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Simulation of Coal Ash Deposition on Modern Turbine Nozzle Guide Vanes
Thesis
Presented in Partial Fulfillment of the Requirements for the Degree of Master of Science
in the Graduate School of The Ohio State University
By:
Brett Jordan Barker, B.S.
Graduate Program in Aeronautical and Astronautical Engineering
The Ohio State University
2010
Master’s Examination Committee:
Dr. Jeffrey P. Bons, Advisor
Dr. James Gregory
Dr. Ali Ameri
Copyright by
Brett Barker
2010
ii
ABSTRACT
A numerical study of the physical mechanisms associated with the deposition of
coal ash particulate on a blade passage of the first stage of the GE-E3 turbine was
performed. The software package FLUENT was used to model the flow through the
turbine passage and predict the trajectory of particles injected into the flow using an
Euler-Lagrangian two phase approach. The standard k-ω turbulence model was used for
its accuracy at modeling flow very near to the wall. A critical impact velocity sticking
model and a critical viscosity sticking model were compared for their usability and
accuracy predicting deposition locations and quantities. The critical impact velocity
sticking model was chosen due to the extensive work that has been done at BYU to
calibrate the model to experimental data. The critical viscosity sticking model proved
simple to use but required additional calibration beyond the initial modeling. The
simulations were compared to experimental results obtained from the Turbine Reacting
Flow Rig (TuRFR) at The Ohio State University. Finally, modifications were made to the
hub end wall geometry to mitigate deposition growth. The hub end wall inlet angle to the
axial was decreased by 30 degrees and showed an over-all 18% decrease in deposition
mass. The area near the hub wall and pressure surface of the vane showed marked
improvement; however, there was a slight increase in deposition along the mid-span
region of the turbine vane.
iii
ACKNOWLEDGEMENTS
I would like to thank Dr. Jeffrey Bons for his guidance and support through my Graduate
career. He provided me with opportunities to learn and encouraged me every step of the
way.
I want to give a special thanks to Dr. Ali Ameri for his patience and availability during
the long nights of computational work. Without him much of this work would not have
been possible. I would also like to thank Dr. James Gregory for being on my committee
and for being a wonderful teacher.
I also want to show my appreciation for all of my fellow Graduate students, Chris Smith,
Josh Webb, Prashanth Shankara, and Kyle Gompertz and the rest of the AARL research
members, for their support and help through this process.
iv
VITA
May 30, 1986 ………………………………….Born in Sidney, Ohio
June 2008 ………………………………………B.S. Aerospace Engineering,
The Ohio State University
2008 – Present ………………………………Graduate Research Associate,
Department of Aerospace Engineering
The Ohio State University
PUBLICATIONS
1. Lewis, S., Barker, B., Bons, J.P., Ai, W., and Fletcher, T.H. "Film Cooling
Effectiveness and Heat Transfer Near Deposit-Laden Film Holes." GT2009-
59567. Presented at the IGTI 2009 conference and recommended for publication
in the Journal of Turbomachinery publication, 2009
2. Smith, C., Barker, B., Clum, C., Bons, J.P. “Deposition in a Turbine Cascade with
Combusting Flow.”Presented at the ASME Turbo Expo 2010: Power for Land,
Sea, and Air (2010)
v
AWARDS
IGTI/ASME Heat Transfer Committee 2009 Best Paper Award for “Film Cooling
Effectiveness and Heat Transfer Near Deposit-Laden Film Holes” S. Lewis, B. Barker,
J.P. Bons, W. Ai, T.H. Fletcher. GT2009-59567
FIELDS OF STUDY
Major Fields of Study: B.S. Aerospace Engineering
M.S. Aerospace Engineering
vi
TABLE OF CONTENTS
ABSTRACT........................................................................................................................ii
ACKOWLEDGEMENTS..................................................................................................iii
VITA...................................................................................................................................iv
FIELDS OF STUDY...........................................................................................................v
TABLE OF CONTENTS....................................................................................................vi
LIST OF FIGURES.............................................................................................................x
LIST OF TABLES...........................................................................................................xiii
LIST OF EQUATIONS....................................................................................................xiv
NOMENCLATURE........................................................................................................xvii
CHAPTER 1........................................................................................................................1
INTRODUCTION...............................................................................................................1
1.1 Background........................................................................................................1
1.2 Literature Review...............................................................................................2
1.3 Other Deposition Research................................................................................3
CHAPTER 2........................................................................................................................5
FLUID MODELING...........................................................................................................5
2.1 Fluid Modeling Package....................................................................................5
2.1.1 Grid Generation..............................................................................................5
2.2 Fluid Equations of Motion.................................................................................9
vii
2.3 Turbulence Modeling.......................................................................................11
2.3.1 Near Wall Flow.............................................................................................12
2.4 Boundary Conditions.......................................................................................13
2.4.1 Inlet Boundary Condition.............................................................................14
2.4.2 Outlet Boundary Condition...........................................................................15
2.4.3 Wall Boundary Condition.............................................................................16
2.4.4 Periodic Boundary Condition.......................................................................16
2.5 Experimental Description................................................................................17
2.6 Operating Conditions.......................................................................................19
2.7 Flow Solution Results......................................................................................20
CHAPTER 3......................................................................................................................23
PARTICULATE MOTION...............................................................................................23
3.1 Particle Trajectory Modeling...........................................................................23
3.1.1 Eulerian-Lagrangian Model..........................................................................23
3.1.2 Particle Forces...............................................................................................26
3.2 Injection Properties..........................................................................................31
3.3 Random Walk Model.......................................................................................32
3.3.1 Perturbation Velocity Corrections................................................................34
3.4 Particle Material Properties..............................................................................35
3.4.1 Ash Composition..........................................................................................36
3.4.2 Size Distribution...........................................................................................37
3.5 Particle Trajectories.........................................................................................40
viii
CHAPTER 4......................................................................................................................42
STICKING MODELING...................................................................................................42
4.1 Impact, Sticking, Capture Efficiency Definitions............................................42
4.2 Wall-Particle Interaction..................................................................................43
4.3 Critical Impact Velocity Sticking Model.........................................................44
4.3.1 Particle Impact Properties.............................................................................45
4.4 Critical Viscosity Sticking Model....................................................................47
4.4.1 Ash Compensation Model.............................................................................48
4.4.2 Viscosity-Temperature Relationship............................................................49
4.4.3 Relationship Limits.......................................................................................51
4.4.4 Critical Viscosity Sticking Criteria...............................................................53
4.5 Pre-Existing Deposition Factor........................................................................55
4.6 Detachment Model...........................................................................................55
4.6.1 Individual Particle Detachment....................................................................56
4.6.2 Large Scale Detachment...............................................................................59
4.7 Sticking Model Comparisons...........................................................................61
4.8 Experimental Comparison...............................................................................67
4.9 Particle Size Deposition Distribution ..............................................................72
CHAPTER 5......................................................................................................................75
END WALL MODIFICATION........................................................................................75
5.1 Reasoning for end wall modification...............................................................75
5.2 Method of adjustment......................................................................................75
ix
5.3 Exit and inlet profile changes..........................................................................78
5.4 Deposition augmentation.................................................................................80
5.5 Recommendations for improvements..............................................................82
CHAPTER 6......................................................................................................................84
CONCLUSION AND FUTURE WORK..........................................................................84
6.1 Conclusion.......................................................................................................84
6.2 Future Work.....................................................................................................85
REFERENCES..................................................................................................................87
APPENDIX........................................................................................................................93
A.1 Critical Impact Velocity Sticking Model UDF...............................................93
A.2 Critical Viscosity Sticking Model UDF..........................................................99
A.3 Random Walk Correction UDF....................................................................105
A.4 Local Flow Properties Injection UDF...........................................................115
x
LIST OF FIGURES
Figure 1 – VKI Grid............................................................................................................6
Figure 2 – E3 Annulus (Colored by Radial Position) .........................................................7
Figure 3 – E3 geometry.......................................................................................................7
Figure 4 – Mid-span of E3 Turbine vane grid....................................................................8
Figure 5 – Modified E3 geometry.......................................................................................9
Figure 6 – Universal laws of the wall, Schlichting and Gersten [35] ...............................12
Figure 7 – Fitted capture efficiencies obtained from 2-D CFD modeling versus measured
values [3] ...........................................................................................................................15
Figure 8 – Image of the TuRFR.........................................................................................17
Figure 9 – Image of vane holder........................................................................................18
Figure 10 – Mach distribution at mid-span........................................................................21
Figure 11 – Pressure distribution at mid-span...................................................................21
Figure 12 – Temperature distribution at mid-span............................................................22
Figure 13 – Total Pressure at Exit Plane............................................................................22
Figure 14 – Classification of flow regimes for gas-solid flows [17] ................................24
Figure 15 – Particle injection locations.............................................................................31
Figure 16 – RMS values of velocity in the boundary layer [13] ......................................34
Figure 17 – Particle size distribution results from Coulter counter for Bituminous coal fly
ash [38] ..............................................................................................................................38
Figure 18 – Size Distribution Modeling............................................................................39
xi
Figure 19 – 1 μm diameter particle trajectory...................................................................40
Figure 20 – 20 μm diameter particle trajectory.................................................................41
Figure 21 – 50 μm diameter particle trajectory.................................................................41
Figure 22 – dimensionless sticking forces versus particle diameter [6] ...........................43
Figure 23 – Transition modified deposition model using critical viscosity approach
[41].....................................................................................................................................48
Figure 24 – Viscosity – Temperature relationship for various ash types [41] ..................52
Figure 25 – Sawtooth wave and random number generated..............................................54
Figure 26 – Geometric features of a spherical particle stuck to a smooth surface [16].....55
Figure 27 – Cone and cylinder modeling...........................................................................59
Figure 28 – Coefficient of drag for a cylinder for vs. Reynolds numbers.........................60
Figure 29 – Large Scale Deposition Removal Comparison...............................................61
Figure 30 – Number of Captured Particles along Axial Chord.........................................63
Figure 31 – Mass % of Captured Particles along Axial Chord..........................................64
Figure 32 – VKI Surface Temperature along Pressure Surface.........................................65
Figure 33 – Sticking Model Comparison of Deposition Mass % locations.......................67
Figure 34 – Low Temperature Comparison between results.............................................68
Figure 35 – Medium Temperature Comparison between results.......................................69
Figure 36 – Time lapse of deposition growth in TuRFR facility.......................................70
Figure 37 – Turbine vane after cool down.........................................................................71
Figure 38 – Turbine vane after cool down with suspended deposition.............................72
Figure 39 – Deposition locations for various particle diameters.......................................73
xii
Figure 40 – Impact and Deposit Mass % locations............................................................76
Figure 41 – Flow visualization of horseshoe vortex for E3 (Vortex magnified) ..............77
Figure 42 – Flow Visualization of horseshoe vortex for E3M (Vortex Magnified) ..........78
Figure 43 – Inlet Mach.......................................................................................................79
Figure 44 – Exit Mach.......................................................................................................79
Figure 45 – Exit flow angle...............................................................................................80
Figure 46 – Impact Mass % Locations..............................................................................81
Figure 47 – Deposition Mass % Locations........................................................................82
xiii
LIST OF TABLES
Table 1 – Experimental Operating Conditions..................................................................19
Table 2 – Simulation Operating Conditions......................................................................20
Table 3 – Particle Forces [37] ...........................................................................................30
Table 4 – Particle Material Properties BYU......................................................................36
Table 5 – Coal fly ash composition (%w.t.) [41] ..............................................................37
Table 6 – Particle size distribution for FLUENT simulation.............................................38
Table 7 – Fitted Values of Coefficients for Compositional Dependence of B[36] ..........50
Table 8 – Coal Ash Chemical Composition used at BYU.................................................51
Table 9 – Viscosity-Temperature relationship constants...................................................52
Table 10 – VKI Sticking Model Comparison Results.......................................................62
Table 11 – E3 Sticking Model Comparison Results..........................................................66
Table 12 – Particle Size Comparison Results....................................................................73
Table 13 – E3 Sticking Model Comparison Results..........................................................81
xiv
LIST OF EQUATIONS
Equation 1 - Mass Conservation Equation..........................................................................9
Equation 2 - Momentum Conservation Equation................................................................9
Equation 3 - Stress Tensor.................................................................................................10
Equation 4 - Energy Conservation Equation.....................................................................10
Equation 5 - Total Energy..................................................................................................10
Equation 6,7 - Transport Equations for Standard k-ω Turbulence Model.........................11
Equation 8 - Non-Dimensional Wall Distance Definition.................................................13
Equation 9 - Non-Dimensional Wall Velocity Definition.................................................13
Equation 10 - Friction Velocity Definition........................................................................13
Equation 11 - Isentropic Pressure Relationship.................................................................14
Equation 12 - Turbulence Intensity....................................................................................15
Equation 13 - Particle Force Balance for Unit Mass.........................................................26
Equation 14 - Spherical Particle Drag Force.....................................................................26
Equation 15 - Summation of Particle Forces.....................................................................27
Equation 16 - Saffman Lift Force......................................................................................28
Equation 17 - General form of Saffman Lift Force...........................................................28
Equation 18 - Deformation Tensor....................................................................................28
Equation 19 - Knudsen Number........................................................................................29
Equation 20 - Mean Free Path...........................................................................................29
Equation 21 - Average Molecule Speed............................................................................29
xv
Equation 22 - Local Flow Velocity....................................................................................32
Equation 23 - Instantaneous Velocity Definition...............................................................32
Equation 24 - RMS Values for Isotropic Flow..................................................................32
Equation 25 - Time Integral...............................................................................................33
Equation 26 - Lagrangian Integral Time............................................................................33
Equation 27 - Time Scale for Eddy....................................................................................33
Equation 28 - Time for Particle to Traverse Eddy.............................................................33
Equation 29 - Rosin-Rammler Distribution.......................................................................39
Equation 30 - Impact Efficiency Definition.......................................................................42
Equation 31 - Sticking Efficiency Definition....................................................................42
Equation 32 - Capture Efficiency Definition.....................................................................42
Equation 33 - Van Der Waals Force..................................................................................44
Equation 34 - Modified Van Der Waals Force..................................................................44
Equation 35 - Critical Impact Velocity..............................................................................45
Equation 36, 37, 38 - Composite Young’s Modulus.........................................................45
Equation 39 - Third-Order Particle Young’s Modulus Relation.......................................46
Equation 40 - First-Order Particle Young’s Modulus Relation.........................................46
Equation 41 - Exponential Particle Young’s Modulus Relation........................................46
Equation 42 - Modified Exponential Particle Young’s Modulus Relation........................46
Equation 43 - Average Temperature..................................................................................46
Equation 44 - Viscosity Sticking Probability.....................................................................47
Equation 45 - Non-Bridging Oxygens to Tetrahedral Oxygens Ratio (NBO/T) ..............49
xvi
Equation 46 - Viscosity Temperature Relationship...........................................................49
Equation 47, 48, 49, 50, 51, 52 - Viscosity Temperature Relationship Coefficient
Equations............................................................................................................................49
Equation 53 - Sticking Probability Exponential Fit...........................................................52
Equation 54 - Sawtooth Function......................................................................................53
Equation 55 - Detachment Model Summation of Moments..............................................56
Equation 56 - Detachment Model Summation of Moments Reduced..............................57
Equation 57 - Attached Spherical Particle Drag................................................................57
Equation 58 - Cunningham Correction Factor...................................................................57
Equation 59 - Stokes Coefficient of Drag Equation..........................................................57
Equation 60 - Modified Attached Spherical Particle Drag................................................58
Equation 61 - Attached Spherical Particle Contact Radius...............................................58
Equation 62 - Composite Young’s Modulus.....................................................................58
Equation 63 - Critical Wall Shear Velocity.......................................................................58
Equation 64 - Large Scale Detachment Moment Balance.................................................60
Equation 65 - Large Scale Sticking Force.........................................................................60
xvii
NOMENCLATURE
A Hamaker Constant
AL Low Temperature Coefficient
AH High Temperature Coefficient
Acp Cross Sectional Area of Particle
BL Low Temperature Coefficient
BH High Temperature Coefficient
CD Coefficient of Drag
CP Heat Capacity at Constant Pressure
Cu Cunningham Correction Factor
CFD Computational Fluid Dynamics
Dp Diameter of particle
DPM Discrete Particle Model
E Total Energy
E Composite Young’s Modulus
Ep Particle Young’s Modulus
Es Surface Young’s Modulus
External Force Vector
FD Drag Force of Particle
Van der Waals Sticking Force
FL Fluid Lift Force
xviii
Gk Generation of Turbulent Kinetic Energy
Gω Generation of ω
I Unit Tensor
I Turbulence Intensity
Diffusion Flow of the Specifies j
Kc Composite Young’s Modulus
Kr Local Velocity Gradient
Knudsen Number
Le Length Scale of Eddy
M Mach Number
P Static Pressure
Po Total Pressure
Ps(T) Probability of Sticking Function
R Specific Gas Constant
ReDH Reynolds Number based on Hydraulic Diameter
Rep Reynolds Number based on Particle Properties
RNG Random Number Generator
S Ratio of Particle Density to Fluid Density
Sh Heat Source
Sk Kinetic Energy Source
Sm Mass Source
Sω ω Source
xix
T Static Temperature
TAVG Average Temperature
TE Time Scale for Eddy
Tg Temperature of Gas
Time Integral
TL Lagrangian Integral Time
Tp Temperature of Particle
Ts Critical Sticking Temperature
TADF Turbine Accelerated Deposition Facility
TuRFR Turbine Reaction Flow Rig
UDF User Defined Function
Vcr Critical Impact Velocity
WA Work of Sticking
Yk Dissipation of k due to Turbulence
Yω Dissipation of ω due to Turbulence
a Random Function Period
Average Molecule Speed
dij Deformation Tensor
dm Mean Particle Diameter
dp Diameter of Particle
f FD Correction Factor for Near Wall
Body Force Vector
xx
h Enthalpy
h Cone Height
hc Height of Cone Center
k Turbulent Kinetic Energy
keff Effective Conductivity
n Spread Parameter
t Time
tcross Time for Particle to Cross Eddy
Velocity Vector
u+ Non-Dimensional Wall Velocity
u* Friction Velocity
Instantaneous Velocity
uc Fluid Velocity at Particle Center
up Particle Velocity
uτc Critical Wall Shear Velocity
y+ Non-Dimensional Wall Distance
z Distance between Particle and Surface
α Relative Approach between Particle and Surface
α2 Volume Fraction of the Dispersed Phase
Ratio of Specific Heats
Normal Distribution Function
Mean Free Path
xxi
μ Molecular Viscosity
μcrit Critical Sticking Viscosity
μTp Viscosity of Particle at Tp
ν Kinematic Viscosity
νp Poisson Ratio for the Particle
νs Poisson Ratio for the Surface
Fluid Density
p Particle Density
Stress Tensor
τx12 Particle Relaxation Time
τk Kolmogorov Time Scale
τt1 Lagrangian Integral Time Scale
τw Wall Shear Stress
ω Specific Dissipation Rate
1
CHAPTER 1: INTRODUCTION
1.1 Background
Traditional turbine engine fuel is becoming less available due to the political and
environmental concerns of oil. To meet with the demand, modern engineers are using
fuels derived from new sources. The direct liquefaction and gasification of coal has
provided an alternative fuel for jet engines and power generators. Coal is widely available
and is inexpensive compared to the highly refined oil based fuels. There are some
drawbacks to using coal derived fuels, however. Coal has negative impacts on the
environment and produces more pollutants during combustion. The products of
combustion contain airborne particulates known as fly ash. This fly ash deposits on the
internal walls of the engine and degrades the system over time. The deposition negatively
affects the power generation of the engine and surface roughness resulting from such
deposition augments the heat transfer of the high temperature fluid to the surfaces.
