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41. Flow of gases and steam through difusers

Introduction
Change state of gas inside diffuser
Flow through diffuser at losses
Efficiency of diffuser ● Diffuser efficiency at flow of liquid
Cone diffuser and similar of them
Ways for decreasing of sensitivity of flow separation ● Shapes of diffuser according requirements on pressure gradient ● Comparison of diffuser with constant pressure gradient and cone diffuser ● The shape of the diffuser with the lowest sensitivity to boundary layer separation
Supersonic diffusers
Problems at non-nominal states of diffusers
Some applications of diffuser theory
Non-nominal states of valve with diffuser ● Diffuser blade passage ● Ejectors and injectors ● Ramjet and Scramjet
References
Appendices – only in Czech language
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www.transformacni-technologie.cz; Copyright©Jiří Škorpík, 2006-2019. All rights reserved. This work was published without any linguistic and editorial revisions.
● 41. Flow of gases and steam through difusers ●

Introduction

Two base concepts of diffusers1.374 Two base concepts of diffusers
(a) subsonic diffuser shortly diffuser; (b) supersonic diffuser. A [m2] flow area of diffuser; c [m·s-1] velocity of gas; Ma [-] Mach number; A* [m2] critical flow area of supersonic diffuser, here velocity of gas is reached speed of sound (critical state of gas). Subscript i denotes state at inlet of the diffuser, subscript e denotes state at exit of the diffuser.

Change state of gas inside diffuser

Change of state quantities inside diffuser 2.723 Change of state quantities inside diffuser
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. Subscript c denotes stagnation state of gas.
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● 41. Flow of gases and steam through difusers ●
Change of state quantities of gas inside the supersonic diffuser 3.727 Change of state quantities of gas inside the supersonic diffuser
i* [J·kg-1critical enthalpy; a [m·s-1] speed of sound.
Mass flow rate through diffuser.
4.513 Mass flow rate through diffuser
v [m3·kg-1] specific volume; ε*c [-] critical pressure ratio; κ [-] isentropic index.

Flow through diffuser at losses

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● 41. Flow of gases and steam through difusers ●
Real compression inside diffusers.
5.98 Real compression inside diffusers
left inside subsonic diffuser; right inside supsonic diffuser. z [J·kg-1] losses – required increasing of inlet kinetic energy of gas for cover of losses*; Δpz [Pa] pressure drop – difference of stagnation pressures. Subscript iz denotes isentropic compression – compression without losses.
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● 41. Flow of gases and steam through difusers ●

Efficiency of diffuser

Efficiency of diffuser 6.405 Efficiency of diffuser
η [-] Efficiency of diffuser – referred to static state of gas*.

Diffuser efficiency at flow of liquid

Energy balance of diffuser at flow of liquid.
7.415 Energy balance of diffuser at flow of liquid
g [m·s-2] gravitational acceleration; yi, e [J·kg-1] specific total energy of liquid at inlet and at exit;
zi-e [J·kg-1] specific internal losses of diffuser; h [m] horizontal level of axis; g·h [J·kg-1] specific potential energy.
Hydraulic efficiency of diffuser 8.411 Hydraulic efficiency of diffuser
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● 41. Flow of gases and steam through difusers ●

Cone diffuser and similar of them

Cone diffuser 9.458 Cone diffuser
r [m] radius; α [°] angle of diverging; l [m] length of diffuser; x [m] axis scale.
Influence angle of diverging of cone diffuser on pressure drop 10.631 Influence angle of diverging of cone diffuser on pressure drop
Graph in scale is shown in [1, p. 382].
Principle of flow separation of boundary layer from diffuser wall and developed of recirculation 11.418 Principle of flow separation of boundary layer from diffuser wall and developed of recirculation
R.P. velocity profile.
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● 41. Flow of gases and steam through difusers ●
Practical solutions of space limited diffusers 12.427 Practical solutions of space limited diffusers

Ways for decreasing of sensitivity of flow separation

Development of velocity profile inside throat of diffuser 13.428 Development of velocity profile inside throat of diffuser
LP laminar flow; PP transition region of flow; TP turbulent flow. xe minimum length of diffuser throat for development turbulent flow of boundary layer.

