Empirical correlations for natural convection, Δ and keff

• Nusselt number (Nu) correlations are used to quantify convection at solid–liquid interface.
• Correlations for simultaneous heat and mass transfer, Nu(Gr,Pr, Sc), are presented.
• Empirical correlations are used to determine Δ(Nu,Pe), and keff. (k0, Δ).
• For CZ growth of 20 cm diameter Si crystals, natural convection is dominant below ~5 RPM.

Natural convection in semiconductor melts is quantified using empirical correlations for Nusselt numbers, Nu, developed in laboratory experiments with liquid metals. Nu is a dimensionless coefficient used to quantify the complex boundary condition of convection, involving advection and diffusion. A dozen of empirical correlations for heat transfer Nusselt numbers, Nu(Gr,Pr), was transformed into correlations for simultaneous heat and mass transfer, Nu(Gr,Pr, Sc), and used to calculate the parameter Δ=Pe/Nu and effective segregation coefficient, keff(Pe, Nu, k0), for horizontal Bridgman and Czochralski growth.