CFD SIMULATION AND ANALYSIS OF PARTICULATE DEPOSITION ON GAS TURBINE VANES

THESIS

Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in

the Graduate School of The Ohio State University

By

PRASHANTH S. SHANKARA, B. TECH

Graduate Program in Aeronautical and Astronautical Engineering

The Ohio State University

2010

Master's Examination Committee:

Dr. Jeffery P. Bons, Advisor

Dr. Ali Ameri

Dr. Jen Ping Chen

Copyright by

Prashanth S Shankara

2010

ii

ABSTRACT Syngas from alternate fuels is used as a fuel in land based gas turbine engines as a low-

grade fuel. The reduction in cost by use of these fuels comes at the cost of deposition

from particulate in the syngas on turbine blades, affecting the turbine performance and

component life. A computational deposition model was developed based on a model

developed at BYU to simulate and study the effects of deposition on gas turbine vanes

with film cooling. The deposition model was built using the CFD software, FLUENT

with User-Defined Functions (UDF) programmed in C language and hooked to FLUENT.

The particle trajectories were calculated by Euler-Lagrange method. The fluid flow and

heat transfer were solved first using RANS and deposition simulations were run as post-

processing in 3 steps moving, sticking and detachment. Improvements to the wall

friction velocity from the BYU model were incorporated and simulations on a bare 3D

domain showed reasonable agreement with experimental results and followed the trend of

capture efficiency decreasing with decreasing temperature. Deposition prediction on 3D

coupon with film cooling showed the relationship between hot-side surface temperature

and capture efficiency at different blowing ratios. Inaccurate prediction of hot-side

surface temperature resulted in higher capture efficiencies. Deposition patterns were

obtained from simulations using User-Defined Memory Locations to show number of

particles depositing at each location on the surface. Simulation of deposition on a VKI

iii

blade in 3D domain showed interesting insight into particle behavior at different

diameters. Smaller particles tended to follow the flow field and as the diameter increased,

the particles showed a tendency to keep their path along the line of injection and not

follow the flow field. Deposition predictions showed higher sticking efficiency at lower

diameter (~1) and very low sticking at higher diameters. A new Young modulus

correlation was developed to account for the dependence of particle Youngs modulus

and deposition on surface temperature. Simulations with new model improved

predictions on a very fine mesh with y+ less than 1 at blowing ratios, M=1 & 2 while

M=0.5 still showed larger capture efficiency due to inaccurate surface temperature

prediction.

iv

Dedication

To my Mom, Dad & Brother for always being there

v

ACKNOWLEDGMENTS

This work would not have been possible without the support of my advisor, Dr. Jeffrey P.

Bons, who offered me the chance to be a part of his wonderful research group and

believed in me just when I was at the crossroads of my graduate studies. Many thanks,

Dr. Bons, for your faith in me and the desire to live up to your expectations has

constantly pushed me to work harder. I would also like to thank my mentor, Dr. Ali

Ameri, for constantly being a source of immense knowledge, advice and most

importantly, for also being so understanding and supportive and always encouraging me

to see the light at the end of the tunnel. My sincere thanks go out to Dr. Jen Ping Chen for

being a part of my thesis committee and providing his valuable insights. I would also like

to thank the University Turbine Systems Research (UTSR) group for their financial

support. The numerical simulations were made possible through the use of

supercomputing resources provided by the Ohio Supercomputer Center (OSC). Special

thanks to Brett Barker who helped with the numerical simulations, Ai Weiguo for passing

on his knowledge of UDFs and also to Trevor Goerig and Curtis Memory for answering

my endless questions and making the lab a fun place to work at. A special note of thanks

to Dr. Gerald M. Gregorek. Our interactions may have been few but you were a source of

inspiration and a major reason behind me choosing to come to Ohio State.

This thesis would not have been possible without the unconditional love and support of

my family back in India. Thank you, Mom and Dad, for letting me follow my dreams and

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to follow my own path in life even though it was light years away from the norm. Special

note of thanks to all my wonderful friends for making everyday life fun at OSU.

vii

VITA

March 2003..SDAV Hr. Sec. School, Chennai, India

March 2007......B.TECH, Mechanical Engineering, SRM University, India

Sep 2007-Present....M.S., Aerospace Engineering, The Ohio State University

FIELDS OF STUDY

Major Field: Aeronautical & Astronautical Engineering

viii

TABLE OF CONTENTS

Abstractii

Acknowledgements..v

Vitavii

List of Tables...x

List of Figures.xi

Nomenclaturexiii

1. Introduction..1

2. Literature Review3

2.1. Eulerian Particle Tracking..3

2.2. Lagrangian Particle Tracking.....4

2.3. Turbulence Models....6

3. Particle Tracking Methodology10

3.1. Carrier Phase.10

3.2. Discrete Phase...12

3.3. Particle Trajectory Calculations...17

3.4. Coupling of discrete & continuous phase.20

3.5. Turbulent Particulate Dispersion..21

4. Particle Deposition Model.26

4.1. Particle-Wall Interaction...27

ix

4.2. Particle Sticking....27

4.3. Particle Detachment..30

4.4. Youngs Modulus Determination.32

5. Deposition Model Development in FLUENT 6.3.26...35

6. Particle Deposition on a Coupon..39

6.1. Boundary Conditions41

6.2. Carrier Phase Simulations.41

6.3. Particle Phase Simulations42

6.4. Results...43

6.A. Particulate Deposition on Coupon with Film Cooling.47

6.1A. Geometry & Grid Generation.47

6.2A. Boundary Conditions & Simulations..51

6.3A. Results.53

7. Application to VKI blade..63

7.1. Boundary Conditions65

7.2. Simulations & Results..66

8. Improvements to the Deposition Simulations...74

9. Conclusions & Recommendations..78

References.82

Appendix: Particle Deposition Model UDF Source Code

x

LIST OF TABLES

Table 3.1: Forces acting on the particle in the dispersed phase.17

Table 6.1: Summary of experiments for the bare coupon case..41

Table 6.2: Ash particle properties..42

Table 7.1: Boundary conditions for the VKI blade66

xi

LIST OF FIGURES

Fig: 3.1. Classification of flow regimes for gas-solid flows, Elgobashi, S.49).............20

Fig 4.1: Forces responsible for particle adhesion on a surface..28

Fig 5.1: Deposition Mode Flowchart.37

Fig. 6.1: Computational domain for the bare coupon simulation..39

Fig. 6.2: Geometry and boundary conditions of the model in 2D view.40

Fig 6.3: Cut section of the mesh along X-plane.41

Fig: 6.4: Impact efficiency vs Particle Diameter at 1453 K..43

Fig: 6.5: Capture Efficiency vs Gas Temperature.44

Fig: 6.6: Schematic of the 3D computational domain...48

Fig. 6.7: View of the tetrahedral volume mesh for the 3D case50

Fig. 6.8: Cut-section view of the volume mesh for the 3D case50

Fig 6.9: Surface mesh on the plate for 3D tetrahedral grid50

Fig. 6.10 Velocity magnitude contours (m/s) for M=0.5, 1, 2 along centerline

plane...54

Fig. 6.11: Surface temperature contours at different blowing ratios on 1 inch diameter

coupon55

Fig 6.12: Comparison of area-averaged surface temperature on coupon at 1453 K..56

Fig 6.13: Comparison of capture efficiency vs blowing ratio at 1453 K...57

Fig. 6.14: Velocity vectors along the centerline plane for M=158

Fig 6.15: Deposition patterns from the model for M=2.60

xii

Fig 6.16: Deposition patterns from the model for M=0.5..60

Fig 6.17: Deposition patterns from the model for M=1.60

Fig 6.18: Comparison of deposition from experimental & simulat ion for M=0.561

Fig 6.19: Comparison of deposition from experiments & simulation for M=0.2..61

Fig 7.1: Computational grid used for the VKI blade.64

Fig 7.2: Computational domain and Internal Region.65

Fig 7.3: Mach number contours for M=0.85 by OSU...66

Fig 7.4: Mach number contours for M=1.02 by OSU...67

Fig 7.5: Mach number contours for M=0.85 by El-Batsh.67

Fig 7.6: Mach number contours for M=1.02 by El-Batsh.67

Fig. 7.7: Particle trajectories through the passage for dp = 0.1m69

Fig. 7.8: Particle trajectories through the passage for dp = 1m....69

Fig. 7.9: Particle trajectories through the passage for dp = 10 m.70

Fig 7.10: Sticking Efficiency vs Particle Diameter at M=0.85..71

Fig. 8.1: Close up view of the boundary layer on the coupon...75

Fig. 8.2: View of the tetrahedral mesh for the computational domain..75

Fig. 8.3: Comparison of capture efficiency from new correlation with earlier results..77

xiii

NOMENCLATURE

Symbol

Ap Surface Area of particle, [m2]