Removal of the deposits is expensive and time consuming as the afflicted parts need to be
removed for repair. The parts require extensive cleaning to remove the deposition.
Understanding how the particulate adheres to the walls and under what conditions is
crucial to reducing the amount of deposition. Reducing the deposition will reduce the
time spent cleaning the engine and prolong the life of the hardware.
2
1.2 Literature Review
There have been many studies looking into the causes and effects of deposition on
turbine hardware. Small scale deposition can increase the surface roughness. Bons
concluded that increases in surface roughness can decrease turbine performance [8].
Abuaf supported the finding that losses are associated with increased roughness and
found that the heat transfer was increased with additional roughness [1]. Large scale
deposition was found to be more damaging to turbine hardware. Kim et. al. conducted
experiments exploring the effects of volcanic ash on turbine hardware [24]. They found
that deposition can clog film cooling holes and can lead to failure of the turbine. Dunn
et. al. also studied the effects of volcanic ash and found that damage due to deposition
was related to the turbine inlet temperature, concentration of particulate and the material
properties of the volcanic ash [14]. Dunn et. al. confirmed that deposition can clog film
cooling holes and that deposition is a major issue for modern engines with high
combustor temperatures. Dunn et. al. also noted that aircraft engines that ingest particle
laden flow have increased difficulty in restarting and that engines exposed to sufficient
levels of particulate can be damaged beyond repair. Sundaram et. al. studied the effects of
deposition on film cooling and found that film cooling effectiveness degrades as
deposition forms near and in film cooling holes [40]. Lewis et. al. found that the location
of deposition around film cooling holes can affect the heat transfer [28]. Deposition that
forms between and downstream of film cooling holes causes the high temperature free
stream to flow into the deposition valleys and increases heat transfer into the surface. In
the rare cases where deposition forms exclusively upstream of the film cooling holes, a
3
cooling benefit was observed. Lawsen et. al. conducted experiments in a low speed wind
tunnel using low melt wax to investigate the sticking mechanism and formation of
deposits on a flat plate [26]. Jensen et. al. and Crosby et. al. constructed the Turbine
Accelerated Deposition Facility (TADF) and observed the growth of deposits on one inch
coupons[11,21]. This facility was operated at actual engine temperatures and used coal
fly ash for particulate. Smith et. al. constructed the Turbine Reaction Flow Rig (TuRFR)
and tested actual turbine hardware at engine temperatures and velocities. Deposition from
coal fly ash was found to form on the pressure surface and leading edge of the turbine
vanes. The negative effects of deposition have demanded that the mechanisms of
deposition formation be better understood yet the difficulties in operating experiments at
engine operating conditions has supported the need for computer modeling of deposition.
1.3 Other Deposition Research
Numerical models of deposition have been the focus of much research due to the
freedoms of computational fluid dynamic (CFD) research. Hossain et. al. developed a
model to track particles through piping with bends [20]. They found that small diameter
particles were less likely to deposit on the walls because they had a higher tendency to
follow the flow. Larger particles with higher inertia were more likely to deposit
downstream of the bends. Longmire looked into the particle trajectories and developed
corrections for particle dispersion when predicting particle flow near surfaces [29]. El-
Batsh et. al. developed a sticking model to quantify the criteria for particle sticking and
detachment [15]. They tested a 2D turbine vane cascade to study the locations where
4
deposition was most likely. Ai et. al. modified the sticking model developed by El-Batsh
and calibrated the sticking model to match experimental results from the TADF [2]. Tafti
et. al. developed a sticking model using the coal ash composition to determine a sticking
probability based on the viscosity-temperature relationship [41]. The focus of the study
presented in this thesis is using the sticking models developed by El-Batsh et. al., Ai et.
al. and Tafti et. al. on a 3D turbine vane with end walls and determining the effects of end
walls on deposition. This study explores the 3D effects on particle trajectories, as well as
the capture efficiencies for particles of varying size. This study also develops
mechanisms for deposition accumulation and removal for transient studies.
5
CHAPTER 2: FLUID MODELING
2.1 Fluid modeling package
The understanding of deposition that was found from experiments was
incorporated into a numerical model. The modeling process is done in two steps. The first
step is the carrier phase simulation step and the second is the discrete particle modeling
(DPM). The modeling was accomplished using the software package FLUENT 6.3.26.
FLUENT is a commercially available computational fluid dynamics (CFD) solver
program and was chosen for its prevalence in industry. The modeling in this study can be
applied using other CFD models as well. FLUENT solves flow equations for finite-
volumes and is used to study the effects of fluid dynamic forces and heat transfer on
aerodynamic bodies. FLUENT was used at the Ohio Super Computer center (OSC).
FLUENT can be modified for specific applications using User Defined Functions
(UDF). UDF’s can be written to adjust certain parameters of the modeling to meet the
requirements of the simulation. Three UDF’s were used in post processing for this study
and are described later.
2.1.1 Grid generation
Definition of the computational domain and production of the required grid for
use in FLUENT was achieved using a grid generator. The number and shape of these
finite volumes affects the quality and accuracy of the solution. There are several
6
programs used for grid generation with each having their strengths. The grid generator
used for this study was Grid Pro. Grid Pro is a commercially available grid generator that
generates the grid automatically from a user defined topology. Surfaces are provided
analytically or from a CAD model.
Three geometries were used in this study. The first is the VKI turbine vane. This
geometry is two dimensional and is an individual blade from a cascade. The grid is
shown below in Fig. 1.
Figure 1 - VKI Grid
This grid has 5168 cells with the near-wall cells having y+ values around 1.
The second geometry is from the first stage of the General Electric Energy
Efficient Engine (GE-E3) turbine. This geometry is three dimensional and is an individual
vane from the annular stage. The geometry includes the casing and hub end walls.
7
Figure 2 – E3 Annulus (Colored by Radial Position)
Figure 3 – E3 geometry
8
The grid for the individual passage has 568512 cells and is show in Fig. 3. A 2D cross
section of the mid-span is shown in Fig. 4.
Figure 4 – Mid-span of E3 Turbine vane grid
The third geometry in this study is the E3 vane, but with a modified hub end wall (E
3M).
The end wall was modified to study the effects it has on the deposition growth. The end
wall surface is a surface of revolution formed by a hub contour (Generatrix) rotated
around the axis of the machine. This hub contour curve was altered as shown in Fig. 5.
9
(a) (b)
Figure 5 – Modified and unmodified hub end wall contour of the E3 geometry, (a)
Analytical function of hub end wall, (b) resultant modified surface
The inlet angle of the hub was changed approximately 30 degrees and the total inlet area
decreased by about 10%.
2.2 Fluid equations of motion
FLUENT solves the equations of fluid motion to determine the flow properties.
The equations of motion are the conservation of mass and momentum and are known as
the Navier-Stokes equations. The differential forms of the mass and momentum
conservation equations are defined below [18]
[1]
[2]
10
where
[3]
where ρ is the fluid density, is the velocity vector, Sm is mass due to a dispersed second
phase (e.g. vaporization of liquid droplets) in the finite volume, is a body force, and
is the external body force term. also contains other source terms dependent on the
model and user-defined sources. The term is a stress tensor where µ is the molecular
viscosity and I is the unit tensor. The momentum equation is a vector equation and has an
equation for each spatial dimension of the case. These equations are solved for each finite
element and their information is passed on to the next element.
For compressible flow and flows with heat transfer, the energy equation is
included. The energy equation is defined below
[4]
where
[5]
where keff is the effective conductivity, is the diffusion flow of the species j, h is the
enthalpy, and Sh is a source of additional heat in the flow (e.g. exothermic chemical
reactions).
11
2.3 Turbulence Modeling
There are several turbulence models that are used in numerical studies; however
only a few have been used with particle transport and deposition studies. Common
turbulence models are of the k-ε family, where k is the turbulent kinetic energy and ε is
the dissipation rate of kinetic energy. A variation of this model is the k-ω turbulence
model which uses ω, the turbulent frequency defined as ω=ε/k. El-Batsh used the
Standard k-ε and the RNG k-ε with Standard wall functions and two-layer zonal model
for studying deposition [16]. This choice was made due to the limitations of turbulence
models available in the earlier versions of FLUENT. Ai et al. used the standard k-ω with
RANS to determine the flow field and heat transfer in their study of deposition on turbine
coupons [2]. k-ω was chosen to avoid using near-wall functions that could incorrectly
model particle trajectories. Ajersh et. al. found that the k-ω turbulence model gave better
predictions of the near-wall flow structures compared to the k-ε model [4]. Wilcox found
that k-ω also handles non-zero pressure gradients better than the k-ε model [47]. For
these reasons, k-ω was used as the turbulence model. The k-ω model is defined by the
equations:
[6]
[7]
12
where Gk represents the generation of turbulent kinetic energy due to mean velocity
gradients, Gω is the generation of ω, Yk and Yω are the dissipation of k and ω due to
turbulence and Sk and Sω are user-defined source terms.
2.3.1 Near wall flow
Flow near a solid surface will form a boundary layer to separate the free stream
flow from the no-slip velocity of the surface. The flow in the boundary layers has been
studied extensively over the years and there has been much development in predicting the
velocity fields. For turbulent boundary layers the boundary layer can be described as two
distinct regions; the viscous sub-layer and the fully turbulent layer. The limits of these
regions are shown in Fig. 6,
Figure 6 - Universal laws of the wall, Schlichting and Gersten [35]
13
The dimensionless terms, u+ and y+, are defined as:
[8]
[9]
where,
[10]
u* is the friction velocity, ν is kinematic viscosity, and τw is the wall shear stress. Several
numerical functions have been developed to describe the two regions of the turbulent
layer for flow over flat plates, channels and tubes.
2.4 Boundary Conditions
The boundary conditions are set to create the flow conditions in the study. The
inlet boundary condition provides the inlet flow. In the case of the turbine vanes, this will
be the exhaust from the combustor. The outlet boundary condition is the exhaust for the
flow through the domain. The outlet for the turbine vanes will normally be the inlet to the
next stage; however, this study has the flow exiting to atmospheric conditions. The wall
boundary conditions are the geometry of the turbine vane. To reduce the computational
load, periodic boundary conditions were used. The periodic boundaries are set in between
adjacent turbine vanes. The periodic boundary conditions as used enforces that the flow
in the annulus between vanes are identical.
14
2.4.1 Inlet boundary conditions
The inlet boundary was set as a “Pressure Inlet” for this study. The
pressure inlet defines the total and static pressure of the flow at the inlet boundary. The
inlet Mach number of the flow is determined by the relation
[11]
The static pressure at the inlet was set to achieve an inlet Mach number of around 0.1.
This value is typical for the exit Mach of a combustor [5]. The total pressure was set to
the operating conditions of the experiment as described below.
The inlet total temperature was set to match the temperatures experienced in
actual engines. The free stream temperature from experimental work done by Crosby et.
al. has been shown to have a significant effect on the accumulation of deposition [11]. Ai
et. al. calibrated their numerical model to match capture efficiencies for various free
stream temperatures as shown in Fig. 7 [3].
15
Figure 7 – Fitted capture efficiencies obtained from 2-D CFD modeling versus measured
experimental values [3]
The combustion process is highly turbulent and can affect the flow turbulence
level entering the turbine stage. The inlet turbulence intensity was set using the relation
for duct flow[18]:
[12]
where ReDH is the Reynolds numbers based on the hydraulic diameter which is set to the
airfoil chord.
2.4.2 Outlet boundary conditions
The outlet boundary was set as a “pressure outlet”. The pressure outlet specifies
the static pressure at the outlet boundary. The Mach number at the exit for the bulk flow
16
can be approximated using equation 11 with the inlet total pressure, outlet static pressure,
and the specific heat ratio, γ, as 1.36 for the high temperatures.
2.4.3 Wall boundary conditions
The wall boundaries guide the flow through the domain and represent the vane
geometry. The walls are smooth and use the “no-slip” condition for near-wall flow. The
wall thermal properties were tested under two conditions. The first condition was a
constant wall temperature. This temperature was set to 90% of the total inlet temperature.
This model is less computationally sensitive but does not describe the physical conditions
experienced with flows with varying heat transfer. The second condition was to set the
wall heat flux to zero. The adiabatic conditions would allow the wall temperature to vary
along the wall. This condition is computationally more difficult to converge but provides
a more accurate model of the vane surface temperatures. Both of these conditions are for
a non-cooled turbine vane. The adiabatic wall case was found to better approximate the
surface temperatures and is presented in this thesis.
2.4.4 Periodic boundary conditions
The periodic boundary condition was used to model the passage flow as a
repeating pattern of identical flow fields for each vane. This is valid when used for a
single row of vanes with circumferentially uniform (or periodic) upstream boundary
conditions. With this, a single turbine vane was modeled.
17
2.5 Experimental Description
This study modeled the Turbine Reacting Flow Rig (TuRFR) as described by
[10]. Fig. 8 shows a schematic of the TuRFR facility. The facility tests turbine hardware
at temperatures and velocities typical in engines.
Figure 8 – Image of the TuRFR
Air flow is brought in through the combustor where fuel and particulate are added. The
fuel is combusted and the hot gases accelerate through the cone. The flow is stabilized in
the equilibration tube. The transition piece changes from circular to rectangular. The flow
is then sent through a rectangular to annular transition. Two turbine vane doublets are
located at the end of this transition. The exit of the turbine vanes open to the atmosphere.
18
The vanes are held in place in the vane holder as shown in Fig. 9. The vanes can be
supplied with coolant for film cooling.
Figure 9 – Image of vane holder
The operation of the facility is described by Smith et. al.[38].
The turbine vanes used in this study are part of the first stage turbine in the
General Electric CFM-56 engine. The CFM-56 turbine vanes feature cooling holes
located near the leading edge and along the pressure surface. The CFM-56 geometry was
unavailable for CFD simulation so the GE-E3 turbine vane was used instead. Due to this
reason, this study will focus on the trends of deposition rather than the specific deposition
locations on the turbine vanes.
19
2.6 Operating conditions
The TuRFR was operated at three temperatures and one mass flow. The operating
conditions of these tests are shown in table 1.
Table 1 – Experimental Operating Conditions
Case Can
Pressure
[Pa]
Inlet Recovery
Temperature
[K]
Inlet
Mach
Exit Static
Pressure [Pa]
Exit Reynolds
Number
CFM56 -1 206842 1284 0.092 98595 360,500
CFM56 -2 144789 1314 0.098 98595 351,900
CFM56-3 193053 1338 0.093 98595 345,500
The simulations utilized the same operating pressures and mass flow and used three
temperatures as shown in Table 2.
