The variation of density with respect to temperature in four metalsâalumi- num, nickel ..... force (CSF) method [20] to model the surface tension and employed Youngs' [21] ..... H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, pp.
Numerical Heat Transfer, Part B, 47: 507–531, 2005 Copyright # Taylor & Francis Inc. ISSN: 1040-7790 print=1521-0626 online DOI: 10.1080/10407790590928964
THREE-DIMENSIONAL MODELING OF DENSITY VARIATION DUE TO PHASE CHANGE IN COMPLEX FREE SURFACE FLOWS M. Raessi and J. Mostaghimi Centre for Advanced Coating Technologies, Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, Ontario, Canada A three-dimensional model of droplet impact and solidification has been modified to include the effects of density variation during phase change. The governing equations for conservation of mass, momentum, and energy, and a volume-of-fluid (VOF) equation are derived by assuming different but constant solid and liquid densities. The equations are solved numerically using a control-volume approach. The model is validated against the Stefan and planar solidification problems. It is then applied to simulate the effects of density variation during solidification of molten tin in a mold and also of an impacting tin droplet on a substrate.
1. INTRODUCTION There are numerous processes in science and industry in which solid–liquid phase change occurs, e.g., metal casting, welding, coating processes, etc. As a result, there have been many attempts to model phase-change phenomena. The earliest mathematical models for heat flow during solidification were developed to predict the rate of polar ice formation [1]. Neumann first presented the well-known error function solution for one-dimensional melting and solidification in 1860 [2]. Other than the simplest problems, modeling the phase-change process is complicated due to the difficulty of accounting for nonlinear phenomena that change in time and space. In the basic case of solidification of a pure material, this involves tracking a sharp, moving solid–liquid interface across which the material properties are discontinuous, and latent heat is evolved. Extensive reviews on numerical methods for phase-change problems can be found in [3–5]. Furthermore, in a material undergoing phase change, there is a change of volume due to the difference in solid and liquid densities. Since most metals and alloys have lower densities in the liquid phase (gallium and bismuth are two exceptions), density variation during phase change, referred to as solidification shrinkage, leads to formation of cavities or porosity. In fact, during solidification of a material, when the solid phase is more dense than the liquid phase, the density difference leads Received 18 November 2004; accepted 30 December 2004. Address correspondence to J. Mostaghimi, Department of Mechanical and Industrial Engineering, University of Toronto, 5 King’s College Road, Toronto, Ont., Canada M5S 3G8. E-mail: mostag@mie. utoronto.ca
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NOMENCLATURE A cp e Ec f ~ fb ~ fb h h hfus hfus k m ^ n p p ~ P Pr ~ q q~0 R Re S t t T Tm T
area specific heat internal energy per unit mass Eckert number ð¼u2o =cpo Tm Þ material volume fraction body force per unit mass dimensionless body force per unit mass ð¼~ f b R=u2o Þ enthalpy per unit mass dimensionless enthalpy ð¼h=cpo Tm Þ heat of fusion dimensionless heat of fusion ð¼hfus =cpo Tm Þ thermal conductivity mass unit normal vector pressure dimensionless pressure ð¼p=qo u2o Þ force vector per unit area Prandtl number ð¼mo cpo =ko Þ heat flux in liquid phase heat flux in solid phase characteristic length Reynolds number ð¼qo uo R=mo Þ surface; function of enthalpy time dimensionless time ð¼uo t=RÞ temperature melting temperature dimensionless temperature ½¼ðT Tm Þ=ðTm Tw Þ
~ u uo ~ u V ~ x ~ x a b C dij dt h k m m n q q r r
velocity vector characteristic velocity dimensionless velocity vector (¼~ u=uo Þ volume position vector dimensionless position vector ð¼~ x=RÞ thermal diffusivity volumetric shrinkage ½¼ðqs ql Þ=ql function of enthalpy Kronecker delta time step solid–liquid volume fraction second viscosity coefficient dynamic viscosity dimensionless viscosity ð¼m=mo Þ kinematic viscosity density dimensionless density ð¼q=qo Þ stress tensor del operator ð¼RrÞ
Subscripts and Superscripts i, j, k l n; n þ 1 o s w
mesh indices liquid time levels initial; reference solid wall nondimensional
the melt to move toward the solid front, and so induces a velocity field in the melt. This flow can affect the solidifying material in several ways: it can change the shape of a final product, cause porosity defects, change the species concentrations in alloys, and consequently lead to poor mechanical properties. More generally, density variations occur in both the solid and liquid phases due to changes in temperature, known as solid or liquid contraction. Liquid contraction can also drive convective motion in the melt, while solid contraction can induce thermal strain in the solid phase, leading to plasticity phenomena as well as to residual stresses. The work presented here is focused on modeling density variation during solidification of droplets impacting a surface, which is of interest in thermal spray coating processes. Thermal sprayed coatings are applied to protect substrates against wear, corrosion, and thermal shock. In thermal spray processes a hot gaseous jet is used to melt and accelerate the powder of a metallic or ceramic coating material. The hot jet draws energy from either a plasma or a combustion source. During these
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processes, a spray of molten (or partially molten) particles, or droplets, is directed at a substrate. As the droplets impact the substrate, they spread and solidify, each forming a so-called splat. A coating forms as a result of the accumulation of many such splats. To understand phenomena related to the deformation, spreading, and solidification of impacting droplets, and therefore to enhance knowledge about the thermal spray coating process, numerical models have been developed to study the impact and solidification of droplets. Bussmann et al. [6] developed a three-dimensional model of droplet impact onto asymmetric surface geometries. The model is based on RIPPLE [7]. Pasandideh-Fard et al. [8] subsequently added heat transfer and phase change to the three-dimensional model, but subject to the assumption that ql ¼ qs . Experiments of droplet impact and solidification have also been done in which effects of density variation have been observed. Images shown in Figure 1, of the impact and solidification of nickel droplets on a stainless steel substrate, serve as an example. Figure 1a illustrates the curling up of the splat edge, which is due to shrinkage-induced stress. This behavior is not desirable in the coating process, since it can result in porosity formation in the coating. Figure 1b illustrates the empty space around the edge of a splat that can remain unoccupied by any coating material even if two splats overlap in an area. This consequently causes porosity in the coating. As well, depending on whether the residual stress is tensile or compressive, such behavior can cause cracks on the coating surface or cause the coating to peel off the substrate. The variation of density with respect to temperature in four metals—aluminum, nickel, tin, and titanium—is shown in Figure 2a. Ni, Ti, and Al are widely used as a base for the alloys used in the thermal spray process. In Figure 2a, it is very well shown that the density variation during phase change is a considerable share of the total volume shrinkage in the material. Therefore, as the first step in modeling the effect of density variation on droplet impact and solidification, the effect of density variation during phase change must be studied.
