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 the
surface of a plate shaped additive immersed in an agitated melt bath. It exhibits the dependence
of this occurrence on independent parameters-the initial temperature, θai of the additive, the bath
temperature, θb , the Biot number, Bi the property ratio, B and the Stefan number, St and yields
closed-form solutions for time variant frozen layer thickness, ξ around the additive and heat
penetration depth, η in the additive. In the solutions, B, Bi, θb and θai appear as a conduction
factor, Cof that ranges from 0 to ∞. The frozen layer thickness per unit St with respect to Cof
takes 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 its
subsequent melting, τct is 4/3 when the heat penetration depth reaches the central axis of the
plate additive, η=1. When Cof →0 signifying highly agitated bath (h→∞) or additive preheated
to the freezing temperature of the bath material, no freezing of the bath material occurs. For the
bath at the freezing temperature of the bath material, the frozen thickness is also obtained. The
model is validated by reducing the present problem to heating of the plate additive subjected to a
constant temperature maintained at the freezing temperature of the bath material.