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40. Flow of gases and steam through nozzles

## Introduction

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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## Converging nozzle

### ● Velocity inside nozzle

The velocity of gas at the exit of the nozzle is function pressure at the inlet pi and at the exit pe (back-pressure) of the nozzle:

416 State change of gas inside the nozzle
A [m2] area of flow; c [m·s-1] velocity of gas; i [J·kg-1] specific enthalpy of gas; s [J·kg-1·K-1] specific entropy; t [°C] temperature of gas; p [Pa] pressure of gas. The subscript i denotes the state at the inlet of nozzle, the subscript e denotes the state at the exit of nozzle and the subscript c denotes the stagnation state of the gas.

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.

101 The flow velocity at the exit of the nozzle in i-s diagram at isentropic expansion
up outlet velocity of gas as function of static state of gas at inlet of nozzle; down formulas of outlet velocity of gas as function of stagnation state of gas at inlet of nozzle. κ [-] heat capacity ratio; r [J·kg-1·K-1] individual gas constant; T [K] absolute temperature of gas; p [Pa] pressure of gas; ε [-] pressure ratio of static pressures (pe·p-1i); εc [-] pressure ratio at stagnation inlet pressure (pe·p-1ic). This equation is called Saint Vénantova-Wantzel equation [2, p. 350]. This equation is derived for flow of ideal gas without friction and at negligible change of potential energy, see Appendix 101.
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40. Flow of gases and steam through nozzles

The velocity of gas ce is function the inlet temperature Ti and pressure pi according the Equation 101, p. 1 and a maximum velocity is reached at expansion in the vacuum pe=0:

514 Velocity of gas at the exit of a nozzle
pat [Pa] atmospheric pressure. Gas properties: κ=1,4, r=287 J·kg-1·K-1, ti=20 °C, pi=pat, ci=0.

### ● Nozzle mass flow rate and critical pressure ratio

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

334 The equation of mass flow rate through the nozzle
m [kg·s-1] mass flow rate of gas through nozzle; v [m3·kg-1] specific volume of gas; χm [-] variable parts of equation that is function of pressure ratio. The derivation of the equation for a calculation of the mass flow rate of gas through the nozzle is shown in the Appendix 334.

According this equation is true, if the pressure on the exit nozzle pe is decreasing then the mass flow rate m is increased only to a pressure ratio ε where the mass flow rate should be decreasing:

515 The maximum of mass flow rate of gas through the nozzle
The curve 1-a-0 corresponds to Equation 4(2). The maximum mass flow rate m* is reached at a pressure ratio ε*c. According Equation 4 should following a mass flow rate fall. In really the mass flow rate is a constant and equals m* from point ε*c to an expansion to vacuum εc=0. The pressure ratio for the maximum mass flow rate of gas through the nozzle is called the critical pressure ratio (symbol of asterisk*). The derivation of the equation for a calculation of the critical pressure ratio of gas is shown in the Appendix 515.

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:

162 Approximate calculation of mass flow rate through the nozzle according Bendemann ellipse
This formula can be use only for pe≥p*. The derivation of the equation of Bendemann ellipse is shown in the Appendix 162.

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.

516 The equations of the critical flow in narrowest area of the nozzle
These quantities are called critically (critical velocity, critical mass flow rate, critical pressure ratio...). χmax these constants are listed in a tables according type of gases and pressure ratio at ci=0; i* [J·kg-1] critical enthalpy (at critical enthalpy reaches isentropic expansion the critical velocity).

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.

Problem 102
The air flows through a nozzle, its velocity is 250 m·s-1, its pressure is 1 MPa, its temperature is 350 °C at the inlet of the nozzle. Surroundings pressure behind the nozzle is 0,25 MPa. The narrowest area of the nozzle has 15 cm2. (a) find if the flow through the nozzle is critical flow. (b) calculate the outlet velocity and (c) the mass flow rate of air. The properties of air are: cp=1,01 kJ·kg-1·K-1, r=287 J·kg-1·K-1, κ=1,4. You do not solve a situation behind the nozzle. The solution of this problem is shown in the Appendix 102.

### ● Ideal contour of converging nozzle

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.

475 Influence contour of the nozzle on the direction of the outlet velocity
(a) cone nozzle; (b) ideal contour of nozzle; (c), (d), (e) usually contours of nozzles; (c) so called Vitoshinsk nozzle or Vitoshinsk converging nozzle [4, p. 320], [16, p. 13] (use for reduction passage between two passages with different diameter or as blower nozzle of wind tunels); (d) contour of nozzle by lemniscate ; (e) contour of nozzle for outlet of bottles (rr≈1,5·re [5, p. 80]); r [m] radius of nozzle; l [m] length of nozzle. The cone nozzle has lower a mass flow coefficient than ideal contour nozzle (see Formula 23).
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40. Flow of gases and steam through nozzles

### ● State at exit of converging nozzle

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 (pe≥p*), or the pressure ratio is less than critical (pe<p*).

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.