Shapes of diffuser according requirements on pressure gradient

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● 41. Flow of gases and steam through difusers ●
Diffuser shape equation14.432 Diffuser shape equation
n [-] polytropic index. Polytropic index is equal isentropic index n=κ for case ideal compression. The equation is derived under simplifying assumptions that velocity of gas has main axis direction – the velocity is deflected from axis direction at walls in realy. The derivation is shown in the Appendix 432.
Design diffuser of circular cross-section corresponding to requirement dp/dx=konst. Parameters at the inlet of diffuser: 80 m·s-1, 110 kPa, 20 °C, dry air. Output parameters: p=114 kPa. The required diffuser length is 100 mm at an input radius of 20 mm. Calculated flow without losses. The solution of this problem is shown in the Appendix 441.
Problem 1.441
Figure at Problem 1 Figure at Problem 1.
(a) calculated profile of radius; (b) cone diffuser about the same length at α=23,18°.
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● 41. Flow of gases and steam through difusers ●

Comparison of diffuser with constant pressure gradient and cone diffuser

Find profile of pressure gradient inside a cone diffuser with lenght 100 mm, in let radius 20 mm, angle of diverging 23,18°. Inlet and outlet state of gass are the same as in Problem 1. Losses at flow are negligable. The solution of this problem is shown in the Appendix 456.
Problem 2.456
Figure at Problem 2 Figure at Problem 2.
Profil of pressure gradient in a cone diffuser. dp/dx [kPa·m-1]; x [mm]. The higher pressure gradient at the beginning of the diffuser is higher than in the case of Problem 1 because there is also a larger angle of expansion.
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● 41. Flow of gases and steam through difusers ●

The shape of the diffuser with the lowest sensitivity to boundary layer separation

Diffuser with linear pressure gradient 15.430 Diffuser with linear pressure gradient
dp/dx [kPa·m-1]; x [mm]. Diffuser on the Figure has parameters: dp/dx=400 kPa·m-1, Ri=10 mm, pi=110 kPa.
Practical design of diffusers with variable diverging 16.831 Practical design of diffusers with variable diverging

Supersonic diffusers

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Supersonic diffusers with stepes deceleration of flow17.552 Supersonic diffusers with stepes deceleration of flow
(a) stepped supersonic diffuser; (b), (c) stepped supersonic diffuser with downstream shock waves - as if reflected from the diffuser wall - which naturally directs the velocity vector in the axial direction and reduces losses [1, p. 409]. RV shock waves.

Problems at non-nominal states of diffusers

Influence of input velocity change on subsonic diffuser function 18.554 Influence of input velocity change on subsonic diffuser function
There are three cases with cia<cib<cic=a. In individual cases, the back pressure also changes, if it were still the same (pe=pea), the diffuser would behave like a short diffuser. At less than the critical pressure p*, a shock wave arises behind the narrowest cross-section and, in addition, when the back pressure decreases below pec the Laval nozzle becomes a diffuser see chapter 40. Flow inside de Laval nozzle at non-nominal states. L.T. Laval nozzle function area.
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● 41. Flow of gases and steam through difusers ●
Influence of input velocity change on supersonic diffuser function 19.654 Influence of input velocity change on supersonic diffuser function
There are three cases with cic<cia<cib>a. In individual cases, the back pressure also changes, if it were still the same (pe=pea), the diffuser would behave like a short diffuser. It is changed so that the subsonic parts of the diffuser do not produce a shock wave(5). In the variant c, the convergent part of the diffuser is not able to accommodate such an amount of gas (it will have a high resistance), therefore, before the diffuser a perpendicular shock wave is generated which increases the pressure above the critical and reduces the velocity to subsonic. Thus, the convergent part of the diffuser will function as a nozzle. The divergent portion of the diffuser will function as a Laval nozzle in a non-design state.

Some applications of diffuser theory

Non-nominal states of valve with diffuser

Valve with diffuser (partially open) 20.110 Valve with diffuser (partially open)
There is subsonic flow inside the valve. Flow control is done by changing the flow cross-section using the valve plug RK, which is either retracted (flow cross-section decreases) or extended (flow cross-section increases). At the narrowest point between the plug and the seat, the flow reaches the maximum speed, which again decreases in the diffuser.
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● 41. Flow of gases and steam through difusers ●
Control valve with diffuser inside steam turbine
21.69 Control valve with diffuser inside steam turbine
Condensing turbine about 25 MW at speed 3 000 min-1, one controlled steam extraction at 0,25 MPa for 80 t·h-1, pressure admission steam is 2 MPa at 390 °C, Made in PBS. Figure: [1].

Blade passage as CD nozzle

Geometric similarity of diffuser blade row with symmetrical diffuser 22.745 Geometric similarity of diffuser blade row with symmetrical diffuser
w [m·s-1] relative velocity or absolute velocity c [m·s-1] for case stator rows.
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● 41. Flow of gases and steam through difusers ●
Development of λ-shock wave in blade row of compressor 23.864 Development of λ-shock wave in blade row of compressor
RV shock wave and separtion of flow.
Example of arrangement of supersonic turbocompressor
24.770 Example of arrangement of supersonic turbocompressor
1 radial compressor impeller radiálního kompresoru; 2 diffuser blades with supersonic profile.

