°. The difference between θ_i and θ_e is called the turn‐back angle. As the gas flow is turned in the opposite direction (between points I and E) oblique compression waves occur. These compression waves are thin surfaces where the flow undergoes a mild shock as it is turned back and its velocity slightly reduced. Each of these multiple compression waves causes a small energy loss. By carefully determining the wall contour (through an analysis termed “method of characteristics”), it is possible to balance the oblique expansion waves with the oblique compression waves and minimize these energy losses. Analyses leading to bell‐shaped nozzle contours are presented in Chapter 20.33 of Ref. 3–3 and also in Refs. 3–8 to 3–11. Most rocket organizations have developed their own computer codes for such work. It has been found that the radius of curvature or contour shape at the throat region noticeably influences the contours in diverging bell‐shaped nozzle sections.

The length of a bell nozzle is usually given as fractions of the length of a reference conical nozzle with a 15° half angle. An 80% bell nozzle has a length (distance between throat plane and exit plane) that is 20% shorter than a comparable 15° cone of the same area ratio. Reference 3–9 shows the original method of characteristics as initially applied to determine bell nozzle contours and later to shortened bell nozzles; a parabola has been used as a good approximation for the bell‐shaped contour curve (Ref. 3–3, Section 20.33), and actual parabolas have been used in some nozzle designs. The top part of Fig. 3–13 shows a parabolic contour that is tangent (θ_i) at the inflection point I and has an exit angle (θ_e) at point E, and that its length L that has to be corrected for the curve TI. These conditions allow the parabola to be determined either from simple geometric analysis or from a drawing. A throat approach radius of 1.5r_t and a throat expansion radius of 0.4r_t were used here. If somewhat different radii had been used, the results would have been only slightly different. The middle set of curves gives the relation between length, area ratio, and the two angles of the bell contour. The bottom set of curves gives the correction factors, equivalent to the λ‐factor for conical nozzles, which are to be applied to the thrust coefficient or the exhaust velocity, provided the nozzles operate at optimum expansion, that is, .

Table 3–3 shows data for parabolas developed from Fig. 3–13 in order to allow the reader to apply this method and check results. The table shows two shortened bell nozzles and a conical nozzle, each for three area ratios. A 15° half angle cone is given as reference. It can be seen that as the length decreases, the losses are higher for the shorter length and slightly higher for the smaller nozzle area ratios. A 1% improvement in the correction factor gives about 1% greater specific impulse (or thrust), and this difference can be significant in many applications. A reduced length is an important benefit usually reflected in improvements of the vehicle mass ratio. Table 3–3 and Fig. 3–13 show that bell nozzles (75 to 85% the length of cones) can be slightly more efficient than a longer 15° conical nozzle (100% length) at the same area ratio. For shorter nozzles (below 70% equivalent length), the energy losses due to internal oblique shock waves can become substantial so that such short nozzles are not commonly used today.

For solid propellant rocket motor exhausts containing small solid particles in the gas (usually aluminum oxide), and for the exhaust of certain gelled liquid propellants, solid particles impinge against the nozzle wall in the reversing curvature section between I and E in Fig. 3–13. While the gas can be turned by oblique waves to have less divergence, any particles (particularly the larger particles) will tend to move in straight lines and hit the walls at high velocities. The resulting abrasion and erosion of the nozzle wall can be severe, especially with commonly used ablative and graphite materials. This abrasion by hot particles increases with turn‐back angle. Only when the turn‐back angle and thus also the inflection angle θ_i are reduced such erosion can become acceptable. Typical solid rocket motors flying today have values of inflection angles between 20 and 26° and turn‐back angles of 10 to 15° with different wall contours in their divergent section. In comparison, current liquid rocket engines without entrained particles have inflection angles between 27 and 50° and turn‐back angles of between 15 and 30°. Therefore, performance enhancements resulting from the use of bell‐shaped nozzles (with high correction factor values) are somewhat lower in solid rocket motors with solid particles in the exhaust.

