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

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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 the Impact 116 - The Effect of Solidification on Droplet Impact ...... 123 5.4.2 Numerical Resuits for Typical Thermal Spray Processes ...... 127

5.5 Summary ...... 132

6 Conclusions 136

References 142

A Grid Size Effect 155 List of Figures

Schematic mode1 of a DC orc plasma spray process (from [8]) ......

Initial configuration of the droplet at the time of impact ......

A finite-difference mesh with variable rectanguiar cells......

Control Volume (shaded area) for finite-difference cont inuity equation ..... Control Volume (shaded area) for finite-difference x-momentum equat ion ...

Control Volume (shaded area) for finite-difference y-momentum equation ... A free-surface ce11 and its neighboring full cell ...... Examples of free-surface shapes used in the advection of F in x-direction

where the donor cell is: (a) a full of Buid cell; (b) a horizontal free-surface ceII; (c-f) a vertical free-surface ce11 . The darker regions shown are the actual arnounts of F advected ......

Computational molecule to construct freesurface in (i. j) ce11 ......

Cylinderical curvature. Jqr ...... Liquid-solid contact line condition ...... 2.1 1 Calculat ion of the init id F values for free-surface cells ...... 2.12 A typical computational mesh ......

2.13 Computer generated images compared with photographs of a 2 mm diameter water droplet-impacting a stainless steel surface with a velocity of 1 m/s . The time of each frame (t) is measured from impact ...... 2.13 Continued...... 2.14 Comparison of photographs with model predictions for impact of droplets of pure water (0 pprn), LOO, and 1000 pprn surfactant solution...... 2.14 Continued...... - . - ......

2.15 Variation of measured spread factor (0,shown by symbols, during impact

of a droplet with 1000 pprn of surfactant, compared with model predictions (solid lines) using surface tension (7)values of 50 and 73 mN/m...... 2.16 Shapes of impacting 1000 pprn surfactant solution droplets calculated using surface tension (7)values of 50 and 73 mN/m...... 2.17 Measured evolution of the contact angle during spreading of droplets of pure

water (0 pprn), 100, and 1000 pprn surfactant solution. This figure is taken from Ref. [20]...... - ...... - . . . 2.18 Variation of the contact angle with contact line velocity. This figure is taken from Ref. [20]...... - ...... 2.19 Evolution of calculated (lines) and measured (symbols) spread factors during

impact of (a) pure water droplet, (b) 100 pprn surfactant solution, and (c) 1000 pprn surfactant solution......

2.20 Predicted droplet shape and velocity distribution at (a)t=0.9 ms, (b) t=l.Zms,

and (c) t=2.4 ms...... - - . - . . . - ......

3.1 An arbitrary control volume of the material containing both liquid and solid.

3.2 Definition of area fractions and q......

4.1 Schematic of the 1D solidification model...... 4.2 Sequence of spreading and simultaneous solidification of an alumina droplet of Case 1 of Table 4.1 (representing a typical RF plasma spray operation). .

4.3 Sequence of spreading and simultaneous solidification of an alumina droplet of Case 2 of Table 4.1 (representing a typical DC plasma spray operation). . Cornparison between experimental observations and simulation results for the spreading and simultaneous solidification of a tin droplet of Case 3 of Table 4.1. Computational results are given for different values of contact resistance at the substrate surface. Experimental results were adapted from Ref. [471. . . .

Variation of the maximum spread factor, &,,, vs. the substrate thermal- contact-resistance per unit area, &, for the tin droplet under consideration on an alumina substrate. Experimental results were adapted from Ref. [47]. .

Spread factor as a function of dimensionless time for : a) an alumina droplet b) a tin droplet ; cases 2 and 3 of Table 4.1, respect ively (arrows indicate where curves will move if Re and Ste are increased)......

Schematic diagram of the droplet and substrate at the time of impact. . . . . Enthalpy against temperature for an isothermd phase change material. . . . Ent halpy agôinst temperat ure for an alloy...... Control Volume (shaded area) for finite-difference enthalpy equation. . . . . Control Volume (shaded area) for finite-difference energy equation in the substrate...... Impact of molten tin drops on a stainless steel surface at an initial temperature of a)25OC, b)150°C, c)240°C. This figure is taken from Ref. [52]...... Liquid-solid contact angle variation during the impact of molten tin drops on a stainless steel surface at initial temperature of This figure is taken from Ref. [52]...... Substrate surface temperature variation during the impact of a molten tin droplet on a stainless steel surface initially at 25OC...... Substrate surface temperature variation during the impact of a molten tin droplet on a stainless steel surface initially at 150°C...... 117 5.10 Evoiution of spread factor during impact of tin droplets on a stainless steel surface initially at 25'C...... 118 5.11 Evolution of spread factor during impact of tin droplets on a stainless steel surface initiallyat 150°C...... 119

5.12 Computer generated images and photographs of the impact of tin droplets on

a surface initially at 240°C...... 120 5.13 Computer generated images and photographs of the impact of tin droplets on

a surface initially at 2j°C...... 121

5-14 Calculated velocity and temperature distributions inside a tin droplet impacting a surface initially at 25OC...... 122 5.15 Cornputer generated images and photographs of the impact of tin droplets on a surface initially at 150°C...... 124 5.16 Calculated velocity and temperature distributions inside a tin droplet impacting a surface initially at 150°C...... 125 5.17 Growth of the dimensionless solid layer thickness inside a tin droplet impact ing

a surface initially at 25OC. Predictions from the numerical model (with the

thickness averaged over the splat diameter), the analytical model of Ref. [115],

and from Eq. (5.52) (which neglects thermal contact resistance and assumes the surface to be isothermal) are shown...... 128

5.18 Computer generated images of the impact of an alumina droplet of Case 1 of Table 4.1 on a cold surface (representing a typical RF plasma spray operation). 130

5.19 Calculated velocity and temperature distributions inside an alumina droplet

irnpacting a cold surface for a typical RF plasma spray process...... 131

5.20 Computer generated images of the impact of an alumina droplet of Case 2 of

Table 4.1 on a cold surface (representing a typical DC plasma spray operation). 133

xii 5.21 Calculated velocity and temperature distributions inside an alumina droplet

impacting a cold surface for a typical DC plasma spray process...... 134

A.1 The effect of grid size on evolution of spread factor during impact of an ah-

mina droplet in a typical RF plasma spray operation (Case 1 of Table 4.1).

Coarse, medium and fine mesh sizes had 10, 20 and 30 cells per radius, re-

spectively...... 156 A.2 The effect of grid size on the dimensionless thickness hg during impact of

an alumina droplet in a typical RF plasma spray operation (Case 1 of Ta-

ble 4.1). Coarse, medium and fine rnesh sizes had 10, 20 and 30 cells per

radius, respectively...... 157

A.3 The effect of gid size on substrate temperature at the center during impact

of an alumina droplet in a typical RF plasma spray operation (Case I of

Table 4.1). Coarse, medium and fine mesh sizes had 10, 20 and 30 cells per radius, respect ively...... 158

A.4 The effect of grid size on substrate temperature at 4.4 ps elapsed after the

impact of an alumina droplet in a typical RF plasma spray operation (Case

1 of Table 4.1). Coarse, medium and fine mesh sizes had 10, 20 and 30 cells per radius, respectively...... 159 List of Tables

4.1 Input parameters for droplet spreading and solidification in typical thermal

spray operations ...... 79

6.1 Dimensionless numbers for typicd cases considered in this study...... 140

xiv Nomenclature

A wetted area of the substrate c specific heat C function of temperature gradient at the substrate d distance between free surface and the center of the neighboring full ce11 4 distance between centers of the free-surface cell and the neighboring full cell D splat diameter, measured at the splat-substrate interface

Do diameter of spherical droplet DIFFH finite-difference representation of diffusion of enthalpy h DIFFS finite-difference representation of diffusion of function S f resultant body force per unit mass F volume of fluid fraction FHX finite-difference representation of advective flux of h in x direction

FHY finite-difference representation of advective flux of h in y direction

FUX finite-difference representation of advective flux of u in x direction

FUY finite-difference representation of advective flux of u in y direction FVX finite-difference representation of advective flux of v in z direction FVY finite-difference representation of advective flux of v in y direction

9 acceleration due to gravity h ent halpy; splat thickness

Hf latent heat of fusion J mean curvature of the liquid-gas interface k thermal conductivity; a freezing parameter

L characterist ic lengt h rn mass P pressure

pressure at the neighboring fui1 cell adjacent to the free-surface ce11

pressure at the free surface inside the fluid

heat flux

principal radius of curvature thermal contact resistance per unit area at the splat-substrate interface

thickness of the solid layer dimensionless thickness of solid layer (= SI&) function of enthalpy h; finite difference representation of continuity equation

time

time for droplet to reach its maximum spread diameter

a function of t and x (=t - t,) dimensionless time (= t Do)

starting time of solidification at position x on the substrate

temperature

heat conduction potential

temperature difference between droplet and substrate

velocity cornponent in radial direction (x)

velocity cornponent in axial direction (y)

volume droplet impact velocity

finite-difference representation of viscous acceleration in x direction

finite-difference representation of viscous acceleration in y direction

work done in deforming the droplet against viscosity radial and axial coordinates, respectively single-dued functions representing free-surface boundq

xvi Greek Symbols a parameter in finitedifference approximation of velocity gradients;

t hermai diffusivi t y angle associated with the interface in cdculation of Jqr

boundary layer thickness; solid layer thickness in Stefan solution

pressure interpolation factor

surface tension

function of enthalpy h

viscosi ty

kinemat ic viscosity volume of viscous Buid; solidification constant

viscous dissipation func tion

area fraction at the right face of the control volume st ream funct ion

area fraction at the left face of the control volume

densi ty

stress tensor representing the resultant surface force liquid-solid contact angle advancing contact angle Lquid-solid volume fraction spread factor (= DIDO)

Dimensionless Numbers Bi Biot number (= Do/(& kr)) Ec Eckert oumber (= h2/(clAT))

xvii Pe Peciet number (= Dolai)

Pr Prandtl number (= v/ul)

Re Reynolds number (= l/o Dolu)

Ste Stefan number (= c(TmP- TWvi)/HI)

We Weber number (= pl D&)

Subscripts A acceptor cell

AD acceptor ce11 or donor ce11 b bottom boundary of the droplet computational domain cyl associated with the azimuthal direction D donor ce11 e east-side face of the control volume

9 g= i ini t id conditions

1 liquid L,R left ( upstream) and right (downstrearn) , respectively

M interphase rnax maximum mP me1 t ing-point condition

S solid w substrate; west-side face of the control volume w,i substrate at initial conditions w,t top boundary of the substrate computational domain

XY associated with x-y plane Chapter 1

Introduction

1.1 Introductory Remarks

Thermal spraying is a particulate deposition process in which solid materials are heated chemically (by combustion) or electricaily (by arcs), then accelerated by gas jets creating molten droplets of 10-100 pm in diameter. These liquid droplets are subsequently propelled towards the surface of a substrate where the individual molten particles impact, cool and solidify to fom a deposit. The microstructure of the coating results from the solidification of the particles [l].

Thermal spray technology is emerging as an important processing tool for both surface protection and advanced materials forming. Despite the technology having been in use for over 100 years, much of its development, driven by aerospace applications, has occurred in the past two decades. Increased understanding of process/structure/property relationships has resulted in a growing application of thermal spray coating technology and new processes; for example, low-pressure plasma spray, high-velocity oxyfuel (HVOF) spray and reactive plasma spray. Thermal spraying is an important method of producing coatings for Wear, thermal, oxidation and corrosion protection. It has been used for seals, dimensional restora- tion and remanufacture, and more recently for biocompatible coatings. Seals and specialized abradable materials, composed of a composite of metal and polymer andior ceramic are used in turbomachinery, pumps or other rotary devices where one part rubs preferentially into another to "wear" a sealing path. Polymers, ceramics, refractory metals, intermetallic doys, and even the new family of superconducton have been deposited onto a wide range of base materials, from plastics to steel and from high-temperat ure jet engine materials to t it anium alloys used in aircraft or axtificial prostheses. Thermal spray coatings have also been used as optical coatings, utilizing oxides that reflect or absorb light, on insulating (ceramics or plastics), or on conduct ing base materials. Jet engine manufact urers, automobile makers, elect ronic fims, industrial machinery producers, chernical plants, steel/met ai working mills, etc., are using thermal spray technologies; however, applications are changing rapidly. These applications are taking advantage of thermal spray's materials flexibility and its ability to form difficult materials, maay of which are too brittle, too refractory, or are alloys that have limited solubility, or axe made from a combination of materials that are not easily processed [l]. Thermd spray technology has grown from a coating to a forming process.

More recent ly, new fabrication met hods have been developed t hat use precisely cont rolled deposit ion of droplets to produce complex, three-dimensional objects, ernploying a computer to manipulate the substrate and spray nozzles [2- 61. Such techniques, known variously as

"spray forming" 12, 31, " microfabrication" [4], "free-form fabrication" [5], or " rnicrocast- hg" [6] offer irnproved metallurgical performance and reduced manufacturing costs.

The future development of thermal spraying is related to the economic requirements of the thermal spraying market. An analysis of this market shows an impressive rate of annual growth which is mainly attributed to new applications of the technology [7]. The need for better materials and coatings is now pushing the thermal spray technology to new Limits that require more precise understanding of processing/structure/property relationships.

In most spraying techniques the gases, combustion gases or plasma are the sources of the thermal and kinetic energy, heating the particles and propeUing them toward the substrate. Figure 1.1: Schematic mode1 of a DC arc plasma spray process (from [SI).

In laser spraying, however, the energy of photons ernitted by a laser is used to heat and melt the particles. A very common spraying technique, DC arc plasma spray process, is shown schematically in Fig. 1.1 [8]. Electric arcs are used as a thermal source to heat gases to temperatures exceeding 10000 K. At these high temperatures, feeding gases (e-g. Ar, Hz, He, N2,etc.) dissociate and ionize into their cornponent "ionsn and "electronsn as energy is pumped into the gases by a confined arc discharge [l]. This "plasman state is what gives the process its name but more importantly gives the process its ability to meit any material that has a stable melting point. The gun consists of an outer housing, a portion of which is conductive, forming the anode of the gun, and which is typically made of copper. The anode is attached to an insulating back plate through which is passed a cathode usually made of tungsten or copper with a tungsten tip. Both the anode and cathode are water cooled. Gases are introduced into the gun, heated to a plasma state by the arc discharge, and expanded through the nozzle to exit at high velocities. Powders are introduced in various locations in the expanding gas jets, depending on their melting point, either interna1 or external to the nozzle exit.

The desired coating properties are dictated by the applications. High density coatings, for exarnple, are required to provide Wear or corrosion resistance, whereas thermal barrier coatings should have a relat ively high percent age of porosity (close porosi ty ) to ensure t hermai shock resistance. The ultirnate goal of research efforts in thermal spraying field is to establish predictive correlations between the processing parameters and the properties (qual- ity) of the coatings. Physical properties such as porosity, microstructure, surface roughness and adhesion strength of coatings or artifacts produced by droplet deposition in a thermal spray process are determined to a large extent by the dynamics of deformation and solidification of the particles impinging on the substrate, the cooiing rate of the resulting splats, and the interactions of these splats with the underlying surface [9]. Better cootrol of the process requires a fundamental understanding of the fluid flow and heat transfer that occurs during the impact, spreading, and solidification of molten droplets.

1.2 Literature Review

1.2.1 Isothermal Droplet-Impact

Analytical Models

Several simple analytical models of droplet-impact have been reported, based on an energy balance that equates initial droplet kinetic energy to change in surface energy due to droplet deformation and work doue in overcorning liquid viscosity during impact. These models assume that the particle initially after impact takes the forrn of a cylinder. Later, the cylinder expands in a radial direction taking the form of a pancake and, at the same time, the solidification process occurs. These processes were analyzed by Jones [lu], Madjeski [ll], and more recently by Trapaga and Szekely [12], Chandra and Avedisian [13] , and Yoshida et ai. [14]. Madjeski's model, which is the most cited approach, accounts for viscous energy dissipation, surface tension effects and simultaneous freezing of the splat, without uncoupling the solidification problem from the flattening process. He tried to find the so-cdled "spread factorn defined as

where Do is the particle diameter prior to impact and A is the wetted area. In general terms, he was solving the equations describing the radial movement of the liquid cylinder (equation of energy balance) and its solidification (Stefan problem). His results for the parameter

( depend on the Re and We numbers, defined as: Re = Do/v and We = plb2D0/y, respectively. is the particle impact velocity prior to impact, pi and u are density and viscosity of the liquid material, respectively, and 7 is the coefficient of surface tension.

Madjeski considered a macroscopic balance for the radial flow of a cylinder representing a spherical droplet of equal volume by using the sirnplest velocity distribution which fulfîlls only the continuity equation and by assuming that the thickness of the liquid layer during the process is only a function of time but not space. He offered the results in the form of asymptotic equations for the maximum degree of spreading, cm,, of the impinging droplets for different cases, one of which accounts for the deformation and simultaneous solidification.

His rnodel for the case where no solidification is involved leads to the following equation [Il]

The equation holds for Re > 100 and We > 100. Bennett and Poulikakos [15] improved

Madjeski's model through combining the treatment of Collings et al. [16] (as well as that of

Chandra and Avedisian [13]) for the contact angle. Fukanuma [l?] also reported a different analytical approach with results similar to Madjeski's model.

Analytical models can only provide an estimate of the maximum degree of the spreading, and no informat ion is available on the transient characteristics of the deformation process or recoil. Experïrnental Studies

Experimentd observations on the spreading of a liquid drop impacting on a flat surface have been reported in the Literature as early as the late nineteenth century by Worthington [la]. A review of these experimental studies on droplet-impact is given by Rein [19]. More recently, a variety of high-speed photography techniques have been used to record the dynarnic shapes of a droplet during the impact. These included the use of film camera, CCD video camera, and electronically controlled short-duration flash light photography. The collision dynamics of a liquid droplet OR a solid rneta.Uk surface was extensively studied by Chandra and Ave- disian [13] using a single-flash-illumination photographie method which greatly improved the clarity of pictures. In a single-shot method only one image of a droplet is recorded at one instant of the impact process. The assumption is that the impact process is sufficiently re- peatable from droplet to droplet that by recording successive stages of the impact of several different drops (one image per drop) the entire impact process can be pieced together from individual images (131. The repeatability of the measurernents are verified by photographing different droplets at the same instant after the impact. This is repeated for different time delays. Key parameters describing the droplet spreading process, i.e. dynamic liquid-solid contact angles and contact diameters (i.e., the diameter of the wetted surface area) can be measured from enlarged recorded splat photographs. The accuracy of the splat size measurernents is in the order of 10 pm 1131. Qiao (201 used the same technique to study the effects of gravity and surfactant on spray cooling of hot surfaces. His results for the effects of adding a surfactant to water droplets impacting on a flat, solid surface will be used to verify the results of the numerical model in Chapter 2. Liquid-solid contact angle was varied in his experiments by adding traces of a surfactant to water. Impacting droplets were then photograp hed and liquid-solid contact diameters and contact angles were measured from enlarged photogaphs by manually drawing a tangent through the liquid-gas interface. His measurernent of the contact angle were reproducible within fY', and of contact diameter within f 0.1 mm (for a 2 mm droplet) [ZO].

An optical holography method was used by Zhao et al. [21] to visualize the droplet deformation. The field distribution resdting fonn the light scat tered by a deforming droplet is recorded on a hologram. The scattered field information recorded on a hologram provides the information about an object. For example, one hologram can readily generote different real images of a deforming splat that correspond to different viewing angles. Zhao et al.

[21] used a photoelectric technique for the fast recording of the splat radius during spreading and recoiling on a flat surface. A simple non-intrusive technique was then used for the measurement of the transient splat radius. The tirne and length scales involved in this technique are of the order of microseconds and microns, respectively [21]. .4 major drawback of the optical holographie technique in visualkation of droplet-impact is the lack of clarity of the recorded images.

Nurnerical Models

The complexity of the fluid dynamic aspects of droplet impingement on a solid surface is exemplified by the extreme deformation of the droplet surface occurring within very short time scaies. Numerical efforts targeting the fluid dynamics of the droplet-impact process initially adop ted oversimpli@ing assumptions in order to facilitate the solution. Hwlow and

Shannon [22] were the first to investigate the flow dynamics of the splash of a liquid droplet onto a flat plate and into a pool of the same liquid, using the then developed "Marker- and-Celln (MAC) [23] finite difference method to solve the Navier-Stokes equations. They attempted to explain experimental observations of the splashing process obtained by high- speed photography [18]. They neglected, however, any effects of Liquid surface tension and viscosity, so that their results are applicable only to the initial stages of droplet-impact when these forces are negligible compared to inertid effects. Their solution could not predict the maximum extent of liquid spread, but proved usefd in research on erosion of turbine blades by high speed impinging droplets to caldate peak liquid pressures immediately after impact.

Modeling heat transfer within the droplet required modifications to the MAC code to include surface tension and viscous effects, which was done by Tsurutani et al. [24] and Watanabe et al. [25].