Figure 1. Curling up of nickel splats due to shrinkage-induced stress. Splats are formed by impact and solidification of nickel droplets on a stainless steel substrate at 400C.
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Figure 2. (a) Density variation versus temperature of aluminum, nickel, tin, and titanium. (b) An arbitrary control volume consisting of liquid and solid phases that undergo phase change.
We conclude this introduction with a brief survey of published models of solidification shrinkage. In general, present algorithms are limited by their inability to model three-dimensional complex moving-boundary problems, such as droplet impact and solidification. Extending these algorithms to general three-dimensional models is extremely difficult, if not impossible, because of the problem complexity and lack of generality in the algorithms. We begin by presenting algorithms that have been developed for modeling solidification shrinkage during phase change in a quiescent bulk of liquid in a simple open geometry. Trovant and Argyropoulos [9] investigated shrinkage in a cylindrical casting of aluminum and magnesium. They presented a two-dimensional axisymmetric fixedgrid numerical model for phase-change problems which solves for heat transfer and fluid flow, and accounts for volumetric shrinkage. At each time step, effective shrinkage was calculated and then material was subtracted from control volumes in the liquid state. A similar model was presented by Beech et al. [10] for solidification shrinkage in cylindrical and T-shaped castings that they incorporated into the Flow-3D [11] software. They only solved the energy equation for the cast and the mold and then predicted the volume and location of shrinkage by evaluating the volume change due to solidification shrinkage in each isolated liquid region of the cast. Kim and Ro [12] performed a similar study on shrinkage formation during solidification of a material in a 2-D rectangular cavity. To account for the solidification shrinkage, they used an overall mass balance to determine explicitly the new locations of the free surface and solid–liquid interface using a geometric analysis. Each of these algorithms is applicable only to a quiescent liquid in a simple geometry. In a complex system, in which the liquid velocity field and the free surface are both complicated (such as a droplet impact and solidification problem), simple algorithms such as these cannot predict the effect of density variation during phase change. The following are examples of more general algorithms to model the effect
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of density variation during the solidification of materials. These algorithms were applied to two-dimensional models of alloy solidification. McBride et al. [13] developed a two-dimensional model of the directional solidification of dendritic binary alloys which included the effect of shrinkage-induced flow. The model was developed under the assumption that the densities in the liquid and solid phases are different but constant. This led to a nonhomogeneous mass conservation equation, which was treated numerically with a variable penalty coefficient in a finite-element formulation. McBride et al. studied the interaction between shrinkage-induced flow and buoyancy flow in solidification of binary alloys at low gravity levels. Similarly, Chiang and Tsai [14] analyzed the fluid flow and domain change caused by solidification shrinkage in a two-dimensional rectangular cavity filled with molten alloy. They modified the continuum equations to include shrinkage-induced fluid flow assuming different but constant densities for the liquid and solid phases. Since the problem involved a distinct free surface at the top of the riser, they placed moving nodes on the free surface to track the physical domain of the riser. However, as they assumed the free surface to remain flat, the movement was one-dimensional. Also, Naterer [15] performed a study on shrinkage-induced flows during solidification of a binary constituent alloy using a two-dimensional model. Utilizing a control-volume-based finite-element method, before solving the fluid flow equations, the continuity equation was replaced by a pressure specification along the free surface. Then, the velocity and pressure fields were obtained from the coupled solution of the fluid flow equations. To conserve mass along the free surface, a mass flow, which was defined as a function of the local free surface area and was equal to the net mass inflows=outflows of the free surface elements, was supplied to displace the free surface. Finally, using a two-dimensional finite-volume model based on volume averaging, Ehlen et al. [16] predicted the transient shape of the melt surface, the shrinkage cavity, and their influence on the final solute distribution of AlSi alloy. They considered source and exchange terms in the equations of conservation of mass, momentum, and energy, as well as an equation of species conservation to reflect the density difference and associated effects. Additionally, to apply the VOF model [17] to the case of nonconstant densities, they modified the classical donor– acceptor flux approximation. This article, then, presents an algorithm applied to a three-dimensional model of droplet impact and solidification, developed by Bussmann et al. [6] and Pasandideh-Fard et al. [8], that accounts for the solidification shrinkage. The algorithm models the volumetric shrinkage and the induced flow due to density variation during phase change of nonquiescent problems with complicated geometries.
2. MATHEMATICAL FORMULATION AND NUMERICAL MODEL 2.1. Mathematical Formulation The fluid flow model that is utilized here is the three-dimensional flow model of Bussmann et al. [6], which is also incorporated in the model of Pasandideh-Fard et al. [8]. A detailed discussion of the flow model is given in [6]. We review that model briefly, and then detail the additional assumptions that are made according to the objective of the current study.