20
Table 2 – Simulation Operating Conditions
Case Inlet Total
Pressure
[Pa]
Inlet Total
Temperature
[K]
Inlet
Mach
Inlet
Turbulence
Intensity %
Exit Static
Pressure
[Pa]
Exit Static
Temperature
[K]
VKI - 1 179263 1283 0.095 4.000 98595 300
VKI - 2 179263 1338 0.095 4.000 98595 300
VKI - 3 179263 1477 0.095 4.000 98595 300
E3 - 1 179263 1283 0.095 4.335 98595 300
E3 - 2 179263 1338 0.095 4.360 98595 300
E3 - 3 179263 1477 0.095 4.418 98595 300
E3M - 1 179263 1283 0.095 4.335 98595 300
E3M - 2 179263 1338 0.095 4.360 98595 300
E3M - 3 179263 1477 0.095 4.419 98595 300
The results of the flow solution and deposition locations will be explored below.
2.7 Flow solution results
Below are the flow properties for each geometry at mid-span. Fig. 10 shows the
Mach distribution. The mach distribution for E3 and E
3 M are very similar despite the
difference in end wall geometry. The static pressure distribution is shown in Fig. 11 and
the static temperature distribution is shown in Fig. 12.
21
(a) (b) (c)
Figure 10 – Mach distribution at mid-span, (a) VKI, (b) E3, (c) E
3M
(a) (b) (c)
Figure 11 – Pressure distribution at mid-span, (a) VKI, (b) E3, (c) E
3M
[Pa]
22
(a) (b) (c)
Figure 12 – Temperature distribution at mid-span, (a) VKI, (b) E3, (c) E
3M
Turbine blade rows are designed so that each row provides the following row the correct
flow angles and magnitudes. Another important parameter is the total pressure loss across
the blade row. The total pressure at the exit plane for the E3 and E
3M cases are shown in
Fig. 13. The E3 and E
3M have very similar exit total pressure maps and the modification
to the end wall does not have a major effect on the over-all performance of the stage. The
mass averaged total pressure at the exit was 166268 Pa and 167281 Pa for the E3 and
E3M respectively.
(a) (b)
Figure 13 – Total Pressure at Exit Plane, (a) E3, (b) E
3M
[K]
[Pa]
H
u
b
W
a
l
l
C
a
s
e
W
a
l
l
Pressure Side
Suction Side
23
CHAPTER 3: PARTICULATE MOTION
3.1 Particle trajectory modeling
The particle trajectory modeling was done using the DPM module in FLUENT.
The DPM uses the Euler-Lagrange approach to track the particle trajectory. The DPM
module allows for flow induced forces on particles to be computed as they determine the
flow path. This method is described below.
3.1.1 Eulerian-Lagrangian model
The Eulerian-Lagrangian model splits the particulate modeling into two phases.
The first phase (Eulerian) is solving the flow solution absent of particulate. The flow
solution phase is treated as a continuum and the Navier-Stokes equations are solved
throughout the fluid domain. The second phase is tracking a large number of dispersed
particles traveling through the flow. The trajectory of each particle is predicted and stored
for analysis. Zhang & Chen found this method to provide accurate results that are easy to
interpret but are computationally expensive [48].
Goesbet et. al. showed in their studies that there can be three types of coupling for
solid phase particles in turbulent flows [19]. The first coupling is One-Way; the
turbulence in the flow affects the particle trajectories and dispersion. The second
coupling is Two-Way and has the particles affecting the turbulence. The third coupling is
24
Four-Way and includes the particles interacting with each other. Fig. 14 shows the
various gas-solid flows where each coupling method is appropriate.
Figure 14 – Classification of flow regimes for gas-solid flows [17]
α2 is the volume fraction of the dispersed phase, τk, is the Kolmogorov time scale in
seconds, τt1 is the Lagrangian Integral time scale in seconds and τ
x12 is the particle
relaxation time.
Typically, the volumetric fraction of particles in an engine is around 0.1 parts per
million by weight (ppmw) [46]. Coal fly ash particulate has a much higher density than
the fluid and the number of particles is low enough for inter-particle collisions to be
neglected. Kulick et. al. and Kaftori et. al. have shown that for volume fractions less than
10E-6 the turbulence modification to particles in the flow is negligible[25]. For these
reasons, One-Way coupling is used in this study and the number of particles injected into
the flow is kept low.
25
The DPM tracking in FLUENT is used for modeling particle trajectories under
One-Way coupling. The particle is tracked from the injection point until it meets with one
of three fates. The first fate is escaped and means the particle exited the domain. For
accurate results, the particle should escape through the outlet boundary. Particles can
escape through the inlet boundary as well but this has been avoided by extending the inlet
boundary far enough upstream to prevent particles bouncing past the inlet.
The second fate is abort and means the particle trajectory has been canceled. This
signifies that the particle was trapped by the flow. The UDF for the sticking models
assigns the abort fate to particles that deposit on the vane surface. Particles that do not
deposit return to the flow and continue their trajectory. Particles that stick can be
removed by the flow as described in Chapter 4 Section 6. Their trajectories were not
computed after leaving the surface.
The third fate is incomplete and means the particle did not reach the other two
fates before reaching the specified number of time steps. This is encountered when
particles are stuck in circulatory flow and cannot escape the domain. FLUENT has a
maximum number of steps a particle can take and the incomplete fate is used to prevent
an infinite loop. For accuracy the number of incomplete particles should be kept less
than 1% of the injected particles.
26
3.1.2 Particle forces
FLUENT predicts the trajectory of a discrete phase particle (or droplet or bubble)
by integrating the force balance on the particle. The particle inertia is balanced with the
forces acting on the particle and can be written per unit mass in the x direction as:
[13]
where u is the fluid phase velocity, up is the particle velocity, μ is the molecular viscosity
of the fluid, ρ is the fluid density, ρp is the particle material density, FD is the drag force
per unit particle mass, and dp is the particle diameter. FD is defined as:
[14]
where CD is the coefficient of drag for the particle. The definition for FD is valid for
Reynolds numbers less than 1. The second term on the right hand side in equation 13 is
the acceleration of fluid around the particle. Fx is an additional acceleration term that
appears under specific situations. These acceleration terms are described in further detail
below.
In this study, the following assumptions were made regarding the interaction
between the fluid phase and the particles. The particles are assumed to be perfect spheres
that are considered as points located at the center of sphere. The various forces acting on
the particle were described by Rudinger as [33]:
27
[15]
These forces have been discussed extensively by El-Batsh et. al. and many others so only
a brief description is provided here.
The drag force is the Stokes drag law and is dependent on the relative velocity of
the particle and the fluid. The Drag force is the most dominant force especially for
particle Reynolds numbers less than 100. The drag force uses the Stokes law for
Reynolds numbers less than 1, the modified Stokes law for 1<Reynolds numbers<500
and Newton’s law for 500<Reynolds numbers<2x105. The range of Reynolds numbers
seen in this study were less than 1for all particles.
The mass effect term is the force of accelerating the fluid around the particle. The
mass that is accelerated is equal to one half of the displaced mass of the fluid. The history
term is the force required to maintain the flow pattern. These effects with the Basset force
are negligible for particle densities much greater than the fluid density.
The Saffman’s lift force is caused by the shear of the surrounding fluid which
causes a non-uniform pressure distribution acting on the particle. If the particle is leading
the fluid motion then the lift force pushes the particle down the velocity gradient towards
the wall. If the particle lags the flow then the lift force moves up the velocity gradient
towards the free stream. This force has been shown to affect the deposition velocity and
Wang et. al. has shown that neglecting this force can reduce the deposition rate slightly
[44]. The Saffman Lift force was originally given by Saffman as [34]:
28
[16]
where Kr is the local velocity gradient. Li and Ahmadi developed a more generalized
version of the Saffman Lift source so that it would apply to three dimensional shear fields
[27]:
[17]
where S is the ratio of particle density to fluid density and K=2.594 is a constant
coefficient. dij is the deformation tensor and is given as:
[18]
The buoyancy force is important for large particles with large changes of position.
Since the particles in this study are small and the domain is small in size, the gravitational
force is expected to be negligible. The buoyancy force is usually negligible for particle
deposition studies and is neglected for this study.
The Magnus force is the lift developed due to rotation of the particle. The lift is
caused by the pressure difference that occurs with the difference in flow on opposite sides
of the sphere. Kallio & Reeks found that this force is an order of magnitude less than the
Saffman lift forces in most cases and could be neglected [23].
The Brownian and Thermophoretic forces are important for sub-micron particles.
The Brownian force is caused by random interactions with agitated gas molecules in the
fluid. The thermophoretic force is caused by the unequal momentum exchange between
29
particle and fluid. Temperature gradients in the flow can cause the momentum exchange
to be higher in the direction of decreasing temperature. Both of these forces are neglected
since the particles used in the study are greater than 0.03 µm.
Another force that can be important is the rarefaction force. The rarefaction force
is the influence of gas particles colliding with the coal ash particle in a non-continuum
regime. The Knudsen number, KN, is the ratio of the mean free path of the gas molecule
and coal ash particle size:
[19]
where λ is the mean free path of gas particles which is defined as:
[20]
[21]
where R is the specific gas constant.
30
Table 3 – Particle Forces [37]
FORCE Domain of Importance Included
1 Drag Dominant force for particle motion; Rep<100, spherical
particles assumed. Stokes' law: Rep < 1; Modified Stokes'
Law: 1<Rep < 500; Newton's Law: 500< Rep< 2 x 105
YES
1<Rep < 500
2 Rarefaction
Effect
Important for sub-micron particles (<1µm) for non-continuum
regime; Continuum: Kn < 0.1; Transition: 0.1 < Kn < 10
Free-molecule: Kn > 10. Important for Kn>0.02
NO
0.1<Kn<10
3 Virtual Mass
Effect
Important for small values of particle material density to gas
density; ρ p / ρ f << 1
NO
ρ p / ρ f >> 1
4 Basset Important for small values of particle material density to gas
density; ρ p / ρ f << 1
NO
ρ p / ρ f >> 1
5 Saffman Lift Non-trivial magnitudes only in the viscous sub-layer; slight
decrease in deposition rate if neglected.
YES
More accurate deposition rate
6 Magnus Lift force due to particle rotation; at least an order of
magnitude lower than Saffman force in most regions;
NO
No particle rotation in the flow
7 Gravity Important in Stokes Regime for large particles NO
Very small particles
8 Thermophoretic Important for sub-micron particles and Kn < 2 NO
Kn > 10
9 Brownian Important for dp < 0.03 µm NO
dp > 0.03 µm
10 Intercollision Important for high-volume fraction of particles NO
Low volume fraction of particles
31
3.2 Injection Properties
The particles in this study were injected at specified locations across the inlet
boundary as shown in Fig. 15. There are 900 injection points located at 30 radial
positions and 30 angular positions. The injection points are grouped into three groups.
Group A is located near the hub end wall, group C is located near the case end wall and
group B is located in between A and C.
Figure 15 – Particle injection locations
Particles were assumed to be in thermal equilibrium with the flow at the inlet and were
injected at the local flow velocity and direction. Each injection point injects one particle
at a time.
32
3.3 Random walk model
The dispersion of particles can be modeled using the stochastic tracking, also
known as random walk, in FLUENT. FLUENT predicts the trajectory of the particles
using the mean velocity parameter, [18]. For turbulent flows, the instantaneous random
velocity, , is used to model dispersion. The local flow velocity is determined by the
relation
[22]
The random effects of turbulence are based on the varying nature of .
The method for determining is based on the random walk model. The model
assumes small scale eddies exist in the flow. These eddies are random and have
instantaneous flow velocities determined by the following relation
[23]
where ζ is a normally distributed random number and the second term is the RMS
fluctuating component. Assuming isotropy the RMS values are defined as
[24]
The time scale of the eddy is used to determine when the instantaneous velocity is
calculated. The integral time, TI, is found using the relation
33
[25]
Infinitely small particles with zero drift velocity will have the integral time become the
fluid Lagrangian integral time, TL. TL is approximated using the relation
[26]
where CL is an empirically determined constant and ω is the specific dissipation rate from
the k-ω turbulence model. CL is set to 0.30 for k-ε and k-ω turbulence models. The time
scale of the eddy, TE, is defined as
TE = 2 TL [27]
The time for the particle to cross the eddy is defined as
[28]
where τ is the particle relaxation time and LE is the eddy length scale. If tcross is shorter
than the lifetime of the eddy, then the particle has left the current eddy into a new one and
is recalculated. If TE passes before the particle has crossed the eddy, the eddy decays.
A new eddy is formed and the instantaneous velocity is recalculated.
34
3.3.1 Perturbation velocity corrections
The assumption of the RMS fluctuating components being isotropic is accurate
for free stream flow. However, near the wall, the RMS values become highly anisoptric.
FLUENT was found to over-predict the wall normal perturbation velocities, v’+
, by more
than an order of magnitude [13]. Fig. 16 shows the magnitudes of the perturbation
velocities near the wall.
Figure 16 –RMS values of velocity in the boundary layer [13]
During preliminary simulations, particles were prone to being trapped in strong eddies
near the leading edge of the turbine vane. These particles would have incomplete
trajectories and would lead to inaccurate particle impact locations.
Attempts to correct this in FLUENT have been suggested with some success.
Dehbi replaces the calculation for u’, v’, and w’ with functions using Y+ values [13].
These functions are based on channel flow and lead to some improvement in the
35
accuracy. Longmire used corrections based on wall functions curve fitted to discrete
Navier-Stokes results [29].
For this study channel flow is too simplistic to model the complex geometries in a
turbine passage. To reduce the error associated with the invalid wall-normal
perturbations, the instantaneous velocities were set to zero for the regions where Y+
values are less than 200. This means the particles will use the mean flow velocities in
equation 13 to determine the particle trajectory in this region. Particles outside of this
limit will have the fluctuating velocity values calculated based on the isotropic
assumption. The Y+ value of 200 was chosen by Longmire as the limit where the error
would still be negligible [29]. A study was performed using a Y+ value of 100 as the limit
and there were no changes in the results. For this study, the interest is in the general
location of deposition and the deviations to the particle trajectories near the wall are
ignored for Y+ values less than 200. The deviations in the free stream are more important
for determining the directions the particle will travel.
3.4 Particle material properties
The material properties of the particle are important in determining how the
particle responds to changes in the flow. The density of the coal fly ash, ρ, the heat
capacity at constant pressure, CP, and the thermal conduction value, K, are used to
determine the thermal diffusivity which is a measure of how quickly the particle changes
temperature with the flow. The material properties were not known for the fly ash used in
36
the TuRFR experiment so the material properties with a similar coal ash particle from
Brigham Young were used [2].
Table 4 – Particle Material Properties BYU [2]
ρ (kg/m3) CP (J/kg-K) K(W/m-K)
990 984 0.5
3.4.1 Ash composition
The chemical composition for the coal fly ash is dependent on the type of coal
used in the combustion process. Coal is found in many geographical regions and has
slight variations in chemical composition. Table 5 shows the chemical properties from
different coal deposits.
37
Table 5 – Coal fly ash composition (%w.t.) [41]
Silicon, Aluminum, and Iron are the common elements in coal ash. The TuRFR
experiment uses the ash from south east (SE) Ohio. Notice that the ash is high in iron
content compared to all other ash types.
3.4.2 Size distribution
The size of the coal ash particle is important in determining the types of forces
that act on its trajectory in a flow. Advanced filters are commonly used in modern
turbines to remove particulate above 10 µm in diameter. The particulate size distribution
for the TuRFR facility is shown in Fig. 17.
Lethabo HN115 KL1 HNP01 WV UF WY ILL Pitt ND SE Ohio
SiO2 60.33 42.3 47.1 50.1 60.43 46.86 35.59 46.62 50.37 23.68 32.9
Al2O3 29.73 34.5 35.3 32.9 30.69 23.24 15.15 14.41 21.04 7.94 20.3
Fe2O3 2.68 6.17 4.72 8.42 3.03 21.02 7.53 26.80 21.23 9.82 40.6
TiO2 1.37 2.24 1.96 1.51 2.02 1.00 1.40 0.73 1.14 0.47 -
P2O5 0.41 0.55 0.19 0.34 0.69 0.69 3.02 1.00 0.90 3.85 -
CaO 3.58 8.55 5.67 1.37 0.42 1.68 18.92 2.74 1.37 18.43 2.93
MgO 1.19 1.00 0.75 0.62 0.67 1.00 4.76 0.72 0.65 7.44 -
Na2O 0.14 0.21 0.26 0.45 0.27 0.27 2.10 0.88 0.53 10.20 -
K2O 0.45 0.76 1.33 2.17 0.97 2.65 1.01 3.15 2.00 1.35 2.48
S 0.10 0.04 0.03 0.01 0.81 1.60 10.53 2.94 0.78 16.82 -
MnO 0.02 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 -
SO3 - - - - - - - - - - 0.83
38
Figure 17 – Particle size distribution results from Coulter counter for Bituminous coal fly
ash [38]
Nearly half of the particles in the TuRFR facility would normally be filtered in modern
turbine engines. Since the TuRFR results are being simulated in this study, the study will
include the larger size particles as well. The distribution of particles used in the
simulation is shown in table 6.
