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Introduction . 1

● Velocity inside nozzle 1 ● Nozzle mass flow rate and critical pressure ratio 2 ● calculation of nozzle gas state [P.102] 3 ● Ideal contour of converging nozzle 3 ● State at exit of converging nozzle 4

● Nozzle as blade passage 11 ● Flow through group of nozzles, flow through group of turbine stages and Stodola law 11 ● Rocket engine 12

References 13

● 88 Linear (conical) Laval nozzle length formula 15 ● 101 Gas velocity at nozzle outlet 15 ● 102 Problem solving 15 ● 104 Problem solving 17 ● 109 Problem solving 18 ● Deriving the equation of Bendemann ellipse 19 ● 334 Mass flow of gas through the nozzle 19 ● 336 Problem solving 20 ● 515 Maximum gas mass flow through the nozzle 21 ● 862 Problem solving 21 ● 993 Formula for calculating the inlet part of the divergent section of the Laval nozzle 25

The article of online continued resource Transformační technologie; ISSN 1804-8293;

www.transformacni-technologie.cz; Copyright©Jiří Škorpík, 2006-2020. All rights reserved. This work was published without any linguistic and editorial revisions.

www.transformacni-technologie.cz; Copyright©Jiří Škorpík, 2006-2020. All rights reserved. This work was published without any linguistic and editorial revisions.

40. Flow of gases and steam through nozzles

A nozzle is a channel with stepless variable of flow area. Flowing of fluid through the nozzle is a process especially with a decrease pressure and an increase kinetic energy.

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The velocity of gas at the exit of the nozzle is function pressure at the inlet *p _{i}* and at the exit

Equation of outlet velocity can be derived from __First law of thermodynamics for open system__. For description of liquid flow through nozzle (change density is negligible) is use Bernoulli equation.

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40. Flow of gases and steam through nozzles

The velocity of gas *c _{e}* is function the inlet temperature

The mass flow rate of gas through the nozzle is calculated from the continuity equation:

According this equation is true, if the pressure on the exit nozzle *p _{e}* is decreasing then the mass flow rate

The curve

The curve *1-a-0* on Figure 515 is similar with an ellipse, therefore this curve is usually substituted by the ellipse for case routine calculations. This ellipse is called Bendemann ellipse:

This formula can be use only for

The critical pressure ratio is a function of the type of gas because the adiabate constant *κ* differs from one gas to another, for example the critical pressure ratio for hydrogen is *0.527*, dry air *0.528*, superheated water vapor *0.546*, saturated water vapor *0.577*. Thus, the critical pressure ratio is around *0.5*.

At a critical or lower pressure ratio, the flow velocity at the narrowest point of the nozzle reaches the speed of sound, this flow condition is called the critical flow condition. Thus, by substituting the critical pressure ratio of Formula 515 in Formula 101 and Formula 334, formulas can be obtained for the narrowest point of the nozzle when the critical pressure ratio is reached or overcome, see Formula 516.

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40. Flow of gases and steam through nozzles

flow condition is called the critical flow condition. Thus, by substituting the critical pressure ratio of Formula 515 in Formula 101 and Formula 334, formulas can be obtained for the narrowest point of the nozzle when the critical pressure ratio is reached or overcome, see Formula 516.

These quantities are called critically (critical velocity, critical mass flow rate, critical pressure ratio...).

3D plot of the equation for mass flow rate of gas as function the inlet pressurea and the back-pressure is called __flow rate cone of the nozzle__.

The air flows through a nozzle, its velocity is

An ideal contour of the nozzle is smooth, parallel with streamlines (on the inlet even the exit to avoid not a rise of turbulence through sudden change of direction of flow velocity at the wall), on the exit must be uniform velocity field (this condition is confirmed by experiments [4, p. 319]). It means the outlet velocity should be in axial direction of the nozzle. This condition must also satisfy the streamlines at the wall of the nozzle. Figure 475 shows the usual converging nozzle contour that can also be applied to non-circular channels and blade channels.

must also satisfy the streamlines at the wall of the nozzle. Figure 475 shows the usual converging nozzle contour that can also be applied to non-circular channels and blade channels.

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40. Flow of gases and steam through nozzles

From the above it is clear that at the outlet of the nozzle into the free surrounding two conditions can occur and the pressure ratio is higher or just critical (*p _{e}≥p^{*}*), or the pressure ratio is less than critical (

If the pressure ratio is greater than the critical, the jet at the nozzle outlet gradually begins to brake and mix with the surrounding gas. At a certain distance from the orifice, the velocity and temperature of the effluent gas will be balanced with the surrounding - it will be in thermodynamic equilibrium with the surrounding.

