XUE JIANQING
136
UNIVERSITY OF CINCINNATI Date:___________________ I, _________________________________________________________, hereby submit this work as part of the requirements for the degree of: in: It is entitled: This work and its defense approved by: Chair: _______________________________ _______________________________ _______________________________ _______________________________ _______________________________
Transcript of XUE JIANQING
UNIVERSITY OF CINCINNATI Date:___________________
I, _________________________________________________________, hereby submit this work as part of the requirements for the degree of:
in:
It is entitled:
This work and its defense approved by:
Chair: _______________________________ _______________________________ _______________________________ _______________________________ _______________________________
COMPUTATIONAL SIMULATION OF FLOW INSIDE PRESSURE-SWIRL ATOMIZERS
A dissertation submitted to the
Division of Research and Advanced Studies of the University of Cincinnati
in partial fulfillment of the requirements for the degree of
DOCTOR OF PHILOSOPHY (Ph. D.)
in the Department of Mechanical, Industrial and Nuclear Engineering of the College of Engineering
2004
by
Jianqing Xue
B.S., Tianjin University, 1989 M.S., Tianjin University, 1992
Dissertation Committee:
Dr. Milind A. Jog, Chair Dr. San-Mou Jeng Dr. Raj M. Manglik Dr. Rupak K. Banerjee
ii
ABSTRACT
Simplex atomizers (pressure-swirl atomizers) are widely used in air-breathing gas
turbine engines as they have good atomization characteristics and are relatively simple
and inexpensive to manufacture. To reduce emissions, it is critical to design fuel
atomizers that can produce spray with a predetermined droplet size distribution at the
desired combustor location (small mean droplet diameters and uniform local air/fuel
ratios). Manufacturing methods are now available where complex atomizer geometries
can be easily obtained. However to use such methods, the influence of atomizer
geometry on its performance must be well understood.
In this dissertation, a two-dimensional axi-symmetric computational fluid
dynamics (CFD) model based on the Arbitrary-Lagrangian-Eulerian (ALE) method to
predict the flow in pressure-swirl atomizers was developed. The Arbitrary-Lagrangian-
Eulerian method was applied so that the free interface between gas and liquid could be
tracked sharply and accurately. The developed code was validated by comparison of
predictions with experimental data for large scale prototype and with semi-empirical
correlations at small scale. The computational predictions agreed well with experimental
data for the film thickness at the exit, spray cone angle, and the pressure drop across the
atomizer as well as velocity field in the swirl chamber.
Using the validated code, a comprehensive parametric study on simplex atomizer
performance was conducted. The geometric parameters of atomizer covered in this study
include: atomizer constant (K), the ratio of length to diameter in swirl chamber (Ls/Ds),
the ratio of length to diameter in orifice (lo/do), the swirl chamber to orifice diameter ratio
(Ds/do), inlet slot angle (β), trumpet angle (θt), trumpet length (lt), and swirl chamber
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convergent angle (θc). The effects of these geometric parameters on the atomizer
performance were studied for a fixed mass flow rate through the atomizer as well as for a
fixed pressure drop across the atomizer. The atomizer performance was described in
term of dimensionless film thickness at the exit (t*), discharge coefficient (Cd) and spray
cone half angle (θ).
To address applications in pharmaceutical and food processing industry, flow of
non-Newtonian power-law fluids through pressure-swirl atomizers was considered.
Detailed flow patterns inside the atomizer for shear-thinning, Newtonian and shear-
thickening fluids were investigated. A range of power law index from 0.7 to 1.3 was
considered. With a fixed flow rate through the atomizer, the shear-thickening fluids
exhibit higher film thickness at exit, lower spray angle, and higher discharge coefficient
compared to Newtonian fluids. For the range of power law index considered in this study,
the atomizer performance parameters for shear-thinning fluids show small change from
Newtonian fluids. The variation of atomizer performance with the atomizer constant was
delineated for different power-law index.
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v
ACKNOWLEDGEMENTS
I am greatly indebted to my advisor, Professor Milind Jog, who not only served as
my supervisor but also encouraged and challenged me throughout my academic program.
His technical and editorial advice was essential to the completion of this dissertation. I
have learned quite a lot from his extensive knowledge and many brilliant ideas. His all-
aspect help has made the past years a good memory in my life.
I would deeply thank my co-advisor, Professor San-mou Jeng, for hosting me in
his group as a PhD student. His supervision and continuous supports laid a smooth way
for my research work. Through rewarding discussions with him, I have learnt much more
knowledge in CFD and atomization. I thank, with the highest respect, Professor Raj
Manglik and Professor Rupak Banerjee for serving on my dissertation committee and for
being there whenever I needed their help.
I am especially grateful to Dr. Erlendur Steighthorsson for his help on automatic
grid generator creating and much valuable advice on CFD code developing. I learnt quite
much CFD knowledge from him. Special thanks go to Dr. Michael Benjamin and Dr.
Adel Mansour, who gave me much encouragement and help during my three-month co-
op work time in Gas Turbine Spray System department, Parker Hannifin Corporation in
1999. I am greatly thankful to Dr. Dexin Wang, one of my best friends, who not only
gave me much advice on my academic work but helped me much in my life too. I am
deeply grateful to Mr. Tolga Sakman, who gave me much help on the ALE method
understanding at the beginning days I was involved in this project. My sincere thanks
should also go to Dr. Zhanhua Ma, Dr. Jun Cai, Dr. Juntao Zhang and Mr. Ibrahim
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Ashraf for their technical suggestion on my research work. Discussion with them for
technical issue was very enjoyable.
I would like to thank my wife, Ms. Li Jia, for her understanding and love during
these years. Her support and encouragement was in the end what made this dissertation
possible. My parents receive my deepest gratitude for their ever-loving supports and
understanding during the years of my study and work.
At last, the financial support of Parker Hannifin Corporation and the National
Science Foundation is greatly appreciated.
1
TABLE OF CONTENTS
ABSTRACT ..............................................................................................ii
ACKNOWLEDGEMENTS .....................................................................v
TABLE OF CONTENTS .........................................................................1
LIST OF TABLES ....................................................................................4
LIST OF FIGURES ..................................................................................5
NOMENCLATURE ..................................................................................8
CHAPTER 1 INTRODUCTION ……………………………….…......11
1.1 Motivation …………………………………………………………………….....11
1.2 Literature Review ………………………………………….………………........14
1.3 Scope of Dissertation ………………………………………………………........18
CHAPTER 2 COMPUTATIONAL MODEL ………………………....21
2.1 Review of Computational Methods ....................................................................21
2.2 Mathematical Formulation …………………………………………………......23
2.3 Stress Calculation ..............................................................................................24
2.4 Discretization ....................................................................................................26
2.5 Pressure Field Iteration ...........................................................................................28
2.6 Boundary Conditions ………………………………………………………........32
2
2.7 Grid Generator ..................................................................................................35
2.8 Code Structure ..................................................................................................36
CHAPTER 3 CODE VALIDATION ……………………………….....37
3.1 Verification of Grid Independence of Results ……………………………........37
3.2 Comparison with Experimental Data and Empirical Correlations .......................37
3.3 Comparison with Correlations at Small Scale ……………………………….....43
3.4 Summary ………………………………………………………………….......49
CHAPTER 4 EFFECTS OF GEOMETRIC PARAMETERS ON
ATOMIZER PERFORMANCE (PART I) ………………………….....50
4.1 Introduction ……………………………………………………………….........50
4.2 Results for Constant Mass Flow Rate through the Atomizer ….……………….52
4.3 Results for Constant Pressure Drop across the Atomizer ………........................61
4.4 Summary …………………………………………………………………….......70
CHAPTER 5 EFFECTS OF GEOMETRIC PARAMETERS ON
ATOMIZER PERFORMANCE (PART II) ……………………….....71
5.1 Introduction ......................................................................................................71
5.2 Effect of Variation in Inlet Slot Angle β ............................................................74
5.3 Effect of Variation in Trumpet Angle θt .............................................................80
5.4 Effect of Variation in Dimensionless Trumpet Length (lt/do) ………………..83
3
5.5 Effect of Variation in Swirl Chamber Convergence Angle θc ………………...85
5.6 Summary …………………………………………………………………….......88
CHAPTER 6 NON-NEWTONIAN FLUID FLOW IN
PRESSURE-SWIRL ATOMIZERS ………………………………......90
6.1 Introduction …………………………………………………………………......90
6.2 Mathematical Model ………………………………………………………........91
6.3 Influence of Power-law Index on the Performance of Atomizers with
Constant Flow Rate across the Atomizer ………………………..……………........92
6.4 Influence of Atomizer Constant on the Performance of Atomizers for both
Newtonian and non-Newtonian Fluids ………………………….………….........107
6.5 Summary .........................................................................................................111
CHAPTER 7 CONCLUSIONS AND RECOMMENDATIONS …...113
7.1 Conclusions ……………………………………………………………….....113
7.2 Recommendations for Future Work ……………………………………….......115
BIBLIOGRAPHY ………………………………………………..........116
APPENDIX INVISCID ANALYSIS FOR PRESSURE SWIRL
ATOMIZERS ………………………………………………………....124
4
LIST OF TABLES
Table 3.1 Results of different grid density …………………………………………......37
Table 3.2 Cases summary ............................................................................................39
Table 3.3 Comparison results of experimental, CFD and correlation data ………….....42
5
LIST OF FIGURES
Figure 1.1 Schematic of a simplex atomizer ……………………………………….......13
Figure 2.1 The assignment of the variables about a cell…………………………..........27
Figure 2.2 The schematic of mass and momentum control volume ………………........27
Figure 2.3 Flowchart of the code ………………………………………………….........36
Figure 3.1 Final grids of different grid density……………………………………........38
Figure 3.2 Comparison of swirl velocity variations at three axial locations………........40
Figure 3.3 Variation of discharge coefficient ……………………………………..........44
Figure 3.4 Variation of dimensionless film thickness …………………………….........46
Figure 3.5 Variation of Spray Angle………………………………………………........48
Figure 4.1 Variation of the dimensionless film thickness at the exit and discharge
coefficient with Ls/Ds under constant flow rate …………………………………..........53
Figure 4.2 Variation of the spray cone half angle and wav/uav with Ls/Ds under
constant flow rate ………………………………………………………………….........54
Figure 4.3 Variation of the dimensionless film thickness at the exit and discharge
coefficient with lo/do under constant flow rate ……………………………………........56
Figure 4.4 Variation of the spray cone half angle and wav/uav with lo/do under
constant flow rate ………………………………………………………………….......57
Figure 4.5 Variation of the dimensionless film thickness at the exit and discharge
coefficient with Ds/do under constant flow rate …………………………………….......59
Figure 4.6 Variation of the spray cone half angle and wav/uav with Ds/do under
constant flow rate …………………………………………………………………........60
6
Figure 4.7 Variation of the dimensionless film thickness at the exit and discharge
coefficient with Ls/Ds under constant pressure drop………………………………........62
Figure 4.8 Variation of the spray cone half angle and wav/uav with Ls/Ds under
constant pressure drop …………………………………………………………….........63
Figure 4.9 Variation of the dimensionless film thickness at the exit and discharge
coefficient with lo/do under constant pressure drop ………………………….......65
Figure 4.10 Variation of the spray cone half angle and wav/uav with lo/do under
constant pressure drop ……………………………………………………………........66
Figure 4.11 Variation of the dimensionless film thickness at the exit and discharge
coefficient with Ds/do under constant pressure drop ………………………………........68
Figure 4.12 Variation of the spray cone half angle and wav/uav with Ds/do under
constant pressure drop …………………………………………………………….........69
Figure 5.1 Schematic of a simplex atomizer with trumpet ………………………........72
Figure 5.2 Variation of atomizer performance parameters with inlet slots angle β ........75
Figure 5.3 Comparison of results for lo/do = 1(solid lines) and lo/do =0.5 (dashed
lines) ...........................................................................................................................76
Figure 5.4 Variation of atomizer performance parameters with trumpet angle θt ….......81
Figure 5.5 Variation of atomizer performance parameters with ratio of trumpet
length to orifice diameter lt/do ……………………………………………………........84
Figure 5.6 Variation of atomizer performance parameters with convergence angle
θc …………………………………………………………………………………........86
Figure 5.7 Streamlines for flow inside the atomizer for different θc. Inlet slot
angle β=900 ......................................................................................................................87
7
Figure 6.1 Initial grid ………………………………………………………………......93
Figure 6.2 Final grid ....................................................................................................94
Figure 6.3 Streamline and pressure distribution…………………………………….......95
Figure 6.4 Variation of film thickness with power index ………………………..........96
Figure 6.5 Variation of discharge coefficient with power index …………………......97
Figure 6.6 Variation of Spray half angle with power index ……………………..….....98
Figure 6.7 Locations for the velocity inside nozzle study ............................................99
Figure 6.8 Axial velocity profiles at the swirl chamber ………………………….......100
Figure 6.9 Axial velocity profiles at the orifice ……………………………................101
Figure 6.10 Axial velocity profiles at the orifice ..........................................................102
Figure 6.11 Swirl velocity profiles at the swirl chamber ..............................................103
Figure 6.12 Swirl velocity profiles at the orifice ..........................................................104
Figure 6.13 Swirl velocity profiles at the exit …………………………………….......105
Figure 6.14 Variation of film thickness with atomizer constant ………………….......108
Figure 6.15 Variation of discharge coefficient with atomizer constant.........................109
Figure 6.16 Variation of spray cone angle with atomizer constant ...............................110
8
NOMENCLATURE
Aa air core area at orifice exit
Ao orifice area
Ap total swirl slot area
At the trumpet end area
Ata air core area at the trumpet end
Cd discharge coefficient, ( ) 5.02 ρPA
m
o ∆
&
Ds spin chamber diameter
do orifice diameter
dt trumpet diameter at atomizer exit
fr
body force
K atomizer constant, Ap/(Dsdo)
K1 Ap/(πrors )
Kt Ap/(πrtrs)
Ls spin chamber length
lo orifice length
lt trumpet length
p static pressure
Q volume flow rate
ri the radial distance from axis to inlet slot
ro orifice radius, do/2
9
rs spin chamber radius, Ds/2
rt dt/2
S(t) surface enclosing control volume V(t)
t film thickness at exit
t* dimensionless film thickness, t/(do/2)
U average total velocity at the end of orifice
u axial velocity component
Ur
arbitrary velocity vector for the control volume V(t)
ur velocity vector
ou average axial velocity at the end of orifice
uoa axial velocity at the liquid-air interface at the end of orifice
tu average axial velocity at the end of trumpet
uta axial velocity at the liquid-air interface at the end of trumpet
V(t) control volume
w tangential velocity component
wi average tangential velocity at the inlet
ow average tangential velocity at the end of orifice
woa tangential velocity at the liquid-air interface at the end of orifice
tw average tangential velocity at the end of trumpet
wta tangential velocity at the liquid-air interface at the end of trumpet
X Aa/Ao
Xt Ata/At
∆p pressure drop across the nozzle
10
β inlet slot angle
θ spray cone half angle
θc spin chamber convergence angle
θt trumpet angle
θ` difference between spray cone half angle and trumpet angle, θ -θt
ρ density
τ viscous stress
Subscript
i parameter at the inlet
o parameter at the end of orifice
oa parameter at the liquid-air interface at the end of orifice
t parameter at the end of trumpet
ta parameter at the liquid-air interface at the end of trumpet
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CHAPTER 1 INTRODUCTION
1.1 Motivation
Liquid atomization, the process of producing a large number of droplets from
bulk liquid, is used in a variety of engineering applications, in pharmaceutical industries,
process industries, fuel injection in combustion applications, and in agricultural sprays,
among others. A number of spray devices have been developed for this purpose, and
they are generally designated as atomizers or nozzles (Lefebvre, 1989). Among these,
pressure-swirl atomizers or simplex atomizers are commonly used for liquid atomization
due to their simple design, ease of manufacture, and good atomization characteristics.