Mean molecular speed, [m/s]

Cc, Cu Cunningham correction factor, [-]

Cd Coefficient of drag, [-]

Cp Specific heat of particle, [J/kg.K]

CL, C Coefficients for eddy lifetime model, [-]

d, D Cooling hole diameter, [m]

dij Deformation tensor, [m/s]

dp, Dp Particle diameter, [m]

E El-Batsh parameter, [Pa]

Ei Mean velocity of fluid, [m/s]

Ep Particle Youngs modulus, [Pa]

Es Surface Youngs modulus, [Pa]

FD Drag force on particle, [N]

Fpo Particle sticking force, [N]

Fs Saffman lift force, [N]

Fx Additional force term in the particle trajectory equation, [N]

gx Acceleration term in the particle trajectory equation, [m/s2]

Gk Generation of turbulence kinetic energy due to mean velocity gradients

G Generation of

xiv

hc Convective heat transfer coefficient, [W/m2K]

I Turbulence intensity, [%]

k Turbulence kinetic energy, [m2 /s2]

k1, k2 El-Batsh parameter constants, [-]

kf Thermal conductivity of the continuous phase, [W/m-K]

ks Sticking force constant, [-]

K Constant coefficient of Saffmans lift force, [-]

Kn Knudsen number, [-]

Kr Local velocity gradient, [m/s]

Le Eddy length, [m]

mp Mass of the particle, [kg]

M Blowing ratio, [-]

M Mach number, [-]

Nu Nusselt number, [-]

P Static pressure of fluid, [Pa]

Pr Prandtl number, [-]

P0 Total pressure of fluid, [Pa]

R Universal gas constant, [J/K.mol]

Rep Reynolds number of the particle, [-]

s Distance between cooling holes, [m]

Sk, S User defined source terms

S Ratio of particle density to fluid density, [-]

t Time, [s]

T Gas temperature, [K]

Tp Particle temperature, [K]

xv

T Free-stream temperature, [K]

Tavg Average between particle and surface temperature, [K]

u Fluid velocity, [m/s]

u Gaussian distributed random velocity fluctuation, [m/s]

u* Wall friction velocity, [m/s]

uj Instantaneous fluid velocity, [m/s]

up Particle velocity, [m/s]

uc Critical wall shear velocity, [m/s]

Uj Coolant velocity at exit of cooling holes, [m/s]

Uf Freestream velocity, [m/s]

vcr Capture velocity, [m/s]

vn Normal velocity, [m/s]

v Gaussian distributed random velocity fluctuation, [m/s]

w Gaussian distributed random velocity fluctuation, [m/s]

WA Work of sticking, [-]

y Distance of first cell center from the wall, [m]

y+ Dimensionless wall distance of first cell center from the wall, [m]

Yk Dissipation of k due to turbulence, [-]

Y Dissipation of due to turbulence, [-]

2 Volume fraction of dispersed phase, [-]

Turbulent dissipation rate, [m2/s3]

Mean free path of the gas molecules, [m]

Dynamic viscosity of fluid, [kg/m.s]

Kinematic viscosity, [m2/s]

p Poissons ratio of particle material, [-]

xvi

s Poissons ratio of surface material, [-]

Specific dissipation rate, [s-1]

, f Density of fluid, [kg/m3]

p Particle density, [kg/m3]

k Effective diffusivity of k, [m2/s]

Effective diffusivity of , [m2/s]

Particle relaxation time, [s]

Wall shear stress, [Pa]

Lifetime of eddy, [s]

1

1. INTRODUCTION

Land-based turbine manufacturers have recently moved towards low-grade fuels in an

effort to reduce the high costs associated with high-quality fuels. These low-grade fuels

are usually gasified to produce syngas which contains a number of impurities. These

impurities are the major causes of the DEC (Deposition, Erosion & Corrosio n)

phenomenon in gas turbine engines. The phenomenon of deposition from syngas fuels on

turbine vanes is of specific interest in this thesis. Various experiments have been

conducted to study the effects of deposition on turbine vanes and blades, especially on

the film cooling and heat transfer in the turbine. Numerical simulation of deposition is

highly important to corroborate the results obtained from the experiments and also to

shed light on various deposition issues that might be hard to decipher in the experiments.

El-Batsh et al., (44), Ai & Fletcher (17), Hamed et al., (62), Brach et al., (59), Soltani &

Ahmadi (57), Greenfield & Quarini (24), Guha (15) and Wang & Squires (14) have all

performed numerical simulations for varying cases to simulate part icle trajectories,

particle impact, sticking, deposition and so forth. Bons et al., (44) have been conducting

deposition experiments on gas turbine material coupons and turbine vanes with particular

interest on the effect of deposition on film cooling. Numerical simulations have been

performed in concurrence with these experiments at various stages to validate the

experimental results and also with a goal to build a numerical model that can effectively

simulate deposition conditions inside a gas turbine. This thesis is an extension of the

earlier numerical simulations and is aimed at building & delivering a particle deposition

2

model that can be applied to future simulations of deposition on a turbine vane with film

cooling.

3

2. LITERATURE REVIEW

Numerical simulation of particle deposition has been performed by various researchers

previously although cases of deposition simulation on gas turbine vanes with film cooling

are few and far between. Initial simulations of particle transport and deposition were

aimed at analyzing the effect of parameters like surface temperature, particle diameter,

particle temperature, turbulent dispersion etc. on deposition. Two different approaches

for particle deposition have been dealt with in literature namely Eulerian and

Lagrangian. The gas phase is always modeled by the Eulerian approach where the gas is

treated as a continuum and can be solved either by RANS simulations (Reynolds

Averaged Navier Stokes) or DNS/LES (Direct Numerical Simulations/Large Eddy

Simulations). The Eulerian-Eulerian method performs particle tracking by focusing on

the control volume while the Eulerian-Lagrangian method focuses on the particle tracks

instead. The Eulerian method considers particles as a continuum and develops the particle

tracks based on the conservation equation applied on a control volume basis with

particles grouped together under various control volumes. The gas and particle phases are

treated as interpenetrating continua and are coupled together by exchange coefficients.

2.1) Eulerian Particle Tracking

The Eulerian particle tracking is the most preferred method for indoor environments as

shown by Murakami et al., (1) and Zhao et al., (2 & 3). Friedlander and Johnstone (18)

and Davies (19) developed the first deposition model based on an Eulerian approach.

4

Menguturk & Sverdrup (4) developed an Eulerian model based on the assumption that

the particles were very small and hence the inertia effect can be ignored. Dehbi (5) noted

that the Eulerian approach is very suitable only for flows with dense particle suspensions

where the particle-particle interaction is too large to ignore. The inter-phase exchange

rates and the closure laws, in addition to the strong coupling between the phases have to

be accurately defined for a proper Eulerian simulation which often presents quite a

challenge. Yau & Young(6), Wood(7) & Kladas(8) used the Eulerian method by solving

all particles on the basis that they were outside the boundary layer, thereby solving the

continuity equations under turbulent flow conditions. Huang et al.,(9) and Ahluwalia et

al.,(10) used the Eulerian deposition model to simulate the deposition of fine particles on

coal-fired gas turbines. They both considered the effects of Brownian diffusion, turbulent

diffusion and thermophoresis on the particles.

