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Effect of shrinkage induced flow on binary alloy dendrite growth: An equivalent undercooling model
International Communications in Heat and Mass Transfer 57 (2014) 216–220
Contents lists available at ScienceDirect
International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt
Effect of shrinkage induced flow on binary alloy dendrite growth: An equivalent undercooling model☆ Anirban Bhattacharya, Pradip Dutta ⁎ Department of Mechanical Engineering, Indian Institute of Science, Bangalore 560012, India
a r t i c l e
i n f o
Available online 8 August 2014 Keywords: Dendrite growth rate Shrinkage flow Undercooling Enthalpy method
a b s t r a c t This paper presents a theoretical model for studying the effects of shrinkage induced flow on the growth rate of binary alloy dendrites. An equivalent undercooling of the melt is defined in terms of ratio of the phase densities to represent the change in dendrite growth rate due to variation in solutal and thermal transport resulting from shrinkage induced flow. Subsequently, results for dendrite growth rate predicted by the equivalent undercooling model is compared with the corresponding predictions obtained using an enthalpy based numerical method for dendrite growth with shrinkage. The agreement is found to be good. Published by Elsevier Ltd.
1. Introduction The morphology of equiaxed dendrites has a significant impact on the properties of a cast product. During the past few decades, several analytical and computational models have been proposed for studying the growth pattern of equiaxed dendrites. Computational models can simulate the six-fold symmetry of equiaxed dendrites [1], predict the formation of secondary arms [2,3], incorporate the presence of alloying elements and consequently their influence on growth morphology [4], calculate the effect of natural and forced convection [5,6] and handle complex scenarios such as the interaction of multiple dendrites [7]. On the other hand, analytical models can calculate the dendrite tip velocity and radius and provide insight into the relative importance of various physical parameters governing the dendrite growth rate and morphology. The initial analytical models focussed on the dependence of tip growth Peclet number on solutal or thermal diffusion assuming a paraboloidal dendrite shape [8,9]. Subsequently, stability criteria were introduced to calculate the relation between tip radii and velocities and obtain the exact operating radius [10–12]. Later, the micro-solvability theory for dendritic growth was developed to obtain unique values for growth radii based on the anisotropy of interface energy [13]. A few analytical models have also incorporated the effects of convection on the growth rate of dendrites [14,15]. However, none of the existing models consider the presence of shrinkage and the resultant melt flow towards the interface.
☆ Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail address: [email protected] (P. Dutta).
http://dx.doi.org/10.1016/j.icheatmasstransfer.2014.08.003 0735-1933/Published by Elsevier Ltd.
In the present work, a theoretical model is developed to represent the effects of shrinkage driven flow on the growth rate of binary alloy dendrites in terms of an equivalent undercooling of the melt. The ratio of the solid phase density ρs to the liquid phase density ρl is used to derive an expression for this equivalent undercooling which takes into account the interaction of shrinkage flow with solutal and thermal transport in the vicinity of the dendrite. The significance of the equivalent undercooling is that the growth rate predicted (without considering any shrinkage effects) is the same as that predicted using the original melt undercooling and shrinkage effects, for any specified density ratio (ρs/ρl) greater than 1. This is similar in nature to the equivalent undercooling proposed in [16] which, however, uses an imposed uniform forced convection instead of shrinkage driven flow. Results for the equivalent undercooling derived using this theoretical model is compared with corresponding predictions obtained using numerical simulations based on the enthalpy model proposed by the present authors [17] with the addition of shrinkage. The main objective of the present work is to relate the effects of unequal phase densities to the different physical parameters governing dendrite growth using simplifying assumptions, instead of deriving a complete and rigorous exact solution.
2. System The model problem consists of a cylindrical dendrite with radius R growing with velocity Va in a binary alloy melt undercooled to a temperature Tb. The tip of the dendrite is assumed to be hemispherical with radius R. The liquid adjacent to the solidification interface has temperature Ti and concentration Cl⁎, while the bulk liquid has a concentration equal to C0. Fluid flow occurs in the domain due to shrinkage with a flow velocity equal to Vf near the interface. A schematic of the system
A. Bhattacharya, P. Dutta / International Communications in Heat and Mass Transfer 57 (2014) 216–220
217
Fig. 1. Schematic diagram of the domain used for the present model.
along with the temperature and concentration conditions is shown in Fig. 1. The following assumptions are made. 1. 2. 3. 4. 5.
Growth occurs only in the x-direction. The density of the solid phase is higher than that of the liquid phase. Flow is present in the domain only due to shrinkage. Shrinkage flow acts uniformly over the entire hemispherical tip. Diffusion of rejected heat and solute from the interface occurs uniformly from the hemispherical tip.
