FLOW OF GASES AND STEAM THROUGH NOZZLES

What are nozzles and other applications of nozzle theory

The nozzle is a channel with a continuously variable flow flow area. Fluid flow in a nozzle is a process in which the pressure decreases and the kinetic energy of the fluid increases. The nozzle theory is well developed and has a wide application in various types of jet machines.

Nozzle shapes depend primarily on required outlet speed

The basic nozzle shapes are the convergent or tapering nozzle, in which subsonic expansion takes place, and the convergent-divergent or de Laval nozzle for supersonic expansion, the shape of which is based on Hugoniot's theorem (area-Mach number realation) for the supersonic flow channel.

Application of nozzle theory

In fact, jet theory can be used to describe some apparently complex flows, for example in blade passages and rocket engines. In addition, a large amount of measured data exists for jets.

Convergent nozzle

Expansion in a nozzle is a frequent problem in engineering, which is why the theory of ideal nozzle expansion was developed in the 19th century [Nožička, 2000]. This theory describes the changes of state variables in a nozzle, especially velocity and mass flow. Furthermore, the occurrence of the so-called critical flow state in the nozzle, at which the nozzle reaches its maximum mass flow, can also be theoretically justified. There are several approaches to nozzle shape design, mainly depending on the purpose of the nozzle, the technological complexity of its production and the required maximum length.

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Calculation outlet velocity of nozzle

From the changes of the state variables in the nozzle plotted in the h-s chart, it can be seen that the gas velocity at the nozzle outlet depends on the inlet pressure p_i and the outlet pressure p_e (backpressure) from the nozzle. Equation 101, p. 4.4 for the outlet velocity can then be derived from the First Law of Thermodynamics equation for an open system. This equation is derived for the perfect expansion of an ideal gas without the effect of gravity.

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Critical flow state in nozzle

The equation for the mass flow, or the outlet coefficient, shows that as the pressure behind the nozzle p_e decreases, the gas mass mass flow m should only increase up to a certain pressure ratio ε_s, then the mass flow should start to decrease, see curve 1-a-0 in Figure 515. In fact, from the ratio ε^*_s until the expansion to vacuum (ε_s=0) the mass flow is constant and equal to m^*, see curve a-b in Figure 515. The pressure ratio at which the maximum gas flow through the nozzle or critical flow state is reached is called the critical pressure ratio (hence the asterisk mark *). The equation for the critical pressure ratio can be derived from the extreme of Equation 334, p. 4 for the mass flow, see Equation 515.

– 515: –

Mass flow through nozzle profile

A* [m²] smallest flow area of nozzle. The derivation of the critical pressure ratio ε*_s equation is shown in Appendix 515.

Calculation of mass flow through a nozzle using critical mass flow

The 1-a-0 curve of Figure 515 is very close in shape to an ellipse, so in engineering practice, to speed up and simplify nozzle calculations, the 1-a segment is often replaced by a portion of the ellipse called the Bendemann ellipse, see Equation 162, whose validity is limited to the range p_e≥p^*.

– 162: –

Bendemann ellipse

The derivation of the equation for the Bendemann ellipse is shown in Appendix 162.

Critical pressure ratio values

The critical pressure ratio is a function of the gas type because the ratio of heat capacities κ varies from gas to gas. The values of the critical pressure ratios for the ideal gas can be calculated from Formula 515. The critical pressure ratios of real gases vary slightly, for example, for hydrogen is 0,527, dry air is 0,528, superheated water vapour is 0,546, and saturated water vapour is 0,577. However, the critical pressure ratio can be expected to be around 0,5.

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Convergent nozzle shapes

Common convergent nozzle shapes are shown in Figure 475. These shapes can also be applied to non-circular channels and blade passage. The ideal nozzle shape is smooth, parallel to the streamlines (at both inlet and outlet to avoid turbulence due to sudden changes in flow direction against the wall) and one in which a uniform velocity profile is achieved at the outlet. That is, the exit velocity should be in the direction of the nozzle axis, as shown by experiments [Dejč, 1967, p. 319]. This condition must also be satisfied by the jet line near the nozzle edge.

– 475: –

(a) conical nozzle; (b) ideal nozzle shape; (c) Vitoshinsky nozzle [Dejč, 1967, p. 320] (equation holds for l≥2·r_e); (d) the shape of the nozzle as a lemniscate ∞; (e) the shape of the nozzles at the outlet of tanks (r_r≈1,5·r_e [Sutton and Biblarlz, 2010, p. 80]). l [m] nozzle length; r [m] nozzle radius; x [m] nozzle axial coordinate.