An ideal (minimum loss through oblique waves) bell‐shaped nozzle is relatively long, equivalent to a conical nozzle of perhaps 10 to 12°, as seen in Fig. 3–11. Because long nozzles are heavy, in applications where vehicle mass is critical, somewhat shortened bell nozzles are preferred.

Two‐Step Nozzles

Two‐position nozzles (with different expansion area ratio sections) can give better performance than conventional nozzles with a fixed single area ratio. This is evident in Fig. 3–9; the lower area ratio nozzle () performs best at low altitudes and the higher area ratio nozzle performs best at higher altitudes. If these two nozzles could somehow be mechanically combined, the resulting two‐position nozzle would perform closer to a nozzle that adjusts continuously to the optimum area ratio, as shown by the thin dashed curve. When integrated over flight time, the extra performance has a noticeable payoff for high‐velocity missions, such as Earth‐orbit injection and deep space missions. Several bell‐shaped nozzle concepts have evolved that achieve maximum performance at more than a single altitude. Figure 3–14 shows three different two‐step nozzle concepts, having an initial low area ratio A₂/A_t for operation at or near the Earth's surface and a larger second area ratio that improves performance at high altitudes. See Ref. 3–4.

Boundary Layers

Real nozzles always develop thin viscous boundary layers adjacent to their inner walls, where gas velocities are much lower than in the free‐stream region. An enlarged depiction of a boundary layer is shown in Fig. 3–15. Immediately next to the wall the flow velocity is zero; beyond the wall the boundary layer may be considered as being built up of successive annular‐shaped thin layers of increasing velocity until the free‐stream value is reached. The low‐velocity flow region closest to the wall is always laminar and subsonic, but at the higher‐velocity regions of the boundary layer the flow becomes supersonic and can be turbulent. Local temperatures at some locations in the boundary layer can be noticeably higher than the free‐stream temperature because of the conversion of kinetic energy into thermal energy that occurs as the velocity is locally slowed down and by heat created by viscous effects. The flow layer immediately next to the wall will be cooler because of heat transfer to the wall. These gaseous boundary layers have a profound effect on the overall heat transfer to nozzle and chamber walls. They also affect rocket performance, particularly in applications with very small nozzles and also with relatively long nozzles with high nozzle area ratios, where a comparatively high proportion of the total mass flow (2 to 25%) can be in the lower‐velocity region of the boundary layer. High gradients in pressure, temperature, or density and changes in local velocity (direction and magnitude) also influence boundary layers. Flow separation in nozzles always originates inside the boundary layers. Scaling laws for boundary layer phenomena have not been reliable to date.

Theoretical approaches to boundary layer effects can be found in Chapters 26 to 28 of Ref. 3–1 and in Ref. 3–18. Even though theoretical descriptions of boundary layers in rocket nozzles are yet unsatisfactory, their overall effects on performance can be small—for most rocket nozzles with unseparated flow, boundary layer derived losses of specific impulse seldom exceeds 1%.

Multiphase Flow

In some propulsion systems, the gaseous working fluid contains small liquid droplets and/or solid particles entrained in the flow. They may heat the gas during nozzle expansion. This heating, for example, occurs with solid propellants (see Section 13–4) or in some gelled liquid propellants, which contain aluminum powder because small oxide particles form in the exhaust. This can also occur with iron oxide catalysts, or solid propellants containing aluminum, boron, or zirconium.

In general, if the particles are relatively small (typically with diameters of 0.005 mm or less), they will have almost the same velocity as the gas and will be in thermal equilibrium with the nozzle gas flow. Thus, even if some kinetic energy is transferred to accelerate the embedded particles, the flow gains thermal energy from them. As particle diameters become larger, the mass (and thus the inertia) of each particle increases as the cube of its diameter; however, the drag or entrainment force increases only as the square of the diameter. Larger particles therefore lag the gas movement and do not transfer heat to the gas as readily as do smaller particles. Larger particles have a lower momentum than an equivalent mass agglomeration of smaller particles and reach the nozzle exit at higher temperatures because they give up less of their thermal energy.

