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# 41. Flow of gases and steam through difusers

Author: Jiří Škorpík, skorpik.jiri@email.cz : last update 2016-03-09

A diffuser is a channel with stepless variable of flow area. Flow of fluid through the diffuser is a process especially with an increase pressure and a decrease kinetic energy. A ideal shape of the diffuser for supsonic flow is not the same as the shape of the diffuser for subsonic flow according the Hugoniot condition: the supsonic flow must be decreased on speed of sound inside convergent section of the diffuser:

1.374 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.

I use frequently terms same as in the article 40.  Flow of gases and steam through nozzles – because process inside the diffuser is oposite to processes inside the nozzle and equations for calculation are the same or similar.

## Change state of gas inside diffuser

Pressure, temeprature and density of gas is increased at flow through the diffuser. Energy for these increasing are obtained form kinetic energy of gas which is decreasing:

 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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The formula of outlet gas velocity of the diffuser is the same as the formula outlet velocity of the nozzle, which is shown in the chapter 40. Converging nozzle. The outlet gas velocity is function is function the inlet pressure pi and back-pressure pe. Last formula can be use for calculation the outlet gas velocity of the supersonic diffuser:

 3.727 Change of state quantities of gas inside the supersonic diffuser.i* [J·kg-1] critical enthalpy; a [m·s-1] speed of sound.

The mass flow rate gas through diffuser is depended to the inlet flow area Ai. On the contrary, the critical area of the supersonic diffuser A* can be calculated from mass flow rate m and state of gas at critical pressure p*, which is calculated from critical pressure ratio of gas:

4.513 Mass flow rate through diffuser.
v [m3·kg-1] spcific volume; ε*c [-] critical pressure ratio; κ [-] isentropic index.

## Flow through fiffuser at losses

In previus chapter is descripted adiabatic compression inside diffuser without losses respectively isentropic compression. Compression inside diffuser is also influenced by internal friction of gas, friction on wall of diffuser and losses heat, which decreasing stagnation pressure and increasing entropy of gas:

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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.
*Remark
There are other definitions of losses of diffuser e.g. [1, p. 387], but this definition is more practis, because is reached back-pressure for isentropic or real compression.

Speed of sound a at real compressin is the same as at isentropic compression, because speed of sound in ideal gas is function temperature only. It means that transition from supersonic to subsonic flow at real comprresion runs at lees pressure than isentropic compression p*<p*iz. This is caused by less velocity of gas at walls of diffuser than in core of flow, therefore mean velocity can be sonic at p* while in core of flow where it is supersonic.

Higher described facts mean, gas reaches speed of sound – mean velocity – in front of the throat of supersonic diffuser.

Construction of i-s diagram for new diffuser is composed under similarities with other diffusers or a calculation pressure drop. The pressure drop is function friction coefficient, which can be derived from equation of adiabatic flow at friction or simple for small change of density, according [2, p. 85].

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## Effecincy of diffuser

There are more definition of diffuser efficiency. Definition as ratio between enthalpy diference at isentropic and real compression is usually use, beacuse these states of gas are simplest found:

 6.405 Efficiency of diffuser.η [-] Efficiency of diffuser – refered to static state of gas*.
*Remark
The effciency refered to stagnation state ie,c,iz, ie,c is higher. It is evident from i-s diagrams.

The description of profile static state of gas inside diffuser or comaparing two different diffuser can be through polytropic index. The equation of mean value of the polytropic index is the same as the equation for case flow throuh the nozzle, which is shown in the chapter 40. Efficiency of nozzle.

At calculation new diffuser can be use similarities polytropic index of diffuser or their models. Acuracy this method depends to degree of similarities of comparating diffusers.

## Diffuser effciency at flow of liquid

In cases liquid, and so gases at small pressure ratio, is used Bernoulli equation for calculation of diffuser. Internal work of diffuser is zero ai=0, therefore total energy of liquid in front of diffuser must be equivalent total energy of liquid and losses behind diffuser:

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.

Efficiency these types of diffusers be can defined as ratio between tatal energy of liquid at outlet and inlet of diffuser, similar as hydraulic effciency of pums:

 8.411 Hydraulic efficiency of diffuser.
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## Cone diffuser and similar of them

Conical shape of the diffusers is simply made even in case of non-circular variants. An angle of diverging α is interval from 6 to 15° according [1, p. 391]. Most cone diffusers is made at interval 1012°. The shape of diffuser is near shape of diverging section of Laval nozzle:

 9.458 Cone diffuser.r [m] radius; α [°] angel of diverginng; l [m] lenght of diffuser; x [m] axis scale.

Influence angle of diverging on pressure drop of diffuser Δpz is evident at compare with stepless extension passage the same flow area:

 10.631 Influence angle of diverging of cone diffuser on pressure drop.Graph in scale is shown in [1, p. 382].

Pressure drop of diverging diffuser can be higher than the stepless extension passage according measurment at some angle of diverging. It is therefore, that internal friction losses decreasin with angle of diverging α but losses through stall and outlet recilculation* increasing with angle of diverging α. There only are vortex at flow through the stepless extension passage [2, p. 88], which caused growth of entropy according principle as at throtlling of flow through a orfice plate.