The applications such as fuel injection systems of the gas turbines engines, crop spraying
in agriculture, the production of powders by spray drying, water sprays for fire
suppression, oil spray for combustors in furnaces and power stations, involve Newtonian
fluids. On the other hand, applications such as atomization of aqueous polymeric
solutions in manufacture of pharmaceutical products, paint sprays, spray drying of food
and detergents, sprays in healthcare products, involve fluids that exhibit non-Newtonian
flow behavior. Figure 1 shows a schematic of simplex atomizer geometry. In a simplex
atomizer, the liquid is forced under high pressure to enter a swirl chamber through inlet
slots at the outer wall. An air core is formed along the centerline due to high swirl
velocity of the fuel. The liquid exits the atomizer through a small orifice with even higher
swirl velocity that forces the liquid to disperse radially outwards to form a hollow-cone.
The thin liquid sheet then becomes unstable and breaks up to form a spray of droplets.
12
Owing to the numerous applications of pressure-swirl atomizers, a large number
of studies are available in published literature. Focus of earlier studies has been on the
phenomena taking place after the liquid exits the atomizers, break-up of simple forms of
bulk fluids, on the statistical nature of fuel sprays and on the development of correlations
and operating parameters relevant to spray characteristics. Internal flows in atomizers are
of interest due to their potential effect on the atomization process that takes place external
to the atomizers. Numerous experimental studies have shown that small changes in
geometric features internal to the atomizers can greatly impact the characteristics of the
spray that is produced. However, due to the difficulties in measurements inside a small
scale atomizer and the challenges in modeling of two-phase flow, the flow phenomena
inside the atomizer have not received much attention. The flow inside the atomizer
involves both air and fuel and the interface between the gas and liquid is not known a
priori and must be determined as part of the solution. Furthermore, the flow is turbulent
and contains regions of recirculating flow. Recent advances in accurate tracking of
gas/liquid interface are used in this thesis to simulate the flow in the atomizer. We have
used the Arbitrary-Lagrangian-Eulerian method to numerically simulate the two-phase
flow inside the atomizer and have determined the characteristics of the liquid sheet
emanating from the atomizer. Other students in our research group (Wang, 1999, and
Ma, 2001) have used large-scale prototype atomizer to measure the velocity field inside
the atomizer, film thickness at exit, and the spray cone angle. The computational code
was validated by using their data. The developed computational code was used to
conduct a comprehensive parametric study of the effect of atomizer geometrical
parameters on its performance under both the constant flow rate and constant pressure
13
drop conditions for Newtonian and non-Newtonian (power-law) fluids. This is first such
computational study that computationally investigates flow inside simplex atomizers in
detail. The study provides useful insights in the flow phenomena inside an atomizer and
the results provide guidance for simplex atomizer designers.
A A
Ls Ds
do
x, Ux
View A-A
dp
θ, Uθ
r, Ur
Air-coredvortex
Thin, hollow, conical liquid sheet
Inlet Inlet
Lp
swirlchambersection
exitorificesection
swirler section
Θ
lo
Figure 1 Schematic of a simplex atomizer
14
1.2 Literature Review
Early studies of simplex atomizers employed analytical and/or experimental
methods to measure and/or predict the external spray characteristics (Rizk and Lefebvre
1983, 1985 and 1986; Suyari and Lefebvre 1986; Dodge and Biaglow 1986; Dodge,
Rhodes and Reitz 1987; Dodge and Schwalb 1989; McDonell and Samuelsen et al 1992;
Yule and Widger 1996; Vanderwege and Hochgreb 1998; Okamoto et al 1998).
Rizk and Lefebvre (1985 and 1986) investigated the internal flow characteristics
of simplex swirl atomizers using approximate analytical treatment of the flow. They
investigated the effects of variation of the individual swirl atomizer geometrical
dimensions on the thickness of annular liquid film at the nozzle exit and the effects of the
variation of the fluid properties on the values of the discharge coefficient, the spray angle
and the liquid film thickness. They developed a general expression for the liquid film
thickness at the exit of the swirl atomizer and stated that the air-core diameter increases
with increasing pressure, decreasing inlet area, increasing swirl chamber diameter,
decreasing swirl chamber length, increasing orifice length, decreasing liquid viscosity
and decreasing liquid density.
Horvay and Leuckel (1985 and 1986) studied the velocity profiles within a
pressure swirl atomizer. The experiment were conducted using three different
convergence configurations (standard, concave, and plain conical) and two different
inlet/swirl chamber configurations (four 20 x 10 mm and four 20 x 5 mm rectangular
inlet slots). The atomizers were manufactured from Plexiglas and have same overall
dimensions: radius of swirl chamber rs = 50 mm, length of swirl chamber ls = 25 mm,
15
radius of orifice ro = 10 mm and length of orifice lo = 20 mm. The measurement of the
liquid velocity components within the atomizer were carried out using LDA and a
refractive index matching fluid, which is a mixture of tetraline, turpentine and castor oil.
The seeding particles are small air bubbles. Radial profiles of the axial and tangential
velocities were taken at six different cross-sections through the atomizer.
De Keukelaere (1995) investigated the pressure drop and diameter of air core
from inlet to outlet of swirl atomizer. He found that the air core was almost fully formed
at a pressure of 15 kPa and observed the fluctuations of the air core. This confirmed
Hsieh and Rajamani’s observation.
Herpfer et al (1996 and 1997) developed a nonintrusive instrumentation called
Streaked Particle Imaging Velocimetry and Sizing (SPIVS) probe and used this probe to
measure the size and velocity of droplets in the spray of a Delavan WDB 45 solid-cone
simplex swirl atomizer. They compared the results of SPIVS with that of PDPA and
concluded that the SPIVS diagnostic technique has the ability to accurately and reliably
characterize the liquid droplet properties from a spray even though the SPIVS system
exhibits a tendency towards sampling bias in favor of the larger drop sizes within a spray.
Holtzclaw et al (1997 and 1998) examined the geometrical effects on the internal
flow field in a large-scale simplex fuel nozzle. They measured the tangential and radial
velocity components using PIV techniques and found that the radial velocity was
significantly less than the swirl velocity at any point within the swirl chamber of a
simplex nozzle. They also derived an empirical equation based on the measured swirl
velocity component. Due to the limitations of the imaging technique and post-processing
16
software of image analysis, Holtzclaw’s PIV measurements are not very accurate but
provide qualitative results.
Benjamin et al (1998) investigated the effects of various geometric and flow
parameters on the performance of large-scale pressure swirl atomizers using optical
methods. They measured the film thickness, droplet size and spray angle of a series of
large-scale pressure-swirl atomizers. After testing and analyzing a large number of
geometric variations covering a wide range of flow capacities, they developed some
correlations on the discharge coefficient, flow number, velocity coefficient, spray angle
and Sauter mean diameter based on their experimental data.
Ma (2001) studied the internal flow characteristics in the swirl chamber for both
large-scale and medium-scale pressure swirl atomizers. The internal flow field was
measured using a two-color PIV system and refractive index matching fluids method.
The spray cone angle and liquid film thickness of the nozzle were also obtained from the
PIV image. The measurements of the droplet size and velocity distribution were carried
out using PDPA. According the experimental data, he gave the relationship between the
internal flow field and the external spray characteristics. A non-dimensional correlation
between the vortex flow pattern in the swirl chamber and atomizer design variables was
presented. A discussion on flow regime (turbulent or laminar) within the atomizer body
was given.
Apart from the experimental measurement, the approximate analytical methods
are also used to study the performance of the pressure swirl atomizers. Giffen and
Muraszew (1953) carried out an inviscid analysis of the flow in a simplex atomizer.
They showed that the atomizer constant, which is the ratio of inlet area to the product of
17
swirl chamber diameter and exit orifice diameter, is the most important geometric
parameter of the atomizer. They derived expressions for liquid film thickness, spray cone
angle, and the discharge coefficient in terms of the atomizer constant. However, in a
practical atomizer, the flow is viscous, and other geometric parameters also influence the
atomizer performance.
Som and Mukherjee (1980) conducted a theoretical investigation on the discharge
coefficient and spray cone angle of a swirl spray nozzle. These two parameters were
theoretically evaluated through the analytical solution of hydrodynamics of flow inside
the nozzle. They found among the nozzle geometries, an increase in the orifice to swirl
chamber diameter ratio, swirl chamber angle or decrease in swirl chamber length to
diameter ratio decreases the discharge coefficient and increases the spray cone angle.
Yule and Chinn (1997) conducted a numerical study by treating the entire
computational domain as single phase, and then guessing the interface by joining grid
points where pressure is found to be atmospheric. Solution was re-calculated by creating
a new grid using the calculated interface and treating the interface as a “with-slip”
boundary. However, the condition of normal stress balance was not applied at the
interface. The velocity and pressure distribution in atomizers were calculated and
discharge coefficient and spray angle were predicted.
Sakman et al (1998) studied the effects of simplex nozzle geometry on its
performance numerically. They concluded that when designing a nozzle, particular
attention should be paid to optimum values of the performance variables like the local
maximum of spray cone half angle at Ds / Do = 4.1 and extreme caution should be taken
when designing a simplex nozzle. The performance variables are dependent on the actual
18
physical dimensions, since these may determine whether the flow is laminar or turbulent,
and may change the trends of the performance parameters.
Steinthorsson and Lee (2000) conducted three-dimensional simulations of
internal, free-surface flow in a pressure-swirl atomizer via commercial software
FLUENT. The atomizer used in the simulations is a large-scale atomizer. The Volume
of Fluid (VOF) method was adopted to capture the formation of the air-core and
Reynolds Stress Turbulence model was used to model the effects of turbulence. The
results were compared to experimental data given by Wang et al (1999, 2000). They
concluded that the effect of the discrete inlet slots disappears before the liquid enters the
orifice.
Hansen (2001) simulated the flow in a scaled model of a Danfoss pressure-swirl
atomizer via commercially available CFX-4.3 code. The Volume of Fluid (VOF) method
was used to track the liquid-gas interface. The simulations were performed in a three-
dimensional curvilinear grid representing the swirl chamber of the atomizer and managed
to capture the overall flow characteristics of a pressure-swirl atomizer with the
formations of an air-core. A simulation using the k-ε turbulence model over-predicted
viscosities and failed to predict a stable air-core in the atomizer. Results from LES and
simulations assuming laminar flow were verified against experimental findings from
LDA and pressure measurements.
1.3 Scope of the Dissertation
In Chapter 2, the details of the numerical method are given. The flow inside a
simplex atomizer is turbulent, unsteady and contains re-circulating regions. Additionally,
19
the shape and the location of the liquid-air interface is not known and must be determined
as part of the solution. This is the primary difficulty in computational modeling of the
flow in a simplex atomizer. To tackle this problem, the Arbitrary-Eulerian-Lagrangian
method is used to track the liquid/gas interface. In this method, the grid points on the
interface always remain on the interface and move with the interface. The advantage of
this approach is that the gas-liquid interface stays sharp at all times. This is especially
important for the flow in a simplex nozzle to accurately determine the film thickness to
evaluate the mean droplet size. In contrast, commercially available general-purpose flow
codes do not have the capability to track a sharp interface.
In Chapter 3, the code validation is presented. Validation of the computational
code is essential before the code can be used to study the flow phenomena inside an
atomizer. As the flow field inside a small scale atomizer is extremely difficult to
measure, a large-scale prototype nozzle was used to measure the detailed flow field for a
number of geometric configurations by Jeng et al. (1998) and Wang (1999). A detailed
comparison with the experimental data was carried out. The comparison showed that for
the cases with four inlet slots the code is able to predict performance parameters very
well (generally within 5% of the experimental measurements). We note that the
differences are less than the uncertainty in the experimental measurements. Grid
independence verification of the results was carried out as well.
In Chapter 4, effects of changes in three non-dimensional geometric parameters
on the atomizer performance are studied. These geometric parameters are - the length to
diameter ratio of the swirl chamber and that of the orifice, and the swirl chamber to
orifice diameter ratio. The variation in the atomizer performance is obtained by keeping
20
the pressure drop across the atomizer or inlet mass flow rate constant and is presented in
terms of the dimensionless liquid film thickness at the exit of the orifice, spray cone half
angle and discharge coefficient.
In Chapter 5, effects on the atomizer performance of varying a number of
geometric parameters that are not considered in available correlations are studied. The
geometric parameters are: inlet angle, smoothing radius between exit orifice and trumpet,
trumpet angle, trumpet length, orifice length, length of the spin chamber, and diameter of
the spin chamber.
