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 * ¹ C di!usion of solute. Depending upon conditions, where 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 , l l R L 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 , 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 c 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 k (1 )kl , (3) similarity solutions that describe how a mushy layer is melted from a hot plate with and without where kl and k 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 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, C 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/<. With these scalings, the the timescale of solute transport interstitially, d/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! " h 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 The equations describing the evolution of the 3. Melting from a hot plate mushy layer are Eqs. (10) and (11). The boundary conditions for t'0 are Consider the one-dimensional system in Fig. 1. h"h " We use a Cartesian coordinate system with its G (z 0), (12) origin "xed in the plate and with the z-axis increas- hPh (zPR), (13) ing into the mushy layer. At times t(0, the plate is in thermodynamic equilibrium with the mushy This implies that h layer at temperature which is taken to be be- !h tween the melting and freezing temperatures of the " " " lim !h (14) mushy layer. At t 0, the temperature of the plate X C h switches discontinuously to a new temperature G. If h '0 then part of the mushy layer will melt The solid fraction of the mushy layer in contact G " "!h !h completely into a melt region with uniform concen- with the plate is (z 0) G/(C G). 'h'h This problem admits a similarity solution tration C.If0 G then the mushy layer will h(z, t)"h(g), where the similarity variable is g"zt\ . Searching for this solution reduces Eq. (10) to an ordinary di!erential equation. The re- duced problem becomes SC !gh 1# "h , (15) A (C!h)B subject to h"h g" G ( 0), (16) hPh gPR ( ), (17) where (g) is given by Eq. (11). Eq. (15) was solved numerically using the method of shooting. In addition, we examined a simple asymptotic limit. Suppose C<1, S<1 with S/C of O(1) and (by assumption) h is of O(1), see [9]. Then, Eq. (15) Fig. 1. A schematic diagram of a mushy layer brought into approximates to contact with a hot plate. At t"0 the plate temperature discon- h h' !gXh "h tinuously switches to G.If G 0 then the melt forms. , (18) 750 D.L. Feltham, M.G. Worster / Journal of Crystal Growth 208 (2000) 746}756 where solution to the full nonlinear problem (obtained numerically) is shown with a solid line, the solution S X"1# . (19) to the asymptotic problem is shown with a dashed C line. The asymptotic problem relies on the limit < < This equation is linear and admits an analytical C 1, S 1 yet the error is less than 10%. solution. Integrating Eq. (18) twice and applying Throughout this paper, we shall present the true the boundary conditions (16) and (17) yields the solutions together with the solutions obtained us- " solution ing the asymptotic limit. Since we take C 1 and S"1 throughout, the di!erence between the true h"h # h !h X g G ( G)erf( /2), (20) and asymptotic solutions can be viewed as an up- per limit to the inaccuracy of the asymptotic ap- where erf(x) is the error function given by proximation. For many materials C<1, S*1 2 V and the asymptotic approximation is excellent. erf(x)" e\Q ds (21) (p Fig. 3 shows solutions to the full nonlinear problem h "! h "! " " with G 0.1, 0.5, S 1 and C 1 and is given by Eq. (11). (solid line), S"10 and C"1 (dashed line), and The solutions for h and are plotted in Fig. 2 for S"1 and C"10 (dotted line). We see that in- h "! h "! " " G 0.1, 0.5, S 1 and C 1. The creasing the Stefan number S decreases the extent Fig. 2. Temperature h and solid fraction within the mushy Fig. 3. Solutions of the full nonlinear problem for temperature h "! h "! " " h layer for G 0.1, 0.5, S 1 and C 1. The solid and solid fraction within the mushy layer for h "! h "! " " " line shows the solution to the full nonlinear problem, the dashed G 0.1, 0.5, S 1 and C 1 (solid line), S 100 line shows