~ ) Pergamon Int. J. Heat Mass Transfer. Vol. 38, No. 6, pp. 1009-1018, 1995 Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0017-9310/95 $9.50+0.00 0017-9310(94) 00221-5 A model of inverse segregation: the role of microporosity V. R. VOLLER and SURESH SUNDARRAJ Department of Civil Engineering, 500 Pillsbury Drive, University of Minnesota, Minneapolis, MN 55455, U.S.A. (Received 11 October 1993 and in final form 13 July 1994) Abstract--A numerical model of inverse segregation in a vertically cast unidirectionally solidified alumi- num-copper binary alloy is presented. The model predicts the solute concentration distribution up the length of the casting. The model is validated on comparison with available analytical solutions. Initial comparisons with experiments show that the model predicts a non-physical region of positive segregation in the upper part of the casting. On accounting for microporosity formation, however, the model predictions show close agreement with experimental measurements. Application of the model also demonstrates the need to correctly account for microsegregation processes. INTRODUCTION Recently, there has been much activity in the modeling ofmacrosegregation [1-4] (the redistribution of solute phases during the solidification of an initially uniform melt of a multicomponent alloy). A commonly studied macrosegregation system is the solidification of a binary material 111-3], initially at a uniform con- centration, against a vertical isothermal wall. In this system, as the solidification proceeds, the solute is rejected and subsequently redistributed by solutal and thermally driven :natural convection flows. An alter- native macrosegregation system results from the unidirectional solidification of a binary alloy, e.g. aluminum-copper, cooled from below. If the solute phase is heavier than the solvent and the partition coefficient k0 < 1, then the system will be both ther- mally and solutally stable. During this solidification, the shrinkage that occurs as the solid + liquid mushy region forms establishes a flow of the inter-dendritic fluid towards the chill face. This flow redistributes the rejected solute phase and forms macrosegregation known as 'inverse segregation' [5-11]. A typical con- centration profile on complete solidification, shown in Fig. 1, is a positively segregated region in the vicinity of the chill, a 'steady-state' region in the mid-section of the ingot, and a negatively segregated region near the top of the ingot. The objective of the current paper is the devel- opment of an inverse segregation model, in an alumi- num-copper system, based on the fundamental heat and solutal transport equations. This work is related to the macrosegregation study reported by Diao and Tsai [12], who used two-dimensional transport equa- tions to describe inverse segregation during the initial stages of solidification. In the current study we include a number of additional and important features in the modeling. In particular : (1) the extent of the simulation is carried out to the completion of solidification ; (2) a full accounting of the density variations within the solid and liquid phases is used [13] ; (3) a treatment of the eutectic reaction [14], where the density change is greatest, is included ; (4) the effects of microporosity formation on the concentration distribution in the finally solidified cast- ing are studied ; and (5) a non-equilibrium treatment is invoked that allows for zero mass diffusion in the solid at the local scale (i.e. at the local scale a Scheil assumption is used, as opposed to an equilibrium lever rule assumption). INVERSE SEGREGATION Consider a dilute aluminum-copper alloy (a binary eutectic alloy), initially in the molten state, at the nominal concentration Co, contained in the insulated mold defined by 0 < x < Xb (see Fig. 1). At time t = 0, the temperature at the bottom face x = 0, is lowered and fixed at a temperature To < T,ut (Teut the eutectic Solute Concentration Insulated Mold D +ve -re Co Water Cooled Chill segregation segregation Fig. 1. Inverse segregation: a schematic of unidirectional solidification; a typical concentration profile after complete solidification. 1009