Table 6 – Particle size distribution for FLUENT simulation
Particle Diameter (µm) 1 10 20 30 40 50
Mass % 15 30 25 20 15 10
The logarithmic form of the Rosin-Rammler distribution equation was used to model the
size distribution. The Rosin Rammler distribution equation is of the form [18]:
0.00%
1.00%
2.00%
3.00%
4.00%
5.00%
6.00%
0 20 40 60 80 100
Ma
ss
%
Diameter (µm)
Mean = 14.117 µmMedian = 12.038 µmSize (µm) Mass %0-5 23.16%5-10 31.15%10-20 25.81%20-40 14.82%40-60 3.82%60-100 1.23%
39
[29]
where dp is the diameter of the particle, dm is the mean diameter and n is the spread
parameter. Yd is the percentage of particles with diameters greater than dp and the
percentage is assumed to be exponential. For the size distribution used in the TuRFR
facility the Rossin-Rammler function was fitted to the experimental data using dm = 12
and n= 2.78. The Rossin-Rammler fit is shown in Fig. 18 with the experimental size
distribution.
Figure 18 – Size Distribution Modeling
0
5
10
15
20
25
30
35
0 20 40 60 80 100
Mas
s %
Particle Diameter (μm)
Experiment Rosin Rammler
40
3.5 Particle Trajectories
Particle diameter has a major effect on the trajectory the particle will take.
Particles with larger diameters have more inertia due to their increased mass. Smaller
particles have less mass and tend to be more susceptible to forces acting on them from the
flow. Fig. 19-21 show the trajectories for particles with diameters of 1, 20, 50 μm.
Particles enter from the inlet on the left and will exit at the outlet to the right. Particles
that pass through the upper/lower bounds will reappear at the opposite side with the same
trajectory.
Figure 19 – 1 μm diameter particle trajectory
41
Figure 20 – 20 μm diameter particle trajectory
Figure 21 – 50 μm diameter particle trajectory
1 μm particles follow the flow closely and tend to not contact the surface. Larger
diameter particles will likely strike the surface of the vane and depending on the
conditions will reflect multiple times before exiting the domain. Particles bouncing
between vanes will increase the number of impacts considerably.
42
CHAPTER 4: STICKING MODELING
4.1 Impact, sticking, capture efficiency definitions
When modeling deposition, it is useful to define three parameters to measure the
performance of the simulations. The impact, sticking and capture efficiencies are defined
below:
[30]
[31]
[32]
The impact efficiency is useful for measuring how likely particles are to hit the surface.
Small particles are more likely to follow the flow and less likely to hit the surface. Larger
particles will have more inertia and will hit the surfaces more. The impact efficiency will
be higher for larger particles than smaller particles if the same number of both particles is
injected. The sticking efficiency shows how likely particles stick to the surface when the
particles contact the surface. The capture efficiency shows how much particulate mass
deposits on the surface. Based on these definitions, impact efficiency is equal to the
sticking efficiency and capture efficiency combined.
43
4.2 Wall-particle interaction
There are several forces that help a particle stick to a surface. The most common
sticking forces are the van der Waals force, electrostatic force and liquid bridging. The
electrostatic force is the attraction of charges that build up on the particle and the a
surface. The van der Waals force is the collection of forces that attract molecules together
that do not include the electrostatic force. The liquid bridge is the force of a semi-molten
particle sticking to a surface. Fig. 22 shows the relation of these forces for various
particle diameters.
Figure 22 – dimensionless sticking forces versus particle diameter [6]
It can be seen that smaller particles will have a stronger sticking force compared to larger
particles. For temperatures below the molten temperature of the particulate, the van der
44
Waals force is the dominant sticking force. The other two forces are neglected for this
study.
The van der Waals force can be expressed in the form:
[33]
where A is the Hamaker constant and z is the distance between the particle and surface.
For particles that contact the surface, the particle deforms and the equation for Fpo
becomes:
[34]
where ka is a constant equal to 3π/4 and WA is the work of sticking that is dependent on
the material [15]. For silicon, WA is 0.039 J and is the value that is used in this study.
4.3 Critical impact velocity model
The critical impact velocity is a model that uses the impact velocity component
normal to the surface to determine if the particle sticks to the surface. The critical impact
velocity is used as criterion for determining whether the particle sticks or reflects and is
affected by various properties of the flow and particle. The critical velocity in this study
is given as [9]:
45
[35]
[36]
[37]
[38]
where Vcr is the critical velocity, E is the composite Young’s modulus, Es is the surface
Young’s modulus, EP is the Young’s modulus of the particle, νs is the surface poisson
ratio and νP is the particle poisson ratio. Particles that impact the surface with a normal
velocity below the critical velocity will deposit. Those particles that impact with
velocities above the critical impact velocity will reflect off the surface and continue their
trajectory until they leave the domain or deposit on the surface further downstream.
4.3.1 Particle impact properties
The Young’s modulus for the particle is difficult to measure experimentally due
to the high temperatures the particles are in. The Young’s modulus is sensitive to the
temperature of the particle and can vary drastically at high temperatures. The material
properties of the particle can also affect the Young’s modulus. Due to the complications
of measuring the Young’s modulus for the particle, it is approximated using empirically
derived functions. El-Batsh used the Young’s modulus as a value that can be determined
empirically from experimental deposition results. El-Batsh developed a function of the
46
Young’s modulus to match his predicted capture efficiencies to experimental results. The
function that El-Batsh used was [15]:
for Tp>1100 K [39]
for Tp>1300 K [40]
Ai developed a function for the particle Young’s modulus based on coupon studies[2].
The function Ai used was [3]:
[41]
where Tg is the free stream gas temperature above the surface. Ai & Fletcher made
modifications to the function to include the surface temperature effects. The new
correlation is[3]:
[42]
[43]
Another modification was made in this study to the function developed by Ai & Fletcher.
The temperature near the surface varies due to the thermal boundary layer. For flows
where the surface is at a lower temperature than the free stream, the flow right above the
surface will be lower temperature than the free stream. Depending on the thermal
properties of the particle, the particle temperature may be at a higher temperature than the
flow near the surface. For this study, TAVG was modified by replacing the Tg in equation
43 with the particle temperature Tp.
47
4.4 Critical viscosity model
One drawback of the critical velocity model is the difficulty in calibrating the
Young’s modulus. Changes in coal ash properties require separate calibrations. Tafti et.
al. developed a sticking model based on the particle viscosity [41]. The particle viscosity
changes with temperature and the relationship can be predicted based on the coal ash
properties.
Tafti et. al. based their model on the probability of the particle sticking. The ash
softening temperature is the critical sticking temperature, TS. Particles above this
temperature will have a sticking probability of one. Particles with temperature much less
than the critical sticking temperature will have a sticking probability of zero. For particles
with temperatures in-between, the probability function is found using [41]:
[44]
where µcrit is the viscosity of the particle at the critical sticking temperature and µTp is the
viscosity of the particle at the current particle temperature. The variation of the
probability function with temperature is shown in Fig. 23.
48
Figure 23 – Transition of modified deposition model using critical viscosity approach
[41]
4.4.1 Ash composition model
Senior et. al. developed a method to determine the coal ash particle viscosity
based on the coal ash chemical composition. Silicate melts are made up of the building
block SiO44-
and can form networks that compose the particle glass structure. There are
three categories of cations and each interacts with the network in different ways. The first
category is known as the glass formers which form the basic anionic polymer unit. Si4+
,
Ti4+
, and P5+
are glass formers. The second category is the modifiers, which disrupt the
polymer chains bonding oxygen. Ca2+
, Mg2+
, Fe2+
, K+ and Na
+ are modifiers and are
effective at terminating chains in the structure. The third category is the amphoterics,
which act either as formers or modifiers. Al3+
, Fe3+
, B3+
are amphoterics. Modifier ions
disrupt the glass structure and tend to lower the viscosity. Amphoterics interact with the
modifiers and can form bonds to balance their charge. If there is an abundance of
amphorterics, they can act as modifier ions as well. The parameter that relates the balance
49
of amphoterics and modifiers is given by the ratio of non-bridging oxygens to tetrahedral
oxygens (NBO/T) given by Vargas[43]:
[45]
4.4.2 Viscosity versus temperature relationship
N’Dala et. al. have shown that the temperature dependence of viscosity of silicate
and aluminosilicate melts can be described by[30]:
[46]
where A and B are constants which depend on the chemical composition. Senior
conducted experimental tests to develop a curve fit using A, B and NBO/T. The model
developed by Senior is split into two temperature regions with AL being A for low
temperatures and AH being A for high temperatures. The functions for A and B developed
by Senior et. al. are [36]:
[47]
NBO/T≥1.3 [48]
0.2≤NBO/T<1.3 [49]
0.0≤NBO/T<0.2 [50]
NBO/T<0.0 [51]
[52]
50
where α is the ratio of mole fraction network modifiers to the sum of network modifiers
and amphoterics, and N is the mole fraction of SiO2 in the glass. The coefficient bi is
shown in table 7.
Table 7 – Fitted Values of Coefficients for Compositional Dependence of B[36]
High Temperature Low Temperature
b0 -224.98 -7563.46
b1 636.67 24431.69
b2 -418.70 -17685.4
b3 823.89 32644.26
b4 -2398.32 -103681.0
b5 1650.56 74541.33
b6 -957.94 -46484.8
b7 3366.61 146008.4
b8 -2551.71 -104306.0
b9 387.32 21904.63
b10 -1722.24 -68194.8
b11 1432.08 48429.31
The coefficients A and B are used in equation 46 to determine the viscosity at the particle
temperature. There will be a viscosity based on the low temperature coefficients and the
high temperature coefficients. The higher of these two viscosities is the value used.
51
4.4.3 Relationship limits
One limitation of the method developed by Senior et. al. is that the model does
not work with ash containing high amounts of iron or no modifier ions. Nowok looked
into finding a correction that would describe the chemical nature of iron in the ash
composition and how to predict the viscosity from the iron content [31]. The limitation of
this correction is knowing the ratio between the Fe3+
and Fe2+
ions at temperature. Due to
the complexities associated with the high iron content, the viscosity-temperature
relationship could not be determined for the ash used in the TuRFR experiment. Instead,
the ash composition from the BYU facility was used due to its lower iron content and the
fact that the sticking model had been calibrated to use the ash [3].
Table 8 – Coal Ash Chemical Composition used at BYU[3]
Mass
Mean
Diameter
Bulk
Density
Apparent
Density
Element Weight %
Na Mg Al Si P S K Ca Ti V Fe Ni
13.4 µm 0.99 g/cc 1.98 g/cc 6.9 3.6 18 47 1.6 1.8 2.6 8.7 1.6 0 6.4 0
For this ash composition, the constants in equation 46 were determined using the method
described by Senior et. al. The viscosity-temperature relationship for the coal fly ash at
BYU is shown in Fig. 24 with fly ash used by Tafti et. al.
52
Table 9 – Viscosity-Temperature relationship constants
A B
High Temperature -11.3059 15.9561
Low Temperature -44.2546 49.8791
Figure 24 – Viscosity – Temperature relationship for various ash types [41]
If the ash composition is unavailable or if there is sufficient experimental data, the
viscosity model can be used by calibrating the constants A and B, and the critical sticking
temperature. Tafti et. al. used this method for studying PVC particles in a heated flow
[41]. The sticking probability used for ash can not be applied to PVC but PVC is assumed
to have the same exponential trend for the viscosity and temperature relationship.
[53]
1.E+00
1.E+01
1.E+02
1.E+03
1.E+04
1.E+05
1.E+06
1.E+07
1.E+08
1.E+09
1250 1300 1350 1400 1450 1500 1550 1600
Vis
cosi
ty (
Pa-
s)
Temperature (K)
BYU Ash
Lethabo
Hn115
KL1
HNPO1
WV
UF
WY
ILL
Pitt
53
The sticking probability at zero Kelvin, PS(T0) is set to have PS equal to 1.00E-6 and at
86% of the melting temperature of the PVC. k is set to have Ps(TMelt) equal to 1.0 at the
melting temperature. This calibration can be done with any particle material as long as
the melting temperature of the material is known.
4.4.4 Critical viscosity model sticking criteria
With the viscosity-temperature relationship the probability of sticking can be
determined by specifying the critical sticking temperature. The softening temperature for
most ash particles was found to be around 1478 K so 1500 K was used as the critical
sticking temperature in this study[42]. Particles with a temperature above this critical
temperature were assumed to always stick.
To determine if a particle sticks for temperatures below the critical sticking
temperature a random number generator (RNG) was used to provide a number between 0
and 1. There are several types of RNG’s used in literature. The randomness of these
RNG’s is dependent on a seed variable being different each time the function is used. A
modified sawtooth wave was used with a flow variable as a seed. The form of the
modified sawtooth wave is given as:
[54]
54
where X is the random number, t is the seed variable and a is the period of the function. a
can be adjusted so that small changes in the seed variable will lead to a nearly random set
of numbers between 0 and 1.
Figure 25 – Sawtooth wave and random number generated
Fig. 25 shows the sawtooth wave with a linearly changing seed and a random number
based on a seed derived from the local flow properties.
When a particle impacts a surface, the viscosity of the particle is determined
based on the particle temperature and the probability of sticking is determined based on
the critical sticking viscosity. The RNG is used to provide a number between 0 and 1. If
the random number generated is greater than the probability of sticking, the particle
reflects off the surface and returns into the flow. If the random number is less than the
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.5 1 1.5 2 2.5 3 3.5 4
X(t
)
Seed
Sawtooth wave Random number
55
probability of sticking, the particle sticks to the surface. Higher particle temperatures will
have higher probabilities of sticking.
4.5 Pre-Existing-Deposition factor
Experiments have shown (see section 4.8) that deposits are slow to grow initially
but grow rapidly over time. Initial deposits act as collectors for future particles to stick.
The Pre-Existing-Deposition (PED) factor is a factor that increases the likelihood of the
particle sticking in the simulation. For the critical velocity model, the critical velocity
was increased by a factor of 10 on cells with pre-existing deposition and for the critical
viscosity model the viscosity of the particle was increased by a factor of 1.1. These
factors were chosen to increase the likelihood of a particle depositing but have not been
calibrated. The result of the PED factor is an increase of deposition near the leading edge
and a decrease near the trailing edge. Particles that would reflect off the leading edge
without the PED factor were more likely to be captured using the PED factor.
4.6 Detachment Model
Deposits can detach from the surface if the shear forces are sufficient to overcome
the sticking force. Many particles stick temporarily before being dislodged from the
surface. The amount of deposit that is on the blade is determined by the balance of
particles sticking and particles detaching.
56
4.6.1 Individual particle detachment
Soltani & Ahmadi studied different mechanisms for particle detachment and
found that for spherical particles they are detached predominantly by rolling [39]. Sliding
and lifting are less important. Das et. al. also found that rolling was the more dominant
removal mechanism [12]. El-Batsh used this reasoning to develop a model for particle
detachment [16]. El-Batsh goes into deep detail about this mechanism so a brief summary
is described below.
Since the primary removal mechanism is the particle rolling off the surface, the
model for determining if a particle detaches is the sum of moments acting on the
spherical particle.
Figure 26 – Geometric features of a spherical particle stuck to a smooth surface [16]
[55]
57
where α is the relative approach between the particle and the surface, FL is the fluid lift
force acting on the particle, and a is the radius of the contact surface. α is small for most
cases and is much smaller than Dp/2. FL is also small compared to the other forces.
Equation 55 becomes:
[56]
For spherical particles and Reynolds number less than 1 FD can be written as:
[57]
where uc is the fluid velocity at the center of the particle, Acp is the cross sectional area of
the particle, f is the correction factor for the particle being near the wall as provided by
Soltani & Ahmadi, and Cu is the Cunningham correction factor which is defined as [39]:
[58]
Using f=1.7 and Acp = (π/4)Dp2 and Stokes drag for CD, FD becomes:
[59]
[60]
58
The contact radius, a, can be found using the relation developed by Soltani & Ahmadi
and Rimai et. al. [32,39]:
[61]
where,
[62]
Kc is the composite Young’s modulus and combines the material properties of the particle
with those of the surface. Using the sticking force, Fpo, from equation 34, equation 56 can
be rewritten to form the critical wall shear velocity uτc:
[63]
If the wall friction velocity is greater than uτc the particle detaches from the surface.
The critical viscosity model proposed by Tafti et. al. does not include a
detachment mechanism. For this study, the detachment model from El-Batsh was
incorporated with the viscosity model to form a hybrid model. The detachment
mechanism is important for determining deposition growth and should be included for all
modeling.
59
4.6.2 Large scale deposit detachment
Experiments (cf. section 4.8) have shown that some large scale deposits are pulled
off the surface. It is believed these deposits are removed by the additional pressure drag
that exists by their large size (Reynolds number 1). A method was developed to
simulate this removal mechanism by modeling the deposits as right circular cones.
Referring to Fig. 27 the deposit accumulation was modeled as a cone for which h is the
total height of the cone, hC is the height to the center of the cone, and B is the area of the
cone base.
Figure 27 – Cone and cylinder modeling
The volume of the cone is based on the amount of particulate mass and particulate
density on a surface face. The base area of the cone is set to the surface area of the cell
face. The height of the cone is determined by the volume of the cone and the base area.
