CFD SIMULATION AND ANALYSIS OF PARTICULATE DEPOSITION ON GAS TURBINE VANES

Transcription

1 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

2 Copyright by Prashanth S Shankara 2010

3 ABSTRACT Syngas from alternate fuels is used as a fuel in land based gas turbine engines as a lowgrade 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 postprocessing 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 ii

4 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 Young s 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. iii

5 Dedication To my Mom, Dad & Brother for always being there iv

6 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 UDF s 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 v

7 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. vi

8 VITA March SDAV Hr. Sec. School, Chennai, India March 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 vii

9 TABLE OF CONTENTS Abstract ii Acknowledgements..v Vita vii List of Tables...x List of Figures.xi Nomenclature xiii 1. Introduction Literature Review Eulerian Particle Tracking Lagrangian Particle Tracking Turbulence Models Particle Tracking Methodology Carrier Phase Discrete Phase Particle Trajectory Calculations Coupling of discrete & continuous phase Turbulent Particulate Dispersion Particle Deposition Model Particle-Wall Interaction...27 viii

10 4.2. Particle Sticking Particle Detachment Young s Modulus Determination Deposition Model Development in FLUENT Particle Deposition on a Coupon Boundary Conditions Carrier Phase Simulations Particle Phase Simulations Results A. Particulate Deposition on Coupon with Film Cooling A. Geometry & Grid Generation A. Boundary Conditions & Simulations A. Results Application to VKI blade Boundary Conditions Simulations & Results Improvements to the Deposition Simulations Conclusions & Recommendations..78 References.82 Appendix: Particle Deposition Model UDF Source Code ix

11 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 blade 66 x

12 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 case 50 Fig. 6.8: Cut-section view of the volume mesh for the 3D case 50 Fig 6.9: Surface mesh on the plate for 3D tetrahedral grid 50 Fig 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 coupon 55 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=1 58 Fig 6.15: Deposition patterns from the model for M=2.60 xi

13 Fig 6.16: Deposition patterns from the model for M= Fig 6.17: Deposition patterns from the model for M=1.60 Fig 6.18: Comparison of deposition from experimental & simulation for M= Fig 6.19: Comparison of deposition from experiments & simulation for M= 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 d p = 0.1μm 69 Fig. 7.8: Particle trajectories through the passage for d p = 1μm...69 Fig. 7.9: Particle trajectories through the passage for d p = 10 μm.70 Fig 7.10: Sticking Efficiency vs Particle Diameter at M= 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 xii

14 NOMENCLATURE Symbol A p Surface Area of particle, [m 2 ] Mean molecular speed, [m/s] Cc, Cu Cunningham correction factor, [-] C d Coefficient of drag, [-] C p Specific heat of particle, [J/kg.K] C L, C µ Coefficients for eddy lifetime model, [-] d, D Cooling hole diameter, [m] d ij d p, D p E E i E p E s F D F po F s F x Deformation tensor, [m/s] Particle diameter, [m] El-Batsh parameter, [Pa] Mean velocity of fluid, [m/s] Particle Young s modulus, [Pa] Surface Young s modulus, [Pa] Drag force on particle, [N] Particle sticking force, [N] Saffman lift force, [N] Additional force term in the particle trajectory equation, [N] g x Acceleration term in the particle trajectory equation, [m/s 2 ] G k G ω Generation of turbulence kinetic energy due to mean velocity gradients Generation of ω xiii

15 h c Convective heat transfer coefficient, [W/m 2 K] I Turbulence intensity, [%] k Turbulence kinetic energy, [m 2 /s 2 ] k1, k2 El-Batsh parameter constants, [-] kf Thermal conductivity of the continuous phase, [W/m-K] ks Sticking force constant, [-] K Constant coefficient of Saffman s lift force, [-] Kn Knudsen number, [-] K r L e m p Local velocity gradient, [m/s] Eddy length, [m] Mass of the particle, [kg] M Blowing ratio, [-] M Mach number, [-] Nu Nusselt number, [-] P Static pressure of fluid, [Pa] Pr Prandtl number, [-] P 0 R Total pressure of fluid, [Pa] Universal gas constant, [J/K.mol] Re p Reynold s number of the particle, [-] s S k, S ω Distance between cooling holes, [m] User defined source terms S Ratio of particle density to fluid density, [-] t T T p Time, [s] Gas temperature, [K] Particle temperature, [K] xiv