the solution to the asymptotic problem. and C"1 (dashed line), and S"1 and C"100 (dotted line). D.L. Feltham, M.G. Worster / Journal of Crystal Growth 208 (2000) 746}756 751 of temperature and phase change, and increasing the motion of the mush}melt interface. We search the compositional ratio C has little e!ect on the for the similarity solution h(z, t)"h(g) with the temperature pro"les but decreases the volume of mush}melt interface position given by f"jt , material undergoing phase change. Increasing S is where j is an eigenvalue determined from equivalent to increasing the latent heat L; more the Stefan condition. In the similarity variable, energy is required to dissolve solid elements and the problem becomes Eq. (15) in the mushy more energy is released on freezing the liquid ele- layer and ments of the mushy layer. This acts to weaken the h !gh "h (26) e!ect of the imposed temperature G. Increasing C is equivalent to steepening the liquidus curve, which in the melt, subject to Eq. (16), implies a smaller amount of phase change is re- quired to keep the interstitial liquid on the liquidus. h"0, (g"j), (27) h K" g"j [ ]l 0( ), (28) 3.2. Melt generated, h '0 G h K" g"j [ ]l 0( ) (29) We now consider the situation in which at t"0, and Eq. (17). The solid fraction is given by Eq. (11). the temperature of the plate is increased discon- The solution in the melt is simply obtained from tinuously to a temperature above the liquidus tem- integration and application of boundary conditions perature at the bulk composition h '0 and a melt G to be region forms with the position of the mush}melt interface at z"f(t). erf(g/2) h"h 1! . (30) The equations describing the mushy layer are GA erf(j/2)B Eqs. (10) and (11). The equation describing conser- vation of heat in the melt is Again, we consider the asymptotic problem with Eq. (15) replaced with Eq. (18). The solution to this Rh Rh approximate system is found from integration and " . (22) Rt Rz application of boundary conditions to be (erf(X g/2)!erf(X j/2)) Since the composition of the mushy layer is uni- h"h , (31) form, there is no concentration variation in the melt erfc(X j/2) formed and thus no solutal di!usion. The boundary where erfc(x) is the complementary error function, conditions at t'0 are Eq. (12), erfc(x)"1!erf(x), and is given by Eq. (11). Ap- h"0(z"f(t)) (23) plication of the Stefan condition (29) yields the transcendental equation [h]K"0(z"f(t)), (24) l X e\XH e\H h "!h , (32) Rf Rh K erfc(X j/2) Gerf(j/2) S " (z"f(t)), (25) Rt CRzD l which determines the eigenvalue j. K We solved the full nonlinear problem consisting and Eq. (13) where [2]l denotes the di!erence in the enclosed quantity between the mushy layer of Eq. (15) subject to Eq. (27) and (17) numerically. m and melt l. Note that (23) combined with (11) We integrate from the initial conditions " implies that 0 at the interface which is consis- h"0, h "g (g"jH), (33) tent with the general set of boundary conditions H hPh gPR derived by Schulze and Worster [10] since, relative to match the condition as . Once we h " jH to the phase boundary, the liquid #ows from mush converge on then gL gH( ) is recorded. This jH "h g"jH to liquid. This reduces the Stefan condition (25) to quantity gH( ) ( ) is the dimensionless continuity of heat #ux; this condition determines di!usional heat #ux from the mushy layer at the 752 D.L. Feltham, M.G. Worster / Journal of Crystal Growth 208 (2000) 746}756 j h h "! " " Fig. 4. Variation of with G for 0.5, S 1 and C 1. The solid line shows the solution to the full nonlinear problem, the dashed line shows the solution to the asymptotic problem. h ( There is no solution shown for G 0 because then no melt region is generated and the model is inappropriate. mush}melt interface and is used in the Stefan condition. The procedure is automated and repeat- j ed to obtain the curve gH( ). In terms of gH, the Stefan condition (29) can be rewritten g "g j h "B Fig. 5. Temperature h and solid fraction in the mushy layer H H( ;S, C, ) , (34) h " h "! " " and melt for G 0.5, 0.5, S 1 and C 1. The solid line shows the solution to the full nonlinear problem, the dashed where line