An alternate algorithm for solving the complete Navier-Stokes equations in a transient fluid flow wit h multiple fie-surface boudaries, the " Volumeof-Fluidn (VOF) met hod, was developed by Nichols et al. [26]. A commercially available code (FLOW3D) [27] which implements the VOF method to model 3-dimensional, unsteady, free-surface flows was used by Trapaga and Szekely [12] to study the isothermd impingement of liquid droplets in spraying processes. The numerical model gave a detailed characterization of the droplet deformation; the free surface was tracked by a record keeping process following the fraction of fluid contained in cells adjacent to the Liquid-gas interface. Liu et al. (281 simulated the deformation and interaction of tungsten droplets wi t h different impact veloci t ies, different viscosities and surface tensions by solving the fdtw-dimensional, axisyrnmetric Navier- Stokes equations and the VOF equation by using the RIPPLE code [29]. A new algorithrn,

FLAIR (flux line-segment model for advect ion and interface recons tmct ion), was developed by Ashgriz and Poo [30] for improving surface advection and its reconstruction in the VOF method. In this technique, the surface is approximated by a set of line segments fitted at the boundary of every two neighboring computational cells. Another method of solution for free-surface flows is the height-flux method (HFM) developed by Mashayek and Ashgriz [31] to study the instability of liquid jets. This method is based on a Gderkin finite element method with penalty formulation, and a flux method for surface advectioo. Fukai et al. (321 presented a finite element-based technique to model the droplet spreading process. This technique accounted for the presence of inertid, viscous, gravitational, and surface tension effects using a varying, rather than fixed, discretization grid to improve solution accuracy when the droplet underwent large deformations. In contrast to other studies of droplet- impact, the Lagrmgian approach was employed to facilitate the accurate simulation of the

motion of the deforming free surface.

An accurate description of Buid flow at the liquid-solid-gas contact line is important in formulating realistic models of droplet impact. Analytical solutions have been derived [33,34]

to predict fluid flow during capillarity driven spreading of droplets deposited gently on a

solid surface. Modeling fluid behavior in the vicinity of a moving contact line is complicated,

because assuming a no-slip boundary condition at the liquid-solid interface leads to a force singularity at the contact line [33]. The problem cm be resolved by replacing the neslip

boundary condit ion wi th a slip mode1 [35]. Although this met hod alleviates mat hematical difficulties, there is no experimental evidence to determine which of several available slip

models is the most appropriate one to use, or whether slip does indeed occur.

Fluid motion in droplets impinging with significant velocity on a surface is controlled by

inertial in addition to capillary forces, and no analytical solution is amilable for the flow

problem. Numerical models of droplet-impact usually specify the contact line boundary condition by assigning a value to the angle between the solid surface and the liguid-gas interface. This apparent contact angle defines the shape of the free liquid surface above the contact line. Though the contact angle CM in principle be rneasured directly, until recently [13, 20) no experimental measurements of contact angle variation during choplet impact and spread were available in the literature. Trapaga and Szekely [12] and Liu et al. [28] therefore assumed contact angles rernain fixed, with arbitrarily selected values ranging from 5' to go0. Rather than assuming an arbitrary contact angle, Fukai et al. [32] neglected capillary forces at the contact line. They noted, however, that capillary forces become increasingly important towards the end of droplet spreading when inertial forces become small, and that mode1 results would be sensitive to capillary effects at this tirne. In a subsequent paper [36], they used experirnentdy determined vatues of advancing and receding contact angles, measured from photographs of droplets sliding down an inclined surface, and found t hat mode1 predictions improved.

1.2.2 Droplet Impact and SolidXcation

Analytical Models

Madjeski [Il]proposed a splat deformation and solidification model for the droplet deposition process in which the liquid phase of the splat assumes the shape of a cylinder. Instead of solving the thermal energy equation, Madjeski used the solution of the Stefan solidification problem to predict the solidification front location. In this treatment [ll] it is assumed that the temperature of the droplet at the time of impact is the solidification temperature of the metal, and that the onset of solidification is simultaneous with the time of impact. Furthemore, it is assumed that the freezing front advances through the melt in the near equilibrium limit controlled by the rate at which latent heat of fusion is transported away from the liquid-solid interface (known as Neumann solution to the Stefan ID solidification problem). Madjeski's investigation yielded an analytical expression for the maximum spread factor of the splat given by [Il]:

where k is a parameter which depends on the freezing constant in Stefan problem of solidification. A major simplification of Madjeski's model of solidification is that the thickness of the liquid layer over the solid layer is only a function of time and not of space. Markworth and Saunders [37] showed that the velocity field used by Madjeski does not satisfy the neshear condition at the upper free surface. They [37] proposed an improved velocity field which sat isfies the upper surface boundary condit ion. An improvement over Madjeski's solut ion presented by Delplanque and Rangel [38] utilizes the velocity profile suggested by Markworth and Saunders [37] and a different derivation of the viscous energy dissipation. Rangel and

Bian [39] presented a model which combines an analyticd solution for the spreading with a numerical solution for the solidification. The combined solution includes the analytical solution of the mech~cdenergy equation (containing kinetic and potential energies as well as viscous dissipation) based on Madjeski's modei and the numerical solution of the transient energy equation for liquid and solid phases using a fînite difference technique. Analytical models do not provide a fundamental understanding of the physical processes controlling molten droplet impact and solidification.

Experimental Studies

The splat formation and cooling rate of plasma sprayed molybdenum particles on different substrates were investigated experimentdy by Moreau et al. [40-421. Their work is concerned with the influence of substrate conditions and materiais on the coating texture and the cooling rate of sprayed particles. There is a large scatter in their measurements which is the result of a very difficult experirnental condition. Their measurements indicate that surface conditions play an important role in the solidification process. Bianchi et al. (431 and Vardelle et al. [44, 451 using a different experimental setup, came to simiiar conclusions.

Only a small number of experimental studies have ben done to directly observe impact dynarnics of molten droplets. Also, Little information is avitilable regarding the value of two important parameters: the liquid-solid contact angle, and the thermal contact resistance between the surface and impacting droplets. hada (461 measured the temperature wiation of a plate on which a molten lead droplet waç dropped, and noted that the droplet cooling rate was a function of impact velocity. Watanabe et al. [25] photographed impact of n- cet ane and n-eicosane (hydrocarbons with melting points of 17OC and 36" C, respectively ) droplets on a cold surface and concluded that in their tests, droplets spread completely before solidifying. Fukânuma and Ohmori [47] photographed the impact of 2 mm-4 mm tin and zinc droplets on a stainless steel surface, and also found freezing had almost no influence on droplet spread. Inada and Yang [48] used holographie interferometry to observe droplet- substrate contact during impact of lead droplets on a quartz plate. Kang et al. [49] studied,

using bot h experiments and simple numerical models, the solidification of two molten lead

droplets impacting sequentidy one on top of the other. They demonstrated that thermal

contact resistance at the splat-substrate interface, and between the two splats, influences

droplet cooling rate and grain structure. Theoretical and experimental studies done by

Bennett and Poulikakos [50] on the impact and solidification of liquid metal droplets on a

cold surface showed that the thermal conductivity of the substrate significantly dects the

cooling rate of the splat. Bhola and Chandra [51] experirnentally studied the impact and

solidification of molten paraffin wax droplets at 0.5 m1c2.7 m/s on an aluminum surface

(23OC-73OC). They found that the extent of droplet solidification was too small to affect

droplet-impact dynamics in their experiments. Bhola and Chandra [52] also studied the

impact and solidification of tin droplets falling on a stainless steel surface. The surface

temperature in their experiments was varied from 25OC to 240°C. Impacting droplets were

photographed and liquid-solid contact diameters and contact angles were measured from

enlarged photographs. These results will be used to verify the results of the numerical

mode1 in Chapter 5. Liu, Wang and Matthys [53] measured the temperature variation on

the upper surface of an impacting metal droplet using a pyrometer, and used these results to estimate the thermal contact resistance under the drop. However, the response time of

their pyrometer (25 ms) was longer than the time taken by the droplet to spread, so that

their results are applicable to the period after the droplet had corne to rest rather than the duration of impact itself.

Most laboratory studies of droplet-impact have been done using metal or hydrocarbon

droplets 2 mm-4 mm in diameter, impacting at velocities of 1 m/s-3 m/s, which are reiatively easy to generate and observe. However, practicd applications encompass a very large range of droplet sizes (from as smd as 10 pm to 1 mm), materials (metals, ceramics, polymers,

etc.), and impact velocities (1 m/s to 500 m/s). Therefore, a need for a numerical study of droplet-impact and solidification in these circumstances becomes cmcid.

Numerical Models

The heat and fluid flow phenomena occurring during the impact and solidification of a single

liquid-metal droplet on a cold substrate are not conventionai or easy to study. There are

several reasons for this fact: two moving boundaries with large deformations within the

cornputat ional domain, i.e. the liquid-gas free-surface and the liquid-solid interphase, should

be treated properly in the numerical model. Contact line issues have to be resolved in the

numericd model for both liquid-substrate-gas and Liquid-solid-gas contact Iines. The heat

transfer model must account for convection in the liquid and conduction in the solid with

steep temperat ure gradients wit hin a severely deforming domain, coupled wit h conduct ion

in the substrate,

Several numerical models have been developed to simulate impact and solidification of

molten droplets on a cold substrate. Bennett md Poulikakos [50] and Kang et al. (491 studied

droplet deposition assuming that solidification starts only after droplet spreading is complete,

when the splat is in the form of a disc. The validity of such an assumption depends on the rate of solidification following droplet-impact . Zhoo et al. [21, 541 st udied, using both numerical models and experiments, heat transfer and fluid dynamics during collision of a liquid droplet on a substrate. They extended the earlier model of Fukai et al [32] to account for the

relevant convection and conduction heat transfer phenomena both in the droplet and in the

substrate in the case that there is no solidification. Their results, therefore, are applicable

to the pre-solidifcation stage of the impact process. Liu et al. [55] used a one-dimensional solidification model in conjunction with a two-phase flow continuum technique to track the moving liquid-solid boundary. The model, however, does not account for the convection

in the liquid and conduction in the substrate. Trapaga et al. [56] used the commercially

amilable code, FLOW-3D [27], to study the heat transfer and solidification phenomena during droplet-impact. They assumed that the substrate was isothermal, and neglected any thermal contact resistance at the liquid-solid interface. Bertagnolli et al. [57] used a finite element approach with an adaptive discretization technique to model the deformation of the droplet geometry as well as the evolution of the themal field within the splat. Their model, however, does not account for solidification and heat transfer to the substrate. The liquid-solid contact angle was assumed to be futed in al1 of the above studies. Waldvogel and

Poulikakos [SI used a finite element model to simulate spreading and solidification during droplet-impact. They neglected capillary forces at the liquid-solid contact line, and assumed a value of the thermal contact resistance.

1.3 Objectives

As stated in the last section, previous models of droplet impact either neglected or used simplifying assumptions when dealing with: capillary effects during droplet impact; sirnultaneous solidification and its effects on arresting droplet spread; and droplet-substrate thermal contact resistance, heat transfer to the substrate, and their effects on the transient defor- rnation and solidification of the splat. It is, therefore, the objective of this study to develop a numerical model of droplet impact and solidification that considers capillary effects at both liquid-su bst rate and liquid-solid interfaces, simulat es simultaneous solidification and heat transfer to the substrate during the impact dynamics, and considers thermal contact resistance at the surface of the substrate. Numerical results in conjunction with analytical models and the concept of dimensionless numbers will then be used to explore the above effects during droplet impact and solidification. The developed numerical mode1 will be used to analyze the formation of a coating layer made from one droplet impact as a function of processing parameters in a thermal spray process. This analysis requires detailed information about droplet shape, pressure, velocity, and temperature distribution within the droplet and substrate during the impact, which can be obtained only by a complete solution of the continuity, momentum, and energy equations.

Droplet impact and solidification is a particdar case of transient free-surface flows, therefore, the developed mathematical model, in general, will be applicable to transient fluid flows and heat transfer including two moving boundaries: a liquid-gas free-surface boundary and a liquid-solid interphase. The rnodel, in particular, will be applied to droplet-impact and solidification in spray processes.

Chapters of this document have ben structured based on the progressing steps under- taken in developing the model on the basis of SOLA-VOF aigorithm. In this chapter, a literature review of relevant andytical and experimental studies as well as numerical models for droplet impact and solidification were discussed. Chapter 2 considers the results of the fiow dynarnic model, SOLA-VOF algorithm, for isothermal droplet-impact where no solidification is involved. The results of the numerical model are compared with available experimental measurements for water droplets under diEerent conditions. The purpose of this cornparison is to validate the accuracy of the model and its underlying assumptions, and to study capillary effects during droplet-impact on a solid surface. The next chapter explains the necessary modifications to the flow dynarnic equations in the presence of a moving liquid-solid interface. In Chapter 4 we will study the development of a one-dimensional solidification method to be included in the Bow dynamic model. Numerical results are next generated to predict the sequence of droplet-impact and solidification in a typical thermal spray process and to investigate the effects of operational parameters and droplet-substrate thermd contact resistance on the transient behavior of the deformation and solidification.

Chapter 5 presents the development of a complete flow dynamic and heat transfer model to include axisymmetric heat transfer for the droplet and substrate and account for sirnultaneous solidification within the droplet. To verify the model, numericd results axe compared with experimental results for tin droplet impact. Values of the thermal contact resist ance at the droplet-substrate interface are estimated by matching numerical predictions of the evolution of substrate temperature and droplet contact diameter with experimentd measurements. Results are then obtained for droplet-impact and solidification in typical thermal spray processes. The effects of solidification on droplet-impact and various aspects of droplet-substrate interactions are studied. The last chapter, Chapter 6, concludes the document and summarizes the achievements of the work. Chapter 2

Isot hermal Droplet-Impact

2.1 Introduction

Droplet impact is a particular case of incompressible transient fluid flow with freesurface

boundaries. Free-surface boundaries are here considered to be surfaces at wbich discontinu-

ities exist in material properties. Modeling free-surface flows poses a significant challenge

because a boundary condition must be applied to a transient, irregular surface (the free

surface) t hat is ideally a discontinuity. Surface tension frequent ly constrains the accuracy with which the resulting boundary condition is applied in numerical models. Several methods are used to approximate free-surface boundaries in finite-difference numerical simulations. A simple, but powerful, method which is based on the concept of a fractional volume of fluid (VOF) is described. This method is shown to be flexible and efficient for treating free-surface Bows. An incompressible hydrodynarnic numerical code, SOLA-VOF [26],that uses the VOF technique to track free fluid surfaces is employed here

to study the fluid deformation during the isothermd impact of a droplet on a solid surface.

The original code was enhanced with special parts for: the implementation of contact angle condition at the contact line, the definition of initial droplet configuration, reading the initial input data, and generation of output files for visualization with Tecplot software. In this su bs trate

Figure 2.1: Initial configuration of the droplet at the time of impact. chap ter, the mat hernat ical formulations of droplet-impact dynamics are first stated. Next , the cornputational treatment of the problem based on the SOLA-VOF algorit hm is presented.

The model is then used to simulate water droplets impacting on a flat solid surface, cases for which experimental observations are available. Through cornparisons between numerical and experimental results, the accuracy of the numerical predictions are verified and capillary effects during isot hermal droplet impact are studied.

2.2 Mat hemat ical Formulations

Figure 2.1 shows the axisymmetric coordinate system used in formulating the numerical rnodell. and the initial configuration of the droplet at the time of impact. t = O. The mat hemat ical mode1 assumes that droplet impingement velocity is normal to the substrate, and that fiuid flow is laminar and incompressible. The mat hematical model of the problem using an Eulerian formulation can be written as follows.

'The radiai direction in an axisymmetric coordinate system iç usually denoted by r instead of x. However, in cornputational codes (for 2D, axisyrnmetric flows) the notation of x is frequently used. In this document, therefore, we use the notation of z to indicate the radial direction. 2.2.1 Governing Equations

The governing flow equations are the classical Navier-Stokes equations:

where u and v are velocity components in the x and y directions, respectively (Fig. 2.1).

P, p, and v are pressure, density, and kinematic viscosity of the fluid, respectively, and g represents gravitationai force per unit mass.

To represent the free boundaries of the droplet, the "fractional volume of fluid (VOF)"[26] scheme is used. In this technique, a function F(x,y, t) is defined whose value is unity at any point occupied by fluid and zero elsewhere. When averaged over the cells of a computationd mesh, the average value of F in a ce11 is equd to the fractional volume of the ce11 occupied by Buid. In particular, a unit value of F corresponds to a ceIl full of Buid, whereas a zero value indicates that the ce11 contains no fluid. Celis with F values between zero and one contain a free surface. The VOF method requires only one storage word for each rnesh cell, which is consistent wit h the storage requirements for al1 other dependent variables. Since F moves with fluid (Le., the total due of F for the droplet is constant) this function satisfies the conservation equat ion:

In addition to defining which ceils contain a free boundary, the F function can be used to define where fluid is located in a boundary ceU. The reconstmcted fiee surface is not necessarily continuous, but is instead represented as a set of discrete, discontinuous line segments. In this study, a Nichois-Hirt algorithm [26]is used to reconstmct the free surface. The free surface is reconstmcted either horizontdy or vertically in each free-surface cell, depending upon its relation to aeighboring ceus, as determined by the gradient of the function

F. The normal direction to the boundary lies in the direction in which the value of F changes most rapidly.

2.2.2 Boundary and Initial Conditions

There are two sets of boundaries for the problem, computationd mesh boundaries and free- surface boundaries-

Mesh Bowidaries. Due to the flow symmetry about the y axis (Fig. 2.1) only one slice of the droplet is considered in the computation. Velocities must be set at the boundaries of the computational domain. The left mesh boundary, the axis of symmetry, is treated as a free-slip wdl; the normal velocity there must be zero and the tangential velocity should have no normal gradient. For the bottom boundary, i.e. the surface of the substrate, we use no-slip boundary conditions; this means that both normal and tangential velocities at the substrate surface are zero. Since the transient flow dynamics of the droplet are under investigation, the computational domain must be large enough such that the fluid does not reach the right and the top boundaries. As a result, the conditions at these boundaries do not count; they are set as free-slip boundary conditions. Based on the above, the left boundary conditions are :

And for the bottom boundary conditions we have :

Free-Surface Boundaries. At a free surface, tangentid stresses are set equal to zero and normal stresses are replaced by an equident surface pressure, calcdated from the interface condit ion given by the Laplace equation: where Pt and P, are pressures inside and outside the draplet, respectively , J is the interface mean curvature, and 7 is the liquid-gas surface tension. This condition must be supplemented wi t h the specification of velocities immediately out side the surface, where these values are needed in the finite-difference approximations for points outside the surface. Velocit ies must be set on every ceil boundary between a surface ce11 and an empty cell.

Describing the liquid-solid contact line requires specid attention. We incorporated the contact angle, 0 in the free-surface boundary condition, Eq. (2.7), by using it to calculate the mean curvature, J, of the liquid meniscus ne= the substrate. The mode1 uses either a fixed value of O, or a dynomic contact angle that varies with time during droplet impact.

When dynamic contact angle values are used, they are updated after each time step. Initial Conditions. Initial conditions are initial dues of function F, and velocity distributions at the time of impact, which is assumed to be t = O. The initial values of the function F are calculated based on the initial fluid configuration. The initial velocities at every point inside the fluid are:

u=O , v=-vo (2-8) where Vo is the impact velocity.

2.2.3 Solution Algorithm

The basic procedure for advancing a solution through one increment in time, bt, consists of the following steps:

1. Explicit approximations of Eqs. (2.2) and (2.3) are used to compute the first guess for

new tirne-level velocities using the initial conditions or previous time-level values of all advective, pressure, and viscous accelerations.

2. To satisfy the continuity equation, Eq. (2.1), pressures are iteratively adjusted in each

ce11 and the velocity changes induced by each pressure change are added to the velocities computed in the previous step. An iteration is needed because the change in pressure

needed in one cell to sati* Eq. (2.1) will upset the balance in the four adjacent cells. Boundary conditions at aU mesh and free-surface boundaries must be imposed after each iteration.

3. The F function definjng fluid regions is updated based on Eq. (2.1) to give the new Buid configuration.

4. Al1 boundary conditions are updated and time is advanced by one increment.

Repetition of these steps will advance a solution through any desired time interval.

2.3 Computational Treatment

The computational treatrnent of the governing equations given in this section is docurnented based on the SOLA-VOF algorithm [26]. We use an Eulerian mesh of rectangular cells having variable sizes, 6xi for the ith column and byj for the jth row, as shown in Fig. 2.2. The goveming and boundary equations are discretized by using a finite difference scheme on a staggered grid. In the staggered grid, the pressure P and the volume-of-fluid function F are defined at the center of the grid while the u and v velocities are defined at the right and top boundaries of the grid, respectively

2.3.1 Finite DifFerence Cont inuity Equation

Figure 2.3 shows the control volume (shaded area) used for constructing a finite-difference approximation for the continuity equation, Eq. (2.1). The finite-difference equation t hen follows Figure 2.2: A finite-difference mesh wit h variable rectangular cells.

Fi,j A

0-1

Figure 2.3: Control Volume (shaded area) for finite-difference continuity equation. Figure 2.4: Cont rol Volume (shaded area) for fini te-difference x-moment urn equation.