In Chapter 6, the numerical code for Newtonian flow described in Chapter 2 is
extended to time-independent purely viscous power-law non-Newtonian flow. A
parametric study is conducted under both constant mass flow rate at the inlet and constant
pressure drop across the atomizer. Detailed flow pattern inside the simplex nozzle is
investigated.
In Chapter 7, the conclusions from this study are listed, the limitations of the
present code are addressed, and suggestions for future work are outlined.
Appendix provides a derivation based on inviscid flow assumption for film
thickness, spray cone angle, and discharge coefficient for nozzles that do not have purely
tangential fuel entry and for nozzles with a trumpet.
21
CHAPTER 2 COMPUTATIONAL MODEL
2.1 Review of Computational Methods
Due to the high swirl velocity in pressure swirl atomizers, an air core exists at the
centerline of the atomizer. The position and geometry of the gas/liquid interface is not
known a priori, and must be determined as a part of the solution. Therefore, the main
difficulty in a numerical scheme that models the interaction of two fluid phases is to
accurately track the interface. Modeling of motion of a liquid gas interface is relevant to
many engineering applications and a number of methods have been used for this purpose.
The easiest approach is to treat the entire computational domain as single phase, and then
guess the interface by joining grid points where pressure is found to be atmospheric. Yule
and Chinn (1997) and Datta and Som (2000) have used such approach using
commercially available software. This is an approximate method as the interface
conditions of stress balance are not applied at the interface. As the liquid gas interface
behaves vary differently from a single-phase region, such method is not likely to provide
accurate results. The volume-of-fluid (VOF) method is widely used in flow simulations
with multiple phases (Steinthorsson and Lee, 2000). In this method the interface is
represented by the fraction of cell volume occupies by the liquid. As the grid points do
not exactly lay on the interface, the interface has to be reconstructed using volume
fractions for each cell. In general, the accuracy strongly depends on the fineness of the
grid around the interface and the method can be computationally very expensive. The
shape of the liquid gas interface greatly affects its stability and breakup. The pressure,
surface tension, and aerodynamic forces determine the breakup behavior of a liquid gas
22
interface. The curvature of the interface can change the pressure variation along the
interface, surface tension forces, and the aerodynamics forces. As such, for the
application being considered here, accurately tracking a sharp interface is very important.
The VOF method does not provide tracking of the sharp interface and as such may not be
the best method for this application. The gradient method does not define the exact
location of the interface but represents the interface as a continuous gradient over several
cells. Although this method is computationally efficient, the accurate determination of
the location is not possible. The most promising approach for such problems, is a fully
Lagrangian scheme (Hirt et al, 1970). In this scheme each node point moves with its
velocity so that the same cell mass is followed in time. Therefore, in a Lagrangian
approach, node points on a two-phase interface always remain on the interface and the
motion and shape of the interface can be evaluated very accurately. Such method works
very well only for well-behaved flows. For domains that include shear, fluid separation,
and recirculation, such as the problem at hand, a Lagrangian scheme would lead to a
highly distorted grid very quickly. When differential operators are calculated on a highly
distorted mesh, the approximations generally lose accuracy. Furthermore, obtaining
converged solutions for a complex flow on a highly distorted grid would be virtually
impossible. To achieve the Lagrangian accurate tracking of the interface and yet
maintain accuracy and obtain convergent solutions, The Arbitrary-Lagrangian-
Eulerian(ALE) method (Hirt et al, 1974) was adopted in this study. The ALE method
comprises of two computational steps that combine the Lagrangian tracking with
Eulerian re-gridding. In the first step, the computational grid vertices move with the
same velocity as the fluid. Therefore there is no mass exchange among computational
23
cells. The continuity and momentum conservation equations are solved using a predictor-
corrector numerical scheme. This is the Lagrangian computational step. In the second
step, a new adaptive grid is generated. The mass and momentum for each new cell is
calculated based on the motion of new grid vertices from their Lagrangian positions.
This is the Eulerian step of the calculation. The Lagrangian step ensures that the points
on the interface remain on the interface and thus the interface is tracked accurately. The
Eulerian step ensures that the grid does not get distorted thereby maintaining solution
accuracy. Among the available methods for interface tracking, the ALE method provides
the most accurate tracking of the interface without numerical smearing.
2.2 Mathematical Formulation
The conservation equations governing the flow are,
Continuity Equation 0. =∇ ur (2.1 a)
Momentum Conservation fPDt
uD rr
ρτρ +∇+−∇= . (2.1 b)
Where ur is velocity, τ is viscous stress term, fr
is a body force, and ρ is
density. Integrating Eqns. 2.1 over a control volume V(t) with a surface S(t) that moves
with an arbitrary velocity, the equations become,
∫∫∫∫∫∫∫∫∫∫
∫∫
=−∇+−−
=−
)()()()(
)(
0.).(
0).(
tAtVtAtV
tA
SdPdVSduUudVudtd
SduU
rrrrrr
rrr
τρρ (2.2)
The Equations 2.2 provides the form in which the equations are used to develop
the code.
24
2.3 Stress Calculation
For an axisymmetric flow ( 0=∂∂θ
) with a swirling velocity, the six components
of sheer stress tensor can be simplified as:
rv
err ∂∂
= µτ 2 (2.3)
rv
eµτθθ 2= (2.4)
zu
ezz ∂∂
= µτ 2 (2.5)
)(rw
rw
er −∂∂
= µτ θ (2.6)
zw
ez ∂∂
= µτθ (2.7)
)(ru
zv
ezr ∂∂
+∂∂
= µτ (2.8)
Here u, v, and w denote the axial, radial and tangential (swirl) velocity
components respectively. µ e is the effective viscosity. It is taken to be the sum of laminar
and turbulent components:
tle µµµ += (2.9)
where the subscripts l and t represent laminar and turbulent, respectively. In this
study, the Baldwin-Lomax two layer turbulence model (Baldwin and Lomax, 1978) is
used to predict value of µe.
The Baldwin-Lomax model is based on the renowned Cebeci-Smith model, which
is a two-layer algebraic model for the turbulent viscosity:
25
⎩⎨⎧
>≤
=co
cit yyfor
yyfor............
µµ
µ (2.10)
where µi is turbulent viscosity within inner layer
µo is turbulent viscosity within outer layer,
and yc is the distance from a wall to the first point at which µi>µo.
A Prandtl-van Driest formulation is used for the inner layer:
ρτν
κωρµ
w
i
u
uyy
AyD
yDll
=
=
⎭⎬⎫
⎩⎨⎧−
−=
=
=
+
+
+
*
*
2
exp1 (2.11)
where ω is the vorticity, ρ is the density, κ is the von Karman constant, and D is the van
Driest damping function with A+=26.
The outer layer formulation is:
⎟⎟⎠
⎞⎜⎜⎝
⎛=
max
2
maxmaxmax ,min.fu
yCfyCK diffwkcpo ργµ (2.12)
where K=0.0168 is the Clauser constant and Ccp=1.6 is an additional constant. In wakes,
D is set to 1 and the lower portion of this equation is used. Cwk was originally reported as
0.25, but is generally taken to be 1.0. udiff stands for the difference between the highest
and the lowest velocities in the profile. The Klebanoff intermittency function γ is given
by:
26
12
max
5.51
−
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+=
yyCKlebγ (2.13)
where CKleb=0.3.
The function f(y) is defined by:
Dyyf ω=)( (2.14)
Noted that fmax = max(f(y)) and ymax = y at fmax
2.4 Discretization
The popular discretizing methods in space consist of Finite Difference Method
(FDM), Finite Volume Method (FVM) and Finite Element Method (FEM). Unlike FDM,
FVM is based on integral formulation. The integral form of conservation laws are
satisfied for each control volume and for the entire domain. Therefore, the finite volume
method is more preferable to the finite difference approach. In code developing, FVM is
much easier to implement compared to FEM. Hence, FVM was chosen here for space
discretization. For time, the explicit time-marching method was selected for simplicity.
The mesh used here consists of a network of quadrilateral cells with vertices
labeled by integral pairs (i, j). Fluid variables are assigned to staggered locations in the
mesh (shown in Figure 2-1). Pressures (p), cell volumes (V), and Masses (M) are all
assigned to cell centers whereas coordinates (z, r) and velocity components (u, v, w) are
all assigned to cell vertices.
Due to the staggered assignment of the fluid variables, the control volume for
mass conservation is different from that for momentum conservation. In particular, the
control volume and control surface for mass continuity are the volume and corresponding
27
Figure 2-1 A schematic showing assignment of the variables for a cell
Figure 2.2 Schematic of mass and momentum control volume
Momentum C. V.
Mass C. V.
J+1
J
J-1
J-2
I-3 I-2 I-1 I I+1
i+1, j
i, j+1
i+1, j+1
p, V, M
i, j
z, r, u, v, w
28
surface of a cell in the mesh, and control volume and control surface for momentum
conservation are the vertex-centered volume and corresponding surface surrounding a
vertex point (shown in figure 2.2). In contrast to a mass cell which has four faces, the
momentum cell has eight faces, each comparable in size to one-half of a regular cell face.
2.5 Pressure Field Iteration
As the time moves from t (time step N) to t +∆t (time step N+1), both velocity
field and pressure field have to be updated. Because velocity and pressure are coupled,
they can not be computed explicitly. Here a predictor-corrector method is applied to
solve this problem. Specifically, the Conjugate-Gradient method (O’Rourke et. al., 1986)
together with Jacobian matrix preconditioning is used to solve the velocity and pressure
implicitly. A brief discussion of this technique is presented here.
The Conjugate-Gradient method is used to solve the problem
0)( =PRrr
(2.15)
Where Rr
is the vector of residuals in the solution of a set of implicit linear finite-
volume equations and Pr
is the vector of unknowns - pressure for our case - we wish to
solve for. Number of components for both Rr
and Pr
are equal to the number of
computational cells. Assuming that vPr
predicts Pr
with volume residual )( vPRrr
, based on
a Taylor expansion neglecting higher order terms;
( ) 0)()( =−⋅⎟⎟⎠
⎞⎜⎜⎝
⎛+= v
Pv PP
PRPRPR
v
rrr
rrrrr
r∂∂ (2.16)
Define a Jacobian matrix A as,
29
vPPRA
rr
r
⎟⎟⎠
⎞⎜⎜⎝
⎛−=
∂∂ (2.17)
The vector vPr
∆ can be defined as:
vv PPPrrr
−=∆ (2.18)
Then,
)( vv PRPArrr
=∆ (2.19)
The problem can now be rephrased as finding the solution of vPr
∆ in Equation
2.19. The convergence criterion is the satisfaction of the Equation 2.15. Note that A has
to be non-singular, and also of rank n where n is the number of components of vPr
in
order to be able to apply the Conjugate-Gradient method. The method is known to be
faster for symmetric A. Assume that A is non-singular at this point.
Define a preconditioning matrix M such that
( ) vv PPRMrrr
∆=)( (2.20)
It is best to take M as close as possible to 1−A , the inverse of A. The iteration
procedure can be explained in four steps:
1) Calculate the current volume residual vRr
on guessed pressure field vPr
at
iteration v ;
2) Calculate
( ))(ˆ vvv PRMPqrrrr
=∆= (2.21)
Which is a guess for the solution to pressure adjustment;
30
3) Orthogonalize vqrˆ with respect to 1−vqr , the previous pressure correction
direction vector, the new pressure correction direction vector is obtained,
1ˆ −−= vvvv qqq rrr λ (2.22)
where
( ) ( )
( ) ( )∑∑
−−
−
=
iiviv
iiviv
v qMAqA
qMAqA
11
1ˆrr
rr
λ (2.23)
4) Find the component 1+∆ vPr
in the direction of vqr ,
vvv qP rrα=∆ +1 (2.24)
Where
( ) ( )
( ) ( )∑∑ −
=
iiviv
iivivB
v qMAqA
qMAPPArr
rrr()(
α (2.25)
Then,
11 ++ ∆+= vvv PPPrrr
(2.26)
Next, return to the first step where 1+= vB PPrr
is the predicted pressure at (N+1)
time step.
To initiate this iteration algorithm for v = 0 at time step (N+1), the pressure fields
of time steps (N) and (N-1) is used to extrapolate linearly. The Conjugate Gradient
method is proved to effectively converge to the solution after L iterations, where L is the
number of computational cells.
31
The key to the implementation of this method into the code is to find the non-
singular matrix A and its pre-conditioning Jacobian matrix M. For given cell (i, j), a nine-
point formula relating residuals and pressures can be expressed as,
),,,,,,,,( 1,11,1,1,11,11,1,1,1,, ++++−−−−−−++= jijijijijijijijijiji PPPPPPPPPfRES (2.27)
It is difficult to include all nine points of the pressure influence into matrix A.
Since the pre-conditioning matrix M needs to be as close as possible to A-1, and also
considering the requirement of computational efficiency, only the main diagonal
elements of A have been used to define an approximation to matrix A, and its pre-
conditioning Jacobian matrix M is given as,
⎟⎟⎠
⎞⎜⎜⎝
⎛= lm
lmaM δ1 (2.28)
Where lma is the respective element of matrix A, and lmδ is the delta function:
⎩⎨⎧
≠=
mlml
lm if 0= if 1
δ (2.29)
This simplification greatly reduces the computational complexity, although it is at
the expense of an increase in the number of iterations.
From Equation 2.15, matrix A can be determined to be:
ji
jiji P
RA
,
,, ∂
∂= (2.30)
Coefficients vα and vλ are evaluated by applying A to the vectors vqArˆ , vqAr , and
)( vB PPArr
− ,
32
1111
1
)()()()(
ˆ
)ˆ()(ˆ
−−−−
−
−=∆−−=−=
−=
+−=
vvvvvBvBv
vvvv
vvvv
qAPRPPPAPPAPR
qAqAqA
qPRPRqA
rrrrrrrrrr
rrr
rrrrrr
α
λ (2.31)
The convergence criterion for each cell (i,j) is based on finding an acceptably
small value of )(PRrr
which obeys
Ajiv VPR ,)( ε≤
rr (2.32)
Where ε is a small number in the order of 710− , and AjiV , is some initial cell
volume.