3.1. Relation between dendrite growth velocity and shrinkage flow velocity The shrinkage flow velocity adjacent to the interface (Vf) is related to the dendrite growth velocity (Va). Per unit time, the volume of flow re2 quired to compensate for shrinkage is given by πR V aρðρs −ρl Þ. This is ball anced by the volume of flow towards the interface which is equal to 2 2 2πR Vf where 2πR is the tip area. Equating the two, the magnitude of shrinkage flow can be stated by the following equation. Vf ¼
3. Derivation of equivalent undercooling The total undercooling of the melt ΔTu is the difference between the undercooled melt temperature Tb and the solidification temperature of the alloy Tm corresponding to the initial concentration C0. The thermal undercooling ΔTt is the difference between the actual interface temperature Ti and the bulk temperature of the liquid Tb, and is the driving force for solidification. The interface temperature Ti differs from Tm due to surface tension effects and partitioning of the solute, which are represented by the curvature undercooling ΔTc and solutal undercooling ΔTs, respectively. Neglecting the effects of kinetics and pressure on the interface temperature, ΔTu can be represented by ΔTu = ΔTt + ΔTs + ΔTc. By definition, the equivalent undercooling ΔTu,eq should be such that the temperature difference at the interface is able to maintain the same growth velocity of the dendrite (without considering any shrinkage effects) as with the original melt undercooling and shrinkage. The procedure for deriving the equivalent undercooling can be described in the following way. At first, equivalent values for the solutal supersaturation and solutal undercooling are derived using solute balance at the interface. Similarly, the equivalent thermal undercooling value is obtained in terms of the original thermal undercooling. Subsequently, a relationship between the thermal undercooling and solutal supersaturation is established. Finally, using the derived relations, the equivalent total undercooling is expressed in terms of the original parameters and the density ratio.
1 ðρs −ρl Þ : V 2 a ρl
ð1Þ
3.2. Solute balance During solidification, solute is rejected at the interface due to partitioning effects. At equilibrium, the rejected solute is carried away by diffusion while the shrinkage induced convection acts in the opposite direction and convects the solute towards the interface. The solute balance at the interface can be stated in the following form. ðC −C 0 Þ 2 2 2 : πR V a ρs 1−kp C l þ 2πR V f ρl C l ¼ 2πR ρl Dl l R
ð2Þ
The first term in Eq. (2) gives the amount of solute rejected per unit time, while the second and third terms represent the convective and diffusive transport of solute, respectively. Dl is the solutal diffusivity, kp is ðC −C 0 Þ is the concentration gradient dethe partition coefficient and l R rived assuming a diffusion length scale equal to the tip radius R. Using Eqs. (1) and (2), we can write the solutal supersaturation Ωc as 2 3 V a R 4ρs ðρs −ρl Þ 5 þ Ωc ¼ 2Dl ρl ρ 1−k l
ð3Þ
p
−C o Þ . where Ωc is defined as Ωc ¼ CðC l 1−k l ð pÞ
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For the equivalent case, the densities of the two phases are equal (i.e. ρs = ρl). Due to the absence of shrinkage driven convection, the only mechanism of solute transport is diffusion. Also, with a modified undercooling ΔTu,eq, the velocity of dendrite growth is the same i.e. equal to Va. Assuming that the tip radius is inversely proportional to the growth velocity, the equivalent tip radius can be assumed to be equal to R. This leads to the following equation for solute balance. 2 2 πR V a ρl 1−kp C l;eq ¼ 2πR ρl Dl
C l;eq −C 0 R
ΔT t ¼ T i −T b ¼ :
ð4Þ
In Eq. (4), Cl,eq⁎ is the equivalent solute concentration at the interface. The subscript ‘eq’ denotes the corresponding equivalent values for each parameter for the same growth velocity and this practice is followed throughout the manuscript. Using Eq. (4), the equivalent solutal supersaturation can be written as Ωc;eq ¼
V aR : 2Dl
ð5Þ
Now, using Eqs. (3) and (5), the following relation can be obtained. Ωc 3: Ωc;eq ¼ 2 ρ ð ρ −ρ Þ s s l 4 þ 5 ρl ρ 1−k l
ð6Þ
p
The equivalent interface concentration Cl,eq⁎ can be derived from Eq. (6) and is given by
C l ;eq ¼ 1−Ωc
C0 2 3: ρs −ρl Þ 5 4ρs þ ð ρl ρ 1−k
=
l
ð7Þ
mC o : 1 1− Ωc 1−kp
ð11Þ
For the equivalent case, the only mechanism of thermal transport is diffusion. A similar heat balance leads to
2
2
πR V a ρl L ¼ 2πR kl
T i;eq −T b;eq R
ð12Þ
and finally to ΔT t;eq ¼ T i;eq −T b;eq ¼
V a R ρl L : 2 kl
ð13Þ
It should be noted that for this case, the interface temperature Ti,eq is different from Ti due to the difference in solute build-up at the interface. Using Eqs. (11) and (13) and noting that the interface temperature Ti is given by Ti = ΔTt + Tb, the equivalent thermal undercooling can be expressed in terms of the original thermal undercooling in the following way. ΔT t : ΔT t;eq ¼ ρs ðρs −ρl Þ cpl þ ðΔT t þ T b Þ ρl L ρl
ð14Þ
3.4. Relation between solutal supersaturation and thermal undercooling The thermal undercooling and solutal supersaturation are coupled together through the growth velocity Va. Using, Eqs. (3) and (11), ΔTt can be written in terms of Ωc as
ð8Þ
Ωc ΔT t ¼
Now, using Eqs. (6) and (8), the equivalent solutal undercooling, ΔT s;eq ¼ −m C l;eq −C o , can be obtained in terms of the original solutal undercooling ΔTs as ΔT s;eq ¼
V a R ρs L ðρs −ρl Þ ρl cpl Ti : þ kl 2 kl ρl
p
It can be observed that, when ρs is greater than ρl, Ωc,eq is less than Ωc ⁎ is smaller than Cl⁎. which means that Cl,eq The solutal undercooling is defined by ΔT s ¼ −mðC l −C o Þ where m is the slope of the liquidus line. ΔTs can be related to Ωc using the following equation. ΔT s ¼
