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Review on pressure swirl injector in liquid rocket engine

Article in Acta Astronautica · April 2018

DOI: 10.1016/j.actaastro.2017.12.038

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Acta Astronautica 145 (2018) 174–198

Contents lists available at ScienceDirect

Acta Astronautica

journal homepage: www.elsevier.com/locate/actaastro

Review on pressure swirl injector in liquid rocket engine

Zhongtao Kang a,*, Zhen-guo Wang b, Qinglian Li b, Peng Cheng b

a Science and Technology on Scramjet Laboratory, CARDC, Mianyang, Sichuan, 621000, Chinab Science and Technology on Scramjet Laboratory, National University of Defense Technology, Changsha, Hunan, 410073, China

A R T I C L E I N F O

Keywords:Pressure swirl injectorLiquid rocket engineReview

* Corresponding author.E-mail address: [email protected] (Z. Kang).

https://doi.org/10.1016/j.actaastro.2017.12.038Received 16 November 2017; Received in revised form 1

0094-5765/© 2018 IAA. Published by Elsevier Ltd. All ri

A B S T R A C T

The pressure swirl injector with tangential inlet ports is widely used in liquid rocket engine. Commonly, this typeof pressure swirl injector consists of tangential inlet ports, a swirl chamber, a converging spin chamber, and adischarge orifice. The atomization of the liquid propellants includes the formation of liquid film, primary breakupand secondary atomization. And the back pressure and temperature in the combustion chamber could have greatinfluence on the atomization of the injector. What's more, when the combustion instability occurs, the pressureoscillation could further affects the atomization process. This paper reviewed the primary atomization and theperformance of the pressure swirl injector, which include the formation of the conical liquid film, the breakup andatomization characteristics of the conical liquid film, the effects of the rocket engine environment, and theresponse of the injector and atomization on the pressure oscillation.

1. Introduction

Liquid-propellant rocket engines have been used as the primarypropulsion systems in most launch vehicles and spacecraft since the late1920's [1,2], such as the planet landers and low-cost engines [3]. Theperformance of liquid rocket engine is determined not only by the pro-pellant selection but also by fuel and oxidizer atomization performance[4,5], evaporation and ignition of droplets [6–8]. The atomization per-formance of propellants is determined by the injector. And there aremany types of injector, for example the liquid centered gas-liquid pintleinjector [9,10], liquid-liquid pintle injector [11], liquid centered swirlcoaxial injector [12,13], etc.

Pressure swirl injectors are extensively used in liquid rocket engines[14], gas turbine engines [15], internal combustion engines [16,17], andmany other combustion applications [18]. The pressure swirl injector canbe divided into hollow cone injector [19], solid cone injector [20], andspill-return injector [21]. And the swirling motion of liquid can beformed by either tangential inlet ports [22] or a swirler [19]. In liquidrocket engine, the injector configuration should been as simple aspossible to ensure the reliability and stability. Thus the pressure swirlinjector with tangential inlet ports is widely used in liquid rocket engine.Commonly, this type of pressure swirl injector consists of tangential inletports, a swirl chamber, a converging spin chamber, and a dischargeorifice [23], as depicted in Fig. 1. And it can be further divided intoconverge-end swirl injector and open-end swirl injector based on

8 December 2017; Accepted 25 Dece

ghts reserved.

whether there is a converging spin chamber, as depicted in Fig. 2. Theliquid is injected through the tangential ports, forming an air core alongthe centerline due to high liquid swirl velocity. The liquid flow at thedischarge end presumes a hollow conical swirling film. Then the swirlingfilm becomes unstable and breaks up into droplets, as shown in Fig. 1.

The pressure swirl injector is used to atomize the liquid propellantsthrough the formation of liquid film, primary breakup and secondaryatomization. Atomization is a process during which the interfacial area ofliquid increases gradually because the bulk liquid is transformed intosmall droplets. So, the evaporation of liquid propellants can be facilitatedby atomization significantly. From the energy point of view, atomizationis a process during which the potential energy of the supplied liquidfinally converts into the needed surface energy, as shown in Fig. 3.Jedelsky and Jicha [21] studied the energy conversion in atomization ofa spill-return pressure swirl injector, they found that 58% of the pressuredrop converts into the kinetic energy of the rotational motion in the swirlchamber, and the energy loss includes hydraulic loss and friction loss.The kinetic energy of the spray near the injector exit is 32–35% of theinlet energy, which contains the droplet kinetic energy (21–26%) and theentrained air kinetic energy (10–13%). Atomization efficiency is definedas the ratio of surface energy and the inlet energy. It decreases with theincrease of pressure drop because the viscous loss increases faster thanthe surface energy. Commonly, the atomization efficiency is less than0.3%. For the pressure swirl injector, most of the energy loss occurs in theswirl chamber.

mber 2017

mailto:[email protected]://crossmark.crossref.org/dialog/?doi=10.1016/j.actaastro.2017.12.038&domain=pdfwww.sciencedirect.com/science/journal/00945765http://www.elsevier.com/locate/actaastrohttps://doi.org/10.1016/j.actaastro.2017.12.038https://doi.org/10.1016/j.actaastro.2017.12.038https://doi.org/10.1016/j.actaastro.2017.12.038

Nomenclature

At Total area of the tangential ports, m2

A Geometry characteristic constantα Converge angle of the swirl chamber, degβ Spray cone angle, degβr Spray cone angle at a distance L from the injector exit, degCd Discharge coefficientdl Ligaments diameter, md Droplet diameter produced by the breakup of the ligaments,

mDt Tangential ports diameter, mD0 Injector diameter, mDs Swirl chamber diameter, mDa Air core diameter, mDat Air core diameter at the tangential inlet, mΔPl Liquid pressure drop, Paη Small disturbance, mηbu Critical amplitude of the disturbance waves, mη0 Initial amplitude of the disturbance waves, mFs Centrifugal force, NFscr1 The critical centrifugal force for judging air core, NFscr2 The critical centrifugal force for judging stable air core, NGSMD Global Sauter Mean Diameter, μmGLR Gas liquid ratioγ Opening coefficientsh Film thickness, mh0 Film thickness at the injector exit, mht Film thickness at the inlet of the tangential ports, mK Injector constantKv Velocity correction coefficientKs Wave number correspond to the maximum growth rateL0 Orifice length, mLs Swirl chamber length, mLbu Slant breakup length, mLv Vertical breakup length of the conical liquid film, mλ Wavelength of the surface wave, m

_ml Liquid mass flow rate, kg/sμl Dynamic viscosity, Pa⋅sνl Kinematic viscosity, m2=sω Angular frequency of the surface wave, Hzωs Maximum surface wave growth rateωi Surface wave growth ratePDA Phase doppler anemometryPc Chamber pressure, PaQ Volume flow rate, m3=sR Radius, mRe Reynolds numberRs Radius of the swirl chamber, mRg Gas constantRt Tangential inlet radius of the injector, mR0 Orifice radius of the injector, mRo Rossby number which characterize the ratio of the axial

velocity and the rotational velocityRet Reynolds number at the tangential inletReth Theoretical Reynolds number of the liquid film at the

injector exitRsw Swirling radius of the pressure swirl injector, mρl Liquid density, kg/m3

ρg Gas density, kg/m3

SMD Sauter Mean Diameter, μmSn Swirl numberσ Surface tension coefficientσcr Critical pressure ratioτbu Breakup time, sφ Filling coefficientϑ Angle of the tangential ports, degWe Webber numberWel Liquid Webber numberW Tangential velocity, m/sWt Velocity of the tangential inlet, m/sX Dimensionless air core diameter

Z. Kang et al. Acta Astronautica 145 (2018) 174–198

The operating principle of liquid rocket engine is quite different fromother combustion applications, making the operation condition of pressureswirl injector in liquid rocket engine quite different. For example, in auto-mobile engine, the spray is pulse and the pressure drop is really high. Andthe research focuses on the atomization of biofuel [24–26], secondary in-jection [27] and the effects of injector geometry [28–30] for energy-savingand emission reduction. However, in liquid rocket engine, the spray isstable, both thepressure drop and the ambient temperaturearehigh.What'smore,whenthrottlingprocess or combustion instabilityoccurs, thepressuredrop and chamber pressure will also vary. These characteristics indicatethat the liquid propellants could be super critical and the pressure in supplysystem and combustion chamber could be oscillating.

Although the pressure swir injector has been reviewed before. Forexample, Vijay et al. [31] reviewed the injector geometrical parameters,fluid properties and operating conditions' influence over the air corestability, breakup length, spray cone angle and Sauter mean diameter.The effects of the rocket operating environments on the atomizationmechanism and spray characteristics of the pressure swirl injector has notbeen summarized before. In this paper, a more comprehensive review onthe pressure swirl injector in liquid rocket engine has been conducted.The internal flow characteristics, conical film formation, primarybreakup and spray characteristics has been discussed. Then the effects ofrocket operating environments on the spray characteristics have beendiscussed further. These operating environments include back pressure,super critical injection and pressure oscillation.

175

2. Discussion

2.1. Formation of conical liquid film

In the pressure swirl injector, the liquid is injected through thetangential ports, forming an air core and an annular liquid film along thecenterline due to high liquid swirl velocity. Then, the annular liquid filmdevelops into hollow conical swirling film when it flows out of theinjector exit. The formation of conical liquid film is strongly related withthe internal flow characteristics which includes the air core formation[31], the boundary layer development [23], the film thickness and thegrowth of the unstable wave at the interface between the gas and liquid,as shown in Fig. 4.