The notation Q:, stands for the value of Q(x,y, t) at time nbt and at a cootrol volume in the it h column in the x-direction and jth row in the y-direction, as shown in Fig. 2.3. Si,j must be driven to zero for al1 mesh cells at al1 times.

2.3.2 Finite Difference Momentum Equations x-Moment um Equation

The control volume (shaded area) used for construct ing a fini te-difference approximation for the x-momentum equation. Eq. (23, is shown in Fig. 2.4. The finite difference equation rnay be written as

n+L - -(S+i,j n+L = ,u?, + at 'GJn+l ) +gz - FUX - FUY + VISX (1.10) 'i.j p62i w here - bxi = (&xi + 6~~+~)/2 (2.11)

FUX and FUY are the advective flux of u in the x and y directions, respectively, and VISX is the viscous acceleration term. These terms are al1 evduated using the old time level n values for velocities (explicit formulation). Because the pressures at time Ievel n + 1 are not known at the beginning of the cycle, Eq. (2.10) cm not be used directly to evaiuate un+', but must be combined with the continuity equation as described later. In the first step of a solution, therefore, the Pn+' values in above equation are replaced by Pn to get a first guess for the new velocities. In the basic solution procedure, the specific approximation chosen for the advective and viscous terms in Eq. (2.10) are relatively unimportant, provided they lead to a numericaily stable algorit hm. Specid care must be exercised, however, when making approximations in a variable mesh like that of Fig. 2.2. In this regard, we use the nonconserva.tive form of the advective terms in the momentum equations, Eqs. (2.2) and (2.3). If the conservative form of the advective terms is used, the variable mesh wiii reduce the order of approximation by one, and in some cases will Iead to an incorrect zeroth order result. The reason is because the control volumes are not centered about the positions where variables are located. Because of this, the advective terms should be corrected to account for the difference in locations of the variables being updated and the centroids of their control volumes; when this is not done, a lower order error is introduced. The stability advantages of the upwinding scheme can be retained in a variable mesh with no reduction in formal accuracy, if the nonconservative form is used for the advection flux. At the sarne time, it is possible to combine the upwinding and centered-differencing approximations into a single relation. The finite-difference expression for FUX may be written as

Fux = ui, (2) at Ui ,J location

where and

The parameter a in the above expression is a variable coefficient, satisfying O 5 a 5 1, that controls the linear combination of upwinding and centered differencing used in the approximation of the velocity gradients. The expression reduces to the first order upwinding scheme when a = 1 and the second order centered differencing when a = O. The basic idea used in Eq. (2.13) is to weight the upstream derivative of the quantity being Buxed more t han the downstream value. The weighting factors are 1+ a and 1- a,for the upstream and downstream derivatives, çespectively. The derivatives are also weighted by cell sizes in such a way that the correct order of approximation is rnaintained in a variable mesh. This type of approximation will be used for dl advective flux terms appearing in momentum equations.

Similarly for FUY we have:

Y = G(g) at u;,, location

w here

and Figure 2.5: Control Volume (shaded area) for finite-difTereoce y-rnomentum equation.

The viscous accelerat ion term is approximated wit h the standard centered-di fferencing approximation (no upwinding); the expression for VISX may be wriritten as:

PU r VISX = V{(*)8x2 + (-)ay + (--)z dx - (5)) at u,,~location where al1 terms are evaluated at ui, location for example

(Z)= &{(ElR-(E)L} Sli? (&) and (e) were defined in Eqç. (2.1 1) and (2.14). y-Momentum Equation

The control volume (shaded xea) used for constructing a finite-difference approximation for the y-momentum equation, Eq. (2.3), is show in Fig. 2.5. The finite difference equation may be written as

w here FVX and FVY are the advective flux of v in the x and y directions, respectively, and VISY is the viscous acceleration term. These terms are al1 evaluated using the old time level n values for velocities (explicit formulation); the finite-difference expressions for t hese tems can be obtained in a fashion quite similar to the one taken for the s-momentum equation.

2.3.3 Pressure Difîerence Equation

Because the velocities appearing in Si,., Eq. (2.9). are evaluated at the new time level, which depend on the n + 1 level pressures according to Eqs. (2.10) and (2.22), this equation is an implicit relation for the new pressures, i.e. S = S(P). A solution may be obtained by the following iterative process. The computational mesh is swept row by row starting with the bottom and working upward. In each cell containing fluid, but not a free surface, the pressure change needed to satisfy Eq. (2.9) may be obtained as

where (aS/aP)is implemented as a relaxation factor. Therefore, for full of fluid cells we have

6Pi, = -S.=J - x (relaxation factor) (2.25)

A similar procedure is used in cells containing a free surface, except that Si,-used in Eq. (2.24) is not the right hand side of Eq. (2.9), but a relation that leads to the proper freesurface boundary condition when driven to zero by the iteration. The boundary condition is satisfied by setting the surface ce11 pressure Pij equal to the value obtained by a linear interpolation between the pressure wanted at the surface P, and a pressure inside the fluid Piv. For this scheme to work, the adjacent cell chosen for the interpolation should be such that the line connecting its center to the center of the surface cell is closest to the normal to the free surface.

Consider Fig. 2.6, where a free-surface cell and its neighboring full celi are shown schematically. Using a pressure interpolation relation we may mite 65 * >

4 free-surface ! ceil 1

! pi s'i A 1 i j 4 1 l i 1 d ldci I neighboring t i full ce11 PN

Figure 2.6: A free-surface ce11 and its neighboring full cell.

where PLVis the pressure at the neighboring full ce11 adjacent to the Free-surface cell, (i, j).

P, is the pressure at the free surface inside the Buid and 7 is called the pressure interpolation factor. Now. the S function giving this result rnay be written as

This is the pressure relation for the free-surface cells which must be driven to zero by the iteration. Therefore, Si,in Eq. (2.24) for free-surface cells must be replaced by the right hand side of Eq. (2.27).

2.3.4 Volume-of-Fluid (VOF) Difference Equation

The governing equation for the VOF function is Eq. (2.4); this equation may be combined with Eq. (2.1) resulting in When this equation, which is in divergence form, is integrated over a computational cell, the

changes in F in a ce11 reduce to fluxes of F across the cell faces. The standard finite-difference approximations of Eq. (2.28) would lead to a smearing of the F function and interfaces would

lose their definition; the cause of and the remedy for this problem are given in the following

discussion. In an Eulerian representation of a fluid, the grid remains fixed and the identity of individual Buid elernents is not maintained. In this technique, to move the fluid elements to the

new positions it is necessary to compute the flow of fluid through the mesh. This Bow, or

convective flux calculation, requires an averaging of the flow properties of al1 fluid elements

that find themselves in a given mesh cell after some period of time. Convective averaging

results in a smoothing of dl variations in flow quantities, and in particular, a smearing of

surfaces of discontinuity such as free surfaces. The way to overcome this loss in boundary

resolution is to introduce some special treatment that recognizes a discontinuity and avoids

averaging across it. Fortunately, the fact that F is a step function with values of zero, one or

in between, permits the use of a flux approximation that preserves its discontinuous nature.

This approximation, referred to as a Donor-Acceptor method [26] is discussed below.

Donor-Acceptor Flux Approximation

The essential idea in this approximation is to use information about F downstream as well

as upstream of a £lux boundary to establish a crude interface shape, and then to use this

shape in computing the flux. The method may be understood by considering the arnount of F to be fluxed through the right hand face of a computational ce11 during a time step of duration 6t. The flux of volume crossing this ce11 face per unit cross sectional area is ubt, where u is the normal velocity at the face. The sign of u determines the donor and acceptor cells, i.e., the cells losing and gaining volume, respectively. For example, if u is positive, the upstream or left cell is the donor and the downstrearn or right cell the acceptor. Depending on F values and free-surface configurations in the donor and acceptor ceus the amount of volume advected across the ceil face is diflerent; a generd relation to compute the transferred volume may be obtained as foliows. Figure 2.7 provides a pictorial exphnation of different fluid configuration in the donor and acceptor cells. For the different cases shown the following expressions rnay be obtained for "the amount of volume advected (per unit iength

nomal to the paper)"

when the donor ce11 is full, Fig. 2.7(a)

a when the donor ce11 is a horizontal free-surface cell, Fig. 2.7(b)

a when the donor cell is a vertical free-surface ceil where the fluid is at the right,

Fig. S.i(c) and 2.7(d)

6yj Min {(uij6t) FD 6xo} (2-31)

when the donor ce11 is a vertical free-surface cell where the Buid is at the left, Fig. 2.7(e)

and 2.7(f)

6yj Max {(ui,6t) - (1 - FD)bxD, 0.0) (2.32) where subscript D denotes the donor cell. It can be seen that the minimum feature in relation (2.31) prevents the advection of more F from the donor cell than it has to give, while the maximum feature in relation (2.32) accounts for an additional F flux which must be considered. For advection of F in the y-direction a similar procedure may be used considering that in this case the horizontal free-surface cell needs special care. To have a general relation applicable to all cases, the above relations may be combined resulting in

"the amount of volume advected (per unit length normal to the paper)" = Figure 2.7: Examples of free-surface shapes used in the advection of F in x-direction where the donor ce11 is: (a) a full of fluid cell; (b) a horizontal free-surface cell; (c-f) a vertical free-surface cell. The darker regions shown are the actual amounts of F advected. The double subscript, AD, refers to either the acceptor A or donor D, depending on the orientation of the interface relative to the direction of the Bow. FA dueis used in place of

FaD when the surface is advected mostly normal to itself; otherwise, the donor ce11 value is used. However, if the acceptor ceIl is empty of F or if the cell upstream of the donor cell is empty, then the acceptor ce11 F value is used in place of FaD regardless of the orientation of the free-surface. This means that a donor ceLl must fill before any F fluid can enter a downstream empty cell.

Once the volume of fluid advected through a ce11 face has been computed, it is divided by the volume of the donor ce11 to get the amount of F fluid to be subtracted from the donor cell. The advected volume of Buid when divided by the volume of the acceptor ce11 gives the amount of F fluid to be added to the acceptor cell. When the process is repeated for ad ce11 faces in the mesh, the resulting F values correspond to the time-advanced values satisS.ing Eq. (2.28) and still sharply define all interfaces.

2.3.5 Free-Surface Construction

For the accurate application of boundary conditions, the free-surface of the fluid has to be constmcted at each time level. In the Volume-of-Fluid (VOF) technique [26], it is assumed that the boundary can be approximated by a straight Line cutting through the cell. By first determining the slope of this line, it con then be moved across the ce11 to a position that intersects the known amount of F volume in the cell.

To determine the boundary slope, it must be recognized that the boundary can be rep resented either as a single-valued function Y(x)or as X(y), depending on its orientation.

To accomplish this in a cell (i,j), eight cells surroundhg it should be considered, i.e., the Figure 2.5: Computational molecule to construct free-surface in (i. j) ce11 cornputational molecule consists of oine adjacent cells (3 x 3) as showo in Fig. 2.8. -4 good approximation to Y(x)is

where Y = O is taken as the bottom edge of the j - 1 row of the molecule. l;+iand x-1 are similarly obtained. Then

where bx is defined by Eq. (2.11 ). Similarly, X(y)may be approximated by

where X = O is taken as the left edge of the i - 1 colurnn of the rnolecule. Having evaluated

X,+l and Xj - 1 similarly, we can write

where iis defined by Eq. (2.23). The free-surface orientation in an (i,j) ceil can be determined by comparing the abs* lute values of the above two derivatives. The derivative with the smdest magnitude gives the best approximation to the dope of the free-surface because the corresponding Y or X approximation is more accurate in that case. Therefore, if IdYldxl is smder than (dX/dyl, the boundary is more nearly horizontal than vertical, otherwise it is more nearly vertical.

The values of the above derintives may also be used to find out where the fluid exists in a free-surface cell. Suppose IdYldxl is srnallest so the interface is more horizontal than vertical. If dX/dy is negative, F fluid lies below the boundary, and ce11 (i, j - 1) is used as the neighboring interpolation cell for the surface ce11 (i, j) in the free-surface pressure difference equation, Eq. (2.27). Had dX/dy been positive, ce11 (i,j + 1) would be chosen for the neighboring interpolation ce11 because fluid would t hen be above the boundary.

Once the boundary dope and the side occupied by fluid have been determined, a Line can be constructed in the ce11 with the correct amount of F volume lying on the F fluid side.

This line is used as an approximation to the actual boundary and provides the information necessary to calculate r) for the application of the free-surface pressure diKerence equation, Eq. (2.27).

2.3.6 Boundary and Init id Condit ions

Mesh Boundaries

To impose the conditions at mesh boundaries a layer of fictitious cells surrounding the actual mesh is assumed. Velocities for the fictitious cells are set such that the proper conditions at the actual wds, Eqs. (2.5) and (2.6), hold. For the Ieft boundary where free-slip conditions exist, t herefore, we rnay write: At the bottom boundary we have neslip conditions, therefore:

where u;,~is set such that u at the bottom wdl would be zero. The conditions at the top and right boundaries are set as a free-slip boundary similar to Eq. (2.39).

Free-Surface Boundaries

There are three sets of boundary conditions at the free surface: surface tension effects based on Eq. (2.7), liquid-solid contact line conditions, and specifications of veloci t ies at and outside the interface. Surface Tension. Two essential steps are needed to include surface tension effects in a calculation. First, it is necessary to compute a local curvature, J, in each free-surface ce11 using Y(r)or X(y) definitions from Eqs. (2.35) and (2.37), so that a surface tension pressure, P., can be evaluated. Once the curvature J is found, this surface tension pressure is caiculated based on Eq. (2.7) and given by

Second, it is necessary to impose this surface force on al1 interfaces based on the pressure relation for the free-surface cells, Eq. (2.27). In the first step, J is given by

where fi& is the principal radius of curvature in the s-y plane and kris the principal radius of curvature associated with the azimuthai direction.

To evaluate J, we use the following equation if the interface is mostiy horizontal Figure 2.9: Cylinderical curvature, Jq1.

When the interface is mostly vertical, the roles of x and y rnust be reversed rvith X(y) replacing Y (x).

To evaluate Jqi, first we calculate the angle /lwhich is the angle between the interface tangent and the positive x axis rvhen the interface is horizontal. If the interface is vertical,

3 is the angle it makes with the positive y axis. For the near horizontal case, as shown in

Fig. 2.9(a), then we may write

For the near vertical case, Fig. 2.9(b), we have Liquid-Solid Contact Line. The liquid-solid contact line needs special attention. We in- corporate the concept of contact angle, 9, in the free surface boundary condition, Eq. (2.41).

6 is the mgle between the solid wall and the Buid interface that includes the fluid; B is assumed to be known. To impose this angle condition we adjust the Y (x) and X(y) values at the wall. For example, consider the boundary cell (i,j) as shown in Figs. 2.10(a) and

2.10(b) with Buid assumed to be below the boundary. The surface tension pressure, P., will be calculated in the same masner as outlined before based on Eq. (2.41), except that the surface is assumed to make an angle, 8, with the wall by adjusting the i + 1 value of Y(x) and the j + 1 value of X(y) as

and

Similar adjustments are made to the appropriate variables for dl other orientations of wds and interfaces. The above equations are used when calculating the mean curvature, J, of the liquid meniscus near the solid wall as was explained previously. It shouid be mentioned that the contact angle might have different values for different solid wds of the computational domain. In addition, the contact angle may be a dynamic contact angle that varies with time in which case it has to be updated after each time step.

Interface Velocities Specification. Velocities immediately outside a free surface must be specified at each time step because their values axe needed in the finite-difference approximation for points outside the surface. Velocities are set on every cell boundary between a surface ce11 and an empty cell. If the surface ceil has only one neighboring empty ceU, the boundary velocity is set to insure the vanishing of Sij, the velocity divergence defined in Eq. (2.9). When there are two or more empty cell neighbors, the individual contributions to the divergence, ;a(xu)/ax and &@y, are separately set to zero. In addition, we have to set the velocities on boudaries between empty cells adjacent to a surface cell. This is wall

fictitious cells 1 fictitious ceh J (a)

Figure 2.10: Liquid-solid contact line condit ion. accomplished by setting zero values for &/ay and âvlâz. i.e., no tangential stresses at the free-surface boundaries.

Initial Conditions.

At the beginning of the first step of the computation, the initial values of volume-of-fluid function, F?,=Oo and the initial velocity distributions within the fluid must be given. When setting the values of FG=O for free-surface cells, first we need to find where a ce11 is cut by the fluid free-surface. Assuming a straight line for the cut through the cell, then the area of the cell occupied by the fluid can be calculated as shown in Fig. 2.11. Now for the cells for which FT-O > O the x-direction velocity, u:~*,is set to zero, aad Figure 2.11: Calculation of the initial F dues for free-surface cells. where is the droplet impact velocity.

2.3.7 Stability Considerations

Numerical calculat ions often have computed quantities t hat develop large, high-frequency oscillations in space, time, or both. This behavior is usually referred to as a numerical instability. Certain restrictions rnust, therefore, be observed in defining the mesh increments,

62 and 6y, the time interval 6t, and the upstream differencing parameter a.

For accuracy. the mesh increments must be chosen small enough to resolve the expected spatial variations in al1 dependent variables. Once a mesh has been chosen. the choice of the time interval necessary for stability is governed by several restrictions. First . material cannot move through more than one cell in one tirne step because the differencing equations assume

Auxes only between adjacent cells. Therefore, the time interml must satisfy the following inequali ty,

where the minimum is with respect to every ce11 in the cornputational mesh.

Second, when a nonzero value of kinematic viscosity is used, momentum must not diffuse more than one ceil in one time step. A linear stability analysis shows that this limitation implies

Third, when surface tension is included there must also be a limit on 6t to prevent capillary waves from traveling more than one ce11 width in one time step. A rough estimate for this stability condition may be written as

The last condition to insure numerical stability is related to the upstrearn differencing parameter a which was used in the fbite-difference expressions (i.e., FUX,FUY, FVX, and FVY) of the advective flux terms appearing in the momentum equations. The proper choice for a is

where a=l reduces the finite-difference expressions of the advective tems to a first order upwinding scheme.

Results and Discussion

The results presented here are those given by Pasandideh-Fard et al. [59]where experimental results have been accomplished by Qiao and Chandra. Since we consider water droplet impact at low We numbers where capillary effects are important and both spreading and recoil exist during the impact, mesh cells are chosen to be fine and uniform dl over the computational domain. This is because the calculation of curnture J is more accurate in a uniform mesh. The droplet was discretized using a grid spacing equai to 1/30 of the droplet radius. Figure 2.12 shows a typical computational mesh used in numericd simulation. Time step was less than one microsecond. Numerical computations were performed on a SGI

(INDIGO2)workstation. The CPU time for a typicd run (until 10 rns elapsed after the impact) was around 8 hours. Figure 2.12: A typical computational mesh.

2.4.1 Validation of Computational Mode1

We compare the results of computations with the experiments for the impact of a water droplet on a flat, solid surface [20]. The details of the experirnental method are given in Refs. [20, 591. For pure water at 25OC we used these values [60]: p = 997 Kg/m3, v = 8.57 x 10-' m2/s, and y = 0.073 N/m. Dynamic contact angles measured from experiments were used in the model. Figure 2.13 shows images of droplet deformation obtained from the numerical model, along with photographs of 2.0 mm diameter droplets of pure water impacting a polished stainless steel surface with a velocity of 1 mis. Both cornputer generated images and photographs are viewed from the same angle (30" from the horizontal), and at the same time (t) after impact. Droplets can be seen reflected in the polished surface in the photographs. A single bubble formed in droplets at their point of impact because of entrapment of air in a cusp at the liquid-solid interface. No bubbles were seen in theoreti- cally predicted droplet shapes, since the model did not consider pressure changes in the air surounding droplets. Droplets did not break up during impact since their kinetic energy was too low to overcome surface tension. A measure of the relative magnitudes of kinetic and surface energies is the Weber number ( We=pDoV,2/$? whose value was 27 for the conditions of Fig. 2.13. The droplet reached its maximum extent at approximately t=2.6 ms, after which surface tension and viscous forces overcame inertia, so that Buid accumulated at the leading edge of the splat and it started pulling back. Surface tension finally caused recoil of droplets off the surface (t=6.2 ms). A good qualitative agreement can be seen between the cornputer simulated images and the photographs in the spreading and also in the recoil. This validated the model and its underlying assumptions. Although both experiments and numerical model predict droplet recoil (t=6.2 ms), there is a discrepancy between the two results at this stage. This is attributed to the fact that droplet recoil is dorninated by surface tension and contact angle effects. Modeling fluid Bow when capillary forces are the only forces driving the fluid presents considerable challenges. Also it is not clear if the measwd values of 0 differ from actual contact angles near the surface.