It is possible to derive simple equations for correcting performance (I_s, c, or ) as shown below and as given in Refs. 3–13 and 3–14. These formulas are based on the following assumptions: specific heats of the gases and the particles are constant throughout the nozzle flow, particles are small enough to move at essentially the same velocity as the gas and are in thermal equilibrium with it, and that particles do not exchange mass with the gas (no vaporization or condensation). Expansion and acceleration occur only in the gas, and the volume occupied by the particles is negligibly small compared to the overall gas volume. When the amount of particles is small, the energy needed to accelerate these particles may also be neglected. There are also no chemical reactions.

The enthalpy h, the specific volume V, the specific heat ratio k, and the gas constant R can be expressed as functions of a particle fraction β, which represents the mass of particles (liquid and/or solid) divided by the total mass. The specific heats at constant pressure and volume are denoted as c_p and . Using the subscripts g and s to refer to gaseous or solid states, the following relationships then apply:

3–35

3–36

3–37

3–38

3–39

Performance parameters for nozzle flows from solid‐propellant combustion products that contain small particles or small droplets in the exhaust gas are briefly discussed at the end of this section.

Equations 3–35 to 3–39 are then used with the formulas for simple one‐dimensional nozzle flow, such as Eq. 2–15, 3–15, or 3–32. Values for specific impulse or characteristic velocity will decrease with β, while the particle percent increases. For very small particles (less than 0.01 mm in diameter) and small values of β (less than 6%) losses in specific impulse are often below 2%. For larger particles (over 0.015 mm diameter) and larger values of β this theory does not apply—the specific impulse can become 10 to 20% less than the value without any flow lag. Actual particle sizes and their distribution depend on the particular propellant combustion products, the actual particle composition, and the specific rocket propulsion system; these parameters usually have to be measured (see Chapters 13 and 20). Thus, adding a metal, such as aluminum, to a solid propellant will increase performance only when the additional heat release can sufficiently increase the combustion temperature T₀ so that it more than offsets any decrease caused by the nonexpanding mass portion of the particles in the exhaust.

With very high area ratio nozzles and low nozzle exit pressures (at high altitudes or the vacuum of space) some gaseous propellant ingredients may condense; when temperatures drop sharply in the nozzle, condensation of H₂O, CO₂, or NH₃ are known to occur. This causes a decrease in the gas flow per unit area while transfer of the latent heat of particle vaporization to the remaining gas takes place. It is also possible to form a solid phase and precipitate fine particles of snow (H₂O) or frozen fogs of other species. Overall effect on performance is small only if the particle sizes are small and the percent of precipitate mass is moderate.

Other Phenomena and Losses

The combustion process can never be totally steady. Low‐ and high‐frequency oscillations in chamber pressure of up to perhaps 5% of rated values are considered to be smooth‐burning and sufficiently steady flow. Gas properties () and flow properties (, V, T, p, etc.) will also oscillate with time and will not necessarily be uniform across the flow channel. We therefore deal with “average” values of these properties only, but it is not always clear what kind of time averages are appropriate. Energy losses due to nonuniform unsteady burning are difficult to assess theoretically; for smooth‐burning rocket systems these losses are neglected, but they may become significant for the larger‐amplitude oscillations.

Gas composition may change somewhat in the nozzle when chemical reactions are occurring in the flowing gas, and the assumption of a uniform or “chemically‐frozen‐composition” gas flow may not be fully valid for all chemical systems. A more sophisticated approach for determining performance with changing composition and changing gas properties is described in Chapter 5. Also, any thermal energy that exits the nozzle () is unavailable for conversion to useful propulsive (kinetic) energy, as shown in Fig. 2–2. The only way to decrease this loss is to reduce the nozzle exit temperature T₂ (with a larger nozzle area ratio), but most often this amounts to other sizable losses.