Photo from [3, p. 5].

If the pressure ratio is less than critical, then beyond the nozzle orifice, the gas further expands and its velocity increases according to Formula 101, p. 1 to supersonic. The gas stream area be must increased according __Hugoniot condition__. The divergent gas stream forms __oblique shock waves__ on border between the stream and the surroundings gas. These shock waves are reflected to the core of gas stream and they are decreased an efficiency of expansion (they cause pressure drop). The expansion is ended when the pressure is equal the surroundings pressure and a next process is similar the previous case (gradually mixes with the surrounding gas).

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For better efficiency of gas expansion behind the narrowest area of the converging nozzle (it is the case *p ^{*}>p_{e}*) is necessary made the appropriate conditions. It means a divergent channel must be added to the converging nozzle behind narrowest flow area of the nozzle (so called critical flow area, because the speed of sound is reached here) – this design is called as De Laval nozzle:

The exit velocity of the Laval nozzle is supersonic, and as it flows into the free space, the flow immediately begins to create shock waves - braking the supersonic jet by the surrounding gas:

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40. Flow of gases and steam through nozzles

The oblique lines inside flow are oblique shock waves, which arise at the exit edge of the nozzle and are reflecting from the boundary of the stream. Photo from [3, p. 23].

The i-s diagram has the same shape as the i-s diagram of the converging nozzle in Figure 416, except that the critical flow parameters are clearly indicated therein, see Figure 517.

Ideal contour of the diverging section of de Laval nozzle is designed by the method of characteristics. There are analytical methods of design of contour diverging nozzle, where the contour of the nozzle is approximated by a polynomial (first-order, second-order and the like).

The nozzles designed by the characteristic method (Figure 993) have the most uniform velocity field at the exit. The contour of the nozzle on interval *t-e* is calculated by the method of characteristics through construction __expansion waves__ inside the nozzle. As boundary condition is used the initial radius *r _{r}* at

of the exit velocity) and the flow area at exit *A _{e}* [4, p. 341], [5, p. 79]. The length of the ideal contour of the nozzle is longer than the nozzle with linear contour, therefore has lower internal efficiency due to internal friction of the working gas. Ideal contour of de Laval nozzle is used in supersonic wind tunels, where is requirement significant uniform velocity field at the outlet:

Conversely, the simplest shape is the linear shape of the Laval nozzle, see Figure 88, p. 6. These nozzles have simple calculation and simple manufactored, becouse has constant the angle *α* for whole length part *t-e*. The de Laval nozzles with cone contour are used as a supersonic blade row of one stage turbine (for cases where other losses of stage are very high and therefore production of complicated contour of nozzle does not economic). This simple contour is also used for small rocket engines, for small nozzles, for nozzle of injectors and ejectors etc. The calculation is composed from the specified of the angle of diverging *α* (usually in interval *8* up *30°*) and from calculated the flow area at exit *A _{e}*. These two parameters are sufficient to a calculation of the length of the diverging section of de Laval nozzle.

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40. Flow of gases and steam through nozzles

nozzle of injectors and ejectors etc. The calculation is composed from the specified of the angle of diverging *α* (usually in interval *8* up *30°*) and from calculated the flow area at exit *A _{e}*. These two parameters are sufficient to a calculation of the length of the diverging section of de Laval nozzle.

The most commonly used shape of the expanding part of the Laval nozzle (especially in rocket engines) is the so-called Bell nozzle (Figure 335). The shape of this nozzle is designed either according to the Rao equation (according to GVR Rao, who compiled this equation based on experiments [6], [8]), or the Allman-Hoffman equation (Allman JG and Hoffman JD) [7]); both equations are polynomials of the second degree (parabola). For case of Rao equation are the boundary conditions for the inlet and the exit angle interdependent (*α _{t}=f(α_{e})*). Choice of optimal pair of intial

about *0,2%* lower axial momentum of flow at expansion to vacuum than the nozzle profiled by Rao method [9]. The Allman-Hoffman equation is used for quicker optimization calculations at wide combinations of the states at the exit. The Bell nozzle is shorter than the linear nozzle but has higher internal efficiency and the axial momentum of flow.

A comparison all method of design of contour of the diverging nozzle are shown in [9].

Calculate a diverging section (cone contour) of the nozzle from the

Water steam flows through a de Laval nozzle. Pressure and temperature of the water steam is

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40. Flow of gases and steam through nozzles

For case good design of de Laval nozzle is reached the pressure *p _{n}* during expansion, which is the same as the back-pressure, this pressure is called the designated pressure of the nozzle. The non-nominalstate are changed the parameters of working gas at the inlet or the exit of nozzle. These parameters are changed from various causes (e.g. a control of the mass flow rate). If

A develop of the normal shock wave inside divergent section of the nozzle be can assumed from the Hugoniot condition. A smooth change the supersonic flow on the subsonic flow is allowed only in throat of the nozzle.