2 A.2 Capillary Effects during Droplet-Impact

Numerical results for different cases of droplet impact are compared with the experimental results. Through t hese comparisons, capillary effects, i .e. the effects of surface tension and contact angle, during droplet impact are invest igated. Liquid-solid contact angles were varied in experiments by adding traces of a surfactant to water2. Impacting droplets were photogaphed and liquid-solid contact diameters and contact angles were measured from photographs. 21t should be mentioned that the purpose of considering a surfactant solution in this work is not to study surfactants but to study the effects of contact angle on droplet impact dynamics. When studying liquid- solid contact angles pure liquids should be considered, however, there is not rnuch information available on dynamic contact angles for different pure liquid droplets irnpacting a substrate. Therefore, available experirnental results for dynamic contact angles of water droplets with ciifferent surfactant concentrations [20] were considered in this work; the rneasured dynarnic contact angle values were used in the contact Line boundary condition in the numericd model. Figure 2.13: Computer generated images compared with photographs of a 2 mm diarneter water droplet-impacting a stainless steel surface with a velocity of 1 m/s. The time of each frame (t) is measured from impact. H2 mm

Figure 2.13 Cont inued, The effect of adding a surfactant on droplet-impact dynamics can be seen in Fig. 2.14.

The surfactant used in experiments was sodium dodecyl sulphate (SDS ) . Computer simulated

images of impacting droplets are compared with photographs taken at the same instant after impact for droplets of pure water (O ppm), and dso for droplets to which 100 and 1000 ppm

of surfactant was added3. The surfactant appeared to have little influence on early stages of

droplet spread: droplet shapes appear similar in a.U three cases for t 5 1.3 ms. The maximum

extent of spread increased as more surfactant was added (see Fig. 2.14, t=2.6 ms). Droplet

shape during recoil was sensitive to surfactant concentration. Adding as little as 100 ppm of

surfactant to water produced significant changes in droplet shape (see Fig. 2.14, t=6.2 ms).

4 simple order of magnitude analysis introduced by Chandra and Avedisian (131 shows that

during the initial period of droplet spreading, inertial forces are much larger than surface

tension and viscous forces; lowering surface tension or contact angle, therefore, has little

influence on fluid Bow. Droplet recoil, though, is controlled by capillary forces, and adding

a surfactant decreases the height of droplet recoil (see Fig. 2.14, t=10.2 ms).

Modeling the effect of a surfactant on the surface tension of a freshly created surface is a complex problem. The surfactant reduces surface tension when it diffuses to the free

liquid surface: dynamic surface tension values therefore depend on the age and history of a

surface [64]. In the experiments, the surfactant was uniforrnly distributed in droplets when they formed. Therefore, the measurement of surface tension, made at the instant when

the droplet detached and started free fa,LLing towards the substrate, represented a lower

3Physical properties of a surfactant solution (such as its surface tension) depend on the surfactant concentration. In this respect the critical micelle concentration (CMC) is very important. Dramatic changes in

the bebavior of a surfactant solution can be seen in the neighbothood of the CMC [61, 621. For an aqueous solution of SDS, the CMC is 8.6 x 10-~moles/liter (0.24% or 2400 ppm) [6l,621 which is well above the SDS concentrations considered in this study (100 and 1000 pprn). The variation of the surface tension with surfactant concentration for this solution is given by Qian and Chandra [63]; the surfactant concentrations (upto

1000 pprn) are considereù low enough that changes in other thermophysicai properties are negligible [63]. 100 ppm 1000 ppm

Figure 2.14: Cornparison of photographs with mode1 predictions for impact of droplets of pure water (O ppm), 100, and 1000 ppm surfactant solution. 100 pprn 1000 ppm

Figure 2.14: Continued. bound on possible surface tension values. However, as the droplet deformed during impact, depletion of surfactant due to expansion of free surface area may have increased surface tension. Experiments [66] on rapid growth of bubbles in aqueous surfactant solutions have shown that dynamic surface tension can equd that of pure Liquid. Surface tension dues during impact of surfactant solution droplets could, therefore, lie between that of pure water

(73 mN/rn) and those measured experimentally (70 mN/m for 100 pprn and 50 mN/m for

1000 ppm surfactant), and also vary from point to point on the droplet surface.

No attempt was made to mode1 transport of surfactant during droplet-impact. Any assumptions made in such a mode1 would have been unverifiable because there was no means of experimentally measuring surface tension distributions during droplet deformat ion. Instead we caiculated droplet shapes duhg impact using the highest and lowest values of surface tension (that of pure water, and the experimentally measured value) to determine if changes in surface tension significantly altered the results. The two sets of results were cornpared with experimental measurements to see which gave better agreement. A quantitative cornparison of experimental and numerical results was done by measuring the diameter of the wetted surface area (D)at successive stages during droplet deformation. Normalizing this quantity by the initial droplet diarneter (Do)yields the 'spread factor', c(t) = D(t)/Do. Experimentally measured values of are shown by symbols in Fig. 2.15 for droplets containing 1000 ppm of surfactant; solid Iines mark mode1 predictions obtained using two different values of -y. Results are shown for both small times after impact (t* < 3.5, where t* = t b/Do)to show details of the impact, and large times (5 < t* < 30) to show the static state. Results obtained from the two calculations showed Little difference. Assuming any surface tension value in the range 50 mN/m 5 7 5 73 mN/m would have produced reasonable predictioos for the evolution of E. Similar calculations for 100 ppm surfactant solution droplets, using surface tension values of 70 mN/m and 73 mN/m, revealed only very minor differences between values of

. However, Fig. 2.15 shows that using 7=73 mN/m gave better predictions for { during Figure 2.15: Variation of rneasured spread factor (c), shown by symbols, during impact of a droplet with 1000 ppm of surfactant, compared with mode1 predictions (solid lines) using surface tension (7) values of 50 and 73 mN/m. the period 3 < t' < 10. Qualitative inspection of predicted droplet shapes showed that they were sensitive to surface tension values during this tirne, when the droplet was recoiling.

Figure 2.16 shows images of impacting LOO0 pprn droplets at t=6.2 ms and 10.2 ms, calculated using the two diKerent values of y. Cornparison with photographs (Fig. 2.14) confims that using 7=73 mN/m gave predictions that were in close agreement with experimental observations, suggest ing t hat dynamic surface tension dues of surfactant solut ions were, indeed, close to those of pure water. Al1 calculations were performed, therefore, âssuming surface tension equal to 73 mN/m.

Since surface tension was assumed constant in the model, adding a surfactant dected impact dynamics only because it reduced the liquid-solid contact angle. Mode1 results shown in Figs. 2.13 and 2.14 were obtained using experimentally measured dynamic contact angle values. Fig. 2.17 shows measured values of contact angles (8) during droplet-impact and rebound. Symbols in Fig. 2.17 mark experimental measurements; linear interpolation was Figure 2.16: Shapes of impacting 1000 ppm surfactant solution droplets cdculated using surface tension (y) values of 50 and 73 mN/m. used in calculôtions to estimate intermediate values. In al1 three cases (0, 100 and 1000 ppm) the advancing contact angle (O,), measured during droplet spreading (t* < 1.5), remained approximately constant ( -- 1IO0), regasdless of surfactant concent ration. Once droplets reached their maximum extension surface tension forces caused recoil ( 1.5 < t* < 3). Splat diameters remained constant while contact angles decreased until t hey reached t heir minimum value, calied the receding contact angle, at t' 3. The periphery of the splat was then drawn inwards, reaching its final position at t* 4. Droplets assumed their static forms, shaped like spherical caps, at t* > 10. Measured static contact angle values were: pure water, 90'; 100 ppm surfactant solution, 57O; and 1000 ppm surfactant solution, 18O. A number of issues relevant to static and dynamic contact lines on solid surfaces, both from macroscopic and microscopic standpoints, have been reviewed in [67-741. To this end, a clear distinction exists between the static contact angle and the dynamic contact angle. Moreover, in the case of the dynamic contact angle, the spreading contact angle and the recoiling contact ongle are different. Three important facts [67-741 that should be considered in the modeling of wetting axe the following: (1)The value of the dynamic contact Tirne, t (ms)

O 4 8 12 16 20 C"" 1 4.- 1 1

Dimensionless the, t*

Figure 2.17: Measured evolution of the contact angle during spreading of droplets of pure water (O pprn), 100, and 1000 ppm surfactant solution. This figure is taken from Ref. [;LOI. angle depends on the veloci ty of the contact line. (2)The advancing contact angle is larger than the receding contact angle. (3)More than one value of the contact angle is possible for a stationary contact line.

Dynamic contact angles are known to increase with the velocity of a moving liquid-solid- air contact line [75, 761. Elliot and Riddiford [75]measured contact angles during liquid flow between two parallel plates, and found that advancing contact angles increased linearly with contact line velocity until finally an upper lirniting value of Ba was reached: contact angles were then independent of further increases in velocity. The same result can be inferred from the work done by Dussan [67]. Elliot and Riddiford [75] also determined that the 0 Oppm A 1OOppm O lOOOppm

- 1 O 1 2 3 4 5 Contact Iine velocity (mis)

Figure 2.18: Variation of the contact angle with contact line velocity. This figure is taken from Ref. [20]. addition of a surfactant to the liquid did not change this maximum value of O,. Contact line veloci ties in the experiments were estimated by differentiating polynomial curves fit ted through measurements of droplet contact diarneter evolution. Fig. 2.18 shows the variation of dynamic contact angles wi t h contact line velocity; positive velocities indicate droplet spreading and negative velocities recoil. The measurements confirm that values of advancing contact angles reach a maximum of approxirnately 110°, independent of both contact Lne velocity and surfactant concentration.

We perfonned calculation using a fixed, as done in previous studies [12,56,77, 781, rather than a dynamic contact angle to see if this increased errors in mode1 predictions. Measured values of for droplets of pure water are indicated in Fig. 2.19(a) by symbols. Numerical predictions are shown by solid lines for simulations done using both measured values of dynamic contact angle, and fixed contact angle set equal to the static value. For pure water the static contact angle (90") was close to the advancing contact angle (- 100" ; see Fig. 2.17).

Consequently, t here was lit t le difference between the results of the two simulations, and both accurately predicted experimentai meaçurements during droplet spreading [Fig. 2.19(a), t' < 21. However, during droplet recoil (2.5 < t* < 10) there was considerable discrepancy between numericd predictions and measured values of c. As the droplet recedes it leaves a very thin liquid füm behind on the surface (Fig. 2.14, t=10.2 rns). Modeling fluid flow realistically in this thin layer presents considerable challenges. It is not clear whether it is appropriate to use a neslip boundary condition in this situation, or whether the measured values of 8, differ from actual contact angles near the surface. When the droplet reached a static shape, predicted values once again agreed well with measurernents [Fig. 2.19(a), t- > 151. A cornparison between measured and predicted duesof c for droplets containing 100 ppm surfactant is shown in Fig. 2.19(b). In this case the static contact angle (57") was much lower than the advancing contact angle (-. 110°, see Fig. 2.17). Results from the mode1 assuming static contact angle, therefore, overestimated [ during droplet spreading. Using dynamic contact angle values gave much more accurate resdts, but both models predicted larger values of contact diameter than seen in experiments. Similar measurements and calculations for droplets with 1000 ppm surfactant are seen in Fig. 2.19(c). The dynamic contact ongle model predicted droplet diameter evolution reasonably accurately during the entire impact process. Assuming static contact angle, however, made the model overpredict by up to 30%.

Using values of static (rather than dynamic) contact angle was found to produce significant erron when modeling droplet-impact in the experiments [59],where impact velocity was low (1 rn/s). However, as impact velocity increases droplet kinetic energy becomes much larger than surface energy (i.e., Weber number becomes large), and surface tension and con- +quilibrium l~~m"'~~~'-"""~~'~-~~lequilibrium contavct angle

dynamic contact angle dynamic contact angle

10 20 30 40 50 60

ih='l.."l'.'.l"..lr...~'.- qdibrium equilibrium contan angle

dynamic contact angle dynamic contact angle

Figure 2.19: Evolution of calculated (lines) and measured (symbols) spread factors during impact of (a) pure water droplet, (b) 100 ppm surfactant solution, and (c) 1000 ppm surfactant solution. tact angle effects eventually become negligible. A criteria to establish conditions under which capillary effects are negligible can be obtained from a simple energy conservation model of droplet spread. Several such models me available in the literature and have been reviewed in detail by Bennett and Pouiikakos [15]. By using detailed numerical results of this study, an analytical model was derived by Qiao and Chandra [20,59] which is an extension of that developed by Chandra and Avedisian [13].

Analyticd models differ in the way they simplify the work done in deforming droplet against viscosity. Chandra and Avedisian [13] approximated this work as

where R is the volume of viscous fluid, t, the time taken for the droplet to spread. and 4 the viscous dissipation function. The magnitude of # is estimated by [13]:

where p is the liquid viscosity and L is a characteristic length in the y-direction. Chandra and Avedisian assumed L equals the splat thickness h. Their results overestimated Dm, values by up to 40%, suggesting t hat L is in fact smaller than h. Detailed numerical results of this study showed that a more appropriate length scale to estimate the magnitude of viscous dissipation is the boundary layer thickness (b)at the liquid-solid interface. Figures 2.20(a)- 2.20(c) show numerical droplet shapes and velocity profiles at three different locations in an impacting water drop, at three different instants during droplet spread. The numerical value of 6 is approximately 0.1 mm, and does not change significantly with position or time while the droplet is spreading. Qiao and Chandra [59], therefore, looked for an anaiyticd expression for the boundary layer thickness. Assuming that liquid motion in the droplet cm be represented by axisymmetric stagnation point Bow, they obtained an expression for boundary layer thickness which resulted in 6=0.09 mm for pure water droplet under consideration. This result is in good agreement with predictions from the numericd model (=O. 1 mm) [see Figs. 2.20(a)-2.20(c)]. (a) radius (mm)

(b) radius (mm)

Figure 2.20: Predicted droplet shape and velocity distribution at (a) t =O.9 ms, (b) t =1.5ms, and (c) t=2.4 ms. Qiao and Chancira's analytical mode1 (591 finally results in the following expression for the maximum spread factor

They tested the accuracy of predictions from Eq. (2.57) by comparison wit h experimental measurements for a variety of droplet-surface combinations, over a wide range of Weber nurnber (26< We<641) and Reynolds number (213

The magnitude of the term (1 - COSO.) in Eq. (2.57) can be at most 2. If ~e/ais large in comparison, the value of contact angle will have Little effect on Q,. Therefore, when modeling droplet impact, capillary effects may be neglected if

If also We » 12, Eq. (2.57) reduces to

Previous analyses [Il, 15, 471 of droplet impact with We + oo have shown tm, to be proportional to Rea, where a is a constant with values ranging from 0.167 to 0.2.

Summary

A finite-difference numerical solution of the Navier-Stokes and the Volume-of-Fluid (VOF) equations based on SOLA-VOF algonthm was employed to study isothemd droplet impact on a solid surface. Impact of water droplets on ô flat, solid surface was studied using the numerical simulation and the results were compared with available experimental observations. Through this cornparison, the idluence of surface tension and contact angle on the impact dynamics of a water droplet falling onto a flat stainless steel surface was investigated. The principle findings were the following:

0 Cornparison of computer generated images with photographs showed t hat the numerical analysis accurately predicts droplet shape during deformation.

0 Adding surfactant, SDS, up to 1000 ppm (which is well below the CMC for a.n aqueous solution of SDS) did not affect droplet spreading significmtly; however, it changed droplet shape during recoil. This phenornenon, observed both in numerical simulations and experiments, was attributed to inertia dominating droplet spread and capillary forces dominating droplet recoil.

0 Adding surfactant did not appear to reduce dynamic surface tension. Using a constant

value of surface tension in the model, equal to that of pure water, gave results that best agreed with experimental observations.

When dynamic contact angle values were used in the numerical model, accurate pre-

dictions were obtained for droplet diameter during spreading and at the static state. The model overpredicted droplet diameters during recoil.

When the contact angle was assumed fixed, equal to the measured static value, model predictions were less accurate. The discrepancy between results obtained using static

and dynamic contact angles was least for pure water drops, where the static and advancing contact angles had values close to each other.

Through an analytical model (that of Qian, and Chandra given in Ref. [59]), capillary effects were found to be negligible during droplet impact if We > 6. Chapter 3

Modification to Flow Dynamic

Equations in the Presence of Solidification

Introduction

Developing a mode1 to simulate the fluid flow and heat transfer in presence of solidification requires solving the momentum and the energy equations while considering phase change.

The velocity computation has to account for a moving and somewhat arbitrary shaped liquid- solid interface (solidification front) on which the relevant boundary conditions have to be applied. Different techniques have been used by a number of researchers to account for the liquid-solid interface in different engineering applications. These techniques and their applicabilities have been reviewed in detail by Salcudean and Abdullah [81]. A well known method to treat the liquid-solid interface in phase change problems is the mesh transformation method. In this method, the moving solidification front is immobilized by transforrning the coordinate systems, but the transformed coordinate systems are, in general, not orthogonal. The purpose of the transformation is to enable the mesh to conform to the liquid-solid interface. A 'guess and correct' procedure is used in which first an interface location is assumed. Then the equations are mapped to the transformed domain and solved by modeling the interface as a heat source proportional to the latent heat of fusion of the materid. The solution is then mapped back to the original domain and the position of the interface is corrected. This procedure is repeated until convergence. The mesh transformation method with different techniques for different applications have been used successfdy by: Sparrow et al. [82], Kroeger and Ostrach [83], Rarnachandran et al. [84],Verwijs et al.

[85],and Lan et al. [86]. Even though the mesh transformation techniques are accurate, their use is rnainly limited to 'quasi one dimensional' solidification in which the interface is const rained to traverse primarily dong one coordinate direction, and is constrained from becoming too convoluted.

These techniques, therefore, can not easily be extended to multidimensional problems. An- other drawback of the mesh transformations techniques is that they are not simple to implement in existing computational codes, and have Limitations in their scope of application [al].

A simpler approach for modeling the solidification front is to use a regdar, non-transformed mesh and treat the solidified regions of the domain using either of the fized velocity, increased vzscosity, or the D 'Arcy source techniques [£Il].In t his approach, if the solidification front has a convoluted shape, it would simply be approximated in a 'stepwise' manner. Advàntages of this formulation are that it can be used to solve multidimensional problems and it can easily be implemented in existing codes. Sindir [87]has shown that a stepwise approximation to a curved boundary can provide results with reasonabh overd accuracy, but in this case the conditions near the boundary are not correctly represented. In this approach, in dl different techniques of treating the solidified regions, a liquid-solid volume fraction 8, as defined in

Eq. (3.1) of the next section, is introduced. When this fkaction in a ce11 approaches zero, the solidified material in the ceIl is assumed to behave as a wall, so the wall boundary conditions have to be applied. In the increased viscosity technique [88, 891, the effective viscosity is coupled to solidification so that it is increased to a very large value as the liquid-solid volume fraction 8 approaches zero. A model is required to couple the liquid viscosity to 0, for example as 8 decreases, the viscosity increases exponentially. As a result, the velocities in the solidi@ing region get suppressed. Boucheron and Smith [go] used this technique to model solidification during impingement of a fluid jet on a plate. This technique sufFers from a lack of information about the viscosity wiation near the freezing point [81].

Another technique, which seems to be especially suited to the modeling of mushy solidification of alloys, is the D'Amy source technique [88,891. During the solidification of eutectic alloys the formation of dendrites takes place which, it is assumed, causes the mushy zone to behave as a porous medium. According to D7Arcy7slaw, the velocity of liquid flowing through porous media is proportional to the pressure gradient. As the volume fraction O in a ce11 decreases, the porosity and, consequently, the pressure gradient goes to zero. In this technique, a mode1 is required to relate the porosity with O. Abdullah and Salcudean [89] have used this technique in a cast iron cylindericd casting by using a linear relationship between the porosity and the volume fraction 8.

In the fized velocit y technique [88, 891, zero normal and tangentid velocit ies are imposed on the walls of the cells when 8 approaches zero. The assumption is that the solidified material in a cell behaves as a wall. This technique has been used by Voller et d. [88] and Abdullah and Salcudean [NI. In their model,however, the liquid-solid volume fraction O in a computational cell can be either one, i.e. the cell is full of fluid, or zero, i-e. the ce11 is completely solidified, but not a value in between. It can be seen that this technique is simple to implement but it lacks accuracy of the results for the solidification front position.

In this work, we treat the liquid-solid interface using a modified version of the fied uelocity technique. To ensure an accurate numerical calculation, the correct position of the solidification front in a ceU has to be accounted for. Therefore, we use a two phase continuum model to rnodifjr the governing flow equations. The mathematical formulation and the implementation procedure of t his modification in the finite difference numerical scheme is given in this chapter.

3.2 Mat hemat ical Formulations

The formulation described in this section is for, but not restricted to, the droplet impact problem. The governing flow equations are the continuity, momenturn, and F transport equations given by Eqs. (2.1)-(2.4). We use a twephase flow continuum model to modify these equations in the presence of solidification. The concept of this model is very similar to the concept of the " fractional volume of fluidn scheme described in Chapter 2. The modified equations obtained in this section have been used by Kothe et al. [29] to model flow dynamics in presence of an obstacle in the computational domain in a finite difference formulations.

However, neither a proof nor a reference has been presented for the modified equations.

