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Review on pressure swirl injector in liquid rocket engine
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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
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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.
Z. Kang et al. Acta Astronautica 145 (2018) 174–198
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.
Z. Kang et al. Acta Astronautica 145 (2018) 174–198
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
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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].
Z. Kang et al. Acta Astronautica 145 (2018) 174–198
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
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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.
Z. Kang et al. Acta Astronautica 145 (2018) 174–198
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
Electrical conductance method Open-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].
Z. Kang et al. Acta Astronautica 145 (2018) 174–198
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-end
Experimental Converge-endExperimental Open-end
Experimental Converge-end
Experimental Converge-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].
Z. Kang et al. Acta Astronautica 145 (2018) 174–198
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].
Z. Kang et al. Acta Astronautica 145 (2018) 174–198
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].
Z. Kang et al. Acta Astronautica 145 (2018) 174–198
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
Experimental Converge-end
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
Z. Kang et al. Acta Astronautica 145 (2018) 174–198
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
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Z. Kang et al. Acta Astronautica 145 (2018) 174–198
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.
Z. Kang et al. Acta Astronautica 145 (2018) 174–198
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
Thinning film Long wave
=3
þ ðD0 � h0Þ3=2Thinning film Long wave
9>>>>>>>>>=>>>>>>>>>;
Thinning film Long wave
Conical film
Conical film
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Z. Kang et al. Acta Astronautica 145 (2018) 174–198
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].
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Z. Kang et al. Acta Astronautica 145 (2018) 174–198
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
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Z. Kang et al. Acta Astronautica 145 (2018) 174–198
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
Z. Kang et al. Acta Astronautica 145 (2018) 174–198
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].
Z. Kang et al. Acta Astronautica 145 (2018) 174–198
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