984 The outflow from a converging nozzle at the critical pressure ratio
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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## De Laval nozzle (converging-diverging nozzle)

For better efficiency of gas expansion behind the narrowest area of the converging nozzle (it is the case p*>pe) 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:

103 De Laval nozzle (CD nozzle)-direction of expansion
(a) converging section of nozzle; (b) divergent section of nozzle. Ma [-] Mach number; l [m] lengh of diverging section of nozzle. The velocity of gas is subsonic Ma<1 in the converging section, and is sonic Ma=1 in the narrowest area (throat), is supersonic Ma>1 in the diverging section.

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
983 Supersonic exhaust of gas from de Laval nozzle
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.

517 i-s diagram used in the description ideal expansion of gas through a CD nozzle

### ● Frequent contours of de Laval nozzles

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 rr at αe=0° (condition of the exit velocity) and the flow area at exit Ae [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:

of the exit velocity) and the flow area at exit Ae [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:

993 Ideal contours of diverging section of de Laval nozzle
α [°] angle of diverging section; t [m] inlet length of diverging section of de Laval nozzle (usually circle contour with radius rr≐0,382·r* [5, p. 80]). The ideal contour of nozzle is designed for maximum momentum of flow in axis direction. The expansion waves are shown on figure also. The derivation of equations for calculation the inlet length of diverging section of de Laval nozzle are shown in the Appendix 993.

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 Ae. 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 Ae. These two parameters are sufficient to a calculation of the length of the diverging section of de Laval nozzle.

88 The linear contour of the Laval nozzle
(a) equation of contour of nozzle for interval t-e; (b) equation of length of nozzle; (c) boundary conditions for calculation of constants a1, a2. The field velocity at the exit is not uniform. The deviation of the velocity from the axis causes losses momentum of the exit flow in axis direction (at the angle α=20° probably 1 % [5, p. 78]). The derivation of equation of the length of linear (cone) de Laval nozzle are shown in the Appendix 88.

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 (αt=f(αe)). Choice of optimal pair of intial αt and the exit angle αe can be from length equivalent linear nozzle at α=30° see tables and charts in [5, p. 80]. The solving of Allman-Hoffman equation requires only the angle αt. This type of nozzle have approximately 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.

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.

16.335 The contour of Bell nozzle
(a) equation of contour of nozzle for interval t-e by Rao; (b) equation of contour of nozzle for interval t-e by Allman-Hoffman; (c) boundary conditions for calculation of constants a1..a4 or b1..b3(8).

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

Problem 104
Calculate a diverging section (cone contour) of the nozzle from the Problem 1. Calculate Mach number on the exit of the diverging section. The flare angle of the diverging section is 10°. The solution of this problem is shown in the Appendix 104.
Problem 336
Water steam flows through a de Laval nozzle. Pressure and temperature of the water steam is 80 bar respectively 500 °C at input to the nozzle. The leaving pressure is 10 bar. The mass flow rate of water steam must be 0,3 kg·s-1. Calculate base dimensions of the nozzle and state of water steam at the exit. The flare angle of the diverging section is α=10°. The solution of this problem is shown in the Appendix 336.
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40. Flow of gases and steam through nozzles

### ● Flow in overexpanded and underexpanded Laval nozzle

For case good design of de Laval nozzle is reached the pressure pn 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 pe>pn (overexpansion nozzle) then the length of the divergent section of the nozzle is longer than it is need. If pe<pn (underexpansion nozzle) then the length of the divergent section of the nozzle is shorter than it is need. At pressure which is the bigger than the designated pressure can arise the normal shock wave inside de Laval nozzle, see Figure 105.

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].

105 De Laval nozzle – characteristics at several back-pressures
Subscript 1 denotes a state in front the normal shock wave; index 2 denotes a state behind the normal shock wave. pe>pb at this back-pressure the velocity at the throat does not reach speed of sound-it does not reach critical state, the curve a. Inside the converging section of the nozzle is done subsonic compression (increasing of pressure and decreasing of velocity-in this case the converging section of the nozzle works as a diffuser) to the pressure pa; pe=pb at this back-pressure the velocity at the throat reaches speed of sound-critical state, the curve b. Inside the converging section of the nozzle is done subsonic compression to the pressure pb; pb>pe>pd at this back-pressure the velocity at somewhere inside converging section of the nozzle occurs non-continuity of the velocity and pressure (is developed the normal shock wave). Behind the normal shock wave is subsonic flow and the gas is being compressed to pressure pc; pe=pd at this back-pressure the normal shock waves arises exactly at the exit of the nozzle; pd>pe>pn at this back-pressure the normal shock waves arises behind the exit of the nozzle. This normal shock wave is unsteady (free flow) and it alternately develops and disappears (similar situation is in case an converging nozzle where back-pressure is smaller than is the critical pressure pe<p*); pn>pe at this back-pressure continued the expand of the working gas behind the exit of the nozzle. Similar as previous case arise sonic effects.
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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.

Problem 862
Find the approximate position of normal shock wave inside de Laval nozzle from Problem 104, p. 6, if the back-pressure is increased about 0,52 MPa. The solution of this problem is shown in the Appendix 862.
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## Flow through oblique cut nozzle

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 p1 on the cross section A-C and the end of expansion is at back-pressure p2 on the cross section A-C'. Therefore the oblique cut Laval nozzle is better for variable backpressure than classic Laval nozzle.