shows the solution to the asymptotic problem. h e\H B"B(j, h )"! G . (35) G (perf(j/2) 4. Mushy layer in contact with a hot liquid Eq. (34) is the transcendental equation determin- j ing the eigenvalue . Since gH is independent of Consider the one-dimensional system in Fig. 6 in h G and B is expressed in closed form, we can calcu- which a mushy layer is in contact with a liquid. j h " late the dependence of on G using repeated root- Before t 0, the mushy layer is of uniform temper- h "nding with the secant method. In Fig. 4, we show ature , bulk composition C and hence solid j h " " h "! ( G) for S 1, C 1 and 0.5. The fraction . There is no melt region. We consider asymptotic solution (the dashed line) was obtained the physical properties of the liquid to be identical from Eq. (32) using a Newton}Raphson root-"n- to the melt generated as the mushy layer becomes h * j* der. For G 0, there exists a solution 0 corre- completely molten. Thus, there is no solutal di!u- sponding to an evolving mush}melt boundary. For sion in the liquid region. We set the far-"eld h ( j h ' G 0, there is no solution for . This is because no temperature in the liquid to be \ 0 and calcu- melt region is generated and the model is inappro- late the evolution of this system after t"0. A Car- priate; where there is no melt region we use the tesian coordinate system is used with the origin of equations of the previous section. We plot the z "xed at the position of the mush}melt}liquid h h " h "! ( solutions and in Fig. 5 for G 0.5, 0.5, interface at time t 0, with z increasing into the S"1 and C"1. mushy layer. D.L. Feltham, M.G. Worster / Journal of Crystal Growth 208 (2000) 746}756 753 j h h "! " Fig. 7. Variation of with \ for 0.5, S 1 and C"1. The solid line shows the solution to the full nonlinear problem, the dashed line shows the solution to the asymptotic problem. Fig. 6. A schematic diagram of a mushy layer brought into contact with a hot melt. The melt may form due to the hot liquid dissolving the mushy layer after t"0. j h The variation of with \ is calculated using j h repeated root-"nding. In Fig. 7, we plot ( \ ) " " h "! We search for a similarity solution in the sim- for S 1, C 1 and 0.5. There exists g" \ h#* j( ilarity variable zt , with the mush}melt a critical temperature \ , below which 0 and interface located at f(t)"jt . The equations de- the mushy layer advances into the liquid by scribing the mushy layer are (15) and (11), the freezing. equation describing the liquid and melt is (26). The We can easily determine the variation of h#* h h#* boundary conditions are \ with . The value of \ is given by Eq. (38) with j"0, rearranging and substituting for hPh gP!R ( ), (36) BI yields Eqs. (27)}(29) and (17). The solution in the melt h#* h "!(p h j" \ ( ) gH( ; 0), (40) region is obtained straightforwardly by integration h j" of the governing equation and applying boundary where gH( ; 0) is obtained numerically using h#* conditions to be shooting. We can estimate \ analytically by con- sidering the asymptotic equation for the mushy (erf(j/2)!erf(g/2)) h"h . (37) layer (18). Solving this equation subject to Eqs. (27) \ 1#erf(j/2) and (17) gives the temperature in the mushy layer to be The problem of obtaining the temperature (and hence solid fraction) in the mushy layer is identical (erf(X g/2)!erf(X j/2)) h"h , (41) to that presented in the preceding section. The erfc(X j/2) equation determining j, however, is modi"ed by the solution in the liquid}melt region to with given by Eq. (11). Application of the Stefan condition (29) yields the transcendental equation " I gH B, (38) h e\H h X e\XH \ # "0. (42) where 1#erf(j/2) erfc(X j/2) h e\H BI "! \ . (39) This equation was used to calculate the asymptotic (p 1#erf(j/2) j h dependence of upon \ shown in Fig. 7 for 754 D.L. Feltham, M.G. Worster / Journal of Crystal Growth 208 (2000) 746}756 h#* h " " Fig. 8. Variation of \ with for S 1 and C 1. The solid line shows the solution to the full nonlinear problem, the dashed line shows the solution to the asymptotic problem. " " h "! j" S 1, C 1 and 0.5. Setting 0, we obtain h#* "!X h \ . (43) h#* h " In Fig. 8, we plot \ versus for S 1 and C"1. The asymptotic problem badly over-esti- mates h#* (h ) at large h but numerical studies Fig. 9. The temperature h and solid fraction in the mushy \ h " h "! " " show that the approximation greatly improves as layer and melt for \ 1.0, 0.5, S 1 and C 1. The solid line shows the solution to the full nonlinear problem, C is increased into the appropriate asymptotic the dashed line shows the solution to the asymptotic problem. regime of C<1. We plot the solutions h and " " h "! in Fig. 9 for S 1, C 1, 0.5 and h " \ 1.0. 