2.6 Boundary Conditions
The governing equations are elliptic in space and require boundary conditions on
all boundaries of the computational domain. There are four kinds of boundary conditions
(BC) encountered in this study: inflow BC, static wall BC, outflow BC and free film BC.
Inflow Boundary Condition
The specified values of the velocity components in axial, radial and tangential
directions are applied here as the inflow boundary condition. The values of the velocity
components are calculated according to the flow rate and the inlet slots geometry. As the
present model is axisymmetric, boundary conditions require an "equivalent"
axisymmetric inlet. The width of this inlet and the velocity components are calculated by
equating the mass flow rate, angular momentum, axial momentum and kinetic energy to
the actual inlet conditions. The equations to calculate the width of the equivalent
axisymmetric inlet and the velocity components vary with nozzle geometry.
33
Static Wall Boundary Conditions:
The rigid no-slip boundary condition is used for the static wall. All three
components of velocity are set to zero on the rigid boundaries.
Outflow Boundary Conditions:
Typically outflow boundary condition of zero velocity gradient along a stream
line is used in many simulations. In the problem at hand, as the liquid exits from the
nozzle in form of a hollow cone, the cross-sectional area for the flow increases as the
liquid spreads out. As such the typical outflow boundary condition introduces slight error
in mass conservation. As a reason, in the present model, a slightly different boundary
conditions is used such that the mass flow, the angular momentum and kinetic energy are
conserved between the grid points on the outflow boundary and their interior neighbors.
According to these three rules, the velocity components at the outflow BC are calculated.
Several cases were run using the typical outflow boundary condition and the new
condition that conserves mass and angular momentum. It was seen that the differences in
the results obtained using the two outflow boundary conditions are slightly different.
However, application of the new condition made the code converge faster and made it
somewhat more stable under extreme conditions. As such the new outflow condition is
better and therefore it is used the present model. The equations for this boundary
condition are:
v(nxp,j) = sqrt[(1-rw**2)*w(nx,j)**2/(1+ru(j)**2)+v(nx,j)**2]
u(nxp,j)=ru(j)*v(nxp,j)
34
w(nxp,j)=rw*w(nx,j)
where,
rw = r1/r2
r1 = (y(nx,1)+y(nx,nyp))/2.
r2 = (y(nxp,1)+y(nxp,nyp))/2
ru(j) = u(nx,j)/v(nx,j)
The node points nxp and nyp indicate the last points in the computational domain
in the z and r directions, respectively.
Free film BC
At a two-phase interface, the normal stress and the shear stress must be
continuous across the interface. As this code deals with an interface between liquid and
air, and as the liquid viscosity is significantly greater than that of air, the shear stress at
the interface is negligible. Such interface is referred to as a "free surface" and the
condition of zero shear stress is imposed at the interface. The surface tension and
pressure is balanced at the interface. It is noted that in certain geometry, the distortion of
the free film inside the atomizer may be large during first few iterations and may lead to
the code to diverge. To overcome this problem, the motion of points on about half of the
free film inside the atomizer is initially constrained in a way that all the points have the
same vertical displacement. Once the initial transients diminish, this constraint is
removed. All of this is done automatically inside the code based on our experience with
nozzle geometries and the user does not need to control this. Additionally, it should be
emphasized that, for most cases, part of inner free film in the swirl chamber has been
35
constrained in order to avoid radial location of any point on the interface becoming zero.
The radial location (r) of any point on the interface going to zero implies that the air core
has collapsed. Should this happen, the boundary condition for balance of surface tension
and pressure is expressed in terms of curvature surface curvature (~1/r) and will become
singular. As such the code would diverge.
2.7 Grid Generator
A pre-processor is developed to generate the initial grid automatically. The entire
calculation domain is divided into 5 blocks. They are: swirl chamber block, convergent
section block, orifice block, trumpet block and free film block. If the geometry has no
trumpet, then there are only 4 blocks. Next, for each block, the boundary is divided into a
set of discrete points. Experience of running the code for a variety of flow condition and
geometries has been used to provide a "reasonable" grid. Finally, the whole domain is
divided based on transfinite Hermite interpolation (Stoer and Bulirsch, 1993). This pre-
processor generates a two-dimensional algebraic grid automatically.
36
2.8 Code Structure
Figure 2.3 Flowchart of the code
Pressure Field Solving Iteration
Time Step ∆t Estimation
Velocity Updating to Take Viscous Stress into Account
Velocity Updating due to the Convective Transport
Velocity Updating with the Pressure Acceleration
Mesh Regridding
Start
Initialization
Converged?
End
Yes
No
37
CHAPTER 3 CODE VALIDATION
3.1 Verification of Grid Independence of Results
For any computational flow simulation, the validation of grid independence is
always necessary (Wilcox, 1993). The code was run under several flow conditions
with different grid density to assure grid independence of results. Two sets of results
with the same flow condition and geometry are shown here with 80X21 and 159X41
grid, respectively. Figure 3.1 shows the final grids. The results shown in Table 3.1
indicate that the difference between coarse grid and fine grid is very small. As such
an 80X21 or similar grid should be sufficient to get accurate results.
Table 3.1 Results of different grid density.
Grid 1
(80X21) Grid 2 (159X41)
t (mm) 0.5378 0.5322 uav(m/s) 24.20 24.51 wav(m/s) 19.15 19.32 Cd 0.4350 0.4505 θ(degree) 38.35 38.25
3.2 Comparison with Experimental Data and Empirical Correlations
Measurement of flow field inside a small scale nozzle (exit orifice diameter ~
1mm) is extremely difficult. Such measurements are essential to validate the
computational model. Limited data for spray angle and film thickness are available in the
literature for small-scale nozzles. Unfortunately the details of the nozzle geometry are
38
Figure 3.1 Final grids of different grid density
(2D
)⏐
29A
ug20
00⏐
cycl
e40
000
0
00
.001
0.00
20.
003
0.00
40.
005
0.0
060
.007
0.0
08X
0
0.0
005
0.00
1
0.0
015
0.00
2
0.0
025
0.00
3
0.0
035
Y
Grid
:80
X21
Rea
l-Siz
eno
zzle
(2D
)⏐
29A
ug20
00⏐
cycl
e5
0000
0
00.
001
0.00
20.
003
0.0
040
.005
0.0
06
0.00
70.
008
X0
0.00
05
0.0
01
0.00
15
0.0
02
0.00
25
0.0
03
0.00
35
Y
Grid
:159
X41
(2D
)⏐
29A
ug20
00⏐
cycl
e5
0000
0
39
generally not provided and therefore the available data can not be used for code
validation. To overcome this difficulty, Jeng et al. (1998), Wang (1999) and Ma (2001)
carried out flow measurements on large-scale prototype nozzle. A brief description of
their setup and measurement technique is described below for sake of completeness. I
have used their data to validate the developed computational code. The large-scale
prototype nozzles provided sufficient spatial resolution for flow measurements. The
body of the prototype nozzle was made of optical quality plexiglass to use optical
diagnostic techniques (Jeng et al, 1998). The Particle-Image-Velocimetry (PIV)
technique, CCD camera, and still photography were used for measurements. In general,
the uncertainty in the measurements is about 2% for discharge coefficient, about 2
degrees in spray angle, and less than 15% in film thickness measurements. Three cases
were simulated for which experimental measurements were available. The details of
three cases are given in Table 3.2.
Table 3.2 Cases summary
Case # Ds
(mm)
Ls
(mm)
d0
(mm)
Inlet
slots
l0
(mm)
Ap
(mm2)
θ Flow rate
(gal./min)
1 76 89 21 4 5 406 90 15
2 76 89 21 4 5 406 45 15
3 76 89 21 4 5 406 45 20
40
Comparison of the Swirl Velocity
Particle-Image-Velocimetry (PIV) technique was used to measure swirl velocity
variation at different axial locations in the large-scale prototype nozzle. For Case # 2
listed in Table 3.2, comparison of experimental data and the CFD predictions of the swirl
velocity profiles at three different axial locations are shown in Figure 3.2. It is seen that
the swirl velocity is the largest close to the free surface and decreases away from the
gas/liquid interface. The variation predicted by the CFD code is seen to match with the
experimental data extremely well. The slight difference near the gas/liquid interface can
be attributed to the viscous effects of the air which are neglected in the CFD model. For
any CFD code, excellent match with local parameters, such as velocity variation at
different points in the flow field, is difficult to achieve compared to good agreement with
global parameters (such as Cd). As such the excellent agreement between the swirl
velocity variation from experiments and CFD, provides a strong validation of the CFD
model.
Figure 3.2 Comparison of swirl velocity variations at three axial locations
41
Comparison of predicted film thickness, spray angle, and discharge coefficient
with experiments and empirical correlations
To further validate the code and to assess its accuracy relative to the available
empirical correlations, comparisons for spray angle, film thickness, and discharge
coefficient were carried out. Several correlations are available in the literature (Lefebvre,
1989).
The correlations listed below are probably the most widely used and as such these
were considered for comparison with CFD predictions. The correlations are listed in two
groups. The first group consists of correlations that are given by Lefebvre and co-workers
(Lefebvre, 1989) and the second group by other researchers (Jones, 1982, Suyari, 1986).
25.0
66.3 ⎟⎟⎠
⎞⎜⎜⎝
⎛∆
=LL
LLo
Pmd
tρ
µ& (3.1)
11.0
2
215.062 ⎟⎟
⎠
⎞⎜⎜⎝
⎛ ∆= −
L
LoLm
dPK
µρ
θ (3.2)
25.05.0
35.0 ⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛=
o
s
os
pd d
DdD
AC (3.3)
Other correlations:
XX
dD
dDA
o
s
os
p
+−
=⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛1
)1(09.035.0
(3.4)
XX
+−
=11cos2 θ (3.5)
23.052.005.003.002.0
45.0 ⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛=
−−
o
s
os
p
s
s
o
o
L
thLod d
DdD
ADL
dlUd
Cµρ
(3.6)
42
Table 3.3 Comparison results of experimental, CFD and correlation data
CFD Lefebvre Other Experimental
Value Error Value Error Value Error
Cases with 4 inlet slots (flow nearly axi-symmetric)
∆p (Pa) 48263 52022 +8%
t (mm) 2.633 2.932 2.434
θ 42.90 41.26 -4% 50.39 +17% 61.22 +43%
CASE 1
Cd 0.2781 0.2678 -4% 0.2435 -12% 0.2446 -12%
∆p (Pa) 46883 58919 +25%
t (mm) 2.68 2.544 -5.1% 2.953 +10% 2.434 -9%
θ 39.35 42.26 +7.4% 50.23 +28% 61.22 +55%
CASE 2
Cd 0.2821 0.2517 -10.8% 0.2435 -14% 0.2446 -13%
∆p (Pa) 89631 105808 +18%
t (mm) 2.64 2.542 +3.7% 2.699 +2% 2.434 -8%
θ 40.35 42.40 +5.1% 53.95 +34% 61.22 +52%
CASE 3
Cd 0.2721 0.2504 -8% 0.2435 -10% 0.2430 -11%
Table 3.3 shows that for all the cases considered here, the CFD code gives a
significantly better prediction for the spray cone angle compared with empirical
correlations. In general, the difference in experimental measurements is about 5%.
Considering that the experimental uncertainty in the measurements is greater than 5%,
this is indeed excellent agreement. In contrast, the correlations deviate from the
experimental values from 20% to 50%. For the free film thickness at the exit and
discharge coefficient, in the cases (case 1, 2 and 3) that have 4 inlet slots, the results from
CFD code are better than those from correlations with deviation from experimental
43
values less than 5%. Once again, this is within the experimental uncertainty and shows
excellent agreement.
3.3 Comparison with Correlations at Small Scale
The results obtained from the CFD code for small scale atomizers were compared
with correlations available in the literature. As mentioned earlier, details of the nozzle
geometry are not available where experimental data for spray angle, film thickness and
discharge coefficient are given in the literature. As such a comparison can only be made
using empirical correlations for small-scale nozzles. Predictions for discharge
coefficient, film thickness, and spray angle were obtained with varying the atomizer
constant. Some of the correlations are known to work well to evaluate the effect of
variation of atomizer constant, and the good agreement with CFD results provides
additional confidence in the computational code.
Discharge coefficient
The discharge coefficient is governed partly by the pressure losses incurred in the
nozzle flow passes and also by the extent to which the liquid flowing through the final
discharge orifice makes full use of the available flow area (Lefebvre, 1989). The
discharge coefficient is the ratio of the actual to the maximum theoretical flow rate that is
determined from the measured pressure drop across the atomizer.
LoL
Ld PA
MCρρ /2∆
=&
(3.7)
44
Atomizer Constant, K
Dis
char
gerC
oeffi
cien
t,C
d
0 0.2 0.4 0.60.1
0.2
0.3
0.4
0.5
CFDBenjaminJonesRizk
Figure 3.3 Variation of discharge coefficient
45
Rizk et al. (1984) derived the following expression:
25.05.0
35.0 ⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛=
o
s
os
pd d
DdD
AC (3.3)
From analysis of experimental data on the discharge coefficient of large-scale
pressure swirl atomizers, Jones (1982) obtained the following correlation:
23.052.005.003.002.0
45.0 ⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛=
−−
o
s
os
p
s
s
o
o
L
thLod d
DdD
ADL
dlUd
Cµρ
(3.6)
Benjamin et al (1998) used the similar equation to Jones’s to correlate his
experimental data and resulted in the following expression:
187.0517.0091.0229.0027.0
466.0 ⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛=
−
o
s
os
p
s
s
o
o
L
thLod d
DdD
ADL
dlUd
Cµρ
(3.8)
As seen in Figure 3.3, there is fair amount of agreement between the different
correlations; and CFD results match with the predicted values quite well.
Film thickness
At small scale, the measurement of film thickness is fairly difficult to accomplish.
A number of techniques have been reported in the literature. Our large-scale nozzle flow
simulations as well as experiments have shown that the film thickness in the exit orifice
is not constant. It varies with axial location and the variation is not monotonic. Film
thickness measurements tend to capture an average value of thickness over a small but
finite axial distance, whereas the film thickness obtained from the CFD code is at the exit
of the orifice. The correlations for film thickness are of two types, first that directly
provide thickness based on fuel properties, pressure drop, and orifice exit diameter.