3.2.1 Liquid-Solid Volume Fraction

To represent the liquid part of the droplet, we define a fmction 8(s1y, t ) whose value is one within liquid and zero within solid. When averaged over the ceils of a computational mesh, the average value of O in a ce11 is equd to the liquid-solid volume fraction of the cell occupied by the material, i.e.,

O= "fluid volumen of C.V. "0uid volume + solid volumen of C.V.

We impose a value of one for O if a cell is empty of both liquid and solid. With the above definition, O is zero if a ce11 contains only solid (not necessarily Ml). If a ceil contains ody liquid (not necessarily fdl) or if a cell is empty of material then 8 will have a value of one. Cells with O values between zero and one contain both liquid and solid. Figure 3.1: An arbitrary control volume of the material containing both liquid and solid.

3.2.2 Modified Continuity Equation

It is probably more convincing to apply the conservation laws to a control volume which always consists of the same material paxticles rather than one through which different material particles pas. For this reason, the Lagrangian coordinate system will be used to derive the basic conservation equations. Since the Eulerian system is the one used in the numerical model, we will convert the derived equations to the Eulerian form usiog Reynolds' Trans- port Theorem ( R.T.T.) [gl]. Consider a specific mass of material containing bot h liquid and solid whose volume V is arbitrarily chosen as shown in Fig. 3.1. Based on the definition of liquid-solid fraction, the liquid and solid volumes are OV and ( 1 - O)V, respectively. If this given mass is followed as it flows, its size and shape will be observed to change but its mass will remain unchanged. This is the statement of mass conservation which may be written mat hematically as

where mi and rn, are liquid and solid mas, respectively, and we have Substituting for ml and rn, into Eq. (3.2) and assuming that pl = p,=constant

This equation may be converted to a volume integral in which the integrand contains ody Eulerian deri~tivesby use of R.T.T.,

where uk and 6, are velocities within the Liquid and solid media, respect ively. Since solid is assumed to be stuck to the solid wall (i.e. the reference frame) particles within the solid have zero velocities, therefore: G, = O for all j. The final form of the modified continuity equation then cm be obtained as

3.2.3 Modified Momentum Equations

The principle of conservation of momentum is an application of Newton's second law of motion to an element of the materid. That is, when considering a given mass of fluid (as in Fig. 3.1) in a Lagrangian frame of reference, it is stated that the rate at which the mass momentum is changing is equal to the net external force acting on the mas. The extemal forces which may act on a mass of fluid are body forces, such as gravitationd forces, and surface forces, such as pressure and viscous forces. Then if f is a vector which represents the resultant body force per unit mass and u is a stress tensor which represents the resultant surface force, the equat ion expressing conservation of momentum becornes (using Gauss' t heorem)

Since the solid part of the volume V is stationary in the reference frame, solid particles have no velocities. Therefore, assuming a uniform density for the liquid and solid we have Using R.T.T. and the fact that the control volume was chosen arbitrady we have

Expanding the second term in the left hand side and using the modified continuity equation,

Eq. (3.6), the final form of the modified momentum equation is obtained as follows

3.2.4 Modified Volume-of-Fluid Equat ion

In the presence of a solid we redefine the function F as follows. Since we assume a constant density for both Iiquid and solid, the solid is modeled as the fluid with zero velocity.

Therefore, the function F can be expressed as " fluid volume + solid volumen of C.V. F = "total volume" of C.V. With this definition, F equals one for cells occupied by the material, Liquid or solid, and zero for empty cells. Cells with values of O < F < 1 axe partidly Wed by the material; these cells contain a free surface, liquid or solid. From the definitions of 8 and F, Eqs. (3.1) and (3.11), I/i and V,, the fluid and solid volumes in the control volume, respectively, may be obtained as follows

Since we assume an incompressible fluid, the total volume of the material is constant. This statement in a Lagrangian system implies that

Substituting for and V, from Eq. (3.12) Using R.T.T. we can write the two sides of this equation in the Eulerian system. Since the control volume is chosen arbitrarily and there is no velocity within the solid we hally get

Expanding the ri& hand side and the second term in the left hand side. and using the modified continuity equation, Eq. (3.6), the find form of the modified volume of tluid equation may be written as

3.3 Computational Treatment

Volume fraction O is defined at the ceU center, but in the nurnerical treatment of the governing equations velocities are defined at the cell boundaries as was shown in Fig. 2.3. For this reason, the numerical mode1 for ai1 computational cells has to be facilitated by defining an area fraction

3.3.1 Finite DWerence Modified Continuity Equation

The finite-difference approximation of the modified continuity equation, Eq. (3.6), can be writteo in a manner very similar to Eq. (2.9) Figure 3.2: Definition of area fractions cP and @.

3.3.2 Finite Difference Modified Momentum Equations

Expanding the first term of Eq. (3.10) and dividing by 8 we will have

The fint term in the right hand side is the only extra term compared to the original momentum equation. The final form of the finite-difference equation in the x-direction may be written as

where al1 terms in the bracket are exactly the same as in the original finite-difference equation of the x-momentum, Eq. (2.10). For the y-momentum equation similarly we have

where al1 terms in the bracket are the same as in Eq. (2.22). 3.3.3 Finite Difference Modified Volume-of-Fluid Equat ion

Using the Donor- Acceptor flux approximation as was explained in Sec. 2.3.4 "the amount of volume advected (per unit length normal to the paper)" through the right face of a cornputational ce11 during a time step 6t is

where CF was defined in Eq. (2.34). When computing the amount of volume advected through y-direction, iI' wil1 be used to adjust for the cell faces open to the fluid flow. Chapter 4

Droplet Impact and Solidification: One-Dimensional Mode1

4.1 Introduction

In Chapter 3 the implementation of the modification to the Bow dynamics equations in the presence of a solid phase was discussed. This modification was based on treating the solid

phase by use of the fied velocity technique. In doing so we defined a liquid/solid volume

fraction O(x,y, t) which must be known at every ce11 of the computational domain and it must be updated at each time step. A cornplete numerical solution of the impact and solidification of a droplet on a cold substrate can be obtained by solving the momentum and the energy equations sirnultane- ously. Based on the solution of the energy equation at each time step the position of the

liquid phase, solid phase, and liquid-solid interphase (solidification front) can be estimated.

O equals one if a ce11 has only liquid phase, and zero if the ce11 has only solid phase. Cells with O values between zero and one contain a solidification front. Several techniques are available for treating the latent heat of Fusion and evaluating 8 in the interphase. These

techniques will be reviewed in Chapter 5 and based on a selected technique the complete solution of the Buid flow, heat transfer, and solidification wili be obtained.

A simpler approach to solve for fluid flow and solidification which is best suited to the droplet-impact and solidification in thermal spray processes is a one-dimensional solidification model. This model is based on two main assumptions. First, it is assumed that the solidification proceeds as a moving front such that the phases are separated by the isotherm corresponding to the melting point temperature. Second, the existence of a principle direction of heat flow within most of the physicd domain leads to a one-dimensional approximation of the actud heat transfer. Under t hese hypotheses, the splat solidification problem reduces to the one-dimensional motion of the liquid-solid interface. The interface can only move orthogonaily to and away from the substrate. It is further assumed that the interface motion can be approximated by Neumann's solution of the Stefan problem (semi- infinite domain) [Il, 921. In this solution, the thickness of the solidified layer at time t at every point on the substrate is cdculated. The liquid-solid volume fraction 8 for the whole computational domain is then evaiuated and the position of the solidification front is deter- rnined. In this chapter we explain this one-dimensional solidification model in conjunction with the existing fluid flow mode1 discussed in Chapter 2 to obtain the numerical solution for the impact and solidification of a droplet on a cold substrate for the cases typical of thermal spray processes.

4.2 Mat hemat ical Formulations

It is reasonable to assume that during spreading, the temperature gradient is much larger in the axial direction than in the radial direction, aad thus, the heat transfer problem can be approximated by a ondimensional model. The mathematical mode1 developed here is based on the assumptions of : laminar and incompressible fluid flow, axisyrnmetric system of coordinates, vertical impingement on the substrate, one-dimensional energy equation for solidification, negligible convection and radiation heat transfer, and negligible viscous dissipation. We also assume that the surface of the substrate is smooth, homogeneous, isotropie and insoluble. It should be noted that in thermal spraying the substrate is usudy rough and oxidized. This represents an additional resistance to heat transfer between droplet and substrate. Based on the above assumptions, the governing flow equations are the modified Navier-Stokes equations given by Eqs. (3.6), (3.10), and the modified volume of fluid equation, Eq. (3.16). In order to calculate the volume fraction 8,the position of the liquid-solid interface (i.e., the solidification front) must be known at each tirne. The liquid-solid interface position can be determined explicitly by solving simplified energy equations similor to those of the Stefan problem of solidification.

4.2.1 Simplified Energy Equations

A schematic diagram of the 1D solidification model is shown in Fig. 4.1. Denoting the liquid phase with an index of 1, the solid phase with s, the substrate with w, and assuming that the solidification problem can be approximated by a one dimensional model similar to the

Stefan problem (921, the equations of heat conduction cm be written as

where T and a are the temperature and thermal difhsivity, respectively, and 6(t) represents the t hickness of the solidified layer at time t .

4.2.2 Boundary and Initial Condit ions

The boundary conditions far from the substrate and the initial conditions are (Fig. 4.1) Substrate

Figure 4.1: Schematic of the ID solidification model.

where Tiand TWuiare the temperature of droplet and substrate before impact. respectively, and Tmprepresents the me1 t ing point temperat ure. At the liquid-solid interface (solidification front) there are two other boundary conditions for the temperature equality and the conduction heat balance given by

where HI is the latent heat of fusion. Similarly, there are two more boundary conditions at the solid-substrate interface as follows When t hennal contact resistance at the substrate is considered, the first boundary condition at y = O, Eq. (4.9) must be rnodified as follows

where R, represents the thermal contact resistance at the substrate per unit area.

4.2.3 Liquid-Solid Interface Position

Based on the formulation given in the previous section there is an analytical solution for the position of the solidification front for the case where there is no contact resistance at the substrate. This solution yields the thickness of the solidified loyer at position x on the substrate at time t as [Il, 921 : qx,t) = 2nJZ7 (4.12) where t' = t - t, and t, is the time at which solidification started at position x on the substrate. 0 is the solidification constant obtained from the solution to the following equation

where b is a constant equal to &z,/al. Sts and Stel are the solid and liquid Stefan numbers, respect ively, given by

When thermal contact resistance is involved, Tsis related to its own gradient with respect to y at y = O as given in Eq. (4.11),therefore, an iteration will be required to obtain R. The iteration procedure is given in the next section. The locai values of b(x, t) can be used to deduce the corresponding liquid/solid volume fraction 8(z, y,t). By calculating the values of Q(x,y,t) over the whole domain of computation, the solidified region and the liquid-solid interface at time t will be known. 4.2.4 Treatment of Thermal Contact Resistance

We first consider the case for which the first boundary condition at the substrate, Eq. (49, is modified to

y = O , T', = 7'' + C(tf) (4.16) where C(tl) is an arbitrary function of tirne. The solution to the probiem with this new boundary condition can be obtained similarly. It is:

where now R is a function of time acquired from the following equation

where Ste, and Stel have the same definition as before.

We now turn our attention to the boundary condition we are looking for, the thermal contact resistance at the droplet substrate interface. From Eqs. (4.11) and (4.16) it can be inferred t hat

knowing already that (Ta),-,- and, consequently, (2) are oniy functions of tirne. To y=o obtain these functions we may consider that the analytical solution to the Stefan problem also gives the distribution of temperature within al1 t hree media of liquid, solid, and substrate. For the modified problem under considerat ion the temperature distribut ion wit hin the solid phase, T,(y, t), is given by

From this solution we cm have Since C(ti) is a function of (2) , given by Eq. (4.19), and (%) itself is a function Y=o y=O of C(tt), R(t') and t', given by Eq. (4.21), therefore, as. iterat ion is needed at each time t to obtain Q(t'). The computational steps are as follows

First assume C(t') = O, i.e. no contact resistance.

Obtain R(ti) from Eq. (4.18).

Evaluate (2)Y=O from Eq. (4.21).

0 Find new C(tf) from Eq. (4.19).

Go back to the second step to evaiuate new fl(ti).

The iteration continues until fl(ti) converges to the required accuracy; a few iterations are usually enough. Finally, 6(x, t ), is evaluated by using Eq. (4.17) and the converged value of

R(tt). The iteration is necessary at each time t and for every position z on the substrate surface.

4.3 Computational Procedure

The modified continuity and Navier-Stokes equations, Eqs. (3.6) and (3.10), are solved on an Eulerian rectitngular, st aggered mesh in axisymmetric geomet ry with the modified SO LA-

VOF numerical method. The finite difference expressions for these equations were obtained in the previous chapter given by Eqs. (3.19), (3.21), (3.22). The basic computational procedure for advancing the solution through one time step is the following:

1. From time Level n values, the finite-difference expressions [i.e. Eqs. (3.2 1) and (3.22)]for advective, pressure, viscous accelerations, and gravitational acceleration terms in the

time discretized rnodified Navier-Stokes equations are used to find provisional values of the new time level velocities. 2. To satisfy the continuity equation, Eq. (3.19), pressures are iteratively adjusted in each ce11 and the velocity changes induced by each pressure change are added to the

velocities computed in the previous step. An iteration is needed because the change in pressure needed in one cell to satisfy the continuity will upset the balance in the four

adjacent cells. Boundory conditions at al1 mesh and free-surface boundaries must be

imposed after each iteration.

3. The VOF function defining Buid regions is updated explicitly based on Eq. (3.16) and the Donor-Acceptor approximation method, Eq. (3.23), to give the new fluid configu-

ration.

4. The starting time of solidification, t,, i.e. the time that solidification starts at a position

x on the substrate, is calculated. This time is assumed to be the time when the fluid

contacts the substrate at x position for the first time.

5. d(x, t ) , the solidified layer t hickness at time t associated with the position z, is calculated based on the procedure explained in previous sections.

6. Having obtained d(x, t) the new values of the liquid/solid volume fraction O for position

x are updated using the definition given in Eq. (3.1).

7. Flow boundary conditions âre imposed at al1 mesh points of free surfaces, solidification

front, and boundaries of the computational domain.

Repetition of these steps allows advancing the solution through an arbitrary interval of t ime.

4.4 Results and Discussion

The results presented in this section are those given by Pasandideh-Fard and Mostaghimi [77]. First, we present the results for two cases representative of RF and DC plasma spraying conditions. Next, the resdts of the present mode1 are compared with available experimental measurements. The effect of contact resistance on droplet impact and the importance of solidification on the spreading will then be investigated.

4.4.1 Results for Typical Spraying Condit ions

Results are presented for three different cases and two different materials, i.e., alumina and tin. Table 4.1 lists the initial droplet diameter (Do),the impinging velocity (V,), and the corresponding Reynolds (Re), Weber ( We) and Stefan (Ste) numbers for each case.

Reynolds, Weber and Stefan numbers are defined as: Re=Dob/v, We=pDoVIr and Ste = c(T- TiK. The initial velocities for cases 1 and 2 are typical of the impinging velocities of particles in soft vacuum (200 torr) radio frequency inductively coupled plasma (RF-ICP) and DC thermal plasma spraying operations, respectively. We assume that the dumina droplets are at 500°C over the melting point temperature and the tin droplet is at 270°C over its melting point temperature at the time of impact. The temperature of the substrate is considered to be 227OC (500 K) for cases 1 and 2 and 20°C (293 K) for case

3. No thermal contact resistance has been used for the first two cases. In addition, for dl cases, we assume a fixed contact angle of 6 = 90°, to represent the wetting behavior. It has been shown elsewhere [93] that under the conditions typical of thermal spray process, the assumption of fixed contact angle is justified. This assurnption can also be supported based on the criterion foz the capillary effects during droplet impact which was obtained in Chapter 2. Under that criterion if We > fi capillary effects can be neglected, i.e. the detailed contact angle information is not important. Based on values of Re and We given in Table 4.1 the criterion is satisfied for all cases, especially for the first two cases which represent typicd plasma spray processes.

Computer-generated images of the spreading of an alumina droplet (Case 1, RF plasma spraying) at a viewing angle of 30" with respect to the horizontal surface are shown in Case Material Dokm] & [mis] Re We Ste 1 aliimina 50 50 245 705 2.16

2 alumina 50 200 975 11305 2.16

------( 3 1 tin 1 2700 1 3.704 1 35339 4417 0.85 /

Table 4.1: Input parameters for droplet spreading and solidification in typicai thermal spray operations

Fig. 4.2. The liquid phase is indicated in gray while the solidified layer is indicated in black.

In the early stages of deformation, the axial velocity of the fluid near the substrate becomes zero while the radial velocity is increased rapidly. A smooth convex shape which is due to the high viscosity of alumina is formed at the splat edge. In the early stages of deformation, the solidified layer is restricted to a very thin layer next to the substrate, t herefore, the Battening process is not considerably afTected by the solidification process. As time increases, the effect of solidification becomes more important. The effect appears as a reduction in the rate of increase in the splat diameter. The spreading of the droplet is finaily arrested after 30 ps when the whole droplet is almost solidified. After nearly 4 fis, the diarneter of the solidified layer rernains constant, i.e. the spreading is completed in a much shorter time. The final diameter of the solidified layer is 0.122 mm and its final thickness is around 0.005 mm (5 pm). The solidification time for an alumina droplet in a typical RF-ICP spraying operation is in the order of 10 p.

Fig. 4.3 shows the spreading of a 50 pm dumina droplet with an initial velocity of 200 m/s impinging on a flat surface (Case 2, DC plasma spraying). In cornparison with Case 1, the spreading develops faster because of the higher impact velocity. However, the basic deformation behavior for the two cases is similar. After nearly 4 ps, the process is completed and a solidified layer of 0.187 mm in diameter and 0.0025 mm (2.5 pm) in thickness is formed. Comparing Fig. 4.2 and Fig. 4.3 shows that the spreading process for Case 2 is approxirnately four tirnes faster than Case 1, a factor which is equd to their impact velocity ratio. We dso note that the lamellae (the coating layer made from one droplet impact) formed in Case 2 has a smooth surface which is a reflection of the high impact velocity.

4.4.2 The Effect of Contact Resistance on Droplet-Impact

To study the effect of contact resistance on droplet-impact, consider Case 3 of Table 4.1, a case for which experirnental results are available in the literature. The transient deformat ion and simultaneous freezing of superheated tin droplets upon their impingement onto a flat substrate was studied experimentally by Fukanuma and Ohmori (471. A single droplet, 2.7 mm in diameter and at 500°C, impacts onto a flat, polished alumina substrate at 20°C with a vertical velocity of 3.704 m/s (Case 3, Table 4.1). The disk diameters in the photographs

(taken by a high-speed camera) were measured as the reference length of the solidified splat diameters. For the purpose of numerical simulation we consider different cases with different substrate conditions.

Fig. 4.4 compares the predicted and measured spread factor against time for the tin droplet under consideration. Simulation results are given for four different conditions at the substrate. When thermal contact resistance (Rc) is lower thon 5 x IOe7 m2K/W (cuve (a) of Fig. 44,solidification behavior is the same as when no contact resistance is considered. As Rc is increased, heat transfer to the substrate is reduced and, therefore, the spread factor is increased. For a contact resistance of 1.2 x m2K/W, the simulation results agree with the experiments. This indicates the approximate value of thermal contact resistance for the case under consideration. Increasing the contact resistance to 1.3 x m2K/W produced a significant change in the shape of the solidified splat. When R, is larger thôn

2 x m2K/W, solidification does not change the spread factor compared to the isothermal case. Fig. 4.5 shows the variation of the maximum spread factor, cm,, vs. the thermd Figure 4.2: Sequence of spreading and simultaneous solidification of an dumina droplet of

Case 1 of Table 4.1 (representing a typical RF plasma spray operation). Figure 4.3: Sequence of spreading and simultaneous solidification of an alumina droplet of

Case 2 of Table 4.1 (representing a typical DC plasma spray operation). O 1 2 3 4 5 6 Time (ms)

Figure 1.1: Corn parison between experimental observations and simulation results for the spreading and simultaneous solidification of a tin droplet of Case 3 of Table 4.1. Compu- tational results are given for different values of contact resistance at the substrate surface.

Experimental results were adapted from Ref. [47]. contact resistance. For the particular case considered here (tin droplet on an alumina substrate), the maximum spread factor, and thus the final splat diameter, is extremely sensitive to the magnitude of the thermal contact resistance (around Rc = 1 x 10-6 m2K/W). This finding is in accordance with the experimental measurements which have shown that the magnitude of the contact resistance for difFerent droplet/substrate materials lies in a range between 5 x W7- 5 x 10-= m2K/W [40-451. The experiments have also confirmed the sensitivity of the final splat diameter to the substrate contact resistance. Figure 4.5: Variation of the maximum spread factor, &,,, vs. the substrate thermal-contact- resistânce per unit area, &, for the tin droplet under consideration on an alumina substrate.

Experimental results were adapted from Ref. [47].

4.4.3 The Effect of Solidification on Droplet-Impact

Most studies have neglected the possible efFects of solidification on the spread factor [15, 49,

501. To study this effect, for the alumina droplet of Case 2 (Table 4.1 ), we considered : 1) no solidification during the spreading and, II) simul taneous solidification and deformation.

