KSME Journal

, Volume 7, Issue 3, pp 203–212 | Cite as

Numerical computation for heat transfer in crystal growth

  • Hiroyuki Ozoe
Article
Received:

Abstract

The numerical analyses for crystallization processes are selected and summarized from the author’s recent works. The importance of heat transfer may be noted in the phase change manufacturing processes from liquid to solid. The effect of convection on the curved interface may be also noted in the floating zone crystallization process. Specifically, the processes presented are as follows. Three-dimensional cylindrical coordinate model for a Czochralski bulk flow of liquid metal in a horizontal or a vertical magnetic field is presented and numerically solved for selected conditions. Then, a floating zone crystallization method is modeled by an axisymmetric coordinate and then solved by an isoparametric finite element method for curved solid/melt interfaces and a gas/melt interface. Sample computational results are also presented.

Key Words

Crystal Growth Czochralski Flow Heat Transfer Magnetic Force Material Processing 

Nomenclature

B0

Bond number=Δϱgr 0 2/σ=Buoyancy force/Surface tension force

Bi*

Biot number=h eq Y 0/k=convective heat transfer rate/conductive rate

b

Magnetic induction vector, kg s−2A−1

C

Dimensionless concentration

Cp

Specific heat, J/kg K

e

Electric field, m kg s−3 A−1

f

Lorentz force, N m−3

Gr*

Modified Grashof number=gβr 0 5 q/(kν 2) force/inertial force

g

Acceleration due to gravity, m/s2

H

Dimensionless height of a crucible

Ha

\(Hartmannumber = \sqrt {\sigma _e /\mu } B_0 h = Lorentz\) force/viscous force

h

Height of a crucible, m

heq

Equivalent heat transfer coefficient, J/m2Ks

j

Electric current density, Am−3

k

Thermal conductivity, J/msK

ls

Radius of a crystal rod, m

Ma

Marangoni number=(C p r 0 3 q/νk 2)(− ∂σ/∂θ)=surface tension force/viscous force

Nu

Nusselt number

P

Lagrange multiplier=Δpr 0/σ

Δp

Dimensionless Lagrange multiplier, N/m2

Pr

Prandtl number=ν/α

q

rate of heat generation, J/m3s

R

Dimensionless radius=r/r 0

Ra

Rayleigh number=Gr·Pr

Re

Reynolds number=l 2 s ω/ν

r

Radial coordinate, m

r0

Radius of a rod, m

t

Time, s

t0

r 0 2/α, s

T

Dimensionless temperature=(θ−θ m )/θ a

Tw

Dimensionless wall temperature

T∞

Dimensionless ambient temperature

U

Dimensionless radial velocity=u/u 0

u

Radial velocity, m/s

u0

α/r 0, m/s

v

Velocity component in the circumferential direction

V

Dimensionless velocity component in the circumferential direction=v/u 0

V

Velocity vector, m/s

W

Dimensionless axial velocity=w/u 0

w

Axial velocity component, m/s

z

Bimensionless axial coordinate=z/r 0

z

Axial coordinate, m

Greek letters

α

Thermal diffusivity, m2/s

β

Volumetric coefficient of expansion, 1/K

Δϱ

Density difference between gas and liquid, kg/m3

θ

Temperature, K

θa

Reference temperature [K]

θm

Melting temperature, K

θw

Wall temperature, K

θ∞

Ambient temperature, K

μ

Viscosity, kg/ms

ν

Kinematic viscosity, m2/s

ϱ

Density, kg/m3

σ

Surface tension, N/m

τ

Dimensionless time=t/t 0

λ

Thermal conductivity, Wm−1K−1

ω

Angular velocity of a crystal rod, rad s−1

ϕ

Circumferential coordinate, rad

φf

Contact angle, rad

Ψ

Dimensionless stream function=ψ/α,−

ψ

Stream function, m2/s

Ω

Dimensionless vorticity=ζr 0 2 /α

ζ

Vorticity, 1/s

σe

electric conductivity, m−3kg−1 s3 A2

Subscript

0

Reference value for a dimensionless variable

c

Cold wall

h

Hot wall

Superscript

T

Transpose of the vector

Operator

∇

[∂(r)/r∂r, ∂/r∂ϕ, ∂/∂z] (dimensional or dimensionless)

∇2

∂2/∂R 2+(∂/∂R)/R+∂2/R 2∂φ2/∂Z 2

D/Dτ

∂/∂t+U∂/∂R+V∂/R∂ƒ+W∂/∂Z

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References

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Copyright information

© The Korean Society of Mechanical Engineers (KSME) 1993

Authors and Affiliations

  • Hiroyuki Ozoe
    • 1
  1. 1.Institute of Advanced Material StudyKyushu UniversityJapan

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