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In the flow model, the liquid is assumed to be incompressible and Newtonian, and the flow is assumed to be laminar. Effects of ambient air on the flow are neglected. As a consequence of these assumptions, the stress on the free surface of the liquid is assumed to be normal and the shear stress is neglected. Solid and liquid densities are assumed to be different but constant. In other words, only the effect of density variation during phase change is considered, and neither solid nor liquid contraction is taken into account. The solidified material is assumed to be immobile. The governing equations for an arbitrary control volume consisting of liquid and solid phases that undergo phase change are derived based on the following definitions. In the presence of a solid phase, the standard volume-of-fluid (VOF) scalar function f is redefined as a material volume fraction. f is defined to be one within the material (be it solid or liquid), and is defined to be zero outside of the material. To treat the presence of a solid phase and the movement of an irregularly shaped solid front, Pasandideh-Fard et al. [8] used a modified version of a fixed-grid method to treat the solid and liquid phases in the domain. To represent the liquid part of the material, Pasandideh-Fard et al. [8] defined a second scalar function, h, as the solid–liquid volume fraction. In this definition, h is equal to one in the liquid and zero in the solid, and in the special case of two-phase flow, the volume fractions of the liquid and solid phases are h and 1 h, respectively. In an arbitrary control volume (Figure 2b) of constant mass that consists of both solid and liquid phases, if m represents the total mass of the control volume, we have D D ðmÞ ¼ ðml þ ms Þ ¼ 0 Dt Dt where the solid and liquid masses are defined as Z Z ml ¼ ql dVl ¼ ql h dV Vl
ms ¼
Z Vs
qs dVs ¼
ð1Þ
ð2Þ
V
Z
qs ð1 hÞ dV
ð3Þ
V
and V represents the volume of the control volume. By substituting Eqs. (2) and (3) in Eq. (1), we obtain Z D ½ql h þ qs ð1 hÞ dV ¼ 0 ð4Þ Dt V Using Reynolds’ transport theorem (RTT) and assigning zero velocity to the solid phase, Eq. (4) becomes Z q q q ðql hÞ þ ðql huk Þ ðqs hÞ dV ¼ 0 ð5Þ qxk qt V qt
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q qh ðhuk Þ ¼ b qxk qt
ð6Þ
which implies that
where b ¼ ðqs ql Þ=ql . Equation (6) reflects conservation of mass. b represents the volumetric variation during phase change and qh=qt represents the rate of phase change. If b > 0 and the material is solidifying ðqh=qt < 0Þ, then the material volume will shrink as solidification occurs and the velocity divergence will be negative in the liquid phase. More discussion of the term bðqh=qtÞ will be presented later. The principle of conservation of momentum, which is basically Newton’s second law, is applied to the above-mentioned control volume. If ~ f b represents the resultant external body forces which may act on the unit mass of the control volume and r represents the stress tensor, we can write: D Dt
Z
ql uj dV ¼
Vl
Z
qrij dV þ Vl qxi
Z ql fbj dV
ð7Þ
Vl
Equation (7) could be written for the solid phase as well, but since the solid has been assigned a zero velocity, the left-hand side of that equation goes to zero (zero acceleration) and so the equation changes to the equilibrium equation. Therefore, the momentum equation is written only for the liquid phase and it is as follows: Z Z Z qrij D q uj h dV ¼ h dV þ ql fbj h dV ð8Þ Dt V l V qxi V Applying the RTT to Eq. (8) and equating the integrands on both sides results in qrij q q ql huj þ hui uj ¼ h þ hql fbj ð9Þ qt qxi qxi in which the stress tensor is quk qui quj rij ¼ pdij þ kdij þm þ qxk qxj qxi
ð10Þ
By taking the derivatives of Eq. (10) and assuming the bulk viscosity to be zero (k ¼ 2m=3) [18], and multiplying both sides by h, we obtain h
qrij q2 uj qp mh qui ¼ h þ mh þ qxj 3 qxi qxj qxi qxi qxi
ð11Þ
Substituting Eq. (11) into Eq. (9), expanding the term qðhui uj Þ=qxi , and replacing the velocity divergence by the corresponding term from Eq. (6) leads to ql
quj q2 uj q qh qp mh q2 ui ðhuj Þ þ b uj þ hui þ mh þ þ hql fbj ¼ h qt qt qxj 3 qxi qxj qui qxi qxi
ð12Þ
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Equation (12) is the equation of conservation of momentum for the abovementioned control volume. Finally, the principle of conservation of energy for a fluid mass implies that D Dt
Z Z Z Z 1 u ~ u dV ¼ ~ qe þ q~ u~ P dS þ ~ u q~ fb dV ~ q ^n dS 2 V S V S
ð13Þ
By converting the surface integrals to volume integrals and relating the force vector ~ P to the stress tensor rij by Pj ¼ rij ni , Eq. (13) in the presence of a solid phase becomes Z Z Z D 1 q ql el þ ql uj uj h dV ¼ ðuj rij Þh dV þ uj ql fbj h dV Dt V 2 V qxi V Z qqj h dV ð14Þ V qxj Using the RTT and equating the integrands on the both sides of Eq. (14) results in q 1 q 1 ql el þ ql uj uj h þ ql el þ ql uj uj huk qt 2 qxk 2 qqj q ¼h ðuj rij Þ þ hql uj fbj h ð15Þ qxi qxj Expanding both sides of Eq. (15) and replacing the corresponding terms by the equations of conservation of mass and momentum leads to quj qqj qel qel 1 2 qh ¼ hrij þ uk hql h ð16Þ þ qs e l u j 2 qt qt qxk qxi qxj But, quj quk ¼ p þU qxi qxk where U is the dissipation function, which in general form is as follows: rij
2 quk qui quj quj þm þ U¼k qxk qxj qxi qxi
ð17Þ
ð18Þ
Transforming the internal energy to enthalpy and pressure by el ¼ hl p=ql , considering heat transfer as qqj q qT ¼ kl ð19Þ qxj qxj qxj and then inserting Eq. (17) into Eq. (16) leads to Dhl 1 qh DðhpÞ q qT ¼ þ hU þ h þ qs hl u2j kl hql 2 qt Dt qxj qxj Dt Equation (20) is the equation of conservation of energy in the liquid phase.