2.1.1. Air core formationAn air core is formed when the centrifugal force of the swirling flow

overcomes the viscous force and a low-pressure area near the injector exitis created by the centrifugal motion of liquid within the swirl chamber[23]. The centrifugal force can be calculated by dFs ¼ dmW2r , and isproportional with the square of the tangential velocityW. The tangentialvelocity at a specific radius positionW is proportional with the velocity atthe tangential inlet ports Wt for the angular momentum conservationWr ¼ WtRsw. It means that the centrifugal force increases with the in-crease of the tangential inlet velocity Wt , the tangential inlet Reynolds

Fig. 1. Schematic of the flow formed by a pressure swirlinjector.

Fig. 2. Schematic of two typical pressure swirl injectors.


number Ret ¼ ρlWtDtμl , the pressure drop and the swirl numberSn ¼ ∫VWr2dr=∫ r0V2rdr.

Table 1 shows the air core formation mechanism of pressure swirlinjector with different operation conditions and geometrical parameters.Datta and Som [32] found the pressure drop rapid increases when itexceeds a critical value. Moon et al. [30] figured out that a stable air coreis formed when the swirl number Sn > 0:6. Typical development of aircore with the increase of Reynolds number is show in Fig. 5. Researchesof Halder et al. [33], Lee et al. [34] and Amini et al. [23] indicate that theflow regime develops from no air core to fully developed stable air corewith the increase of liquid Reynolds number, and a transition stage with adeveloping air core locates between these two regimes. It means thatthere are two critical centrifugal forces: Fscr1 and Fscr2. When the

176

centrifugal force is smaller than Fscr1, no air core is formed. When thecentrifugal force locates between Fscr1 and Fscr2, a developing unstable aircore is formed. When the centrifugal force is larger than Fscr2, a fullydeveloped stable air core is formed. The fully developed stable air core iscylindrical and bulges in diameter at the entrance of the discharge orifice[33]. The geometrical parameters and liquid viscosity have great influ-ence on the formation of air core. When the ratio of the swirl chamberlength to the swirl chamber diameter is large enough and the initialangular momentum is smaller than the needed one for a stable air core, adouble-helical unstable air core is formed [35,36], as shown in Fig. 6.Liquid viscosity hinders the formation of air core [37], because the vis-cosity decreases the velocity of the swirling flow [38].

The diameter of the fully developed air core increases with the in-crease of liquid Reynolds number and tends to a constant [23,32,33].And among the geometrical parameters of pressure swirl injector, theinjector constant K ¼ AtðDs�Dt ÞD0 has the greatest influence on the internalflow characteristics. For example, the air core diameter decreases withthe increase of injector constant K [23]. That's why the air core diameterincreases with the increase of injector diameter D0, the decrease oftangential ports diameter Dt and the increase of swirl chamber diameterDs[33]. Halder [33] obtained the empirical equation of air core diameterfrom the experimental data:

DaD0

¼ 0:338�1� e�1:45�10�4Re

�α0:073

�D0Ds

�0:424�DtDs

��0:732�L0Ds

��0:252(1)

It is clear that besides the injector constant K, the converge angle ofswirl chamber α and the orifice length L0 also have great influence on theair core. The air core diameter Da increases with the increase of the swirlchamber converge angle and decreases with the increase of orificelength.

2.1.2. Film thicknessThe film thickness at the injector exit is an initial parameter of the

conical liquid film, thus have great influence on the breakup of conicalfilm. There are three methods to measure the film thickness at theinjector exit, as shown in Table 2.

As the radius of the air core plus the film thickness equal to the radiusof the injector, the variation of film thickness shows an opposite tendency

Fig. 3. Sankey diagram for energy balanceof the atomization process [21].

Fig. 4. Schematic of the inner flow of a pressure swirlinjector.


with the variation of air core. In the orifice of the injector, the filmthickness is about 1000 μm and decreases along stream [47]. The filmthickness is influenced by both the operation condition and injector ge-ometry significantly. On the point of view of the operation condition, thefilm thickness decreases with the increase of liquid Reynolds number andpressure drop, and tends to a constant value [23,33,37]. And it increaseswith the increase of liquid viscosity [37]. On the point of view of theinjector geometry, injector constant K has the most significant influence,the length of orifice and the converge angle of the swirl chamber alsohave great influence. The film thickness increases with the increase ofinjector constant K [23,32,33,48], increases with the increase of orificelength L0 and decreases with the increase of swirl chamber convergeangle α [23,32,33].

Using theoretical and experimental methods, researchers obtainedseveral empirical equations to predict the film thickness at the injectorexit, as shown in Table 3. Where h0 is the film thickness at the injectorexit, μl is the liquid viscosity, _ml is the liquid mass flow rate, D0 is theinjector diameter, ΔPl is the pressure drop, ρl is the density of liquid andL0 is the orifice length. These equations indicate that the film thickness h0

is proportional with the term�

_mlμlD0ρlΔPl

�0:25, while the coefficient varies

between different injectors. Among these empirical equations, the

177

equation proposed by Kim et al. [35,36] includes the most geometricalparameters of the injector and predict the film thickness well. However,these equations is derived based on the assumption that the azimuthaldistribution of the film thickness is uniform and is different with theactual situation. In fact, Inamura et al. [57,58] found that the azimuthaldistribution of the film thickness is nonuniform. Moreover, the filmthickness fluctuates with peaks and troughs, and the number of peaks iscorrespond to the number of tangential inlet ports.

2.1.3. Boundary layerOnce the fully developed stable air core is formed, the viscous internal

flow of the pressure swirl injector can be divided into boundary layer andpotential core [23], as shown in Fig. 4. The boundary layer contains axialflow, while whether the potential core contains axial flow has not beensettled. Amini [23] has compared three theoretical models: inviscid flow,viscous flowwithout the axial flow in the potential core, and viscous flowwith the axial flow in the potential core, and found the viscous flow withthe axial flow in the potential core agrees better with the simulationresults. The dimensionless thickness of the boundary layer obtained byAmini with the axial flow in the potential core considered is shown inFig. 7. It is clear that the thickness of the boundary layer in the swirlchamber and the orifice of the injector increases gradually along the

Table 1Regimes of air core.

Author Type of data andReference

Regimes Constraints Fluidmedium

Datta and Som[32]

Theoretical 1. Increasing air core radius The pressure drop ΔPl keeps constant2. Stable air core The pressure drop ΔPl rapid increases when it exceeds a critical

valueMoon et al. [30] Experimental 1. Fluctuating air core Sn < 0:6 Gasoline

Numerical 2. Stable air core Sn > 0:6Halder et al. [33] Experimental 1. No air core Ret < 800 Water

0:25 < D0Ds < 0:38

2. Transition stage with 800 < Ret < 2400a developing air core 0:25 < D0Ds < 0:38

3. Fully developed Ret > 2400stable air core 0:25 < D0Ds < 0:38

Lee et al. [34] Experimental 1. No air core Reth ¼ ρlVthD0μl < 2550 DieselΔPl < 0:5MPa

2. Transition stage with 2550 < Reth < 3450 bunker-Aa developing air core 0:5MPa < ΔPl < 0:9MPa3. Fully developed Reth > 3450stable air core ΔPl > 0:9MPa

Amini et al. [23] Theoretical 1. No air core Ret < 80 WaterExperimental 2. Transition stage with a developing air

core80 < Ret < 144

Numerical 3. Fully developed stable air core Ret > 186Kim et al. [35,36] Experimental 1. Cylindrical stable air core 0:7 < LsDs < 1:06 Water

2. Double-helical unstable air core 1:27 < LsDs < 3:06

Fig. 5. Air core develops with the increaseof Reynolds number: (a)Re¼ 101, (b)Re¼ 144, (c)Re¼ 186 [23].


stream, while that in the convergent section decreases along the stream.Dumouchel et al. [61] found that the boundary layer inside the injectorare both functions of the injector geometrical parameters and the pres-sure drop. When the pressure drop is small, the thickness of the boundarylayer is in the same order of magnitude with the film thickness. And theinfluence of boundary layer on the flow decreases with the increase oftangential inlet ports diameter Dt , injector diameter D0, and pressuredrop.

2.1.4. Discharge coefficientThe discharge coefficient directly reflects the discharge characteris-

tics of pressure swirl injector, and is defined as the ratio of the actual

mass flow rate to the theoretical mass flow rate: Cd ¼ _ml=�

πD204

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ρlΔPl

p �.

For inviscid flow, the discharge coefficient is only related with thegeometrical parameters. Giffen and Abramovich [62] derived the equa-tion of discharge coefficient based on the inviscid assumption:

Cd ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1� XÞ31þ X

s¼

ffiffiffiffiffiffiffiffiffiffiffiffiφ3

2� φ

s(2)

where X is the dimensionless air core area, φ ¼ 1� X is the filling co-efficient and can be calculated with the geometry characteristics constantA by

A ¼ ð1� φÞffiffiffi2

p

φffiffiffiφ

p (3)

the geometry characteristics constant A is the most important parameterof pressure swirl injector, and is defined as:

Fig. 6. Typical inner flow of pressure swirl injector [35,36].