*Losses through stall and outlet recilculation
Stall and outlet reculation is caused by pressure drop inside boundary layer and next a flow separation from wall of diffuser. The flow separation is caused by decreasing of stagnation pressure inside boundary layer under static backpressure of diffuser. Decreasing total pressure inside boundary layer is caused losses of kinetic energy:
 11.418 Principle of flow sepearation of boundary layer from diffuser wall and developed of recirculation.R.P. velocity profile.
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Flow separation is developed by all cases, therefore short diffusers with higher angle of diverging (they are use at requiments on small size) is better their shape is a combination the diverging tube and the stepless extension no oposite, then vortex be develompmeted at the inlet even the outlet:

 12.427 Practis ?provedení? of diffuser inside small volume.

## Ways for incresing of sense on flow separation

The losses at flow separation is increasing with ?vzdálenost? from the exit of diffuser. Position of the flow separation be can infunce e.g. by increasing momentum at the walls of difuser, therefore turbulent flow is "méně" infunce than laminar flow – at turbulent flow is runn transport of momentum between perifery and core of flow. For tubulent flow is necesery development ?plně vyvinuté proudění? at outlet of diffuser. It is usually reached through a throat in front of diffuser:

 13.428 Develepment of velocity profile inside throat of diffuser.LP laminar flow; PP ?přechodová oblast?; TP turbulent flow. xe minimal lenght of diffuser throat for development turbulent flow of boundary layer.

Turbulation of flow be can also higher through variable ?vestavby? inside diffuser see [1, p. 395], [3]. Some ?vestaby? ?acts? tangential velocity of flow and ?odstředivá? force incresing pressure at wall of diffuser see Eulerova n-rovnice.

## Shapes of diffuser according requirements on pressure gradient

This shape is designed according require axis pressure gradient, respectively according function dp/dx=f(x). Equation of diffuser shape according ?navrhnout? function f(x) can be derived from equation of outlet velocity and continuity equation:

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14.432 Equation of diffuser shape.
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.

The solving of equation of diffuser shape is simple, e.g. ?diferenční počet? , if pressure gradient is known. In the first step is claculated the pressure for any point on the axis and the velocity and in the third step values for left side of the equation:

Designed of a diffuser with circle area for a require dp/dx=konst. Parameters at the inlet of diffuser: 80 m·s-1, 110 kPa, 20 °C, dry air. Outlet parameters: p=114 kPa. Requirement leght of diffuser is 100 mm. Radius at inlet is 20 mm. Calculated flow without losses.
Problem 1.441
```x [mm]  p [Pa]  c [m·s-1]  r [mm]      x [mm]  p [Pa]  c [m·s-1]  r [mm]
0       110000  80         20          50      112000  57,97      23,34
5       110200  78,07      20,23       60      112400  52,51      24,5
10      110400  76,08      20,48       65      112600  49,56      25,2
15      110600  74,05      20,75       70      112800  46,42      26,02
20      110800  71,97      21,03       75      113000  43,07      27
25      111000  69,82      21,34       80      113200  39,43      28,19
30      111200  67,61      21,67       85      113400  35,43      29,72
35      111400  65,33      22,03       90      113600  30,92      31,79
40      111600  62,97      22,43       95      113800  25,64      34,88
45      111800  60,52      22,86       100     114000  18,95      40,51 ```
Problem 1: results.
Pressure gradient is constant at lengthof diffuser dp/dx=40 kPa·m-1.
 Figure at Problem 1.(a) calculated profile of radius; (b) cone diffuser about the same length at α=23,18°.

The pressure gradinet of cone diffuser can be variable at comparison with shape of diffuser at constatn pressure gradient.

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## Comparison of diffuser properties with constant(najít synonymum) pressure gradient and cone diffuser

Pressure gradient of diffusser be can calculated through right side of Equation 12 as shown next problem:

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.
Problem 2.456
 Figure at Problem 2.Profil of pressure gradient in a cone diffuser. dp/dx [kPa·m-1]; x [mm]. Higher pressure gradient at the inlet of the cone diffuser ?je způsoben? by higher angle of diverging than the case of diffuser with constant pressure gradient at Problem 1.

The diffuser with constant pressure gradient contain ?prudké? diverging, therefore higher sentens on flow separtion be can predicted than for case the cone diffusers. Measurments kvitation these fact only long diffuser, at shrot diffuser (α>18°) it is ?naopak? [1, p. 392]. ?To je způsobeno? therefore the most higher pressure incresing is at outlet short cone diffuser and at outlet diffuser is small pressure difference between boundary layer and backpressure. Therefore ?je výhodné? use short diffuser with constant pressure gradient respectively (high) pressure gradient at the exit.

*Remark
These diffusers are similar to short diffuser on Figure 14 without ?skokové, odstupňované? chnage of the flow area.
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## Shape of diffuser with small ?sensitive? on flow separtion

Kinetic energy of boundary layer increasing step by step from the inlet to the outlet of diffuser, therefore ideal diffuser should be higher angle of diverging at the inlet and smaller at the exit [1, s. 388]. Diffuser with linear pressure gradient is near ideal shape of the diffuser:

 15.430 Diffuser with linear pressure gradientdp/dx [kPa·m-1]; x [mm]. Dffuser on the Figure has parameters: dp/dx=400 kPa·m-1, Ri=10 mm, pi=110 kPa.

Slowly change of diffusers contour are dificulty on made and therefore they are substituted by cone sections with diferent diverging or by combined diffusers with stepless change of dimaters [1, p. 393]:

 16.831 Practis design of diffusers with variable diverging.

## Supersonic diffusers

A supersonic diffuser ?should be? contain a converging section in front of a diverging section for efficiencies proces.

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Ideal design of the supersonic diffuser is ?problematický/sporný?. In ideal case compression inside diffuser should be run through compression waves, which are ?opakem? expnasion funs. The compresson eaves shoul be developent inside convergent section of diffuser, which is eqvivalent of mirror of ideal Laval nozzle. These supersonic diffusers are not ?ale? use in praxis. A problem, according [1, s. 405], these supersonic diffusers is obligue schock waves, which are developed at inlet edges or inside the converging section in real aplications.

The most stable flow at real conditions is reached through supersonic diffuser with stepless ?zbrzděním? of flow. They are contured through way at which obligue shock waves are rised. These waves ?na sebe navazují? and grove its ?sklon?, then last waves is ?kolmá?. Design of supersonic stepless diffusers is simple because the shock waves are very good ?probádány? and description. In these cases must be calculated losses through shock waves, also.

17.552 Supersonic diffusers with stepless ?zbrzdění? of flow.
(a) stepless supersonic diffuser; (b), (c) stepless supersonic difusser with ?navazující? shock waves – ?jako by se? similar reflextion from wall of diffuser – it ?usměrňuje? vector of velocity to axial direction and decreasing losses [1, p. 409].RV shock waves.

Shapes of supersonic diffuser are complicate, therefore diffusers for Ma<1,5 are designed without converging section. In front of converging section is only a throat with constant flow area as is showed on Figure 13. At these diffusers is expected development ?kolmá? shock wave at the inlet [1, p. 406], through its is increased velocity under sonic speed. If ?kolmá? shock waves is developed at the outlet of the throat are developed oblique shock waves inside the throat. The losses inside the throat are not ?výraznější? than for cases complicatet shapes of the converging section at this Mach number.

## Problems at non-nominal states of diffusers

Každý difuzor je navržen na konkrétní stav plynu před a za difuzorem. Jestliže se tento stav změní změní se i proudění v difuzoru. Takový stav se nazývá nenávrhový. Při nenávrhových stavech se snižuje účinnost difuzoru a může se i stát, že se difuzor změní na Lavalovu trysku:

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# 42. Some applications of nozzle theory