ð20Þ
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For the solid phase, the energy equation reduces to D Dt
Z V
qs es ð1 hÞ dV ¼
Z
qq0j V
qxj
ð1 hÞ dV
ð21Þ
By using Reynolds’ transport theorem and equating the integrands of both sides and assigning zero velocity to the solid phase, we obtain qq0j qes qh es qs ð1 hÞ ¼ ð1 hÞ qt qt qxj
ð22Þ
The internal energy in the solid phase can be assumed to be equal to its enthalpy, and heat transfer can be considered as qq0j q qT ¼ ks qxj qxj qxj Equation (22) then changes to qhs qh q qT hs ks qs ð1 hÞ ¼ ð1 hÞ qt qxj qxj qt
ð23Þ
ð24Þ
Now, by considering the fact that at melting point hs ¼ hl hfus , Eqs. (20) and (24) can be combined: qh qh 1 2 qh ½q þ ql uk þ qs hfus uj qt qxk 2 qt DðhpÞ q qT q qT þh ¼ kl ks þ ð1 hÞ þ hU ð25Þ Dt qxj qxj qxj qxj where ½q ¼ hql þ ð1 hÞqs . Equation (25) is the conservation-of-energy equation, which is valid in both the solid and liquid phases of the control volume. The term qs ðhfus 12 u2j Þ qh=qt is related to the phase change in the control volume, qs hfus qh=qt represents the latent heat release (or absorption) during phase change, and the term 12 qs u2j qh=qt represents the kinetic energy loss (or gain) in the moving liquid due to solidification (or melting). The general form of the VOF equation, which is basically an advection equation for f , is as follows: qf qf þ uk ¼0 qt qxk
ð26Þ
However, in the presence of a solid phase, Eq. (26) must be modified. By employing the modified definition of material volume fraction f and using Eqs. (1)–(5) and the RTT, after equating the integrands of corresponding integrals and expanding
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the terms, we obtain qh qf q qf qf qh þh þf ql f ðhuk Þ þ huk ¼0 þ qs ð1 hÞ f qt qt qxk qxk qt qt
ð27Þ
From Eq. (6), the term qðhuk Þ=qxk can be replaced by bðqh=qtÞ: ½hql þ ð1 hÞqs
qf qf þ hql uk ¼0 qt qxk
ð28Þ
or qf qf qf þ huk ¼ bðh 1Þ qt qxk qt
ð29Þ
This concludes the derivation of the governing equations. Nondimensional variables have been defined in the Nomenclature. In nondimensional form, equations of conservation of mass, momentum, and energy, and the VOF equation [Eqs. (6), (12), (25), and (29)], are as follows: qh ~ ðh~ r u Þ ¼ b qt ql
qðh~ u Þ qh ~ ~ r Þ~ u þ b u þðh~ u qt qt hm ~ ~ ~ p þ hm r2~ fb r ðr ~ u Þ þ hql ~ u þ ¼ hr Re 3 Re qh 1 qh 2 ~ u j u r Þh þ qs hfus Ec j~ ½q þ ql ð~ 2 qt qt Dðhp Þ hTm ~ ~ T Þ r ðkl r ¼ Ec þ Dt Pr Re ð1 hÞTm ~ ~ T Þ þ Ec h U þ r ðks r Re Pr Re
ð30Þ
ð31Þ
ð32Þ
where Tm ¼ ðTm Tw Þ=Tm . qf ~ Þf ¼ bðh 1Þ qf þ ðh~ u r qt qt
ð33Þ
We now briefly consider the order of magnitude of nondimensional numbers for two problems: 1. Impact and solidification of tin droplets of 0.3- and 1.35-(mm) radius impacting a substrate at 1, 10, and 40 (m/s) 2. Solidification of stationary molten aluminum in an 8-(cm)-diameter cylindrical mold The results are listed in Tables 1 and 2.
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For the tin droplet impact, Ec=Re is the smallest of the coefficients in Eq. (32), and so the dissipation function term associated with Ec=Re is neglected. In addition, at low impact velocities Ec is one order of magnitude less than the rest of the coefficients. However, it should be noted that at high velocities ½40 ðm=sÞ, Ec is an order of magnitude greater than 1=Pr Re. Therefore, although it is generally recommended to consider the terms associated with Ec, for the problems of interest in this work it is reasonable to neglect the terms associated with Ec in Eq. (32) (as they were in [8]). Finally, the dimensionless heat of fusion ðhfus Þ is of the order of 101 in this problem. Hence, its associated term in Eq. (32) will be considered. For the problem of aluminum solidification, the order of magnitude of hfus is 1 10 . Therefore, the dominant coefficients in the equation are 1=Pr Re and hfus , and the terms associated with Ec and Ec=Re can be neglected. It should be mentioned that the velocity of liquid aluminum flow driven by density variations during phase change was estimated by McBride et al. [13] to be of order 103 (m=s) under the normal gravitational field. However, a wider range of characteristic velocities is presented in Table 2 to provide a more general analysis of the order of magnitudes in Eq. (32).
2.2. Numerical Model A three-dimensional fluid flow model [6, 19] has been developed based on a 2-D fixed-grid Eulerian code called RIPPLE [7], which was developed specifically for solving free surface flows with surface tension. In the 3-D model, fluid flow is modeled by using a finite-volume method for Eqs. (6), (12), and (29) in a threedimensional Cartesian coordinate system. In each computational cell, velocities and pressure are defined at the center of cell faces and at the cell center, respectively. To obtain the material volume fraction in a cell ðfi;j;k Þ, the scalar function f is integrated over the cell volume Vi;j;k : fi; j; k ¼
Z
1 Vi; j; k
ð34Þ
f dV Vi; j; k
It then follows that for a cell fully occupied with material, fi;j;k ¼ 1, for an empty cell, fi;j;k ¼ 0, and for a cell on the free surface, 0 < fi;j;k < 1.
Table 1. Nondimensional numbers in the tin droplet impact and solidification problem Radius (mm) 0.3
1.35
Impact velocity (m=s)
Re
Ec
1=Pr Re
Ec=Re
1 10 40 1 10 40
1,091 10,910 43,640 4,909 49,090 196,360
8 106 8 104 1:3 102 8 106 8 104 1:3 102
6:5 102 6:5 103 1:6 103 1:5 102 1:5 103 3:6 104
7 109 7 108 3 107 1:6 109 1:6 108 7 108
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M. RAESSI AND J. MOSTAGHIMI Table 2. Nondimensional numbers in the solidification of stationary aluminum melt in an 8-(cm)-diameter cylindrical mold Radius (m) 4 102
Characteristic velocity (m=s)
Re
Ec
1=Pr Re
Ec=Re
105 103 1
102 1 1,136
9 1017 9 1013 9 107
100 1 8 104
9 1015 9 1013 7 1010
Also, for each cell, the solid–liquid volume fraction is defined as hi;j;k ¼ Vl =ðVl þ Vs Þ. Therefore, cells in which h is between zero and one are solidifying (or melting) cells and are termed ‘‘partial-flow cells.’’ The solid is treated as a liquid of infinite density and zero velocity. Hence, in a partial-flow cell only the h portion of the material volume inside the cell is open to flow and the remaining portion (1 h) that is occupied by the solid is closed to flow [8]. Equations (6) and (12) are solved by using a two-step projection method that is discussed by Bussmann et al. [6, 19]. In this method, a time discretization of the momentum equation is split into two steps: n ~ h~ u ðh~ uÞ n qh t ~ ~ ~u~n þ thn r2 u~n þ h~ ¼ ½ðh~ uÞ n r f b b u~n þ hnr ðr ~ uÞ qt 3 dt
~ ðh~ uÞnþ1 ðh~ uÞ hn ~ nþ1 ¼ nr p dt q
ð35Þ
ð36Þ
~ In the first step, an interim velocity ~ u is computed explicitly by considering the effects of the convective, viscous, body force, and density change terms. In the second step, Eq. (6), which is the nonzero divergence of the velocity field, is combined with Eq. (36) at time level n þ 1 to yield an implicit equation for pressure: " # n 1 qh nþ1 ~ h ~ nþ1 ~ ~ b r ðh~ uÞ ¼ r rp dt qt qn
ð37Þ
A solution for pressure is obtained at each time step from Eq. (37), and finally the new velocity field ~ unþ1 is evaluated via Eq. (36). The term bðqh=qtÞnþ1 =dt in Eq. (37) acts as a source term in the implicit pressure equation. This term has a significant role in accommodating the effects of density variation during phase change (e.g., volume shrinkage). In a cell which is undergoing phase change, this source term will result in a negative or positive pressure gradient that depends on whether the material is solidifying or melting and on the ratio of solid to liquid phase densities. For example, if a material with qs > ql solidifies, this source term will result in a negative pressure gradient that causes liquid to be pulled toward the solidification front. Therefore, in a solidifying cell a velocity field will be induced toward the solid front, due solely to the density variation during phase change (solidification shrinkage).