The normal shock wave inside the nozzle is not usually stable [4, p. 363] therefore it can cause a vibration of the nozzle and connected machines, and increases noise.

A operation back-pressure has an influence on length of the nozzle of a rocket engine. During flight of a rocket through atmosphere is changed surrounding pressure with the altitude. Therefore the nozzles of first stage are designed on atmospheric pressure (pressure near ground) and the next stages are designed on smaller pressure (according the ignition altitude). The engines of last stage are designed on expansion to vacuum [1].

Subscript

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40. Flow of gases and steam through nozzles

The position of normal shock wave inside the diverging section of the Laval nozzle is can calculate through the __Rankine-Hugoniot equations__ for stable normal shock wave.

Find the approximate position of normal shock wave inside de Laval nozzle from Problem 104, p. 6, if the back-pressure is increased about

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If there is supersonic flow inside a oblique cut nozzle then the stream of the gas is deviated through an __expansion fan__ from the nozzle axis (Figure 106). This expansion fan is developed on shortly side of the nozzle. The flow through oblique cut Laval nozzle is the same as the supersonic flow around an obtuse angle. The start of expansion is at pressure *p _{1}* on the cross section

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40. Flow of gases and steam through nozzles

Similar situation arises for case a blade passage at the end blade passage (see a subchapter lower Nozzle as blade passage p. 11).

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In the previous paragraphs is described isentropic expansion of the working gas in the nozzle. But the expansion in the nozzle is influenced by a friction or also __internal loss heat__ that arises through inner friction of the gas and a friction of gas on a wall of the nozzle. This friction heat decreases of the kinetic energy of the gas at the end of the nozzle. The friction heat is a loss between the kinetic energy at the nozzle exit for case isentropic expansion and the actual kinetic energy at the nozzle exit.

In addition, turbulence are formed in the stream in which the compressed energy is transformed into kinetic and vice versa, so that the vortices are at a different temperature than the surrounding gas and the heat transfer causes irreversible losses associated with the growth of entropy (a known effect from throttling gases).

These losses increase gas entropy inside the nozzle, see Figure 108.

At pressure *p ^{*}_{iz}* can be the velocity in core of the stream equal as sound velocity. But the velocity of flow can be subsonic on the periphery of the flow area due to boundary layer at the wall of the nozzle. Therefore the mean velocity at the throat of the nozzle is smaller than sound velocity respective the mean kinetic energy of the stream is smaller than kinetic energy for case sound velocity. The mean velocity stream is equal the sound velocity at the pressure

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40. Flow of gases and steam through nozzles

core of the stream equal as sound velocity. But the velocity of flow can be subsonic on the periphery of the flow area due to boundary layer at the wall of the nozzle. Therefore the mean velocity at the throat of the nozzle is smaller than sound velocity respective the mean kinetic energy of the stream is smaller than kinetic energy for case sound velocity. The mean velocity stream is equal the sound velocity at the pressure *p ^{*}* see chapter viz kapitola

The loss can be calculated through the velocity coefficient **φ** and a nozzle efficiency **η**:

The description of the static state profile inside the nozzle or comparing two different nozzles can be through a polytropic index. Mean value of the polytropic index can be derived from equation for difference of specific enthalpy between two states referred to in subsection of the gas 13. Adiabatic expansion inside heat turbine:

Calculate the throat area and the exit area of the de Laval nozzle and its efficiency. Through the de Laval nozzle flows the water steam saturation. The mass flow rate is

The mass flow is increased not only under internal friction of fluid but also under a contraction of flow behind narrowest area of the nozzle [15 p. 14]. The contraction of flow is caused by inertia flow and it has the same impact as an decreasing flow area of the nozzle:

Real mass flow of the nozzle is calculated by discharge coefficient, which takes into account influence of internal losses and the contraction of flow. The mass flow coefficient is ratio the real mass flow of the nozzle and isentropic mass flow without any contraction:

Values of the mass flow coefficients any types of the nozzles and the orffice plates are shown in [4], [15].

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40. Flow of gases and steam through nozzles

The theory of the nozzles can be aplicated on various type of flow devices. Through theory of the nozzles be can descripted complicated flow system.