46
Atomizer Constant, K
Dim
ensi
onle
ssTh
ickn
ess
atE
xit,
t*
0 0.2 0.4 0.60.1
0.2
0.3
0.4
0.5
CFDRizkGiffenBenjamin
Figure 3.4 Variation of dimensionless film thickness
47
Rizk et al. (1984) gave the following correlation,
25.0
66.3 ⎥⎦
⎤⎢⎣
⎡∆
=P
mdt
L
LLo
ρµ&
(3.1)
Benjamin et al. (1998) derived their correlation as,
25.0
5.0)(78.3 ⎥
⎦
⎤⎢⎣
⎡∆
=P
FNdt
L
Lo
ρµ
(3.9)
Another type of correlation was given by Suyari and Lefebvre (1986) where the
film thickness is obtained by equating discharge coefficient to the discharge coefficient
obtained by Giffen and Muraszev (1953) by inviscid analysis.
XX
dD
dDA
o
S
oS
p
+−
=⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡1
)1(09.035.0
(3.4)
Suyari and Lefebvre (1986) have compared small scale film thickness
measurements to a number of correlations and found that there is large disagreement
between different correlations. They found that Equation 3.4 provides the best fit for
available data. Figure 3.4 indicates that the CFD predictions match very well predictions
by Equation 3.4. Also, the CFD results are significantly better than other correlations.
This provides further validation of the CFD code.
Spray cone half angle
There is no unanimity for the spray cone angle predictions by using different
correlations. The following three correlations are compared with CFD results.
Rizk’s correlation 1 (Rizk and Lefebvre, 1984)
XX
+−
=11cos2 θ (3.5)
48
Atomizer Constant, K
Spr
ayC
one
Hal
fAng
le,θ
0 0.2 0.4 0.6
20
30
40
50
60
70
CFDATAN(wav/uav)Rizk2BenjaminRizk1
Figure 3.5 Variation of Spray Angle
49
Rizk’s correlations 2 (Rizk and Lefebvre, 1984)
11.0
2
215.062
⎥⎥⎦
⎤
⎢⎢⎣
⎡∆= −
L
LoPdK
µρ
θ (3.2)
Benjamin gave the following correlation (Benjamin et al, 1998),
0647.0
2
2237.075.92
⎥⎥⎦
⎤
⎢⎢⎣
⎡∆= −
L
LoPdK
µρ
θ (3.10)
Of these correlations, Benjamin's correlations was developed using data for large
scale prototype nozzle and as such the difference between that and the CFD results is not
surprising. It is interesting that the other two correlations have significant disagreement.
Figure 3.5 shows the CFD results match well with one of Rizk's correlations.
3.4 Summary
The grid independence study was conducted and the result indicates an 81X21 or
similar grid should be sufficient to get accurate simulation for typical simplex nozzle.
A detailed comparison with the experimental data was carried out. The
comparison showed that for the cases with four inlet slots the code is able to predict
performance parameters very well (generally within 5% of the experimental
measurements). We note that the differences are less than the uncertainty in the
experimental measurements.
The results obtained from the CFD code for small scale atomizers were compared
with correlations available in the literature. Predictions for discharge coefficient, film
thickness, and spray angle were obtained with varying the atomizer constant. The good
agreement with CFD results provides additional confidence in the computational code.
50
CHAPTER 4 EFFECTS OF GEOMETRIC PARAMETERS ON
ATOMIZER PERFORMANCE (PART I)
4.1 Introduction
Pressure-swirl atomizers used in gas turbine engines and other spray applications
have to operate over a wide range of flow conditions. It is very important to be able to
predict spray performance based on the relevant input parameters into the nozzle. Since
predictive breakup models are not available with the level of accuracy required for the
design process, design engineers still have to struggle with the cut-and-try methodology
in order to fine-tune the nozzle into the required spray properties. Presently, semi-
empirical correlations are used to provide guidance in designing simplex nozzles
(Benjamin et al 1998). However, the semi-empirical correlations do not always predict
the effects of changes in the length-scales on spray properties. Furthermore, not all the
geometric parameters are included in the currently available correlations. The validated
CFD code is used in this study to perform such parametric study to understand the effect
of atomizer geometry on atomizer performance.
For pressure-swirl atomizers, it is now generally accepted that the dimension of
most importance for atomization is the thickness of the liquid sheet as it leaves the final
orifice (Lefebvre 1983). Theory predicts, and experiment confirms, that mean drop size
is roughly proportional to the square root of liquid sheet thickness at the exit (Lefebvre
1989). Discharge coefficient is another important parameter for simplex nozzle, because
51
this coefficient not only affects the flow rate of any given nozzle but also can be used to
calculate its velocity coefficient and spray cone angle (Lefebvre 1989). The spray angles
produced by pressure-swirl nozzles are of special importance in their application. One
important reason is that normally any increase in spray cone angle will lead to improved
atomization (Lefebvre 1989). Therefore, the dimensionless film thickness at the exit,
discharge coefficient and spray cone half angle are used here to describe the performance
of pressure-swirl atomizers.
In this chapter, the effects of the length to diameter ratio of the swirl chamber
(Ls/Ds) and of orifice (lo/do), the swirl chamber to orifice diameter ratio (Ds/do) on the
performance of a simplex atomizer are numerically investigated. The definitions of these
geometric parameters are described in Figure 1.1. These three parameters are generally
not included in correlations for film thickness and spray angle. Two flow situations are
important – constant mass flow rate through the nozzle and constant pressure drop across
the nozzle. Here results are presented for both working conditions. A range of Ls/Ds from
0.1 to 1.5, a range of lo/do from 0.2 to 2.0, a range of Ds/do from 2 to 6 have been covered.
The base geometric parameters are Ls/Ds = 0.25, lo/do = 0.5, Ds/do = 4.0, and K = 0.47.
While studying the effects of changes in Ls/Ds, only Ls is varied and all other dimensions
are kept as base geometry. Similarly, lo is varied to change lo/do. While studying the
effects of changes in Ds/do, in order to hold K as constant, both Ds and do are varied such
that Ds*do is kept constant. The atomizer performance is monitored in terms of
dimensionless film thickness at the exit, spray cone half angle and discharge coefficient.
52
4.2 Results for Constant Mass Flow Rate through the Atomizer
Effect of variation in Ls/Ds
For given inlet mass flow rate, eight different cases, which Ls/Ds is 0.1, 0.2, 0.25,
0.5, 0.75, 1 and 1.5 respectively, are investigated numerically here. Noted that the
change of Ls/Ds results from the variation of Ls whereas Ds is held constant.
Figure 4.1 gives the variations of the dimensionless film thickness at the exit and
discharge coefficient with changing Ls/Ds. As Ls/Ds increases, both t* and Cd increase
monotonously. Because Ds is held constant, larger Ls/Ds corresponds to longer swirl
chamber length and results in an increasing decay of swirl energy and consequently, a
smaller air core, i.e., a bigger film thickness at the exit. As the inlet flow rate keeps
constant, the thicker the film is, the smaller the pressure drop throughout the atomizer and
consequently the larger Cd is.
Figure 4.2 indicates that with an increase of Ls/Ds, the spray cone half angle
diminishes slightly. Theoretically, as Ls increases, both the swirl energy and axial energy
are deceased at the exit. However, Figure 4.2 shows that wav/uav decreases with increase
in Ls/Ds. That means the decay of swirl energy is larger than that of axial energy
throughout the nozzle. Therefore, a decrease in the spray angle results from a decrease in
the ratio of exit swirl to axial velocity components.
53
Ls/Ds
Dim
ensi
onle
ssTh
ickn
ess
atE
xit,
t*
Dis
char
geC
oeffi
cien
t,C
d
0 0.5 1 1.50.3
0.35
0.4
0.3
0.35
0.4
0.45
Cdt*
Figure 4.1 Variation of the dimensionless film thickness at the exit and discharge
coefficient with Ls/Ds under constant flow rate
54
Ls/Ds
Spr
ayC
one
Hal
fAng
le,θ
wav
/uav
0 0.5 1 1.530
35
40
0.6
0.7
0.8
0.9
θwav/uav
Figure 4.2 Variation of the spray cone half angle and wav/uav with Ls/Ds under constant
flow rate
55
Effect of variation in lo/do
In order to conduct the influence of variation in lo/do on the atomizer performance,
eight different values of lo/do are considered here, which are 0.2, 0.3, 0.4, 0.5, 0.75 1, 1.5
and 2. Similar to the variation of Ls/Ds, only lo is changed to change the value of lo/do
and do is constant for the all eight cases.
Figure 4.3 shows that as lo/do increases from 0.2 to 0.5, t* decreases dramatically;
when lo/do = 0.75, t* is minimum; and as lo/do changes from 0.75 to 2, t* increases
slightly. This behavior of film thickness matches the trends discussed in Benjamin et al
(1998).
Figure 4.3 also indicates that as lo/do increases, Cd decreases with lo/do. This is
similar to the variation of Cd with Ls/Ds and because of the similar reason. It is noted that
the trend is rather steep at lower lo/do, while its gradient decreases at higher lo/do.
The variations of spray cone half angle and wav/uav with changing lo/do are given
in Figure 4.4. With increasing lo/do, θ initially diminishes with a high gradient when lo/do
is less than 0.75; as lo/do is greater than 0.75, the gradient of θ decreasing diminishes
rapidly. The varying tendency of wav/uav with increase in lo/do is similar to that of θ.
56
lo/do
Dim
ensi
onle
ssTh
ickn
ess
atE
xit,
t*
Dis
char
geC
oeffi
cien
t,C
d
0 0.5 1 1.5 20.3
0.35
0.4
0.3
0.35
0.4
0.45
Cdt*
Figure 4.3 Variation of the dimensionless film thickness at the exit and discharge
coefficient with lo/do under constant flow rate
57
lo/do
Spr
ayC
one
Hal
fAng
le,θ
wav
/uav
0 0.5 1 1.5 230
35
40
0.6
0.7
0.8
0.9
θwav/uav
Figure 4.4 Variation of the spray cone half angle and wav/uav with lo/do under constant
flow rate
58
Effect of variation in Ds/do
Eight values of Ds/do are considered 2, 2.5, 3, 3.5, 4, 4.5, 5 and 6. Figures 3.5, 3.6
and 3.7 show that the atomizer performance is strongly influenced by atomizer constant
K. In order to extract the effect of variation in Ds/do independently, K has to keep
constant. This is achieved by keeping Ds*do as constant when varying the Ds/do.
Figure 4.5 illustrates that with increasing Ds/do, the dimensionless film thickness
at the exit diminishes when Ds/do is less than 3; as Ds/do is greater than 3, t* becomes to
increase. Since K (and Ds*do) is constant in this study, to keep Ds*do constant, do has to
be decreased with increasing Ds/do. Actually, computational result indicates that the
absolute film thickness at the exit decreases monotonously with increasing Ds/do.
Figure 4.5 also shows the discharge coefficient variation with changing Ds/do. It
can bee seen that with increasing Ds/do, Cd decreases at smaller value of Ds/do, eventually
becoming almost constant at larger values of Ds/do.
The variation of spray cone half angle and wav/uav with changing Ds/do are given
in Figure 4.6. With increase in Ds/do from 2 to 6, the spray cone half angle increase
initially and decreases at larger values of Ds/do but the maximum relative change is only
about 4%. This indicates that Ds/do has little influence on θ. With increasing in Ds/do,
wav/uav shows the similar behavior to Ds/do.
59
Ds/do
Dim
ensi
onle
ssTh
ickn
ess
atE
xit,
t*
Dis
char
geC
oeffi
cien
t,C
d
1 2 3 4 5 60.3
0.35
0.4
0.3
0.35
0.4
0.45
Cdt*
Figure 4.5 Variation of the dimensionless film thickness at the exit and discharge
coefficient with Ds/do under constant flow rate
60
Ds/do
Spr
ayC
one
Hal
fAng
le,θ
wav
/uav
1 2 3 4 5 630
35
40
0.6
0.7
0.8
0.9
θwav/uav
Figure 4.6 Variation of the spray cone half angle and wav/uav with Ds/do under constant
flow rate
61
4.3 Results for Constant Pressure Drop across the Atomizer
Effect of variation in Ls/Ds
The variations of the dimensionless film thickness at the exit and discharge
coefficient with changing Ls/Ds are shown in Figure 4.7. With the increase in Ls/Ds, both
t* and Cd increase monotonously. Larger Ls/Ds corresponds to longer swirl chamber
length and results in an increasing decay of swirl energy and a smaller air core at the exit.
The same trend was also observed in the experiment (Kutty et al 1978). As the pressure
drop is held constant, the thicker the film is, the larger the mass flow rate and
consequently the larger Cd is.
Figure 4.8 shows that with an increase of Ls/Ds, the spray cone half angle and
wav/uav decrease almost linearly but the diminishment of the spray angle is somewhat
small (only about 9% in the given condition and atomizer). This tendency coincides with
the experimental observation (Kutty et al 1978). Theoretically, this also results from
increasingly decaying swirl energy at the exit for a longer swirl chamber. A decrease in
the ratio of swirl to axial velocity components at the exit results in a decrease in the spray
angle.
62
Ls/Ds
Dim
ensi
onle
ssTh
ickn
ess
atE
xit,
t*
Dis
char
geC
oeffi
cien
t,C
d
0 0.5 1 1.50.3
0.35
0.4
0.45
0.3
0.35
0.4
0.45
Cdt*
Figure 4.7 Variation of the dimensionless film thickness at the exit and discharge
coefficient with Ls/Ds under constant pressure drop
63
Ls/Ds
Spr
ayC
one
Hal
fAng
le,θ
wav
/uav
0 0.5 1 1.530
35
0.6
0.7
0.8
θwav/uav
Figure 4.8 Variation of the spray cone half angle and wav/uav with Ls/Ds under constant
pressure drop
64
Effect of variation in lo/do
Figure 4.9 shows that as lo/do increases from 0.2 to 0.5, t* decreases dramatically;
when lo/do = 0.75, t* is minimum; and as lo/do changes from 0.75 to 2, t* slightly
increases. .
Figure 4.9 also shows that as lo/do increases, Cd decreases with lo/do. This is
similar to the variation of Cd with Ls/Ds and because of the similar reason. Note that the
trend is rather steep at lower lo/do, while its gradient decreases at higher lo/do.
