Skip to main content
SpringerLink
Log in

Numerical computation for heat transfer in crystal growth

  • Published:
KSME Journal Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Abbreviations

B 0 :

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

C p :

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

h eq :

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

r 0 :

Radius of a rod, m

t :

Time, s

t 0 :

r 0 2/α, s

T :

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

T w :

Dimensionless wall temperature

T ∞ :

Dimensionless ambient temperature

U :

Dimensionless radial velocity=u/u 0

u :

Radial velocity, m/s

u 0 :

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

α:

Thermal diffusivity, m2/s

β:

Volumetric coefficient of expansion, 1/K

Δϱ:

Density difference between gas and liquid, kg/m3

θ:

Temperature, K

θ a :

Reference temperature [K]

θ m :

Melting temperature, K

θ w :

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

φ f :

Contact angle, rad

Ψ:

Dimensionless stream function=ψ/α,−

ψ:

Stream function, m2/s

Ω:

Dimensionless vorticity=ζr 20 /α

ζ:

Vorticity, 1/s

σ e :

electric conductivity, m−3kg−1 s3 A2

0:

Reference value for a dimensionless variable

c :

Cold wall

h :

Hot wall

T :

Transpose of the vector

∇:

[∂(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

References

  • Abe, T. et al., 1962, “A Monthly Publication of the Japan Society of Applied Physics,” Vol. 31, pp. 58.

    Google Scholar 

  • Brown, R.A., 1988, “Theory of Transport Processes in Single Crystal,” AIChE Journal, Vol. 34, pp. 881–911.

    Article  Google Scholar 

  • Coriell, S.R. and Cordes, M.R., 1977, “Theory of Molten Zone Shape and Stability,” J. Crystal Growth, Vol. 42, pp. 466–472.

    Article  Google Scholar 

  • Duranceau, J.L. and Brown, R.A., 1986, “Thermal Capillary Analysis of Small-Scale Floating Zones; Steady State Calculations,” J. Crystal Growth, Vol. 75, pp. 367–389.

    Article  Google Scholar 

  • Hirt, C.W. and Cook, J.L., 1972, “Calculating Three-Dimensional Flows around Structures and over Rough Terrain,” Journal Computational Physics, Vol. 10, pp. 324–340.

    Article  Google Scholar 

  • Hjelming, L.N. and Walker, J.S., 1986, “Melt Motion in a Czochralski Crystal Puller with an Axial Magnetic Field: Isothermal Motion,” Journal of the Fluid Mechanics, Vol. 164, pp. 237–273.

    Article  Google Scholar 

  • Imai, K. and Ozoe, H., 1992, “Numerical Computation of Fz Crystallization with Internal Heat Generation,” Kagaku Kogaku Saga Meeting.

  • Kobayashi, S., 1986, “Effect of an Axial Magnetic Field on Solute Distribution in Czochralski Grown Crystals-a Theoretical Analysis,” Journal of Crystal Growth, Vol. 75, pp. 301–308.

    Article  Google Scholar 

  • Langlois, W.E., 1982, “A Parameter Sensitivity Study for Czochralski Bulk Flow Silicon,” Journal of Crystal Growth, Vol. 56, pp. 15–18.

    Article  Google Scholar 

  • Mihelcic, M., Wingerath, K. and Pirron, C., 1984, “Three-Dimensional Simulations of the Czochralski Bulk Flow,” Journal of Crystal Growth, Vol. 69, pp. 473–488.

    Article  Google Scholar 

  • Ozoe, H. and Toh, K., 1990, “Numerical Computation for a Czochralski Bulk Flow of Liquid Metals under a Vertical External Magnetic Field.” The 9th International Heat Transfer Conference, Jerusalem, Vol. 6, pp. 311–316.

    Google Scholar 

  • Ozoe, H. and Tanaka, S., 1991, “Finite Element Analyses of a Simplified Model for Floating Zone Crystallization Method,” Proc. of the 4th International Symposium on Transport Phenomena in Heat and Mass Transfer, Sydney.

  • Smith, M.K., 1986, “Thermocapillary and Centrifugal-Buoyancy-Driven Motion in a Rapidly Rotating Liquid Cylinder,” J. Fluid Mech., Vol. 166, pp. 245–264.

    Article  MATH  Google Scholar 

  • Toh, K. and Ozoe, H., 1992, “Three-Dimensional Czochralski Flow of Liquid Metal in a Lateral Magnetic Field.” 1st International Conference on Transport Phenomena in Processing, Hawaii.

  • Utech, U.P. and Fleming, M.C., 1966, “Elimination of Solute Banding in Indium Antimonide Crystals by Grown in a Magnetic Field,” Journal of Applied Physics, Vol. 37, pp. 2021–2024.

    Article  Google Scholar 

  • Witt, A. F., Hertman, C.J. and Gatos, H.C., 1970, “Czochralski-Type Crystal Growth in Transverse Magnetic Fields, The Observation of Magnetic Domain Structures of the Intermetallic Compounds SnCo5, LaCo5 and CeCo5,” Journal of Materials Science, Vol. 37, pp. 822–824.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Advanced Material Study, Kyushu University, Japan

    Hiroyuki Ozoe

Authors
  1. Hiroyuki Ozoe
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ozoe, H. Numerical computation for heat transfer in crystal growth. KSME Journal 7, 203–212 (1993). https://doi.org/10.1007/BF02970965

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02970965

Key Words

Search

Navigation