Eq. (10) represents the latent heat released at the interface per unit time, while the second and third terms represent the convective and diffusive transport, respectively. As in the case of solute diffusion, the heat diffusion term is also derived assuming a diffusion length scale equal to R. Eq. (10) leads to the following expression for the thermal undercooling ΔTt
mC 0 2 3: mC 0 4ρs ðρs −ρl Þ 5 1− 1− þ ΔT s ρl ρ 1−k l
ð9Þ
L 1 1 ρs ðρs −ρl Þ cpl Tb þ L cpl Le Aρ ρl ρl L 1 1 ðρs −ρl Þ cpl 1−Ωc L cpl Le Aρ ρl
ð15Þ
where Le is the Lewis number given by Le = kl/(ρlcplDl), and Aρ ¼ ρs ðρs −ρl Þ . þ ρl ρl ð1−kp Þ 3.5. Equivalent total undercooling
p
The equivalent total undercooling can now be derived using the previous relations. From Eq. (14), 3.3. Heat balance At equilibrium, the latent heat of solidification at the interface is removed by diffusion, while the shrinkage flow convects the heat in the opposite direction, i.e., towards the interface. The overall heat balance can be written as 2
2
2
πR V a ρs L þ 2πR V f ρl cpl T i ¼ 2πR kl
ðT i −T b Þ R
ð10Þ
where L, cpl and kl are the latent heat of solidification, specific heat and thermal conductivity of the liquid, respectively. The first term in
ΔT t : ΔT u;eq −ΔT s;eq −ΔT c;eq ¼ ρs ðρs −ρl Þ cpl þ ðΔT t þ T b Þ ρl L ρl
ð16Þ
Now, the curvature undercooling ΔTc is given by ΔT c ¼ 2Γ R , where Γ is the Gibbs–Thomson coefficient and R is the radius of curvature of the interface. The radius of curvature is equal for both the original and the equivalent cases which leads to ΔT c;eq ¼ ΔT c ¼ ΔT u −ΔT t −ΔT s :
ð17Þ
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219
4. Comparison with numerical simulations
Fig. 2. Comparison of analytical and numerical predictions for equivalent undercooling.
Finally, using Eqs. (9) and (17), the equivalent total undercooling ΔTu,eq can be written as 2 3 1 ΔT u;eq ¼ ΔT u þ ΔT t 4 −15 ρs =ρl þ ðρs −ρl Þcpl =ðρl LÞ ðΔT t þ T b Þ mC 0 2 3 −ΔT s : þ mC 0 4ρs ðρs −ρl Þ 5 1− 1− þ ΔT s ρl ρ 1−k l p
ð18Þ
Noting that ΔTt and ΔTs are both functions of Ωc (Eqs. (15) and (8)), it can be observed that Eq. (18) is an expression for the equivalent total undercooling ΔTu,eq in terms of the original undercooling ΔTu, the density ratio ρs/ρl and the solutal supersaturation Ωc. ΔTu,eq represents the degree of undercooling that needs to be provided to the melt to maintain the same growth rate as in the original case with shrinkage effects. Eq. (18) reduces to ΔTu,eq = ΔTu for ρs/ρl = 1. When ρs is greater than ρl, ΔTu,eq is less than ΔTu, which means that a lower undercooling is sufficient to maintain the same growth rate without the presence of shrinkage.
The equivalent undercooling values given by the present model are compared with corresponding values obtained from numerical simulations. The numerical simulations are performed based on the enthalpy method presented in Bhattacharya and Dutta [17] with the addition of solidification shrinkage. Dendrite growth is tracked by solving the energy and solute transport equations using an explicit formulation as given in [4]. The flow field is calculated using the SIMPLER algorithm [18]. All the governing equations are formulated based on density as a function of liquid fraction. For each time step, at first the progress of solidification is obtained using the previous time step values for the velocity field. An iterative scheme [17] is implemented for maintaining the consistency between the interface temperature and concentration as per the phase diagram. The new densities at each computational node are then calculated based on the updated liquid fraction field, and subsequently, the momentum equations are solved to obtain the new velocities. Simulations are performed for three sets of undercooling values ΔTu = 0.6, 0.65, 0.7 with density ratio ρ s/ρl varying from 1.0 to 1.16 in steps of 0.02. The values for the other parameters are specified as kp = 0.1, mC0 = 0.1, Le = 1.0, cps = cpl = 1.0, ks = kl = 1.0 and L = 1.0. A domain size of 1000 × 1000 with a grid size equal to Δx* = Δy* = 4 and a time step equal to Δt* = 1.0 is used for the simulations. For each simulation, when the growth rate becomes steady, the average tip velocity of the four dendrite arms is calculated. Additional simulations are performed without shrinkage effects (ρs/ρl = 1) with undercooling varying by small steps of 0.005. From these simulations, the undercooling values are noted for which the tip velocities are equal to those with shrinkage. This gives the numerical equivalent undercooling values. The analytical equivalent undercooling values are obtained using Eq. (18) and compared with the corresponding numerical values in Fig. 2. For calculating the analytical values, Ωc is obtained from the simulations. It can be observed that the equivalent undercooling values match closely for both the methods and linearly decrease with increasing density ratio. This is expected, because the presence of shrinkage decreases the growth rate due to a number of factors. Shrinkage results in convection towards the interface which hinders the movement of solute into the bulk liquid, thus increasing the concentration of solute at the interface. This decreases the interface temperature, reducing its difference with the bulk liquid temperature, and consequently slows the growth velocity. Additionally, the shrinkage driven flow also opposes the
Fig. 3. Comparison of dendrite shapes at t = 20, 000 from numerical studies with shrinkage and without shrinkage to that with simulations using equivalent undercooling: (a) For density ratio ρs/ρl = 1.04 (b) For density ratio ρs/ρl = 1.16.
