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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.

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__:

^{(1)}Remark- 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

3.514 Velocity of gas at the exit of a nozzle.p [Pa] atmospheric pressure. Gas properties: _{at}κ=1,4, r=287 J·kg, ^{-1}·K^{-1}t, _{i}=20 °Cp, _{i}=p_{at}c._{i}=0 |

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

5.515 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. According ^{*}_{c}Equation 4 should following a mass flow rate fall. In really the mass flow rate is a constant and equals m from point ^{*}ε to an expansion to vacuum ^{*}_{c}ε. The pressure ratio for the maximum mass flow rate of gas through the nozzle is called the _{c}=0critical 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. |

^{(2)}Bendemann ellipse- The curve
*1-a-0*is similar with an ellipse, therefore this curve is usually substituted by the ellipse for case routine calculations. This ellipse is called Bendemann ellipse:

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6.162 Approximate calculation of mass flow rate through the nozzle according Bendemann ellipse.This formula can be use only for p. The derivation of the equation of Bendemann ellipse is shown in the Appendix 162._{e}≥p^{*} |

The heat capacity ratio *κ* can be various for individual types of gases and therefore their critical pressure ratios can be also various:

gas ε*c [-] gas ε*c [-] ------------ ------------------------- H 0,527 air (dry) 0,528 He 0,487 superheated steam 0,546 CO2 0,540 saturated steam 0,577

Gas velocity reaches the __speed of sound__ at the critical and a lower pressure ratio in the nozzle throat (the narrowest area of the nozzle), this state of the flow is called a critical state of flow. The equations of flow for narrowest area of the nozzle be can derived by substituting equations for the critical pressure ratio (*Equation 5*) to the above equations for the gas velocity and mass flow rate:

These quantities are called critically (critical velocity, critical mass flow rate, critical pressure ratio...). **χ**_{max} these constants are listed in a tables according type of gases and pressure ratio at *c*_{i}=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__.

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 cm*^{2}. (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: *c*_{p}=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.

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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:

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These types of contours be can use for non-circular passages and blade passages.

The free flow at the exit of the nozzle can have two states^{(3, 4)}:

^{(3)}The velocity at the exit corresponds sub-critical or critical pressure ratio, p_{e}≥p^{*}.- The outlet gas stream is deformed and gradually mixes with the surrounding gas. In this case the state of the outlet gas (velocity, temperature, pressure) will be the same as the surrounding gas at some distance from the exit of the nozzle – the outlet gas is in thermodynamic balance with the surrounding gas:

10.984 The outflow from a converging nozzle at the critical pressure ratio.Photo from [3, p. 5]. |

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^{(4)}The pressure ratio is less than the critical ratio, p_{e}<p^{*}.- Expansion behind the nozzle continues and the velocity of gas increases to supersonic according the
*Equation 2*. 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).

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

11.103 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. |

12.983 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]. |

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

Ideal contour of the diverging section of de Laval nozzle is designed by the **method of characteristics**^{(5)}. 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)^{(6, 7)}.

^{(5)}Ideal contour of de Laval nozzle- This type of the nozzle 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*at_{r}*α*(condition of the exit velocity) and the flow area at exit_{e}=0°*A*[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:_{e}

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^{(6)}Linear (cone) contour of de Laval nozzle- 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*. These two parameters are sufficient to a calculation of the length of the diverging section of de Laval nozzle:_{e}

^{(7)}Bell nozzle- is the most common contour of the diverging section of de Laval nozzles for wide types of applications especially for rocket engines. This contour is descripted by an equation Rao (according Rao, G.V.R., which derived on base experiments [6], [8]) or by an equation Allman-Hoffman (according Allman J. G. and Hoffman J. D., which it was derived through a simplification of the Rao equation [7]). The Bell nozzle is shorter than the linear nozzle but has higher internal efficiency and the axial momentum of flow:

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For case of Rao equation are the boundary conditions for the inlet and the exit angle interdependent (

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 *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.