16 T T avg u u u * u j u p u τc U j U f v cr v n v w Free-stream temperature, [K] Average between particle and surface temperature, [K] Fluid velocity, [m/s] Gaussian distributed random velocity fluctuation, [m/s] Wall friction velocity, [m/s] Instantaneous fluid velocity, [m/s] Particle velocity, [m/s] Critical wall shear velocity, [m/s] Coolant velocity at exit of cooling holes, [m/s] Freestream velocity, [m/s] Capture velocity, [m/s] Normal velocity, [m/s] Gaussian distributed random velocity fluctuation, [m/s] Gaussian distributed random velocity fluctuation, [m/s] W A Work of sticking, [-] y y + Distance of first cell center from the wall, [m] Dimensionless wall distance of first cell center from the wall, [m] Y k Dissipation of k due to turbulence, [-] Y ω Dissipation of ω due to turbulence, [-] α 2 Volume fraction of dispersed phase, [-] ε Turbulent dissipation rate, [m 2 /s 3 ] λ Mean free path of the gas molecules, [µm] µ Dynamic viscosity of fluid, [kg/m.s] ν Kinematic viscosity, [m 2 /s] ν p Poisson s ratio of particle material, [-] xv

17 ν s Poisson s ratio of surface material, [-] ω Specific dissipation rate, [s -1 ] ρ, ρ f Density of fluid, [kg/m 3 ] ρ p Particle density, [kg/m 3 ] Γ k Γ ω Effective diffusivity of k, [m 2 /s] Effective diffusivity of ω, [m 2 /s] Particle relaxation time, [s] Wall shear stress, [Pa] Lifetime of eddy, [s] xvi

18 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 particle 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 1

19 model that can be applied to future simulations of deposition on a turbine vane with film cooling. 2

20 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. 3

21 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 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: 4

22 (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. 5

23 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-ε model and the RNG k-ε 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-ω turbulence 6

24 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-ω model gave better prediction of the near-wall flow structures compared to the k-ε model. Silieti et al., (28) predicted film-cooling effectiveness using 3 different turbulence models: the Realizable k-ε model (RKE), Standard k-ω model and the v 2 -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 literature. 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-ω (SKW) model 7

25 best predicted laterally averaged adiabatic effectiveness, and that the realizable k-ε model was best along the centerline. El-Batsh et al., (16) showed that the Std. k-ε model with standard wall functions over-predicts the deposition velocity for particles with relaxation time less than 10. The std k-ε model has been used to simulate indoor flow field successfully by Zhang & Chen (30). Jovanovic et al (31) et al used std k-ω model for their two-phase flow modeling of air-coal mixture channels with single blade turbulators. Theodoridis (32) et al used a std k-ε 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-ε 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-ω model with RANS to compute flow field and heat transfer for their analysis of particle deposition on a turbine coupon. The k-ω 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-ε 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-ε 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-ε turbulence model over-predicts the turbulent kinetic energy 8

26 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-ω 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. 9

27 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 FLUENT 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 CPU s simultaneously. DNS is expensive for the current problem and not available in FLUENT. The conservation equations for mass and 10

28 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: u u p u i i i ( u ) ( ) ( u u ) j i j t x x x x x j i j j j (3.1) where u is the Reynolds stress. Standard k-ω turbulence model will be used for the u i j closure equations, which is based on the Wilcox k-ω model. The turbulence kinetic energy, k, and the specific dissipation rate ω are obtained from the following transport equations: k () ku i k ) ( ) GYS k k k k t x x x ( i j j (3.2) and () ui ( ) t x i ( ) GY S k x x j j (3.3) Where k is the turbulent kinetic energy, ω is the specific dissipation rate. In these equations, G k represents the generation of turbulence kinetic energy due to mean velocity gradients. G ω represents the generation of ω. Γ k and Γ ω represent the effective diffusivity of k and ω, respectively. Y k and Y ω represent the dissipation of k and ω due to turbulence. S k and S ω are user-defined source terms. 11

29 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 F x 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. 12