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removal of latent heat from the interface, further reducing the growth rate. Also, since the density of the solid is higher than that of the liquid, for the same mass of solid formed due to certain amount of latent heat removal, the volume of solid formed is less, resulting in the reduction of dendrite size. From Fig. 2, it can also be observed that there is a significant change in undercooling for high density ratios which proves that shrinkage is an important factor in determining the growth rate of dendrites. The actual shapes of the growing dendrites are compared using two sets of results from the previous simulations. Fig. 3(a) and (b) shows the dendrite shapes at time t = 20, 000 for ΔTu = 0.6 and ρs/ρl = 1.04 and ρs/ρl = 1.16, respectively. The equivalent undercooling values for the two cases are calculated using Eq. (18) and found to be equal to 0.581 for ρs/ρl = 1.04 and 0.537 for ρs/ρl = 1.16. Dendrite growth from simulations assuming no density change (ρs/ρl = 1) and the appropriate equivalent undercooling values are shown by the bold lines while the numerical results with the prescribed undercooling and density difference are represented by the dashed lines. The growth of the dendrite without any shrinkage effects is shown by the dash–dot lines. It can be observed that the presence of shrinkage considerably slows down the growth rate. This decrease in growth rate is captured quite accurately using the equivalent undercooling model and the shapes predicted using the equivalent undercooling without any shrinkage are very similar to the original cases with shrinkage effects. It should be noted here that simulating the growth of a dendrite in presence of shrinkage necessitates the solution of the mass and momentum conservation equations (to track the shrinkage driven convection) in addition to the energy and species conservation equations. The use of an equivalent undercooling for the numerical simulations removes the requirement of solving the mass and momentum conservation equations while maintaining a high level of accuracy in predicting the dendrite shapes and growth rates. As a result, the computational effort for performing the numerical simulations is significantly reduced. 5. Conclusions A theoretical analysis was presented for studying the effects of shrinkage on the growth of dendrites. The presence of shrinkage flow and the resultant variation in the solutal and thermal transport mechanisms were represented by an equivalent undercooling of the melt. The model predictions show excellent agreement with corresponding results from numerical simulations. The present work provides an insight
into the relative importance of shrinkage in determining the growth rate and morphology of dendrites as compared to other factors such as melt undercooling, solute concentration and so forth. Additionally, the use of an equivalent undercooling for numerical simulations removes the necessity of solving for the shrinkage driven convection, thus resulting in significant reduction of the required computational effort.