The variations of spray cone half angle and wav/uav with changing lo/do are given
in Figure 4.10. With increasing lo/do, both θ and wav/uav initially have a high gradient
which decreases rapidly at lo/do greater than 0.75.
65
lo/do
Dim
ensi
onle
ssTh
ickn
ess
atE
xit,
t*
Dis
char
geC
oeffi
cien
t,C
d
0 0.5 1 1.5 20.3
0.35
0.4
0.3
0.35
0.4
Cdt*
Figure 4.9 Variation of the dimensionless film thickness at the exit and discharge
coefficient with lo/do under constant pressure drop
66
lo/do
Spr
ayC
one
Hal
fAng
le,θ
wav
/uav
0 0.5 1 1.5 230
35
40
0.6
0.7
0.8
0.9
θwav/uav
Figure 4.10 Variation of the spray cone half angle and wav/uav with lo/do under constant
pressure drop
67
Effect of variation in Ds/do
From Figure 4.11, we can see that with increasing Ds/do, the dimensionless film
thickness at the exit first decreases, then increases for Ds/do greater than 4.5. Note that in
this case K (and Ds*do) is constant. To keep Ds*do constant, do has to be decreased with
increasing Ds/do. It is found that the absolute film thickness at the exit decreases
monotonously with increasing Ds/do. Such behavior has been reported in several
experimental observations (Lefebvre 1989).
The discharge coefficient variation with changing Ds/do is also given in Figure
4.11. Cd decreases with increasing Ds/do, eventually becoming almost constant at larger
values of Ds/do.
The variation of spray cone half angle and wav/uav with changing Ds/do are shown
in Figure 4.12. With increasing Ds/do from 2 to 6, the spray cone half angle decreases by
about 15%.
68
Ds/do
Dim
ensi
onle
ssTh
ickn
ess
atE
xit,
t*
Dis
char
geC
oeffi
cien
t,C
d
1 2 3 4 5 60.3
0.35
0.4
0.3
0.35
0.4
0.45
Cdt*
Figure 11 Variation of the dimensionless film thickness at the exit and discharge
coefficient with Ds/do under constant pressure drop
69
Ds/do
Spr
ayC
one
Hal
fAng
le,θ
wav
/uav
1 2 3 4 5 630
35
40
0.6
0.7
0.8
θwav/uav
Figure 12 Variation of the spray cone half angle and wav/uav with Ds/do under constant
pressure drop
70
4.4 Summary
The effects of changes in the ratio of length to diameter in swirl chamber, the ratio
of length to diameter in orifice and ratio of orifice to swirl chamber diameters were
numerically investigated. The investigation was conducted under both constant inlet flow
rate and constant pressure drop across the atomizer. For both working conditions, the
dimensionless film thickness at the exit increases with increasing the ratio of length to
diameter in swirl chamber, the ratio of length to diameter in orifice when its value is
small and ratio of swirl chamber to orifice diameters when its value is small and with
decreasing the ratio of length to diameter in orifice and the ratio of swirl chamber to
orifice diameters when their values are large; the discharge coefficient increases with
increasing the ratio of length to diameter in swirl chamber and with decreasing the ratio
of length to diameter in orifice and the ratio of swirl chamber to orifice diameters; the
spray cone half angle increases with decreasing the ratio of length to diameter in swirl
chamber and the ratio of length to diameter in orifice. The spray angle has little change
with changing the ratio of swirl chamber to orifice diameters under constant flow rate
whereas it decreases with increase in the ratio of swirl chamber to orifice diameters under
constant pressure drop.
71
CHAPTER 5 EFFECTS OF GEOMETRIC PARAMETERS ON
ATOMIZER PERFORMANCE (PART II)
5.1 Introduction
With the advent of new manufacturing techniques, simplex atomizers with
different geometries can be produced easily. However to design such atomizers, the
effect of all geometric parameters on the performance of the atomizer must be well
understood. Newer designs of simplex atomizers may include inlets that are not
perpendicular to the axis and inlet slot angle (angle β in Figure 5.1) may not be 90
degrees. The geometry may include a trumpet at the end of the exit orifice to limit the
spray cone angle. It is noted that, to achieve rapid fuel air mixing, air flow may be
introduced around the fuel injector through a swirler cup. In such an arrangement, it is
desired to impinge the atomized fuel on the wall of a swirler cup to undergo further
atomization due to shearing action of air flow along the swirler cup wall. Therefore, in
such configurations, a particular spray angle is desirable rather than the maximum
possible spray angle. Hence a trumpet is introduced in the atomizer geometry to limit the
cone angle of the spray. Also a particular trumpet length may be needed to inject the
atomized fuel at the right location in the air flow for rapid mixing. The internal
convergence angle (angle θc in Fig. 5.1) may be different from 45 degrees. These
geometrical parameters have not been considered in studies available in literature and the
effect of variation in these parameters on the performance of simplex atomizers has not
72
Figure 5.1 Schematic of a simplex atomizer with trumpet
β
θc
θt do
dt
DS
lo
LS
Free Film
Inlet Slot
Interface Liquid
Gas
lt
73
been reported in literature. None of the available semi-empirical correlations address
these geometric parameters and as such the present correlations do not provide any
guidance about these parameters to atomizer designers.
The knowledge of flow in a pressure-swirl atomizer and the effect of geometric
parameters on the characteristics of the liquid sheet emanating from the atomizer are
extremely important to design a fuel injector. The characteristics of the liquid sheet
emanating from the atomizer depend on the geometrical parameters of the atomizer
which can be expressed in dimensionless form as atomizer constant (K), ratio of length to
diameter of the swirl chamber (Ls/Ds), that of the orifice (lo/do), ratio of swirl chamber to
orifice diameter (Ds/do), inlet angle (β), trumpet length, trumpet angle (θt), and the
convergence angle (θc). The detailed study of first four parameters listed above was
presented in Chapter 3 and Chapter 4. An atomizer designer must choose a combination
of geometric parameters to achieve the desired spray angle, film thickness, and discharge
coefficient. Therefore, it is important to know how each geometric parameter affects the
atomizer performance. Semi-empirical correlations have been developed by varying the
atomizer constant, Ls/Ds, lo/do, and Ds/do and measuring their effect on atomizer
performance parameters. Also, the influence of K, Ls/Ds, lo/do, and Ds/do on atomizer
performance has been studied computationally (Jeng et al 1998, Sakman et al 2000, Xue
et al 2002). However, no studies are currently available that provide any guidance on the
change in atomizer performance with a change in inlet slot angle, swirl chamber
convergence angle, trumpet angle and trumpet length.
74
In this chapter, the effect of four geometric parameters viz., inlet slot angle β,
swirl chamber convergence angle θc, trumpet angle θt, and ratio of trumpet length to
orifice diameter lt/do, on the performance of pressure-swirl atomizers is studied. The
variation in the atomizer performance is presented in terms of the dimensionless liquid
film thickness at the exit of the orifice (t*), spray cone half angle (θ) and discharge
coefficient (Cd). All cases were run keeping mass flow rate constant. When one
parameter was changed, all other geometric parameters were held constant. Also,
analytical solutions are developed based on inviscid approximation to determine the film
thickness, spray cone angle, and discharge coefficient for the atomizer.
5.2 Effect of Variation in Inlet Slot Angle β
To investigate the effect of inlet slot angle, An atomizer without a trumpet is used.
The dimensionless film thickness, spray cone half angle, and discharge coefficient are
plotted in Figures 5.2 and 5.3 with variation in inlet slot angle. The film thickness was
determined from the CFD results at the orifice exit. In the numerical method used here,
the interface between liquid and gas is tracked as a sharp interface and therefore the CFD
results directly provide the film thickness. The discharge coefficient is calculated based
on the pressure drop between in inlet and exit of the atomizer obtained in the numerical
solution.
75
Inlet Slots Angle β
Dim
ensi
onle
ssTh
ickn
ess
t*an
dD
isch
arge
Coe
ffici
entC
d
Spr
ayC
one
Hal
fAng
leθ
40 50 60 70 80 90
0.2
0.3
0.4
30
35
40
45
50
55
t* (CFD)t* (Equation 5.3)Cd (CFD)Cd (Equation 5.2)Cd (Equation 5.5)θ (CFD)θ (Equation 5.4)
Figure 5.2 Variation of atomizer performance parameters with inlet slots angle β
76
Inlet Slots Angle β
Dim
ensi
onle
ssTh
ickn
ess
t*an
dD
isch
arge
Coe
ffici
entC
d
Spr
ayC
one
Hal
fAng
leθ
40 50 60 70 80 90
0.2
0.3
0.4
30
35
40
45
50
55
t* (lo/do=1)t* (lo/do=0.5)Cd (lo/do=1)Cd (lo/do=0.5)θ (lo/do=1)θ (lo/do=0.5)
Figure 5.3 Comparison of results for lo/do = 1(solid lines) and lo/do =0.5 (dashed lines)
77
With a fixed mass flow rate through the atomizer, a change in inlet slot angle
results in a change in the ratio of inlet axial to swirl velocity components. With β = 90o
the fuel has no axial velocity component at the inlet. Hence for the same mass flow rate,
β = 90o gives greater swirl velocity compared to β = 40o case.
Figure 5.2 shows that with an increase of β from 40o to 90o, dimensionless
thickness t* decreases about 25%, discharge coefficient Cd decreases about 35% and
spray cone half angle θ increases by about 15%. The higher swirl velocity that
corresponds to a higher inlet slot angle pushes the liquid to the atomizer walls and the
liquid film thickness in the exit orifice decreases. The spray cone angle is governed by
the ratio of axial to swirl velocity components at the exit. The cone angle is seen to
increase with inlet slot angle. This is due to the larger centrifugal force caused by the
higher swirl velocity component. As the mass flow rate is kept constant, thinner liquid
film in the exit orifice section corresponds to higher axial velocity. A combination of
increased swirl and axial velocity leads to a higher pressure drop across the atomizer and
the discharge coefficient decreases. As evident from Figure 5.2, the changes in spray
cone angle, discharge coefficient, and film thickness are large, indicating that inlet angle
is an important parameter that significantly influences the performance of the atomizer.
It should be noted that empirical correlations available in the literature do not include the
inlet slot angle and are unable to predict its influence on atomizer performance.
Analysis can be carried out using inviscid theory to examine the effect of the inlet
slot angle. We have modified the inviscid analysis of Giffen and Muraszew (1953) to
78
include the inlet slot angle. The details are given in the appendix. Only the final
equations are given here.
βπ 22
322 sin)1(
32 XXK −
= (5.1)
5.03
1)1(
⎥⎦
⎤⎢⎣
⎡+−
=XXCd
(5.2)
From the definition of X, we get,
Xt −= 1* (5.3)
⎟⎟⎠
⎞⎜⎜⎝
⎛
+=
s
id
rr
XKC
)1(sin)2/(
sinβπ
θ (5.4)
It is interesting to note that Equation 5.2 is similar to one obtained by Giffen and
Muraszew (1953), however the effect of inlet slot angle is apparent in Equation 5.1. The
results Equations 5.2, 5.3, and 5.4 are plotted in Figure 2(a) using dashed lines. For film
thickness, comparing with the CFD results, it is seen that the analytical inviscid solution
provides the correct qualitative variation but predicts lower values. This is to be expected
as the inviscid theory does not account for the viscous effects that would tend to decrease
the velocity and increase film thickness.
Equation 5.2 predicts lower values for discharge coefficient than the
computational results. Giffen and Muraszew (1953) also observed this trend in
comparison with available experimental data. They modified their equation by
introducing a coefficient (A) as shown below and suggested A = 1.17 based on
experimented data.
79
0.53(1 )1d
XC AX
⎡ ⎤−= ⎢ ⎥+⎣ ⎦
(5.5)
It is seen in Figure 5.2 that our computational results match well with this
modified equation (Equation 5.5).
It is noted that there is significant difference in the prediction of spray cone angle
using inviscid approximation and numerical results. The numerical results are based on
solution of Navier-Stokes equations which fully account for viscous forces. Due to the
accounting of viscous forces, difference in the predictions of CFD results and those with
inviscid approximation (theoretical results) is to be expected. Furthermore, the spray
angle is governed by the ratio of the average swirl to the average axial velocity at the
atomizer exit. More precisely it is related to the tan inverse of the average swirl to axial
velocity ratio. A change in this ratio will change the spray angle. For the cases
considered here, the decrease in swirl velocity compared to inviscid prediction is larger
compared to the decrease in the axial velocity. Hence the inviscid results over-predict the
spray angle. The experiments (Jeng et al, 1998) also show considerable difference
between experimental measurements and inviscid analysis for spray angle, with inviscid
theory over-predicting the spray angle compared to experimental measurements,
consistent with the CFD results.
It is important to note that the equations derived using inviscid approximation
(Equations 5.1-5.4) do not account for variation in all geometric parameters. For
example, the ratio of length to diameter of the swirl chamber (Ls/Ds), that of the orifice
80
(lo/do) influence the spray angle, however these parameters do not appear in Equations
5.1-5.4 at all. Numerical results obtained with a different value of Ls/Ds (or lo/do) will be
different, but the inviscid predictions will remain the same. Therefore, for different value
of Ls/Ds (or lo/do) the difference between numerical results and inviscid approximation
will be different. This is seen in Figure 5.3 where results for spray cone angle, discharge
coefficient, and film thickness at exit, are presented for lo/do = 0.5 and lo/do = 1. It is
noted that the variations in atomizer performance parameters with β, for the two values of
lo/do are essentially parallel. This makes the results very useful in designing simplex
atomizers.
5.3 Effect of Variation in Trumpet Angle θt
Note in this study, the length of the trumpet lt has been held constant as θt is
varied. Figure 5.4 indicates that as θt increases from 10o to 40o, dimensionless thickness
t* decreases about 42%, spray cone half angle θ increases from 22o to 40o but discharge
coefficient Cd remains almost constant. This shows that the trumpet can be very effective
in controlling the spray cone angle and the film thickness without significantly affecting
the pressure drop across the atomizer. As pressure drop across the atomizer remains
relatively steady, the axial velocity at the end of the exit orifice is expected to exhibit
little variation. However, trumpet diameter dt increases with increase in θt so the film
thickness t* decreases when the flow rate through the atomizer is kept constant.