When operating durations are multiple and short (as with antitank rockets or pulsed attitude control rockets which start and stop repeatedly), start and stop transients become a significant fraction of total operating time. During these transient periods the average thrust, chamber pressure, and specific impulse will be lower than during steady operating conditions. This situation may be analyzed with a step‐by‐step process; for example, during start‐up the amount of propellant flowing in the chamber has to equal the flow of gas through the nozzle plus the amount of gas needed to fill the chamber to a higher pressure; alternatively, an empirical curve of chamber pressure versus time, if available, could be used as the basis of such calculations. Transition times can be negligibly short in small, low‐thrust propulsion systems, perhaps a few milliseconds, but they become longer (several seconds) for large propulsion systems.

Performance Correction Factors

In addition to the theoretical cone‐ and bell‐shaped nozzle correction factors discussed in Section 3–4, we discuss here another set of empirical correction factors. These factors represent a variety of nonideal phenomena (such as friction, imperfect mixing and/or combustion, heat transfer, chemical nonequilibrium, and two‐ and three‐dimensional effects) that are unavoidably present; refer to Fig. 2–2 and Ref. 3–4. Correction factors are arbitrarily defined in rocket analysis and may differ from the more conventional efficiencies—there are no “universal performance correction factors.” For specific propulsion systems, where accurately measured data are available, they allow for simple predictions of actual performance. For example, a velocity correction factor of 0.942 means that the velocity or the actual specific impulse is about 94% of theoretical (a commonly used value might be closer to 0.92).

Corrections factors are used by engineers to determine performance ahead of testing and for preliminary designs, informal proposals, and health monitoring systems. In all these, a set of accepted or nominal values is needed to estimate performance together with some useful formula recipes. For accurate calculations, the industry relies extensively on computer programs which are largely proprietary.

Correction factors are ratios of measured or actual (subscript “a”) to formulated or ideal (subscript “i”) values. In the ordinary testing of rocket propulsion systems, the combustion chamber pressure, the propellant mass flow rates, the thrust force, and the throat and exit areas are typically measured. These measurements yield two direct ratios, namely, the thrust correction factor () and the discharge correction factor and, using measured chamber pressures, the product [(p₁)_a(A_t)_a] that has units of force. This product enters into the formulation of two other correction factors discussed below. The nozzle area ratio [] in real nozzles is found from measurements and this ratio may differ from the ideal or calculated value as will be shown in Example 3–7. In liquid propellant rocket engines, the propellant mass flow rate is measured during ground tests using calibrated flowmeters, but it is practically impossible to directly measure mass flow rates in solid propellant motors. Only an effective is determined from the initial and final weights at ground testing. Therefore, the discharge correction factor for solid propellant rocket motors represents only an averaged value.

The thrust correction factor (ζ_F) is found from the ratio of thrust measurements with the corresponding ideal values given by Eq. 3–29. The discharge correction factor (ζ_d) can be determined from the ratio of mass‐flow‐rate measurements with the corresponding theoretical values of Eq. 3–24. Unlike incompressible flows, in rocket propulsion systems the value of ζ_d is always somewhat larger than 1.0 (up to 1.15) because actual flow rates may exceed ideal flow rates for the following reasons:

1. Incomplete combustion (a lower combustion temperature), which results in increases of the exhaust gas densities.
2. Cooling at the walls that reduces boundary layer temperatures and thus the average gas temperature, especially in small thrust chambers.
3. Changes in the specific heat ratio and molecular mass in an actual nozzle that affect the flow rate and thus the discharge correction factor (see Eq. 3–24).

The efficiency or correction factor represents a combined effectiveness of the combustion chamber and the injector design. It can be determined from the ratio of measured values of (from Eq. 2–17) with the corresponding ideal value of the right‐hand side of Eq. 3–32. In well‐designed combustion chambers, the value of correction factor is over 95%. The correction factor, also known as the C_F efficiency, represents the effectiveness of the nozzle design at its operating conditions. It can be determined using measured values of F_a/[(p₁)_a(A_t)_a] (from Eq. 3–31) with the corresponding ideal value of Eq. 3–30. In well‐designed nozzles, the value of the correction factor is above 90%.

