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Numerical determination of liquid flow permeabilities for equiaxed dendritic structures
Acta Materialia 50 (2002) 1559–1569 www.actamat-journals.com
Numerical determination of liquid flow permeabilities for equiaxed dendritic structures S.G.R. Brown a, J.A. Spittle a,*, D.J. Jarvis b, R. Walden-Bevan c a
Department of Materials Engineering, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, UK b ESTEC-MSM-GA-P, Keplerlaan 1, PO Box 299, 2200 AG Noordwijk ZH, The Netherlands c Enviros QRA, Culham Science Park, Abingdon OX14 3DB, UK Received 31 July 2001; received in revised form 18 December 2001; accepted 19 December 2001
Abstract Darcy’s law has been applied to the 3D finite difference numerical determination of the influence of solid fraction and geometry on the permeability of equiaxed dendritic structures. The micro-model computes the permeability for flow through a domain equivalent to the volume ultimately occupied by a single solid solution dendritic grain in an Al3Cu3Si alloy. Evolution of the dendrite shape during solidification was simulated using a novel cellular automatonfinite difference technique. Numerically determined permeabilities compare well with reported experimental data for aluminium alloys. For solid fractions in excess of 苲20%, there is also reasonable correlation with the Kozeny–Carman (KC) expression for a KC constant of unity. A significant feature of the micro-model is that it is able to account for the isolation of interdendritic liquid pools in calculating the effective values of the solid–liquid interfacial area and of the fraction liquid. 2002 Published by Elsevier Science Ltd on behalf of Acta Materialia Inc. Zusammenfassung Das Darcysche Gesetz ist bei den 3D numerischen Finite-Differenz Berechnungen angewendet worden, um den Einfluß des Festko¨rperanteils und der Geometrie auf die Permeabilita¨t von dendritischen Strukturen zu bestimmen. Das mikroskopische Modell berechnet die Permeabilita¨t durch ein repra¨sentatives Volumen wa¨hrend des Fließens. Die Gro¨ße dieses Volumens entspricht der Endgro¨ße eines alleinstehenden prima¨rphasigen dendritischen Kornes einer Al3Cu3Si Legierung. Die Entwicklung der Dendritenmorphologie wa¨hrend der Erstarrung ist mittels einer neuartigen ZellularAutomaten-Finite-Differenz Methode simuliert worden. Die numerisch bestimmten Werte von der Permeabilita¨t stimmen mit vero¨ffentlichten Versuchsergebnissen fu¨r Aluminiumlegierungen gut u¨berein. Daru¨ber hinaus gibt es eine angemessene Korrelation mit der Kozeny–Carman (KC) Gleichung, wenn die KC Konstante der Eins gleicht und der Festko¨rperanteil gro¨ßer als 苲20% ist. Eine wichtige Besonderheit des mikroskopischen Modelles liegt darin, daß es beim Berechnen der effektiven Werte von der Grenzfla¨che Festko¨rper/Schmelze und dem Schmelzanteil, auch die Isolierung von interdendritischen Schmelztro¨pfchen beru¨cksichtigen kann. 2002 Published by Elsevier Science Ltd on behalf of Acta Materialia Inc. Re´sume´ La loi Darcy a e´te´ applique´e a` la methode de differences finies 3D afin de determiner l’influence de la fraction solide et la ge´ometrie sur la permeabilite´ des structures dendritiques e´quiaxe´es. Le micro-mode`le simule la permeabilite´ de 1359-6454/02/$22.00 2002 Published by Elsevier Science Ltd on behalf of Acta Materialia Inc. PII: S 1 3 5 9 - 6 4 5 4 ( 0 2 ) 0 0 0 1 4 - 9
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l’e´ coulement a` travers un domaine equivalent au volume occupe´ par un seul grain de solution solide dendritique dans un alliage Al3Cu3Si. Pendant la solidification, l’e´ volution de la forme dendritique a e´ te´ simule´ e a` l’aide d’une nouvelle technique cellulaire automaton couple´ e avec les diffe´ rences finies. Les permeabilite´ s determine´ es numericalement sont comparables aux donne´ es experimentales pour les alliages d’aluminium. Pour les fractions solides en exce`s de 20%, une bonne correlation existe egalement avec l’expression de Kozeny–Carman (KC) pour une unite´ de KC constante. Une caracte´ ristique importante du mode`le est sa capacite´ a` prendre en conside´ ration l’isolation de poches de liquide interdendritiques en calculant les valeurs correctes de la region interfaciale solide–liquide et de la fraction solide. 2002 Published by Elsevier Science Ltd on behalf of Acta Materialia Inc. Keywords: Modelling; Transport; Microstructure; Permeability; Aluminium
1. Introduction Modelling of various solidification phenomena, related to fluid flow in the interdendritic liquid channels of shaped castings and continuously cast alloys, requires a knowledge of the way in which permeability changes with local solidification conditions and the associated geometry of the solid dendrite. Such phenomena include macrosegregation (due either to convection or to liquid flow to compensate for shrinkage) [1] and microporosity formation [2]. In the absence of experimental permeability data, many modellers [1,2] resort to using the Kozeny–Carman (KC) expression for permeability, K. This expression arises from a consideration of viscous flow through a bundle of parallel capillaries [3] where K ⫽ (1⫺gS)3 / kS2V