2.2) Lagrangian Particle Tracking

The Lagrangian approach treats the particles as a dispersed phase and tracks individual

particles. The particle volume fraction is usually assumed negligible compared to the

carrier phase volume and particle-particle interactions are usually neglected. Kallio &

Reeks (11) calculated the deposition of particles in a simulated turbulent flow field using

the Lagrangian model in a turbulent duct. They solved the equation of motion for

particles with relaxation time ranging from 0.3 to 1000. The particle relaxation time is a

measure of particle inertia and denotes the time scale with which any slip velocity

between the particles and the fluid is equilibrated. The relaxation time is usually the time

required by the particle to respond to changes in fluid velocity and depends on particle

size, particle density and fluid viscosity. The particle relaxation time is:

5

(2.1)

Their model showed very good agreement with the experimental data of Liu & Agarwal

(20). Ounis et al., (12) and Brooke etal., (13) used the Lagrangian approach whilst

solving the carrier phase flow by DNS method while Wang & Squires (14) used LES to

simulate the flow field in their Eulerian-Lagrangian calculations. Guha (15) noted that

when particle motion is significantly affected by turbulence and the fluctuating flow field

velocities become important, Lagrangian calculations are needed. Lagrangian approach

provides a more detailed and realistic model of particle deposition because the

instantaneous equation of motion is solved for each particle moving through the field of

random fluid eddies. This method is valid for all particle sizes as particles are treated

individually. Moreover, it provides information about particle collision at the surface

which is helpful while incorporating the sticking model. El-Batsh et al., (16) pioneered

the Lagrangian DPM (Discrete Phase Method) modelin the CFD software, FLUENT by

developing a deposition model based on Eulerian-Lagrangian approach and successfully

demonstrated the model for various experimental cases. This deposition model was based

on three processes: particle transport, particle sticking and particle detachment and serves

as the basis for the development of the OSU model. Ai et al., (17) employed the El-Batsh

study the particle-wall interaction in the previous phase of this research study. They

developed and validated the deposition model with experimental results of deposition on

a bare and TBC coated coupon with film cooling. The OSU model is an extension of the

model used by Ai et al., (17) and is intended to extend the applicability of the deposition

model to actual turbine vane geometries with film cooling.

6

The turbulence model used in the Eulerian simulation of the flow field is usually chosen

based on the flow physics. RANS simulations assume isotropic turbulence which is not

the case near-wall and hence the accuracy is affected due to empiricism in the turbulence

model. El-Batsh et al., (16) and Ai et al., (17) both used the RANS in their simulations.

DNS solves the exact Navier-Stokes equations without any empiricism or modeling and

hence is very computationally extensive. LES is mainly used for unsteady flows as it uses

small and more universal scales and minimizes empiricism in turbulence modeling. The

relevant turbulent eddies are resolved and the unsteady flow is represented accurately.

Shah (21) used LES to study the particle transport in an internal cooling ribbed duct.

Iacono et al., (22) used LES with one-way coupling for particle deposition onto rough

surfaces with good results. Mazur et al (23) used LES with the CFD software, FLUENT

for particle deposition simulations on turbine vane successfully. Greenfield & Quarini (24

& 25) modeled the turbulence as a series of random eddies with a lifetime of their own

and associated random fluctuating velocities. Abuzeid et al., (26) modeled the transport

of particles in turbulent flow field using both Eulerian & Lagrangian simulations. They

found that the Lagrangian simulation was more accurate than Eulerian for various particle

sizes. In addition, Lagrangian particle tracking calculations provided information about

number of particles impinging on the surface, impinging velocity and particle direction

relative to the surface.

2.3) Turbulence Models:

El-Batsh et al., (16) used the Standard k- model and the RNG k- model in conjunction

with the Standard wall function and the two-layer zonal model. This choice was based

more on the fact that the CFD software, FLUENT did not offer the k- turbulence

7

models at that time. The turbulence model affects the particle trajectory through the

turbulent kinetic energy which is used to calculate the fluctuating velocities. There is not

enough literature on the effect of various turbulence models on particle transport,

although selection of the right turbulence model based on the flow field characteristics

will ensure accurate flow field prediction which affects particle transport. Ajersch et al.,

(27) reported that the k- model gave better prediction of the near-wall flow structures

compared to the k- model. Silieti et al., (28) predicted film-cooling effectiveness using 3

different turbulence models: the Realizable k- model (RKE), Standard k- model and

the v2-f model. They also compared the same for different mesh groups, namely

hexahedral, hybrid and tetrahedral grids. They noted that tetrahedral grids needed enough

near wall resolution to accurately predict the film-cooling effectiveness which has been

found to influence the particle deposition greatly in litera ture. The deposition on a surface

depends on the particle and the surface temperature. Higher film cooling effectiveness

leads to lesser deposition due to the cooler temperatures on the surface. Particle

deposition also affects the film cooling effectiveness. Presence of surface roughness and

blockage of film cooling holes due to deposition seriously affect film cooling

effectiveness and performance. Bogard et al., (67) conducted deposition experiments

using molten wax materials and found that leading edge film cooling reduced the

deposition compared to no film cooling. They also found that deposition decreased

cooling effectiveness by as much as 25%. Their simulations showed that the RKE model

better predicted the film-cooling effectiveness in the region of 2 x/D 6. Harrison and

Bogard (29) found that the standard k- (SKW) model

8

best predicted laterally averaged adiabatic effectiveness, and that the realizable k- model

was best along the centerline. El-Batsh et al., (16) showed that the Std. k- model with

standard wall functions over-predicts the deposition velocity for particles with relaxation

time less than 10. The std k- model has been used to simulate indoor flow field

successfully by Zhang & Chen (30). Jovanovic et al (31) et al used std k- model for

their two-phase flow modeling of air-coal mixture channels with single blade turbulators.

Theodoridis (32) et al used a std k- model with std wall function for simulation of

turbine blade film cooling without lateral injection. Turbulence intensity predicted in the

stagnation region was not realistic and an anisotropy correction was applied for better

prediction. York and Laylek (33) used a realizable k- model which over predicted the

results in the region between stagnation line and the second row of holes. The general

idea from the literature is that accounting for anisotropic effects is important when using

the standard turbulence models in FLUENT. Ai et al., (17) used Std. k- model with

RANS to compute flow field and heat transfer for their analysis of particle deposition on

a turbine coupon. The k- model was used to eliminate the use of wall functions, thereby

eliminating the approximation of particle trajectory near the wall and resolving the actual

trajectory equations for better prediction. The k- model depends on isotropic turbulence

assumption which is not the case near wall and hence leads to over-prediction of

deposition velocity for particles with relaxation time less than 10. Jin (64) noted from his

simulations that the standard wall functions used with the k- model generate unrealistic

large steady state velocities within the boundary layer leading to large deposition

velocities for particles with relaxation time less than 10. So et al (63) proved in their

simulations that the k- turbulence model over-predicts the turbulent kinetic energy

9

within the viscous sub- layer where the turbulence energy damps out much faster. This

causes some regions in the viscous sub-layer to acquire abnormally high fluctuating

velocities normal to the wall. Particles with low relaxation time change quickly with the

flow field changes and the high normal velocities in the flow cause these particles to

acquire a high normal velocity, leading to over-prediction of deposition velocity. The k-

turbulence model can be applied throughout the boundary layer provided the near wall

mesh resolution is sufficient. This model requires no special near-wall treatment and

hence will be used in this study.

10

3. PARTICLE TRACKING METHODOLOGY

The particle tracking methodology used in the OSU model is based on the Discrete Phase

Model (DPM) of the CFD simulation software, FLUENT 6.3.26. FLUENT 6.3.26 was

chosen after a careful consideration of various commercial CFD packages. Zevenhoven

(34) compared 6 different CFD packages with particle tracking capabilities. Although

STAR-CD was found to be the most versatile, he noted that FLUENT is very capable in

cases where particle-particle interaction is negligible. Hence, FLUENT was used for

particle tracking simulations through the Ohio Supercomputer Center (OSC). Particle

tracking in FLUENT is divided into two phases: Carrier phase and Discrete phase. The

DPM model follows Lagrangian particle tracking and hence the carrier phase flow field is

solved initially and allowed to reach a steady state before the discrete phase is injected

into the carrier phase. The particles are considered to be in the discrete phase since the

particle loading volume is considerably negligible in all cases compared to the carrier

phase volume.

3.1) Carrier Phase:

The flow field is assumed to be single phase, incompressible and Newtonian. The effect

of particles on the flow field is negligible and is not taken into account. The RANS

equations will be used as the governing equations to transport the flow field quantities.

LES is computationally intensive and needs several computers using the same jobs to

process different datasets on different CPUs simultaneously. DNS is expensive for the

current problem and not available in FLUENT. The conservation equations for mass and

11

momentum are solved for all flows with an additional energy equation solved for cases

with heat transfer or compressibility. The initial cases of validation of the model with

results from Ai et al., (17) are incompressible flows and the later cases with turbine vanes

are compressible flows and solved accordingly.

The governing equations for the carrier phase are from the FLUENT manual (35) as:

)()()(ji

jj

i

jij

i

j

i uuxx

u

xx

p

x

uu

t

u

(3.1)

where ji

uu is the Reynolds stress. Standard k- turbulence model will be used for the

closure equations, which is based on the Wilcox k- model. The turbulence kinetic

energy, k, and the specific dissipation rate are obtained from the following transport

equations:

kkk

j

k

ji

iSYG

x

k

xx

ku

t

k

)()

)((

(3.2)

and

SYGxxx

u

tj

k

ji

i

)()

)((

(3.3)

Where k is the turbulent kinetic energy, is the specific dissipation rate. In these

equations, Gk represents the generation of turbulence kinetic energy due to mean velocity

gradients. G represents the generation of . k and represent the effective diffusivity

of k and , respectively. Yk and Y represent the dissipation of k and due to turbulence.