81
Trumpet Angle θtDim
ensi
onle
ssTh
ickn
ess
t*an
dD
isch
arge
Coe
ffici
entC
d
Spr
ayC
one
Hal
fAng
leθ
10 20 30 40
0.1
0.2
0.3
0.4
20
25
30
35
40
45
50
t* (CFD)t* (Equation 5.7)Cd (CFD)Cd (Equation 5.2)Cd (Equation 5.5)θ (CFD)θ (Equation 5.8)
Figure 5.4 Variation of atomizer performance parameters with trumpet angle θt
82
Considering inviscid flow through the atomizer, Analytical solutions for film
thickness, spray cone half angle, and the discharge coefficient for atomizer geometry with
trumpet have been developed. Detailed analysis and a schematic of the geometry are
provided in the appendix. The final equations are:
22
2
222
21sin1
)1(cos1
ds
i
ttttt
o
Crr
XKXAA
=⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+
−⎟⎟⎠
⎞⎜⎜⎝
⎛β
θ (5.6)
Where,st
pt
t
tat rr
AK
AA
Xπ
== ,
to
tt r
rXt θcos)1(*
⎟⎟⎠
⎞⎜⎜⎝
⎛−= (5.7)
'θθθ += t (5.8)
Where, ts
i
t
t
rr
KX
θβθ cossin)1(2
tan '⎟⎟⎠
⎞⎜⎜⎝
⎛−=
Results using Equations 5.6-5.8 are shown in Figure 5.4 with dashed lines. Once
again the trends shown by the analytical solutions are seen to match well with
computational results. Similar to Figure 5.2, the analytical inviscid solutions give lower
values of film thickness and discharge coefficient, and higher values of spray cone half
angle. The results of Equation 5.2 under-predict the discharge coefficient. The modified
Equation 5.5 provides improved match with the numerical results. As stated earlier, the
differences in inviscid solutions and numerical results can be attributed to omission of
viscous forces in the inviscid treatment. It is interesting to note that compared to inviscid
flow through the atomizer, the decrease in average axial velocity due to viscous forces is
83
less than the reduction in average swirl velocity resulting in lower values of spray cone
half angle.
5.4 Effect of Variation in Dimensionless Trumpet Length (lt/do)
To study the effect of variation trumpet length, both orifice diameter (do) and
trumpet angle θt are held constant as lt is varied. Figure 5.5 indicates that with an
increase in lt/do, dimensionless thickness t* decreases about 14%, discharge coefficient
shows virtually no change, whereas spray cone half angle decreases about 9% for the
range of dimensionless trumpet length considered here. For constant flow rate across the
atomizer, the pressure drop across the atomizer, and the axial velocity at the end of the
exit orifice are not expected to change significantly with a change in the trumpet length.
As such the discharge coefficient shows very little change with trumpet length.
However, as the trumpet length increases, to keep the flow cross-sectional area relatively
constant, the film thickness decreases. Also, as the trumpet diameter increases, the
principle of conservation of angular momentum dictates that the liquid swirl velocity
would decrease as it moves along the trumpet length. Therefore, the spray cone angle is
seen to decrease with increasing trumpet length. Equations 5.6-5.8 outlined above
provide atomizer performance parameters for inviscid flow through the atomizer.
Although the trumpet length does not directly appear in Equations 5.6-5.8, it is noted that
for a fixed trumpet angle, Xt is a function of the trumpet length. The effects of variation
in trumpet length for inviscid flow are plotted in Figure 5.5 with dashed lines. Similar to
84
lt/doDim
ensi
onle
ssTh
ickn
ess
t*an
dD
isch
arge
Coe
ffici
entC
d
Spr
ayC
one
Hal
fAng
leθ
0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1
0.2
0.3
0.4
30
35
40
45
50
55
60
t* (CFD)t* (Equation 5.7)Cd (CFD)Cd (Equation 5.2)Cd (Equation 5.5)θ (CFD)θ (Equation 5.8)
Figure 5.5 Variation of atomizer performance parameters with ratio of trumpet length to
orifice diameter lt/do
85
inviscid flow solutions for variations in inlet slot angle and trumpet angle, the values for
film thickness and discharge coefficient predicted with computational model are higher
than analytical solutions and those of spray angle are lower than analytical solutions. The
results for discharge coefficient using modified Equation 5.5 are significantly closer to
computational results.
5.5 Effect of Variation in Swirl Chamber Convergence Angle θc
In this case, the geometry of the atomizer is one without trumpet and all
geometric parameters have been kept fixed except for θc. As seen from Figure 5.6, for the
atomizer configuration considered in this study, with an increase in θc from 45o to 90o,
dimensionless thickness t* increases about 16%, Cd increases about 34% and spray cone
half angle θ decreases about 9%. Earlier studies (Jeng et al 1998, Xue et al 2002) have
shown that the flow field inside a simplex atomizer has a region of recirculating flow.
The majority of the liquid entering from the inlet slots flows through a region close to the
liquid gas interface and enters the orifice. The location and size of the recirculating flow
affects the axial velocity variations in the swirl chamber and consequently in the exit
orifice. As the convergence angle changes, the re-circulating region in the swirl chamber
can change considerably. This is illustrated in Figures 5.7(a) and 5.7(b) which show flow
streamlines for two atomizer geometries with convergence angle 60o and 90o,
respectively. As the flow structure changes, the velocity variation in the atomizer is
altered and consequently the film thickness at the exit, spray cone half angle, and
86
Spin Chamber Convergence Angle θcDim
ensi
onle
ssTh
ickn
ess
t*an
dD
isch
arge
Coe
ffici
entC
d
Spr
ayC
one
Hal
fAng
leθ
40 50 60 70 80 900.2
0.3
0.4
36
38
40
42
t*
Cdθ
Figure 5.6 Variation of atomizer performance parameters with convergence angle θc
87
(b) θc = 90o
Exit
AirC
ore
Inlet
(a) θc = 60o
Exit
AirC
ore
Inlet
Figure 5.7 Streamlines for flow inside the atomizer for different θc. Inlet slot angle β=900.
88
discharge coefficient change. Although the spray angle is lowest and the film thickness
is largest at convergence angle of 90o, this geometry may be preferred in some cases as it
is easier and less expensive to manufacture compared to the geometry with smaller
convergence angle. It is noted that the effects due to changes in swirl chamber
convergence angle can not be predicted by inviscid analysis.
5.6 Summary
By using a computational model based on the Arbitrary-Lagrangian-Eulerian
method, the flow in simplex atomizers was predicted. The effect of four geometric
parameters on the atomizer performance was investigated. The four geometric
parameters considered were the inlet slot angle, trumpet angle and length, and the swirl
chamber convergence angle. The atomizer performance was shown in terms of variations
of dimensionless film thickness, spray cone half angle, and discharge coefficient. Results
indicated that increase in inlet slot angle results in lower film thickness and discharge
coefficient and higher spray cone angle. The swirl chamber convergence angle has an
opposite effect on performance parameters. With an increase in convergence angle, film
thickness and discharge coefficient increase whereas the spray cone angle decreases. The
trumpet angle has very little influence on discharge coefficient. However, the film
thickness decreases and spray cone angle increases with increasing trumpet angle. The
discharge coefficient is insensitive to trumpet length whereas both the spray cone angle
and the film thickness decrease with increasing trumpet length. The effect of the
89
geometric parameters has not been reported in the literature and as such the results
presented here will provide useful guidance in designing simplex atomizers.
90
CHAPTER 6 NON-NEWTONIAN FLUID FLOW IN PRESSURE-
SWIRL ATOMIZERS
6.1 Introduction
Although the fluid flow in the pressure-swirl atomizers used in combustors
involves Newtonian fluids, the fluid to be atomized is likely to have non-Newtonian
viscous characteristics in many other atomization applications, such as paint spray,
agricultural spray, pharmaceutical and food processing. Compared to the large number of
publications on the Newtonian fluid sprays from simplex nozzles, only two studies by
Som and co-workers on non-Newtonian fluids are available (Som, 1983, Biswas and
Som, 1986). They determined both theoretically and experimentally the discharge
coefficient and spray cone angle of swirl nozzle using a time-independent purely viscous
power-law non-Newtonian fluid. In their papers, theoretical predictions were made
through an approximate analytical solution of the hydrodynamics of flow inside the
nozzle and experiments were carried out with aqueous solutions of CMC powder of
various concentrations as the working fluid. Their experiments were restricted to shear-
thinning fluids. Their experiment data indicated that at fixed values of generalized
Reynolds number based on tangential velocity at the inlet, ReGi, an increase in the power
index n increases the discharge coefficient and the power index n has a negligible
influence on the spray cone angle. Biswas’s experimental result showed that for fixed
values of ReGi, an increase in the power index n decreases the spray cone angle and the
power index n has a negligible influence on the discharge coefficient. The experiment
were conducted on two atomizer geometries one with pure tangential inlet (Som) and
91
with combined axial and tangential entry (Biswas and Som). Note that in their study,
with the decrease in value of n, the condition of constant ReGi requires much higher
tangential velocity at the inlet. That means the mass flow rate varies considerably.
Furthermore, the fluid consistency (K) of the fluids used was significantly higher for low
power law index. These limitations make direct comparison of our predictions with their
results difficult.
We have extended the CFD model developed earlier for Newtonian flow by using
a two-parameter non-Newtonian power law fluid model, the flow of time-independent
purely viscous power-law fluids in pressure-swirl atomizers is investigated.
6.2 Mathematical Model
The generalized Newtonian fluid model is (Bird et al., 2002):
γηητ &−=∇+∇−= + ))(( vv (6.1)
Where,
)(γηη &= (6.2)
+∇+∇= )( vvγ& (6.3)
According to two-parameter power law model:
1−= nmγη & (6.4)
Where n is power index (fluid behavior index) and m is fluid consistency index.
So the shear stress tensor can be expressed as,
nmγτ &−= (6.5)
For n < 1, the fluid is shear-thinning non-Newtonian fluid; n = 1, is Newtonian
fluid; for n > 1, the fluid is shear-thickening non-Newtonian fluid. When Equation 6.5 is
92
used to compute the shear stress term in the Navier-Stokes equations, the model can be
used to handle both Newtonian and non-Newtonian fluid flow.
6.3 Influence of power-law index on the performance of atomizers with constant
flow rate across the atomizer
The effect of different power law index on atomizer performance was investigated
keeping the mass flow rate across the atomizer constant. In most applications the
atomizers operate in a flow regime where the Reynolds number has little effect on the
atomizer performance and resulting spray characteristics. To make sure that we consider
an appropriate flow rate for such a regime, the results were obtained with constant inlet
flow rates of 300lbm/h and 400lbm/h. The range for the power index n is from 0.7 to 1.3.
The atomizer performance is shown in terms of variations of dimensionless film
thickness at exit, spray cone half angle, and discharge coefficient.
Figure 6.1 shows the geometry of atomizer used in this study. The final grid is
shown in Figure 6.2. Figure 6.3 gives the simulation result of typical velocity field and
pressure distribution.
Figure 6.4 shows the variation of dimensionless film thickness at the exit with the
power index and indicates that as power index increases, the film thickness increases.
Due to the increase in shear stress with the power index, the air core diameter at the exit
should decrease. The variation of discharge coefficient with the power index is given in
Figure 6.5. The discharge coefficient increases slightly with increase n power index.
The increase in film thickness will decrease the exit velocity and consequently decrease
the pressure drop across the atomizer and increase the discharge coefficient. Figure 6.6
93
Radial Direction (r)A
xial
Dire
ctio
n(z
)
Exit
Air
core
Figure 6.1 Initial grid
94
Radial direction (r)A
xial
Dire
ctio
n(z
)
Exit
Air
core
Figure 6.2 Final grid
95
Figure 6.3 Streamline and pressure distribution
Axialdirection
(z)
Radial Direction (r)
P1.17198E
+061.09096E
+061.00994E
+0692891584789376687268585160483052380844278736176628074419972311870237680.3
ExitA
ircore
96
Power index (n)
Dim
ensi
onle
ssTh
ickn
ess
(t* )
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.40.3
0.31
0.32
0.33
0.34
0.35
0.36
0.37
400 lbm/h300 lbm/h
Figure 6.4 Variation of film thickness with power index
97
Power index (n)
Dis
char
geco
effic
ient
(Cd)
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.40.36
0.37
0.38
400 lbm/h300 lbm/h
Figure 6.5 Variation of discharge coefficient with power index
98
Power index (n)
Spr
ayco
nean
gle
(θ)
0.6 0.8 1 1.2 1.430
31
32
33
34
35
400 lb/h300 lb/h
Figure 6.6 Variation of Spray half angle with power index
99
Radial Direction (r)A
xial
dire
ctio
n(z
)
Exit
Air
core
AA
B B
CC
Figure 6.7 Locations for the velocity inside nozzle study
100
r/rSA
xial
Vel
ocity
(m/s
)0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
n=1.3n=0.7n=1.0
Locaion A-A
Figure 6.8 Axial velocity profiles at the swirl chamber
101
r/rSA
xial
Vel
ocity
(m/s
)0.4 0.6 0.8 10
10
20
30
40
50
n=1.3n=0.7n=1.0
Locaion B-B
Figure 6.9 Axial velocity profiles at the orifice
102
r/rSA
xial
Vel
ocity
(m/s
)0.6 0.8 1
0
10
20
30
40
50
n=1.3n=0.7n=1.0
Locaion C-C
Figure 6.10 Axial velocity profiles at the exit
103
r/rSS
wirl
Vel
ocity
(m/s
)0 0.2 0.4 0.6 0.8 1
0
10
20
30
40
50
n=1.3n=0.7n=1.0
Locaion A-A
Figure 6.11 Swirl velocity profiles at the swirl chamber
104
r/rSS
wirl
Vel
ocity
(m/s
)0.4 0.6 0.8 10
10
20
30
40
50
n=1.3n=0.7n=1.0
Locaion B-B
Figure 6.12 Swirl velocity profiles at the orifice
105
r/rSS
wirl
Vel
ocity
(m/s
)0.6 0.8 1
0
10
20
30
40
50
n=1.3n=0.7n=1.0
Locaion C-C
Figure 6.13 Swirl velocity profiles at the exit
106
indicates the spray cone angle decreases when power index increases. This behavior is
consistent with measurements of Som (1983) and Biswas (1986) of discharge coefficient
and spray cone angle with variation in power index n. However, in their experiment, the
swirl and axial velocity decreased with increase in n whereas here the mass flow rate is
constant.