(1)
gS is the volume fraction of solid and SV is the solid–liquid interfacial area per unit volume. This is based on an earlier expression due to Kozeny [4], which can also be written as in Eq. (1), whence k is the Kozeny or KC constant. Alternatively, K ⫽ (1⫺gS)3 / kS20g2S
(2)
where S0 is the solid/liquid interfacial area per unit volume of solid. Equations (1) and (2) assume that there is a uniform distribution of liquid in the solid, that the
* Corresponding author. Fax: +44-1792-295244. E-mail address: [email protected] (J.A. Spittle).
liquid channels have the same cross-section, that the channels are of uniform cross-section and that k remains constant as gS changes. For simplicity, it is often assumed in numerical models of interdendritic liquid flow that k has a value of 5 and that S0 remains constant as gS increases. These assumptions may not be valid for interdendritic fluid flow for the following reasons. First, the value of 5 is typical, for example, of fluid flow through a randomly packed bed of spheres. However, for other solid geometries, k will be governed by the channel cross-sectional shape and the uniformity of the dimensions of the channels. Experimental observations on flow along paralleloriented fibres for porosity fractions ranging from 0.984 to 0.093 [5] have shown that k decreases continually to very low values with increasing fraction solid instead of approaching a constant value, i.e. the Kozeny equation fails. Similarly, the equation does not hold for flow through a bundle of capillaries of widely different cross-sections. Second, S0 will not remain constant as solidification proceeds but varies continuously with gS. Finally, as solidification progresses, i.e. gS increases, some interdendritic liquid regions will become enclosed by solid and hence isolated from flow channels. Since these regions do not contribute to the permeability, the effective values of gL (the volume fraction of liquid), SV and S0 are all lowered. A number of investigators, using a variety of permeameter designs, have attempted to determine experimentally the permeability of semi- solid dendritic materials [6–14]. Significant problems with experimentation include preparing test samples that initially exhibit uniformity of structure, main-
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taining a uniform temperature throughout a sample for the duration of a test, and ensuring that the solid dendritic grains remain interconnected and static during the test. An additional major source of error is the natural tendency of the microstructure to change during the test period due to coarsening and back diffusion in the solid. Most investigators have also attempted to relate their experimentally observed permeabilities to values calculated from expressions based on structural geometry, such as Eqs. (1) and (2). Often this has been achieved by comparing plots of KSV2 versus gL. Many of the experimental studies referred to in the paragraph above have been performed on structures having equiaxed primary grains in an attempt to study isotropic structures. Murakami and Okamoto [9] measured the permeabilities of mushy zones in borneol–paraffin mixtures for gL values in the range 0.27–0.48. The permeabilities were measured just above the eutectic temperature which allowed the SV values to be measured at room temperature on structures consisting of equiaxed borneol grains in a eutectic matrix. At a fraction liquid of 0.523 the solid in the mushy zone moved with the liquid, resulting in a higher permeability than expected. The investigators found fairly good agreement between measured permeabilities and values calculated from Eq. (1) for k=5. Several investigators have measured the permeabilities of equiaxed structures, for alloys containing eutectic, at temperatures just above the eutectic temperature. By using a metallostatic head of molten eutectic alloy, the liquid flowing into the structures has the same composition as the interdendritic liquid. This approach was first used by Streat and Weinberg for Pb–Sn alloys and demonstrated that the permeabilities of equiaxed structures are less than those for columnar structures [8]. A number of recent investigations have concentrated on Al–Cu alloys [10,13,14]. Poirier and Ganesan [10], from measurement of permeabilities for an Al–15.6 wt% Cu alloy with equiaxed structures, investigated the influence of the morphology of the dendritic solid for samples whose microstructures exhibited a range of values of grain size, of specific surface area S0 and of gL. They recognized that K represented an average measured value and that the microstructure and hence per-