Ejectors and injectors

Main parts of the ejector
25.112 Main parts of the ejector
vlevo general scheme of ejector or injector; vpravo example of a steam ejector as the vacuum pump of a condenser [6]. p motive steam; v suction gas (mix of steam and inertial gas); 1 suction section; 2 diffuser throat (mixture section); 3 exit diffuzer. The function of the ejectors or injectors is based on the transmission of a portion of the kinetic energy of the motive fluid to the suction fluid. This happens approximately in the throat of the diffuser. Prior to this, however, it is necessary to suck in the suction fluid into the motive fluid jet exiting the nozzle (in this case the Laval nozzle), which happens at the boundary of the suction and mixing zone due to turbulence at the stream interface. In the diffuser part of the machine, kinetic energy is transformed into pressure energy.
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Energy balance of ejectors and injectors
26.404 Energy balance of ejectors and injectors
μ [-] ejection respectively injection ratio [1, p. 419]. u [J·kg-1] the specific internal thermal energy. The index p indicates the motive and the index v the sucktion working fluid. Δ is a change mark, for example, Δ(p/ρ)p means a change in the pressure energy of the motive between the inlet and the outlet. The derivation of the equation when neglecting the change in potential energy is given  Appendix 404. The ejector and injector are also calculated in [7], [1], [8], [9].
Design the basic dimensions of the jet pump for pumping water (injector) from an open tank at a temperature of 90 °C to a pressure of 0,54 MPa. The required water flow is 60 kg·h-1. You consider the efficiency of the diffuser part 80 %. The efficiency value of the nozzle also includes the efficiency of transferring kinetic energy from the steam to the pumped water and is 10 %. The steam velocity at the pump inlet is 20 m·s-1. The water velocity at the pump outlet is 3 m·s-1. Do not consider pressure losses in the boiler and in the piping. The solution of this problem is shown in the Appendix 410.
Problem 3.410

Ramjet and Scramjet

Simple ramjet operation.
27.114 Simple ramjet operation
a inlet throutling area; b outlet throutling area. 1 supersonic diffuser; 2 combustion chamber and fuel injection to subsonic flow of compressed air; 3 expansion of exhaust gas inside nozzle.
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Scramjet.
28.512 Scramjet
(a) description of function; (b) experimental unmanned aircraft X-43A with Scramjet*. 1 supersonic diffuser; 2 combustion chamber inside throutling and inlet of fuel to sonic flow; 3 expansion of exhaust through nozzle; 4 shock waves; 5 expansion waves.

References

1. DEJČ, Michail. Technická dynamika plynů, 1967. Vydání první. Praha: SNTL.

2. MAŠTOVSKÝ, Otakar. Hydromechanika, 1964. 2. vydání. Praha: Statní nakladatelství technické literatury.

3. JAPIKSE, David a N BAINES. Diffuser design technology. Norwich, VT: Concepts ETI, 1995. ISBN 0933283083.

4. MICHELE, F. a kol. Historie a současnost Parní turbíny v Brně, 2010. 3. rozšířené a doplněné vydání. Brno. ISBN: 978-80-902681-3-5.

5. KADRNOŽKA, Jaroslav. Tepelné turbíny a turbokompresory I, 2004. 1. vydání. Brno: Akademické nakladatelství CERM, s.r.o., ISBN 80-7204-346-3.

6. NOŽIČKA, Jiří. Osudy a proměny trysky Lavalovy, Bulletin asociace strojních inženýrů, 2000, č. 23. Praha: ASI, Technická 4, 166 07.

7. HIBŠ, Miroslav. Proudové přístroje, 1981. 2. vydání-přepracované. Praha: SNTL – Nakladatelství technické literatury, n. p., DT 621.694.

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8. KADRNOŽKA, Jaroslav. Tepelné elektrárny a teplárny. 1. vyd. Praha: SNTL-Nakladatelství technické literatury, 1984.

9. NECHLEBA, Miroslav, HUŠEK, Josef. Hydraulické stroje, 1966. Vydání první. Praha Státní nakladatelství technické literatury.

10. GOROŠČENKO, B. T. Aerodynamika rychlých letounů, 1952. Vydalo Technicko-vědecké vydavatelství. Překlad z Ruštiny.
  • GOROŠČENKO, B. T. Aerodynamika rychlých letounů, 1952. Vydalo Technicko-vědecké vydavatelství. Překlad z Ruštiny.
  • Citation this article

    This document is English version of the original in Czech language: ŠKORPÍK, Jiří. Proudění plynů a par difuzory, Transformační technologie, 2016-03, [last updated 2018-11-26]. Brno: Jiří Škorpík, [on-line] pokračující zdroj, ISSN 1804-8293. Dostupné z http://www.transformacni-technologie.cz/41.html. English version: Flow of gases and steam through diffusers. Web: http://www.transformacni-technologie.cz/en_41.html.

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