30 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: ΣF p = 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 Re p <1, modified Stokes law when 1<Re p <500 and the Newton s law when 500<Re p <2x10 5. The second term describes the acceleration of the fluid near the particle surface from fluid velocity to the particle velocity. Mass of fluid that undergoes this acceleration is called carried mass which is equal to one half of the displaced mass of the fluid. The acceleration of the fluid near the particle causes the flow around the particle to differ from that in the steady motion. The force required to maintain the flow pattern was approximated by Basset and 13

31 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 (ρ p /ρ f << 1). The added mass effect was also found to be true in the same regime. Saffman s lift force is caused by the shear of the surrounding fluid which results in a non-uniform pressure distribution around the particle. This force assumes non-trivial magnitudes only in the viscous sublayer. If a particle leads the fluid motion, then the lift force is negative and the particle moves down the velocity gradient towards the wall. Conversely, if the particle lags the fluid, then the lift force is positive and it moves up the velocity gradient away from the wall. This force usually enhances deposition velocity and Wang et al (40) have shown that neglecting this force results in a small decrease in the deposition rate. The gravitational force is the body force acting on the particle and is only important for large particles in a Stokes regime. The particles used in this study are in the region of 1-15 µm and hence this force is neglected. Basset Force, Buoyancy Force and added mass effect are usually negligible for particle deposition studies and hence neglected. Intercollision force is the force exerted due to inter-particle collisions. It is usually important when the volume fraction of the dispersed phase is high. Magnus force is the lift developed due to rotation of the particle. The lift is caused by the pressure dfference between both sides of the particle resulting from the velocity dfference due to rotation. Kallio & Reeks (11) noted that it is an order of magnitude lesser than the Saffman force in most regions of flow field and hence neglected. Brownian and Thermophoretic forces are 14

32 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 μm. 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 submicron 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. 15

33 All experimental cases in this study have 0.1<Kn<10. Talbot et al., (42) have shown in their experiments that there is considerable reduction of drag coefficient due to rarefaction effect for Kn>0.02 and hence this effect is included. The reduction in the drag coefficient is accounted for by the Cunningham correction factor described by Talbot et al., (42) as: (3.8) Shah et al., (21) used the drag force and the gravitational force only and considered other forces to be negligible based on previous literature. Dehbi (5) considered only the drag force in his Eddy Interaction Model. Hamed et al (43) also considered only the drag and gravitational force in addition to their model for rotating machinery and the force due to rotation can be ignored. Iacono et al., (22) used drag force, gravitational force and Saffman Lift Force as the effect of Saffman Lift on particle deposition is high in the viscous sublayer. Ai et al., (17) used the drag force at steady state and Saffman Lift Force in their simulation of particle deposition on high pressure turbine vane. Table 3.1 summarizes the various forces considered and identifies those that are incorporated in the current model. 16

34 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<100, spherical particles assumed. Stokes' law: Rep < 1; Modified Stokes' YES 1<Rep < 500 Law: 1<Rep < 500; Newton's Law: 500< Rep< 2 x 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 YES 0.1<Kn<10 though sub-micron particles are less 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 5 Saffman Lift Non-trivial magnitudes only in the viscous sub-layer; slight decrease in deposition rate if neglected. 6 Magnus Lift force due to particle rotation; atleast an order of magnitude lower than Saffman force in most regions; NO ρ p / ρ f >> 1 YES More accurate deposition rate 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 3.3) Particle Trajectory Calculations Based on the particle characteristics and previous literature, the forces that are considered to be acting on the particle throughout this study are as follows: 17

35 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 C d 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 λ is the molecular mean free path. The Saffman Lift force was initially given by Saffman (47) as: (3.12) where K r 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: 18

36 (3.13) where, is the velocity of the particle, d is the particle diameter, S is the ratio of particle density to fluid density, ν is kinematic viscosity, K = is the constant coefficient of Saffman's lift force and is the instantaneous fluid velocity with u j = E j + u:, where E j 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 m p is the particle mass, C p is the particle specific heat, T p is the particle temperature, A p is the surface area of the particle, and h c 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) 19

37 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 5μm, for example, was 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) 20

38 Fig 3.1 shows the various flow regimes and the type of coupling suitable between carrier and dispersed phase. α 2 is the volume fraction of the dispersed phase, τ k is the Kolmogorov time scale in seconds, τ 1 t is the Lagrangian Integral time scale in seconds and τ x 12 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 transport. 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. 21

39 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 22

40 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 t o, a particle with velocity u p 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) 23

