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A Thermal Spray Process DROPLET IMPACT AND SOLIDIFICATION IN A THERMAL SPRAY PROCESS Moh;i.mmad Pasandideh-Fard .A t hesis su bmi t ted in conformi ty wi t h the requirements for the degree of Doctor of Philosophy Graduate Department of Mechanical and Industrial Engineering University of Toronto @ Copyright by Mohammad Pasandideh-Fard 1998 National Library Bibliothéque nationale du Canada Acquisitions and Acquisitions et Bibiiographic Services services bibliographiques 395 Wellington Street 395, me WeUington OtiawaON K1AW OttawaON K1AW canada Canada The author has granted a non- L'auteur a accordé une licence non exclusive licence allowing the exclusive permettant à la National Library of Canada to Bibliothèque nationale du Canada de reproduce, loan, distribute or sell reproduire, prêter, distriiuer ou copies of this thesis in microform, vendre des copies de cette thèse sous paper or electronic formats. la forme de microfiche/fïlm, de reproduction sur papier ou sur format électronique. The author retains ownership of the L'auteur conserve la propriété du copyright in this thesis. Neither the droit d'auteur qui protège cette thèse. thesis nor substantial extracts hmit Ni la thèse ni des extraits substantiels may be printed or otherwise de celle-ci ne doivent être imprimés reproduced without the author's ou autrement reproduits sans son permission. autorisation. To Fatemeh and Farid DROPLET IMPACT AND SOLIDIFICATION IN A THERMAL SPRAY PROCESS kioharnmad Pasandideh-Fard Ph.D. Thesis Graduate Department of Mechanicd and Industrial Engineering University of Toronto Abstract A numerical model Ras developed on the basis of SOLA-VOF algorithm to study the impact and solidification of a liquid droplet upon its impingernent on a substrate. The model, in generd, is applicable to transient fluid flows and heat transfer including two moving boundaries: a liquid-gas free-surface boundary and a liquid-solid interphase. The model, in particular, was used to analyze the formation of a coating layer made from one droplet impact as a function of processing paxameters in a thermal spray process. The numerical model was developed step by step by first considering an isothermal droplet impact and then modifying the flow dynamic model to include heat transfer and simultaneous solidification. The modification of the fluid dynamic equations in the presence of solidification was based on the improved fied velocity technique. The first solidification model used was a 1D model well suited for plasma spray operations. A 2D, axisymmetric enthalpy model was finally employed for heat transfer and simultaneous solidification in the droplet and conduction heat transfer in the substrate. Previous models of droplet impact either neglected or used simplifying assumptions when dealing wit h: capillary effects, sirnultaneous solidification, droplet-subst rate thermal contact resistance, and heat transfer to the substrate. The model developed in this study. however, considered capillary effects at bot h liquid-su bstrate and liquid-solid interfaces, simulated simultaneous solidification and heat transfer to the substrate during the impact dynamics. and considered thermal contact resistance at the surface of the substrate. By comparing detailed numerical predictions with available experirnental rneasurements and by considering analytical models in conjunction with the concept of dimensionless numbers. it was found that: capillary effects during droplet impact is negligible if CCé » fi.and the effect of solidification on droplet impact dynarnics is negiigible if \l~te/~r « 1. Yumerical predict ions were compared and verified wi t h available experirnent al results for two sets of water and tin droplets impacting a Bat stainless steel surface. The model predictions were then obtained for two typical thermal spray processes: RF and DC plasma spray operations. For water and tin droplets. capillary effects were important: for typical plasma spray operations, however. capillary effects were negligible. i-e., no knowledge of contact angle is required when studying the impact dynamics in plasma spray conditions. For tin droplets considered in this study, we found t hat simultaneous solidification considerably affected the impact dynamics and maximum droplet spread. For typical plasma spray cases, however, solidification effects were much lower. When capillary and solidification effects are more important the numerical modeling of the problem is more challenging; a fine and uniform computational mesh, therefore, must be used. Verification of the mode1 results for droplet impact cases with high capillary and solidification effects, therefore, verifies its results for the cases where these effects are less important. Acknowledgments 1 would like to take this opportunity to express my thds and appreciations to my supervisor, Professor J. Mostaghimi, for his guidance, encouragement and financiai support during the course of this study at the University of Toronto. My sincere gratitude goes to the Ministry of Culture and Higher Education of Iran for the financial assistance in the form of PhD Scholarship, without which this work would not have been possible. In addition, 1 wish to acknowledge the hancial support provided by the University of Toronto in the forms of Open Fellowship and Ontario Graduate Scholarship. 