References [1] A. Karma, W. Rappel, Quantitative phase field modeling of dendritic growth in two and three dimensions, Phys. Rev. E. 57 (1998) 4323–4434. [2] N. Al-Rawahi, G. Tryggvason, Numerical simulation of dendritic solidification with convection: three-dimensional flow, J. Comput. Phys. 194 (2004) 677–696. [3] D. Pal, J. Bhattacharya, S. Chakraborty, P. Dutta, An enthalpy model for simulation of dendritic growth, Numer. Heat Transfer B 50 (1) (2006) 59–78. [4] V.R. Voller, An enthalpy method for modeling dendritic growth in a binary alloy, Int. J. Heat Mass Transfer 51 (2008) 823–834. [5] C. Beckermann, H.J. Diepers, I. Steinbach, A. Karma, X. Tong, Modeling melt convection in phase-field simulations of solidification, J. Comput. Phys. 154 (2) (1999) 468–496. [6] D.M. Anderson, G.B. McFadden, A.A. Wheeler, A phase-field model of solidification with convection, Physica D 135 (2000) 175–194. [7] S. Karagadde, A. Bhattacharya, G. Tomar, P. Dutta, A coupled VOF–IBM–enthalpy approach for modeling motion and growth of equiaxed dendrites in a solidifying melt, J. Comput. Phys. 231 (2012) 3987–4000. [8] G.P. Ivantsov, Temperature around a spheroidal, cylindrical and acicular crystal growing in a supercooled melt, Dokl. Akad. Nauk SSSR 58 (1947) 565–567. [9] R. Trivedi, Growth of dendritic needles from a supercooled melt, Acta Metall. 18 (3) (1970) 287–296. [10] J.S. Langer, H. Müller Krumbhaar, Theory of dendritic growth—I. Elements of a stability analysis, Acta Metall. 26 (1978) 1681–1687. [11] W. Kurz, D.J. Fisher, Dendrite growth at the limit of stability: tip radius and spacing, Acta Metall. 29 (1) (1981) 11–20. [12] J. Lipton, M.E. Glicksman, W. Kurz, Dendritic growth into undercooled alloy metals, Mater. Sci. Eng. 65 (1) (1984) 57–63. [13] D.A. Kessler, J. Koplik, H. Levine, Steady-state dendritic crystal growth, Phys. Rev. A 33 (1986) 3352–3357. [14] R. Ananth, W.N. Gill, Dendritic growth with thermal convection, J. Cryst. Growth 91 (1988) 587–598. [15] R. Ananth, W.N. Gill, Self-consistent theory of dendritic growth with convection, J. Cryst. Growth 108 (1991) 173–189. [16] A. Bhattacharya, S. Karagadde, P. Dutta, An equivalent undercooling model to account for flow effect on binary alloy dendrite growth rate, Int. Commun. Heat Mass Transfer 47 (2013) 15–19. [17] A. Bhattacharya, P. Dutta, An enthalpy-based model of dendritic growth in a convecting binary alloy melt, Int. J. Numer. Methods Heat Fluid Flow 23 (7) (2013) 1121–1135. [18] S.V. Patankar, Numerical Heat Transfer & Fluid Flow, Hemisphere, New York, 1980.
Contents lists available at ScienceDirect
International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt
Effect of shrinkage induced flow on binary alloy dendrite growth: An equivalent undercooling model☆ Anirban Bhattacharya, Pradip Dutta ⁎ Department of Mechanical Engineering, Indian Institute of Science, Bangalore 560012, India
a r t i c l e
i n f o
Available online 8 August 2014 Keywords: Dendrite growth rate Shrinkage flow Undercooling Enthalpy method
a b s t r a c t This paper presents a theoretical model for studying the effects of shrinkage induced flow on the growth rate of binary alloy dendrites. An equivalent undercooling of the melt is defined in terms of ratio of the phase densities to represent the change in dendrite growth rate due to variation in solutal and thermal transport resulting from shrinkage induced flow. Subsequently, results for dendrite growth rate predicted by the equivalent undercooling model is compared with the corresponding predictions obtained using an enthalpy based numerical method for dendrite growth with shrinkage. The agreement is found to be good. Published by Elsevier Ltd.
1. Introduction The morphology of equiaxed dendrites has a significant impact on the properties of a cast product. During the past few decades, several analytical and computational models have been proposed for studying the growth pattern of equiaxed dendrites. Computational models can simulate the six-fold symmetry of equiaxed dendrites [1], predict the formation of secondary arms [2,3], incorporate the presence of alloying elements and consequently their influence on growth morphology [4], calculate the effect of natural and forced convection [5,6] and handle complex scenarios such as the interaction of multiple dendrites [7]. On the other hand, analytical models can calculate the dendrite tip velocity and radius and provide insight into the relative importance of various physical parameters governing the dendrite growth rate and morphology. The initial analytical models focussed on the dependence of tip growth Peclet number on solutal or thermal diffusion assuming a paraboloidal dendrite shape [8,9]. Subsequently, stability criteria were introduced to calculate the relation between tip radii and velocities and obtain the exact operating radius [10–12]. Later, the micro-solvability theory for dendritic growth was developed to obtain unique values for growth radii based on the anisotropy of interface energy [13]. A few analytical models have also incorporated the effects of convection on the growth rate of dendrites [14,15]. However, none of the existing models consider the presence of shrinkage and the resultant melt flow towards the interface.
☆ Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail address: [email protected] (P. Dutta).
http://dx.doi.org/10.1016/j.icheatmasstransfer.2014.08.003 0735-1933/Published by Elsevier Ltd.
In the present work, a theoretical model is developed to represent the effects of shrinkage driven flow on the growth rate of binary alloy dendrites in terms of an equivalent undercooling of the melt. The ratio of the solid phase density ρs to the liquid phase density ρl is used to derive an expression for this equivalent undercooling which takes into account the interaction of shrinkage flow with solutal and thermal transport in the vicinity of the dendrite. The significance of the equivalent undercooling is that the growth rate predicted (without considering any shrinkage effects) is the same as that predicted using the original melt undercooling and shrinkage effects, for any specified density ratio (ρs/ρl) greater than 1. This is similar in nature to the equivalent undercooling proposed in [16] which, however, uses an imposed uniform forced convection instead of shrinkage driven flow. Results for the equivalent undercooling derived using this theoretical model is compared with corresponding predictions obtained using numerical simulations based on the enthalpy model proposed by the present authors [17] with the addition of shrinkage. The main objective of the present work is to relate the effects of unequal phase densities to the different physical parameters governing dendrite growth using simplifying assumptions, instead of deriving a complete and rigorous exact solution.