Sk and S are user-defined source terms.

12

3.2) Discrete Phase

FLUENT predicts the trajectory of a discrete phase particle (or droplet or bubble) by

integrating the force balance on the particle, which is written in a Lagrangian reference

frame. This force balance equates the particle inertia with the forces acting on the

particle, and can be written (for the direction in Cartesian coordinates) as:

(3.4)

Where the first term on the right hand side is the drag force on the particle per unit

particle mass, the second term is the effect of gravity on the particle and Fx indicates all

other additional forces.

In all the simulations in this study, the following assumptions were made regarding the

dispersed phase based on the experimental conditions and the particle characteristics used

in the experiments:

The particles are rigid spheres and they are considered as points located at the

center of the sphere.

The particle density is substantially larger than the fluid density.

Inter-collision forces are neglected due to low volume fraction of the particles.

Particles do not affect fluid turbulence. Experiments by Kulick et al., (37) and

Kaftori et al., (38) have shown that for low volume fractions the turbulence

modifications are negligible. Also, in the near-wall region where the particle

concentration may be locally large, the turbulence intensities are modified by a

very small amount and can be neglected.

For the particle sizes considered in the study, sub-grid scales have a negligible

effect on particle trajectories.

13

The effect of material roughness is not considered when bouncing the particles

from the wall.

The particles in the flow field are assumed to be spherical particles throughout this study

and are subjected to various forces as explained by Rudinger (36) as follows:

Fp = drag force + added mass effect + history effect + gravitational force + Buoyancy

force + Lift force + Intercollision force + Brownian force + Thermophoresis force +

Magnus force + Basset Force

These forces have been discussed extensively by El-Batsh et al., (17) and many others in

previous literature and hence only a brief description of these forces is provided here.

Various deposition models in literature have used either one or a combination of the

forces mentioned above based on the characteristics of the particle flow expected.

Identification of the forces that affect the particle regime for a particular case is extremely

important for accurate tracking of the particle trajectory.

Drag force is the Stokes drag that acts on the particle due to the relative velocity between

the fluid and the particle and acts in the direction of the flow. The drag force is the most

dominant force for particle motion, especially when the particle Reynolds number is less

than 100. The drag force is based on the Stokes law when Rep

14

is represented by the third term. This force depends on the history of the particle

trajectory and hence is called the history effect. The effect of the added mass and

Basset forces is negligible for particles with density substantially larger than the fluid

density. Sommerfeld (41) and Elgobashi and Truesdell (39) showed that the Basset forces

are only important for particles with (p/f

15

important for sub-micron particles. Brownian force is caused by the random impact of

particles with agitated gas molecules. Talbot et al., (42) showed that the thermophoretic

force is caused by the unequal momentum exchange between the particle and the fluid.

This force is caused by the unequal momentum exchange between the particle and the

fluid. The higher molecular velocities on one side of the particle due to the higher

temperature give rise to more momentum exchange and a resulting force in the direction

of decreasing temperature. Both these forces are neglected as the particles considered in

this study are larger than 0.03 m. El-Batsh et al (16) noted that based on the results of

Talbot et al., (42), rarefaction effects are important when the particles are in the sub-

micron region as there is a reduction in the drag coefficient. In such a situation, the gas

flow around the particle cannot be regarded as a continuum. Instead, the particle motion

is induced by collisions of gas molecules with the particle surface. The rarefaction is

important in the non-continuum regime which is decided by the Knudsen number (Kn).

The Knudsen number is defined as the ratio of the mean free path of the gas molecules to

the particle size.

(3.5)

And

(3.6)

and the mean molecular speed are given by:

(3.7)

where R is the gas constant.

16

All experimental cases in this study have 0.1

17

Table 3.1: Forces acting on the particle in the dispersed phase

FORCE Domain of Importance Included

1 Drag Dominant force for particle motion; Rep

18

Drag Force with Cunningham correction factor for rarefaction effect

Saffman Lift Force

Accordingly, eq. 3.4 can be re-written as follows to calculate the particle trajectory by

integrating the following equation of motion (in the x direction):

(3.9)

where Cd is the drag coefficient. The first term on the right hand side represents the drag

force per unit particle mass and the second term contains only the Saffman Lift Force.

FLUENT provides controls to include the Cunningham correction factor and the Saffman

force in the particle trajectory calculations. A user-defined subroutine can also be used to

include these forces. The drag force is given by:

(3.10)

where the Cunningham correction factor is

(3.11)

where is the molecular mean free path.

The Saffman Lift force was initially given by Saffman (47) as:

(3.12)

where Kr is the local velocity gradient.

However, this expression was originally derived for an unbounded shear flow and does

not include the effects due to proximity of the wall and finite Reynolds numbers. A more

accurate representation of this force is given by Li and Ahmadi (48) who used the

following generalized expression of the force for three-dimensional shear fields:

19

(3.13)

where, is the velocity of the particle, d is the particle diameter, S is the ratio of particle

density to fluid density, is kinematic viscosity, K = 2.594 is the constant coefficient of

Saffman's lift force and is the instantaneous fluid velocity with uj = Ej + u:, where Ej is

the mean velocity of the fluid, and u is its fluctuating component. is the deformation

tensor and is given by:

(3.14)

The expression for the Saffman lift force is restricted to small particle Reynolds number.

In addition, the particle Reynolds number based on the particle-fluid velocity difference

must be also smaller than the square root of the particle Reynolds number based on the

shear field. The calculation of heat transfer to or from the particles in this study

considered only heating or cooling of the particles and neglected any phase changes or

particle radiation. The particle energy equation in terms of particle temperature is given

by:

) (3.15)

where mp is the particle mass, Cp is the particle specific heat, Tp is the particle

temperature, Ap is the surface area of the particle, and hc is the convective heat transfer

coefficient. The assumption is made that the particle has no effect on the fluid flow due to

the dilute particle flow. The convective heat transfer coefficient is evaluated using the

correlation given by FLUENT (35) and Crowe et al (45):

(3.16)

20

where Nu is the Nusselt number and Pr is the Prandtl number. The Biot number in these

experiments in less than 0.1 and there is negligible internal resistance to heat transfer at

these Biot number values. The Biot number for 5m, for example, was 0.043. The body

has high internal conductivity at these values and the temperature change remains the

same. Hence, the body or particle has uniform temperature throughout and the lumped

mass system approximation can be used to solve the Heat Transfer

3.4) Coupling of discrete and continuous phase:

Goesbet et al. (46) showed in their studies that there can be three types of coupling for

solid particles in turbulent flows as follows:

One-way coupling: Effect of turbulence on particle trajectories and dispersion

Two-way coupling: Effect of particles on turbulence

Four-way coupling: Effect of particles on each other

Usually, the particulate flow in compressor or turbine regimes is very dilute flow. The

present problem can be modeled with one-way coupling as the particle volume is very

low compared to the flow volume and hence the effect of particles on turbulence and on

each other is very negligible.

Fig: 3.1. Classification of flow regimes for gas-solid flows, Elgobashi, S. (49)

21

Fig 3.1 shows the various flow regimes and the type of coupling suitable between carrier

and dispersed phase. 2 is the volume fraction of the dispersed phase, k is the

Kolmogorov time scale in seconds, t1 is the Lagrangian Integral time scale in seconds

and x12 is the particle relaxation time in seconds.

3.5) Turbulent Particulate Dispersion

One of the prominent characteristics of turbulent flows is their diffusivity. Turbulence is

able to mix and transport species, momentum and energy much faster than is done by

molecular diffusion. Turbulent dispersion is best studied from a Lagrangian viewpoint by

following the motion of fluid elements. Kuo (50) noted that turbulent dispersion can be

accounted for by either a deterministic or Stochastic model. Deterministic models take

into account the slip velocity and calculate the interface mass/heat transport rates using

the slip velocity by taking into account the Reynolds number and the Sherwood/Nusselt

number. Stochastic models are similar to the deterministic models but they also take into

account the effect of turbulent fluctuations on particle motion and interface t ransport.