As more particulate mass is added to the surface face, the cone grows higher from the
surface. The drag on the cone is modeled by representing the cone as a series of cylinders
with variable radius. The coefficient of drag was determined for each cylinder based on
h
Fpo
FD
dh
hC
B
60
the Reynolds number for the cylinder. The relation between Reynolds number and CD is
shown in Fig. 28.
Figure 28 – Coefficient of drag for an infinite cylinder vs. Reynolds numbers [45]
The drag on each cylinder was then integrated for the cone to determine the pressure
drag. The same methodology for the detachment of a single particle was applied to the
entire cone. Equation 56 would become:
[64]
where Fpo is now,
[65]
If the sum of moments due to the pressure drag exceeds the moment due to the sticking
force, the cone will be removed. If the sticking force prevailed, then the cone would
remain. This model does not take into account deposits merging together from adjacent
61
cells. This model is a first step at predicting the complex removal mechanisms that occur
with large scale deposits and neglects the complex geometries of deposits that form in
real life. The effects of this model are shown below.
(a) (b)
Figure 29 – Large Scale Deposition Removal Comparison, (a) No Cone Removal, (b)
Cone Removal
Tall deposits with small bases are likely to be removed by the addition of pressure drag
acting on the deposit. The results from Fig. 29 show that the large scale deposits would
detach. Deposits that remain are likely to merge together and increase their likelihood of
remaining attached.
4.7 Comparison between sticking models
The major difference between the sticking models is that the critical impact
velocity model accounts for the particle velocity as well as the particle temperature. The
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critical viscosity model is more simplistic by being dependent entirely on the particle
temperature. For flows with wide ranges in velocity magnitudes the sticking models are
expected to produce different results. The critical impact velocity and critical viscosity
sticking models were compared for the VKI and the E3 turbine vanes.
For the VKI case, particles were injected at the inlet boundary with the local
temperature and velocity. The sticking models were compared for the high temperature
VKI case as shown in table 10. 0.0001 kg of particulate was injected with 5 particle
diameters as shown in table 2. The particles either passed through the domain or
deposited on the vane. The injection results for both sticking models are shown in table
10.
Table 10 - VKI Sticking Model Comparison Results
Critical Velocity Critical Viscosity
Total Injected # 600 600
Aborted # 216 224
Incomplete # 0 0
Escaped # 384 376
Impact Efficiency % 77 70
Sticking Efficiency % 75 35
Capture Efficiency % 58 45
Table 10 shows that there are a large number of impacts. This is likely due to large
particles reflecting multiple times before depositing or escaping. The critical viscosity
63
model shows that more of the injected particles deposited on the surface; however the
capture efficiency is lower. The lower capture efficiency shows that smaller particles
were more likely to be captured. Fig. 30 shows the number of particles, regardless of size,
that were captured along the pressure surface of the vane at midspan. Notice that the
viscosity model predicts more captures near the leading edge while the velocity model
predicts more deposits near the trailing edge.
Figure 30- Number of Captured Particles along Axial Chord
The low capture efficiency near the leading edge is due to the particles impacting the
leading edge with high normal velocities. Although the viscosity model shows more
captured particles near the leading edge, the percentage of mass depositing near the
leading edge is similar for both sticking models. Fig. 31 shows the percentage of injected
mass that deposited on the surface.
0
5
10
15
20
25
30
0 0.2 0.4 0.6 0.8 1
#
Axial Distance/Axial Chord
X vs Deposit #
Viscous model Velocity Model
64
Figure 31 – Mass % of Captured Particles along Axial Chord
Fig. 31 shows higher amounts of mass that deposited near the trailing edge which is due
to large particles favoring the trailing edge. The differences in the models can be
explained by the sticking criteria. The temperature near the trailing edge drops
significantly. The viscosity model is more sensitive to changes in temperature so a
decrease in captures is expected.
012345678
0 0.2 0.4 0.6 0.8 1
Inje
cte
d M
ass
%
Axial Distance/Axial Chord
X vs Deposit Mass %
Viscous model Velocity Model
65
Figure 32 – VKI Surface Temperature along Pressure Surface
Fig. 21 shows that larger particles tend to bounce between the adjacent vanes and there
are a high number of impacts near the trailing edge. The particles lose some momentum
due to the drag forces resisting cross-streamline flow. Reflecting particles that impact the
trailing edge have a lower normal velocity and are more likely to deposit according to the
velocity sticking model.
For the E3 case, particles were injected at the locations shown in Fig. 15 at the
local temperature and velocity. The particle mass flow was again 0.0001 and was
distributed among 5 particle diameters given by table 2. The E3 high temperature
adiabatic wall case was used for the comparison. The injection results are given in table
11.
1450
1455
1460
1465
1470
1475
1480
0 0.2 0.4 0.6 0.8 1 1.2
Surf
ace
Te
mp
era
ture
(K
)
Axial Distance/Axial Chord
66
Table 11 – E3 Sticking Model Comparison Results
Critical Velocity Critical Viscosity
Total Injected # 22525 20115
Aborted # 9283 7197
Incomplete # 77 52
Escaped # 13165 12866
Impact Efficiency % 63 69
Sticking Efficiency % 92 49
Capture Efficiency % 58 34
The critical velocity and critical viscosity sticking models differed significantly for this
test capture and sticking efficiencies although the impacts were similar. The difference is
likely due to the calibration that was performed by Ai et. al. for the critical velocity
model. Ai et. al. calibrated the model for temperatures ranging from 1300 K to 1450 K.
The critical viscosity model was created using the known properties of coal ash
particulate and no calibration was performed. The simplicity of the viscosity model
requires that the constants A, B and the critical sticking temperature be calibrated using
experimental results. Using the ash composition model is a good initial start for the
values of A and B. The critical sticking temperature can also have a significant effect on
the amount of deposition at a given temperature. The critical sticking temperature of the
coal ash used in the simulation was not known and likely led to the discrepancy between
the models. Fig. 33 shows the deposition locations for the two models.
67
(a) (b)
Figure 33 – Sticking Model Comparison of Deposition Mass % locations, (a) Critical
Velocity, (b) Critical Viscosity
The deposition locations for the two models show a majority of the deposition being on
the pressure surface near the trailing edge. The viscosity model predicts lower overall
deposition. Due to time constraints, the critical impact velocity sticking model will be
used for studying the end-wall modification in Chapter 5.
4.8 Experimental comparison
The simulations results were compared with the TuRFR facility operating at the
conditions listed in table 1. The amount of deposition on the turbine vanes in the
experiment was very difficult to quantify due to the frailty of the deposition.
Measurements were limited to optical scans of the topography. This work is ongoing and
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as of this study is unavailable. The results with the simulations will be compared
qualitatively with the experiment to study the deposition locations and relative amounts.
The low temperature case is compared in Fig. 34. The low temperature was found
to be at the lower limit of deposition growth in the facility.
(a) (b)
Figure 34 – Low Temperature Comparison between results, (a) Simulation, (b) TuRFR
Experiment
The deposition in the simulation was less than 0.1% of the injected mass. The experiment
shows a light dusting of particulate but no major deposits.
The medium temperature case is compared in Fig. 35. This experimental test
shows a great amount of deposition along the leading edge and pressure side.
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(a) (b)
Figure 35 – Medium Temperature Comparison between results, (a) Simulation, (b)
TuRFR Experiment
The simulation shows increased amounts of deposition compared to the low temperature
case. The simulation also predicts a significant amount of deposition near the trailing
edge which is in contrast to the experimental results. There are multiple explanations for
this discrepancy. One limitation of the current modeling is the deposition growth. The
simulation predicts the particulate trajectories at the time of injection. Transient results
are not modeled. The cone removal mechanism is the first step in exploring the time after
the initial injection of particulate. The E3 geometry in this study does not have film
cooling holes which serve as deposition sites in the experiment.
A time lapse video of the formation of deposition was taken with the TuRFR
facility for a test at 1310 K with no film cooling. The video shows the film cooling holes
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acting as the starting locations of deposits and the growth of the deposition. Fig. 36
shows the growth of the deposits on the pressure surface of the vane.
t=0 t=25 s t=29 s
t=35 s t=79 s t=86 s
Figure 36 – Time lapse of deposition growth in TuRFR facility
At t= 25 seconds deposition has formed in the film cooling holes. By t=35 seconds the
deposition has filled in the regions between the film cooling holes. The majority of the
deposition is half way to the trailing edge. At t= 79 seconds, large deposits begin flaking
FC
hole
Dep.
Detach
Lost
Dep.
near
T.E.
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off. By t= 86 seconds large amounts of the initial deposition near the trailing edge is
gone. Fig. 37 shows the vane after the conclusion of the test.
Figure 37 – Turbine vane after cool down
The deposition in the trailing edge is significantly less than the amount seen during the
test at operating conditions. This is likely due to the thermal expansion of the blades at
the temperature. During cool down the blades contract and the brittle deposits flake off
during this time. The amount of deposition lost during this time is currently unknown;
however, it is clear that the deposition near the trailing edge is most susceptible to being
removed. Inspection of the remaining deposition shows some deposits that are suspended
above the vane leading edge surface as seen in Fig. 38. The fragility of the deposits
requires that the deposits be compared while at temperature.
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Figure 38 - Turbine vane after cool down with suspended deposition
4.9 Particle Size Deposition Distribution
An important part of this study is the effect of the size distribution of particles.
Particles follow different trajectories based on their size and mass and the likelihood of
the particle depositing can be affected by its diameter. Filtration advancements have
made the filtration of large particles possible. Understanding the effects of particle size is
important for optimizing filtration techniques to eliminate the particles most susceptible
to depositing. A study was performed using the E3 high temperature adiabatic wall case
to determine the location and amount of deposition for a single particle size. A range of
particle sizes were injected and their deposition locations are shown in Fig. 39. Table 12
shows the capture efficiencies for various particle sizes.
73
(a) (b) (c)
Figure 39 – Deposition locations for various particle diameters, (a) 1μm, (b) 5μm,
(c) 10μm
Table 12 – Particle Size Comparison Results
1μm 2μm 5μm 8μm 10μm 20μm
Total Injected # 900 900 900 900 900 900
Impact Efficiency % 0.77 2.33 18.9 43.7 68.7 119.9
Sticking Efficiency % 100 100 100 100 100 41.2
Capture Efficiency % 0.77 2.33 18.9 43.7 68.7 49.3
It can be seen from these results that increasing particle diameter above 1μm greatly
increases the amount of particulate that deposits. The particles above 2μm in diameter are
more likely to impact the surface and due to the high surface temperature are more likely
to deposit. There is a cut off above 10 μm where large particles tend to reflect off the
surface more, but there is still a significant amount of deposition. For the 20μm particles
the impact efficiency is above 100% which indicates that some of the particles reflect
# of Particles
74
multiple times before being captured. It can be seen from this study that particles above
2μm should be filtered to reduce the amount of deposition significantly.
75
CHAPTER 5: END WALL MODIFICATION
5.1 Reasoning for end wall modification
A significant amount of time has gone into the design and manufacturing of
turbine vanes to achieve the power requirements of the turbine. The turbine hub and case
walls are far less complex than the turbine vane and modifying the end walls is seen as
the most ideal form of preventing deposition growth. The E3M was tested under similar
conditions as the TuRFR experiments to determine the benefits of end wall modification.
5.2 Method of adjustment
An observation of the simulation results indicates a distinct pattern of deposition
on the pressure surface of the vane close to the hub end wall. A line of deposits was
observed running parallel with the end wall as shown in Fig. 40. The number of impacts
in this region is also high, which indicates the increased deposition was related to the
flow field.
76
(a) (b)
Figure 40 – Impact and Deposit Mass % locations
Streamlines were injected into the flow to visualize the flow in this region. It became
clear from Fig. 41 that a powerful horseshoe vortex existed on the junction of the turbine
vane and hub wall. The horseshoe vortex can pull particles into the wall increasing the
likelihood that particles will stick.
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Figure 41 – Flow visualization of horseshoe vortex for E3 (Vortex magnified)
The deposition on the casing side was significantly less. The large inlet angle associated
with the hub wall causes stronger spanwise flow to develop in the pressure surface
boundary layer. This spanwise flow pulls more particles to the surface. The case wall has
a less significant inlet angle and the spanwise flow is far less powerful. This limits the
number of particles being pulled to the surface. The hub end wall was adjusted to have a
less severe inlet angle in the E3M case. The case end wall was left the same. The vortex
for this new configuration is shown in Fig. 42.
Streamwise
Vorticity
78
Figure 42 – Flow Visualization of horseshoe vortex for E3M (Vortex Magnified)
5.3 Exit and Inlet profile changes
The low Mach flow exiting the combustor means that modification of the end
walls will affect the flow upstream and downstream of the modification. This will affect
the design of the combustor as well as the later stages of the turbine. Therefore,
modification of the end wall should be limited to reduce excessive changes to the
neighboring engine subsystems. The inlet Mach number is shown in Fig. 43. The changes
are limited to the region near the hub end wall. The majority of the fluid remains
unaffected.
Streamwise
Vorticity
79
(a) (b)
Figure 43 – Inlet Mach, (a) E3, (b) E
3M
Downstream of vane, the Mach number, total pressure and exit flow angle are important
for the performance of later turbine stages. The exit total pressure was shown in Fig. 13
and showed little difference between the E3 and modified E
3. The exit Mach number is
shown in Fig. 44.
(a) (b)
Figure 44 – Exit Mach, (a) E3, (b) E
3M
The exit Mach values are similar indicating that the modified hub has little effect on the
Mach number downstream. Fig. 45 shows the flow angle relative to the axial axis. The
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angles are very similar across the span. The modified hub end wall has little effect on the
flow properties. This satisfies the requirement that the end wall modification be minute so
that it can be applied to engines without extensive redesign of the engine components.
(a) (b)
Figure 45 – Exit flow angle, (a) E3, (b) E
3M
5.4 Deposition augmentation
The deposition result for the E3M is shown in table 13 and Figs. 46 and 47. The
capture efficiency for the E3M was less than the standard E
3. The impacts were
significantly less as well. The impacts in the region near the hub were much less,
although there was a slight increase in the mid-span of the vane.
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Table 13 – E3 Sticking Model Comparison Results
E3 E
3M
Total Injected # 22525 18105
Aborted # 9283 8826
Incomplete # 77 260
Escaped # 13165 9019
Impact Efficiency % 63 54
Sticking Efficiency % 92 87
Capture Efficiency % 58 47
(a) (b)
Figure 46 – Impact Mass % Locations, (a) E3, (b) E
3M
H
u
b
W
a
l
l
C
a
s
e
W
a
l
l
Pressure Side
82
(a) (b)
Figure 47 – Deposition Mass % Locations, (a) E3, (b) E
3M
The deposition decreased in the area of the vortex when the hub wall was modified. The
decrease in capture efficiency also shows that there was an improvement in the
deposition amount. There is a slight increase in deposition in the mid-span region.
Although there is an over-all decrease in deposition the new location of the deposition
could be the source of future problems.
5.5 Recommendations for improvements
The modification of the hub end wall was very simple and appeared to mitigate
the formation of deposits. The modification was the first step in exploring methods of
reducing deposition and there is plenty of room for optimization. Further decrease of the
inlet angle could yield additional results, although there are limitations to how much the
H
u
b
W
a
l
l
C
a
s
e
W
a
l
l
Pressure Side
83
angle can be changed before major changes would need to be made to the combustor
outlet. Modification of the casing could also be explored but it is unclear if the benefit
would be as great as the hub wall modification.
84
CHAPTER 6: CONCLUSIONS AND FUTURE WORK
6.1 Conclusion
A numerical study of the physical mechanisms associated with the deposition of
coal ash particulate on a blade passage of the first stage of the GE- E3 turbine was
performed. The flow was calculated using the software package FLUENT and particulate
was tracked using a Lagrangian approach. Two sticking models were developed and
compared with experimental data. The critical viscosity sticking model was found to
require additional calibration apart from using the viscosity-temperature relationship
derived from the ash composition. The viscosity model was simple to implement and
with proper calibration could be quickly applied to different particulate-laden flows. For
this study, the basic calibration described by Tafti et. al. was insufficient; however, the
experimentally specific calibration described by Tafti et. al. for unknown or complicated
viscosity-temperature relationships may yield better results. The critical impact velocity
sticking model was found to match the initial deposition observed from experimental
tests in the TuRFR facility. The initial deposition locations and quantities were similar
upon injection of particulate but diverged from the experiment as the test advanced.
Modification of the fluid flow due to deposition growth and experimental operation
limitations are likely the source of these errors. Finally, the hub end wall of the turbine
passage was modified to observe the effects of the end walls on deposition. Adjustment
of the hub end wall showed an 18% decrease in over-all deposition on the turbine vane
85
and reduced a significant amount of deposition near the hub and vane interface. The end
wall adjustment also caused a slight increase in deposition in the mid-span region of the
turbine vane. These findings support end wall modification as a viable means of reducing
deposition growth. This paper has shown that:
The 3D flow features in turbine passages can greatly affect the particle trajectories
and deposition amount
The balance of deposition removal and accumulation mechanisms is important for
determining how much and how quickly deposition forms on a surface
The size of particles in the flow can significantly affect the amount of deposition,
with particles less than 2μm in diameter being less likely to deposit while
particles above are highly likely to deposit
6.2 Future Work
Future studies will further develop the transient effects of deposition modeling.