1 would also Lke to extend rny special acknowledgment to my wife Fatemeh and my parents for their interest and support throughout my years as a student. Contents List of Figures ix List of Tables xiv Nomenclature xv 1 Introduction 1 1.1 Introductory Remarks .............................. 1 1.2 Literature Review ................................. 4 1.2.1 Isothennal Droplet-Impact ........................ 4 1.2.2 Droplet Impact and Solidification .................... 10 1.3 Objectives ..................................... 14 2 Isothermal Droplet-Impact 17 2.1 Introduction .................................... 17 2.2 Mathematical F~rmulations ........................... 18 2.2.1 Governing Equations ........................... 19 2.2.2 Boundary and Initial Conditions ..................... 20 2.2.3 Solution Algorithm ............................ 21 2.3 Computational Treatment ............................ 22 2.3.1 Finite Difference Continuity Equation .................. 22 2.3.2 Finite Difference Momentum Equations ................. 24 2.3.3 Pressure Difference Equation ....................... 28 2.3.4 Volume-of-Fluid (VOF) Difference Equation .............. 29 2-35 Free-Surface Construction ........................ 33 2.3.6 Boundary aad Initial Conditions ..................... 35 2.3.7 S t ability Considerations ......................... 40 2.4 Results and Discussion .............................. 41 2.4.1 Validation of Computational Mode1 ................... 42 2.4.2 Capiilaq EEects during Droplet-Impact ................ 43 2.5 Summary ..................................... 58 3 Modification to Flow Dynarnic Equations in the Presence of Solidification 60 3.1 Introduction .................................... 60 3.2 Matbernatical Formulations ........................... 63 3.2.1 Liquid-Solid Volume Fraction ...................... 63 3.2.2 Modified Continuity Equation ...................... 64 3.2.3 Modified Momentum Equations ..................... 65 3.2.4 Modified Volume-of-Fluid Equation ................... 66 3.3 Computational Treatment ............................ 67 3.3.1 Finite Difference Modified Cont inuity Equation ............ 67 3.3.2 Finite Difference Modified Momentum Equations ........... 68 3.3.3 Finite Difference Modified Volume-of-Fluid Equation ......... 69 4 Droplet Impact and Solidification: One-Dimensional Mode1 70 4.1 Introduction .................................... 70 4.2 Mathematical Formulations ........................... 71 4.2.1 Sirnplified Energy Equations ....................... 72 4.2.2 Boundary and Initial Conditions ..................... 72 4.2.3 Licpid-Solid Interface Position ...................... 74 4.2.4 Treatment of Thermal Contact Resistance ............... 75 4.3 Computationd Procedure ............................ 76 4.4 Resiilts and Discussion .............................. 77 4.4.1 Results for Typical Spraying Conditions ................ 78 4.4.2 The Effect of Contact Resistance on Droplet-Impact .......... 80 4.4.3 The ERect of Solidification on Droplet-Impact ............. 84 5 Droplet Impact and Solidification: 2D. Axisymmetric Enthalpy Model 88 1 Introduction .................................... 88 5.2 Mathematical Formulations ........................... 91 5.2.1 Energy Equation in Droplet Region ................... 92 5.2.2 Enthalpy Transforming Model ...................... 93 5.2.3 Energy Equation in Substrate ...................... 97 5.2.4 Boundary and Initial Conditions ..................... 97 5.3 Cornputational Treatment ............................ 101 5.3.1 Finite Difference Enthalpy Equation in Droplet Region ........ 101 5.3.2 Finite Difference Energy Equation in Substrate ............ 104 5.3.3 Boundaxy and Initial Conditions ..................... 105 5.3.4 Evaluation of Liquid-Solid Volume Fraction .............. 108 5.3.5 Computationd Steps ........................... 108 5.3.6 Stability Considerations ......................... 110 5.4 Results and Discussion .............................. 110 5.4.1 Cornparison of Numerical and Experimental Results .......... 110 - Review of Experimental Results ................... 111 - Estimation of Thermal Contact Resistance ............. 113 vii - Droplet Shapes and Temperature Distributions during
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