5. Investigation of internal phase change and discussion We consider the case of a mushy layer in contact with a hot liquid since this system exhibits behav- iour which encompasses that of the simpler sys- tems. We investigate the extent and volume of the h Fig. 10. Variation of the volume of phase change with \ for internal phase change within the mushy layer and h "! " " 0.5, S 1 and C 1. The solid line shows the total contrast the melting behaviour with that of a pure j phase change, M2; the dashed line shows ; and the dashed- j material. A full exploration of the realistic para- dotted line shows . meter regimes would be infeasible; we concentrate here upon the e!ect of varying the far-"eld temper- " ! g ature in the hot liquid. where M' H ( )d is that melted inter- Consider Fig. 10, which is plotted for S"1, nally and is calculated using shooting and numer- " h "! j C 1 and 0.5. The solid line shows the ical quadrature. The dashed line shows which, " #j h j total volume of phase change M2 M' , for the melting values of \ ( positive), equals the D.L. Feltham, M.G. Worster / Journal of Crystal Growth 208 (2000) 746}756 755 g Fig. 12. Variation of the extent of internal phase change C with h h "! " " Fig. 11. Variation of the volume of internal phase change \ for 0.5, S 1 and C 1. h h "! " " M' with \ for 0.5, S 1 and C 1. volume of mushy layer that becomes completely ishes its strength and can allow break-up under the j molten. Negative values of correspond to stresses imposed by the wind and ocean. A decrease freezing of the hot liquid, which increases the depth in the solid fraction of sea ice also increases its of the mushy layer. The di!erence between the permeability (a measure of its resistance to internal j total phase change curve and is the volume of #ows). This may promote signi"cant brine #ows internal phase change, M'. From Fig. 11, we see and #ushing of fresh water through the sea ice from that the volume of internal phase change within the the melting of snow and ice at its surface. These h mushy layer decreases as \ increases. The extent #ows desalinate the sea ice and enhance the fresh- of the internal phase change within the mushy water #ux into the North Atlantic. More details g layer is also of interest; we de"ne C from about sea ice can be found in [8]; a discussion of g"j#g " ( C) , the depth within the mushy mushy layers applied to sea ice can be found in layer at which the solid fraction reaches of its Refs. [11,12]. far-"eld value. From Fig. 12, we see that the extent h of internal phase change decreases as \ in- creases. In Fig. 10, the dashed-dotted curve is 6. Conclusions j j , where is the position of a pure solid}melt j interface. The value of is determined as part of In this paper, we have presented some solutions the classical melting Stefan problem given by (10) of the equations describing local conservation of with CPR subject to Eqs. (36), (23)}(25), with heat and solute in a mushy layer and melt in the P h j 1 and Eq. (13), see [4]. As \ increases, absence of #ow. This section summarises our main j approaches , this is because the volume and conclusions. extent of phase change within the mushy layer A mushy layer can be partially melted from h decreases as \ increases. From these results it is a plate without generation of a completely molten clear that internal melting can be signi"cant for region by altering its solid fraction. Increasing the both melting and freezing mushy layers and be- Stefan number S decreases the region of substan- comes less signi"cant in strong melting conditions tial phase change and temperature variation. In- h