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Modelling Transport Phenomena during Spreading
and Solidification of Droplets in Plasma
Projection
Dominique GOBIN CNRS France
NGU Seminar
Nova Gorica (November 5, 2009)
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Contents

• Motivation
• Equations
• Isothermal spreading
• Spreading with solidification
• Perspective

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Building up a coating
The functional properties of the coating depend
on the cohesion and adhesion of the
splats
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Characteristic times Spatial scales
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Modelling issues
Modelling the plasma
Spreading and solidification of droplets on a
cold substrate
In-flight melting (vaporization) of particles
Building-up the coating
Define and control the process parameters
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Droplet spreading and solidification
Impacting Particle
T0 gt Tm V0 ? 100 m/s 20 lt d0 lt 50 µm
Tsplat and dsplat time evolution
Ts
Substrate
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2. Equations
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Modelling spreading
Pure fluid dynamics problem. The substrate is a
boundary condition
Mass Conservation
Momentum Conservation
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Modelling spreading
Non-dimensionalizing variables (choosing
reference values d0, V0, etc) yields the
dimensionless parameters of the problem
Mass Conservation
Momentum Conservation
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Modelling spreading with solidification
Coupling the equations of fluid dynamics with
the heat transfer equations
Mass Conservation
Momentum Conservation
Energy Conservation - in the splat -
in the substrate
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Modelling spreading with solidification
During solidification two phases (solide and
fluid) are present. A phase function is defined

1 if liquid 0 if solid

Mass Conservation
Momentum Conservation
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Modelling spreading with solidification
Heat transfer and enthalpy formulation
Energy Conservation
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Conservation equations
Mass Conservation
Momentum Conservation
Energy Conservation
1 liquid 0 solid
Liquid Fraction
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Physical parameters of the problem
Parameters of the particles at impact Nature
Size Velocity
Temperature and state of
melting Parameters of the substrate
Nature Rugosity Initial temperature Surface
chemistry (wettability)
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Spreading and solidification of a splat
1. Operation parameters

• Dynamic contact angle

2. Adjustable parameters
Splat
Substrate

• Contact thermal resistance

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Numerical tool
Simulent-Drop a software developed at the
University of Toronto (J. Mostaghimi et al.)
Main hypotheses

• Newtonian fluid
• Constant properties (surface tension, contact
  resistance, conductivities, viscosity, )
• Equilibrium solidification

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Numerical tool main features

• Finite difference method
• Fixed regular grid (Eulerian formulation)
• Boundary condition using dynamic contact angles
• Interface reconstruction VoF method
• 3-D Geometry (computational domain a quarter
  of the domain)

Typical grid
Symmetry
Full domain
Computational domain
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Scales
Micrometric droplets (Conditions of plasma
projection)
Millimetric droplets (Free fall conditions)
Similitude ?
Re identiques We 10 à 100 fois plus grand
d
1 mm
gt 10 µm
Vimpact
1 m/s
gt 100 m/s
Characteristic times
ms
µs
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3. Isothermal spreading
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Isothermal impact of a water droplet
Simulation F. Loghmari
Water droplet spreading d0 2,75 mm , V0 1.18
m/s on soft wax q (105,95) Rioboo et al.
(2001)
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Simulation Experiments
Spreading factor d(t)/do
Reduced time t t Vo/ do
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Wettability effect
?a
Substrat
Forward dynamic contact angle
Backward angle effect (?a 105)
Forward angle effect (?r 95)
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4. Spreading with solidification
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mm-size droplet simulation
Simulation Nabil Ferguen
Copper droplet on steel substrate d 3 mm V
4 m/s Ts 25C
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Impact velocity influence
Vp8 m/s
Vp4 m/s
Vp2 m/s
Time evolution of the spreading factor
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Impact velocity influence
Vp 8 m/s
Vp8 m/s
Vp4 m/s
Vp2 m/s
Vp 4 m/s
Time evolution of the spreading factor
Vp 2 m/s
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Contact thermal resistance
Non perfect contact between the drop and a rugous
substrate gt resistance to the heat flux
temperature discontinuity at the interface
CTR Model
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Influence of the contact thermal resistance


10-5 m²K.W-1
5.10-6 m²K.W-1
2.10-6 m²K.W-1
10-6 m²K.W-1
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High contact resistance
RTC 10-5 m²K.W-1
Simulation Nabil Ferguen
Copper droplet on steel substrate d 3 mm V
4 m/s Ts 400C
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Low contact resistance
RTC 10-6 m²K.W-1
Simulation Nabil Ferguen
Copper droplet on steel substrate d 3 mm V
4 m/s Ts 400C
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Influence of the initial substrate temperature
Ti Cr Cu
To 300 K
To 673 K
From Fukumoto et al. (1995)
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Splat formation
Morphological transition temperature Tt
Alumina on steel 304L
 Splat  Pre-heated substrate Tsubgt Tt Better
adhesion (? 30 MPa)
 Splash  Cold substrate Tsublt Tt Poor
adhesion of the coating (? 4 MPa)
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Influence of the substrate temperature
Vp 4 m/s dp 2 mm T0 1100 C, Tf
1080 C
Re 23900 , We 191
Ts 1084 C
Ts 800C
Ts 400C 
Ts 25C
Pre-heating of the substrate higher final splat
diameter
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Transition Temperature ?

• Desorption of adsorbates and condensates
• Modification of wettability of the substrate
• Modification the thermal resistance
• Possible evolution of the surface state of the
  substrate

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5. Further developments
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Non equilibrium Solidification

• Basic hypothesis solidification at equilibrium
• Most models do not take into account
  undercooling, nucleation and growth
  problem of multi-scale
  (micro macro) simulation
• But in plasma projection, the cooling velocity
  measured in the experiments reaches from 106 to
  5.108 K/s
• undercooling about 0,1 to 0,2 Tm.
• ? Include rapid solidification

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Experiments on mm-size droplets
Film S. Goutier M. Vardelle
Alumina droplet on steel substrate d 5 mm V
10 m/s Ts 400C
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Thank you for your attention

• Special Thanks to
• Nabil Ferguen SPCTS Laboratory
• Simon Goutier SPCTS Laboratory
• Fahmi Loghmari FAST Laboratory

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Isothermal impact of a water droplet
Simulation F. Loghmari
Water droplet spreading d0 2,75mm , V0
1.18m/s on soft wax q (105,95) Rioboo et
al. (2001)
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