178

Table 2Measurement method for the film thickness at the injector exit.

Method Principle Author Advantages and Disadvantages

Electricalconductancemethod

1. Two electrodes are fixed inside the orifice; 2. The voltagefor the liquid film thickness was measured; 3. Calculate thefilm thickness with the calibrationed relation between thefilm thickness and the voltage.

Fu et al. [39–41] Khil et al.[42–44] Chung et al. [45] Kimet al. [35,36]

Advantages: 1. High accuracy of measurement, transientmeasurement. Disadvantages: 1. The relation between thefilm thickness and the voltage should be calibrationed before,and demand high accuracy of the calibration. So, the standardposts used for calibration should be concentric with theinjector, because eccentric setup have great influence on theaccuracy [46]. 2. The frequency of the film thickness variationis limited because the distance between the electrodes isfinitude. If the distance between the electrodes is much largerthan the wavelength of surface wave, only the average filmthickness can be measured.

Opticalmeasurement

Transparent acrylic injector: 1. Injector or the orifice ofthe injector is made by transparent acrylic material; 2. Highresolution instantaneous images of the internal flow arecaptured with high speed camera; 3. The film thickness isdirectly measured and calibrationed.

Feikema [47] Jeng et al. [48]Yao and Fang [37] Kenny et al.[49,50] Moon et al. [51]

Advantages: 1. Transient measurement, can measure the filmthickness varying with high frequency. Disadvantages: 1.Three-dimensional overlay effects the visual demarcation andthe threshold gray value chosen also causes deviation. 2.Refraction occurs among transparent nozzle, fuel and air,direct measured film thickness is not accurate and opticalcalibration is needed.

Opticalmeasurement

PLIF: 1. The orifice of the injector is made by transparentacrylic material; 2. Seed the flow with a fluorescent dye; 3.Raw PLIF images are obtained by the camera when thefluorescent dye is illuminated by a laser sheet; 4. Opticalcalibration is conducted with the raw PLIF image and thefilm thickness is obtained.

Zadrazil et al. [52] Haber et al.[53] Schubring et al. [54,55]

Advantages: 1. Without the three-dimensional overlay effect,thus this method has high accuracy and can measure transientfilm thickness; 2. Surface wave structure on the gas-liquidinterface can be captured clearly. Disadvantages: 1. Theexperimental facility is complex; 2. The threshold valuechosen causes deviation; 3. Refraction occurs amongtransparent nozzle, fuel and air, direct measured filmthickness is not accurate and optical calibration is needed.

Image processing: 1. Capture the raw images with longexposures and deconvolute the digital images with an Abeltransform technique; 2. Choose a threshold value and obtainthe film thickness.

Eberhart et al. [56] Advantages: 1. Film thickness downstream the injector exitcan be measured. Disadvantages: 1. Only the average filmthickness is measured; 2. The choose of threshold value causesdeviation.

Contact needleprobe method

1. Move the contact needle probe to touch the liquid filmand record the position when the needle touches the film; 2.Calculate the film thickness with the position of the filmsurface.

Inamura et al. [57,58] Disadvantages: 1. The needle probe would influence theinternal flow; 2. Only the average film thickness is measured.


A ¼ ðDs � DtÞD04nR2

¼ Q=nπR2t

Q=πR2Ds � Dt

D0¼ Vtangential

VaxialCopen (4)

t

� �0

For viscous flow, the discharge coefficient of the pressure swirlinjector with tangential ports decreases with the increase of pressuredrop, and tends to a constant [63]. The reason is that during the for-mation of air core, the diameter of the air core increases gradually, whichdecreases the filling coefficient. Lee et al. [34] found that the dischargecoefficient decreases with the increase of Reynolds number Re at the inletof the tangential ports. And when the flow develops from the unstablestage without air core to the stable stage with air core, the dischargecoefficient decreases the fastest, as shown in Fig. 8. What's interesting isthat the discharge coefficient of a pressure swirl injector with a swirlerhas the inverse trend with the pressure swirl injector with tangentialports. Namely, the discharge coefficient of a pressure swirl injector with aswirler increases with the increase of pressure drop [64]. It is becausethat the larger the pressure drop, the larger the proportion of the axial

Table 3Empirical equations of the film thickness at the injector exit.

Author Equation

Rizk and Lefebvre [59]he ¼

"1560μl _mlDcΔPl

1þϕð1�ϕÞ2

#0:5;ϕ ¼

�1� 2heDc

�2Rizk and Lefebvre [59]

he ¼ 3:66�

Dc _mlμlρlΔPl

�0:25Suyari and Lefebvre [60]

he ¼ 2:7�

Dc _mlμlρlΔPl

�0:25Fu et al. [40]

he ¼ 3:1�

Dc _mlμlρlΔPl

�0:25Kim et al. [35,36]

he ¼ 1:44Dc�

_mlμlρlΔPlD3c

�0:25�LcDc

�0:6

179

momentum and the larger the discharge coefficient.When the discharge coefficient becomes constant, it is no longer

influenced by the operation condition and only influenced by the injectorgeometry. The most important geometrical parameters of pressure swirlinjector includes the geometry characteristics constant A and the injectorconstant K. And the injector constant K is defined as:

K ¼ AtðDs � DtÞD0 (5)

While the geometry characteristics constant A and the injector con-stant K satisfies A ¼ π4K, the variation trends of discharge coefficient withthe increase of these two parameters are on the contrary. Namely, thedischarge coefficient decreases with the increase of the geometry char-acteristics constant A [65], while it increases with the increase of theinjector constant K [48]. Giffen et al. [15], Rizk and Lefebvre [59],Abramovich [62], Fu [66], Liu [67], Jones [68] and Benjamin et al. [69,70] proposed the empirical equations of the discharge coefficient from

Method to obtain Injector type

Theoretical derivation Converge-end

Theoretical derivation Converge-end

Electrical conductance method Open-end



Fig. 7. Dimensionless thickness of the boundary layer in the injector [23].

Fig. 8. Discharge coefficient variation with different Reynolds number at thetangential inlet port [34].


experimental data, and are listed in Table 4. It is clear that these equa-tions all predict the variation trend of the discharge coefficient well.

Besides with the geometry characteristics constant A and the injectorconstant K, other important geometrical parameters includes the injectordiameter D0, the orifice length L0, the swirl chamber length Ls, the

Table 4Empirical equations of the discharge coefficient.

Author Equation

Giffen et al. [15] Cd ¼ 1:17ffiffiffiffiffiffiffiffiffiffiffiffið1�XÞ31þX

qRizk and Lefebvre [59]

Cd ¼ 0:35�

AtDsD0

�0:5�DsD0

�0:25Abramovich [62] Cd ¼ 0:432A0:64Fu [66,71]

Cd ¼ 0:19�

AtD20

�0:65γ�2:13, Cd ¼ 0:4354A0:877

Hong et al. [62]Cd ¼ 0:44

�AtD20

�0:84γ�0:52γ�0:59, γ ¼ Ds�DtD0 < 2:3

Liu [67]Cd ¼ 0:721

�1A

�0:416�L0D0

��0:0558�DsD0

�0:147Jones [68]

Cd ¼ 0:45�D0ρlV0

μl

��0:02�L0D0

��0:03�LsDs

�0:05��

AD

Benjamin et al. [69,70]Cd ¼ 0:466

�D0ρlV0

μl

��0:027�L0D0

�0:229�LsDs

�0:091��

180

converge angle of swirl chamber α, the area of tangential ports At , the

opening coefficient γ ¼ Ds�DpD0 , the inlet port angle and so on. The variationpattern of the discharge coefficient with these parameters are as follow:

1. The discharge coefficient decreases with the increase of injectordiameter D0[32,72], because the increase of injector diameter D0 willdecrease the injector constant K and increase the geometry charac-teristics constant A, which finally promote the swirling flow anddecrease the filling coefficient;

2. The discharge coefficient increases with the increase of the area oftangential ports At[32,73], because the swirling flow is weakened bythe increase of the tangential ports area, which increases the fillingcoefficient;

3. The discharge coefficient decreases with the increase of the inlet portangle [15,74], because the increase of the inlet port angle can in-crease the tangential velocity and decrease the axial velocity, namely,the swirling flow is enhanced;

4. The discharge coefficient decreases with the increase of the openingcoefficients γ [75];

5. A lot of research found the discharge coefficient decreases with theincrease of the orifice length L0[67,68,75], because the longer theorifice length, the larger the friction loss. However, the equation ofBenjamin et al. [69,70] indicates that the discharge coefficient in-creases with the increase of the orifice length. The reason may be thatthe range of the length to diameter ratio is too small (1.4�2.28);

6. A lot of research found the discharge coefficient increases with theincrease of the swirl chamber converge angleα [15,23,74], becausethe larger the converge angle, the shorter the converge section andthe smaller the friction loss. Only the results of Datta and Som [32]shows that the discharge coefficient decrease with the converge angleα;

7. The effects of the swirl chamber length Ls is not clear, Yule andWidger [72] found that increase the swirl chamber length increasesthe discharge coefficient. However, Sakman et al. [75] found thatincrease the length to diameter ratio of the swirl chamber increasesthe discharge coefficient initially and then decreases;

8. The trumpet at the injector exit has little influence on the dischargecoefficient [15,74].

In conclusion, the orifice length L0, swirl chamber length Ls, swirlchamber converge angle α and the trumpet have little influence on thedischarge coefficient. Thus the variation trends can be easily affected byother factors. That's why the contradictions exist in literature. In injectordesign, it is acceptable to consider the geometry characteristics constantA and the injector constant K only, because the influence of otherimportant parameters (injector diameter D0, area of tangential inlet portsAt and opening coefficient γ) are included in these two parameters.