Fig. 4.6(a) shows the variation of the spread factor, (, for these considerations. The spread factor is somewhat smaller when simultaneous solidification is considered. The decrease in

( is, however, not substantial (about 10%) and the effect of solidification on the spreading behavior is negligible for this particular case. Similar calculations were carried out for the tin droplet of Case 3 (Table 4.1). Fig. 4.6(b) shows that in this Case, the effect of solidification on arresting the spread of the droplet is quite substantial and neglecting this effect will result in a large error. solidification

2 4 6 8 Dimensionsless time, f *

2 4 6 8 Dimensionsless time, t *

Figure 4.6: Spread factor as a function of dimensionless time for : a) an alumina droplet b) a tin droplet; cases 2 and 3 of Table 4.1, respectively (arrows indicate where curves will move if Re and Ste are increased). The relative importance of solidification on the spreading behavior depends on both

Reynolds and Stefan numbers. Higher Re indicates faster spreading, while higher Ste re-

flects faster solidification. Thus, for conditions of high Re and/or high Ste numbers, the solidification process plays a crucial role in arresting the spreading of the droplet, and its

effect can not be neglected. A cornparison between Case 2 and Case 3 shows that while Ste

numbers are close, Re number for Case 2 is almost two orders of magnitude smaller than

Case 3.

4.5 Summary

A mathematical model for predicting the deformation and simultaneous solidification process of a droplet impinging on a Bat substrate was developed in this chapter. Thermal contact

resistance at the substrate was permitted. The flow model is based on the full Navier-Stokes equations, while the solidification model is based on a one-dimensionai energy equat ion. The main features and findings are as foiIows:

a The results were compared with the available experimental data for spreading and simultaneous solidification of superheated tin droplets on an alumina substrate. Close

agreement was obtained when a thermal contact resistance of 1.2 x 10-~ m2K/W was assumed .

a When the thermal contact resistance was around 1 x IOw6 m2K/W, the final splat

diameter was extremely sensitive to the magnitude of the contact resistance.

Simultaneous solidification causes a reduction in the final splat diameter. This effect is

more pronounced for hi& Re and/or high Ste numbers. Previous studies [15,49, 50) in

which the solidification was considered only after the completion of droplet spreading are only valid for low Re and/or low Ste cases. 0 The spreading process in DC plasma spraying is approximately four times faster than

in RF plasma spraying, a factor which is equal to their impact velocity ratio. a The time required for spreading and solidification in DC plasma spraying was typicdy

one order of magnitude smaJer than in RF plasma spray process. a Compared to RF plasma spraying, the solidified splats in DC plasma spraying had

larger diameters and were of uniform thickness. Chapter 5

Droplet Impact and Solidification: 2D, Axisymmetric Enthalpy Mode1

5.1 Introduction

In Chapter 4 a simple one-dimensional approach to solidification during droplet-impact was used to obtain the solidified layer configuration during the spreading process. This model, however, is not applicable to the cases in which convection heat transfer during the spreading is significant. The simple model also neglects the conduction along the spreading direction. At the initial stages of the impact the conduction to the substrate is the dominant heat transfer mode. At later stages of deformation the convection in the fluid and the conduction along the spreading direction in both fluid and solid become important and in some cases dominate the heat transfer of the process. A complete numerical solution of the impact and solidification, therefore, requires solving the momentum and the fdenergy equations simultaneously.

The latent heat treatment is a major heat transfer problem associated with phase change.

Cao et al. [94] stated that phase change problems are solved using two main groups of numerical methods: strong solutions and weak solutions. In strong numerical solut ions: the mesh transformation method, as described in Chapter 3, is used to treat the liquid-solid interface; the liquid and solid regions are considered separately; the shape of the phase front is detennined by an iteration scheme; and the latent heat of fusion is treated as a heat source at the Liquid-solid interface. The strong solutions are not easily extendable to mult idimensional problems.

In weak solutions it is not necessary to separately consider the liquid and the solid regions: the phases and the shape of the phase front can be determined later from the solutions. These solutions are important because certain numerical schemes converge to the weak solution of the original boundary value problem, despite discontinuities of the temperature gradient at the phase front [95]. Methods which utilize weak solutions can easily be extended to solve multidimensional problems. Some methods which are widely used in technical applications are the apparent capacity, heat integration, and enthalpy methods. A cornparison of these and other methods for one and two dimensional water and liquid metal problems has been published by Poirier and Salcudean [96]. It was concluded by Poirier and Salcudean [96]that none of the weak methods perform well for ice water problems. Weak methods, by definition, do not specifically account for the discontinuities in a problem, therefore, they cannot be expected to be accurate in the region near the discontinuity. -4 discontinuity occurs at the phase front in solidification or melting problems. The more severe the discontinuity, the poorer the performance of the weak methods. In the case of water the ratio of latent heat to sensible heat is about an order of magnitude greater than that for iron. The discontinuity is much more severe, resulting in a poorer performance of these methods.

In the apparent capacity method the latent heat is included in the specific heat of the material. An apparent capacity is defined in the range of phase change temperatures to account for the entire enthalpy change, including the sensible as well as the latent heat. In the case of pure materids an artificial temperature range has to be used, over which the latent heat is assumed to be released. This method has been used by Hashemi and Sliepcevich [97] in a finite ciifference formulation, and Comini et ai. [98] in a finite element formulation. A major shortcoming of the apparent capacity method is that if, for a solidification problem, a nodal temperature fds from above the liquidus to below the solidus temperature in one time step, the latent heat release for the ce11 corresponding to that node in not accounted for. A similar deficiency exists if the method is applied to melting problems. Poirier and

Salcudean [96] found that, in generd, this method performed poorly when compared with other formulations. This formulation is simple and computationally economical. However, it does not perform well for isothermal solidification or phase change over a narrow temperature range. In this case extremely small time steps are required or an assurnption of a large fictitious mushy range has to be made which does not correctly reflect the physics of the problem.

In the heat integration method the temperatures of al the nodes are monitored. If the temperature of any node drops below the solidus temperature (or liquidus temperature in the case of a mushy region), the material in the control volume associated with the node is assumed to experience phase change. An accounting for the energy loss by the node at each cycle is carried out. When the total energy lost by the control volume is geater than or equai to the latent heat content, a flag is set indicating that the material in the control volume has solidified; the associated nodal temperature is then reset such that the extra heat loss from the node is accounted for. In subsequent cycles the temperature of the node is allowed to fa11 normally. This method was used by Rolph and Bathe [99]in a finite element formulation. Alt hough this method can be easily used in multidimensional modeling and it is computationally economical, the solution in the region of phase change is inaccurate as the phase front is not tracked. The accuracy of the results depends on the time step. Also, a rather elaborate system of accounting and indexing has to be maintained for each node. Another formulation, enthalpy method, has continued to enjoy widespread application

(Boucheron and Smith [SOI ,Crawley [100],Beil and Wood [101],Shamsunder and Spamow [102], Jerome [103], Vouer [104], Voiler and Cross [105], Chabchoub [106], Chabchoub et al. [107]).

The basic method consists of writing the transient part of the energy equation in terms of the enthalpy instead of the temperature. The enthalpy method is reasonably accurate for metals solidifying over a mushy range. The solution is relatively independent of the time step and mushy range. The method is somewhat more cornplex and expensive than the other approaches. The computational cost increases rapidly with mesh refinement . This technique performs pooriy close to the phase front for isothermal solidification [104, 1051. Reviews of enthalpy formulations were done by Voller [104]; the limitations of the method of modeling isothermal phase change were recognized and a technique to improve the accuracy of the results was presented.

In this work we use the enthafpy method with Voller's formulation [104, 1051 to treat the phase change during the impact of a droplet on a substrate because this formulation tracks the phase front and utilizes this knowledge of its position to improve the accuracy of the solution. That is why this formulation cannot be strictly classified as a weak solution. The solution of this formulation includes the temperature distribution ail over the material and the position of the liquid-solid interface. The knowledge of the position of the interface will t hen be used to obtain the liquid-solid volume fraction 8 as defined in Chapter 3. To treat the solidification in the fluid dynamics computation we will use the improved jbed uelocity technique as described in detail in Chapter 3. In this chapter we study the enthalpy method in conjunction with the existing Buid flow mode1 discussed in Chapter 2 by using a finite difference formulation.

5.2 Mat hemat ical Formulations

A complete solution of the energy equations for droplet impact must include the solution for the droplet, both liquid and solid, and the substrate regions. The important boundary conditions are the free surface of the droplet, liquid-solid interface or solidification front and Figure 5.1: Schematic diagram of the droplet and substrate at the time of impact.

the droplet su bst rate interface. In particular, thermal contact resistance at the droplet- substrate interface highly controls the heat transfer between the two media which in turn affects the final soiidified shape of the splat to a very high extent. The formulations givec

in this chapter, therefore, include energy equations in the droplet computational domain and in the substrate with special attention to the droplet-substrate boundary condit ion. An axisymmetric enthalp y met hod for the droplet domain (liquid, solid. and interphase zones) and an axisymrnetric conduction heat transfer mode1 for the substrate will be explained in this section. We use the Eulerian formulation to construct the mathematical mode1 for the conjugate heat transfer process in the droplet and substrate. A schematic diagram of the droplet and substrate at the time of impact, t = O, is shown in Fig. -5.1.

5.2.1 Energy Equation in Droplet Region

Physical and thermal properties are assumed to be constant but with different values in each phase. We assume no viscous dissipation in the energy equation. This assumption can be justified by considering Eckert number, Ec=I(-*/(ci AT) where ci is the liquid specific heat and AT is the temperature difference between droplet and substrate. For the cases under consideration in this study, the Eckert number is in order of 10e3 indicating that the viscous dissipation compared to the boundary layer enthalpy difference is in fact negligible. Using

these assumptions, for the axisymmetric coordinate system (Fig. 5.1) we have

Ik = k,, solid phase

k=k, liquid phase where h is enthalpy, T is temperature and k is material thermal conductivity. Subscripts s, I, and m refer to the solid, liquid, and liquid-solid interphase, respectively. Carslaw and

Jaeger [92] and Shamsunder and Sparrow [102] have shown how the above equation may be derived from the heat conduction equations in the liquid and the solid regions and a heat conservation condition at the phase front. Due to the existence of both liquid and solid phases in the domain of calculation, the situation presents a conjugate heat transfer problem where convection in the liquid and conduction in the solid must both be considered. Since the energy equation is written in ternis of two dependent variables, temperature T and enthalpy h, we wiIi use the Enthalpy Transforming Model [94] to convert the energy equation to one with only one dependent variable; the enthalpy. The transformation procedure is given in next section.

5.2.2 Enthalpy Transforming Model

The enthalpy Transforming Model has been developed by Cao et al. [94]. The application of this mode1 to the energy equation, Eq. (5.1), is discussed in this section. From the thermodynamic relations of properties, the temperature is related to enthalpy via the following state equation

where c is the specific heat coefficient of the materid. A variation between the total (sensible plus latent) enthalpy and temperature is defined, based on the latent heat release charac- Figure 5.2: Enthalpy against temperature for an isothermal phase change materia teristics of the phase change material. This nriation is in the fom of a step function for isothermal solidification. Fig. 5.2 shows the enthalpy as a function of temperature for an isothermal phase change rnaterial with constant specific heat in each phase. The temperature

T for this case is given by

where HI is the latent heat of fusion. In Fig. 5.2 we choose a new coordinate system where h = O corresponds to phase-change materials in their solid state at temperature Tm.

Consequently, Eq. (5.4) can be written as

In the energy equation, Eq. (5.1), k is in the argument of partial derivatives on the right hand side. Therefore, we use Kirchhoff's temperature, T*, defined as [IO81 T* is knom as the heat conduction potential and it is used to solve the energy equation in

arc-dischorge plasma problems. For the different phases, T' can be written as

Transforming Eq. (5.5) with the definition given in Eq. (5.7) results in

The energy equation, Eq. (5.1), can be written in tems of T*as follows

From Eq. (5.8) it can be seen t hat T* is a linear function of h as

where I' and S are functions of h. Upon substituting Eq. (5.10) into Eq. (5.9) finally we will have

a 3 a a ah a ah -(ph) + u-(ph) + v-(ph) = [-(r-)+ -(r-) + at d~ 3~ ax al ay ay z ax

r = kS/cs, S = O, hSO solid phase S = O, O < h < Hf interphase (5.12)

S=-H1k~/ci, hlHf liquid phase Figure 5.3: Enthalpy against temperature for an doy.

The energy equation has been written in terms of only one dependent variable, the enthalpy h and a new source term S. Note that this equation is non-lineax because r is a function of h. The non-linear behavior is a common characteristic of phasechange problems because of the moving interface [log]. However, in the Iiquid and solid regions away from the rnoving phase front, the gradient of the source terms vanishes and Eq. (5.11) reduces to a normal linear energy equation. It should be rnentioned that Eqs. (5.11) and (5.12) were obtained using an isothermal phase change (Le. the phase change occurs at a single temperature). If phase change occurs over a temperature range (such as alloys), as shown in Fig. 5.3 with constant specific heats for each phase, the resulting energy equation from the Enthalpy Transforming Mode1 will still be Eq. (5.11), however, functions r(h) and S(h) will have different expressions (Le.

Eq. (5.12) will be different) [94].

Eq. (5.1l), as is, considers the density changes due to the phase change. In this work, however, we assume the droplet liquid and solid phase densities to be identical. For the cases under consideration the materials experience Little contraction upon freezing. 5.2.3 Energy Equation in Substrate

We treat the substrate as a solid wd with no melting. The energy equation, therefore, is the conduction heat transfer equation which in an axisymmetric coordinate system can be written as

where subscript w refers to the substrate and y, is in the opposite direction of y as shown in Fig. 5.1.

5.2.4 Boundary and Initial Conditions

Droplet Region

To satisfy the temperature boundôry conditions, first we need to express temperature T in tems of enthalpy h and the new functions r and S. The relation can be obtained from Eqs. (5.5) and (5.12) as follows.

h50 solid phase where k = km, O < h < HI interphase ik = kl, h 2 Hf Liquid phase Similar to the flow dynamics problem, Chapter 2, two sets of boundaries have to be considered: computational mesh boundaries and free-surface boundaries. The problem is symrnetric about y axis as shown in Fig. 5.1, therefore, only one slice of the droplet is considered in the comput ation. Mesh Boundaries. Enthalpy h and functions I' and S must be set at the boundaries of the computational domain. The left mesh boundq, the axis of symmetry, is treated as an adiabatic boundary; the temperature there must have no normal gradient. For the bottom boundary, the surface of the substrate, we use adiabatic boundary conditions for the non- wetted area. However at the wetted area of the substrate, i.e. the area which is in contact with the droplet in either liquid or solid forms, there is heat interaction with the substrate. The thermal contact resistance at the substrate surface, which is an important factor in heat interaction between droplet and substrate, must be considered in the formulations. Since the transient flow dynamics and heat interactions of the droplet and substrate are under investigation, the computational domain must be large enough such that the material does not reach the right and the top boundaries. As a result, the conditions at these boundaries do not count; they are set as simple adiabatic conditions. Based on the above, the condition at the left boundary is : (aT/ax)= O; adiabatic condition also requires that no heat of fusion is released at the boundary. Therefore, we must have ah - = O at the left boundary dx

Since r and S are functions of h based on Eq. (5.12), the above condition will also resuit in

dï as -=O - = O at the left boundary dx ' da:

For the bottom boundary at the non-wetted area, similady we have ah ar as - = O and consequently : - = O - = 0 3~ ay ' ay

At the wetted area of the bottom boundary, there will be a heat interaction with the substrate. If q is the rate of heat transfer from droplet to substrate per unit area as shown in

Fig. 5.1, then this boundary condition may be written as

where the positive direction of q is assumed to be from droplet to substrate (Fig. 5.1). The connection between the solutions of the energy equations for droplet and substrate domains is through p. The surface thermal contact resistance affects the calculation of q, as described later in this chapter. Eq. (5.19) must be translated in terms of enthalpy h and functions ï and S. The easiest way is to first obtain the temperature of the boundary by using Eq. (5.19)

and then apply Eq. (5.14) to get enthdpy h; functions l? and S can be found at the same

time as follows. Frorn Eq. (5.19) we can write

When the temperature at the ce11 adjacent to the substrate is known, the bottom-boundzuy

temperature Tb may be obtained using Eq. (5.20). Equation (5.14) then is applied to get hb as follows 1 hb = - [kb(Tb - Tm) - Sb] rb if Tb

Free Surface Boundaries. At a free surface of the droplet in either phase, we use an adiabatic boundary condition. This condition must be supplemented with the specification of ent halpy h and functions r and S immediateiy outside the surface, where t hese values are needed in the fini te-difference approximations for points outside the surface. This condition cm easily be modified to a convective or radiative boundary condition. For the cases under consideration, the dominant heat transfer is due to conduction to the substrate at the initial stages of impact, and conduction and convection within the droplet at the later stages of impact. Therefore, the adiabatic condition at the free surface is reasonable.

Initial Conditions. Initiai conditions are initial values of h (and consequently: ï, S, and k) at the time of impact which is assumed to be t = O. These initiai values are calculated based on the initial temperature distribution within the material; any initial distribution in either of the liquid or soiid phases is permitted. For-the cases under consideration, the droplet is assumed to be at a unifnrm temperature above the melting point in a fully liquid phase.

Substrate Region

Mesh Boundaries. Temperature must be set at the boundaries of the substrate computational domain. The left mesh boundary, the axis of symmetry as shown in Fig. 5.1, should be treated as an adiabatic boundary where the temperature has no normal gradient. The top boundary is the surface of the substrate, therefore, it is treated similar to the bottorn boundary of the droplet computational domain. The substrate domain must be large enough such that the heat transfer does not affect the right and bottom boundaries of the substrate domain. As a result, we use a constant temperature condition at these two boundaries. The mathematical formulation of the above explanations is as follows. At the left boundary we have : (aTW/ax)= O; similarly at the non-wetted area of the top boundary we have an adiabatic condition : (aTW/ayw)= O. At the wetted area of the top boundary we have heat interaction with the droplet. The condition, therefore, may be written similar to Eq. (5.19)

where positive direction of q (droplet to substrate) now is in the positive direction of y, in the substrate region as shown in Fig. 5.1. The above equation cm be used to obtain the substrate top-boundary temperature by empioying a similar treatment applied to get the droplet bottom-boundary temperature Tb. At the right and bottorn boundaries of the substrate we have : T, = TWvi,where TWviis the initial substrate temperature at the time of impact.

Initial Conditions. Initial condition at the substrate can be any temperature distribution at the time of impact (t = O). For the cases considered in this work, we assume a uniform temperature of TWvithroughout the substrate as the initial substrate condition. Droplet-Substrate Interface

The connection between the solutions of the energy equations for the droplet and substrate regions is through the interface heat interaction, q. At the non-wetted area of the interface there is no heat transfer between the two media and therefore : q = O. At the wetted axea, however, q has a value equal to

where 6T here is the temperature difference between the two adjacent cells, one above the substrate surface in the droplet region and another under the substrate surface in the substrate region; Rt is the total thermal resistance per unit area exists between the two ceils.

Rt includes the surface thermal contact resistance per unit area, &, which in general can be a function of both time and space.

5.3 Computational Treatment

For the droplet domain, we use the same Eulerian mesh applied to the flow dynamic problem as described in Chapter 2 and shown in Fig. 2.2. For the substrate domain, the mesh discretization in the x direction should be the çame as for the droplet domain. This is because of the boundary condition at the droplet-substrate interface. We use a finite-difference scheme to discretize the governing equations and satisS the boundary and initial conditions.

The enthaipy h and btions r and S are defined at the center of the grid. Similarly, for the substrate computational mesh, T, is dehed at the grid center.

5.3.1 Finite Difference Ent halpy Equation in Droplet Region

The control volume (shaded axea) used for constructing a finite-difference approximation for the enthalpy equation, Eq. (5.11), is shown in Fig. 5.4. The finite ùifference equation may be written as Figure 5.4: Control Volume (shaded ârea) for finite-difference ent halpy equat ion.

6t hn+'ZJ = htj + - {DCFFH + DIFFS) - 6t {FHX + FHY) (5.25) P where DIFFH and DIFFS are the diffusion terms of enthalpy h and function S. and FHX and FHY are the advective flux of enthalpy h in the x and y directions, respectively. These terrns are al1 evaluated using the old time level n values for velocities. enthalpy, and functions r and S (explicit formulation). This is because the numerical scheme for the flow dynamic problem is also explicit.

The diffusion terms are approximated with the standard centered-differencing approximation (no upwinding); the expression for DIFFH is given as

(-5.26) at h,,) location where al1 terrns ore evaluated at hij location. The expression for (E(I'2))is very similar to (g)in the flow dynamic problern; we cm have

where subscripts e and w refer to the east and west of the control volume as indicated in Fig. 5.4. Functioo r is weighted by ce11 sizes in such a way that the correct order of approximation is maintained in a variable mesh. Therefore, r, may be written as

The expression for I' on the west boundary face of the control volume, ï,, is obtained in a very similar fashion. The enthalpy derivatives at the right and left hand sides of hij are obtained by a linear approximation as

where

The next component of the enthalpy difision terms is (&(rg)),the diffusion in y direction. The expression for this term is obtained in a way quite similar to the one taken for x direction; the role of east and west faces of the control volume are now replaced by the nort h and sout h faces.