One feature recently made available in FLUENT is the real-time grid adaptation. This
feature can be used to adjust the geometry based on user-defined criteria. The fluid flow
is automatically updated as the geometry varies due to deposition growth and the
performance of the vane can be evaluated after the injection of particulate. The large
scale deposition removal mechanisms will also be developed further to more accurately
predict the locations of deposition formation. Additionally, film cooling will be modeled
on the turbine vane to study the effects of film cooling on deposition. The film cooling
benefit of reducing surface temperatures is offset by the tendency of film cooling holes to
86
serve as sites for deposition collection. The blockage of film cooling holes is also a major
issue with deposition growth. Finally, further modifications to the end walls will be
explored. This study was the first step in exploring the effects of end wall modification
and allow for optimization.
87
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22) Kaftori, D., Hetsroni, G., and Banerjee, S., “Particle behavior in the turbulent
boundary layer. I. Motion, deposition, and entrainment”, Phys. Fluids 7, 1095-1106,
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23) Kallio GA, Reeks MW. 1989. A numerical simulation of particle deposition in
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24) Kim, J., Dunn, M.G., and Baran, A.J. et al, “Deposition of Volcanic Materials in the
Hot Sections of Two Gas Turbine Engines,” J. Engr. Gas Turbines & Power. Vol 115,
Jul 1993, pp 641-651.
25) Kulick, J.D., Fessler, J.R., and Eaton, J.K., “Particle response and turbulence
modification in fully developed channel flow”, J. Fluid Mech., 277, 109-134, 1994
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93
APPENDIX
A.1 Critical Impact Velocity Sticking Model UDF
#include "udf.h"
#include "dpm.h"
#include "mem.h"
#include "sg.h"
#define nu_s 0.27
#define nu_p 0.27
#define YMIN 0.0
#define YMAX 0.4
#define UMEAN 1.0
#define B 1./7.
#define DELOVRH 0.5
#define VISC 1.7894e-05
/*#define RGAS (UNIVERSAL_GAS_CONSTANT/MW)*/
#define Tdatum 288.15
#define NUM_UDM 6
real ParticleTotalMass;
real P_Mass[6];
real P_Impact_Mass[6];
real P_Stick_Mass[6];
DEFINE_DPM_BC(udfdeposition8,p,t,f,f_normal,dim)
{
#if !RP_HOST
real crit_vel,alpha,Tavg,Ep,vcr=0.,Es,k1,k2,calc,E,utau1,dudz,wall_shear_stress;
real Cu,cbar,lamda,kn,val,kc,ucws,utc,h,del,ufr,ff,utau,area,wall_shear_force,utaunew;
real vn=0.;
real utau2, utc1,vpabs=0.,MassImpact;
real nor_coeff=0.8;
real tan_coeff=0.4;
real R=287.;
real tem_Mass=0.;
real tem_Particle_Dia=0.;
real A[ND_ND],ds,es[ND_ND],A_by_es,dr0[ND_ND],ivu=0.,jvv=0.,kvw=0.,du=0.;
real tauwall1=0.,tauwall2=0.,tauwall3=0.,wallfricv1=0.,wallfricv2=0.,wallfricv3=0.;
real yplus,wallfricv4=0.,wallfricv5=0.,uvel,vvel,wvel,dumag=0.;
FILE *fp;
Thread *tcell=P_CELL_THREAD(p);
cell_t c=P_CELL(p);
/*Message("boo1\n");*/
Domain *d;
real normal[3];
94
int i,idim=dim;
real Wa = 0.039;
d=Get_Domain(1);
real x[ND_ND];
for (i=0.; i<idim; i++)
normal[i] = f_normal[i];
/*Message("boo2\n");*/
C_UDMI(c,tcell,11) += 1.0;
/*Message("boo3\n");*/
if(p->type==DPM_TYPE_INERT)
{
/*||||||||||||||||||||||||||||||||computing normal velocity|||||||||||||||||||||||||||||*/
for(i=0.;i<idim;i++)
{
vn += p->state.V[i]*normal[i];
vpabs += pow(p->state.V[i],2.);
}
vpabs = pow(vpabs,.5);
/*Message("boo4\n");*/
/*|||||||||||||||||||||||||||||computing critical velocity||||||||||||||||||||||||||||||*/
Tavg = (F_T(f,t)+P_T(p))/2.;
/*Avg temp is Tavg)*/
/*particle temp is P_T(p))*/
/*face temp from tcell is F_T(f,tcell));*/
/*normal face temp is F_T(f,t));*/
/*NEW CORRELATION*/
Ep=(3.*(pow(10.,20.)))*exp(-0.02365*Tavg);
/*Young's Modulous is for particle is Ep*/
Es=Ep;
k1 = (1.-(nu_s*nu_s))/(3.14*Es);
k2 = (1.-(nu_p*nu_p))/(3.14*Ep);
/*particle density is P_RHO(p))*/
/*Ai particle density is p->init_state.rho);*/
calc = (5.*3.14*3.14*(k1+k2))/(4.*(pow(P_RHO(p),1.5)));
E = 0.51*(pow(calc,(2./5.)));
/*young's modulus is E*/
vcr = pow(((2.*E)/P_DIAM(p)),(10./7.));
/*Message("boo5\n");*/
/*capture velocity is vcr*/
/*particle diameter is P_DIAM(p))*/
/*Ai particle dia is p->init.state.diam*/
/*|||||||||||||||||||||||||||Calculating mass of particles impacting||||||||||||||||||||*/
/*particle mass is P_MASS(p)*/
MassImpact=P_FLOW_RATE(p)*pow(10,9);
C_UDMI(c,tcell,12)+=MassImpact;
/*Message("boo6\n");*/
/*Work of sticking - assumed Wa of silicon */
95
cbar = sqrt(((8.*C_RGAS(c,tcell)*C_T(c,tcell))/3.14));
lamda = (2.*C_MU_L(c,tcell))/(C_R(c,tcell)*cbar);
/*Ai gas viscosity is VISC, C_MU_EFF(c,tcell)*/
kn = (2.*lamda)/P_DIAM(p);
Cu = 1.+kn*(1.2+(0.41*exp(-0.88/kn))); /*Cunningham Correction Factor*/
Tavg = (F_T(f,t)+C_T(c,tcell))/2.; /*Avg of particle & surface temperature*/
/*Young's Mod of part & surf equal*/
/*Calculating composite Young's Modulus*/
val = ((1.-(pow(nu_s,2.)))/Es)+((1.-(pow(nu_p,2.)))/Ep);
kc = (4./3.)/val;
/*Calculating critical wall shear velocity*/
ucws = ((Cu*Wa)/(C_R(c,tcell)*P_DIAM(p)))*(pow((Wa/(P_DIAM(p)*kc)),(1./3.)));
utc = sqrt(ucws);
/*wall shear vel is utc*/
utc1=pow(ucws,0.5);
/*Message("boo7\n");*/
/*critical wall shear vel is ,utc,utc1); */
/*Calculating wall friction velocity*/
/*new wall friction calculation*/
/*Message("Got to here...\n");*/
Thread *t0=THREAD_T0(t);
cell_t c0=F_C0(f,t);
BOUNDARY_FACE_GEOMETRY(f,t,A,ds,es,A_by_es,dr0);
/*wall distance is ds*/
tauwall1=C_MU_EFF(c,tcell)*(du/ds);
wallfricv1=sqrt(tauwall1/C_R(c,tcell));
/*wallfrictionvelocity 1 wallfricv1*/
/*Message("Got this far...\n");*/
/*NEW 2*/
/*Message("boo8\n");*/
utaunew = sqrt((2.*C_MU_L(c,tcell)*vpabs)/(C_R(c,tcell)*P_DIAM(p)));
/*Message("boo10\n");*/
/*wallfrictionvelocity of weiguo is utaunew*/
/*fp=fopen("Impact2.txt","a");*/
/*fprintf(fp,"%6.3f %6.2f %6.2f %f %f %f %f\n",
P_DIAM(p),vn,vcr,Ep,MassImpact,ucws,utaunew);*/
/*fclose(fp)*/
tem_Particle_Dia=P_DIAM(p)*pow(10,6);
if(tem_Particle_Dia<1)
P_Impact_Mass[0]+=P_FLOW_RATE(p)*pow(10,9);
else if(tem_Particle_Dia>1 && tem_Particle_Dia<3)
P_Impact_Mass[1]+=P_FLOW_RATE(p)*pow(10,9);
else if(tem_Particle_Dia>3 && tem_Particle_Dia<5)
96
P_Impact_Mass[2]+=P_FLOW_RATE(p)*pow(10,9);
else if(tem_Particle_Dia>5 && tem_Particle_Dia<7)
P_Impact_Mass[3]+=P_FLOW_RATE(p)*pow(10,9);
else if(tem_Particle_Dia>7 && tem_Particle_Dia<10)
P_Impact_Mass[4]+=P_FLOW_RATE(p)*pow(10,9);
else
P_Impact_Mass[5]+=P_FLOW_RATE(p)*pow(10,9);
/*Message("boo11\n");*/
fp=fopen("Impact_Location.txt","a");
fprintf(fp,"%f %f %f %E\n",P_POS(p)[0],P_POS(p)[1],P_POS(p)[2],P_MASS(p));
fclose(fp);
/*Message("E is:%f\n",E);
Message("Ep is:%f\n",Ep);*/
Message("vn is:%f\n",vn);
Message("vcr is:%f\n",vcr);
if(C_UDMI(c,tcell,15)!=0.0)
{
/*Message("Vcr:%f\n",vcr);*/
vcr=10*vcr;
/*Message("Vcr:%f\n",vcr);*/
}
if(abs(vn)>vcr)
{
C_UDMI(c,tcell,13) += 1.0;
C_UDMI(c,tcell,14) += P_FLOW_RATE(p)*pow(10,9);
for(i=0;i<idim;i++)
p->state.V[i]-=vn*normal[i];
/*Apply tangential coefficient of restitution. */
for (i=0; i<idim; i++)
p->state.V[i]*=tan_coeff;
/*Add reflected normal velocity. */
for (i=0; i<idim; i++)
p->state.V[i]-=nor_coeff*vn*normal[i];
/* Store new velocity in state0 of particle*/
for (i=0; i<idim; i++)
p->state0.V[i]=p->state.V[i];
return PATH_ACTIVE;
}
else
/*particle deposits*/
97
if(wallfricv1 < utc)
{
/*num of particles deposited or num of hits*/
C_UDMI(c,tcell,15) += 1.0;
/* mass of particles deposited*/
C_UDMI(c,tcell,16) += P_FLOW_RATE(p)*pow(10,9);
tem_Mass=P_FLOW_RATE(p)*pow(10,9);
/*Message("boo13\n");*/
if(tem_Particle_Dia<1.)
P_Stick_Mass[0]+=P_FLOW_RATE(p)*pow(10,9);
else if(tem_Particle_Dia>1 && tem_Particle_Dia<3)
P_Stick_Mass[1]+=P_FLOW_RATE(p)*pow(10,9);
else if(tem_Particle_Dia>3 && tem_Particle_Dia<5)
P_Stick_Mass[2]+=P_FLOW_RATE(p)*pow(10,9);
else if(tem_Particle_Dia>5 && tem_Particle_Dia<7)
P_Stick_Mass[3]+=P_FLOW_RATE(p)*pow(10,9);
else if(tem_Particle_Dia>7 && tem_Particle_Dia<10)
P_Stick_Mass[4]+=P_FLOW_RATE(p)*pow(10,9);
else
P_Stick_Mass[5]+=P_FLOW_RATE(p)*pow(10,9);
/*Message("boo14\n");*/
/*fp=fopen("Stick2.txt","a");*/
/*fprintf(fp,"%f %f %6.2f %f %6.2f %6.2f %f\n",
tem_Particle_Dia,tem_Mass,vn,vcr,Ep,ucws,utaunew);*/
/*fclose(fp);*/
fp=fopen("Deposition_Location.txt","a");
fprintf(fp,"%f %f %f %E\n",P_POS(p)[0],P_POS(p)[1],P_POS(p)[2],P_MASS(p));
fclose(fp);
/* real volume,Re,r,R,h,Cd,D,theta,BA,CA;
volume=C_UDMI(c,tcell,16)/(pow(10,9)*P_RHO(p));
BA=F_AREA(A,f,t);
h=3*volume/BA;
R=sqrt(BA/3.14);
theta=atan(h/R);
D=0;
Message("height:%E\n",h);
Message("Base Radius:%E\n",R);
Message("Angle:%f\n",theta*57.295);
for (i=1; i<=3; i++)
{
r=h/(tan(theta))*(1+(1-i)/3);
98
Re=C_R(c,tcell)*sqrt(pow(C_U(c,tcell),2)+pow(C_V(c,tcell),2)+pow(C_W(c,tcell),2))*2*r/C_MU_EFF(c,
tcell);
if(Re<100)
Cd=24/(pow(Re,0.7));
else
Cd=3/(pow(Re,0.25));
CA=(r+(r-h/(3*tan(theta))))*h/3;
D=D+0.5*Cd*(pow(C_U(c,tcell),2)+pow(C_V(c,tcell),2)+pow(C_W(c,tcell),2))*CA;
}
Message("Drag is:%E\n",D);
Message("Fpo is:%E\n",0.75*3.14*Wa*R*R*3/h);
if(D>=0.75*3.14*Wa*R*R*3/h)
{
C_UDMI(c,tcell,15)=0.0;
C_UDMI(c,tcell,16)=0.0;
Message("!!!!!Deposit Removed!!!!!\n");
}
*/
}
}
/*Message("boo15, did it stick?");*/
return PATH_ABORT;
#endif
}
/*Setting UDM locations and initializing them to 0*/
DEFINE_ON_DEMAND(reset_UDMsnew)
{
int i=0;
Thread *t;
Domain *d;
cell_t c;
face_t f;
d=Get_Domain(1);
Message("Setting UDMs \n");
thread_loop_c(t,d)
{
begin_c_loop(c,t)
{
for(i=11;i<17;i++)
C_UDMI(c,t,i)=0.0;
}
end_c_loop(c,t)
}
99
}
/*Macro to sum up all the UDM values and display*/
A.2 Critical Viscosity Sticking Model UDF
#include "udf.h"
#include "dpm.h"
#include "mem.h"
#include "sg.h"
#include "stdlib.h"
#include "stdio.h"
#include "math.h"
#define nu_s 0.27
#define nu_p 0.27
#define YMIN 0.0
#define YMAX 0.4
#define UMEAN 1.0
#define B 1./7.