2. System The model problem consists of a cylindrical dendrite with radius R growing with velocity Va in a binary alloy melt undercooled to a temperature Tb. The tip of the dendrite is assumed to be hemispherical with radius R. The liquid adjacent to the solidification interface has temperature Ti and concentration Cl⁎, while the bulk liquid has a concentration equal to C0. Fluid flow occurs in the domain due to shrinkage with a flow velocity equal to Vf near the interface. A schematic of the system
A. Bhattacharya, P. Dutta / International Communications in Heat and Mass Transfer 57 (2014) 216–220
217
Fig. 1. Schematic diagram of the domain used for the present model.
along with the temperature and concentration conditions is shown in Fig. 1. The following assumptions are made. 1. 2. 3. 4. 5.
Growth occurs only in the x-direction. The density of the solid phase is higher than that of the liquid phase. Flow is present in the domain only due to shrinkage. Shrinkage flow acts uniformly over the entire hemispherical tip. Diffusion of rejected heat and solute from the interface occurs uniformly from the hemispherical tip.
3.1. Relation between dendrite growth velocity and shrinkage flow velocity The shrinkage flow velocity adjacent to the interface (Vf) is related to the dendrite growth velocity (Va). Per unit time, the volume of flow re2 quired to compensate for shrinkage is given by πR V aρðρs −ρl Þ. This is ball anced by the volume of flow towards the interface which is equal to 2 2 2πR Vf where 2πR is the tip area. Equating the two, the magnitude of shrinkage flow can be stated by the following equation. Vf ¼
3. Derivation of equivalent undercooling The total undercooling of the melt ΔTu is the difference between the undercooled melt temperature Tb and the solidification temperature of the alloy Tm corresponding to the initial concentration C0. The thermal undercooling ΔTt is the difference between the actual interface temperature Ti and the bulk temperature of the liquid Tb, and is the driving force for solidification. The interface temperature Ti differs from Tm due to surface tension effects and partitioning of the solute, which are represented by the curvature undercooling ΔTc and solutal undercooling ΔTs, respectively. Neglecting the effects of kinetics and pressure on the interface temperature, ΔTu can be represented by ΔTu = ΔTt + ΔTs + ΔTc. By definition, the equivalent undercooling ΔTu,eq should be such that the temperature difference at the interface is able to maintain the same growth velocity of the dendrite (without considering any shrinkage effects) as with the original melt undercooling and shrinkage. The procedure for deriving the equivalent undercooling can be described in the following way. At first, equivalent values for the solutal supersaturation and solutal undercooling are derived using solute balance at the interface. Similarly, the equivalent thermal undercooling value is obtained in terms of the original thermal undercooling. Subsequently, a relationship between the thermal undercooling and solutal supersaturation is established. Finally, using the derived relations, the equivalent total undercooling is expressed in terms of the original parameters and the density ratio.
1 ðρs −ρl Þ : V 2 a ρl
ð1Þ
3.2. Solute balance During solidification, solute is rejected at the interface due to partitioning effects. At equilibrium, the rejected solute is carried away by diffusion while the shrinkage induced convection acts in the opposite direction and convects the solute towards the interface. The solute balance at the interface can be stated in the following form. ðC −C 0 Þ 2 2 2 : πR V a ρs 1−kp C l þ 2πR V f ρl C l ¼ 2πR ρl Dl l R
ð2Þ
The first term in Eq. (2) gives the amount of solute rejected per unit time, while the second and third terms represent the convective and diffusive transport of solute, respectively. Dl is the solutal diffusivity, kp is ðC −C 0 Þ is the concentration gradient dethe partition coefficient and l R rived assuming a diffusion length scale equal to the tip radius R. Using Eqs. (1) and (2), we can write the solutal supersaturation Ωc as 2 3 V a R 4ρs ðρs −ρl Þ 5 þ Ωc ¼ 2Dl ρl ρ 1−k l
ð3Þ
p
−C o Þ . where Ωc is defined as Ωc ¼ CðC l 1−k l ð pÞ
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For the equivalent case, the densities of the two phases are equal (i.e. ρs = ρl). Due to the absence of shrinkage driven convection, the only mechanism of solute transport is diffusion. Also, with a modified undercooling ΔTu,eq, the velocity of dendrite growth is the same i.e. equal to Va. Assuming that the tip radius is inversely proportional to the growth velocity, the equivalent tip radius can be assumed to be equal to R. This leads to the following equation for solute balance. 2 2 πR V a ρl 1−kp C l;eq ¼ 2πR ρl Dl
