A LUMP-INTEGRAL MODEL FOR FREEZING AND MELTING OF A BATH MATERIAL ONTO A PLATE SHAPED SOLID ADDITIVE IN AN AGITATED BATH

Abstract

A lump integral model is developed for freezing and melting of the bath material onto thesurface of a plate shaped additive immersed in an agitated melt bath. It exhibits the dependenceof this occurrence on independent parameters-the initial temperature, θai of the additive, the bathtemperature, θb , the Biot number, Bi the property ratio, B and the Stefan number, St and yieldsclosed-form solutions for time variant frozen layer thickness, ξ around the additive and heatpenetration depth, η in the additive. In the solutions, B, Bi, θb and θai appear as a conductionfactor, Cof that ranges from 0 to ∞. The frozen layer thickness per unit St with respect to Coftakes time τcmax=1/3 for its maximum growth whereas this maximum thickness ξ*cmax becomes(1- θai)/3. The total time of the growth of the maximum frozen layer thickness with itssubsequent melting, τct is 4/3 when the heat penetration depth reaches the central axis of theplate additive, η=1. When Cof →0 signifying highly agitated bath (h→∞) or additive preheatedto the freezing temperature of the bath material, no freezing of the bath material occurs. For thebath at the freezing temperature of the bath material, the frozen thickness is also obtained. Themodel is validated by reducing the present problem to heating of the plate additive subjected to aconstant temperature maintained at the freezing temperature of the bath material.