The particle dispersion in the turbulent flow field can be accounted for by two methods in

FLUENT (35):

(1) Stochastic tracking/Discrete Random Walk (DRW) model

(2) Particle cloud approach

For a case of steady state particle tracking, FLUENT simulates particle streams rather

than individual particles. The one-way coupling method is generally used to simulate the

particle tracks. Information about the discrete phase concentration can only be obtained

by two-way coupling.

22

As explained by Tian & Ahmadi (66), in turbulent flow field, turbulence diffusion by

instantaneous flow fluctuations is the main mechanism for particle dispersion and

depositions. This is in addition to the other mechanisms such as molecular diffusion,

convective transport and gravitational sedimentation. Therefore, it is critical to

incorporate appropriate model for simulating turbulence fluctuations for accurate analysis

of particle transport and deposition processes. The most faithful simulation of fluctuation

velocity should be able to capture the details of the turbulence eddy structures. Currently,

this is only possible by DNS that is only practical for low Reynolds number duct flows.

For practical applications, however, turbulence fluctuation is mainly estimated using a

variety of stochastic approaches.

In the DRW model, each injection is tracked repeatedly to obtain a statistically

meaningful sampling. The number of tries option in the Injections panel in FLUENT

is used to set the number of times every injection needs to be tracked. Mass flow rates

and exchange source terms for each injection are divided equally among the multiple

stochastic tracks. Without Stochastic Tracking, only one particle trajectory is calculated

for each injection point and the effects of turbulence are ignored which is not a valid

assumption. FLUENT uses a probability distribution function (PDF) for calculation of the

perturbation in flow field velocities. For n number of stochastic tries, n values of

perturbation are calculated for n different regions in the PDF and n different particle

tracks are generated from the same injection point. But the mass flow rate for the

injection at that point will be divided equally among the n particles, thus matching the

total mass flow rate through the inlet while accounting for the particle dispersion. This

accounts for the dispersion effect and ensures the deposition calculation is performed for

23

n different trajectories instead of just one particle track, thus being more representative

of deposition. The DRW model moves each particle through the medium using the

velocity field obtained from the solution of the flow equation to simulate advection and

adds a random displacement to simulate dispersion. Hence, the transport equations are

not solved directly and the approach is free of numerical dispersion and artificial

oscillations.

The DRW model is also popularly known as the Eddy Interaction Model and was

developed by Gosman & Ioannides (51). The EIM is a stochastic random walk treatment

in which particles are made to interact with the instantaneous velocity field u+u(t),

where u is the mean velocity and u(t) the fluctuating velocity. By computing the paths of

a large enough number of particles, the effects of the fluctuating flow field can be taken

into account. In essence, the EIM aims at reconstructing the instantaneous field from the

local mean values of velocity and turbulent intensity. The EIM models the turbulent

dispersion of particles as a succession of interactions between a particle and eddies which

have finite lengths and lifetimes. It is assumed that at time to, a particle with velocity up is

captured by an eddy which moves with a velocity composed of the mean fluid velocity,

augmented by a random instantaneous component which is piecewise constant in time.

When the lifetime of the eddy is over or the particle crosses the eddy, another interaction

is generated with a different eddy, and so forth. One drawback of the EIM/DRW model is

that it does not account for the strong anisotropic nature of turbulence inside the

boundary layer as it is based on an assumption of isotropic turbulence. Based on the

model of Gosman and Ioannides (51), the eddy has the following length and lifetime:

(3.17)

24

(3.18)

where k and are respectively the turbulent kinetic energy and dissipation rate, while Cs

are constants. In FLUENT, the fluid velocity fluctuations are assumed isotropic and the

rms values of the velocity are obtained from the following relationship:

(3.19)

Where (3.20)

Each eddy is characterized by a Gaussian distributed random velocity fluctuation and a

time scale in this model. Dehbi (5) successfully included a boundary layer model which

models the turbulence differently inside and outside the boundary layer. Although this

would be a much more accurate representation of particle dispersion in turbulent flows,

previous deposition studies have used the default isotropic FLUENT model with success

for particle deposition studies.

The particle cloud model considers the statistical evolution of a particle cloud about a

mean trajectory. A particle cloud is required for each particle type in this model. The

concentration of particles about the mean trajectory is represented by a Gaussian

probability density function (PDF) whose variance is based on the degree of particle

dispersion due to turbulent fluctuations. The mean trajectory is obtained by solving the

ensemble-averaged equations of motion for all particles represented by the cloud. The

cloud enters the domain either as a point source or with an initial diameter. The cloud

expands due to turbulent dispersion as it is transported through the domain until it exits.

As mentioned before, the distribution of particles in the cloud is defined by a probability

density function (PDF) based on the position in the cloud relative to the cloud center. The

25

value of the PDF represents the probability of finding particles represented by that cloud

with residence time t at location x i in the flow field. This model is computationally less

expensive but is less accurate since the gas phase properties like temperature are

averaged within a cloud. Hence, the Stochastic DRW model was chosen to model the

turbulent dispersion of particles.

26

4. PARTICLE DEPOSITION MODEL

The OSU deposition model based on the previous deposition models of El-Batsh et al.,

(16) and the BYU model by Ai et al., (17). The main goal of the deposition model was to

accurately model the particle-wall interaction and to improve upon the BYU model,

while extending the applicability to simulate deposition on a 3D turbine vane with film

cooling. FLUENT has built- in conditions and offers the following boundary conditions

when a particle strikes a boundary face:

Reflect elastic or inelastic collision

Trap particle is trapped at the wall

Escape particle escapes through the boundary

Wall-jet particle spray acts as a jet with high Weber number & no liquid film

Wall- film stick, rebound, spread & flash based on impact energy & wall

temperature

Interior particle passes through an internal boundary zone

Since none of these boundary conditions accurately represent the particle-wall interaction

in the compressor and turbine regimes, a deposition model was built- in FLUENT using

User Defined Functions (UDF) which would serve as the boundary condition for

modeling particle-wall interaction. A UDF is a routine (programmed by the user) written

in C using standard C functions and pre-defined FLUENT macros that can be

dynamically linked with the solver. The source files containing UDFs can either be

interpreted or compiled by the user in FLUENT.

27

4.1) Particle-Wall Interaction:

Interaction of a particle with a surface usually results in sticking and buildup (deposition),

impact removal of the surface (erosion) and chemical buildup (corrosion). This study is

primarily concerned with modeling the deposition under the assumption of smooth

surfaces. The modeling process will deal with the build-up of deposition and the

subsequent effect on film cooling effectiveness due to this buildup in the next stage.

The particle-wall interaction leading to deposition is a two-step process, involving a

purely mechanical interaction and a fluid dynamic interaction. The mechanical

interaction called the sticking process is the determination of whether the particle sticks

to the surface when it comes into contact with a wall. The sticking model is based on the

previous adhesion models in literature which consider the elastic properties of the particle

and the surface only under dry conditions. Once the particle sticks, the next process is to

determine whether the particle remains stuck to the surface or is removed from the

surface based on the critical moment theory. This step is called the detachment process

and is the fluid dynamic interaction.

4.2) Particle Sticking:

Extensive reviews of particle adhesion/sticking have been provided in literature by Corn

(52), Krupp (53), Visser (54), Tabor (55) and Bowling (56). There are three main forces

that contribute to particle adhesion as shown in fig 3.2 and they are:

Van der Waals force Arises due to molecular interaction between solid surfaces

Electrostatic force Caused by charging the particles electrically in the gas

stream

28

Liquid Bridge force Caused by the formation of a liquid bridge between

particles and surface. In gas turbines, the use of low-grade fuels containing alkali

components gives rise to alkali vapor in addition to ash. If the temperature in the

thermal boundary layer is lower than the dew point of the alkali vapors, the alkali

vapors condense and form the liquid bridge.

Fig 4.1: Forces responsible for particle adhesion on a surface

From the literature mentioned above, Soltani & Ahmadi (57) concluded that the Van der

Waals force is the major contributor to surface adhesion under dry conditions. The Van

der Waals force was calculated by either a microscopic or a macroscopic approach. The

microscopic approach was based on the interactions of the individual molecules, while

the macroscopic approach dealt directly with the bulk properties of the interacting bodies.

One shortcoming of these early theories was that the effect of contact deformation on the

adhesion force was neglected. Johnson, Kendall, and Roberts (58) used the surface

energy and surface deformation effects to develop an improved contact model. This

model was nicknamed the JKR theory. According to this model, at the moment of

separation, the contact area does not disappear entirely; instead, a finite contact area

exists. Soltani & Ahmadi (57) used the JKR theory as a basis to form the evaluation of

the minimum critical shear velocity to be used in the critical moment theory for particle

29

detachment. Based on all the previous literature on deposition, El-Batsh et al., (16) put

together a complete deposition model to model the sticking and detachment process. The

JKR model gives the sticking force based on the particle size and material properties with

constants being derived from experiments.