C l;eq −C 0 R
ΔT t ¼ T i −T b ¼ :
ð4Þ
In Eq. (4), Cl,eq⁎ is the equivalent solute concentration at the interface. The subscript ‘eq’ denotes the corresponding equivalent values for each parameter for the same growth velocity and this practice is followed throughout the manuscript. Using Eq. (4), the equivalent solutal supersaturation can be written as Ωc;eq ¼
V aR : 2Dl
ð5Þ
Now, using Eqs. (3) and (5), the following relation can be obtained. Ωc 3: Ωc;eq ¼ 2 ρ ð ρ −ρ Þ s s l 4 þ 5 ρl ρ 1−k l
ð6Þ
p
The equivalent interface concentration Cl,eq⁎ can be derived from Eq. (6) and is given by
C l ;eq ¼ 1−Ωc
C0 2 3: ρs −ρl Þ 5 4ρs þ ð ρl ρ 1−k
=
l
ð7Þ
mC o : 1 1− Ωc 1−kp
ð11Þ
For the equivalent case, the only mechanism of thermal transport is diffusion. A similar heat balance leads to
2
2
πR V a ρl L ¼ 2πR kl
T i;eq −T b;eq R
ð12Þ
and finally to ΔT t;eq ¼ T i;eq −T b;eq ¼
V a R ρl L : 2 kl
ð13Þ
It should be noted that for this case, the interface temperature Ti,eq is different from Ti due to the difference in solute build-up at the interface. Using Eqs. (11) and (13) and noting that the interface temperature Ti is given by Ti = ΔTt + Tb, the equivalent thermal undercooling can be expressed in terms of the original thermal undercooling in the following way. ΔT t : ΔT t;eq ¼ ρs ðρs −ρl Þ cpl þ ðΔT t þ T b Þ ρl L ρl
ð14Þ
3.4. Relation between solutal supersaturation and thermal undercooling The thermal undercooling and solutal supersaturation are coupled together through the growth velocity Va. Using, Eqs. (3) and (11), ΔTt can be written in terms of Ωc as
ð8Þ
Ωc ΔT t ¼
Now, using Eqs. (6) and (8), the equivalent solutal undercooling, ΔT s;eq ¼ −m C l;eq −C o , can be obtained in terms of the original solutal undercooling ΔTs as ΔT s;eq ¼
V a R ρs L ðρs −ρl Þ ρl cpl Ti : þ kl 2 kl ρl
p
It can be observed that, when ρs is greater than ρl, Ωc,eq is less than Ωc ⁎ is smaller than Cl⁎. which means that Cl,eq The solutal undercooling is defined by ΔT s ¼ −mðC l −C o Þ where m is the slope of the liquidus line. ΔTs can be related to Ωc using the following equation. ΔT s ¼
Eq. (10) represents the latent heat released at the interface per unit time, while the second and third terms represent the convective and diffusive transport, respectively. As in the case of solute diffusion, the heat diffusion term is also derived assuming a diffusion length scale equal to R. Eq. (10) leads to the following expression for the thermal undercooling ΔTt
mC 0 2 3: mC 0 4ρs ðρs −ρl Þ 5 1− 1− þ ΔT s ρl ρ 1−k l
ð9Þ
L 1 1 ρs ðρs −ρl Þ cpl Tb þ L cpl Le Aρ ρl ρl L 1 1 ðρs −ρl Þ cpl 1−Ωc L cpl Le Aρ ρl
ð15Þ
where Le is the Lewis number given by Le = kl/(ρlcplDl), and Aρ ¼ ρs ðρs −ρl Þ . þ ρl ρl ð1−kp Þ 3.5. Equivalent total undercooling
p
The equivalent total undercooling can now be derived using the previous relations. From Eq. (14), 3.3. Heat balance At equilibrium, the latent heat of solidification at the interface is removed by diffusion, while the shrinkage flow convects the heat in the opposite direction, i.e., towards the interface. The overall heat balance can be written as 2
2
2
πR V a ρs L þ 2πR V f ρl cpl T i ¼ 2πR kl
ðT i −T b Þ R
ð10Þ
where L, cpl and kl are the latent heat of solidification, specific heat and thermal conductivity of the liquid, respectively. The first term in
ΔT t : ΔT u;eq −ΔT s;eq −ΔT c;eq ¼ ρs ðρs −ρl Þ cpl þ ðΔT t þ T b Þ ρl L ρl
ð16Þ
Now, the curvature undercooling ΔTc is given by ΔT c ¼ 2Γ R , where Γ is the Gibbs–Thomson coefficient and R is the radius of curvature of the interface. The radius of curvature is equal for both the original and the equivalent cases which leads to ΔT c;eq ¼ ΔT c ¼ ΔT u −ΔT t −ΔT s :
ð17Þ
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4. Comparison with numerical simulations
Fig. 2. Comparison of analytical and numerical predictions for equivalent undercooling.
Finally, using Eqs. (9) and (17), the equivalent total undercooling ΔTu,eq can be written as 2 3 1 ΔT u;eq ¼ ΔT u þ ΔT t 4 −15 ρs =ρl þ ðρs −ρl Þcpl =ðρl LÞ ðΔT t þ T b Þ mC 0 2 3 −ΔT s : þ mC 0 4ρs ðρs −ρl Þ 5 1− 1− þ ΔT s ρl ρ 1−k l p
ð18Þ
Noting that ΔTt and ΔTs are both functions of Ωc (Eqs. (15) and (8)), it can be observed that Eq. (18) is an expression for the equivalent total undercooling ΔTu,eq in terms of the original undercooling ΔTu, the density ratio ρs/ρl and the solutal supersaturation Ωc. ΔTu,eq represents the degree of undercooling that needs to be provided to the melt to maintain the same growth rate as in the original case with shrinkage effects. Eq. (18) reduces to ΔTu,eq = ΔTu for ρs/ρl = 1. When ρs is greater than ρl, ΔTu,eq is less than ΔTu, which means that a lower undercooling is sufficient to maintain the same growth rate without the presence of shrinkage.