41 (3.18) where k and ε are respectively the turbulent kinetic energy and dissipation rate, while C s 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 24

42 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. 25

43 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. 26

44 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 27

45 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 28

46 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 k s is a constant equal to 3π/4. The Work of Sticking, W A is a constant which depends upon the material properties of the particle and of the surface and has the units of J/m 2. 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 Young s modulus which is determined based on the Young s modulus of the particle and the surface. The particle normal velocity (v n ) 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. v n < v cr - particle sticks; v n > v cr - particle rebounds 29

47 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 Young s 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], ν s is the Poisson's ratio of surface material, Ep is the Young's modulus of particle material [Pa], ν p is the Poisson's ratio of particle material, d p is the particle diameter [m] and ρ 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 u τc 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 τ w 30

48 is the wall shear stress) which is larger than u τc. 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 31

49 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 m 2 /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) Young s 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 32

50 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 timeperiod 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 Young s 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 33

51 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. 34

52 5. DEPOSITION MODEL DEVELOPMENT IN FLUENT 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 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 35

53 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 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

54 Fig. 5.1: Deposition model flow chart 37

55 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. 38

56 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 Young s 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 45º 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 39

57 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 mm. The coupon is placed at an angle of 45º 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 and 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 used in the BYU model. Detailed sections of the mesh are shown below: 40

58 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) Mass mean diameter (μm) 13.4 Gas Temperature (K) Surface Temperature (K) Capture Efficiency (%) ) 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 41

59 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 secondorder upwind scheme and the discretization of the k and ω equations in the k- ω 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 (<10-6 ), energy (<10-7 ), and turbulence quantities (<10-5 ). Convergence monitors were set up at various points inside the domain to monitor the flow variables and full convergence was deemed to achieved only after the monitors of the flow variab les became steady. 6.3) Particle Phase Simulations The simulations were performed with group injection in FLUENT. In group injections, 5000 ash particles were released at the center of the inlet surface and impinged on the target surface. The particles are in equilibrium with the gas phase at the inlet of the domain and hence had the same temperature and velocity as the carrier phase. Although this is not a fair representation of the particle distribution at the inlet, all previous modeling results by Ai & Fletcher were performed using group injection and hence it was carried out to validate the deposition model. The properties of the ash particle are given below: Table 6.2: Ash particle properties from Ai et al., (17) d(um) ρ(kg/m^3) Cp(J/kg.K) k(w/m.k)

60 Impact Efficiency (%) 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 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. Impact Efficiency vs Particle Diameter at 1453K Impact % by OSU Impact % by Ai et al Particle diameter, in μm Fig: 6.4: Impact efficiency vs Particle Diameter at 1453 K 43

61 Capture Efficiency (%) 9 Capture Efficiency vs Gas Temperature Experimental OSU model Ai et al Gas Temperature (K) 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 44

62 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 to be in conjunction with the y+ of used in the simulations at BYU. The k-ω 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 Young s 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 45

63 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. 46

64 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, 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 (U j ) to the free-stream velocity (U f ). 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 47

65 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. 48

66 The cooling holes diameter is 1 mm and the hole spacing is 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 The hexahedral mesh generated using GRIDPRO is shown in fig 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 (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. 49

67 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 50

68 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 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 51

69 '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 and 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-ω 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 52

70 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 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 These are the results from the tetrahedral mesh with y+ of Initial deposition simulations were performed on this mesh to determine the effect of the new wall friction velocity calculation on the efficiency. The hexahedral mesh will be used with a new Young s modulus correlation obtained from the OSU model. The formulae for calculating the wall friction velocity depending on the non-dimensionless distance away from the wall is:, (6.2) The velocity contours below show that at higher blowing ratios, the velocity of the coolant coming out of the cooling holes is higher, thereby pushing more particles away 53

71 from the wall resulting in lesser deposition. This is shown in fig 6.3 where the capture efficiency is lower as the blowing ratio increases. Also, the presence of more coolant reduces the surface temperature which in turn increases the particle Young s modulus (E p ) as per eq. 4.10, resulting in lesser capture velocity. As a result, more particles have normal velocity greater than the capture velocity and hence do not stick to the surface, giving low capture efficiency. These findings have already been reporter by Ai & Fletcher (17). M=0.5 M=1 M=2 Fig Velocity magnitude contours (m/s) for M=0.5, 1, 2 along centerline plane 54