This sticking force is given by the JKR model as:

(4.1)

where ks is a constant equal to 3/4. The Work of Sticking, WA is a constant which

depends upon the material properties of the particle and of the surface and has the units of

J/m2. This constant is obtained experimentally for some materials. For any particle, the

co-efficient of restitution is defined as the ratio of the particle rebound velocity to the

particle normal velocity. As the particle normal velocity decreases, the particle rebound

velocity decreases and eventually reaches a point where no rebound occurs and the

particle is captured. This velocity at which capture of a particle occurs is known as the

capture/critical velocity. Brach and Dunn (59) formulated an expression to calculate the

capture velocity of a particle using a semi-empirical model. In this model, the capture

velocity of the particle was calculated based on the experimental data and is given as

follows:

(4.2)

where E is the composite Youngs modulus which is determined based on the Youngs

modulus of the particle and the surface. The particle normal velocity (vn) is then

compared to the capture velocity. If the particle normal velocity is less than the capture

velocity, the particle sticks to the surface; else, it rebounds.

vn < vcr - particle sticks; vn > vcr - particle rebounds

30

Once the particle rebounds, it continues on its trajectory until it leaves the domain or

impacts the surface again. The El-Batsh parameter is based on the Youngs modulus of

the particle and the surface and is given as:

E (4.3)

and (4.4)

and (4.5)

where vcr is the particle capture velocity [m/s], Es is the Young's modulus of surface

material [Pa], s is the Poisson's ratio of surface material, Ep is the Young's modulus of

particle material [Pa], p is the Poisson's ratio of particle material, dp is the particle

diameter [m] and p is the particle density [kg/m3].

4.3) Particle Detachment

A particle may be detached from a surface when the applied forces overcome the

adhesion forces. Therefore, particles may lift-off from the surface, slide over it or roll on

the surface. These detachment mechanisms have been discussed by Wang [60], among

others. The critical moment theory of Soltani & Ahmadi (57) is used to determine the

detachment of particle from the surface. Here, the critical wall shear velocity is defined

as:

(4.6)

where uc is the critical wall shear velocity, Cu is the Cunningham correction factor, dp is

the diameter of particle and Kc is the El-Batsh parameter. The particle will be removed

from the surface if the turbulent flow has a wall friction velocity ( ) where w

31

is the wall shear stress) which is larger than uc. Detachment occurs when the fluid

dynamic moment in the viscous sublayer exceeds the moment exerted o n the particle by

the sticking force. A user-defined function (UDF) was created in the C programming

language using the various UDF macros available to create a deposition model which

would determine the capture velocity and critical wall shear velocity of every particle that

hits the surface and create a dataset for all particles that deposit on the surface. The

development of the UDF and its incorporation into FLUENT will be dealt with in the

next section. The BYU model calculated the wall friction velocity of the particle based on

the particle velocity instead of the gas velocity. This was based on the assumption that the

particle and the gas phase are in equilibrium. The wall shear stress is usually given by:

(4.7)

Where u is the time-averaged velocity at the wall and the shear stress is the shear stress

calculated at the wall.

The BYU model used the particle velocity at the center of the first cell near the wall and

the corresponding distance of the cell center from the wall to calculate the velocity

gradient resulting in the following formulation:

(4.8)

The particles are considered to be spherical particles and hence the distance of the cell

center from the wall was considered to be the distance of the center of the particle from

the wall, assuming that the particle is in contact with the surface. The OSU model has

32

done away with this assumption and calculates the wall friction velocity from the y+

formulation.

The wall friction velocity in the model was calculated as follows:

(4.9)

where is the dimensionless wall distance, y is the distance of the first grid point from

the wall and is the local kinematic viscosity in m2/s. The effects of viscosity were not

accounted for in these simulations as using the same flow conditions as the BYU model

would help identify the areas of concern in the model. Still, in the next phase, viscosity

will be used as a function of temperature and this can be specified in the FLUENT -

MATERIALS panel. This is expected to be a more robust method of calculating the wall

friction velocity as the y+ is calculated inherently in the FLUENT code based on the wall

shear stress as is the universal method rather than being dependent on the particle

velocity. The deposition model differed from the BYU model in the calculation of the

wall friction velocity and was validated against the previous simulation results from

BYU.

4.4) Youngs Modulus Determination

El-Batsh et al., (16) used a deposition model to calculate the deposition for the impact of

an Ammonium Fluorescein sphere against a Molybdenum surface. The Young's modulus

of the Ammonium Fluorescein sphere and the Molybdenum surface were known from

experimental results previously. One of the problems encountered when calculating the

capture velocity in the BYU model is the information on the material properties of the

particle and the surface. These properties were not available in literature for the fly-ash

material. Also, to study the effect of surface temperature on particle sticking, the

33

dependence of material properties like Young's modulus and Poisson's ratio on

temperature was required. This led to a correlation between the material property,

Young's modulus and the temperature in El-Batsh model. The El-Batsh parameter is

needed to calculate the capture velocity and this information is obtained from a

correlation by fitting the experimental data. For every gas temperature, the value of E was

changed in the eq.4.2 until the capture efficiency matched that from the experiments. The

assumption was made that the particle sticking properties represent the target surface

properties as well Richards et al., (68) performed deposition experiments for a time-

period that would build a monolayer on the surface. They found that the surface

properties were not changing as the monolayer developed. The experiments at BYU were

run for a period of time long enough to let a monolayer to build on the surface and hence,

the majority of the particles interacting with the surface would be interacting with the

monolayer and hence the assumption of same properties for the particle and the surface is

valid. The El-Batsh parameter was calculated by assuming a constant value of 0.27 for

the Poisson ratio of both particle and the surface based on experimental results. Using the

E values obtained for each gas temperature, a correlation was developed by Ai & Fletcher

(17) as follows:

(4.10)

Soltani & Ahmadi (57) showed that as the Youngs modulus increases, the capture

velocity and subsequently, the capture efficiency decreases. Though the gas temperature

was used to achieve the correlation, using the average temperature of the particle and the

surface instead resulted in better agreement with the experimental results. This

34

correlation was used in the current deposition modeling initially before obtaining our own

correlation to account for the change in the calculation of the wall friction velocity.

Capture efficiency is defined as the ratio of the mass of the particles deposited on the

surface to the total mass of particles entering the domain. It can also be defined as the

product of the impact efficiency and the sticking efficiency. Impact efficiency is the

ratio of the mass of the particles impacting the surface to the total mass of the particles

entering the domain. Sticking efficiency is the ratio of the mass of the particles

deposited on the surface to the mass of the particles impacting the surface. These 3

efficiencies are the most important parameters in the deposition calculations.

35

5. DEPOSITION MODEL DEVELOPMENT IN FLUENT 6.3.26

The previous section provided a detailed description of the Lagrangian particle tracking

methodology and the particle deposition model to be used in this study. This section will

deal with the programming and development of the deposition model and the integration

of the model using User Defined Functions (UDF) in the commercial CFD software,

FLUENT 6.3.26. The deposition model was programmed using the C language and is

shown in appendix.1. User Defined Memory Locations (UDML) were used to store the

deposition results in order to enable post-processing of the results and simulate images of

deposition. The process of running the FLUENT DPM model with the deposition model

is shown below:

1. Create the geometry and mesh in a pre-processor (GAMBIT for tetrahedral grids

and GRIDPRO for hexahedral grids)

2. Load the mesh in FLUENT and solve the flow-field and heat transfer

3. Save the case and data file

4. Open the case file

5. Set the number of User Defined Memory Locations (UDML) using Define User

Defined Memory

6. Initialize the flow field

7. Use Display Contours to display the UDML on the wall surface where the

deposition model is used as a boundary condition to initialize the UDML values

to zero

36

8. Compile and load the UDF through Define User Defined Function Compile

option. The UDF should be in the same folder as the case and data files

9. Open the data file

10. Set User-defined memory locations through the Execute-on-demand function

using Define User Defined Execute on Demand

11. Set up the DPM model using Define Models Discrete Phase. This panel

enables setting up the parameters for steady particle tracking and also the

injection parameters

12. Choose Stokes-Cunningham as the drag force parameter and set the value of the

Cunningham Correction Factor to 1.2

13. Use the Physical Models tab and enable the Saffman Lift Force option

14. Use the Injections option in this panel to setup the particle injections.

15. Choose inert as the particle type for all simulations. Injection type can be either

group or surface depending on the simulation

16. Set injection parameters like location, velocity, temperature, diameter, etc. Also,

set the number of iterations for the stochastic particle tracking.