The axial and swirl velocity profiles at three different locations inside the
atomizer are investigated here. The locations are shown in Figure 6.7, and include a
location in the swirl chamber (A-A), in the orifice (B-B) and at the orifice exit (C-C).
The velocity profiles for three different power index values, viz. n= 0.7, 1 and 1.3, are
given in Figure 6.8-6.13.
Figure 6.8 shows that in the swirl chamber, the axial velocity is larger near the
wall and near the liquid/gas interface. The axial velocities are relatively weak in the
middle. Such velocity variations have also been observed in experiments (Wang et al
1999). Among the three fluids, the shear-thickening fluid (n= 1.3) shows the largest
variation in the velocity magnitude throughout the flow cross-section. Figure 6.9
indicates that at the orifice, the profiles of the axial velocities are quite flat except for the
zone near the wall. The velocity gradients for the n = 0.7 and 1 are greater than that of
the n = 1.3 in the zone near the wall but the shear-thickening fluid has greater maximum
axial velocity. The axial velocity profiles at the exit are shown in Figure 6.10. At the
exit, the locations of the maximum axial velocities are more close to the wall than the
107
interface. Comparing to the axial velocities at the orifice, the average values become
larger due to the decrease of the film thickness.
The swirl velocity profiles at the swirl chamber are given in Figure 6.11. All
three profiles show the typical free-vortex swirl flow pattern. The shear-thickening fluid
has slightly smaller velocity magnitude. Figure 6.12 indicates that at the orifice, the
shear-thinning fluid and Newtonian fluid have the combined vortex structure, a solid
vortex in the inner region connected with a free vortex in the outer region and the
velocity variation in the shear-thickening fluid closely resembles a free vortex structure.
The swirl velocity profiles at the exit are shown in Figure 6.13. The flow patterns at the
exit are similar to those at the orifice.
6.4 Effect of atomizer constant on the performance of atomizers for both
Newtonian and non-Newtonian fluids
In this section, the influence of atomizer constant on the performance of atomizers
for both Newtonian and non-Newtonian fluid is investigated. The working condition is
constant pressure drop throughout the atomizer. Three different power index values, viz.
0.8, 1.0 and 1.2, are considered here.
The variation of dimensionless film thickness with atomizer constant is shown in
Figure 6.14. As atomizer constant K changes from 0.2 to 0.6, the film thickness increases
about 50% for n = 0.8 and 1, and about 46% for n = 1.2. Increase in K corresponds to
increase in inlet area and consequently a decrease in swirl velocity. Therefore, with
108
Atomizer Constant K
Dim
ensi
onle
ssTh
ickn
ess
t*
0.2 0.3 0.4 0.5 0.60.2
0.25
0.3
0.35
0.4
t* (n = 0.8)t* (n = 1.0)t* (n = 1.2)
Figure 6.14 Variation of film thickness with atomizer constant
109
Atomizer Constant K
Dis
char
geC
oeffi
cien
tCd
0.2 0.3 0.4 0.5 0.60.2
0.25
0.3
0.35
0.4
0.45
Cd (n = 0.8)Cd (n = 1.0)Cd (n = 1.2)
Figure 6.15 Variation of discharge coefficient with atomizer constant
110
Atomizer Constant K
Spr
ayC
one
Hal
fAng
leθ
(deg
)
0.2 0.3 0.4 0.5 0.632
34
36
38
40
42
44
46
θ (n = 0.8)θ (n = 1.0)θ (n = 1.2)
Figure 6.16 Variation of spray cone angle with atomizer constant
111
increasing K, the film thickness increases. We note that for given atomizer constant
value, as power index increases, the film thickness increases.
Figure 6.15 gives variation of discharge coefficient with atomizer constant. It
indicates that the discharge coefficient increases about 60% for n = 0.8 and 1, and about
65% for n = 1.2. With increasing K, the decrease in swirl results in thicker fluid film.
With a constant pressure drop across the atomizer, the thicker fluid film leads to higher
mass flow rate and hence increases in the discharge coefficient. Also note that the
discharge coefficient increases with power index for a given atomizer constant.
The variation of spray cone half angle with atomizer constant is shown in Figure
6.16. As atomizer constant varies from 0.2 to 0.6, the spray angle approximately changes
from 44 to 36 for n = 0.8 and n = 1.0, from 44 to 32 for n = 1.2. The spray angle
decreases with increase in power index for a given atomizer constant. Once again, this is
to be expected as lower swirl with higher K would lead to smaller spray cone angle.
6.5 Summary
Using a two-parameter power law model, the flow of power-law fluids inside
pressure-swirl atomizers was investigated. Under constant mass flow rate through the
atomizer, the dimensionless film thickness at the exit increases and the spray cone angle
decreases with increase in the power index. The discharge coefficient exhibits only a
small influence of power-law index. Detailed flow pattern inside the atomizer for shear-
thinning, Newtonian and shear-thickening fluids were investigated.
112
Under constant pressure drop across the atomizer, the dimensionless film
thickness at the exit and the discharge coefficient increase, and the spray cone angle
decreases with increase in the atomizer constant. For the given atomizer constant value,
the dimensionless film thickness at the exit and the discharge coefficient increase, and the
spray cone angle decreases with increase in the power index.
113
CHAPTER 7 CONCLUSIONS AND RECOMMENDATIONS
7.1 Conclusions
A 2D CFD code for simulation of flow in pressure-swirl atomizers was
developed. The code can track the free interface between gas and liquid accurately via
ALE method. The code incorporated a method for grid generation at each time step. The
code was validated by comparison of computational predictions with experimental data
for large scale prototype and with semi-empirical correlations at small scale. The CFD
predictions of spray angle were closer to the experimental measurements on large scale
prototype atomizer than those obtained with semi-empirical correlations for all the cases
considered. The CFD prediction agreed well with experimental data of the film thickness
at the exit and the pressure drop across the nozzle. The agreement was found to be
substantially better than the predictions using correlations.
Among the geometric parameters considered here, atomizer constant was found to
be the most dominant parameter. With the atomizer constant (K) increasing from 0.1 to
0.6, dimensionless film thickness at the exit increases by about 0.15; the discharge
coefficient increases by about 0.2; and the spray cone half angle decreases by about 25%.
With other parameters, the dimensionless film thickness and spray cone half angle
variations are not always monotonic and exhibit optimal conditions (small thickness and
large cone angle) at certain parameter values. This may be of interest in designing
atomizers.
114
The effect on the performance of the nozzle with variations with Ls/Ds, lo/do, and
Ds/do were considered under constant inlet flow rate as well as constant pressure drop
across the atomizer. The code could predict these results easily, whereas the present
correlations can not provide such guidance for nozzle designers.
There are several parameters that are not considered in the correlations. Four
such parameters were investigated here, viz., inlet slot angle, trumpet angle, trumpet
length and convergent angle. The result illustrated that the film thickness increases with
decreasing inlet angle, trumpet angle and trumpet length and increasing convergent angle;
discharge coefficient increases with decreasing inlet angle and increasing convergent
angle; spray cone angle increases with increasing inlet angle and trumpet angle and
decreasing trumpet length and convergent angle; trumpet angle and trumpet length have
little effect on discharge coefficient.
The code was extended to model the flow of time-independent purely viscous
power-law fluids in pressure-swirl atomizers. A range of power-law index from 0.7 to
1.3 was considered. Detailed flow patterns inside the atomizer for shear-thinning,
Newtonian and shear-thickening fluids were investigated. The effect of variation in
atomizer constant on the atomizer performance was studied for fluids with different
power index values for a fixed pressure drop across the atomizer. The results indicated
that the dimensionless film thickness at the exit and the discharge coefficient increase,
and the spray cone angle decreases with increase in the atomizer constant. For a given
115
atomizer constant, the dimensionless film thickness at the exit and the discharge
coefficient increase, and the spray cone angle decreases with increase in the power index.
7.2 Recommendations for future work
For the atomizers with very short swirl chamber and two inlet slots, the flow may
be have three-dimensional features that can not be determined by the present
axisymmetric approach. In order to simulate such flows, a fully three dimensional
treatment may be needed. The present code can be extended to model three dimensional
flows.
In this study, the performance of atomizers was expressed indirectly in term of
dimensionless film thickness at the exit, the spray cone half angle and discharge
coefficient. If a model to simulate the break-up of the liquid sheet can be developed and
is integrated in current code, the spray characteristics, such as mean drop size, drop size
distribution, can be predicted directly from atomizer geometry and input flow parameters.
Such a break-up model, combined with the present code, will represent a major advance
in developing predictive tools for fuel injection.
116
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124
APPENDIX INVISCID ANALYSIS FOR PRESSURE SWIRL
ATOMIZERS
The analysis closely follows the development by Giffen and Muraszev (1953)
with appropriate additions for the new geometric parameters considered here. First we
consider the effect of inlet slot angle for an atomizer without trumpet. Here we consider
a more general case where the inlet is not necessarily at r = rs (shown in Figure A-1).
Figure A-1 Schematic of pressure swirl atomizer
In pressure swirl atomizer, the liquid flow can be considered as a spiral motion as
a result of an axial flow being superimposed on a free vortex.
For a free vortex:
Gas/Liquid interface
ri
roa ro
β
rs
125
== ii rwwr constant (A-1)
where:
pi A
Qw βsin= (A-2)
According to Bernoulli’s equation, the total pressure (injection pressure) in the
liquid flowing through the orifice is:
=++=∆ 22
21
21 wupP ρρ constant (A-3)
Where p is the static pressure.
At the air core, p = 0. So,
22
21
21
oaoa wuP ρρ +=∆ (A-4)
The axial velocity can be calculated as:
)( oaooa AA
Qu−
= (A-5)
According to Eq. (A-1) and Eq. (A-2),
oap
ioa rA
rQw
βsin= (A-6)
Substituting into Eq. (A-3) u and w from Eq. (A-5) & (A-6),
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛−
=∆22
sin21
oap
i
oao rAQr
AAQP
βρ (A-7)
From the definition of Cd, we get,
5.02
⎟⎟⎠
⎞⎜⎜⎝
⎛ ∆=
ρPACQ od (A-8)
Substitution of Q from Eq. (A-8) into Eq. (A-7),
126
β22
21
22 sin1)1(
11⎟⎟⎠
⎞⎜⎜⎝
⎛+
−=
s
i
d rr
XKXC (A-9)
Where,so
p
o
oa
rrA
KAA
Xπ
== 1,
Assume that the size of the air core in the orifice will be such as to give maximum
flow. That means Cd is maximum:
β22
2
32
1
2
sin2
)1(0
1
⎟⎟⎠
⎞⎜⎜⎝
⎛−=⇒=
⎟⎟⎠
⎞⎜⎜⎝
⎛
s
id
rr
XXK
dXC
d (A-10)
Substitute Eq. (A-10) into Eq. (A-9),
5.03
1)1(
⎥⎦
⎤⎢⎣
⎡+−
=XXCd
(A-11)
14KK
ddA
Kos
p π=⇒=Q (A-12)
Substitute Eq. (A-12) into Eq. (A-10),
βπ 22
2
322 sin)1(
32 ⎟⎟⎠
⎞⎜⎜⎝
⎛−=
s
i
rr
XXK (A-13)
From the definition of X, we get,
Xt −= 1* (A-14)
Following Giffen and Muraszew[12], for the spray cone half angle θ, we have,
Uw
oruw o
o
o == θθ sintan (A-15)
odod AC
QUUACQ =⇒=Q (A-16)
At the exit, the total angular momentum in the orifice,
127
∫−
=o
oa
r
rp
oaoio
p
io A
rrrQurA
Qrdrur β
πρβρπ sin
)(2sin)2( (A-17)
The total mass flow in the orifice,
)( oaoo AAu −ρ (A-18)
⇒ mean tangential velocity at the exit,
)(sin)(2
oaop
oaoio AAA
rrQrw
−−
=βπ (A-19)
Substituting Eq. (A-16) & Eq. (A-19) into Eq. (A-15), we get,
⎟⎟⎠
⎞⎜⎜⎝
⎛
+=
s
id
rr
XKC
)1(sin)2/(
sinβπ
θ (A-20)
Now we consider an atomizer geometry with trumpet.
Figure A-2 Schematic of orifice with trumpet
Following the foregoing analysis, consider the air core at the trumpet end, we
have,
do
doa
dt
dta
128
ttatta AA
Quθcos)( −
= (A-21)
tap
ita rA
rQw
βsin= (A-22)
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛−
=∆⇒22
sincos)(2
1
tap
i
ttat rAQr
AAQP
βθ
ρ (A-23)
Substitution of Q from Eq. (A-8) into Eq. (A-23),
22
2
222
21sin1
)1(cos1
ds
i
ttttt
o
Crr
XKXAA
=⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+
−⎟⎟⎠
⎞⎜⎜⎝
⎛β
θ (A-24)
where,st
pt
t
tat rr
AK
AA
Xπ
== ,
to
tt r
rXt θcos)1(*
⎟⎟⎠
⎞⎜⎜⎝
⎛−=⇒ (A-25)
For the spray cone half angle θ, approximately we have,
'θθθ += t (A-26)
where t
t
uw
='tanθ (A-27)
the mean velocity along with the trumpet direction at the trumpet end,
ttatt AA
Quθcos)( −
= (A-28)
At the trumpet end, the total angular momentum,
∫−
=t
ta
r
rp
tatitt
p
itt A
rrrQurA
Qrdrur β
θπρβθρπ sin
)(cos2sin)cos2( (A-29)
The total mass flow at the trumpet end,
129
)(cos tattt AAu −θρ (A-30)
⇒ mean tangential velocity at the trumpet end,
)(sin)(2
tatp
tatit AAA
rrQrw
−−
=βπ (A-31)
Substituting Eq. (A-28) & Eq. (A-31) into Eq. (A-27), we get,
ts
i
t
t
rr
KX
θβθ cossin)1(2
tan '⎟⎟⎠
⎞⎜⎜⎝
⎛−= (A-32)