Advantages and disadvantages of individual nozzle shapes

Cone nozzles are simple to manufacture, but they do cause flow contraction (see chapter Flow through nozzle with losses, p. 14) than nozzles of the shape shown in Figure 475b. The most uniform velocity field at the outlet has nozzles of the lemniscate shape, which can be approximately described by Vitoshinsky's formula (Figure 475c) - such nozzle shapes are used as a transition channel between two channels and for blowing nozzles in wind tunnels.

– 983: –

Figure from [Slavík, 1938, s. 23].

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Laval nozzle shapes

The ideal shape of a Laval nozzle is the shape constructed by the so-called characteristic method, but this shape is very demanding to calculate and produce. On the other hand, the simplest shape is the cone nozzle, while bell-shaped nozzles are common in rocketry.

Laval nozzles designed by method of characteristics

Shape of the Laval nozzles modelled by the method of characteristics (Figure 993) is the ideal shape. This is because the nozzles designed by this method have a uniform velocity profile at the outlet. The method of characteristics is based on the successive construction of expansion waves, these waves are plotted in blue in Figure 993. The boundary condition of this method is a given initial radius r_r at α_e=0° (the condition of the outlet velocity in the axial direction) and the flow area at the outlet A_e [Dejč, 1967, p. 341], [Sutton and Biblarlz, 2010, p. 79]. The disadvantage is that the length of such a nozzle is much greater than that of a conical nozzle, so that due to internal friction, its efficiency may be lower than that of a conical nozzle, so this nozzle shape is practically only used where a uniform velocity field at the outlet is very important.

– 993: –

α [°] diverging nozzle angle; t [m] inlet nozzle length (usually a circular contour with radius r_r≈0,382·r* [Sutton and Biblarlz, 2010, p. 80]). Derivations of the equations for rt and t are shown in Appendix 993.

Bell Laval nozzles

The bell nozzle is primarily the shape of rocket engine nozzles. The shape of this nozzle is designed either according to the Rao equation (following G.V.R. Rao, who developed this equation based on experiments [Rao, 1958]) or the Allman-Hoffman equation (following Allman J. G. and Hoffman J. D., who derived the equation by simplifying the Rao equation [Allman and Hoffman, 1981]); both equations are second degree polynomials (parabolas), see Figure 335. In the case of boundary conditions for the Rao equations, the outlet and input angles are interdependent (α_t=f(α_e)). The selection of the optimal pair of input α_t and outlet angle α_e is possible from the length of the equivalent conical nozzle at α=30°, see tables and graphs in [Sutton and Biblarlz, 2010, p. 80]. In the case of the Allman-Hoffman equation, only the input angle α_t is sufficient to solve. A nozzle designed according to the Allaman-Hoffman equation has about 0.2% less exit gas momentum in the axial direction when expanding into vacuum than a nozzle designed according to the Rao equation [Haddad, 1988], but it is easier to work with in finding the optimal nozzle shape for a large number of combinations of working gas input parameters. The bell nozzle is shorter than the conical nozzle, yet has greater efficiency and momentum in the axial direction.

– 335: –

(a) Rao nozzle contour equation; (b) Allman-Hoffman nozzle contour equation; (c) boundary conditions for calculating constants a₁..a₄ or b₁..b₃.

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Non-design conditions of Laval nozzles

The discharge pressure p_e,n for which the nozzle is designed is called the design pressure. Thus, the non-design nozzle condition means a condition where the inlet gas parameters or the outlet gas parameters or both parameters are changed. These parameters may change for various reasons (flow control through the nozzle, etc.). In total, there are two basic cases of Laval nozzle overexpanded state and underexpanded state.

Underexpanded state of nozzle

If the back pressure p_e is lower than pressure p_en, then the nozzle is said to be underexpanded (the nozzle was designed for a "shorter" expansion than the actual expansion). In cases where the backpressure is lower than the design pressure, the expansion behind the nozzle will continue, similar to a convergent nozzle.

Underexpanded state of rocket engine nozzles

The change in backpressure also affects the design of rocket engine nozzles. During the rocket's flight in the atmosphere, the external pressure varies with altitude, so the first stage nozzles are designed to expand to atmospheric pressure and the last stage is designed to expand to vacuum. The greater the thrust range offered by a rocket engine, the more its nozzle must be under-expanded. Therefore, for a perpendicular landing of a rocket with a poorly underexpanded nozzle, the calculation of the ignition of the landing engine must be very precise, because its thrust is de facto constant, so that the rocket acceleration and the engine thrust must be equal just at the rocket contact with the ground.

Flow in beveled nozzle

In a supersonic flow in an beveled nozzle, the flow is deflected from the axial direction δ due to an expansion wave that originates at the edge of the shorter side of the nozzle, see Figure 106. The situation for an beleved Laval nozzle is identical to the flow around an obtuse angle at supersonic speed. The procedure for calculating the deflected flow from the axial direction δ is given, for example, in [Kadrnožka, 2004, Equations 3.6-10] or Prandtl-Meyer function can also be used.