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In the flow model, Bussmann et al. [6, 19] implemented the continuum surface force (CSF) method [20] to model the surface tension and employed Youngs’ [21] volume-tracking algorithm to solve the VOF equation. These methods are well addressed in previous publications [6, 20, 21] and will not be repeated here. Based on the order-of-magnitude analysis presented previously, the conservation-of-energy equation [Eq. (25)] reduces to ½q
qh qh qh q qT q qT þ ql u k ¼h þ qs hfus kl ks þ ð1 hÞ qt qxk qt qxj qxj qxj qxj
ð38Þ
Since this equation has two dependent variables (temperature T and enthalpy h), the enthalpy-transforming model of Cao et al. [22] is used [23] to convert Eq. (38) into an equation strictly for enthalpy. The main advantage of this method is that it solves the energy equation for both phases simultaneously. Equation (38) then becomes ½q
qh qh qh q2 q2 S þ ql u k þ qs hfus ¼ 2 ðChÞ þ 2 qt qxk qt qxk qxk
ð39Þ
where 8 > < ks =cps 0 CðhÞ ¼ > : kl =cpl
h0 0 < h < hfus h hfus
ð40Þ
and 8 > : hfus kl =cpl
h0 0 < h < hfus
ð41Þ
h hfus
Thus, the energy equation has been transformed into a nonlinear equation with a single dependent variable, enthalpy (h). It should be mentioned that Eq. (39) was derived assuming that phase change occurs at a single temperature Tm (isothermal phase change), the case for pure materials. For alloys in which phase change occurs over a range of temperature, Eq. (39) is still valid but functions C and S have different forms [22]. The temperature in the solid or liquid phase can be calculated from the enthalpy by the following relationship: 1 T ¼ Tm þ ðCh þ SÞ k
ð42Þ
The solid–liquid volume fraction (h) in a computational cell is evaluated by using the algorithm of Voller and Cross [24]. In a cell that is undergoing phase change, the rate of change of the enthalpy of the cell is proportional to the velocity of the phase-change front, with the latent heat of phase change as the constant of
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proportionality: hfus
qhi;j;k qhi;j;k ¼ qt qt
t1 < t < t2
ð43Þ
t1 and t2 correspond to h ¼ 1 (fully liquid; hi; j;k ¼ hfus ) and h ¼ 0 (fully solid; hi; j;k ¼ 0), respectively. Since hfus is assumed to be constant, Eq. (43) leads to hi; j;k ¼
hi; j;k hfus
0 < hi; j;k < hfus
ð44Þ
Equation (44) does not take into account the variation in density with phase change. To account for that, the solid–liquid volume fraction calculated from Eq. (44) must be adjusted in the following way: h0 hi; j;k ¼ 0 ð45Þ h þ ð1 hn Þ þ ðql =qs Þðhn h0 Þ i; j;k where h0 is the solid–liquid volume fraction evaluated from Eq. (44) and hn is the solid–liquid volume fraction of the previous time step. Flow and thermal boundary conditions must be applied at fluid=solid interfaces (solidification front or substrate), at the fluid free surface, and at any symmetry boundaries. The applications of boundary conditions are discussed in detail in [6, 8] and will not be repeated here. The numerical procedure of Pasandideh-Fard et al. [8] for solving the governing equation has been modified to include the effect of density variation during phase change. The new procedure is as follows: 1. Given the values at the time level n, the interim velocity ~ u~ is calculated as described earlier [Eq. (35)]. 2. Based on the known values of ~ un , hn , and hn , the energy equation [Eq. (39)] is solved for the enthalpy field and the solid–liquid volume fraction values at time level n þ 1. 3. The implicit pressure equation [Eq. (37)] is solved for the pressure field at time level n þ 1, and then the new velocity field is calculated from Eq. (36). 4. The appropriate flow and thermal boundary conditions are applied on the free surface, the solid front, and the boundaries of the computational domain. 5. Equation (29) is solved via Youngs’ [21] method for the configuration of the free surface at time level n þ 1. Repetition of these steps advances the solution through an arbitrary time interval. To maintain stability of the solution, restrictions must be placed on the explicitly evaluated sections of the numerical model [8, 19]. These restrictions are related to the Courant number, surface tension, viscosity, and thermal diffusivities in the liquid and solid phases. The particular time step chosen at any time level is then the minimum value allowed by any of these restrictions.