## Non-nominal states of valve with diffuser

The valve with diffuser is used for governing of flow rate. Inside the valve is subsonic flow. The regulation of mass flow rate is made through a change of the flow area. This area is changed by motion of valve cone upder down (decreasing of flow area) or upwards (increasing of the flow area). The maximal flow velocity is reached between the cone and seat of the valve, in the diffuser this velocity is decreased.

Obrázek 1/id110. Valve with diffuser (partially open).

The shape of diffuser and the diverging nozzle is the same, therefore the velocity of flow is increased inside diffuser at low back-pressure. This non-nominal states of valve can be developed at the opening valve. In these cases, the valve has properties as CD nozzle at non-nominal states, therefore the shock waves can rise inside or at the exit of the diffuser. These effects can be reason for development of vibration of the valve and surrounding accessories, higher losses and or for damage of the valve.

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Figure 2/id69. Control valve with diffuser inside steam turbine.
Condensing turbine about 25 MW at speed 3000 min-1, one controlled steam extraction at 0,25MPa for 80 t·h-1, pressure admission steam is 2 MPa at 390 °C, Made in PBS. Figure: [1].

## Blade passage as CD nozzle

The diffuser blade passage of a compressor can has properties of the CD nozzle at non-design states if the velocity on the suction side of blade is higher than sound velocity. Nevertheless the pressure at the exit is higher than on the inlet, and Figure 8 is evident the supersonic velocity is must increasing to subsonic by a shock wave. This shock wave is dveloped local near the airfoil and it has resemblance of λ-shock wave:

Figure 4/id864. Development of λ-shock wave in blade rov of compressor.
w [m·s-1] velocity of attack; RV shock wave and separtion of flow.
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Remark
A maeasure for increasing of influnce of the shock wave in blade passge is descripted in [3, p. 136].

## Ejector and injector

Ejectors and injectors are used as a vacuum pump or a feed pump. The working gas at the exit nozle has high kinetic energy, which is use for transport suction fluid:

5.id112. A steam ejector as the vacuum pump of a condenser.
1 motive steam; 2 suction gas (mix of steam and inertial gas); 3 mixture section; 4 exit diffuzer. The motive steam at the exit nozzle creates a low pressure zone that draws in and entrains the suction gas. The suction gas and the motive steram are mixed in mixture section and the kinetic energy of the motive steam is transformed of kinetic energy of the suction gas. The kinetic energy of the mix is transformed into the pressure energy (compression of the mix). The sectional view of the ejector from [2].
Ejector
The pressure at exit of the ejector is lower than the inlet pressure of the motive fluid. For example, the ejectors are used as the vacuum pump of the steam condensers, in these cases is water steam as the motive fluid.
Injector
The pressure at the exit of injector is higher than the inlet pressure of the motive fluid (in this case a vapouar, p4>p1). The kinetic energy of the motive steam must be higher than is equvaelnt the kinetic energy at expansion the pressure of the suction fluid p2. The higher kinetic energy of the motive steam is reached under its condensation at temperature of the suction fluid inside the mixture section, therefore for the ejectors is significant low temperature of the suction fluid. Under these condition is condensing pressure lower then pressure of the suction fluid. From these reasons has temperature of the suction fluid higher influence on the effciency of the injector. The injectors are used as feed pump of steam locomotives. In this case is evident the pressure at the exit of the ejector must be higher than the pressure of the motive steam (about pressure drop).

The efficiency of the ejectors interval from 10 to 30% [4, p. 16]. The efficiency of the injectors must be lower than 10%. The calculation of the ejectors and the injectors is shown in [5, p. 416].

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## Ramjet and Scramjet

These jet engines re using shock waves to compress air at supersonic motion inside their inlet parts. The compressed air is combisting inside the combustion chamber with fuel and hot exhaust gas expans inside the nozzle and develops the thrust. The ramjets and scramjets do not conten any turbocompressors and turbine parts as turbo jet:

Figure 8/id114. 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.

For design of the ramjet engine is typical two throutles, first contens supersocnic diffuser and second contens nozzle. The flow rate through the nozzle is hogher than the mas flow rate of air through the throutling of the diffuser b, the difference is flow rate of the fuel. Therefore a control this engines is difficult (at a increasing of the flow rate is increasing pressure inside the combustion chamber).

The ramjet engines can alone to work at high speed only. For example the british misle GWS-30 Sea Dart use ramjet engine with combination starting rocket engine ?na tuhé palivo? The ramjet engine reaches the more high eficiency at Ma=5.

Wide regulation power can be reched through ?sloučením? throutling of diffuser and the nozzle, this type of jet engine is called Scramjet. Injection of fuel and its burnning is performed inside throutling. Scramjet be able working in more wide rate od speed than the ramjet engine, but for alone work need very high speed than is sound velocity. It maximum efficiency reached about Ma=9:

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29.id512. Scramjet.
(a) description of function; (b) experimental "bezpilotní" aeroplan 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.
*Experimental ?bezpilotní? aeroplan X-43A
This aeroplan flighted under drive of the Scramjet with speed 6,83 Ma. The flight was ?trval? aproximatel 10 minutes at the spring 2005. The work velocity was arised through ?pomocná? rocket engine at alitude 30 000 m. The set X-34A and the rocket engine stared from a bomber B-52B. The X-34A engine use oblique cut nozzle effects [11].
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