72 Fig 6.11 shows the surface temperature contours for all three blowing ratios. The area of interest is not the entire rectangular plate, rather a circular area of diameter one inch similar to the coupon in the experiments. The images show the higher surface temperature for the M=0.5 case, thereby resulting in higher capture efficiency. M=0.5 M=1 M=2 Fig. 6.11: Surface temperature contours at different blowing ratios on 1 inch diameter coupon; Flow Direction is from left to right 55

73 Front-side surface Temperature (K) The initial deposition simulations have been carried out with group injection where 5000 particles are injected from the center of the hot gas inlet. This has been done so as to compare the deposition results with the new wall friction velocity calculation with those of Ai & Fletcher (17) who used a similar injection. A much more representative injection type is the surface injection option, where a particle is injected from the center of every face in the surface mesh at the inlet. All simulations shown in this section have been carried out with the same Young s modulus correlation as obtained at BYU. These results will give an indication of any potential changes that need to be incorporated in the model, most particularly revisiting the Young s modulus correlation and the effect of wall shear stress and in turn, the non-dimensionless wall distance (y+) on the capture efficiency prediction. Fig 6.12 and 6.13 show the comparison of the front-side surface averaged temperature and the capture efficiency obtained with the experimental results and the results of Ai & Fletcher (17) Front side surface Temperature vs Blowing Ratio at 1453K Blowing Ratio, M Experimental OSU model Ai et al Fig 6.12: Comparison of area-averaged surface temperature on coupon at 1453 K 56

74 Capture Efficiency (%) Capture % vs Blowing Ratio at 1453 K Experimental OSU model Ai et al Blowing Ratio (M) Fig 6.13: Comparison of capture efficiency vs blowing ratio at 1453 K Fig 6.12 shows that the front-side surface temperature from both simulations shows a reasonable agreement with the experimental surface temperature. Particle capture velocity and capture efficiency are directly proportional to the hot-side surface temperature of the coupon. A higher hot-side surface temperature at M=0.5 is seen in both OSU and BYU models. This has a direct correlation to the higher capture efficiency obtained at this blowing ratio in the next figure. At M=1, the OSU model had a closer agreement with the experimental hot-side surface temperature than the BYU model which is reflected in the better agreement with the capture efficiency. The change in calculation of the wall friction velocity also played a part in the improvement prediction of the capture efficiency at M=1. At M=2, the OSU model exhibits a higher hot-side surface temperature, resulting in a higher capture efficiency. These results show that modeling the conjugate heat transfer properly plays an important role in the prediction of 57

75 capture efficiency. The increased value of surface temperature predicted could be a result of not enough mesh resolution near the wall resulting in fluid flow field not being resolved properly. This directly affects the surface temperature on the plate and also the particle transport near the wall. It has been shown in literature that boundary layers act as a deterrent to the particles reaching the wall surface, thereby reducing the chances of particles sticking to the surface. The figure 6.15 shows the near wall velocity vectors with a y+ value of in the OSU model. As observed, prominent boundary layers are absent near the wall which is a potential cause of more particles reaching and sticking to the surface. Fig. 6.14: Velocity vectors along the centerline plane for M=1 The images of the deposition patterns on the plate at different blowing ratios provide interesting insight into the effects of film cooling on deposition. These images were obtained from the OSU deposition model using the same group injection conditions as 58

76 used by Ai et al. Deposition patterns from surface injection will be obtained for these cases too which will provide better comparison with the actual experimental deposition patterns obtained at BYU due to surface injection being more representative of experimental conditions. The images show that deposition is concentrated around the cooling hole in the middle and there is no deposition near or in the flow path of coolant from the cooling holes at the end. Although the images give rise to this conclusion, it should be noted that the injection type was group injection with all particles released from the center of the inlet. This is a fair assumption for obtaining capture efficiency but cannot be used to compare the deposition patterns with images from experiments. The experiments had uniform distribution of particles throughout the entire inlet area rather than injection from the center and hence the patterns obtained would show deposition all over the one inch diameter coupon. Still, the following images can be used as a validation of the model as they show almost no deposition in the flowpath right ahead of the cooling holes. The deposition is mainly concentrated near the outer edges of the coupon. This is also seen in the experimental images where deposition is extremely low in the flowpath right ahead of the cooling holes. The images also show the higher capture efficiency at M=0.5 compared to M=2 as seen in the experiments. The low blowing ratio at M=0.5 results in lower velocities coming out of the cooling holes. The particle normal velocity near the walls is closer to the fluid velocity and the lower values of fluid velocity reduce the particle normal velocity. This results in more particles having lesser normal velocities than capture velocity leading to more sticking. 59