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3. MODEL VALIDATION AND RESULTS The Stefan problem, for which an analytical solution is available [2], was used as the first validation problem for the numerical model. In this problem, a semi-infinite expanse of molten metal is suddenly brought into contact with an isothermal wall at a temperature below the melting point of the molten metal. In our simulation, molten tin initially at 233C (melting point of tin: 232C) was brought into contact with an isothermal wall at 0C. The contact resistance between the wall and the molten tin was assumed to be zero. The analytical solution, known as Neumann’s solution, for the solid-front position is given by the expression [2] pffiffiffiffiffiffi X ðtÞ ¼ 2k as t
ð46Þ
where k is a constant that satisfies the following relationship: pffiffiffiffiffiffiffiffiffiffiffi
pffiffiffi expðk2 Þ kl ðT1 Tm Þ as =al expðk2 q2s as q2l al Þ khfus p qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ cps Tm erfðkÞ k T erfcðk q2 a =q2 a Þ s
m
s s
l
ð47Þ
l
The physical properties of tin in the solid and liquid phases are listed in Table 3 [25, 26]. For this problem, k was calculated from Eq. (47) to be 0.6002. Therefore, the location of the solid front is known from Eq. (46). To calculate the same problem numerically, a cubical column of molten tin was considered. The schematic of the numerical domain is depicted in Figure 3a. The initial height of the molten tin was considered high enough (10 times longer than the portion of the domain monitored for the solid-front location) to prevent the solution from being affected by the free surface boundary. Free-slip and adiabatic boundary conditions were applied on the side walls of the domain, and a constant temperature (0C) was applied at the interface between the molten tin and the wall. Figure 3b illustrates a comparison of the numerical and analytical solutions. The solid line shows the analytical solid-front location and the dots are the numerically predicted solid-layer thickness at different time sequences. The numerical result agrees well with the analytical solution. The second validation problem is the planar (one-dimensional) solidification of molten aluminum. In this problem, a slab of molten aluminum of a finite thickness Table 3. Physical properties of tin and aluminum in the solid and liquid phases
Material Tin
Phase
Liquid Solid Aluminum Liquid Solid
Thermal Specific Density conductivity (W=m K) heat (J=kg K) (kg=m3) 6,970 7,200 2,368 2,550
33.6 62.2 98.1 237
244 243 1,176.7 903
Kinematic viscosity (m2s) 2:7 107 — 2:07 107 —
Surface Latent tension heat Melting (N=m) (kJ=kg) point (C) 0.526 — 0.914 —
60.9 387
232 660
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M. RAESSI AND J. MOSTAGHIMI
Figure 3. (a) Schematic of the numerical domain for the Stefan problem. (b) Analytical and numerical results for the Stefan problem.
Ho is initially at a constant temperature Ti > Tm . The slab is cooled from below by contact with an isothermal wall at temperature Tw < Tm . Solidification begins at the bottom and the solid front moves toward the free surface. Since aluminum is denser in the solid phase than in the liquid phase, the liquid free surface is pulled toward the solid front by solidification shrinkage. Thus, from conservation of mass, the final height of the aluminum slab H after complete solidification is H¼
ql Ho qs
ð48Þ
To simulate this problem by the proposed numerical model, a 10-(mm)-high cubical column of molten aluminum at 661C was considered. The bottom surface of the molten aluminum was put in contact with an isothermal wall at 25C. It was assumed that there was no contact resistance between the wall and the molten aluminum. Free-slip and adiabatic boundary conditions were imposed on the side walls of the numerical domain. The physical properties of aluminum in the solid and liquid phases are presented in Table 3 [26, 27]. From Eq. (48), the exact value for the final height is H ¼ 9:2863 (mm). The numerical results and the numerical error, which is defined as: jHNumerical HAnalytical j=HAnalytical , are presented in Table 4 for different mesh resolutions. As can be seen in Table 4, even on a coarse grid, the error is very small, and
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Table 4. Numerical results for planar solidification problem of molten aluminum for different mesh resolutions [the exact value H ¼ 9:2863 (mm)] Mesh size (mm) 1 0.5 6:25 102
H (mm)
Error (%)
9.2967 9.2880 9.2867
0.112 0.019 0.005
it decreases quickly as the mesh resolution increases. There is very good agreement between the analytical and numerical values. To demonstrate the performance of the numerical model in a three-dimensional problem, the model was used to simulate solidification and fluid flow driven by density variation during phase change of molten tin in a rectangular mold. The advantage of this problem is that because of its simple geometry, all the phenomena related to density variation during phase change can be easily observed. In this problem, molten tin initially at 233C is in a rectangular mold of size 2 2 1:5 (cm). The initial height of the molten tin is 1.35 (cm). From time zero, the temperature of the side walls and of the bottom of the container is specified to be 25C. The free surface of the molten tin is assumed to be adiabatic. Of interest is how the density variation during phase change affects the liquid. What is the final shape of the material predicted by the numerical model after complete solidification? Because of symmetry, only a quarter of the rectangular mold was solved numerically. Thus, the computational domain was defined as a rectangular cube of size 1 1 1:5 (cm), discretized by 18 18 27 cells. No-slip and no-penetration flow boundary conditions as well as a constant-temperature thermal boundary condition were applied to the fluid adjacent to the mold walls. The liquid free surface was assumed to be adiabatic. The symmetry walls were treated with free-slip and no-penetration flow boundary conditions in addition to an adiabatic thermal boundary condition. The numerical results for the free surface geometry and for the solid-front location at different times are presented in Figure 4. In this figure, the cross section of the mold is depicted and the solid and liquid phases are shown in black and gray, respectively. The mesh on the free surface illustrates the volumetric shrinkage. It is observed that solidification starts from the bottom surface and from the side walls of the mold, and that the solid front moves toward the free surface and center of the container. The material in the vicinity of the mold center and near to the free surface is the last liquid to solidify. Thus, the liquid in this region is most affected by the solidification shrinkage-induced flow. The free-surface profile of the material initially (t ¼ 0) and after complete solidification is depicted in Figure 5a in a cross-section view. Because of symmetry, only half of the mold is shown. The dashed line corresponds to the liquid free surface at t ¼ 0 and the solid line represents the surface profile of material after complete solidification. The volumetric shrinkage predicted by the numerical model is well seen in Figure 5a. For the above-mentioned problem, the initial volume of the material Vo , as well as the volume of the completely solidified material V , were calculated. Vo
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Figure 4. Solidification and shrinkage of molten tin in a rectangular mold. Bottom and side walls are at 25C. Solid and liquid phases are shown in black and gray, respectively.
and V are related to the solid and liquid densities through the following equation: Vo V qs ql ¼ ¼b V ql
ð49Þ
ðVo V Þ=V was calculated from the numerical results as 0.0328571, while b is equal to 0.032998 based on the solid and liquid densities of tin. Therefore, the numerical error in material volume is about 0.43%, which is good for such a coarse mesh resolution. The temperature contours in different time sequences are presented in Figure 5b. The observed cooling of the liquid by the side walls and the bottom surface is as expected. In Figure 6, the pressure and velocity fields in the molten tin are depicted at different times. The solid–liquid interface and the free surface are also shown in this figure. The negative pressure gradient induced by the density variation during phase
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Figure 5. (a) Free-surface profiles of tin in the rectangular mold at the initial state, shown by the dashed line, and after complete solidification, depicted by the solid line. (b) Temperature contours in the molten tin at different time sequences.
change is clearly observed in this figure. It is seen that as solidification begins, a negative pressure gradient is formed in the liquid. The induced pressure field pulls the liquid toward the solidifying and shrinking regions and so induces a velocity field. The results show that the numerical model predicts the effect of density variation during phase change in three-dimensional problems very well. We considered the same problem again, but this time the domain was initially filled with the material ( fi; j;k ¼ 1 for all the cells), so that no free surface existed. Effectively, this is a case of solidification in a sealed box. In this case, at the moment at which solidification begins, the flow pressure solution was unable to converge to a solution that satisfies continuity in the absence of a vent.