106 Flow through an oblique cut nozzle-for a case the critical state in the throat
left converging nozzle; right de Laval nozzle. μ [°] Mach angle; δ [°] deviation of stream from nozzle axis. p2>p* in the throat of the nozzle is the pressure p2 because the flow is subsonic. The direct of the outlet stream is the same as axis of the nozzle; p2=p* if the back-pressure p2 is the critical pressure then this pressure must be in the throat of the nozzle. The velocity stream is equal the sound velocity at this location. The expansion is stopped and the stream does not deviate from the axis of the nozzle; p2<p* the critical pressure is in the throat (cross section A-C) and the back-pressure is at the cross section A-C'. Between the cross section A-C and A-C' there is an expansion fan. The stream is deviated from the axis of nozzle with angle δ through the expansion fan.
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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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## Flow through nozzle at losses

### ● Losses through friction and turbulence

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.

108 Flow through the nozzle at the losses
z [J·kg-1] specific loss in the nozzle. Index iz denotes the state for case the isentropic expansion.

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 p* see chapter viz kapitola 38. Flow of gas through channel with constant flow area. In addition, if the heat capacity ratio κ is not the same as at isentropic expansion, then konetic energy of speed sound is not the same as isentropic expansion (see speed of sound formula). It means, enthalpy is changed i*≠i*iz.

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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 38. Flow of gas through channel with constant flow area. In addition, if the heat capacity ratio κ is not the same as at isentropic expansion, then konetic energy of speed sound is not the same as isentropic expansion (see speed of sound formula). It means, enthalpy is changed i*≠i*iz.

### ● Efficiency of nozzle

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

569 The energy parameters of the nozzle
φ [-] velocity coefficient; η [-] nozzle efficiency. The values of the velocity coefficients φ are shown in [4, p. 328] (for convergent nozzle including the cone nozzles) a [4, p. 348] (for Laval nozzles).

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:

883 The equation for calculation mean value of polytropic index between two state of gas
n [-] polytropic index.
Problem 109
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 0,2 kg·s-1. The stagnation pressure at the inlet of the nozzle is 200 kPa and back-pressure is 20 kPa. The velocity coefficient of the nozzle is 0,95. The solution of this problem is shown in the Appendix 109.

### ● Contraction of flow and discharge coefficient

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:

761 Contraction of flow in the nozzle
A'min [m2] flow area of nozzle at contraction. The contraction for perfectly contour nozzles is small (Amin≈A'min), it is the bigger for case orfice plates.

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:

478 Mass flow coefficient
μ [-] discharge coefficient; miz [kg·s-1] mass flow rate at flow without loss (isentropic expansion).

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

## Some applications of nozzle theory

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.

### ● Nozzle as blade passage

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:

111 The supersconic flow at the exit of blade passage
(a) convergent blade passage; (b) blade passge for supersonic flow. A [m2] flow area; δ [°] deflection of supersonic flow from axis of passage. The procedure of calculation of angle δ is shown in [3, Equation 3.6-10] or be can use Prandtl-Meyer equation. Photos of high velocity flow of gas through blade rows are shown in chapter 16. Aerodynamics of airfoils and blade rows at compressible flow.

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

### ● Flow through group of nozzles, flow through group of turbine stages

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 (pe=0), see Formula 996, p. 12.

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40. Flow of gases and steam through nozzles
994 Approximate formula of change of flow mass rate through the group of turbine stages
(a) profile of specific volume inside stages; (b) profile of specific volume inside stage according second assumption. n [-] polytropic index of flow through group of stages; x [m] length of group of stages. Indexes: R rotor, S stator, j nominal state; z number of stages; k number of stage. Derivation of the equation of approximate calculation of change of flow rate through the group of turbine stages is shown in [14, p. 315].
995 Approximate formula of change of flow rate through the group of turbine stages which is derived use Bendeman ellipse
The derivation is shown in [13, p. 181].
996 Flow through the group of stages at critical pressure ratio of the last of blade row
Derived from Formula 995 for pe=0.

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

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 TH2O=3517 K at molar mass MH2O=18 kg·mol-1) are the suitable fuels of the rocket engines:

113 The liquid propellant rocket engine and formula of exhaust gas velocity at the exit
1 oxidizer; 2 fuel; 3a pump oxidizer; 3b fuel pump; 4 combustion chamber; 5 exit of Laval nozzle; 6 resource of hot gasess for turbine; 7 turbine; 8 turbine exhaust. T [N] thrust; R [J·mol-1·K-1] Avogadro gas constant (8314 J·mol-1·K-1); M [kg·mol-1] molar mass of exhaust gas.

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):

511 Solid propellant rocket engine
1 combustion chamber; 2 mix of fuel and oxidizer; 3 throat of nozzle; 4 diverging nozzle. Thrust vector is usually regulated through an oblique shock wave. Star cross-section allows gradual burning of fuel mixture and stable combustion. This shape was consistently developed during World War II in England and culminated in the construction of a ballistic missile on the TPL Sergant type [18, pp. 94-110].
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## References

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