The equivalent undercooling values given by the present model are compared with corresponding values obtained from numerical simulations. The numerical simulations are performed based on the enthalpy method presented in Bhattacharya and Dutta [17] with the addition of solidification shrinkage. Dendrite growth is tracked by solving the energy and solute transport equations using an explicit formulation as given in [4]. The flow field is calculated using the SIMPLER algorithm [18]. All the governing equations are formulated based on density as a function of liquid fraction. For each time step, at first the progress of solidification is obtained using the previous time step values for the velocity field. An iterative scheme [17] is implemented for maintaining the consistency between the interface temperature and concentration as per the phase diagram. The new densities at each computational node are then calculated based on the updated liquid fraction field, and subsequently, the momentum equations are solved to obtain the new velocities. Simulations are performed for three sets of undercooling values ΔTu = 0.6, 0.65, 0.7 with density ratio ρ s/ρl varying from 1.0 to 1.16 in steps of 0.02. The values for the other parameters are specified as kp = 0.1, mC0 = 0.1, Le = 1.0, cps = cpl = 1.0, ks = kl = 1.0 and L = 1.0. A domain size of 1000 × 1000 with a grid size equal to Δx* = Δy* = 4 and a time step equal to Δt* = 1.0 is used for the simulations. For each simulation, when the growth rate becomes steady, the average tip velocity of the four dendrite arms is calculated. Additional simulations are performed without shrinkage effects (ρs/ρl = 1) with undercooling varying by small steps of 0.005. From these simulations, the undercooling values are noted for which the tip velocities are equal to those with shrinkage. This gives the numerical equivalent undercooling values. The analytical equivalent undercooling values are obtained using Eq. (18) and compared with the corresponding numerical values in Fig. 2. For calculating the analytical values, Ωc is obtained from the simulations. It can be observed that the equivalent undercooling values match closely for both the methods and linearly decrease with increasing density ratio. This is expected, because the presence of shrinkage decreases the growth rate due to a number of factors. Shrinkage results in convection towards the interface which hinders the movement of solute into the bulk liquid, thus increasing the concentration of solute at the interface. This decreases the interface temperature, reducing its difference with the bulk liquid temperature, and consequently slows the growth velocity. Additionally, the shrinkage driven flow also opposes the
Fig. 3. Comparison of dendrite shapes at t = 20, 000 from numerical studies with shrinkage and without shrinkage to that with simulations using equivalent undercooling: (a) For density ratio ρs/ρl = 1.04 (b) For density ratio ρs/ρl = 1.16.
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removal of latent heat from the interface, further reducing the growth rate. Also, since the density of the solid is higher than that of the liquid, for the same mass of solid formed due to certain amount of latent heat removal, the volume of solid formed is less, resulting in the reduction of dendrite size. From Fig. 2, it can also be observed that there is a significant change in undercooling for high density ratios which proves that shrinkage is an important factor in determining the growth rate of dendrites. The actual shapes of the growing dendrites are compared using two sets of results from the previous simulations. Fig. 3(a) and (b) shows the dendrite shapes at time t = 20, 000 for ΔTu = 0.6 and ρs/ρl = 1.04 and ρs/ρl = 1.16, respectively. The equivalent undercooling values for the two cases are calculated using Eq. (18) and found to be equal to 0.581 for ρs/ρl = 1.04 and 0.537 for ρs/ρl = 1.16. Dendrite growth from simulations assuming no density change (ρs/ρl = 1) and the appropriate equivalent undercooling values are shown by the bold lines while the numerical results with the prescribed undercooling and density difference are represented by the dashed lines. The growth of the dendrite without any shrinkage effects is shown by the dash–dot lines. It can be observed that the presence of shrinkage considerably slows down the growth rate. This decrease in growth rate is captured quite accurately using the equivalent undercooling model and the shapes predicted using the equivalent undercooling without any shrinkage are very similar to the original cases with shrinkage effects. It should be noted here that simulating the growth of a dendrite in presence of shrinkage necessitates the solution of the mass and momentum conservation equations (to track the shrinkage driven convection) in addition to the energy and species conservation equations. The use of an equivalent undercooling for the numerical simulations removes the requirement of solving the mass and momentum conservation equations while maintaining a high level of accuracy in predicting the dendrite shapes and growth rates. As a result, the computational effort for performing the numerical simulations is significantly reduced. 5. Conclusions A theoretical analysis was presented for studying the effects of shrinkage on the growth of dendrites. The presence of shrinkage flow and the resultant variation in the solutal and thermal transport mechanisms were represented by an equivalent undercooling of the melt. The model predictions show excellent agreement with corresponding results from numerical simulations. The present work provides an insight
into the relative importance of shrinkage in determining the growth rate and morphology of dendrites as compared to other factors such as melt undercooling, solute concentration and so forth. Additionally, the use of an equivalent undercooling for numerical simulations removes the necessity of solving for the shrinkage driven convection, thus resulting in significant reduction of the required computational effort.
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