Method Injector type

Experimental Converge-end


Experimental Converge-endExperimental Open-end



t

sD0

�0:52�DsD0

�0:23 Experimental Converge-endAt

DsD0

�0:517�DsD0

�0:187 Experimental Converge-end

Fig. 9. Air core in the swirl chamber. (Two tangential ports) [78].


2.1.5. Surface waveIn fact, the air core is not cylindrical but spiral, as shown in Fig. 9.

Wang et al. [76] and Huo et al. [77] simulated the internal flow of apressure swirl injector at supercritical conditions, and found the tem-perature and density iso-surfaces present a spiral structure. Chinn et al.[78] found a helical air core occurs in the injector, and the number of thestriation is related with the number of tangential ports. Inamura et al.[57,58] measured the film thickness with the contact needle probemethod, and found three film thickness peaks corresponding to thenumber of the tangential ports. The spiral air core is formed by tworeasons. The first one is the nonuniform azimuthal distribution of thetangential velocity, because the larger the tangential velocity, the largerthe centrifugal force, and then the larger the air core diameter. As thenonuniform azimuthal distribution of the tangential velocity is producedby the azimuthal distribution of tangential ports, the number of thestriation equals the number of tangential ports. The second one is the

Fig. 10. Stationary pulsating waves in the pressure injector [78].

181

axial delay of the nonuniform azimuthal distribution of the tangentialvelocity, because the liquid flow towards the exit with a axial velocity. Ifthe azimuthal distribution of the tangential velocity is uniform, the aircore should be cylindrical.

Besides with the spiral air core, stationary pulsating waves also existon the gas liquid interface in the pressure swirl injector, as shown inFig. 10. Normally, small disturbances with different frequencies on thegas liquid interface will grow, and the wave with the largest growth ratewill dominate the interface. Chinn et al. [78] found the stationary pul-sating waves exist on the gas liquid interface. Richardson [79] conducteda linear stability analysis on a confined swirling annular liquid film, andfound that the swirling and surface tension are stabilizing forces whilegas liquid density ratio is destabilizing force. When the Rossby NumberRo (the ratio of axial liquid velocity to circumferential velocity) is largerthan one, the larger the Rossby Number, more unstable the film is. Fuet al. [80] investigated the linear temporal instability of a confinedswirling annular liquid film. And found that the injector diameter, Rossbynumber, and liquid Weber number destabilize the liquid film, while alarger liquid-to-gas density ratio and velocity ratio stabilize the liquidfilm. Fu [66] further figured out that the film would develop from con-vectively unstable to absolutely unstable when decrease the liquid-to-gasdensity ratio, velocity ratio, and liquid Weber number or increase thenon-dimensional film thickness and Rossby number Ro.

2.2. Breakup of conical liquid film

The breakup of conical liquid film, also called as the primary breakup,is due to the growth of the unstable wave at the interface between the gasand the liquid film [81]. And the surface waves have already beenobserved, as shown in Fig. 11. However, the spray pattern and breakupmechanism is much more complex in most cases, because the flowpattern is much more complex than a laminar conical liquid film flowwith the influence of the injector geometry and the turbulivity of theflow. The most important parameters of primary breakup are the spraycone angle and the breakup length. The spray cone angle characterize thespace distribution range of the liquid film, while the breakup lengthcharacterize the breakup position of the liquid film.

2.2.1. Surface waveThere are two independent modes of unstable waves exist on the

liquid-gas interface: sinuous wave and varicose wave. While the distor-tion on two liquid-gas interfaces is same-phase, it is called the sinuouswave (Fig. 12a). While the phase contrast of distortion on two interfacesis 180�, it is called the varicose wave (Fig. 12b). Linear stability analysishas long been used to analyze the development of the unstable waves onthe liquid-gas interface [82]. A lot of stability analysis have been focusedon round jet, planar liquid film and annular liquid film, while the stabilityanalysis on conical liquid film was limited. Yue and Yang [83] derivedthe dispersion equation of a conical liquid film and solved the equationnumerically. In his derivation, the spray cone angle, the injector diam-eter, the film thickness at the injector exit and the film thinning alongstream has been considered. Wang [84] also derived the dispersionequation of a conical liquid film, but the film thinning is not considered.Fu [66,81] obtained the same dispersion equation of conical liquid filmwith Yue and Yang [83], and solved the equation numerically. Hisresearch indicates that the conical liquid film is dominated by sinuouswave, and the maximum growth rate and dominant wave number in-crease with the increase of pressure drop, which in turn decrease thebreakup time and breakup length. Hosseinalipour et al. [85] furtherderived the dispersion equation of a conical liquid film with coaxial gasflow, and found that the liquid swirling and gas liquid density ratio arethe destabilize force. What's more, a larger spray cone angle destabilizesthe conical liquid film and decreases the breakup length. However, thefilm thinning along stream is also not considered.

When the spray cone angle decreases to zero, the conical liquid filmcan be looked as an annular liquid film. So the annular liquid film is a

Fig. 11. Surface waves on the conical liquid film [78].

Fig. 12. Wave modes on the conical liquid film [81].


special conical liquid film and the linear stability analysis on annularliquid film could help understanding the growth of surface waves.Chauhan et al. [86] found that the annular liquid film is absolutely un-stable for small velocities and convectively unstable for larger velocities.And with the increase of liquid velocity, temporal and spatial instabilityanalysis tend to coincide. Yan et al. [87] and Lin et al. [88] also foundthat the liquid velocity destabilize the surface and promote the breakupof liquid film. In some cases, the film is swirl. Liao et al. [89] found theliquid swirl not only destabilize the surface but also shifts the dominantwave from axisymmetric mode to helical mode. Lin et al. [88] also foundthat the liquid swirl could greatly promote the destabilization anddisintegration of liquid film, and the dominant mode of annular liquidfilm is nonaxisymmetric mode. Ibrahim and Jog [90]. further figured outthat the liquid swirl could destabilize the outer surface and stabilize theinner surface of the annular liquid film.

Besides the liquid flow, the radius of the annular film, the liquidviscosity, gas liquid density ratio and the surface tension all have greatinfluence on the instability of surface waves. Crapper et al. [91] and Shenand Li [92] found that both the growth rate of axisymmetric and non-axisymmetric disturbances rapid increase with the decrease of the radiusof the annular liquid film. Yan and Xie [93] also found that the smallerthe liquid film radius to film thickness ratio, the more unstable theannular liquid film. Jeandel and Dumouchel [94], Shen and Li [95] andHerrero et al. [96] all found that the liquid viscosity can stabilize theliquid film and suppress the growth of surface waves. Shen and Li [95],Lin et al. [88], Yan and Xie [93], Ibrahim and Jog [90], and Shen and Li[92] found that the gas liquid density ratio can destabilize the liquid film.As for the effects of surface tension, Shen and Li [95] figured out that thesurface tension is the stabilize force. However, Shen and Li [92] foundthat there exists a critical Webber number. The surface tension destabi-lize the gas liquid interface when the Webber number is smaller than the

182

critical Webber number, while it stabilize the gas liquid interface whenthe Webber number is larger than the critical one. It means that theaerodynamic force determines whether the surface tension destabilize orstabilize the gas liquid interface.

2.2.2. Spray pattern and breakup mechanismIn most cases, the spray is not regular cone because the spray pattern

is influenced by the centrifugal force and surface tension. And thebreakup of liquid film is not solely due to the growth of the unstable waveat the gas liquid interface, because the flow at the injector exit could beturbulent. In fact, the breakup mechanism and the spray pattern arerelated with each other. Typical spray patterns and breakup processes ofa conical liquid film are shown in Fig. 13. Prakash et al. [97] found fivespray patterns: dribbling stage, distorted pencil stage, onion stage, tulipstage and fully developed stage. And these spray patterns occur in turnwith the increase of liquid Webber number. Ghorbanian et al. [19]observed another wavy stage locates between the tulip stage and fullydeveloped cone stage. Ramamurthi et al. [98] further figured out that thespray will transfer from tulip to cone shape when the liquid Webbernumber exceeds about 150. Then, Dumouchel [99] found that there existanother perforated cone spray. Kang et al. [100] further figured out thatthe perforated cone stage followed the wavy stage with the increase ofpressure drop, and the spray finally develops into turbulent cone spray(also called as fully developed cone spray).

Above all, there are seven spray patterns: dribbling, distorted pencil,onion, tulip, wavy cone, perforated cone and fully developed cone. Thesespray patterns occur in turn with the increase of pressure drop, and onlythe last five spray patterns have cone shape. Furthermore, if the liquid isgelled, the spray patterns transfer to swirling jet, twisted ribbon, fluidweb and fully developed hollow cone [65].