17. Use Define Boundary conditions to select the wall on which the deposition

model has to be applied. In the DPM tab in the Boundary Condition panel, choose

user-defined as the boundary condition and select the UDF file (*.c)

18. Click Display Particle tracks to display the particle tracks and run the deposition

model

A flow-chart detailing the process is shown in fig. 5.1

37

Fig. 5.1: Deposition model flow chart

38

The UDF was programmed using built- in macros in FLUENT for the DPM model. The

DEFINE_DPM_BC macro enables the user to specify a boundary condition that is

different from the default boundary conditions for the particle-wall interaction. The

EXECUTE_ON_DEMAND macro is used to execute any process at any time during the

simulation. In this UDF, this macro is used to set the user-defined memory locations in

the data file. One major point to be noted in the current simulations is that the c hange in

the geometry due to the deposition and its effect on the fluid flow and cooling

effectiveness is not considered. Still, various methods to incorporate the changes in

geometry and the subsequent changes in the flow field have been analyzed and a

framework on this has been created for the next user.

39

6) PARTICLE DEPOSITION ON A COUPON

Ai et al., (17) used their deposition model on two different cases for a bare coupon,

without and with film cooling. They obtained their Youngs modulus correlation from

simulations on a bare coupon without film cooling in a 2D domain. The bare coupon has

a thermal conductivity of 9 W/m.K. The OSU model was validated against results from

the experiments on the bare coupon without film cooling initially. The results from these

simulations and comparisons with the BYU model are detailed in this section. The initial

experiments were conducted with a coupon made of Inconel. The coupon was set at an

angle of 45 to the flow field. The backside of the coupon was insulated with ceramic

material, resulting in nearly adiabatic conditions. The initial computational model was an

extension of the BYU 2D simulations in a 3D domain. The computational domain in 3D

space and the schematic and boundary conditions for the simulations are shown below:

Fig. 6.1: Computational domain for the bare coupon simulation

40

Fig. 6.2: Geometry and boundary conditions of the model in 2D view

The high temperature circular gas jet has a diameter of 25.4 mm, the same as the

equilibrium duct diameter in the experiments. The whole domain is a cylindrical section

of 508 mm in diameter and height. The coupon is 25.4 mm in diameter, cylindrical and

has a thickness of 3.556 mm. The coupon is placed at an angle of 45 to the mainstream

gas flow as in the experiments. The coupon holder from the experiments is neglected

since it does not affect the deposition and flow field to a large extent and also due to the

ease of modeling and meshing the domain by neglecting the holder. The geometry a nd

mesh were generated using GAMBIT's unstructured tetrahedral topology grids consisting

of tetrahedral cells. The total number of computational cells was 1,260,184. The accuracy

of the computational model and deposition model are strongly influenced by the quantity,

quality and location of grids resolving the flow physics. The y+ value of the mesh was

between 15 & 40, in conjunction with the y+ of 12-300 used in the BYU model. Detailed

sections of the mesh are shown below:

41

Fig 6.3: Cut section of the mesh along X-plane

6.1) Boundary Conditions:

The fluid enters through the velocity inlet at 173 m/s and at a gas temperature varying

from 1293 K to 1453 K. The table below gives the experimental conditions and the

capture efficiency obtained from the experiments, along with the coupon surface

temperature. All other sides of the coupon were considered to be adiabatic. All walls of

the mainstream duct are considered as pressure outlets with a temperature of 300 K,

simulating atmospheric conditions. The walls of the inlet equilibrium duct were

considered to be adiabatic. The gas was modeled as incompressible air using the ideal gas

law, with gas density a function of the fluid temperature.

Table 6.1: Summary of experiments for the bare coupon case

Inlet Velocity (m/s) 173.0

Mass mean diameter (m) 13.4

Gas Temperature (K) 1453 1425 1408 1374 1352 1293

Surface Temperature (K) 1311 1281 1270 1234 1232 1191

Capture Efficiency (%) 7.87 4.47 2.938 0.932 0.517 0.0001

6.2) Carrier Phase Simulations

The continuous phase flow field was solved first and then the discrete phase model was

used to track the trajectory of the particles. The fluid/carrier phase was solved using the

Reynolds-Averaged Navier-Stokes (RANS) simulations governing the transport of the

42

averaged flow quantities. The SIMPLE algorithm couples the pressure and velocity.

Pressure and Momentum equations are discretized by the PRESTO and QUICK scheme

respectively. The discretization of the energy equation is performed using the second-

order upwind scheme and the discretization of the k and equations in the k-

turbulence model uses the first-order upwind scheme. Convergence was determined by

reduction of normalized residuals for each parameter as follows: continuity (< 10 -4),

velocity (

43

Particle trajectories and temperatures were modeled on a particle-by-particle basis using

the stochastic random-walk model as explained before. All simulations with group

injections were run with a minimum of 10-20 tries in the stochastic model to obtain a

better representation of each particle's behavior. The Runge Kutta method was used to

integrate the particle equations.

6.4) Results

Fig 6.5 and 6.6 show the comparison of the impact and capture efficiency from the OSU

model with previous results from the BYU model and the experiments. This initial

comparison was necessary to identify the shortcomings of the OSU model and to validate

the model against well-established results before making improvements to the model. The

OSU model contained the new formulation of wall- friction velocity and the results will

shed light on whether the new model improves upon the capture efficiency prediction.

Fig: 6.4: Impact efficiency vs Particle Diameter at 1453 K

0

20

40

60

80

100

120

0 2 4 6 8

Imp

act

Effi

cie

ncy

(%)

Particle diameter, in m

Impact Efficiency vs Particle Diameter at 1453K

Impact % by OSU

Impact % by Ai et al

44

Fig: 6.5: Capture Efficiency vs Gas Temperature

The capture efficiency is based mainly on the impact & sticking efficiency. Fig. 6.4

shows that the OSU model shows extremely good agreement with the BYU model for

impact efficiency at 1453 K. This shows that the particle tracking methodology and

parameters used in the OSU model work well since the trajectory of all particles have

been calculated for calculating the impact efficiency and this value agreed well with the

BYU results.

Fig. 6.5 shows the comparison of capture efficiency with different gas temperatures for a

mass mean diameter of 13.4 um. The OSU model agrees reasonably well with Ai et al.

and this improved model was expected to give better results when we move on to cases

with film cooling and vanes. One thing of note in the OSU model is that the capture

efficiency does not agree well with the experiments at lower temperatures. On further

analysis of the model, it was decided that the different methodology for calculating the

wall friction velocity is the cause for this. The newer wall friction velocity was supposed

0

1

2

3

4

5

6

7

8

9

1250 1300 1350 1400 1450 1500

Cap

ture

Eff

icie

ncy

(%)

Gas Temperature (K)

Capture Efficiency vs Gas Temperature

Experimental

OSU model

Ai et al

45

to provide improved predictions but is based solely on the y+ at the wall which in turn is

dependent on the shear stress of the wall surface. The y+ value in these simulations was

kept at 15-40 to be in conjunction with the y+ of 12-300 used in the simulations at BYU.

The k- turbulence model usually needs the mesh to be refined as close to the wall as

possible with a y+ of around 1 giving better results than a higher y+ value (35). Wang et

al., (61) showed that the boundary layer usually acts as a barrier for particles to reach the

wall, thereby reducing the capture efficiency. Hence, proper modeling of the boundary

layer is extremely important for accurate capture efficiency prediction. A high value of

y+ as used in the BYU model will not resolve the boundary layer sufficiently to track

particle behavior inside this layer. The higher capture efficiency shown by both the

computational models can be attributed to not resolving the boundary layer completely to

the wall. Dehbi (5) used a separate model to account for the dispersion of particles inside

the boundary layer. The wall friction velocity in the OSU model is directly proportional

to the dimensionless wall distance (y+) and inversely proportional to the distance of the

center of the first cell away from the wall to the wall (y). Resolution of the mesh all the

way to the wall will enable more accurate representation of the wall shear stress, possibly

correcting for errors in the y+. A lower y+ would also mean a change in the value of y.