#define DELOVRH 0.5
#define VISC 1.7894e-05
/*#define RGAS (UNIVERSAL_GAS_CONSTANT/MW)*/
#define Tdatum 288.15
#define NUM_UDM 6
real ParticleTotalMass;
real P_Mass[6];
real P_Impact_Mass[6];
real P_Stick_Mass[6];
DEFINE_DPM_BC(udfdeposition8,p,t,f,f_normal,dim)
{
#if !RP_HOST
/*Message("boo\n");*/
real crit_vel,alpha,Tavg,Ep,vcr=0.,Es,k1,k2,calc,E,utau1,dudz,wall_shear_stress;
real Cu,cbar,lamda,kn,val,kc,ucws,utc,h,del,ufr,ff,utau,area,wall_shear_force,utaunew;
real vn=0.;
real utau2, utc1,vpabs=0.,MassImpact;
real nor_coeff=0.8;
real tan_coeff=0.4;
real R=287.;
real tem_Mass=0.;
real tem_Particle_Dia=0.;
real A[ND_ND],ds,es[ND_ND],A_by_es,dr0[ND_ND],ivu=0.,jvv=0.,kvw=0.,du=0.;
real tauwall1=0.,tauwall2=0.,tauwall3=0.,wallfricv1=0.,wallfricv2=0.,wallfricv3=0.;
real yplus,wallfricv4=0.,wallfricv5=0.,uvel,vvel,wvel,dumag=0.;
real RANDOM;
real Ac,Bc,Sticky,StickyCr,Sticky_ratio;
FILE *fp;
Thread *tcell=P_CELL_THREAD(p);
cell_t c=P_CELL(p);
100
/*Message("boo1\n");*/
Domain *d;
real normal[3];
int i,idim=dim;
real Wa = 0.039;
d=Get_Domain(1);
real x[ND_ND];
for (i=0.; i<idim; i++)
normal[i] = f_normal[i];
/*Message("boo2\n");*/
C_UDMI(c,tcell,11) += 1.0;
/*Message("boo3\n");*/
if(p->type==DPM_TYPE_INERT)
{
/*||||||||||||||||||||||||||||||||computing normal velocity|||||||||||||||||||||||||||||*/
for(i=0.;i<idim;i++)
{
vn += p->state.V[i]*normal[i];
vpabs += pow(p->state.V[i],2.);
}
vpabs = pow(vpabs,.5);
/*Message("boo4\n");*/
/*|||||||||||||||||||||||||||||computing critical velocity||||||||||||||||||||||||||||||*/
Tavg = (F_T(f,t)+P_T(p))/2.;
/*Avg temp is Tavg)*/
/*particle temp is P_T(p))*/
/*face temp from tcell is F_T(f,tcell));*/
/*normal face temp is F_T(f,t));*/
/*NEW CORRELATION*/
Ep=(3.*(pow(10.,20.)))*exp(-0.02365*Tavg);
/*Young's Modulous is for particle is Ep*/
Es=Ep;
k1 = (1.-(nu_s*nu_s))/(3.14*Es);
k2 = (1.-(nu_p*nu_p))/(3.14*Ep);
/*particle density is P_RHO(p))*/
/*Ai particle density is p->init_state.rho);*/
calc = (5.*3.14*3.14*(k1+k2))/(4.*(pow(P_RHO(p),1.5)));
E = 0.51*(pow(calc,(2./5.)));
/*young's modulus is E*/
vcr = pow(((2.*E)/P_DIAM(p)),(10./7.));
/*Message("boo5\n");*/
/*capture velocity is vcr*/
/*particle diameter is P_DIAM(p))*/
/*Ai particle dia is p->init.state.diam*/
/*|||||||||||||||||||||||||||Calculating mass of particles impacting||||||||||||||||||||*/
/*particle mass is P_MASS(p)*/
MassImpact=P_FLOW_RATE(p)*pow(10,9);
C_UDMI(c,tcell,12)+=MassImpact;
101
/*Message("boo6\n");*/
/*Work of sticking - assumed Wa of silicon */
cbar = sqrt(((8.*C_RGAS(c,tcell)*C_T(c,tcell))/3.14));
lamda = (2.*C_MU_L(c,tcell))/(C_R(c,tcell)*cbar);
/*Ai gas viscosity is VISC, C_MU_EFF(c,tcell)*/
kn = (2.*lamda)/P_DIAM(p);
Cu = 1.+kn*(1.2+(0.41*exp(-0.88/kn))); /*Cunningham Correction Factor*/
Tavg = (F_T(f,t)+C_T(c,tcell))/2.; /*Avg of particle & surface temperature*/
/*Young's Mod of part & surf equal*/
/*Calculating composite Young's Modulus*/
val = ((1.-(pow(nu_s,2.)))/Es)+((1.-(pow(nu_p,2.)))/Ep);
kc = (4./3.)/val;
/*Calculating critical wall shear velocity*/
ucws = ((Cu*Wa)/(C_R(c,tcell)*P_DIAM(p)))*(pow((Wa/(P_DIAM(p)*kc)),(1./3.)));
utc = sqrt(ucws);
/*wall shear vel is utc*/
utc1=pow(ucws,0.5);
/*Message("boo7\n");*/
/*critical wall shear vel is ,utc,utc1); */
/*Calculating wall friction velocity*/
/*new wall friction calculation*/
/*Message("Got to here...\n");*/
Thread *t0=THREAD_T0(t);
cell_t c0=F_C0(f,t);
BOUNDARY_FACE_GEOMETRY(f,t,A,ds,es,A_by_es,dr0);
/*wall distance is ds*/
tauwall1=C_MU_EFF(c,tcell)*(du/ds);
wallfricv1=sqrt(tauwall1/C_R(c,tcell));
/*wallfrictionvelocity 1 wallfricv1*/
/*Message("Got this far...\n");*/
/*NEW 2*/
/*Message("boo8\n");*/
utaunew = sqrt((2.*C_MU_L(c,tcell)*vpabs)/(C_R(c,tcell)*P_DIAM(p)));
/*Message("boo10\n");*/
/*wallfrictionvelocity of weiguo is utaunew*/
/*fp=fopen("Impact2.txt","a");*/
/*fprintf(fp,"%6.3f %6.2f %6.2f %f %f %f %f\n",
P_DIAM(p),vn,vcr,Ep,MassImpact,ucws,utaunew);*/
/*fclose(fp)*/
tem_Particle_Dia=P_DIAM(p)*pow(10,6);
if(tem_Particle_Dia<1)
102
P_Impact_Mass[0]+=P_FLOW_RATE(p)*pow(10,9);
else if(tem_Particle_Dia>1 && tem_Particle_Dia<3)
P_Impact_Mass[1]+=P_FLOW_RATE(p)*pow(10,9);
else if(tem_Particle_Dia>3 && tem_Particle_Dia<5)
P_Impact_Mass[2]+=P_FLOW_RATE(p)*pow(10,9);
else if(tem_Particle_Dia>5 && tem_Particle_Dia<7)
P_Impact_Mass[3]+=P_FLOW_RATE(p)*pow(10,9);
else if(tem_Particle_Dia>7 && tem_Particle_Dia<10)
P_Impact_Mass[4]+=P_FLOW_RATE(p)*pow(10,9);
else
P_Impact_Mass[5]+=P_FLOW_RATE(p)*pow(10,9);
/*Message("boo11\n");*/
fp=fopen("Impact_Location.txt","a");
fprintf(fp,"%f %f %f %E\n",P_POS(p)[0],P_POS(p)[1],P_POS(p)[2],P_MASS(p));
fclose(fp);
/*||||||||||||||||||||||||Random number Generator|||||||||||||||||||||||||||||||||||||||||||||||||||||||*/
/*RANDOM=sqrt(sin(P_T(p)*100)*sin(P_T(p)*100));*/
RANDOM=(2*(pow(sin(C_T(c,tcell)*C_U(c,tcell)),2)-
floor(pow(sin(C_T(c,tcell)*C_U(c,tcell)),2)+0.5))+1)/2;
Message("RANDOM is:%f\n",RANDOM);
Message("p is:%f\n",p);
fp=fopen("Random.txt","a");
fprintf(fp,"%f\n",RANDOM);
fclose(fp);
/*||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||*/
/*|||||||||||||||||||||||||||||||||||Viscosity Model||||||||||||||||||||||||||||||||||||||||||||||||||||*/
/*Ash Compisition Model - Weiguo, Hot>1037, critical viscosity is in Hot range */
/*Critical Temp Coefficients*/
Ac=-11.3059025;
Bc=15.95606;
StickyCr=pow(10,Ac+pow(10,3)*Bc/1500);
if(P_T(p)>1500)
Sticky_ratio=1;
else if(P_T(p)<1037)
{
/*Cold Coefficients*/
Ac=-44.25456;
Bc=49.87914;
Sticky=pow(10,Ac+pow(10,3)*Bc/P_T(p));
Sticky_ratio=(StickyCr*1500)/(Sticky*P_T(p));
}
else
{
Ac=-11.3059025;
Bc=15.95606;
103
Sticky=pow(10,Ac+pow(10,3)*Bc/P_T(p));
Sticky_ratio=(StickyCr*1500)/(Sticky*P_T(p));
}
if(C_UDMI(c,tcell,15)!=0.0)
{
Sticky_ratio=1.1*Sticky_ratio;
}
if(RANDOM>Sticky_ratio)
{
C_UDMI(c,tcell,13) += 1.0;
C_UDMI(c,tcell,14) += P_FLOW_RATE(p)*pow(10,9);
for(i=0;i<idim;i++)
p->state.V[i]-=vn*normal[i];
/*Apply tangential coefficient of restitution. */
for (i=0; i<idim; i++)
p->state.V[i]*=tan_coeff;
/*Add reflected normal velocity. */
for (i=0; i<idim; i++)
p->state.V[i]-=nor_coeff*vn*normal[i];
/* Store new velocity in state0 of particle*/
for (i=0; i<idim; i++)
p->state0.V[i]=p->state.V[i];
return PATH_ACTIVE;
}
else
/*particle deposits*/
if(wallfricv1 < utc)
{
/*num of particles deposited or num of hits*/
C_UDMI(c,tcell,15) += 1.0;
/* mass of particles deposited*/
C_UDMI(c,tcell,16) += P_FLOW_RATE(p)*pow(10,9);
tem_Mass=P_FLOW_RATE(p)*pow(10,9);
/*Message("boo13\n");*/
if(tem_Particle_Dia<1.)
P_Stick_Mass[0]+=P_FLOW_RATE(p)*pow(10,9);
else if(tem_Particle_Dia>1 && tem_Particle_Dia<3)
P_Stick_Mass[1]+=P_FLOW_RATE(p)*pow(10,9);
else if(tem_Particle_Dia>3 && tem_Particle_Dia<5)
P_Stick_Mass[2]+=P_FLOW_RATE(p)*pow(10,9);
else if(tem_Particle_Dia>5 && tem_Particle_Dia<7)
104
P_Stick_Mass[3]+=P_FLOW_RATE(p)*pow(10,9);
else if(tem_Particle_Dia>7 && tem_Particle_Dia<10)
P_Stick_Mass[4]+=P_FLOW_RATE(p)*pow(10,9);
else
P_Stick_Mass[5]+=P_FLOW_RATE(p)*pow(10,9);
/*Message("boo14\n");*/
/*fp=fopen("Stick2.txt","a");*/
/*fprintf(fp,"%f %f %6.2f %f %6.2f %6.2f %f\n",
tem_Particle_Dia,tem_Mass,vn,vcr,Ep,ucws,utaunew);*/
/*fclose(fp);*/
fp=fopen("Deposition_Location.txt","a");
fprintf(fp,"%f %f %f %E\n",P_POS(p)[0],P_POS(p)[1],P_POS(p)[2],P_MASS(p));
fclose(fp);
}
}
return PATH_ABORT;
#endif
}
/*Setting UDM locations and initializing them to 0*/
DEFINE_ON_DEMAND(reset_UDMsnew)
{
int i=0;
Thread *t;
Domain *d;
cell_t c;
face_t f;
d=Get_Domain(1);
Message("Setting UDMs \n");
thread_loop_c(t,d)
{
begin_c_loop(c,t)
{
for(i=11;i<17;i++)
C_UDMI(c,t,i)=0.0;
}
end_c_loop(c,t)
}
}
/*Macro to sum up all the UDM values and display*/
105
A.3 Random Walk Correction UDF(excerpt from Longmire, [Longmire])
/* The function do_stochastics shows an implementation of the standard stochastic
* tracking in FLUENT and an additional function for boundary layer treatment.
* Limitations:
* - Uses only spherical drag
* - tested for steady state simulations
* - does not work with accuracy control of particle tracks
* - stochastic tracking is activated for ALL defined injections
*
* The function do_wall_stoch also takes into account the effect of fluid
* boundary layer damping of turbulent fluctuations.
* Eddy lifetime is stored in scalar p->user[1]. The residence time of a particle
* in a given eddy is stored in p->user[0]
*
* Usage:
* - compile and load udfs
* - activate drag law
* - activate time step routines
* - for walls with reflection activate DPM boundary condition
* - activate 11 particle scalars and hook SCALAR_UPDATE function
* - hook DEFINE_INIT function
* - activate 11 UDM locations
* - after calc. of flow field execute the set_udm function for setting
* Y_plus values for particles in wall nearest cell (coarse grid), or
* if the sublayer is resolved choose set_udm_finegrid (may take a long time to execute)
* - finally perform particle tracking. The DPM_DRAG function evaluates the particle Y_plus
* and chooses the appropriate time scale for eddy interaction.
*
*
*
*/
#include "udf.h"
#include "dpm.h"
#include "cxsurf.h" /* for screen output with CX_Message(...) */
#include "surf.h"
#include "sg.h"
#define C_MU_FACTOR(I) (I->time_scale_constant)
#define C_MU 0.09
#define MY_WALL_SHEAR_X(f,t) F_STORAGE_R_N3V(f,t,SV_WALL_SHEAR)[0] /* FORCES!!
divide by face area */
#define MY_WALL_SHEAR_Y(f,t) F_STORAGE_R_N3V(f,t,SV_WALL_SHEAR)[1]
#define MY_WALL_SHEAR_Z(f,t) F_STORAGE_R_N3V(f,t,SV_WALL_SHEAR)[2]
#define THERMO 0
#define INTERPOLATION 1
#define DEBUGGER 0
cxboolean do_stochastics = TRUE;
cxboolean do_wall_stoch = FALSE;
cxboolean random_eddy_lifetime = FALSE;
cxboolean do_thermophoretic = TRUE;
real y_T = 11.8;
int update_interval = 100001;
int update_interval_up = 1000000;
106
int update_interval_old = 5000000;
static real therm_diff_factor = 100.;
static real yplus_thresh = 200.0;
static real dt_min = 1.0e-06;
static real te_min = 1.0e-05;
static int msg_count = 0;
/* global function declarations */
static real get_yplus_particle(cell_t c, Thread *t, real *pp);
static real get_tl_plus(real);
static void get_rms_vel_fluc(real,real,real*,real*,real*,real*,cphase_state_t*,cell_t c0, Thread *t0);
static real half_zero(real,real);
enum udm_names{
Y_PLUS,
TAU_WALL,
N_VEC_X,
N_VEC_Y,
N_VEC_Z,
FACE_X,
FACE_Y,
FACE_Z,
TWALL,
QWALL,
WALL_HIT
};
enum p_names{
E_PLUS,
T_EDDY,
Y_PLUS_P,
LAMBDA1,
LAMBDA2,
LAMBDA3,
T_FORCE_X,
T_FORCE_Y,
T_FORCE_Z,
N_EDDY,
NEW_FORCE
};
/********************/
real get_tl_plus(real yplus)
{
real TL_plus = 1.0;
#if 0
if (yplus>5.0 && yplus<yplus_thresh)
{
TL_plus = 7.122 + 0.5731*yplus - 0.00129*yplus*yplus;
}
else if (yplus <= 5.0)
{
TL_plus = 10.0;
}
#endif
TL_plus = 1./(4.529 + 0.0116*pow(yplus,1.75) + 0.768*sqrt(yplus) );
return TL_plus;
107
}
/********************/
void get_rms_vel_fluc(real yplus, real vel_fric, real *Vprime_vec, real *I, real *J, real *K,cphase_state_t*
c,cell_t c0, Thread *t0)
{
real I_mag;
Vprime_vec[0] = 0.0;
Vprime_vec[1] = 0.0;
Vprime_vec[2] = 0.0;
/*Message("yplus is: %E\n",yplus);*/
/* calc. anisoptropic local fluctuation velocities ...*/
/* and assign the local coordinate system vectors ... */
I[0] = c->V[0];
I[1] = c->V[1];
#if RP_3D
I[2] = c->V[2];
#endif
I_mag = NV_MAG(I);
NV_S(I,/=,I_mag); /* normalize */
J[0] = C_UDMI(c0,t0,N_VEC_X);
J[1] = C_UDMI(c0,t0,N_VEC_Y);
#if RP_3D
J[2] = C_UDMI(c0,t0,N_VEC_Z);
#endif
NV_CROSS(K,I,J); /* K = I x J */
#if DEBUGGER
CX_Message("I0 = %f, I1 = %f, I2 = %f\n",I[0],I[1],I[2]);
CX_Message("J0 = %f, J1 = %f, J2 = %f\n",J[0],J[1],J[2]);
CX_Message("K0 = %f, K1 = %f, K2 = %f\n",K[0],K[1],K[2]);
#endif
/* ... finally transform everything back into cartesian coordinates */
NV_S(I,*=,Vprime_vec[0]); /* x-component stored in I*/
NV_S(J,*=,Vprime_vec[1]); /* y-component stored in J*/
NV_S(K,*=,Vprime_vec[2]); /* z-component stored in K*/
/*re-assign fluctuation velocities*/
Vprime_vec[0] = I[0] + J[0] + K[0];
Vprime_vec[1] = I[1] + J[1] + K[1];
Vprime_vec[2] = I[2] + J[2] + K[2];
#if 0
Vprime_vec[0] = 0.*vel_fric;
Vprime_vec[1] = 0.*vel_fric;
Vprime_vec[2] = 0.*vel_fric;
#endif
}
/* function to determine a particle's dimensionless wall distance */
real get_yplus_particle(cell_t c,Thread *t,real *pp)
{
real Y_plus_p,Y_p;
real p_dist[ND_ND];
real A[ND_ND];
real Xf[ND_ND];
108
if(C_UDMI(c,t,Y_PLUS) < 0.0)
return -1.0;
A[0] = C_UDMI(c,t,N_VEC_X);
A[1] = C_UDMI(c,t,N_VEC_Y);
#if RP_3D
A[2] = C_UDMI(c,t,N_VEC_Z);
#endif
Xf[0] = C_UDMI(c,t,FACE_X);
Xf[1] = C_UDMI(c,t,FACE_Y);
#if RP_3D
Xf[2] = C_UDMI(c,t,FACE_Z);
#endif
NV_VV(p_dist,=,Xf,-,pp);
Y_p = fabs(NV_DOT(A,p_dist));
Y_plus_p = Y_p*sqrt(C_UDMI(c,t,TAU_WALL)*C_R(c,t))/C_MU_L(c,t);
return Y_plus_p;
}
/* distribute user RP variables */
…
DEFINE_DPM_DRAG(dpm_stochastic_drag,Re,p)
{
cphase_state_t *c = &(p->cphase); /* pointer to continuous phase properties, velocity etc. (dpm.h)
*/
real Vrel;
real drag_factor;
real k,eps,eplus,TL_plus = 1.0; /* dimensionless time scale */
real Y_plus_p = 200.0; /* dimensionless particle dist. */
cell_t c0 = RP_CELL(&(p->cCell)); /* index of cell in which currently tracked particle is in */
Thread *t0 = RP_THREAD(&(p->cCell)); /* cell thread to which above cell belongs */
real coeff;
cxboolean check_time = FALSE;
int dim = ND_ND;
real pv[3] = {0.,0.,0.}; /* particle velocity vector */
real pp[3] = {0.,0.,0.}; /* particle position vector */
real I[3] = {0.,0.,0.}; /* streamwise direction vector at particle location */
real J[3] = {0.,0.,0.}; /* wall normal vector */
real K[3] = {0.,0.,0.}; /* spanwise direction vector */
int i;
real Vprime_rms;
real Vprime_vec[3]; /* fluct. vel. vector */
real vel_fric; /* u_tauwall , frictional velocity */
real y_dist;
NV_V(pv, =, p->state.V);
NV_V(pp, =, p->state.pos); /* position of particle, x,y,z-coord. */
#if !INTERPOLATION
c->V[0] = C_U(c0,t0);
c->V[1] = C_V(c0,t0);
#if RP_3D
c->V[2] = C_W(c0,t0);
#endif
#endif /* INTERPOLATION */
109
#if !PARALLEL
update_interval_old = (int)RP_Get_Real("update_old");
update_interval_up = (int)RP_Get_Real("update_new");
yplus_thresh = RP_Get_Real("yplus_threshold");
dt_min = RP_Get_Real("delta_t_min");
te_min = RP_Get_Real("t_eddy_min");
#endif
Y_plus_p = get_yplus_particle(c0,t0,pp);
/* CX_Message("Y_plus_p = %f, px = %f, py = %f, pz = %f\n",Y_plus_p,pp[0],pp[1],pp[2]); */
if(Y_plus_p < yplus_thresh && Y_plus_p > -0.1 )
{
do_stochastics = FALSE;
do_wall_stoch = TRUE;
}
else
{
do_stochastics = TRUE;
do_wall_stoch = FALSE;
if(p->user[1]>0.)