The breakup mechanism of a conical liquid film mainly includes

Fig. 13. Typical spray patterns and breakup processes of apressure swirl injector [97].


surface wave instability, perforation and turbulence. The surface waveinstability regime think that the conical liquid film is disintegrated by thegrowth of surface wave, the perforation regime think that the liquid filmis disintegrated by the perforation, while the turbulence regime thinkthat droplets are produced from the liquid film by its own turbulence.Yue et al. [101] found that the conical liquid film breakup at halfwavelength. Both short and long wavelength waves exist on the liquidfilm and influence the breakup simultaneously. Moon et al. [102] foundthat long-wave regime dominant the breakup under atmospheric pres-sure while short-wave regime play a dominant role under high ambientpressure. Santolaya et al. [103] analyzed the surface wave instabilityregime and perforation regime, and figured out that the surface waveinstability regime significantly increase the atomization performance andradial distribution of droplets. However, these research mentionedbefore did not connect the breakup regimes with the spray patterns, butin fact, they are interrelated with each other. For example, the breakupregime of a wavy cone film is the surface wave instability while that of aperforated cone film is perforation. Ramamurthi et al. [98] compared thebreakup process of a tulip film and a wavy cone film, and found that thesurface wave grow rapidly when the spray pattern changes from tulip towavy cone. The disintegration of wavy cone film is caused by the growthof surface wave and is different from the tulip film. Kang et al. [100]further figured out that when the spray has an onion shape, the growth of‘Impact Wave’ produced by the impingement will finally pinch off thecollapsed liquid jet. When the spray has a tulip shape or fully developedwavy cone shape, the breakup regime is surface wave instability. Whenthe spray has a perforated cone shape, the breakup regime is mainlyperforation and surface wave instability as a supplement. When the sprayhas a turbulent cone shape, the breakup regime is mainly turbulence,with perforation and surface wave instability as a supplement.

Besides with these breakup regimes, Garcia et al. [104] and Le et al.[105] investigated the breakup of liquid film from the fractal view, andfound that the liquid film is a fractal object and the breakup of liquid filmis a fractal process.

2.2.3. Spray cone angleThe spray cone angle is one of the most important parameters of spray

characteristics. It controls the space distribution of the spray and hassignificant influence on the primary breakup of conical liquid film. Forinviscid flow, Abramovich [106] derived the equation of the spray coneangle:

183

tan β ¼ 2 2ð1� φÞffiffiffiφ

p ð1þ ffiffiffiffiffiffiffiffiffiffiffiffi1� φp Þ (6)

ffiffiffip

where β is the spray half angle. Giffen and Muraszew [107] derivedanother equations:

tan β ¼ π�1� ffiffiffiffiXp �2K

sin β ¼ πCd2K�1þ ffiffiffiffiXp �

(7)

Xue et al. [15] introduced the effects of the tangential ports angle ϑ,and obtained the calculation equation, as listed in Table 5. Rizk andLefebvre [59] introduced the correction factor of velocity Kv and derivedthe following equation:

cos β ¼ CdKvð1� XÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffi1� X1þ X

r(8)

All the equations above are based on the inviscid flow in which the aircore diameter at the injector exit equals to the air core diameter in theinjector. However, the air core diameter at the injector exit is larger thanthat in the injector, because the flow conservation is satisfied whenacross the injector exit, and the liquid film is accelerated by the dynamicpressure that transferred from the centrifugal overpressure. Orzechowski(Chinn [107] and Moon et al. [102]) considered this effect and derivedan equation of the spray cone angle:

tan β ¼ 2CdAffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ SbÞ2 � 4C2dA2

q (9)

where Sb ¼ RaR0 ¼ air core radius at the injector exit/injector radius.Besides the theoretical derivation, there are several methods to esti-

mate the spray cone angle by image processing in experiments. Usuallystraight lines are fitted through parts of the spray edge and the anglebetween them defines the spray cone angle [110], as shown in Fig. 14.Pastor et al. [111] defined the angle of two straight lines which are fittedto the first 60% of the spray closest to the nozzle as the spray cone angle.Kang et al. [112] defined the spray cone angle as the angle of an imag-inary spray cone that possesses the same magnitude of cross-sectionalarea to the spray area when the height of spray cone is spray

Table 5Equations of the spray cone angle.

Author Equation Type Injector type

Rizk and Lefebvre [59]cos β ¼ CdKvð1�XÞ, Cd ¼ Kv

"ð1�XÞ31þX

#0:5 Theoreticalcos2β ¼ 1�X1þX

Rizk(Liu et al. [14]) cos β ¼ CdKvð1�XÞ ¼0:35K0:5 ðDs=D0Þ0:25

Kvð1�XÞTheoretical

Kv ¼ 0:00367K0:29�

ΔPlρlμl

�0:2Liu [67]

cos β ¼ 0:302ð1þ tanϑÞ0:414�1A

�0:35��L0D0

�0:043�DsD0

�0:026þ 0:612


Giffen and Muraszew sin β ¼ πCd2Kð1þ ffiffiffiXp Þ ¼ ðπ=2Þð1�XÞ1:5Kð1þ ffiffiffiXp Þð1þXÞ0:5 Theoretical

(Santangelo [108], Couto et al. [63] and Jeng et al. [48])Cd ¼

"ð1�XÞ31þX

#0:5, K2 ¼ π2ð1�XÞ332X2

Xue et al. [15]sin β ¼ πCdsinϑ

2Kð1þ ffiffiffiffiXp Þ�RswRs

�

K2 ¼ π2ð1� XÞ332X2

�RswRs

�2sin2ϑ

Rsw ¼ Rs � Rt

Theoretical

Giffen and Muraszew (Liu et al. [22]) tan β ¼ 2φ1þ

ffiffiffiffiffiffiffi1�φ

p � RsR0nR2t TheoreticalOrzechowski tan β ¼ 2CdAffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð1þSbÞ2�4C2dA2p Theoretical

(Chinn [107] and Moon et al. [102]) Sb ¼ a0R0 ¼Air core diameter at the injector exit/Injector diameterFu et al. [71] tan β ¼ 0:033� A0:338 � Re0:249t Experimental Open-endInamura et al. [57,58] tan β ¼ kffiffiffiffiffiffiffiffi

1�k2p Theoretical

βr ¼ β affiffiffiffiffiffiffiffiffiffiLhtcosβ

p exp�� bRe�

a ¼ 18:9, b ¼ 670, k ¼ DatD0Dat : Air core diameter at the axial position of the tangential portsβr : Spray half angle at a distance L to the injector exitht : Film thickness at the axial position of the tangential ports

Rizk and Lefebvre (Ma [70], van Banning et al. [109] and2β ¼ 6

�At

DsD0

��0:15�ΔPlD20ρl

μ2l

�0:11 Experimental Converge-endKhil et al. [43,44])

2β ¼ 6K�0:15�

ΔPlD20ρlμ2l

�0:11Benjamin et al. [69,70]

2β ¼ 9:75�

AtDsD0

��0:237�ΔPlD20ρl

μ2l

�0:067 Experimental Converge-end


penetration. Daviault et al. [110] used two orthogonal lines to the nozzleaxis at two axial distances from the injector tip to define four points onthe edge of the spray. And then used these four points to define two linesyielding two half angles relative to the injector axis. The sum of the twohalf angles is defined as the spray cone angle.

For a pressure swirl injector with given geometrical parameters, thespray cone angle is determined by the operation condition. The mostimportant operation parameter is the pressure drop. The spray cone angleincreases with the increase of the pressure drop [73,101], and the spraypattern also varies with the increase of pressure drop. It means that

Fig. 14. Definition of the spray cone angle [71].

184

variation of the spray cone angle is related with that of the spray pattern.Ghorbanian et al. [19] found that the spray cone angle increases rapidlywith the increase of pressure drop in the onion stage, and it increasesslower in the tulip stage and much slower in wavy stage. When the sprayis fully developed, the spray cone angle becomes constant and pressureindependent. Besides with the pressure drop, liquid viscosity, backpressure and surface tension also have great influence on the spray coneangle. The spray cone angle decreases with the increase of liquid vis-cosity and back pressure when it becomes pressure independent [18,23,102,113,114]. And if the spray cone angle is small enough, the surfacetension converges the spray along the axis [58].

The most important geometrical parameter that influence the spraycone angle is the injector constant K. Any geometrical variation that in-crease the injector constant K decreases the spray cone angle [48]. Forexample, the decrease of injector diameter and the increase of thetangential ports area decrease the spray cone angle [32,115]. The in-crease of the tangential ports number, the geometry characteristicsconstant A and the tangential ports angle ϑ all increase the spray coneangle [15,65,73,74]. Other geometrical parameters includes the orificelength and the converge angle of the swirl chamber. The spray cone angleis found to decrease with the increase of the orifice length [75] and theconverge angle of the swirl chamber [15,23,74,75,115]. Only Datta andSom [32] found that the spray cone angle increases with the increase ofthe convergent angle of the swirl chamber. The reason could be that theconvergent angle of the swirl chamber in their research is too small thatthe increase of the convergent angle could significantly decrease thelength of the converge section and then decrease the friction loss. Besides


with these geometrical parameters, the trumpet at the injector exit couldalso influence the spray cone angle, but the influence is still not clearnow. Xue et al. [15,74] found that the spray cone angle increases with theincrease of the trumpet angle. However, Liu et al. [14] found it decreaseswith the increase of the trumpet angle because this geometry variationincreases the axial velocity and decreases the circumferential velocity ofthe liquid film.