The 3D simulations throw light on the shortcomings of the previous model and on ways

to improve the capture efficiency prediction. Another area of concern is the Youngs

modulus correlation in the BYU model that was developed based on a 2D simulation of a

3D domain. The 2D simulation cannot capture the exact flow field over the coupon as in

a 3D simulation due to the circular nature of the coupon. This correlation will be revisited

after the initial simulations to validate the model. The impact efficiency graph shows that

46

the smaller the particle, the lesser the impact efficiency, thus corroborating the results of

Ai & Fletcher (17). Similar calculations for capture efficiency of the model showed a

trend similar to the one observed by Ai & Fletcher where the capture efficiency decreased

with decreasing gas temperatures. The OSU model captures the trend expected from the

experiments and further improvements will be made in the next simulations with film

cooling holes, thereby ensuring the changes to the model hold well for future cases with

film cooling in a vane.

47

6-A) CONJUGATE HEAT TRANSFER AND PARTICULATE DEPOSITION ON

A COUPON WITH FILM COOLING

The deposition model has been tested with previous test results from Ai & Fletcher (17)

on a 2D geometry with no film cooling and no conjugate heat transfer. The results as

mentioned above capture the various trends in deposition effectively and hence the model

is validated as fit to be applied to the 3D case with conjugate heat transfer and film

cooling. The experiments were carried out for various geometries with s/d being 3, 3.375

and 4.5 and the blowing ratios ranging from M=0.5 to M=2.0 with each of these cases

being modeled by Ai & Fletcher (17). Our modeling pertains to the single case of

s/d=3.375 where d=1 mm with blowing ratios of M=0.5, 1.0, 2.0. The distance between

the cooling holes is denoted by s/d with s being the actual distance between the holes

given with reference to the hole diameter (d). The blowing ratio (M) is defined as the

ratio of the coolant velocity at the exit of the cooling holes (Uj) to the free-stream

velocity (Uf).

6.1A) Geometry & Grid Generation

A schematic of the computational domain is given in figure 6.6. The computational

domain is the same as the one used by Ai & Fletcher (17) and the simulation has been

carried out with the exact same conditions as in their modeling. This has been done in the

view that any areas of concern in the deposition model can be easily identified and

rectified if the simulation is carried out in the same way as in (17). The computational

48

domain includes a mainstream duct, the coolant plenum and the solid plate with film

cooling holes completely occupying the area between the two ducts.

Fig: 6.6: Schematic of the 3D computational domain

The cooling holes are 3 in number and are cylindrical in shape. The mainstream duct has

a mixture outlet through which the mixture of gas and coolant flows. In the actual

experiments, the solid coupon is inclined at an angle of 45 to the mainstream flow and

the cooling holes are at an angle of 30 to the solid coupon surface and this has been

replicated in the domain geometry. The mainstream section is 81 mm in length, 39 mm in

width and 36 mm in height. The row of 3 film cooling holes are located inclined at an

angle of 30 to the plate surface and their centers are located 36 mm from the flat plate

leading edge and 45 mm upstream of the mixture outlet. The coolant plenum is 81 mm in

length, 39 mm in width and 40.5 mm in height. The flat plate has a thickness of 3 mm.

49

The cooling holes diameter is 1 mm and the hole spacing is 3.375 mm. The mainstream

gas enters the duct through an inlet of diameter 25.5 mm while the coolant enters the

plenum through an inlet of diameter 13.5 mm.

Two different grids were generated for this geometry, a tetrahedral mesh using GAMBIT

and a hexahedral mesh using GRIDPRO. The mesh generated was an unstructured

tetrahedral mesh as shown in fig 6.7 and a close-up of the mesh on the plate is shown in

fig. 6.9. The hexahedral mesh generated using GRIDPRO is shown in fig. 6.8. The total

number of computational cells was 746,554 for the tetrahedral case. The accuracy of the

computational model and the deposition model depends on the quality and location on

grids fine enough to resolve the flow physics in the areas of interest. Keeping this in

mind, the grid was created with fine cells near the coupon surface and the film cooling

holes where there are reasonably large gradients of the flow variables. The y+ was still

maintained at 12-300 (tetrahedral mesh) as in the BYU model to gain more insight on

how the y+ is affecting the deposition with the new wall friction calculation. The

deposition model was run on both grids for better insight into whether any one type of

mesh has an advantage over another for future cases. Skewness in the grid was kept to a

maximum of 0.86 while meshing and is later brought down to 0.75 in FLUENT using

polyhedral cells.

50

Fig. 6.7: View of the tetrahedral volume mesh for the 3D case

Fig. 6.8: Cut-section view of the volume mesh for the 3D case for hex mesh

Fig 6.9: Surface mesh on the plate for 3D tetrahedral grid

51

6.2A) Boundary Conditions & Simulations

The boundary conditions for the case were obtained from experimental values from Ai &

Fletcher (17). The mainstream gas enters the inlet at a temperature of 1453K and a

velocity of 173 m/s. The turbulence intensity at the mainstream inlet was specified as a

value of 4.25% based on the flow conditions. The temperatures on the top and side walls

of the mainstream duct were specified as 900K while the temperature of the wall close to

the inlet was 300K. These values were obtained from the experiments. The viscosity of

the fluid was kept as a constant at 1.79e-05 kg/m-s. The hot and cold sides of the coupon

were designated as a coupled boundary in FLUENT. This eliminates the need to specify

the heat flux or any other boundary conditions and this facilitates conjugate heat transfer

between the solid and fluid domains. The side walls of the coupon plate were set to be

adiabatic, thereby making the heat flux to flow in only one direction inside the solid

plate. A no-slip condition was applied to all the walls. The coolant inlet conditions were

derived at from the blowing ratio and the velocity and the temperature expected at the

entry to the film cooling holes. A density ratio was chosen such that the entry conditions

into the film cooling holes were satisfied. The coolant inlet had a temperature of 293K

and a velocity of 0.592 m/s for the M=1 case which would give the desired conditions at

the cooling holes entry. The coolant velocity for the other cases of M=0.5 and M=2 were

arrived at from the blowing ratio formulation using the free-stream velocity. The walls of

the coolant plenum were set to be adiabatic. The turbulence intensity is:

(6.1)

Since prior knowledge of the flow velocity at both inlets was available, the turbulence

intensity was easily calculated. The initial simulations were made based on

52

'Incompressible ideal gas law' in FLUENT for the density of air as stated in Ai & Fletcher

(17). The inlets were mentioned as Velocity Inlets and the outlet was a Pressure Outlet

for the incompressible case. The incompressible ideal gas law calculates the density

based only on the temperature. The density does not depend on the local relative pressure

field.

The chosen case was initially run as a compressible flow using the ideal gas law for

density as per Ai and Fletcher(17). The thermal conductivity of the solid plate was set to

9 W/m-K. The Mach number at the gas inlet was 0.2263 and 0.00172 at the coolant inlet.

Compressible flow simulations for gases with Mach numbers in the incompressible range

are extremely difficult to converge and though the results of temperature and velocity

field on the plate were in the vicinity of the previous experimental and modeling results,

it was observed that more than 20,000 iterations will be required for the case to converge

fully. One factor contributing to this is the use of mass flow inlet in FLUENT which

significantly takes longer to converge compared to velocity and pressure inlets. Also, the

mesh was found to be too coarse to capture the flow physics effectively and the y+ values

were adapted on the hot and cold side to around 1, which is the standard y+ value region

for the k- model. This was found to bring the temperature down considerably. Low

surface temperatures were observed near the cooling hole exits. The surface temperature

is the area-weighted average temperature on the plate's hot side over a circle of diameter

one inch which is the diameter of the actual coupon used in the experiments. The

deposition model will also take into account only those particles that are deposited within

this circular area. The residuals for continuity were less than 10e-4, the residuals for

velocity was 10e-6, the residuals for energy were 10e-7 and the turbulence quantities had

53

a residual of 10e-5. Although the residuals indicated convergence, the monitors of

pressure, temperature and velocity at various points inside the domain did not converge.

Even after 20,000 iterations, the flow still did not achieve complete convergence and

correspondence with the FLUENT support center also reiterated the theory that mass

flow inlets for this case will take longer to converge. Initially, we were not able to

achieve agreement with experimental data for the average surface temperature on the two

sides of the plate using compressible flow solver. This led to the conclusion that

incompressible analysis is needed because the cold side flow has a Mach number of

0.002 for a compressible solver is not suitable. A compressible solver may be used if

combined with a pre-conditioner for low Mach number flows. The final deposition

simulation has been performed on the flow-field solution from the incompressible solver.

6.3A) Results

The flow-field results and the surface temperature profiles for all 3 blowing ratios are

shown below in fig. 6.10. These are the results from the tetrahedral mesh w