The last cornponent of Eq. (5.26) is (5%) which has to be evaluated at hi,location. Using the sarne scheme of weighting by the ce11 sizes we can find the expression as

The next group of the diffusion associated with the enthalpy equation corresponds to function

S and is denoted by DIFFS as used in Eq. (5.25). The finite difference approximation of DIFFS is very sirnilar to the one taken for DIFFH, Eq. (5.26), and is not repeated here.

The finite difference expressions of the advective terms, FHX and FHY, are writ ten in a way similar to the one used to express the advective terms of the momentum equations. As described in Chapter 2, for a variable mesh the nonconsenstive form of the advection flux is used. We then apply a combination of the upwinding and centered-differencing approximations to obtain FHX as the following

FHX = 2 Figure 5.5: Control Volume (shaded area) for finite-difference energy equation in the substrate. w here

and (g) and (g) were defined in Eq. (L29). The expression for the advective term in y direction, FHI: can be obtained similarly.

5.3.2 Finite Difference Energy Equation in Substrate

The control volume used for const ruct ing a fini te-difference approximation for the energy equation in the substrate, Eq. (5.13), is shown in Fig. 5.5. If substrate properties are assumed to be constant the finite difference equation is written as

"+l= T" + Twt ., Wt .) (5.34) at Tw,.l location where a11 diffusion terms in the right hand side are evaluated using the old time level n values for substrate temperatures. The finite difference expressions for t hese terms are obtained with standard centered-differencing approximation (no upwinding) in a manner quite similar to the one thken for the diffusion terms of the enthalpy equation in the dropiet region. The variable mesh considerations are also similar to those of the enthalpy equation in the droplet. 5.3.3 Boundary and Init id Condit ions

Droplet Region

Mesh Boundaries. Similar to the %ow dynamic problem, to impose boundary conditions a layer of fictitious cells surrounding the actual mesh is assumed. Enthalpy, and functions

r and S for the fictitious cells are set such that the proper conditions at the actual wds, Eqs.(5.16)-(5.22), hold. For the left boundq where the adiabatic conditions exist , t herefore,

we have

h1.j = h2,j and consequently : rivj= r2,j , Sij = S2.j (5.35)

At the non-wetted area of the bottom boundary we have the adiabatic conditions, therefore

hiVl= hi,? and consequently : rivl= riv2 , Si,1 = SiVl (5.36)

At the wetted area of the bottom boundary, where there is heat interaction with the sub-

strate, we need to first obtain T,,l from Eq. (5.20) as

where the positive direction of q is from droplet to substrate. Since the ce11 adjacent to the

wall, (i, 2), may not be full of material, FiV2has been multiplied by qi; Fi,* is the volume of

fluid function in the ce11 (i,2). We can now set the required boundary conditions based on Eqs. (5.21) and (5.22) os foUows

As it is clear from this equation, for the fictitious cells where there is no real materid the imaginary phase at temperature Tm is assumed to be liquid. In other words, no interphase is considered for these cells. Free Boundaries. At a free-boundary of material (in either of liquid or solid phase), the adiabatic condition is applied. This is accomplished by setting zero for ahli3z and ahlay and consequently for gradients of r and S in two directions. Therefore, the dues of h, î and S are assigned to the empty neighboring cells of the freboundary cells. These values will be exactly the same as for the corresponding freboundary ceils. The free-boundary condit ion is easiIy modifiable to a convect ive or radiat ive boundary condition.

At a Liquid-substrate contact Lne, the concept of contact angle, 8, as explained in Chap ter 2 is used. This condition must be applied on the liquid contact line over the previously solidified layer as well. Fully solidified cells are, therefore, assumed to act as a wall for the liquid moving around their faces. The contact angle for a rnolten materid wetting its own solidified layer is usually smdl.

Initial Conditions. At the beginning of the first step of computation, the initial distribution of temperature within the material, Trl=o, must be given. This condition is then expressed in terms of h, r and S as follows.

It shouid be noted that the calculation is skipped over a ce11 (2, j) if it contains no material. The values of T, h, r and S for these cells will be unimportant and, therefore, are set to zero at al1 tirnes.

Substrate Region

Mesh Boundaries. Similar to mesh boudaries of the droplet computational domain, we introduce a layer of fictitious cells surrounding the actual substrate mesh domain. For the left boundary and the non-wetted area of the top boundâry where the adiabatic conditions exist, therefore, we have

TWl,= T-, (left) , TW,,= T,, (top; non wetted area) (5.42)

At the right and bottom boundaries which are taken far from droplet-substrate heat transfer zone, constant temperature boundary conditions are assumed. At the wetted area of the top boundary there is heat interaction with the droplet. For this boundq form Eq. (5.23) we have

This condition is quite similar to the condition for the droplet bottom boundary, Eq. (5.37), except that here the second term of the right hand side is positive. This is because the positive direction of q was assumed to be in the same direction as y,.

initial Conditions. Any temperat ure distribution at the time of impact (t = O) is permitted in the substrate. For the cases considered in this work where there is initidy a uniform temperature throughout the substrate

where Twviis the initial substrate temperature.

Droplet-Substrate Interface

Since both droplet and substrate computatiood domains are surrounded by the fictitious cells, the actual mesh of both domains starts from the second row, i.e. (2,2). Therefore, 6T in Eq. (5.24) is the difference between Tiqaand Twi,2.As a consequence, Rt in Eq. (5.24) is the total resistance between the (i,2) cells in droplet and substrate domains. This condition, which applies to the wetted area of the interface, can be written as When ZV2in this equation is replaced based on Eq. (5.14) we will have

k.2 = k, h.2 < O soiid phase where kiV2=km7 O < hiV2< Hf interphase I ki.2 = k~, k.2 L Hf liquid phase At the non-wetted axea of the interface where there is no heat transfer between the two media qi is set to zero.

5.3.4 Evaluat ion of Liquid-Solid Volume Fraction

To evaluate the liquid-solid volume fraction 8 at the interphase (solidification front) we use an algorithm introduced by Voller [104, 1051. In this algorithm, while the phase change is occurring in the subregion about node (i, j), the rate of change in the nodal enthaipy hij is proportional to the velocity of the phase change front across the subregion, with the latent heat of fusion the constant of proportionaiity. This can be described mathematically by the following

where tl and ta correspond to O = O (Mysolid; h = 0) and O = 1 (fully liquid; h = H!), respectively. Since HI is assumed to be constant, this equation leads to

The computational steps of the complete numerical solution for flow dynamics, heat transfer and solidification are as follows. 1. From time level n values, the velocity and pressure fields as well as F are calculated at tirne level n + in accordance with the modified SOLA-VOF algorithm described in detail in Chapter 4 (first three steps of computations in Sec. 4.3).

2. Given the droplet enthalpy and substrate temperature fields at time level n, Eqs. (5.25)

and (5.34) are used to obtain the new enthalpy field in the droplet and the new tem-

perature field in the substrate (explicit formulations). Temperatures in the droplet can then be calculated from Eqs. (5.14) and (5.15).

3. New values of the liquid-solid volume fraction, 0, are calculated from the enthalpy field in the droplet by using Eq. (5.15) in conjunction with Voller's algorithm, Eq. (5.49).

4. Flow and thermal boundary conditions are imposed on the free surface, at the se lidification front, and the boundaries of the computational domain for both droplet

and substrate. In particular, the thermal contact resistance at the droplet-substrate interface is applied by using Eqs. (5.46) and (5.47) to calculate heat flux from the

droplet. This value of q is then used to update temperature boundq conditions along

the bottom surface of the droplet and the upper plane of the substrate according to

Eqs. (5.37-5.39) and Eq. (5.43), respectively.

Repetition of these steps allows advancing the solution through an arbitrary time interval.

The droplet was discretized using a uniform computational mesh, with a grid spacing equal to

1/30 of the droplet radius. The substrate mesh had the same resolution, and was extended far enough that its bottom and right boundaries could be assumed to be at constant temperature.

Numericai cornputations were performed on a SGI (INDIGO2)workstation. Typicd CPU times ranged from 2-5 hrs. 5.3.6 Stability Considerations

In addition to the stability considerations for flow dynômics given in Sec. 2.3.7 we have to consider the stability of the numerical solution with regard to energy equations in droplet and substrate. Since these equations were treated explicitly, there are restrictions on bt as follows. When a nonzero value of material thermal diffusivity is used, heat must not diffuse more than one cell in one time step. A lineu stability analysis shows that this limitation implies 1 ax;ay; dt < 2 max{ar,a,, a,) 6xt + 6yt where rr indicates diffusivity and subscripts 1, s and w represent liquid, solid and substrate, respectively.

5.4 Results and Discussion

5.4.1 Cornparison of Numerical and Experimental Result s

The results presented here axe t hose given by Pasandideh-Fard et al. [Il01 where experimen- ta1 results have been accomplished by Bhola and Chandra [52]. We consider the impact and solidification of tin droplets on a Bat stainless steel plate. The details of the experimental method and apparatus are given in Refs. [52, 1101. Tin droplets, 2.1 mm in diameter, impact onto a stainless steel surface whose temperature is varied from 25OC to 240°C. The impact velocity is 1.6 m/s. In the experiments [52, 1101, impact of droplets are photographed, and evolution of droplet spread diameter and liquid-solid contact angle measured from ph* tographs. Substrate temperature variation under an impinging droplet is measured. For the mode1 predictions, the complete numerical solution of the Navier-Stokes and energy equations, based on the development given in this chapter, is used to model droplet deformation and solidification and heat transfer in the substrate. Measured dues of liquid-solid contact angle are used as a boundary condition for the numerical model. The thermal contact resistance at the droplet-substrate interface is estimated by rnatching numerical predictions of the variation of substrate temperature with measurements. Cornparison of cornputer generated images of impacting droplets with photographs shows that the numericd mode1 correctly models droplet shape during impact as it simultaneously deforms and solidifies.

Review of Experiment al Results

Bhola and Chandra's experimental results for tin droplet impact are briefly presented in this section. These results are those given in Refs. [52, 1101. Figure 5.6 shows photographs of the impact of 2.1 mm diameter tin droplets, with an initial temperature of approximately

240°C and velocity of 1.6 m/s, impacting on a stainless steel surface [52, 1101. Each column of photographs in Fig. 5.6 shows successive stages of impact on surfaces at temperatures of 25"C, 150°C and 240°C, respectively. The time of each image (t), measured from the instant of first contact with the surface, is shown. The reflection of each drop in the polished stainless steel surface can be seen in the photographs.

Droplets falling on a surface at 25OC spread after impact to their maximum extension at approximately t = 3.0 ms (seFig. 5.6), after which surface tension forces prevented any further spread. The edges of the drop were drawn back by surface tension for t > 4.5 ms, decreasing the splat diameter. The edges of the droplet solidified by t=7 ms, after which the splat diarneter did not change. However, the center of the splat was still liquid, and continued to flow until t=20 ms. When the initial surface temperature (TW,i)was raised to 150°C, the initial stages of impact appeared qualitatively similar to that with TW,i= 25OC

(see Fig. 5.6). However, solidification was slower on the hotter surface, and the droplet rernained liquid for a longer period of time, so that the maximum splat diameter was slightly

Iarger. Surface tension forces caused a slight recoil of the drop off the surface (t=l 1 ms), and also smoothed out the surface. At Tw,i = 240°C, the surface was at the same temperature as the initial droplet temperature: impact was isothermd and the droplet remained liquid Figure 5.6: Impact of molten tin drops on a stainless steel surface at an initial temperature of a)25OC, b)150°C, c)240°C. This figure is taken from Ref. [52]. throughout impact. After a droplet spread on the surface to its maximum extent (t=3 ms), surface tension and viscous forces overcame liquid inertia, so that fluid accurnulated at the leading edge of the splat and it started pulling back, eventually rising off the surface. It then fell back and reached its static state, shaped Like a truncated sphere, after t > 500 ms. Figure 5.7 shows the evolution of contact angle, measured from photographs of impacting droplets [52, 1101. The advancing contact angle, during outward spreading of the droplet

(t < 4 ms), was approximately constant at 140' f IO0. The receding contact angle, during droplet recoil, was somewhat smder (- 125') on a surface at 240°C (see Fig. 5.7). The definition of a receding contact angle was rather arnbiguous in the case of = 25°C and

TWpi= 150°C, since the layer in contact with the hot surface solidified shortly after impact and stopped moving (see Fig. 5.6). However, the liquid above this solidified layer continues to move. Therefore a contact angle was defined between a line drawn tangentid to the edge of the liquid portion and the plane of the solid substrate: these values are shown in Fig. 5.7.

This contact angle, for molten tin wetting a solidified tin layer, was smail (- 10") during recoil, and relatively constant.

Estimation of Thermal Contact Resistance

Predictions from the computer model of droplet-impact are sensitive to the values of two input parameters: the liquid-solid contact angle (O), and the thermal contact resistance (%) at the droplet-substrate interface. For the liquid-solid contact angle we used the measured dynamic contact angles given in Fig. 5.7. In the recoil (for t > 10 ms), molten tin wetting a solidified tin layer, a constant value of 0 = 10' was used in the model. The thermal contact resistance (&) was estimated by comparing the computationai substrate temperature variation during droplet-impact with results from the experiment, and adjusting the thermal contact resistance value to obtain the best agreement. The measured evolution of surface temperature, set prîor to droplet deposition at 25OC and 150°C, respec- O 2 4 6 8 10 12 14 16 18 20 The (ms)

Figure 5.7: Liquid-solid contact angle variation during the impact of molten tin drops on a stainless steel surface at initial temperature of Tu,;. This figure is taken from Ref. [52]. tively, is shown in Figs. 5.8 and 5.9 [110]. The substrate temperature was measured by a thermocouple placed at the point the drop impacted. A surface initially at 25OC was heated to 160°C in less than 1 ms (Fig. 5.8); its temperature remained nearly constant for the next few milliseconds, after which it began to cool. When the initial substrate temperature was set at 150°C the peak temperature rose to 225'C (Fig. 5.9). The predicted temperature variation is shown by solid lines in Figs. 5.8 and 5.9. At an initial surface temperature of

25OC satisfactory agreement between the measured and predicted values wâs obtained using a single, constant value of the thermal contact resistance, Rc = 1 x 10-~m2K/W for t < 3 ms

(see Fig. 5.8). However, at later tirnes (t > 6 rns) a higher value of Rc = 5 x 1V6 m2K/W gave better predictions of substrate temperature. Liu et al. [53] have shown that the thermal contact resistance increases as a droplet solidifies, because of higher resistance to heat transfer at a solid-solid interface than at a liquid-solid interface. It should aiso be noted that the above temperature measurement (Figs. 5.8 and 5.9) was done at a single point: it is likely that the local thermal contact resistance varies with position under the splat. With an initial surface temperature of 150°C the surface temperature variation was best simulated using a low thermal contact resistance, R, = 1 x IO-' m2K/W (see see Fig. 5.9). This result was expected, since droplet solidification would be delayed on the hotter surface, reducing contact resist ance.

Cornparison of computed droplet shapes during impact with photographs showed that using the range of thermal contact resistance values obtained from temperature measurements resulted in reasonably good predictions. The rate of droplet spreading was quantified by measuring the splat diameter (D) at successive stages durhg droplet deformation. Nor- malizing D by the initial droplet diameter ( Do)yields the "spread factor," { = D(t)/Do.

Measured values of ( during the impact of droplets on surfaces with initiai temperatures of 25°C and 150°C are shown in Figs. 5.10 and 5.11, respectively. Using a single value of thermal contact resistance R, = 1 x 10-6 m2K/W gave good predictions for ( for t < 1 ms. At that tirne, however, the simulated droplet solidified completely and did not spread further. Increasing & to 5 x IO-' m2K/W in the model reduced the droplet solidification rate, and improved agreement between predicted and measured values of e. The same thermal contact resistance also gave good predictions for the spread of a droplet on surface at 150°C (Fig. 5.1 1). The range of thermal contact resistance obtained by these two different methods

- matching either the measured surface temperature variation or the droplet spread rate with results from the model - appears to confirm earlier observations [53]that Rc varies during impact. Bet ter agreement between experimental md numerical results could have been obtained by using local thermal contact resistances that mied with time and position, rather Figure 5.8: Substrate surface temperature variation during the impact of a molten tin droplet on a stainless steel surface initially at 25OC. than an average, constant due. However, we did not attempt to do this since we had no independent met hod of j ustifying any assumed variation of &. A11 calculat ions hereafter were performed, therefore, using a constant thermal contact resistance value of 5 x 10-~m2K/W.

Droplet Shapes and Temperature Distributions during the Impact

Cornputer generated images of impacting droplets are compared with photographs taken at the same time after impact in Figs. 5.12-5.16. Both computer generated images and ph* Time (ms)

Figure 5.9: Subst rate surface temperature variation during the impact of a molten tin droplet on a stainless steel surface initialiy at 150°C. tographs are viewed from the same angle (30' from the horizontal). impact on a surface with initial temperature TWti= 240°C is shown in Fig. 5.12. In this case impact was isothermd, and there was no solidification. Agreement between the predicted droplet shapes and ph* tographs was in general good during droplet spreading (t < 1 ms). Fingers formed around the periphery of the drop (se t=9.5 ms) as a result of the Rayleigh-Taylor instability. The model. which was twedimensiond and assumed axisyrnmetry, was not capable of simulating such an instability. It did, however, accurately predict droplet recoil off the surface

( t = 15 ms). The Rayleigh-Taylor instability occurs when t here is an accelerated interface Time (ms)

Figure 5.10: Evolution of spread factor during impact of tin droplets on a stainless steel surface initially at %OC. between two fluids of different densities (here liquid tin and air). The amplitude of this instability, which is caused by an initial perturbation, may grow exponentially [Ill, 1121.

Figure 5.13 shows computer predictions of the shape of droplets impacting on a surface with Tws = 25"C, in which case the extent of solidification during impact was significant.

Calculated velocity and temperature distributions inside droplets, at the same times following impact as those in Fig. 5.13, are displayed in Fig. 5.14. The solidified layer (seen in yellow in Fig. 5.14) was thickest at the center of the splat, which first contacted the surface, and along its edge, which was nearest the surrounding cold plate. Results from the computation Time (ms)

Figure 5.11: Evolution of spread factor during impact of tin droplets on a stainless steel surface ini tially at 150° C. showed that the heat flux from the droplet to the substrate increases with radial distance from the center. Solidification along the edge prevented further spread of the drop (see

Fig. 5.14, t=4.5 ms). However, there remained a film of molten tin above the solid layer, which recoiled and flowed back towards the center of the splat. This movement can be seen in the photographs of Fig. 5.13 (see t=%5 and 12 ms). By t=12 ms almost al1 the tin had frozen. Solidification was rapid enough that the splat surface did not have enough time to become even, but had a number of craters left in it.

Similar photographs and computer generated images of a droplet laoding on a surface with Figure 5.12: Cornputer generated images and photographs of the impact of tin droplets on a surface initidy at 240°C. Figure 5.13: Computer generated images and photographs of the impact of tin àroplets on a surface initidy at 25OC. Figure 5.14: Cdculated velocity and temperature distributions inside a tin droplet impacting a surface initially at 25OC. Tw,i = 150°C are shown in Figs. 5.15 and 5.16. Growth of the frozen tayer was much siower in this case, because of the hotter substrate. However, the periphery of the splat solidified by t=8.5 ms. Close inspection of the photographs in Fig. 5.15 at t=8.5 ms confirms that the splat edges were solid; surface tension would have evened out the irregularities seen around the fringes if they remained liquid. Most of the droplet remained liquid as late as t= 15 ms, leading to a large recoil towards the center. The surface of the splat was smooth in this case, because of the slower solidification (see Fig. 5.15, t =15 ms).

The Effect of Solidification on Droplet Impact

The effect of solidification on droplet impact was studied nurnerically in Chapter 4, Sec. 4.4.3 and a conclusion was made that for high Re andfor high Ste numbers this efFect is more pronounced. An analytical study of solidification effect performed by Bhola and Chandra

(given in Ref. [110]) is briefly presented in this section.

A correlation was developed by Bhola and Chandra (given in Ref. [110]) to predict the diameter of the splat formed after droplet solidification by extending the simple droplet- impact model of Qiao and Chandra given in Ref. [59]. The model, which assumes that al1 the kinetic energy stored in the solidified layer is lost, findy yields:

-- We 12 &,,= = -= + (318)We - s* + 3(1 - cos 6,) + 4( ~e/fi) where Re is Reynolds nurnber (Re=l/oDolu), We is Weber nurnber ( We=pqDo/7)and s* is the dimensionless solid layer thickness (s' = s/Do);s is the solid layer average thickness.