77 Total Impact Sticking Not Sticking Fig 6.15: Deposition patterns from the model for M=2 Total Impact Sticking Not Sticking Fig 6.16: Deposition patterns for the model for M=0.5 Total Impact Sticking Not Sticking Fig 6.17: Deposition patterns for the model for M=1.0 60

78 Fig 6.18: Comparison of deposition from experimental & simulation for M=0.5 Fig 6.19: Comparison of deposition from experiments & simulation for M=2 The above images show the comparison of the simulation pattern developed from the OSU model with the actual deposition images from the experiments. The deposition simulation for M=2.0 predicted a higher capture efficiency than in the experiments and hence there are more deposits around the edges. Still, the deposits in the film cooling hole downstream is almost negligible compared to the total amount deposited which is the main observation from the experimental images. These images do not show the deposition but a representation of the number of particles deposited at each location. The 61

79 accumulated number of particles at each face location on the surface is shown as the value of the cell to which the face belongs to. In the next phase of the research, generation of images with realistic deposition heights will be handled. 62

80 7. APPLICATION TO VKI BLADE El-Batsh et al. (16) used their deposition model on a VKI transonic turbine inlet guide vane. Experimental results were not available but since the deposition model was already validated for various cases and agreed well with experimental data, a simulation on this vane with similar boundary conditions as expected in actual vanes would provide insight into how deposition works on vanes. This vane was chosen as there are well documented experimental results obtained at the Von Karman Institute (VKI) for flow field measurements. The velocity values used in these experiments satisfied the requirements of modern day gas turbines. Validation on this 2D vane is essential before moving onto a 3D vane with film cooling. All previous validation of the OSU model has been against experimental data from coupons made of turbine material and it is essential to test the model s performance on an actual turbine vane. The surface velocity and the downstream total pressure distribution are known from the experiments. The model will be applied for 2 Mach numbers, namely M=0.85 and M=1.02. The vane geometry and the mesh are given below: Chord Length, (mm): Pitch, (mm) : 57.5 Throat, (mm) :

81 Fig 7.1: Computational grid used for the VKI blade A hexahedral mesh was created in a 2D domain with 5168 cells. El-Batsh (17) used a structured body-fitted 2D hexahedral grid to accurately resolve the pressure distribution behind the vane. The mesh at OSU was generated using GRIDPRO by breaking up the domain into multiple regions of interest, thereby preserving the fineness of the mesh in the important areas of the flow field. The passage behind the vane is crucial due to larger pressure distribution variations and so is the leading edge of the vane. A body-fitted mesh was generated to better resolve the boundary layer properties along the surface of the vane. The mesh at OSU was generated with time constraints in mind and hence is much coarser than the mesh used by El-Batsh. There are not enough cells in the region near the leading edge of the blade and also in the aft region of the trailing edge where the flow leaves the vane. Also, enough mesh cells are not available in the blade suction surface to account for the difference in length between the pressure and suction surfaces. Even with these considerations in mind, the mesh generated was deemed to be suitable enough to accurately represent the trend of the sticking efficiency with particle diameter as noticed 64

82 by El-Batsh et al (17). Since the exact injection properties used by El-Batsh are not known, it would be futile to spend time on matching his exact flow field conditions and wiser to concentrate simply on validating whether the deposition model follows the trend for deposition on a turbine vane. 7.1) Boundary Conditions The inlet and exit boundary conditions were specified using a Pressure Inlet and Exit boundary condition in FLUENT. The inlet Mach number was set to 0.85 and then to 1.02 and the flow was solved using the compressible solver in FLUENT. The k-ε turbulence model was used with enhanced wall treatment. Fig 3.2 shows the schematic of the computational domain and table 3.1 lists the boundary conditions used. The turbulence length scale was set using l=0.07l where L is the characteristic length and was set to be the blade pitch. Fig 7.2: Computational domain and Internal Region The total pressure was used at the inlet and the static pressure was used at the exit. The total pressure was calculated from the following relation using the standard air properties: 65