2.2.4. Breakup lengthThere are two types of breakup length: slant breakup length Lbu and

vertical breakup length Lv. The slant breakup length characterize thedistance between the breakup position of liquid film and the injector exit,while the vertical breakup length characterize the vertical distance be-tween them. These two breakup lengths are related with Lv ¼ Lbucosβ.

The liquid film is pinched off when the amplitude of the surface waveexceeds a critical value ηbu. And the critical amplitude ηbu can be calcu-lated by ηbu ¼ η0eωsτbu , where η0 is the initial amplitude, ωs is themaximum growth rate of the surface wave which can be calculated by thelinear instability analysis, τbu is the breakup time of the liquid film. Basedon this equation, the breakup time of the liquid film can be calculated by:

τbu ¼ 1ωs ln�ηbuη0

�(10)

Once the velocity of the liquid film is known, the breakup length canbe calculated by:

Lbu ¼ τbuV0 ¼ V0ωs ln�ηbuη0

�(11)

It is clear that three parameters are needed to calculate the breakuplength of liquid film: lnðηbu=η0Þ, the maximum surface wave growth rateωs and the liquid film velocity V0. First, the liquid film velocity V0 can beobtained from experiment data or derived from inviscid theory. Second,the maximum surface wave growth rate ωs can be calculated by the linearinstability analysis. However, neither the dispersion equation of conicalliquid film nor that of annular liquid film has analytical solution. Thus theanalytical solution of a planar film has long been used to predict thebreakup length of a conical liquid film. As the breakup of a conical liquidfilm is dominated by the sinuous waves [81], it is more reasonable to usethe analytical solution of this mode of surface waves to calculated thebreakup length. Senecal et al. [116] derived the dispersion equations of aviscous planar liquid film with varicose and sinuous waves separately.And obtained the growth rate of the sinuous long waves:

185

ωi ¼ �2νlk2 þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4ν2l k4 þ

2QU2k � 2σk2

s(12)

h ρlh

the growth rate of the sinuous short waves:

ωi ¼ �2νlk2 þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4ν2l k4 þ QU2k �

σk2

ρl

s(13)

The above equations indicate that the growth rate of the sinuous shortwaves has nothing to do with the film thickness while that of the sinuouslong waves is related with the film thickness. Once the analytical solutionof the sinuous wave growth rate is derived, the maximum surface wavegrowth rate ωs can be obtained. Third, lnðηbu=η0Þ is an empirical coeffi-cient. Dombrowski and Hooper [117] found that lnðηbu=η0Þ ¼ 12 forplanar liquid film agrees well with experiment data, and it has the samevalue for liquid jets in literature. Since then, this value has been widelyused for breakup length prediction. Senecal et al. [116], Moon et al.[102], Laryea and No [118], Inamura et al. [57,58], Hosseinalipour et al.[85], Sivakumar et al. [119], Xiao and Huang [120], Tratnig and Brenn[121] all set lnðηbu=η0Þ ¼ 12. However, Kim et al. [122] found that thisvalue is not a universal constant and should be obtained from experi-ment. He found that the breakup length predicted with lnðηbu=η0Þ ¼ 6.9agrees well with experiments. Fu et al. [81] also think that lnðηbu=η0Þ isnot a universal constant, and found that lnðηbu=η0Þ ¼ 2.5 agrees well withexperiments. Furthermore, Clark and Dombrowski [123] found that thisvalue is also related with the operation condition. When the liquidReynolds number is larger than 9000, lnðηbu=η0Þ ¼ 12, otherwiselnðηbu=η0Þ ¼ 50.

The above breakup length prediction is based on the assumption thatthe liquid film thickness keeps constant along the stream, but in fact thefilm thickness of a conical liquid film is thinning along the stream, asshown in Fig. 15. Thus the growth rate of long waves is time-dependent,and the time-averaged growth rate is more appropriate for breakup timecalculation. Namely

ln�ηη0

�¼ ∫ t0ωdt (14)

As the viscosity has little influence on the growth rate of the longwave [116], Equation (12) can be simplified:

ωi ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2QU2k2 � σk3=ρl

kh

r(15)

where h is the conical liquid film thickness downstream, as shown inFig. 16. It can be calculated by

Fig. 15. Schematic of the breakup of a thinning film [124].

Fig. 16. Schematic of the film thickness of a thinning film [102].

Table 6Equations of the breakup length.

Author Equation

Senecal et al. [116]Lbu ¼ V0

3ln�

ηbuη0

�2=3�Jσ

Q2U4ρl

�1=3h ¼ J=t, J is constant, lnðηbu=η0Þ ¼ 12

Kim et al. [122]

τbu ¼ R0V0 tanβ

266640B@C h0tanβR0

�ρgρl

��1We�1=2l

þ1

1CA

�R0

h0tanβ

�"�Lbu tanβ

R0þ 1�3=2

� 1#¼ C

�ρgρl

��1C is a constant that is related with lnðηbulnðηbu=η0Þ ¼ 6.9 correspond to C¼ 14.78

Clark and Dombrowski [123]Lbu ¼ C �

"ρlσKlnðηbu=η0 Þ

ρ2g U2

#13

, K ¼ hx

Moon et al. [102]Lbu ¼ 12sinβ

"3V0sinβln

�ηbuη0

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih0ðD0�h0 Þ

QU2k�σk2=ρl

q #2

lnðηbu=η0Þ ¼ 12Sivakumar et al. [119]

Lbu ¼ 12sinβ

8>>>>>>>>><>>>>>>>>>:

3ffiffiffi2

pV0sinβln

�ηbuη0

�

"�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiσh0ðD0 � h0Þcosβ

Q2U4ρl

s #2=3

þðD0 � 2h0Þ3=2

�ðD0 � 2h0Þlnðηbu=η0Þ ¼ 12

Fu et al. [81]Lbu ¼ 0:82

"ρlσlnðηbu=η0 Þh0cosβ

ρ2g U2

#0:5

lnðηbu=η0Þ ¼ 2.5Inamura et al. [57,58]

Lbu ¼ 0:2175"ρlσlnðηbu=η0Þh0cosβ

ρ2g U2

#0:3

lnðηbu=η0Þ ¼ 12, U: gas liquid relative velocity.


186

h ¼ h0ðD0 � h0ÞD0 � h0 þ 2V0t sin β (16)

Substitute Equation (16) into Equation (15) and integrate Equation(14), then substitute the results into Equation (11) and obtained thebreakup length of a conical liquid film:

Lbu ¼ 12 sin β

32ln�ηbuη0

� ffiffiffi2

psin βV0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiρlh0ðD0 � h0ÞðQU2ρl � kσÞk

sþ ðD0 � h0Þ3=2

!2=3

� D0 þ h0!

(17)

where the most unstable wave number k ¼ QU2ρl2σ .It is clear that the conical liquid film is thinning film. The film thin-

ning influences the growth rate of surface waves and makes the predic-tion equation different from that of planar liquid film. The breakuplength prediction equation (17) also contains some parameters thatshould be obtained from experiment. In literature, theoretical derivationsand empirical coefficients are bond to predict the breakup length, andthese semi-empirical equations are listed in Table 6.

These semi-empirical equations indicate that the breakup length isstrongly related with the flow characteristics. And the flow characteris-tics include the liquid film velocity, the gas-liquid relative velocity, thebackpressure, the gas density, the liquid density, the surface tension andthe spray cone angle. A larger liquid film velocity or gas-liquid relativevelocity, a larger backpressure or gas density and a larger spray coneangle would produce smaller breakup length. And a larger liquid densityor surface tension would produce a thicker liquid film at the injector exit,which in turn increase the breakup length. In experiment, these flow

Film type Wave mode

Thinning film Long wave

2=3

� 1

37775

Thinning film

We�1=2l

=η0Þ


=3

þ ðD0 � h0Þ3=2Thinning film Long wave

9>>>>>>>>>=>>>>>>>>>;


Conical film

Conical film


characteristics are influenced by the operation condition and thegeometrical parameters of the injector. The operation conditions mainlyinfluence the liquid film velocity V0 or the gas-liquid relative velocity U,the gas density, the spray cone half angle β and the film thickness at theinjector exit h0. A lot of experiments show that the increase of the liquidviscosity increases the breakup length [18,113,114]. And the increase ofthe backpressure decreases the breakup length [122,125,126], becausethe intensified aerodynamic force increases the amplitude of the surfacewaves, which in turn promote the breakup of the liquid film [122]. Theinjector geometrical parameters also influence the liquid film velocity V0or the gas-liquid relative velocity U, the spray cone half angle β and thefilm thickness at the injector exit h0. The larger the geometry charac-teristics constant, the larger the breakup length [65].

Besides with the flow characteristics, the spray patterns and the tur-bulence in the liquid film also have great influence on the breakuplength. Ghorbanian et al. [19] found that the breakup length increaseswith the increase of pressure drop in the onion stage, and then decreaseswith the increase of pressure drop in the next stages. Inamura et al. [58]found that the breakup length increases with the increase of liquid

Fig. 17. Typical spray structure of

187

velocity when the flow is laminar and decreases with that when the flowis turbulent.

2.3. Atomization characteristics of the conical liquid film

Atomization of the conical liquid film is quite complicate, whichinclude primary atomization and secondary atomization. The primaryatomization defines the breakup of the liquid film and the furtherbreakup of the produced ligaments. And the secondary atomization de-fines the breakup of the big droplets. Typical spray structure of pressureswirl injector is shown in Fig. 17.

