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Heat flow model of rapid solidification with nonhomogeneous thermal contact

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Abstract

A heat flow model is presented of the solidification process of a thin melt layer on a heat conducting substrate. The model is based on the two-dimensional heat conduction equation, which was solved numerically. The effect of coexisting regions of good and bad thermal contact between foil and substrate is considered. The numerical results for thermal parameters of the Al-Cu eutectic alloy show considerable deviations from one-dimensional solidification models. Except for drastic differences in the magnitude of the solidification rate near the foil-substrate interface, the solidification direction deviates from being perpendicular to the substrate and large lateral temperature gradients occur. Interruption of the thermal contact may lead to back-melting effects. A new quantity, the effective diffusion length, is introduced which allows some conclusions to be drawn concerning the behaviour of the frozen microstructure during subsequent cooling.

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Abbreviations

ā i ,a i :

Thermal diffusivityā i =\(\bar k\) i /c i ρ i ,a i =ā i /ā 1

c i :

Specific heat capacity

d :

Foil thickness

D :

Solid state diffusion coefficient

ex, ez :

Unit vectors

H :

Latent heat of fusion

h α,h β :

Foil-substrate heat transfer coefficients

i :

Index: 1, melt; 2, solidified foil; 3, substrate

\(\bar k\) i ,k i :

Thermal conductivityk i =\(\bar k\) i /\(\bar k\) 1

n:

Normal unit vector

Nu α,Nu β :

Nusselt numbers for regions of badNu(x,τ) and good thermal contact, respectivelyNu α =h α Nu d /\(\bar k\) 1,,Nu(x, τ)=h(x,τ)d/\(\bar k\) 1

R :

Universal gas constant

\(\bar s\), s:

Position of the liquid-solid interface ¯s/d=s=s xex+s zez

\(\dot s\) :

Local solidification rate\(\bar s\)/d = s =s xex +s zez

t :

Real time

T i :

Temperature field

T 0 :

Ambient temperature

T f :

Melting temperature

u i :

Dimensionless temperature fieldu i (x, z,τ)=T i (x,z,τ)/T f

u 0 :

Dimensionless ambient temperatureu 0=T 0/T f

\(\dot u\) i :

Local cooling rate within the foil\(\dot u\) i = du i /dτ

W :

Stefan numberW=H/c 1 T f

\(\bar x\),x :

Cartesian coordinate parallel to the foil-substrate interfacex=\(\bar x\)/d

\(\bar x\) 0,x 0 :

Lateral extension of foil sectionx 0=\(\bar x\) 0/d

\(\bar x\) 1,x 1 :

Lateral contact lengthx 1=\(\bar x\) 1/d

\(\bar z\),z :

Cartesian coordinate perpendicular to the foil-substrate interfacez=\(\bar z\)/d

\(\bar z\) 0,z 0 :

Substrate thicknessz 0=\(\bar z\) 0/d

ΔE :

Activation energy of diffusion

ΔT :

Initial superheat of the melt

Δu :

Dimensionless initial superheat ΔuT/T f

θ(x):

Step function\(\theta (x) = \left\{ {\begin{array}{*{20}c} 0 \\ 1 \\ \end{array} {\text{ if }}\begin{array}{*{20}c} {x{\text{< 0}}} \\ {x{\text{ }} \geqslant {\text{ 0}}} \\ \end{array} } \right.\)

λeff :

Dimensionless effective diffusion length

ρ i :

Mass density

τ :

Dimensionless timeτ= 1/d 2

τ f,τ f(x, z):

Total and local dimensionless freezing time, respectively

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Sellger, R., Löser, W. & Neumann, W. Heat flow model of rapid solidification with nonhomogeneous thermal contact. J Mater Sci 19, 2145–2152 (1984). https://doi.org/10.1007/BF01058090

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