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Figure 6. Pressure and velocity fields in the molten tin induced due to density variation during phase change at different time sequences: (a) t ¼ 0:02 s; (b) t ¼ 0:42 s; (c) t ¼ 1:0 s.
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In another problem, molten tin initially at 233C is in a rectangular mold of size 2 2 1 (cm). The computational domain size is exactly equal to the mold size and the mold is fully filled with molten tin. Again, because of symmetry only a quarter of the mold was solved numerically, and the size of the computational domain was exactly equal to the quarter of the mold size. A cavity was generated inside the volume of molten tin by setting fi;j;k ¼ 0 in one cell located approximately at the center of the mold. The flow and thermal boundary conditions were exactly the same as for the last case except for the top surface of molten tin. The top surface of the molten tin was also set to be an isothermal surface at 25C. The results at different times are presented in Figure 7 in cross-section view. Because of symmetry, only half
Figure 7. Growing of cavity size due to solidification shrinkage of molten tin in a rectangular mold completely filled with the melt. The computational domain size is exactly equal to mold size. Solid and liquid phases are shown in black and gray, respectively.
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of the mold is shown. As can be seen, as a result of solidification shrinkage, the cavity size grows as the material solidifies until complete solidification occurs. Nevertheless, the location and time of initiating the cavity inside the material are still in question. This is especially of interest in casting problems in which cavities are generated inside the solidifying material as a result of solidification shrinkage. Finally, the numerical model was used to simulate the impact and solidification of a tin droplet onto a stainless steel substrate, and to study the effects of solidification shrinkage in this problem. Pasandideh-Fard et al. [8] modeled the corresponding problem without the density variation during phase change. The results of the current study are compared with these results to determine the effect of solidification shrinkage. In this problem a tin droplet of 2.7 (mm) diameter, which is initially at 240C, impinges at 1 (m=s) onto a cold stainless steel substrate initially at 25C. The contact resistance between the surface and the droplet was assumed to be 5 106 (m2 K=W) [8]. Because of symmetry, only a quarter of the droplet was simulated. The droplet was discretized using a computational mesh with a uniform grid spacing equal to 1=22 of the droplet radius. The simulation results of the current study, in which density variation during phase change is taken into account, along with the results of the simulation done by Pasandideh-Fard et al. [8], in which the solid and liquid densities were assumed to be the same, are presented on the right and left sides of Figure 8. The gray color corresponds to the liquid phase and the black to the solidified material. Although the problem was modeled three-dimensionally, a cross-sectional (two-dimensional) view is presented to allow the detailed study of the droplet free surface and the solidified layer-surface profile, to determine the effect of solidification shrinkage on the impacting and solidifying droplet. It is observed that, following the impact, the tin droplet spreads on the substrate and flattens out into a disk with a raised rim. Meanwhile, a solid layer begins to form at the droplet–substrate interface. A film of molten tin remains above the solid layer. Surface-tension forces prevent the molten tin from spreading farther, so it recoils and flows backwards. When the recoiling ring of liquid meets in the center of the splat, a void is formed. The size of the void appears to be bigger when solidification shrinkage is taken into account. The cross-sectional area of the void in the shrinkage case is 2:58 102 (mm2), as compared to the nonshrinkage case in which the area is 2:09 102 (mm2). The liquid recoils above the surface, reaching a maximum height at about t ¼ 10 (ms). Gravity and surface tension pull back the liquid until it subsides on the solid layer and forms a splat with a rounded upper surface. As the solid layer grows, the area of the solid–liquid interface is gradually reduced. Surface tension and contact angle dictate the liquid shape at the periphery of the solid–liquid interface. The liquid is pushed up by surface tension and by the effect of the contact angle, until it is completely solidified. Finally, it can be seen that the splat surface profile in the results of the current study is apparently depressed due to solidification shrinkage. Because of the large ratio of area to volume in a spreading droplet, the effect of solidification shrinkage is subtle; however, even small shrinkage effect is of interest in a thermal sprayed coating. Referring to Figure 1 again, even small shrinkage effects such as porosity and void formation will substantially damage the thermal and mechanical performance of a coating and weaken its bond to the substrate.
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Figure 8. Comparison of numerical results of a tin droplet impacting and solidifying onto a stainless steel substrate. The right half of each image is the result of the current study, in which density variation during phase change was considered. The left half is the simulation result when the solid and liquid densities were assumed to be the same. Solid and liquid phases are shown in black and gray, respectively.
