Journal of Crystal Growth 208 (2000) 746}756 Similarity solutions describing the melting of a mushy layer Daniel L. Feltham!, M. Grae Worster",* !Department of Chemical Engineering, Institute of Science and Technology, University of Manchester, P.O. Box 88, Manchester M60 1QD, UK "Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK Received 27 March 1999; accepted 20 August 1999 Communicated by D.T.J. Hurle Abstract A model of the melting of a mushy region in the absence of #uid #ow is presented. Similarity solutions are obtained which are used to describe melting from a hot plate with and without the generation of a completely molten region. These solutions are extended to describe the melting of a mushy region in contact with a hot liquid. A signi"cant feature of melting mushy regions is that the phase change occurs internally by dissolution. Our solutions for melting of a mushy region are used to investigate this internal phase change and are compared with the classical Neumann solutions for melting of a pure substance. ( 2000 Elsevier Science B.V. All rights reserved. 1. Introduction to the large di!erence in the di!usion rates of heat and the components of the alloy. A discussion of A mushy region is a mixture of solid and liquid mushy layers with many examples of where they elements coexisting in thermodynamic equilibrium occur may be found in Ref. [1]. and typically takes the form of a porous matrix of There have been many studies of solidifying the solid phase bathed in interstitial liquid. Mushy mushy regions [2]. By contrast, we present a funda- regions form in a variety of alloys such as mental study of a melting mushy layer owing iron}carbon (steel), copper}zinc (brass) and salt to internal dissolution. Dissolution is driven by solutions such as sodium-chloride}water. The con- thermodynamic disequilibrium resulting from voluted geometry of a mushy region is created by compositional variations, and is rate limited by morphological instabilities that enhance the expul- compositional di!usion. Melting, resulting from sion of one or more components from the solid thermal disequilibrium, proceeds much more rap- phase as it grows. These instabilities are typically idly, at a rate dictated by heat transfer [3]. Within caused by constitutional supercooling, which is due a mushy layer, internal dissolution tends to keep the interstitial concentration on the local liquidus by solute transport on the microscale of the inter- nal morphology. On the macroscale, therefore, the * Corresponding author. Fax: #44-1223-337918. change of phase from solid to liquid is still control- E-mail address: [email protected] (M.G. Worster) led by rates of heat transfer. We shall, therefore, 0022-0248/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 2 4 8 ( 9 9 ) 0 0 4 5 6 - X D.L. Feltham, M.G. Worster / Journal of Crystal Growth 208 (2000) 746}756 747 refer to phase change on the macroscopic scale, as assume that the liquidus is linear, melting (or freezing) though it is to be understood ¹"¹ (C)"¹ !CC, (1) that the change of phase on the microscale is due to L 0 ¹ C di!usion of solute. Depending upon conditions, where 0 and are positive constants, and that the melting of a mushy layer can create a completely solidus is equal to zero. A more detailed description molten region or simply reduce its solid fraction. of the model we now introduce can be found in Ref. A signi"cant issue is the extent to which partial as [5]. Within a mushy zone the alloy consists of opposed to complete melting takes place. Internal a mixture of solid and liquid elements coexisting in phase change plays an important role in "elds as thermodynamic equilibrium. In the absence of #uid diverse as the reprocessing of composite materials, #ow, local conservation of heat can be expressed by heat transfer during welding, and to the seasonal R¹ R/ melting of sea ice. [o c /#o c (1!/)] "+ ) (k +¹)#o , 4 4 l l R . 4L R We restrict our attention to binary alloys since t t this simpli"es the mathematical description of the (2) mushy layer, and, in any case, the behaviour of / alloys is often dominated by two major compo- where the variables (r, t) and ¹(r, t) represent the nents. We obtain similarity solutions which can locally averaged solid fraction and temperature, be used to describe the melting of a mushy layer respectively, r is a position vector in the mushy when heat and mass transport is e!ected solely by layer and t is time. The latent heat released as liquid di!usion. solidi"es is L per unit mass of solid, the densities of o o In Section 2, we introduce the mathematical the liquid and solid are l and 4, and the speci"c model and describe the various assumptions we heat capacities of the liquid and solid are cl and make in order to simplify the analysis and avoid c4 respectively. The thermal conductivity of the details that are extraneous to the fundamental pro- mushy layer is approximated by cesses. In Section 3, we use our model to determine "/ # !