Global heat transfer analysis in Czochralski silicon furnace with radiation on curved specular surfaces

Numerical analysis of global heat transfer with coupled thermal radiation and heat conduction is investigated in Czochralski silicon crystal growth furnace with curved diffuse and specular surfaces. The ®nite element method and the radiation element method are adopted to solve the global heat transfer and the radiative heat exchange, respectively. The emphasis focuses on the discussion of the in ̄uence of silicon surface radiative characteristics, i.e., either diffuse or specular, on the global heat transfer and the crystal growth process. When the specular character of the silicon crystal and melt surfaces is considered, it is found that the temperature of the melt is obviously decreased and the crystal pulling rate is enhanced. List of symbols A area of surface element cp speci®c heat capacity êz unit vector in the axial direction F ij absorption view factor from surface element i to surface element j F ij diffuse re ̄ection view factor from surface element i to surface element j DLs latent heat of solidi®cation n̂ normal unit vector Pe Peclet number Q heat generation rate QJ diffuse radiation heat transfer rate QX net rate of radiative heat loss QT heat generation rate of emission, Aer T q radiative heat ̄ux Rc radius of crystal St Stefan number, DLs/cpsT0 T temperature T0 temperature at melting point of silicon V pulling velocity Greek symbols e emissivity k thermal conductivity q density q diffuse re ̄ectivity q specular re ̄ectivity r Stefan-Boltzmann constant