Analytical technique for the determination of solidification rates during the inward freezing of cylinders

An analytical/experimental approach which permits the determination of solidification rates during the inward solidification of cylinders is proposed. The technique is based on a previous analytical solution that treats the generalized problem of solidification of slabs. This solution is modified by a geometric correlation to compensate for the cylindrical geometry. A number of experiments have been carried out with a special experimental set-up, designed to simulate the inward solidification of cylinders in a water-cooled mould. A series of comparisons of experimental results, numerical predictions and calculations furnished by the proposed technique were made, showing good agreement for any case examined.Nomenclature a _s Thermal diffusivity of solid metal = k _s/c _s d _s (m² sec^–1) - A _i Internal surface area of the mould (m²) - b _s Heat diffusivity of solid metal = (k _s c _s d _s ^1/2(J m^–2 sec^–1/2 K^–1) - c _s Specific heat of solid metal (J kg^–1 K^–1) - d _s Density of solid metal (kg m^–3) - h Newtonian heat transfer coefficien (W m^–2 K^–1) - H Latent heat of fusion (J kg^–1) - k _s Thermal conductivity of solid metal (W m^–1 K^–1) - q Heat flux (W m^–2) - r Radial position (m) - r _o Radius of cylinder (m) - r _f Radius of solid/liquid interface (m) - S Thickness of solidified metal (m) - S _o Thickness of metal side adjunct (m) - t Solidification time (sec) - T Temperature (K) - T _i Surface temperature (K) - T _f Freezing temperature of metal (K) - T _o Temperature of the coolant (K) - T _s Temperature at any point in the solidified metal (K) - V ₁ Volume of remaining liquid metal during the solidification (m³) - V _s Volume of solidified metal (m³) - V _T Total volume of metal in the mould (m³) - x Distance from metal/mould interface (m) - Dimensionless solidification constant.