Atomization characteristics are the parameters that characterize theatomization efficiency. And they are the most direct indexes to evaluatethe performance of the pressure swirl injector. The atomization charac-teristics include the diameter distribution, the velocity distribution, thediameter-velocity distribution and the mass flow rate distribution.Limited by the experimental technique, it is hard to obtain the diameterand velocity of all the droplets in the spray at a specific time. Thus thediameter, velocity and mass flow rate are statistical results of the droplets

pressure swirl injector [127].


accumulated over some time at one point in the spray. And these pa-rameters are the space distributions of the statistical diameter, the sta-tistical velocity and the statistical mass flow rate.

2.3.1. SMD distributionThe conical liquid film first breakup into ligaments, and then these

ligaments further breakup into droplets. Assume that the produceddroplets have the same diameter, thus the theoretical diameter of thedroplets d can be calculated with the diameter of the ligaments dl.However, the practical atomization process is much more complicatethan the theoretical case, because the droplets with different diameterscan be produced by multiple ways. For example, the intense gas liquidinteraction can peel droplets from the film, the turbulence energy canshake out droplets from the film and the satellite droplets are producedaccompany with the production of ligaments. So, the theoretical analysisis insufficient to reflect the practical diameter of the spray. And experi-ment is still the most important approach to obtain the statistical diam-eter of the spray.

Sauter mean diameter (SMD) and its' distribution is the most impor-tant parameter that characterize the statistical diameter of the spray. Andthe definition of SMD is:

SMD ¼P

D3i NiPD2i Ni

(18)

The SMD can be measured with Mie scattering technique (Malvernsizer), Phase Doppler Anemometry (PDA), laser holography and imageprocessing of instantaneous spray. And empirical equations of SMD isthen obtained with these measurement methods. However, the

Table 7SMD calculation equations.

Author Equation

Sivakumar et al. [119] dl ¼ffiffiffiffiffiffi4h0Ks

q,

Moon et al. [102] d ¼ 1:88dlðDombrowski and

dl ¼ 0:9614

Johns(Xiao and Huang [120]) d ¼ 1:88dlðCouto et al. [63,128]

dl ¼ 0:961

d ¼ 1:89dlRadcliffe(Xiao and Huang [120], Santangelo et al. [129]) SMD ¼ 7:3Jasuja(Xiao and Huang [120], Yang [130]) SMD ¼ 4:4Ballester(Xiao and Huang [120]) SMD ¼ 0:4

At : Area ofLefebvre(Santolaya et al. [103], Park and Heister [131]) SMD ¼ 2:2Van Banning et al. [109]

SMD ¼

σμρ

Liu et al. [14]SMD ¼ 0:5

where f ¼Wang and Lefebvre (Sivakumar et al. [119], Semiao [132])

SMD ¼ A

where A ¼

B ¼ 0:635½cWang and Lefebvre (Xiao and Huang [120], Santangelo [108])

SMD ¼ 4:5

Davanlou et al. [133]SMD ¼ 2:1

Xiao and Huang [120]SMD ¼ 22

"

188

application of these empirical equations is limited because these equa-tions are strongly related with the geometry and machining accuracy ofthe given pressure swirl injector. To overcome the deficiencies of theo-retical analysis and experimental derived prediction equations, boththeoretical analysis and experimental results are combined together. Anda semi-empirical prediction equation of SMD is then derived. All thetheoretical, empirical and semi-empirical prediction equations of SMDare listed in Table 7.

It is clear that SMD is significantly influenced by the injector geom-etry, operation condition and physical parameters of liquid. The injectorgeometry includes the geometry characteristics constant K, the injectordiameter D0, tangential ports area At , diameter of the swirl chamber Dsand the orifice length L0. The operation condition includes the pressuredrop ΔPl and the mass flow rate _ml. And the physical parameters containthe surface tension σ, liquid density ρl, ambient gas density ρg and liquidviscosity μl. Besides with these geometry parameters, operation param-eters and physical parameters, some parameters determined by bothinjector geometry and operation condition also have great influence onSMD. For example the film thickness at the injector exit h0, the spray conehalf angle β and the velocity of the liquid film V0.

First, the influence of the operation condition on SMD. As the pressuredrop ΔPl and the liquid mass flow rate _ml are related with each other,only the variation trend of SMD with pressure drop is given. And thevariation trends of SMDwith the liquid mass flow rate and liquid velocityare similar with the variation trend of SMD with pressure drop, becausethe liquid mass flow rate _ml and liquid film velocity V0 increase with theincrease of pressure drop ΔPl. The results through extensive experimentshow that the SMD decreases with the increase of pressure drop [19,63,

Method

where Ks ¼ ρg V20

2σTheoretical

1þ 3OhÞ16, where Oh ¼ μlffiffiffiffiffiffiffiffiρlσdl

p

k20σ2

ρgρlV20

!1=61þ 2:6μl

k0ρ4g V

70

72ρ2l σ5

!1=3 1=5375264

Theoretical

1þ 3OhÞ16, Oh ¼ μlffiffiffiffiffiffiffiffiρlσdl

p , k0 ¼ h0x

4cosβ

h40σ

2

ρgρlV40

!1=6�

h20ρ

4g V

70

72ρ2l σ5

!1=3 1=53524 Theoretical

σ0:6μ0:2l ρ�0:2l m

0:25l ΔP

�0:4l Empirical

σ0:6μ0:16l ρ�0:16l m

0:22l ΔP

�0:43l Empirical

36μ0:55l ΔP�0:74l D

�0:050 A

�0:24t Empirical

the tangential ports5σ0:25μ0:25l m

0:25l ρ

�0:25g ΔP�0:5l Empirical

l _mlg

!0:25ð3:29� 0:06Þ � 10000� ΔP�0:5l

Empirical

536σ0:25μ0:25l ρ0:125l D

0:50 � ρ�0:25g ΔP�0:375l f

�DsD0

;L0D0

; β

�Empirical

�DsD0

�0:33�L0D0

�0:122ð1þ tanβÞ1:38

σμ2lρgΔP2l

!0:25ðh0cosβÞ0:25 þ B

σρl

ρgΔPl

!0:25ðh0cosβÞ0:75

Empirical

2:11½cos2ðβ � 30Þ�2:25�

3:4�10�4D0

�0:4

os2ðβ � 30Þ�2:25�

3:4�10�4D0

�0:2

2

σμ2l

ρgΔP2l

!0:25ðh0cosβÞ0:25 þ 0:39

σρl

ρgΔPl

!0:25ðh0cosβÞ0:75

Empirical

1

σμ2l

ρgΔP2l

!0:25ðh0cosβÞ0:25 þ 0:62

σρl

ρgΔPl

!0:1ðh0cosβÞ0:9

Empirical

KD0ð1þffiffiffiX

p Þ1�X

#12"h1:170 ðD0�h0 Þ0:67

_m0:67l

# σ2ρ3lρg

!16 Semi-empirical


109,120,129,130,134–140]. It is because that the increase of pressuredrop can accelerate the liquid film. And the accelerated liquid film havelarger turbulence energy and larger growth rate of surface waves, whichin turn promote the breakup of conical liquid film. What's more, the in-crease of pressure drop can decrease the film thickness at the injector exith0 and increase the spray cone half angle β, both the variations of h0 and βpromote the breakup of liquid film and decrease SMD [120]. Eberhartet al. [134] found that a 22% mass flow rate sub-nominal throttle in-creases the surface wave length by 30%, and a 23% mean axial velocitydecrease increases the SMD by 9%.

Second, the influence of the physical parameters on SMD. The SMDincreases with the increase of surface tension σ and liquid viscosity μl,which indicates that the surface tension and liquid viscosity suppress thebreakup and atomization of the conical liquid film [114,135]. The reasonis that the increase of surface tension suppresses the growth of the surfacewaves, and the increase of liquid viscosity increases the film thickness atthe injector exit and decreases the spray cone angle. Both the increase offilm thickness and the decrease of spray cone angle increase the SMD[120]. Besides with the surface tension and liquid viscosity, the tem-perature of the liquid and the ambient gas density also have great in-fluence on SMD. Kim et al. [136,137,141] and van Banning et al. [109]found that the SMD decreases with the increase of liquid temperatureespecially when the liquid is superheated, because the increase of liquidtemperature decreases the surface tension, which in turn promote thebreakup of the conical liquid film. What's more, with the increase ofliquid temperature, the perforation breakup seems to be more frequentand distinct, and onset of holes on the conical liquid film occurs earlier[136]. And the perforation also promotes the breakup and atomization ofthe conical liquid film. The increase of the ambient gas density ρg de-creases SMD, because it intensify the gas liquid interaction and promotethe breakup of the conical liquid film.

Third, the influence of the injector geometry on SMD. The mostimportant geometrical parameter is the injector constant K, and it isrelated with the tangential ports area At . Xiao and Huang [120] foundthat SMD increases with the increase of tangential ports area, because theincrease of tangential ports area increases the injector constant. And thefilm thickness at the injector exit increases and the spray cone angledecreases with the increase of injector constant. What's more, the smallerthe spray cone angle and the thicker the film at the injector exit, thelarger the SMD [120]. On the effects of the orifice length L0, Xiao andHuang [120] and Liu et al. [14] found that SMD increases with the in-crease of orifice length, because it decreases the spray cone angle andincreases the film thickness at the injector exit. No matter how thegeometrical parameters vary, as long as it decreases the film thickness atthe injector exit and increases the spray cone angle, it decreases the SMD.