83 (7.1) Table 7.1: Boundary Conditions for the VKI Blade Mach Number Total Inlet Pressure, [Pa] Total Inlet Temperature, [K] Freestream Turbulence [%] 1 1 Turbulence Characteristic Length, [mm] 4 4 Static Outlet Pressure, [Pa] Wall Temperature, [K] Incidence Angle, [deg] ) Simulations and Results The pressure-velocity coupling was performed using the SIMPLE discretization scheme. The Quadratic Upwind Interpolation (QUICK) scheme was used as the discretization scheme to provide higher order accuracy. The flow-field results for the two Mach number cases are shown below. Fig 7.3: Mach number contours for M=0.85 by OSU 66

84 Fig 7.4: Mach number contours for M=1.02 by OSU Fig 7.5: Mach number contours for M=0.85 by El-Batsh Fig 7.6: Mach number contours for M=1.02 by El-Batsh 67

85 The Mach number contours show the pressure distribution expected in a vane passage. The pressure side distributions are very smooth with smooth transition in the Mach numbers too. The Mach number distribution from OSU simulations on the suction side, especially from the region between the mid-chord and the trailing edge differ from the results from El-Batsh in that the shock observed is less prominent in the OSU simulations. Also to be noted is the flow leaving the trailing edge, the OSU simulations not showing accurate Mach number as in the simulations by El-Batsh. These differences can be attributed to not enough mesh resolution in these passages in the mesh generated at OSU. Still, the variations are not considerable enough to affect the particle trajectories to a great extent. Hamed et al. (62) performed erosion simulations on a GE E 3 first stage LP turbine with similar conditions for particle sticking. Their simulations provide insight into the particle trajectory behavior in a turbine, specially the stator vane. These simulations also give valuable information on particle tracking and modeling the particlewall interaction in the flow-field of a turbine. The figures below show the particle trajectories for 3 different particle diameters of 0.1, 1 & 10 μm with surface injectio n at M=

86 Fig. 7.7: Particle trajectories through the passage for d p = 0.1μm Fig. 7.8: Particle trajectories through the passage for d p = 1μm 69

87 Fig. 7.9: Particle trajectories through the passage for d p = 10 μm Fig. 7.7 shows the particle pathlines at a particle diameter of 0.1 μm. The smaller particles tend to follow the carrier phase flow since the effects of inertia are less prominent. The image shows that the particles move very close to the wall and hence a lesser chance of impacting the wall as they are almost parallel to the wall. Particles that stick to the wall and are sheared away by the friction forces are still influenced greatly by the flow field and are carried away by the fluid flow in its path. The particles that do tend to impact the surface have very low normal velocities. This gives a greater chance of their normal velocity being lesser than the capture velocity and hence sticking to the wall. This explains the lower impact efficiency but higher sticking efficiency at low particle diameters. This affirms the notion that smaller particles tend to impact the wall less than the larger particles and also tend to have greater capture efficiency. Fig. 7.8 shows similar pathline results for the 1μm diameter particles. The impact on the pressure side is similar 70

88 Sticking Efficiency (%) to the earlier case but the number of impacts on the suction surface is considerably lesser. Most of the impacts on the suction surface are caused by the particles reflected from the pressure surface of the neighboring blade in the passage. In this case, the particles deviate away from the suction surface very early and hence are moving away from the suction side of the next blade. Fig. 7.9 shows an even more prominent behavior of the particle trajectory based on the particle size. The 10μm particles are inertia-dominated and hence are not greatly influenced by the flow field, resulting in trajectories completely different from the fluid flow path. Larger particles tend to almost centrifuge after impacts and also are prone to more impacts due to a change of angle from the fluid flow and reach the wall surface almost perpendicular to it, leading to more impacts. This in turn causes greater normal velocities, leading to more chance of the normal velocity being greater than the capture velocity resulting in greater number of particles not sticking to the surface. All this leads to the lesser sticking efficiency usually noted at large diameters. Sticking Efficiency vs Particle Diameter at Wall Temperature of 1273K OSU model El-Batsh model Particle Diameter (μm) Fig 7.10: Sticking Efficiency vs Particle Diameter at M=

