Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Converging nozzle . . . . . . . . . . . . . . . . . . . . . 1
Ideal contour of converging nozzle . . . . . . . . . . . . 3
State at exit of converging nozzle . . . . . . . . . . . . 4
Frequent contours of de Laval nozzles . . . . . . . . . 6
Flow through oblique cut nozzle . . . . . . . . . . 10
Flow through nozzle at losses . . . . . . . . . . . . 11

Efficiency of nozzle . . . . . . . . . . . . . . . . . . . . . 12
Some applications of nozzle theory . . . . . . . . 13
Nozzle as blade passage . . . . . . . . . . . . . . . . . 13
Rocket engine . . . . . . . . . . . . . . . . . . . . . . . . . 14
Flow through group of nozzles, flow through group of turbine stages . . . . . . . . . . . . . . . . . . . . . . . . 15
References . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Appendices . . (13 pages – only in Czech language)

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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 p_{e} (backpressure) of the nozzle:
Equation of outlet velocity can be derived from First law of thermodynamics for open system:
The velocity of gas c_{e} is function the inlet temperature T_{i} and pressure p_{i} according the Equation 2 and a maximum velocity is reached at expansion in the vacuum p_{e}=0:
3.514 Velocity of gas at the exit of a nozzle. p_{at} [Pa] atmospheric pressure. Gas properties: κ=1,4, r=287 J·kg^{1}·K^{1}, t_{i}=20 °C, p_{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 m^{•} is increased only to a pressure ratio ε where the mass flow rate should be decreasing:
5.515 The maximum of mass flow rate of gas through the nozzle. The curve 1a0 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 ε_{c}=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. 
6.162 Approximate calculation of mass flow rate through the nozzle according Bendemann ellipse. This formula can be use only for p_{e}≥p^{*}. The derivation of the equation of Bendemann ellipse is shown in the Appendix 162. 
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 0,5777.699 The critical pressure ratios of some gases.
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:
3D plot of the equation for mass flow rate of gas as function the inlet pressurea and the backpressure is called flow rate cone of the nozzle.
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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:
These types of contours be can use for noncircular passages and blade passages.
The free flow at the exit of the nozzle can have two states^{(3, 4)}:
10.984 The outflow from a converging nozzle at the critical pressure ratio. Photo from [3, p. 5]. 
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:
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]. 
13.517 is 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 (firstorder, secondorder and the like)^{(6, 7)}.
A comparison all method of design of contour of the diverging nozzle are shown in [9].
For case good design of de Laval nozzle is reached the pressure p_{n} during expansion, which is the same as the backpressure, this pressure is called the designated pressure of the nozzle. The nonnominalstate 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 p_{e}>p_{n} (overexpansion nozzle) then the length of the divergent section of the nozzle is longer than it is need. If p_{e}<p_{n} (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.
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 position of normal shock wave inside the diverging section of the Laval nozzle is can calculate through the RankineHugoniot equations for stable normal shock wave:
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 backpressure 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):
A change of the backpressure p_{2} at the end of the converging nozzle influences the direct of the outlet stream through three possible ways^{(1517)}:
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 AC and the end of expansion is at backpressure p_{2} on the cross section AC'. Therefore the oblique cut Laval nozzle is better for variable backpressure than classic Laval nozzle.
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. 
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^{(18)}. 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.
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. 
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:
22.761 Contraction of flow inside the nozzle. A'_{min} [m^{2}] flow area of nozzle at contraction. The contraction for cases perfectly made nozzles is very small (A_{min}≈A'_{min}), it is the bigger for case 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^{•}_{iz} [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].
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 onestage 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 M_{H2O}=18 kg·mol^{1}) are the suitable fuels of the rocket engines:
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. 
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.
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]. 
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 nonquadratic 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.
This document is English version of the original in Czech language: ŠKORPÍK, Jiří. Proudění plynů a par tryskami, Transformační technologie, 200602, [last updated 20180410]. Brno: Jiří Škorpík, [online] pokračující zdroj, ISSN 18048293. Dostupné z http://www.transformacnitechnologie.cz/40.html. English version: Flow of gases and steam through nozzles. Web: http://www.transformacnitechnologie.cz/en_40.html.