Freezing and melting of a bath material onto a cylindrical solid additive in an agitated bath

In melting and assimilation of a cylindrical shaped additive in an agitated hot melt bath during the process of preparation of cast iron and steel of different grades, an unavoidable step of transient conjugated conduction-controlled axisymmetric freezing and melting of the bath material onto the additive immediately after its dunking in bath occurs. Decreasing the time of completion of this step is of great significance for production cost reduction and increasing the productivity of such preparations. Its suitable mathematical model of lump-integral type is developed. Its nondimensional format indicates the dependence of this step upon independent nondimensional parameters- the bath temperature, θb the modified Biot number, Bim denoting the bath agitation, the property-ratio, B and the heat capacity-ratio, Cr of the melt bath-additive system, the Stefan number, St pertaining to the phase-change of the bath material. The model provides the closed-form expressions for both the growth of the frozen layer thickness, ξ onto the additive and the heat penetration depth, η in the additive. Both are functions of these parameters, but when they are transformed to the growth of the frozen layer thickness with respect to the heat capacity ratio per unit Stefan number; and the time per unit property-ratio, B, their expressions become only a function of single parameter, the conduction factor, Cof consisting of the parameters, B, Bim and θb. The closed-form expression for the growth of the maximum thickness of the frozen layer, its time of growth, the time of the freezing and melting; the heat penetration depth are also derived. When the heat penetration depth approaches the central axis of the cylindrical additive in case of the complete melting of the frozen layer developed Cof≤11/72. It is found that the decreasing Cof reduces both the time of this unavoidable step and the growth of the maximum frozen layer thickness and at Cof=0, the frozen layer does not form leading to zero time for this step. If the bath is kept at the freezing temperature of the bath material, only freezing occurs. To validate the model, it is cast to resemble the freezing and melting of the bath material onto the plate shaped additive. The results are exactly the same as those of the plate.