An effective exhaust velocity correction factor may now be introduced using that velocity's definition given in Eq. 2–16 (or from Eq. 3–32, ) as

3–40

This further suggests a correction factors equation, a form equivalent to Eq. 3–33, written as , which has meaning only within existing experimental uncertainties and implies that in steady flow all four correction factors cannot be arbitrarily determined but must be so related.

Additional useful relations may also be written. The actual specific impulse, from Eq. 2–5, may now be calculated from

3–41

The thermodynamic nozzle efficiency η_n which is traditionally defined as the ratio of ideal to the actual enthalpy changes (see Eqs. 3–15a and 3–16) under a given pressure ratio becomes

3–42

The approximate sign above becomes an equality when . This nozzle efficiency will always have a value less than 1.0 and represents losses inside the nozzle (Ref. 3–3). Small nozzles being used for micropropulsion can have quite low η_n's because of their relatively large frictional effects which depend in the surface‐to‐volume ratio (some cross sections being rectangular thus having additional sharp‐corner losses).

Under strictly chemically‐frozen‐composition flow assumptions, the right‐hand side of Eq. 3–32 leads to a useful relation between the ideal and actual stagnation temperatures [here T₁ ≡ (T₀)_i]:

3–43

With conditions at the inlet of a supersonic nozzle readily calculable, one task for the analyst is to determine the actual throat area required to pass a specified mass flow rate of gaseous propellant and also to determine the area and the flow properties at the nozzle exit plane. Across the nozzle heat losses modify the local stagnation temperatures and, together with friction, the stagnation pressures (both of which begin as and T₁ in the combustion chamber). Because ideal formulations are based on the local stagnation values at any given cross section, the challenge is to work with the appropriate flow assumptions as discussed below.

When the applicable correction factors values are available, the overall ratio of the product of the stagnation pressure with the throat area may be inferred using

3–44

In order to arrive at an actual nozzle throat area from such available parameters, additional information is needed. The key question is how to relate stagnation pressure ratios to their corresponding temperature ratios across the nozzle. When known, a polytropic index “n” provides a relation between gas properties (Ref. 3–3); otherwise, we may apply the isentropic relation given as Eq. 3–7 because nozzle efficiencies are typically high (around 90%). Real nozzle flows are not isentropic but the net entropy change along the nozzle may be insignificant, though noticeable decreases of the stagnation pressure and temperature do occur as Example 3–7 will show next. When dealing with real nozzles with given expansion ratios, it becomes necessary to increase somewhat the stagnation pressure beyond (p₁)_i to achieve any previously defined ideal nozzle performance.

Four Performance Parameters

When using or quoting values of thrust, specific impulse, propellant flow, and other performance parameters one must be careful to specify or as bare minimum qualify the conditions under which any particular number applies. There are at least four sets of performance parameters, and they are different in concept and value, even when referring to the same rocket propulsion system. Parameters such as F, I_s, c, , and/or should be accompanied by a clear definition of the conditions under which they apply. Not all the items below apply to every parameter.

1. Chamber pressure; also, for slender chambers, the location where this pressure prevails or is measured (e.g., at nozzle entrance).
2. Ambient pressure or altitude or space (vacuum).
3. Nozzle expansion area ratio and whether this is an optimum.
4. Nozzle shape and exit angle (see Fig. 3–11 and Eq. 3–34).
5. Propellants, their composition and mixture ratio.
6. Initial ambient temperature of propellants, prior to start.