4. SUMMARY AND CONCLUSIONS We have presented a numerical model to study the effects of density variation during phase change, which is applicable to problems characterized by complex geometries and velocity fields. In the present study, a three-dimensional model of droplet impact and solidification, developed by Bussmann et al. [6] and Pasandideh-Fard et al. [8], was modified to include the effects of density variation during phase change. The governing equations of conservation of mass, momentum, and energy, and the VOF equation were derived by assuming different but constant solid and liquid densities. In the numerical model, a fixed-grid control-volume discretization of the fluid flow and energy equations was combined with a volume-tracking algorithm to track any free surfaces of the liquid. The Stefan problem, for which an analytical solution is available, was used to test and validate the model. For this problem, the numerical results were in good
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agreement with the analytical solution. Furthermore, the numerical model was tested against the problem of planar solidification of a finite extent of molten pure aluminum in which the final height of the completely solidified layer is known analytically. There was good agreement between the numerical and analytical solutions for this problem as well. To assess the performance of the numerical model in a three-dimensional problem, the model was used to simulate solidification and volumetric shrinkage during solidification of molten pure tin in an open rectangular mold. It was observed that as the solidified layer grows in the mold, the free surface of the molten tin is depressed due to volumetric shrinkage. The volume of the completely solidified tin predicted by the numerical model was compared with the analytical value, and the error was very small. Results were also presented of solidification shrinkage of molten tin in a closed rectangular mold. In this case, the flow pressure solution was unable to converge to a solution that satisfies continuity in the absence of a free surface. However, in a similar case in which a void cell was initially seeded at the center of the mold, as the material was solidifying, the cavity size grew because of density variation due to phase change. Finally, the numerical model was used to simulate impact and solidification of a tin droplet on a stainless steel substrate. The main application of this model is the thermal spray coating process, which is commonly used by the aerospace, automotive, and power-generation industries. Simulation results of the numerical model were compared with the results of simulation done by Pasandideh-Fard et al. [8], in which solid and liquid densities were assumed to be equal. Effects of density variation during phase change were predicted well for this problem. During recoil of the droplet a small cavity is formed within the splat. As a result of solidification shrinkage, the size of the cavity is predicted to be bigger than when shrinkage is ignored. Furthermore, the splat surface is seen to be depressed as a result of volumetric shrinkage during phase change. REFERENCES 1. M. C. Flemings, Solidification Modeling, Past and Present, in B. G. Thomas and C. Beckermann (eds.), Proc. Eighth Int. Conf. on Modeling of Casting, Welding and Advanced Solidification Processes, pp. 1–13, San Diego, CA, 1998. 2. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, pp. 282–296, Oxford University Press, London, 1959. 3. A. J. Dalhuijsen and A. Segal, Comparison of Finite Element Techniques for Solidification Problems, Int. J. Numer. Meth. Eng., vol. 23, pp. 1807–1829, 1986. 4. A. A. Samarskii, P. N. Varbishchevich, O. P. Iliev, and A. G. Churbanov, Numerical Simulation of Convection-Diffusion Phase Change Problems—A Review, Int. J. Heat Mass Transfer, vol. 36, pp. 4095–4106, 1993. 5. V. R. Voller, An Overview of Numerical Methods for Solving Phase Change Problems, Adv. Numer. Heat Transfer, pp. 341–380, 1997. 6. M. Bussmann, J. Mostaghimi, and S. Chandra, On a Three-Dimensional Volume Tracking Model of Droplet Impact, Phys. Fluids, vol. 11, pp. 1406–1417, 1999. 7. D. B. Kothe, R. C. Mjolsness, and M. D. Torrey, Ripple: A Computer Program for Incompressible Flows with Free Surfaces, Los Alamos National Laboratory Rep. LA-12007-MS, 1991.
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8. M. Pasandideh-Fard, S. Chandra, and J. Mostaghimi, A Three-Dimensional Model of Droplet Impact and Solidification, Int. J. Heat Mass Transfer, vol. 45, pp. 2229–2242, 2002. 9. M. Trovant and S. A. Argyropoulos, Mathematical Modeling and Experimental Measurements of Shrinkage in the Casting of Metals, Can. Metall. Quart., vol. 35, pp. 77–84, 1996. 10. J. Beech, M. Barkhudarov, K. Chang, and S. B. Chin, Computer Modelling of the Formation of Macro Shrinkage Cavities during Solidification, in B. G. Thomas and C. Beckermann (eds.), Proc. Eighth Int. Conf. on Modeling of Casting, Welding and Advanced Solidification Processes, pp. 1071–1078, San Diego, CA, 1998. 11. M. R. Barkhudarov, Enhancement to Heat Transfer and Solidification Shrinkage Models in Flow-3D, Flow Science, Inc., Rep. FSI-95-TN43, 1995. 12. C. J. Kim and S. T. Ro, Shrinkage Formation during the Solidification Process in an Open Rectangular Cavity, ASME J. Heat Transfer, vol. 115, pp. 1078–1081, 1993. 13. E. McBride, J. C. Heinrich, and D. R. Poirier, Numerical Simulation of Incompressible Flow Driven by Density Variation during Phase Change, Int. J. Numer. Meth. Fluids, vol. 31, pp. 787–800, 1999. 14. K. C. Chiang and H. L. Tsai, Shrinkage-Induced Fluid Flow and Domain Change in Two-Dimensional Alloy Solidification, Int. J. Heat Mass Transfer, vol. 35, no. 7, pp. 1763–1770, 1992. 15. G. F. Naterer, Simultaneous Pressure-Velocity Coupling in the Two-Phase Zone for Solidification Shrinkage in an Open Cavity, Modell. Simul. Mater. Sci. Eng., vol. 5, pp. 595–613, 1997. 16. G. Ehlen, A. Schweizer, A. Ludwig, and P. R. Sahm, Free Surface Model to Predict the Influence of Shrinkage Cavities on the Solute Redistribution in Castings, in P. R. Sahm, P. N. Hansen, and J. G. Conley (eds.), Proc. Ninth Int. Conf. on Modeling of Casting, Welding and Advanced Solidification Processes, p. 632, Aachen, Germany, 2000. 17. B. D. Nichols, C. W. Hirt, and R. S. Hotchkiss, SOLA-VOF: A Solution Algorithm for Transient Fluid Flow with Multiple Free Boundaries, Los Alamos Scientific Laboratory Rep. LA-8355, 1980. 18. I. G. Currie, Fundamental Mechanics of Fluids, 2nd ed., pp. 26–28, McGraw-Hill, New York, 1993. 19. M. Bussmann, A Three-Dimensional Model of an Impacting Droplet, Ph.D. thesis, University of Toronto, Toronto, Canada, 2000. 20. J. U. Brackbill, D. B. Kothe, and C. Zemach, A Continuum Method for Modeling Surface Tension, J. Comput. Phys., vol. 100, p. 335, 1991. 21. D. L. Youngs, An Interface Tracking Method for a 3D Eulerian Hydrodynamics Code, Atomic Weapons Research Establishment (AWRE) Rep. 44=92=35, 1984. 22. Y. Cao, A. Faghri, and W. S. Chang, A Numerical Analysis of Stefan Problems for Generalized Multi-dimensional Phase-Change Structures Using the Enthalpy Transforming Model, Int. J. Heat Mass Transfer, vol. 32, no. 7, pp. 1289–1298, 1989. 23. M. Raessi, Modelling Density Variation due to Phase Change during Droplet Impact, Master’s thesis, University of Toronto, Toronto, Canada, 2003. 24. V. R. Voller and M. Cross, An Explicit Numerical Method to Track a Moving Phase Change Front, Int. J. Heat Mass Transfer, vol. 26, no. 1, pp. 147–150, 1983. 25. F. P. Incropera and D. P. DeWitt, Fundamentals of Heat and Mass Transfer, 3rd ed., pp. A3–A6, Wiley, New York, 1990. 26. V. Zinovev, Handbook of Thermophysical Properties of Metals at High Temperature, pp. 140, 153, Nova Science, New York, 1996. 27. T. Iida and R. I. Guthrie, The Physical Properties of Liquid Metals, pp. 134, 183, Oxford University Press, London, 1988.
