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Modeling of Rapid Solidification with Undercooling Effect During Droplet Flattening on a Substrate in Coating Formation

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Abstract

The coating deposit on the substrate in thermal spray coating process develops by solidification of individual molten particle which impacts, flattens and solidifies on the surface of the substrate. Droplet flattening and solidification typically involves rapid cooling. In this paper, a model for non-equilibrium rapid solidification of a molten droplet spreading onto a substrate is presented. Transient flow during droplet impact and its subsequent spreading is considered using the volume of fluid surface tracking method which was fully coupled with the rapid solidification model. The rapid solidification model includes undercooling, nucleation, interface tracking, non-equilibrium solidification kinetics and combined heat transfer and fluid flow as required to treat a non-stagnant splat formed from droplet flattening. The model is validated with the literature results on stagnant splats. Subsequently, using the model the characteristics of the rapidly solidifying interface for non-stagnant splat, such as interface velocity and interface temperature, are described and the effect of undercooling and interfacial heat transfer coefficient are highlighted. In contrast to the stagnant splat, the non-stagnant splat considered in this study displays interesting features in the rapidly solidifying interface. These are attributed to droplet thinning and droplet recoiling that occur during the droplet spreading process.

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Abbreviations

c :

Specific heat (J kg−1 K−1)

C :

Constant related to Darcy source term (kg m−3 s−1)

D 0 :

Initial droplet diameter (m)

f s :

Weight fraction of solid

f l :

Weight fraction of liquid

F :

Volume of fluid function

F vol :

Continuum surface tension force (N m−3)

\(\vec{g}\) :

Acceleration due to gravity vector (m s−2)

\(\Delta G_{\text{am}}\) :

Activation energy for molecular migration (\({\text{J}}\,{\text{mol}}^{ - 1}\))

h :

Heat transfer coefficient (\({\text{W}}\,{\text{m}}^{ - 2} \,{\text{K}}^{ - 1}\))

H :

Enthalpy (J)

\(\Delta H_{\text{m}}\) :

Heat of fusion (\({\text{J}}\;{\text{mol}}^{ - 1}\))

k :

Thermal conductivity (W m−1 K−1)

L d :

Latent heat of fusion (J kg−1)

P :

Pressure (Pa)

T :

Temperature (K)

T i :

Interface temperature (K)

t :

Time (s)

\(t_{\text{s}}^{*}\) :

Dimensionless spreading time

\(T_{\text{m}}\) :

Equilibrium melting temperature (K)

\(T_{\text{N}}\) :

Nucleation temperature (K)

\(T_{0}\) :

Initial temperature (K)

\(\Delta T_{\text{i}}\) :

Interface undercooling (K)

\(U_{0}\) :

Droplet’s initial impact velocity (m s−1)

\(\vec{u}\) :

Continuum mixture velocity vector (m s−1)

\(V_{\text{i}}\) :

Interface velocity (m s−1)

μ :

Dynamic viscosity (kg m−1 s−1)

\(\mu_{k}\) :

Linear kinetic coefficient (\({\text{m s K}}^{ - 1}\))

\(\theta\) :

Contact angle

\(\beta\) :

Kinetics constant

\(\kappa\) :

Boltzmann’s constant (\({\text{J}}\;{\text{K}}^{ - 1}\))

\(\delta\) :

Solid–liquid interface thickness (m)

ξ:

Maximum spreading ratio

air:

Air

eff:

Effective

i:

Solid–liquid interface

l:

Liquid

d:

Droplet

l, d:

Liquid droplet

0:

Initial

sub:

Substrate

s, d:

Solid droplet

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Authors and Affiliations

  1. Department of Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur, 208016, India

    Rajesh Kumar Shukla, Virendra Patel & Arvind Kumar

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  1. Rajesh Kumar Shukla
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  2. Virendra Patel
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Correspondence to Arvind Kumar.

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Shukla, R.K., Patel, V. & Kumar, A. Modeling of Rapid Solidification with Undercooling Effect During Droplet Flattening on a Substrate in Coating Formation. J Therm Spray Tech 27, 269–287 (2018). https://doi.org/10.1007/s11666-017-0666-y

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  • DOI: https://doi.org/10.1007/s11666-017-0666-y

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