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.

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.

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Subscript **1** denotes a state in front the normal shock wave; index **2** denotes a state behind the normal shock wave.

^{(9)}p_{e}>p_{b}- 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*p*._{a}

^{(10)}p_{e}=p_{b}- 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*p*._{b}

^{(11)}p_{b}>p_{e}>p_{d}- 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
*p*._{c}

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^{(12)}p_{e}=p_{d}- At this back-pressure the normal shock waves arises exactly at the exit of the nozzle.

^{(13)}p_{d}>p_{e}>p_{n}- 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
*p*)_{e}<p^{*}

^{(14)}p_{n}>p_{e}- At this back-pressure continued the expand of the working gas behind the exit of the nozzle. Similar as previous case arise sonic effects.

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 2*, if the back-pressure is increased about *0,52 MPa*. The solution of this problem is shown in the Appendix 862.

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].

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. This expansion fan is developed on shortly side of the nozzle. Similar situation arises for case a blade passage at the end blade passage (see a chapter lower Nozzle as blade passage):

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A change of the back-pressure *p _{2}* at the end of the converging nozzle influences the direct of the outlet stream through three possible ways

^{(15)}p_{2}>p*- In the throat of the nozzle is the pressure
*p*because the flow is subsonic. The direct of the outlet stream is the same as axis of the nozzle._{2}

^{(16)}p_{2}=p*- If the back-pressure
*p*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._{2}

^{(17)}p_{2}<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.

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

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:

19.108 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. |

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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

If the heat capacity ratio

The nozzle loss is can calculated through energy parameters of the nozzle as a **velocity coefficient** **φ** and a **nozzle efficiency** **η**:

The description of the static state profile inside the nozzle or comparing two different nozzles cam be through a **polytropic index**. Mean value of the polytropic index can be derived from equation for difference of specific enthalpy between two states of the gas:

21.883 The equation for calculation mean value of polytropic index between two state of gas.n [-] polytropic index. |

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.

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:

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22.761 Contraction of flow inside the nozzle.A' [m_{min}^{2}] flow area of nozzle at contraction. The contraction for cases perfectly made nozzles is very small (A), it is the bigger for case _{min}≈A'_{min}orfice plates. |

Real mass flow of the nozzle is calculated by **mass flow coefficient** (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:

23.478 Mass flow coefficient.μ [-] mass flow coefficient.; m [kg·s^{•}_{iz}^{-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].

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:

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

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 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):

26.511 Solid propellant rocket engine.1 combustion chamber; 2 mix of fuel and oxidizer (its surface infuences mass flow rate of exhaust gas); 3 throat of nozzle; 4 diverging nozzle. Thrust vector is usually regulated through an oblique shock wave. |

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● 40. Flow of gases and steam through nozzles ●

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__.

The blade passages of one stage of the turbine can be compared with two nozzles which working at series^{(19)}. 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.

^{(19)}The flow through the stage of the turbine as the flow through two nozzle in row- 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:

Last equation be can simplification through Bendeman ellipse :

28.995 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]. |

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The *Equation 28* has a less accuracy than the *Equation 27*, but is simpler and its solve is the same as a quadratic equation. The *Equation 27* has non-quadratic solve.

The *Equations 27* and *28* 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*):

29.996 Flow through the group of stages at critical pressure ratio of the last of blade row.Derived from Equation 28 for p._{e}=0 |

Last blade row with critical state are use e.g. by condensing turbine. An example of calculation change of flow rate through steam turbine is shown in chapter __25. Consumption characteristics of steam turbines at change state of steam__.

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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 2018-04-10]. Brno: Jiří Škorpík, [on-line] pokračující zdroj, ISSN 1804-8293. Dostupné z http://www.transformacni-technologie.cz/40.html. English version: Flow of gases and steam through nozzles. Web: http://www.transformacni-technologie.cz/en_40.html.

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