89 The above graph shows the comparison of the particle sticking efficiency predicted from the OSU model in comparison with El-Batsh et al. The Young s modulus correlation was changed according to the material used by El-Batsh but the correlation was still dependent on the gas temperature. Samples of 4100 particles were injected with a uniform distribution over the blade pitch. The particle density was considered to be 1700 kg/m 3 based on the experimental results of Ahluwalia et al., (10). The specific heat of ash from the corresponding experiments was found to be 710 J/KgK. Three particle diameters were considered for this case d=0.1, 1 and 10 µm. The particle velocity and temperature at injection were considered to be the same as the gas properties at the inlet. The thermal conductivity of the ash particle used by El-Batsh was unknown and was considered to be the same as the one used in the previous simulations. The Young s modulus was assumed as per the following correlation: E p = 120 (1589 T p ) 3 (7.2) The trend of the sticking efficiency decreasing with increasing diameter as seen by El- Batsh is reflected by the OSU model. The sticking efficiencies calculated by the OSU model are not the same as El-Batsh at higher diameters. The mach number contours show that El-Batsh simulations show a much more prominent shock than the OSU simulations. There is no experimental data for these cases to compare with and the wall friction velocity calculation used by El-Batsh is not known. There is no proof that the sticking efficiencies calculated by El-Batsh are accurate values for these cases considering the lack of experimental data to support these predictions. Hence, the values obtained from the OSU model cannot be dismissed altogether. Also, at 10μm, the sticking efficiency is so low in both the models that the number of particles sticking might be anywhere 72

90 between one and five particles out of a total of The DPM results from FLUENT cannot be expected to accurately predict the exact number of particles sticking due to the various assumptions that go into the deposition model and the calculation of model parameters by different methods in different models. Incidentally, the simulations from the BYU model predict a sticking efficiency of between for temperatures in the range of 1400 K for a particle diameter of 1μm for simulations on a bare coupon. The OSU model shows a sticking efficiency of 0.8 for 1μm for wall temperature of 1476K which is in the range predicted by BYU model. Still, the deposition model matches the trend and this is a desirable result and a validation checkpoint before moving onto the OSU turbine vane simulations with film cooling. Also, the deposition model will be similarly validated for the GE-E 3 vane in a 3D domain and with the time constraints in mind, these results for the VKI vane are considered enough for validation. 73

91 8. IMPROVEMENTS TO THE DEPOSITION SIMULATIONS In this section, a description of changes and improvements to the deposition simulations is given. One of the major changes from the BYU model is the change in the ca lculation of wall friction velocity from being based on the particle velocity to being based on the non-dimensionless distance to the wall (y+) and in turn, the shear stress at the wall. All BYU simulations were performed on computational grids with y+ of and the k-ω turbulence model. Although this y+ lies outside the viscous sub-layer zone, the k-ω turbulence model usually requires the mesh to be resolved all the way up to the wall. This, in turn, requires the y+ value to be less than 1 to achieve a fine mesh near the wall surface of interest. A better resolution of the flow-field is achieved, especially of the viscous boundary layer near the wall with a finer mesh. Also, as shown by Wang et al., (61) boundary layers ensure lesser number of particles reach the surface which is an actual phenomena occurring in the experimental cases. Further, Ai & Fletcher (17) developed their Young s modulus correlation based on the older method of calculating the wall friction velocity and also on a mesh with a y+ of It was deemed fit to arrive at a newer correlation using the OSU model, ensuring the flow-field results for obtaining the new correlation are obtained from a finely resolved near-wall mesh. A 2-D mesh for the same geometry as in the 3-D simulations on the bare coupon in Section.6 was generated with enough near-wall resolution to give a y+<5 which is the acceptable region for the k-ω turbulence model used in these simulations. The mesh was generated with 165,258 cells in the 2-D domain. A hexahedral boundary layer was fitted to the 74

92 coupon wall and the remaining areas of the domain were meshed with tetrahedral cells. All boundary conditions and flow-field simulation conditions were the same as in the bare coupon case. Fig. 8.1: Close up view of the boundary layer on the coupon Fig. 8.2: View of the tetrahedral mesh for the computational domain Once the flow-field solution was obtained, the OSU deposition model was applied to all 6 temperatures as in the earlier case. The BYU Young s modulus correlation was not used here. Instead, the value of the Young s modulus was iterated with the OSU model until 75