1. Theoretical performance values as defined in Chapters 2, 3, and 5 generally apply only to ideal rockets. Conditions for ideal nozzles are given in Section 3.1. The theoretical performance of an ideal nozzle per se is not used, but one that includes corrections is used instead; analyses of one or more losses may be included as well as values for the correction factors described in the previous section and these will yield a lower performance. Most of these modifications are two dimensional and correct for chemical reactions in the nozzle using real gas properties, and most correct for exit divergence. Many also correct for one or more of the other losses mentioned above. For example, computer programs for solid propellant motor nozzles can include losses for throat erosion and multiphase flow; for liquid propellant engines they may include two or more concentric zones, each at different mixture ratios and thus with different gas properties. Nozzle wall contour analyses with expansion and compression waves may apply finite element analysis and/or a method of characteristics approach. Some of the more sophisticated programs include analyses of viscous boundary layer effects and heat transfer to the walls. Typically, these computer simulations are based on finite element descriptions using the basic Navier–Stokes relationships. Most organizations doing nozzle design have their own computer programs, often different programs for different nozzle designs, different thrust levels, or operating durations. Many also use simpler, one‐dimensional computer programs, which may include one or more correction factors; these programs are used frequently for preliminary estimates, informal proposals, or evaluation of ground test results.
2. Delivered, that is, actually measured, performance values are obtained from static tests or flight tests of full‐scale propulsion systems. Again, the conditions should be defined (e.g., p₁, A₂/A_t, p₃, etc.) and the measured values should be corrected for instrument deviations, errors, or calibration constants. Flight test data need to be corrected for aerodynamic effects, such as drag. Often, empirical values of the thrust correction factor, the velocity correction factor, and the mass discharge flow correction factors are used to convert theoretical values from item 1 above to approximate actual values, something that is often satisfactory for preliminary estimates. Sometimes subscale thrust chambers are used in the development of new large thrust chambers and then scale factors are applied to correct the measured data to full‐scale values.
3. Performance values at standard conditions are the corrected values of items 1 and 2 above. These standard conditions are generally rigidly specified by the customer reflecting commonly accepted industrial practice. Usually, they refer to conditions that allow evaluation or comparison with reference values and often they refer to conditions that can be easily measured and/or corrected. For example, to allow for a proper comparison of specific impulse for several propellants or rocket propulsion systems, the values are often corrected to the following standard conditions (see Example 3–6):
   1. psia (6.894 MPa) or other agreed‐upon value.
   2. psia (0.10132 MPa) sea level.
   3. Nozzle exit area ratio, optimum, .
   4. Nozzle divergence half angle ° for conical nozzles, or other agreed‐upon value.
   5. Specific propellant, its design mixture ratio, and/or propellant composition and maximum allowed impurities.
   6. Propellant initial ambient temperature: 21°C (sometimes 20 or 25°C) or boiling temperature, if cryogenic.

   A rocket propulsion system is generally designed, built, tested, and delivered in accordance with certain predetermined requirements or customer‐approved specifications found in formal documents (often called the rocket engine or rocket motor specifications). These may define the performance as shown above along with detailing many other requirements. More discussion of these specifications is given as a part of the selection process for propulsion systems in Chapter 19.

4. Rocket manufacturers are often required by their customers to deliver rocket propulsion systems with a guaranteed minimum performance, such as minimum F and/or minimum I_s. The determination of such values can be based on a nominal value (items 1, 2, or 3 above) diminished by all likely losses, including changes in chamber pressure due to variation of pressure drops in injector or pipelines, losses due to nozzle surface roughness, propellant initial ambient temperatures, ambient pressure variations, and manufacturing variations from rocket to rocket (e.g., in grain volume, nozzle dimensions, or pump impeller diameters, etc.). Minimum values can be determined by probabilistic evaluations of these losses, which are then usually validated by actual full‐scale static and flights tests.

Greek Letters

α	half angle of divergent conical nozzle section
β	mass fraction of solid particles
ε	nozzle expansion area ratio A2/At
ηn	nozzle efficiency
	thrust coefficient correction factor
	correction factor
ζd	discharge or mass flow correction factor
ζF	thrust correction factor
	velocity correction factor
λ	divergence angle correction factor for conical nozzle exit

Subscripts