/ k. k4 (1 )kl , (3) similarity solutions that describe how a mushy layer is melted from a hot plate with and without where kl and k4 are the thermal conductivities in generation of an entirely molten region. In Section the liquid and solid, respectively. This expression is 4, we extend these solutions to describe how exact for a parallel laminated medium in the case a mushy layer in contact with a hot liquid may that the heat #ux is aligned with the lamellae [6]. melt. The solutions that we present are extensions Local conservation of solute can be written to binary alloys of the Neumann solution [4], of the RC R/ classical Stefan problem for melting. In Section 5, (1!/) "+ ) (D +C)#(C!C ) , (4) Rt . 4 Rt we use our solutions to investigate the extent of internal phase change within the mushy layer and where C(r, t) represents the locally averaged con- compare our results with those for the melting of centration of the interstitial liquid, C4 is the com- a pure substance. The possible implications of in- position of the solid formed due to freezing, and ternal dissolution are brie#y explored using the D. is the solute di!usion coe$cient in the mushy example of sea ice. Section 6 summarises our main layer. For the sake of clarity, we assume that the conclusions. thermal conductivites of the solid and liquid ele- ments of the mushy layer are identical and equal " to the constant k; thus k. k. Since di!usion of 2. Mathematical formulation the solute within the solid phase is neglected, " !/ D. (1 )D, where D is the solute di!usion coef- We consider the melting of a simple binary eutec- "cient in the liquid. We assume that local thermo- tic. Without loss of generality, we formulate the dynamic equilibrium prevails within the mushy problem for the case that the bulk composition of layer. This is justi"ed provided that the rate of the alloy is less than the eutectic composition. We phase change within the mushy layer is su$ciently 748 D.L. Feltham, M.G. Worster / Journal of Crystal Growth 208 (2000) 746}756 slow that thermodynamic equilibrium can be main- ivity i"k/(oc). Lengths are scaled with i/< and tained between the solid elements. This is satis"ed if times are scaled with i/<2. With these scalings, the the timescale of solute transport interstitially, d2/D, conservation equations combined with the liquidus where d is the solid-element spacing in the mushy constraint and the assumption of no solutal di!u- layer, is short compared to the timescale of temper- sion become ature variations within the mushy layer. During the R freezing of a mushy layer, solute transport is en- h! / "+2h R ( S ) (5) hanced by the side-branching of dendrites resulting t d from morphological instabilities, which reduces . and During melting, however, d corresponds to the pri- mary dendrite spacing and the condition is more R [(1!/)h#C/]"0. (6) restrictive. When similarity solutions apply, the Rt heating rate scales with the elapsed time t and We have introduced the dimensionless temperature thermodynamic equilibrium is therefore main- and concentration tained once t<d2/D. In a typical metallic mushy d+ ] ~4 + ] ~9 ¹!¹ ! layer, with 3 10 m and D 3 10 L(C0) C C0 2 ~1 h" " , (7) m s [7], we expect thermodynamic equilibrium *¹ *C to be maintained after about 30 s. For mushy layers *¹"¹ !¹ * " ! of ice crystals formed from aqueous solutions of where L(C0) E and C C0 CE. sodium chloride, for example, d+10~3 m and The dimensionless melting temperature of the D+10~9 m2s~1 so that thermodynamic equilib- mushy layer is h"0; above this temperature the rium is maintained after about 1000 s. This time is mushy layer becomes completely molten. The di- still small compared to the timescales of variation mensionless eutectic temperature of the mushy in various applications such as the diurnal vari- layer is h"!1; below this temperature the mushy ations within a sea-ice layer on the polar oceans layer forms a eutectic solid. The dimensionless [8]. parameters are the Stefan number "i The Lewis number Le /D of liquids and L solids is very much greater than unity; for example, S" (8) c*¹ Le+100 in aqueous solutions.