– 106: –

left-convergent nozzle; right-convergent-divergent nozzle. α [°] nozzle cutting angle; μ [°] Mach angle; δ [°] jet deflection from nozzle axis. The expansion of gas from pressure p₁ starts at line A-C and finishes at line A-C' where pressure p₂ is set.

Influence of internal friction on nozzle efficiency

The loss can be calculated from the energy parameters of the nozzle, which are the velocity coefficient φ and the nozzle efficiency η, these two quantities are defined by Formula 569.

– 569: –

φ [1] velocity coefficient; η [1] nozzle efficiency. The values of the velocity coefficient φ for nozzles are given in [Dejč, 1967, p. 328] for divergent nozzles and in [Dejč, 1967, p. 348] for Laval nozzles.

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Reduction in mass flow through nozzle due to flow contraction

The mass flow through the nozzle can be reduced not only due to internal friction in the fluid, but also due to contraction of the flow behind the nozzle throat (vena contracta) [Jarkovsky, 1958 p. 14]. This contraction is due to jet inertia, environmental effects, an increase in the thickness of the boundary layer in the throat and has the same effect on flow as the reduction in nozzle flow area, see Figure 761. In well-designed nozzles, the contraction of the jet is very small (A_min≈A'_min), whereas it is significant in orifices.

– 761: –

A'_min [m²] flow area in the constriction of the stream.

Calculation of mass flow through a nozzle using the nozzle flow coefficient

The actual mass flow through the nozzle is calculated using a flow coefficient that includes the effect of internal friction as well as stream contraction. The flow coefficient is defined as the ratio of the actual flow to the flow at isentropic expansion without contraction, see Equation 478. Values of flow coefficients of nozzles and orifices are given in [Dejč, 1967], [Jarkovský, 1958].

– 478: –

μ [1] nozzle flow coefficient; m^•_is [kg·s^-1] flow through nozzle at lossless flow.

Simplifying assumptions for using nozzle theory to calculate mass flow change through group of turbine stages

Two simplifying assumptions are introduced for approximate calculations of the change in flow through a larger group of stages. The first assumption is the adiabatic expansion and its constant value of the polytropic exponent even when the mass flow changes. The second assumption is to simplify the gradual change of the specific volume of gas v in the stage to a step change, where the specific volume always changes stepwise at the outlet of the blade passage, see Figure 1273. Based on these simplifications, Formula 994 can be derived.

– 1273: –

(a) evolution of specific volume change in multistage turbine; (b) specific volume change in multistage turbine under the simplifying assumption. v [m³·kg^-1] specific volume of working gas; x [m] length of stage group under investigation.

– 994: –

Indexes: nom-name state. The derivation of the formula for the approximate calculation of the change of flow through a large group of turbine stages is given in [Ambrož, et al., 1956, p. 315].

Application of Bendeman ellipse in calculating the flow change through group of turbine stages

The general Formula 994 has the disadvantage that it is necessary to look for the root of a polynomial with a general (non-integer) exponent. The solution is to simplify Formula 994 by applying Bendemann ellipse to the simpler quadratic Equation 995.

– 995: –

The derivation is shown in [Kadrnožka, 1987, s. 181].

Calculation of mass flow change through group of turbine stages at critical flow

If the critical pressure ratio occurs on the last blade cascade of a stage group, then the findings for critical nozzle flow can be applied to that stage group. This means that the equation for the flow should be the same as for the vacuum discharge (p_e=0), see Formula 996.

– 996: –

Derived from Equation 995 for vacuum expansion p_e=0.

Stodola's law

The above equations were first derived by Auler Stodola and are therefore referred to as Stodola's law.

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Solid propellant rocket engines

There are also solid rocket propellant (SRP), in which the fuel mixture is gradually burned off, producing very hot exhaust gases (Figure 511). The thrust vector in SRP engines is often controlled by an oblique shock wave generated by injecting liquid into the inside of the nozzle. The star-shaped cross-section of the fuel charge allows for the gradual burning of the fuel mixture and stable combustion. This star shape was systematically developed during World War II in England and culminated in the design of the Sergant SPR ballistic missile [Holt, 2017, p. 94-110].

– 511: –

1-combustion chamber; 2-fuel and oxidizer mixture; 3-critical nozzle flow area; 4-Laval nozzle.

Thrust control options for rocket engines on SRP

The disadvantages of SRP engines are limited ability to control thrust and the engine can only be ignited once. On the other hand, they are simpler than liquid fuel engines and, above all, more responsive (no refueling before launch) and have a significantly longer storage life, which is important for military use. There are also hybrid rocket engines, where the fuel is in solid form and the oxidizer is supplied from an external tank, this way it is possible to better control thrust. TPL engines can also be used repeatedly, for example, the first stages of the Space Shuttle, the so-called SRB engines.

References