On a nonlinear stefan problem arising in the continuous casting of steel

Summary The continuous casting process employed in the steel industry is a many faceted big industrial problem which has given rise to many sub-problems. Here, we examine the problem involving the determination of the solid-liquid steel interface and we develop and extend a previously proposed model, which incorporates heat transfer through two layers of solid and liquid mould powder and the interface between the solid powder and the mould wall. The problem simplifies to the classical Stefan problem except that the condition on the boundary is nonlinear. Integral formulation procedures are used to establish the normalized pseudo steady state temperature as an upper bound to the normalized actual temperature. The pseudo steady state approximation yields an upper bound on the interface position, which an independent numerical enthalpy scheme confirms to be an extremely accurate approximation for the parameter values occurring in practice. The present work is important since it provides a simple method for the prediction of the solid-liquid steel interface and a bounding procedure which can be used to validate other estimates.List of symbols D flux thickness atz^*=0 - H enthalpy - L latent heat of steel - M the half thickness of the cast steel - Q heat flux - R interface thermal contact resistance - S_m^* melting temperature of steel - T^* temperature - T normalized temperature - T_m^* melting temperature of mould powder - T^* temperature of cooling water - T_w^* temperature on mould wall - T_u^* temperature of solid flux on its interface with mould wall - T₀^* temperature on casting surfaceT^*(0,z^*) - U casting speed - X^*(z^*) physical coordinate of the steel phase change boundary - X(z) non-dimensional coordinate of the steel phase change boundary - c specific heat of steel - h(z^*) thickness of liquid flux layer - k thermal conductivity of steel - ks thermal conductivity of solid flux layer - k_l thermal conductivity of liquid flux layer - m surface heat transfer coefficient - s(z^*) thickness of solid flux layer - t time - , , positive constants given by (3.2) - constant given by (3.5) - coefficient of linear thermal expansion of steel - angle shown in Figure 2 - positive constant defined by (M-D)/2 - (z) positive parameter - (z^*) amount of contraction of steel - density - (z) positive parameter used in (5.7) and (5.8)