The blade passage can have same shape as convergent nozzle or de Laval nozzle. The blade passage with de Laval profile is used for case supersonic velocity of working gas at the exit (decreasing of enthalpy between the inlet and the exit is under critical enthalpy *i**). This type of blade passage has properties as oblique cut CD nozzle:

Blade passage with supersonic flow are occured usually in small one-stage turbine and last stages of condesing turbines.

The theory of the nozzles be can use for calaculation of a flow through a group of turbine stages at change of state of the gas in front of or behind this group of the stages. There are a few theories of calculation (e.g. v [14], [13]). These theories are not in use currently, because are used numerical method. I describe here only the simplest method. The method is usefull for aproximate calculation see also __25. Consumption characteristics of steam turbines at change state of steam__.

usefull for aproximate calculation see also __25. Consumption characteristics of steam turbines at change state of steam__.

The blade passages of one stage of the turbine can be compared with two nozzles which working at series. It means the mass flow rate through both nozzles is the same. The same assumption can be applied to the group of with a few stages or on group a few nozzles which are in row, wherein the flow through the blade passages of the rotor must be calculated from the relative velocity.

Sufficient solving of calculation of change flow mass rate through the group of the stages be can reached by use only two simplifying assumptions. The assumption of adiabatic expansion and its constant polytropic index at change of mass flow rate is the first assumption. The change of specific volume at flow of the working gas through one stage is negligible and specific volume is suddenly changed at the exit of stage, it is the second assumption, see Formula 994, p. 12 – using Bendeman ellipse, this formula can be simplified to Formula 995, p. 12. The Formula 995 has a less accuracy than the Formula 994, but is simpler and its solve is the same as a quadratic equation. The Formula 994 has non-quadratic solve. Both formulas are also accurate for saturation vapour, but the reading of specific volume is not sufficient at gas near his saturation.

If critical state is indicated at the last blade row of the group of stages, then be can use knowledge for critical flow through nozzle. It means that the equation for mass flow for these conditions must be same as the equation at expansion to vacuum (*p _{e}=0*), see Formula 996, p. 12.

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40. Flow of gases and steam through nozzles

The derivation is shown in [13, p. 181].

Derived from Formula 995 for

Last blade row with critical state are use e.g. by condensing turbine. The formulas for flow through a group of nozzles were first derived by Auler Stodola and are therefore referred to as Stodol's rule.

Rocket engine is a reaction engine. The thrust of the engine is equal the momentum of exhaust gas flow at the exit. Main parts of the rocket engine is a combustion chamber and deLaval nozzle which is fastened at the exhaust of the combustion chamber. Inside the combustion chamber is burning an oxidizer and the fuel at develepment of the exhaust gas, which expands through the nozzle. Significant requirement is high velocity of the exhaust at the exit nozzle, because this is way reach of higher ration between the thrust and consumation of the fuel (ratio is called specific impulse). From a rearrange of the equation for the velocity at the exit of nozzle is evident, substances with high burning temperature and low molar weight, e.g. hydrogen (burning temperature of hydrogen is to *T _{H2O}=3517 K* at molar mass

The performance of the rocket engine is then given by the pressure in the combustion chamber and its size. For example, the required pressure in the combustion chamber of the Space Shuttle SSME engine was *20.3 MPa* and the turbine pump turbine power was *56 MW*. Oxidizer and fuel pumps are powered by combustion turbines that use fuel and engine oxidant as fuel or have other fuel [16, s. 25], [5].

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40. Flow of gases and steam through nozzles

example, the required pressure in the combustion chamber of the Space Shuttle SSME engine was *20.3 MPa* and the turbine pump turbine power was *56 MW*. Oxidizer and fuel pumps are powered by combustion turbines that use fuel and engine oxidant as fuel or have other fuel [16, s. 25], [5].

The solid propellant rocket engine use solid fuel. The hot exhaust gas is being arised at burning of the solid fuel. Thrust and burning of these engines not possible governing. Other side they are simpler than the liquid propellant rocket engines. There are hybrid solid propellant rocket engines with combination solid fuel and liquid oxidizer (this way be can regulation of thrust). The solid propellant rocket engines can be repeatedly use (e.g. the first stage of Space shuttle so called SRB):

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40. Flow of gases and steam through nozzles

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This document is English version of the original in Czech language: ŠKORPÍK, Jiří. Proudění plynů a par tryskami, *Transformační technologie*, 2006-02, [last updated 2020-01-31]. Brno: Jiří Škorpík, [on-line] pokračující zdroj, ISSN 1804-8293. Dostupné z https://www.transformacni-technologie.cz/40.html. English version: Flow of gases and steam through nozzles. Web: https://www.transformacni-technologie.cz/en_40.html.

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