SMD is a kind of statistical diameter of the spray. And each spray canbe characterized by one SMD, but it not means that the droplets in thespray is uniform. In fact, the droplets in the spray are quite nonuniformand have a diameter distribution. Typical diameter distribution of a sprayis shown in Fig. 18. There are three methods to model the droplet

Fig. 18. Typical diameter distribution of a spray [109].

189

diameter distribution: the Maximum Entropy (ME) method, the DiscreteProbability Function (DPF) method and the empirical method. Babinskyand Sojka [142] have thorough reviewed the modeling methods of dropsize distribution. The maximum entropy method have been widelyinvestigated by Dumouchel et al. [28,143–148], and Mondal et al. [149,150]. The idea of the discrete probability function method is simple.First, it assumes that the conical liquid film initially breakup into liga-ments, which eventually collapse into drops. This process is determin-istic, and the diameter of the ligaments and droplets can be calculatedwith the linear analysis. Then, the diameter distribution is producedwhen the input initial conditions (pressure drop, mass flow rate, velocity,physical properties, etc.) fluctuate with a probability density function.The empirical method is the classical method to model the diameterdistribution. It uses a curve to fit to the data obtained in a wide range ofoperation condition with all kinds of injectors. Babinsky and Sojka [142]summarized seven empirical distributions, as listed in Table 8.

Among these three droplets diameter distribution predictionmethods, the empirical method is hard to apply to other injectors,because the distribution function is strongly related with the injectorgeometry and operation condition. The maximum entropy method focuson the initial and final states of the spray and neglect the primarybreakup and secondary atomization. The discrete probability functionmethod is suitable for the spray produced by the primary breakup. Thespray with strong gas-liquid interaction is dominated by secondary at-omization, and is more appropriate to use the maximum entropy method.As the present paper places emphasis on the physical process of primarybreakup and atomization, the maximum entropy method is not discussedtoo much. For interested readers, Dumouchel's research is recommended.

If the measuring volume is small enough, the SMD calculated with thedroplets in this volume can represent the diameter of droplets at this‘point’ of the spray. Thus the SMD distribution along radius and axis canbe obtained by moving the measuring volume. A large amount ofexperimental results indicate that the SMD distribution of the pressureswirl injector shows a ‘single peak’ shape [21,102,103,127,134,139,151–153], as shown in Fig. 19. It is because that the droplets around thestream line of the conical liquid film are directly breakup from ligaments,while the droplets in the center or at the periphery of the spray areentrained from the stream-line region of the liquid film. Secondary at-omization occurs during the entrainment, which produces smallerdroplets. Normally, the SMD decreases along the axis. But if the spray istense enough, the SMD initially decreases and then increases slightlyalong the axis [108,137], because the coalescence of droplets due tocollision is significant [133], as shown in Fig. 20. Santangelo [108] foundthat the secondary atomization governs the droplet size until a criticallocation is reached, then the coalescence of the droplets increases thediameter of the droplets. Sivakumar et al. [119] found that the liquid filmundergoes primary breakup and secondary atomization until 35–45mmdownstream the injector exit. After that the SMD increases for the coa-lescence of the droplets. Santolaya et al. [103,127,153] investigated theeffects of droplets collision on the diameter distribution. And found thatthe maximum collision rate occurs at the densest zone where the relativevelocity of droplets is also high. He figured out that the increase ofpressure drop increases the collision rate and the increase of collisionWebber number promote the separation of droplets with satellite dropletformation.

2.3.2. Velocity distributionDroplet velocity is another important parameter of the atomization

characteristics. Similar with the SMD, the mean axial velocity of dropletsalong radius shows a ‘single peak’ shape [19,21,127,151,153], as shownin Fig. 21. It is clear that the droplet's velocity is negative at the spraycenter near the injector exit, which indicates that there exists a recircu-lation zone [134]. The minimum mean axial velocity of the dropletsoccurs at the center of the spray because these droplets are entrained bythe recirculation zone. And the maximum mean axial velocity of thedroplets occurs around the stream line of the liquid film because these

Table 8Empirical diameter distributions.

Distribution name Distribution function Distributiontype

The log-normaldistribution f0ðDÞ ¼ 1DðlnσLN Þ ffiffiffiffi2πp exp

�� 12lnðD=DÞlnσLN

2� Numberdistribution

D: logarithmic mean size of the distribution, σLN : width of the distributionThe upper-limit distribution

f3ðDÞ ¼ δDmaxDðDmax�DÞ ffiffiπp exp�� δ2

ln�

aDDmax�D

�2� Volumedistribution

a ¼ DmaxD, δ ¼ 1ðlnσULÞ ffiffi2p

D: representative diameter, Dmax : maximum drop diameter, σUL: distribution widthThe root-normaldistribution f3ðDÞ ¼ 12σRN ffiffiffiffiffiffi2πDp exp

(� 12" ffiffiffi

Dp �

ffiffiffiD

pσRN

#2) Volumedistribution

D: mean diameter, σRN : distribution widthThe Rosin-Rammlerdistribution

f0ðDÞ ¼ qD�qDq�1expf � ðD=DÞqg Numberdistribution

f3ðDÞ ¼ 1� expf � ðD=DÞqg Volumedistribution

D: mean diameter, q: distribution widthThe Nukiyama-Tanasawadistribution

f0ðDÞ ¼ aDpexpf�bDqg, b; p; q adjustable parameters, a is a normalizing constant Numberdistribution

The Log-hyperbolicdistribution

f0ðx; α; β; δ; μÞ ¼ aðα; β; δÞ � expf � αffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiδ2 þ ðx � μÞ2

qþ βðx � μÞg Number

distribution

a ¼ffiffiffiffiffiffiffiffiffiffiα2�β2

p2αδK1ðδ

ffiffiffiffiffiffiffiffiffiffiα2�β2

pÞis a normalizing constant, K1 is the modified Bessel function of the third kind and first order, δ is the scale

parameter, μ is the location parameter, α and β describes the shape of the PDFThe three-parameter log-hyperbolic distribution f0ðxÞ ¼ Aexp

�a

a2cos2θ�sin2θ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiða2cos2θ � sin2θÞ þ ðx þ μ0 � μÞ2

q� ða2þ1Þsinθcosθa2cos2θ�sin2θ ðx þ μ0 � μÞ

� Numberdistribution

where α, θ, μ are shape parameters, A is the normalizing constantμ0 ¼ � ða2þ1Þsinθcosθffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a2�ða2þ1Þ2 sin2 θcos2 θa2 cos2 θ�sin2 θ

q A ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2�ða2þ1Þ2sin2θcos2θp2affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2cos2θ�sin2θ

pK1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2�ða2þ1Þ2 sin2 θcos2 θ

a2 cos2 θ�sin2 θ

q


droplets directly breakup from the ligaments and inherit the filmvelocity.

2.3.3. Diameter-velocity distributionBesides with SMD and mean axial velocity, diameter-velocity distri-

bution is also an important atomization characteristics. It shows thecorresponding relation of the droplet diameter and the droplet velocity.At present research on diameter-velocity distribution is seldom, Belhadefet al. [154] and Santolaya et al. [103] shows the diameter-velocity dis-tribution in there research, and the typical diameter-velocity distributionmeasured by PDA is shown in Fig. 22. Ayres et al. [155] proposed anprediction model for diameter-velocity distribution based on themaximum entropy formalism, and the typical predicteddiameter-velocity distribution is shown in Fig. 23. The predictions agreewell with the available data for the velocity distribution.

Fig. 19. Typical SMD distribution along radius [134].

190

2.3.4. Mass flow rate distributionThe mass flow rate distribution have great influence on the mixing,

which in turn influence the combustion. Similar with the distribution ofSMD and mean axial velocity, the mass flow rate distribution (or themass/volume flux) shows ‘single peak’ shape, as shown in Fig. 17. It isbecause that the primary breakup and secondary atomization produces alarge amount of droplets around the stream line of the liquid film. Andthese droplets have larger SMD and mean axial velocity than the centerand the periphery of the spray. Santolaya et al. [103,127,153] found thatthe liquid flow rate is concentrated in a small region where the dropletspresented a high dispersion of sizes. And the high collision rates ofdroplets in the dense spray could redistribute the liquid mass flow rate.Koh et al. [156] measured the mass flow rate from laser induced

Fig. 20. Typical SMD distribution along axis [108].

Fig. 21. Typical mean axial velocity distribution of the pressure swirlinjector [134].

Fig. 22. Typical diameter-velocity distribution of droplets in one measuringvolume [154].

Fig. 23. Diameter-velocity distribution predicted by the theoreticalmodel [155].


191

fluorescence images, and found it agrees well with the mass flow ratemeasured by PDPA.

2.4. Effects of the rocket engine environment

In the liquid rocket engine, the chamber pressure and the combustiongas temperature are really high, which indicates that the atomizationoccurs with the back pressure and high temperature. If the pressure ex-ceeds the critical pressure and the temperature exceeds the criticaltemperature, the propellant is supercritical. The atomization of propel-lant at supercritical conditions is quite different from that at normalcondition. In this section, the effects of high back pressur

review on pressure swirl injector in liquid rocket engine · liquid-propellant rocket engines have...

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