{
p->eddy_time=DPM_SMALL;
p->user[T_EDDY] = DPM_SMALL;
}
}
y_dist = Y_plus_p/sqrt(C_UDMI(c0,t0,TAU_WALL)*C_R(c0,t0))*C_MU_L(c0,t0);
if(y_dist < 0.5*P_DIAM(p) && Y_plus_p > -0.1)
{
do_stochastics = FALSE;
do_wall_stoch = FALSE;
Trap_Particle(p);
}
#if RP_2D
if (rp_axi_swirl)
{
dim = 3;
pv[2] = p->state.V[2];
}
else
{
pv[2]=0;
pp[2]=0.;
}
#endif
/* toggle for comparison - remove later*/
/* do_stochastics = TRUE; */
/* do_wall_stoch = FALSE; */
if (do_wall_stoch)
{
if (p->eddy_time <= DPM_SMALL)
{
check_time = TRUE;
p->user[N_EDDY] ++;
p->user[E_PLUS]=0.; /* reset interaction time */
110
vel_fric = sqrt(C_UDMI(c0,t0,TAU_WALL)/C_R(c0,t0));
get_rms_vel_fluc(Y_plus_p,vel_fric,Vprime_vec,I,J,K,c,c0,t0);
#if DEBUGGER
CX_Message("I0 = %f, I1 = %f, I2 = %f\n",I[0],I[1],I[2]);
CX_Message("J0 = %f, J1 = %f, J2 = %f\n",J[0],J[1],J[2]);
CX_Message("K0 = %f, K1 = %f, K2 = %f\n",K[0],K[1],K[2]);
CX_Message("U' = %f, V' = %f, W' =
%f\n",Vprime_vec[0],Vprime_vec[1],Vprime_vec[2]);
#endif
/*TL_plus = get_tl_plus(Y_plus_p);*/ /* calc. dimensionless time scale based on
particle y_plus */
eplus = get_tl_plus(Y_plus_p);
eps = eplus*vel_fric*vel_fric*vel_fric*vel_fric*C_R(c0,t0)/C_MU_L(c0,t0);
k =
0.5*(Vprime_vec[0]*Vprime_vec[0]+Vprime_vec[1]*Vprime_vec[1]+Vprime_vec[2]*Vprime_vec[2]);
/* calc. modified time scale */
p->eddy_time = TL_plus*C_MU_L(c0,t0)/C_UDMI(c0,t0,TAU_WALL); /* eq.
5 of proposal*/
p->eddy_time = 0.15*k/eps;
if (random_eddy_lifetime)
p->eddy_time *= -log(dpm_uniform_random(p)); /* Eq. 23.2-32 User's
Guide, characteristic eddy
lifetime*/
else
p->eddy_time *=2; /* Eq. 23.2-31 User's Guide, characteristic eddy
lifetime */
if (p->state.time == 0.0) p->eddy_time *= dpm_uniform_random(p);
if (p->eddy_time < te_min)
p->eddy_time = te_min;
/*store eddy time*/
p->user[E_PLUS] = eplus; /* time scale constant */
p->user[T_EDDY] = p->eddy_time; /* store eddy lifetime */
p->user[Y_PLUS_P] = Y_plus_p;
p->user[NEW_FORCE] = 1.;
/* velocity fluctuations are calculated using a Gaussian deviate random number
*/
for (i=0; i<3; i++) p->user[3+i] = dpm_gauss_random(p); /* random
fluctuations constant for t_eddy */
/*for (i=0; i<3; i++) p->user[LAMBDA1+i] = 1.0; /* random
fluctuations constant for t_eddy */
for (i=0; i<3; i++) p->V_prime[i] = p->user[LAMBDA1+i] *
Vprime_vec[i]; /******************/
if (msg_count%update_interval_up==0)
{
CX_Message("NEW: particle Y+ = %f , t_eddy = %g
, eplus = %f , dt = %g\n",Y_plus_p,p->user[T_EDDY],p->user[E_PLUS],P_DT(p));
CX_Message("NEW: u' = %f , v' = %f , w' = %f\n"
,p->V_prime[0],p->V_prime[1],p->V_prime[2]);
msg_count++;
}
}
else
{
111
check_time = FALSE;
/* p->eddy_time = p->user[1] - p->user[0]; */
vel_fric = sqrt(C_UDMI(c0,t0,TAU_WALL)/C_R(c0,t0));
get_rms_vel_fluc(Y_plus_p,vel_fric,Vprime_vec,I,J,K,c,c0,t0);
for (i=0; i<3; i++) p->V_prime[i] = p->user[LAMBDA1+i] * Vprime_vec[i];
/*****************/
if (msg_count%update_interval_old==0)
{
CX_Message("OLD: particle Y+ = %f , remaining eddy-
lifetime = %g ( = %3.2f \% ), dt = %g\n",Y_plus_p,p->eddy_time,100*p->eddy_time/p-
>user[T_EDDY],P_DT(p));
CX_Message("OLD: u' = %f , v' = %f , w' = %f\n",p-
>V_prime[0],p->V_prime[1],p->V_prime[2]);
msg_count++;
}
}
msg_count++;
for (i=0; i<dim; i++) c->V[i] += p->V_prime[i];/* add fluctuating vel. to mean cont.
phase vel*/
#if 0
CX_Message("u = %f ; v = %f ; w = %f ; t = %f ; Y_plus_p = %f ; Y =
%f\n ",c->V[0], c->V[1],c->V[2],p-
>state.time,Y_plus_p,Y_plus_p*C_MU_L(c0,t0)/sqrt(C_UDMI(c0,t0,TAU_WALL)*C_R(c0,t0)));
#endif
}
/* standard stochastic tracking outside of boundary layer */
if (do_stochastics)
{
if (p->eddy_time <= DPM_SMALL)
{
check_time = TRUE;
p->user[9] ++;
p->eddy_time = C_MU_FACTOR(p->injection) * c->tke / c->ted; /* Eq. 23.2-
23 User's Guide, fluid
Lagrangian integral time */
if (random_eddy_lifetime)
p->eddy_time *= -log(dpm_uniform_random(p)); /* Eq. 23.2-32 User's
Guide, characteristic eddy lifetime */
else
p->eddy_time *=2; /* Eq. 23.2-31 User's Guide, characteristic eddy
lifetime */
if (p->state.time == 0.0) p->eddy_time *= dpm_uniform_random(p);
if (p->eddy_time < te_min)
p->eddy_time = te_min;
/* CX_Message("tke %e, ted %e, eddy_time %e\n", c->tke, c-
>ted, p->eddy_time); */
/* velocity fluctuations are calculated using a Gaussian deviate
random number */
if (sg_rsm)
{
real random[3];
real chc11=0., chc12=0., chc13=0.;
real chc22=0., chc23=0., chc33=0.;
112
chc11 = sqrt(C_RUU(c0,t0));
chc12 = C_RUV(c0,t0)/chc11;
chc22 = sqrt( MAX(SMALL, C_RVV(c0,t0)-chc12*chc12) );
#if RP_2D
if (sg_swirl)
{
#endif
chc13 = C_RUW(c0,t0)/chc11;
chc23 = (C_RVW(c0,t0)-chc12*chc13)/chc22;
chc33 = sqrt( MAX(SMALL, C_RWW(c0,t0)-
chc13*chc13-chc23*chc23) );
#if RP_2D
}
else
chc33 = sqrt(C_RWW(c0,t0));
#endif
for (i=0; i<3; i++) random[i] = dpm_gauss_random(p);
p->V_prime[0] = random[0] * chc11;
p->V_prime[1] = random[0] * chc12 + random[1] * chc22;
#if RP_2D
if (rp_axi)
#endif
p->V_prime[2] = random[0] * chc13 + random[1] * chc23
+random[2] * chc33;
#if RP_2D
else
p->V_prime[2] = 0.;
if (rp_axi_swirl)
{
/* tracking done in 3-D, rotate fluctuation
velocity to particle */
/* position. No need for this if isotropic
fluctuations */
real R = sqrt(SQR(p->state.pos[1]) +
SQR(p->state.pos[2]));
real over_R =
R/SQR(MAX(R,DPM_SMALL));
real Vr=p->V_prime[1];
real Vt=p->V_prime[2];
p->V_prime[1] = (Vr*p->state.pos[1] -
Vt*p->state.pos[2])*over_R;
p->V_prime[2] = (Vr*p->state.pos[2] +
Vt*p->state.pos[1])*over_R;
}
#endif
}
else
{
Vprime_rms = sqrt(2./3. * c->tke); /* Eq. 23.2.27 User's
Guide, RMS fluctuating components */
for (i=0; i<3; i++) p->V_prime[i] = dpm_gauss_random(p) *
Vprime_rms; /* Eq. 23.2.28-30 User's Guide, velocity fluctuations */
}
113
}
else
{
/* rescale the remaining portion of eddy life time and rms values */
p->eddy_time *= (p->old_ted/p->old_tke) * (c->tke/c->ted);
for (i=0; i<dim; i++) p->V_prime[i] *= sqrt(c->tke/p->old_tke);
check_time = FALSE;
}
p->old_tke = c->tke;
p->old_ted = c->ted;
for (i=0; i<dim; i++) c->V[i] += p->V_prime[i];/* add fluctuating vel. to mean
cont. phase vel*/
}
Vrel = 0.0;
for (i=0; i<dim; i++)
Vrel += SQR(c->V[i]-pv[i]);
Vrel = sqrt(Vrel);
if (rp_visc)
p->Re = c->rho * P_DIAM(p) * Vrel / c->mu;
else
p->Re = 0.;
coeff = SphereDragCoeff(p->Re);
if (P_DIAM(p) != 0.0)
drag_factor = coeff * c->mu / ( P_RHO(p) * P_DIAM(p) * P_DIAM(p));
else
drag_factor = 1.;
if (check_time && (do_stochastics||do_wall_stoch) ) /* !!! */
{
real drag_f = MAX(drag_factor, ACCURACY);
/* limit the eddy_time by the eddy crossing time too */
real Le = C_MU_FACTOR(p->injection) / 1.225 * pow(c->tke,1.5)/ c->ted;
if (Le < Vrel/drag_f)
p->eddy_time = MIN(p->eddy_time,-2*log(1. -
Le/(Vrel/drag_f))/drag_f); /* Eq. 23.2.33 User's Guide, particle eddy crossing time */
/*CX_Message("std.-tracking: TL = %g \n",p->eddy_time);*/
}
return coeff;
}
/* set particle trajectory integration time step size */
DEFINE_DPM_TIMESTEP(dpm_ts_by_eddy_time,p, dt)
{
#if !PARALLEL
dt_min = RP_Get_Real("delta_t_min");
te_min = RP_Get_Real("t_eddy_min");
#endif
if (do_stochastics)
{
if (0) /* (msg_count%update_interval==0) */
{
CX_Message("dt %e, eddy_time %e\n", dt, p->eddy_time);
msg_count++;
}
if (dt>p->eddy_time)
114
return p->eddy_time;
}
if (do_wall_stoch)
{
if (0) /* (msg_count%update_interval==0) */
{
CX_Message("dt %e, eddy_time %e\n", dt, p->eddy_time);
msg_count++;
}
/*if (dt>p->eddy_time) return p->eddy_time;*/
dt = p->user[T_EDDY]/10.;
if (dt < dt_min) dt = dt_min;
}
return dt;
}
/* name particle scalars for post-processing */
DEFINE_DPM_SCALAR_UPDATE(dpm_sc_interact,cell,thread,initialize,p)
{
cell_t c = RP_CELL(&p->cCell); /* Get Cell and Thread from */
Thread *t = RP_THREAD(&p->cCell); /* Particle Structure using new macros*/
real pp[3] = {0.,0.,0.}; /* particle position vector */
NV_V(pp, =, p->state.pos); /* position of particle, x,y,z-coord. */
if (initialize)
{
/* this is the initialization call */
p->user[E_PLUS] = 0.; /* e_plus */
p->user[T_EDDY] = 0.; /* eddy lifetime */
p->user[Y_PLUS_P] = get_yplus_particle(c,t,pp); /* y_plus value at particle location */
p->user[LAMBDA1] = dpm_gauss_random(p);
p->user[LAMBDA2] = dpm_gauss_random(p);
p->user[LAMBDA3] = dpm_gauss_random(p);
p->user[T_FORCE_X] = 0.; /* thermophoretic body force x */
p->user[T_FORCE_Y] = 0.; /* thermophoretic body force y */
p->user[T_FORCE_Z] = 0.; /* thermophoretic body force z */
p->user[N_EDDY] = 0.; /* number of eddies */
p->user[NEW_FORCE] = 0.; /* flag for thermophoretic model */
strcpy(user_particle_vars[E_PLUS].name,"e_plus");
strcpy(user_particle_vars[E_PLUS].label,"e_Plus");
strcpy(user_particle_vars[T_EDDY].name,"t_eddy");
strcpy(user_particle_vars[T_EDDY].label,"t_Eddy");
strcpy(user_particle_vars[Y_PLUS_P].name,"y_plus");
strcpy(user_particle_vars[Y_PLUS_P].label,"Y_Plus");
strcpy(user_particle_vars[LAMBDA1].name,"lambda_1");
strcpy(user_particle_vars[LAMBDA1].label,"Lambda_1");
strcpy(user_particle_vars[LAMBDA2].name,"lambda_2");
strcpy(user_particle_vars[LAMBDA2].label,"Lambda_2");
strcpy(user_particle_vars[LAMBDA3].name,"lambda_3");
strcpy(user_particle_vars[LAMBDA3].label,"Lambda_3");
strcpy(user_particle_vars[T_FORCE_X].name,"t_force_x");
strcpy(user_particle_vars[T_FORCE_X].label,"t_Force_x");
strcpy(user_particle_vars[T_FORCE_Y].name,"t_force_y");
strcpy(user_particle_vars[T_FORCE_Y].label,"t_Force_y");
strcpy(user_particle_vars[T_FORCE_Z].name,"t_force_z");
115
strcpy(user_particle_vars[T_FORCE_Z].label,"t_Force_z");
strcpy(user_particle_vars[N_EDDY].name,"n_eddy");
strcpy(user_particle_vars[N_EDDY].label,"N_Eddy");
strcpy(user_particle_vars[NEW_FORCE].name,"new_force");
strcpy(user_particle_vars[NEW_FORCE].label,"New_Force");
}
else
{
/* not needed */
}
}
/* UD boundary condition for trapping particles on a wall and mark position of impact */
…
A.4 Local Flow Property Injection UDF
/*********************************************************************/
/* UDF that initializes particle injection properties */
/*********************************************************************/
#include "udf.h"
#include "cxiface.h"
DEFINE_DPM_INJECTION_INIT(injection_velocity,I)
{
Particle *p;
loop(p,I->p) /* Standard Fluent Looping Macro to get
particle streams in an Injection */
{
cell_t c;
Thread *t;
CX_Cell_Id cx_cell;
cx_cell=*CX_Find_Cell_With_Point(P_POS(p));
c=RP_CELL(&cx_cell);
t=RP_THREAD(&cx_cell);
P_VEL(p)[0] = C_U(c,t);
P_VEL(p)[1] = C_V(c,t);
#if RP_3D
P_VEL(p)[2] = C_W(c,t);
#endif
P_T(p)=C_T(c,t);
}
}
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