Using the simplifying assumptions t hat : heat transfer is by one-dimensional heat conduct ion; the substrate is isot hemal; thermal contact resistance is negligible; and the Stefan nurnber (Ste=ci(Tm- TWvi)/HI)is srnall; s' may be written as [113]

where t* (= t Do) is the dimensionless time. Using this result, Bhola and Chandra came Figure 5.15: Cornputer generated images and photographs of the impact of tin droplets on a surface initidy at 150°C. Figure 5.16: Calculated velocity and temperature distributions inside a tin droplet impacting a surface initidy at 150°C. to the following equation for {,, [110]:

The magnitude of the term ~ed(3~te) /(4 ~e ) in Eq. (5.53) determines whet her solidification iduences droplet spread. Cornparison with the other two terms in the denominator shows that the kinetic energy ioss due to solidification will be too smdl to affect the extent of droplet deformation if @te/ ~r < 1 (where the Prandtl nurnber Pr=Pe/Re=v/u). This

result is in accordance with the numerical hding for the solidification effect described in

Chapter 4, Sec. 4.4.3.

For a tin droplet impact ing on a surface at W°C, St e=O.83, Pr= 14 x IO-^, and t herefore \l~te/~r=7.7:clearly solidification may influence droplet spread. However, substituting

the experimental values (1101 of We=71, Re=1.2 x IO4 and 8. = 140" in Eq. (5.53) gives

Q, = 0.4, much srnailer than the predicted dueof 2.6 (and the measured value of 2.9) obtained from Fig. 5.10. The discrepancy arises because the assumptions made in deriving Eq. (5.52), that thermal contact resistance at the interface is negligible, and that the sub-

strate is isothemal, are not valid. The Biot nurnber (Bi = h,Do/kl = Do/(k&)is relatively

small (approximately 6.8 for Rc = 5 x m2K/W), showing that thermal contact resistance

is significant. Analysis of splat cooling (1141 has shown that the effects of thermal contact

resistance are negligible only if Bi > 30. Equation (5.53) therefore is based on an upper

bound on the thickness of the soiidified layer; the actual thickness may be significantly lower and consequently the actual 6, may be considerably higher than predicted by Eq. (5.53). A more realistic value of sa was presented by Bhola and Chandra (given in Ref. [110]) which is calculated using an analytical model developed by Garcia et al. [115] that predicts the rate of solidification of molten metal in contact with a cold surface. The modei assumes: heat transfer in the droplet is by one dimensional heat conduction; the droplet and substrate are semi-infinite bodies suddedy brought into contact; and thermal contact resistance at the droplet-substrate interface is constant. The calculated variation of 8,using a constant value of Rc = 5 x 10-~m2K/W, is shown in Fig. 5.17. The thickness of the solid layer predicted by the numerical mode1 (see Fig. 5.14), averaged over the splat diameter, is also shown. Results from the analytical model, which now includes the influence of thermal contact resistance and substrate heating, agree well with predictions from the numericd calculation. Values of su calculated from Eq. (5.52), where the above effects are neglected, are seen to be much larger .

5 A.2 Numerical Results for Typical Thermal Spray Processes

Results are presented for two cases representative of RF and DC plasma spraying processes.

These cases and the conditions are exactly the same as were given in Chapter 3, Sec. 4.4. Table 4.1 (first two cases) lists the initial droplet diameter (Do),the impinging velocity (b), and the corresponding Reynolds, Weber and Stefan numbers for typical RF and DG plasma spray operations, cases 1 and 2 of the table, respectively. The substrate material is stainless steel.

Computer-generated images of the spraying of a 50 prn dumina droplet with an initiai velocity of 50 m/s on a Bat surface (Case 1 of Table 4.1, RF plasma spraying) at a viewing angle of 30' with respect to the horizontal surface axe shown in Fig. 5.18. The liquid phase is indicated in gray while the solid layer is indicated in black. The droplet spreads to its maximum extent at 1.5 ps, after which further spreading is arrested by the solidification.

Up until 10 ps a small layer of fluid exists at the top of the splat, therefore, the solid layer is not visible. At t =3O ps the whole splat is solidified (the last image of Fig. 5.18). Compaxing

Fig. 5.18 to the results from the 1D model, Fig. 4.2, shows that the deformation behavior are quite similar; only small differences exist between the two figures in the soIidification behavior of the splat at the late stages of deformation. The final diameter of the solid layer from Fig. 5.18 is 0.125 mm while from the 1D model we had 0.122 mm (Fig. 4.2). The final thickness here is around 4.9 Pm, from the ID model we had 5 Fm. This cornparison shows Dimensionless time, t'

Figure 5.17: Grcwth of the dimensionless solid layer thickness inside a tin droplet impacting a surface initially at 25OC. Predictions from the numerical model (with the thickness averaged over the splat diameetr), the analytical model of Ref. [115], and from Eq. (5.52) (which neglects thermal contact resistance and assumes the surface to be isothemal) axe shown. that for plasma spraying conditions, the 1D model resdts in good estimation of the splat shape and solid layer configuration. Calcuiated velocity and temperature dist ributioos inside the droplet , at three times following impact as those in Fig. 5.18, are displayed in Fig. 5.19. In the early stages of deformation, the axial velocity of the fluid near the substrate becomes zero while the radial velocity is increased rapidly. A smooth convex shape which is due to the high viscosity of alumina is formed at the splat edge. The solidified layer (melting temperature of alumina is 2050°C) has almost a uniform thickness all over the splat. This explains why the results from the 1D model of solidification for plasma spraying conditions are in good agreement with the results from the enthalpy model. Results from the computations showed that the heat flux from the droplet to the substrate increases with radial distance from the center. Substrate temperature which was initially 227"C (500 K) is increased to above 1000"C. The substrate heat affected depth is more than 0.025 mm, the droplet initial radius. More computational results for typicai RF plasma spray processes are presented in Appendix A where the grid size effects on numerical calculations are examined.

Similar computer-generated images of a 50 Pm alumina droplet wi t h an initial velocity of 200 m/s impinging on a flat surface (Case 2 of Table 4.1, DC plasma spraying) are shown in Figs. 5.20 and 5.21. Compared to RF plasma spraying, the spreading develops faster; a sheet jet of fluid is formed at the splat edge close to the substrate as seen in both figures. After nearly 4 (rs, the solidification is completed and a solid layer of 0.196 mm in diameter and 0.0023 mm (2.3 pm) in average thickness is formed. From the 1D model

(Fig. 4.3) we obtained very close values, 0.187 mm for diarneter and 0.0025 mm (2.5 pm) for thickness. This is another indication that employing the ID model of solidification is justified for plasma spraying conditions. Cornparison of Figs. 5.20 and 4.3 also shows that the deformation and solidification behavior of the splat resulting from bot h ent halpy and ID models are quite similar. From Fig. 5.21 it can be seen that the solid layer has a uniform Figure 5.18: Cornputer generated imagea of the impact of an alumina droplet of Case 1 of

Table 4.1 on a cold surface (representing a typicd RF plasma spray operation). Figure 5.19: Calcdated vebocity and temperature distributions inside an durnina dr impacting a cold surface for RF plasma spray process. thickness during the process. Compared to RF plasma spraying (Fig. 5. N),the substrate heat dected zone coven a larger area of the substrate with a smder depth (0.015 mm). Extensive computationd results and discussion for typical DC plasma spray processes have been presented elsewhere (1161.

Summary

A complete numerical solution of the Navier-Stokes and energy equations, based on a modified SOLA-VOF method, was used to model droplet deformation and simultaneous solidification and heat transfer to the substrate- Thermal contact resistance at the substrate was permitted. The phase change problem was treated by an axisymmetric enthalpy method.

The modification of the Buid dynamics equations was based on the improved fized velocity technique given in Chapter 3. The numericd results were compared with anilable experimental measurements for tin droplets (2.1 mm diameter) impacting with a velocity of 1.6 m/s on a staidess steel whose temperature was varied from 25°C to 240°C. The major findings are:

a Cornparison of cornputer generated images of deformirtg droplet s with experimental

photographs showed that the model correctly modeled droplet shape during the impact.

The value of contact resistance between the tin droplet and stainless steel substrate

was estimated by matching numerical predictions of su bstrate temperature with exper-

imental measurements. Reasonably accurate simulations of droplet impact dynamics

could be done using a constant value of contact resistance equal to 5 x m2K/W.

However, predictions of the maximum splat diameter were sensitive to the value as-

sumed. Therefore, accurate information regarding thermal contact resistance during the early stages of droplet impact is required to mode1 droplet impact and solidification. Figure 5.20: Cornputer generated images of the impact of an dumina droplet of Case 2 of

Table 4.1 on a cold surface (representing a typical DC plasma spray operation). Figure 5.21: Calculated velocity and temperature distributions inside an alurnina droplet impacting a cold surface for a typical DC plasma spray process. The solidified layer was thickest at the center of the splat, which fmst contacted the

surface, and dong its edge, which was nearest to the surrounding cold plate. a The heat flux from the droplet to the substrate increased with radial distance from the center.

Good agreement was obtained between the results of the enthdpy model with those

obtained from a 1D model of solidification (as given in Chapter 4) for plasma spraying conditions. This cornparison cohedthat the 1D solidification model results in good

estimation of the splat shape and solid layer configuration for plasma spray operations.

Through a simple anaiyticd model (that of Bhola and Chandra given in Ref. [110]),

the effect of solidification on droplet impact dynamics was found to be negligibie if

@te/ PT < 1. This resdt is in accordance with the numerical finding for the solidification effect described in Chopter 4. Chapter 6

Conclusions

A numerical model was developed on the basis of SOLA-VOF dgorithm to study the impact and solidification of a liquid droplet upon its impingement on a substrate. The model, in general, is applicable to transient Buid 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 parameters in a thermal spray process. This analysis required the complete solution of flow dynarnics as well as solidification and heat transfer within the droplet and substrate during the impact. The numericd model was developed step by step by first considering an isothermai droplet impact and then modifying the flow dynamic rnodel to include heat transfer and simultaneous solidification. The first solidification model used was a 1D model weD suited for plasma spray operations. A 2D, axisyrnmetric 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 with: capillary effects during droplet impact; simultaneous solidification and its effects on arresting droplet spread; and droplet-substrate thermal contact resistance, heat transfer to the substrate, and their effects on the transient deformation and solidification of the splat. The model developed in this study, however, considered capillq effects at both liquid-substrate and liquid-solid interfaces, simulated sirnultaneous solidification and heat trader to the substrate during the impact dynamics, and considered thermal contact resistance at the surface of the substrate. By using numerical predictions in conjunction with andytical rnodels, the above effects on the impact dynamics have been extensively studied. A finite-difference numerical solution of the Navier-Stokes and the Volume-of-Fluid (VOF) equations based on the SOLA-VOF algorithm was employed to study the isothermal droplet impact on a solid surface. Impact of water droplets on a flat, solid surface was studied using numerical simulations and the results were compared with amilable experimental observations (201. Through this cornparison, the influence of surface tension and contact angle on the impact dynamics of a water droplet falling onto a flat stainless steel surface was investigated. Cornparison of computer generated images with photographs showed that the numerical analysis accurately predicted droplet shape during deformation. Liquid-solid contact angles were varied in experiments by adding traces of a surfactant, sodium dodecyl sulphate (SDS), to water'. Adding surfactant up to 1000 ppm, which is well below the critical micelle concentration (CMC) of an aqueous solution of SDS, did not affect droplet spreading significantly; however, it changed droplet shape during recoil. This phenornenon was attributed to inertia dominating droplet spread and capillary forces dominating droplet recoil. Adding surfactant did not appeûr to reduce dynamic surface tension; using a constant value of surface tension in the model, equd to that of pure water, gave results that best

'It should be mentioned that the purpose of considering a surfactant solution in this work was not to study surfactants but to study the effects of contact angle on droplet impact dynarnics. When studying liquid- solid contact angles pure liquids should be considered, however, there is not much information avdable on dynamic contact angles for different pure liquid droplets irnpacting a substrate. Therefore, available experimental results for dynamic contact angles of water droplets wit h different surfactant concentrations [20] were considered in this work; the rneasured dynamic contact angle values were used in the contact line boundary condition in the numericd model. agreed with experimental observations. When dynamic contact angle values were used in the

numerical model, accurate predictions were obtained for droplet diameter during spreading

and at the static state. The model overpredicted droplet diameters during recoil. When the

contact angle was assumed fixed, equal to the measured static value, model predictions were

las accurate. The discrepancy between results obtained using static and dynamic contact

angles was least for pure water drops, where the static and advancing contact angles had

values close to each other. Through on analytical model (that of Qiao and Chandra given in Ref. [59]), capillary effects were found to be negiigible during droplet impact if We > fi. In the presence of a solid phase, the Navier-Stokes and VOF equations were modified

to account for a moving and somewhat arbitrary shaped liquid-solid interface. This modi-

fication was done using a two phase continuum model based on the improved fized uelocity

technique. The hydrodynamic rnodel was then rnodified to account for fluid deformation and simultaneous solidification using a one-dimensional solut ion of the energy equation which is

well suited to the droplet-impact and solidification in plasma spray processes. Thermal contact resistance was permitted in the model. The modified model was used to simulate the

impact and simultaneous solidification of superheated tin droplets on an alumina substrate.

The results were compaxed with available experimental data [NI; close agreement was obtained when a thermal contact resistance of 1.2 x 10-6 m2K/w was assumed. When the thermal contact resistance was around 1 x m2K/W, the final splat diameter was extremely sensitive to the magnitude of the contact resistance. Sirnultaneous solidification was found to cause a reduction in the final splat diameter. This effect was more pronounced for high Re and/or high Ste numbers. Previous studies [15, 49, 501 in which solidification was considered only after the completion of droplet spreading are only valid for low Re andior low Ste cases. Mode1 predictions were then obtained for typical RF and DC plasma spray processes. The spreading process in DC plasma spraying was approximately four times faster than in RF plasma spraying, a factor which is equal to their impact velocity ratio. The time required for spreading and solidification in DC plasma spray process was typically one order of magnitude smaller than in RF plasma spray process. Compared to RF plasma spraying, the solidified splats in DC plasma spraying had larger diameters and were of uniform t hickness.

A complete numerical solution of the Navier-Stokes and energy equations, based on a modified SOLA-VOF method, was used to model droplet deformation and simultaneous solidification and heat trander to the substrate. The phase change problem was treated by an axisymmetric enthalpy method. The modification of the fluid dynarnic equations was based on the improved fized velocity technique. Thermal contact resistance at the substrate was permitted. The numericd results were compared with amilable experimental measurements [52] for tin droplets (2.1 mm diameter) impacting with a velocity of 1.6 m/s on a stoinless steel whose temperature was varied from 25°C to 240°C. Cornparison of cornputer generated images of deforming droplets with experimental photographs showed that the model correctly modeled droplet shape during the impact. The value of contact resistance between the tin droplet and stainless steel substrate was estimated by matching numerical predictions of substrate temperature with experimental measurements. Reasonably accurate simulations of droplet impact dynamics could be done using a constant value of contact resistance equal to 5 x m2K/W. However, predictions of the maximum splat diameter were sensitive to the value assumed. Therefore, accurate information regarding thermal contact resistance during the early stages of droplet impact is required to model droplet impact and solidification. The solidified loyer was thickest at the center of the splat, which first contacted the surface, and dong its edge, which was nearest to the surrounding cold plate. The heat flux from the droplet to the substrate increased with radial distance from the center. Good agreement was obtained between the results of the enthalpy model with those obtained from the ID model of solidification for RF and DC plasma spray conditions.

This comparison confirmed that the 1D solidification model results in a good estimate of typical cases Do [mm] h [ni/s] Re We Ste Pr @ water droplets 2.0 1.O 2334 27.3 - - 0.56 -

tin droplets 2.1 1.6 12200 71 0.33-0.83 0.014 0.64 4.8-7.7

RF plasma spray 0.05 50 245 705 3.2 8.7 45.0 0.5

DC plasma spray 0.05 200 975 11305 3.2 8.7 362 0.5

Table 6.1: Dimensionless numbers for typical cases considered in this study. the splat shape and solid layer configuration for plasma spray operations. Through a simple analytical mode1 (that of Bhola and Chandra given in Ref. [110]), the effect of solidification on droplet impact dynamics was found to be negligible if @te/ PT-« 1. This result was in accordance wi t h the numerical hding for the solidification effect . The effects of processing parameters (such as impact velocity, droplet diameter, material properties, initial substrate temperature, etc.) as well as the effects of capillary and simultaneous solidification on the impact dynamics was studied by using the concept of dimensionless numbers. These numbers are Reynolds (Re), Weber ( We), Stefan (Ste) and

Prandtl ( Pr) numbers defined as

Tw,i) u Ste = ~(Tmp- , Pr=- Hf a1 Table 6.1 summarizes the values of these numbers for typical cases considered in this study. These cases cover a wide range of Re, We, Ste and Pr numbers. For the two sets of water and tin droplets, where numerical predictions were compaxed and verified with âvailable experimental results, We is lower than 6and, therefore, capillary effects are important. For typical RF and DC plasma spray operations (alumina droplets), however, We is rnuch larger than 6;for these cases capillazy effects are negligible, i.e. no knowledge of contact angle is required in studying the impact dynamics of plasma spray conditions. Checking soiidification effects on droplet spread, we observe that the value of 4% for tin droplets is almost one order of magnitude higher than its value for RF and DC plasma spray cases. This indicates that for tin droplets considered in this study, simultaneous solidification considerably affects the impact dynamics and maximum droplet spread. For

typical plasma spray cases, however, solidification effects are much lower.

When capillary and solidification effects are more important the numerical modeling of

the problem is more chdenging; a fine and uniform computational mesh, therefore, must

be used. Verification of the model results for droplet impact cases with high capillary and solidification effects (water and tin droplets), therefore, verifies its resuits for the cases where t hese effects are less important (plasma spray cases).

Future work on the numerical model developed in this study requires further improve- ments to: the contact line condition, the free-surface boundary condition in presence of tangentid stresses caused by a gas flow around the droplet, the consideration of material properties' temperature dependency, and the density change in presence of solidification.

Materid properties such as density, viscosity, and surface tension vary with temperature especially neas the melting point; the model, therefore, may be modified to take these effects into consideration. The free-surface boundôry condition has to be modified if there is a gas flow around the droplet or if there is a surface tension gradient at the free-surface (due to the surface tension's ternperat ure dependency ) ; the Iat ter which causes a shear stress discont i- nuity at the surface is called Marangoni effect. The model, as is, considers 2D, axisymrnetric problems such as vertical droplet impact. A three dimensional extension of the model is, therefore, required when simulating a general 3D problem such as inclined droplet impact or multiple simultaneous droplet impacts. References

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43-50, July 1994.

[2] R.E. Lewis and A. Lawley, "Spray forming of metallic materials: an overview," P/M in Aerospace and Defense Technologies, Metal Powders Industries Federation, Princeton, 173-184 (1991).

[3] C.-A. Chen, J.-H. Sahu, J.-H. Chun and T. Ando, "Spray forming with uniform

droplets," Science and Technology of Rapid Solidification and Processing, edited by M. A. Otooni, Kluwer Academic Publishers, Netherlands, 123-134 ( 1995).

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Grid Size Effect

The grid size effect was investigated by perfonning calculations for the case of RF plasma spray operation (as described in Chapter 5) with a coarse mesh size, a medium mesh size, and a fine mesh size. In the coarse mesh size, the computational domain was discretized such that the droplet had 10 cells in radius. For medium and fine mesh sizes, the droplet had 20 and 30 ceils in radius, respectively. The results for the spread factor 5, and the dimensionless thickness h* (=hl Do,where h is the splat thickness at the center) are shown in Figs. A.1 and A.2. It can be seen that as the mesh size increases (Le., approaches the fine mesh) its effect becomes insignificant. The dimensionless thickness h' is practically insensitive to the three mesh sizes (Fig. A.2); whereas, the spread factor 5 exhibits a relatively greater sensitivity as shown in Fig. A.1. To study the grid size effect on the enthalpy model, the results for substrate temperat ure at the center during the process and substrate temperature at a certain time elapsed after the impact are given in Figs. A.3 and A.4, respectively. These two figures show the sarne behavior with respect to the grid size effect; the effect becomes insignificant as the mesh sizes increases. Therefore, the increased computational effort (fiom medium to fine mesh size) is not justified since no benefit is gained. As a compromise, we can use the medium mesh size and accept the obtoined accuracy for its reasonable cornputational effort. Figure A.1: The effect of grid size on evolution of spread factor during impact of an alumina droplet in a typical RF plasma spray operation (Case 1 of Table 4.1). Coarse, medium and fine mesh sizes had 10, 20 and 30 ceils per radius, respectively. O 1 2 3 4 5 Dimensionless the, t *

Figure -4.2: The effect of grid size on the dimensionless thickness h' during impact of an alumina droplet in a typicd RF plasma spray operation (Case 1 of Table 4.1). Coarse, medium and fine mesh sizes had 10, 20 and 30 cells per radius, respectively.

0.00 0.02 0.04 0.06 0.08 0.10 Radius (mm)

Figure A.4: The eifect of grid size on substrate temperature at 4.4 ps elapsed after the impact of an alumina droplet in a typical RF plasma spray operation (Case 1 of Table 4.1).

Coarse, medium and fine mesh sizes had 10, 20 and 30 cells per radius, respectively. IMAGE EVALUATION TEST TARGET (QA-3)

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