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meability changed continuously during a test. They identified a time tm for each test during which the permeability data were assumed to apply. Depending on length of test, the morphologies changed from dendritic to dendritic-globular to globular with time. They also recognized that liquid inclusions could be present and that these do not participate in fluid flow. The microstructures corresponding to the measured K values for particular times tm were characterized from either quenched test specimens or samples that were separately isothermally coarsened. They observed for gL values in the range 0.166–0.434 that permeability in equiaxed structures is very structure sensitive. In addition, they found that the measured dimensionless permeability data KS02 compared reasonably well with equivalent theoretical data for flow through arrays of uniform spheres using a theoretical expression due to Zick and Homsy [15]. The experimental S0 values were corrected to account for the liquid inclusions. Duncan et al. [13] examined the permeabilities of Al–15.42 and Al–8.68 wt% Cu alloys at just above the eutectic temperature. Recognizing that Poirier and Ganesan had determined an average value for K over a period of time, they used a modified technique for calculating experimental permeability as a function of time. They demonstrated that permeability initially increases continuously to a maximum value as coarsening proceeds and then falls due to blockages at the sample inlet surface. From a comparison of microstructures from isothermal coarsening experiments and from quenched samples during permeability tests, they also found that samples coarsen significantly faster (in the presence of fluid flow) in the permeability tests. They also observed from both sets of samples that gL decreases with time to the equilibrium value due to back diffusion. Using this more accurate method of calculating K and the more accurate values for SV and gL determined from quenched permeability test samples, they found from a plot of KSV2 versus gL for k=5 that the Kozeny–Carman equation gave an accurate representation of permeability. No mention was made, however, of allowing for entrapment of liquid pools in the calculation of SV. Nielsen et al. [14] measured the permeability of
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equiaxed Al–Cu alloys for gL values in the range 0.09 to 0.32 to more rigorously examine the match between experimental data and the Kozeny–Carman equation. Whereas the latter equation is based on a single length scale (SV)⫺1 (which equates to the arm spacing), Wang et al. [16] developed a relationship for permeability that also takes into account the equiaxed grains, arguing that liquid flow is partitioned between flow through the dendrite grains and flow around the grains. Permeabilities were therefore related to five parameters defining the solid morphology [14]. On completion of each permeability test (lasting about 50 s) the sample was quenched and the microstructures analysed for the five features and an average value for K calculated for the 50-s period. From plots of KS02 versus gS it was concluded that the measurements constitute experimental validation of the model of Wang et al. [16]. However, any microstructural coarsening and any liquid pool entrapment occurring during a test were ignored. Also, as noted by the authors [14] and illustrated in several representative micrographs, in a number of the tests individual grains had been completely surrounded by liquid. This would allow solid movement as well as fluid flow. It is certainly unclear at the present time whether either the Kozeny–Carman [3], Zick and Homsy [15] or Wang et al. [16] relationships are applicable to the equiaxed solidification of alloys over a wide range of values of gL. Even if one or more of them are, their implementation for the accurate prediction of interdendritic flow would require a quantitative knowledge of the geometry of the solid as a function of gL. Experimental determination of permeability requires not only precise measurement of the permeability but also specification of the microstructure corresponding to the permeability measured. Since microstructure changes continuously during a permeameter test (and these microstructures can only be subsequently examined two dimensionally) and tests can only be performed if there is a rigid interconnected solid, the accuracy of experimentation is questionable. The only alternative to the above approaches for obtaining permeability values is to employ a numerical method to model the permeability of 3D
microstructures. In this way an attempt can be made to overcome the problems relating to maintaining microstructural rigidity and stability experienced during experimentation and to evaluating SV, S0, gL or any other required geometrical parameters for 3D structures. To date, a variety of numerical methods have been applied to determining the permeability of dendritic structures, using either simplified dendrite-like shapes or images of actual dendrites [17– 19]. A numerical simulation has also been performed of the influence of Ostwald ripening of an Al– 4% Cu solid–liquid mixture on permeability [20]. All of these studies [17–20] are limited by the fact that they were performed for 2D structures and for large liquid fractions. The purpose of the present research was to develop a comprehensive 3D finite difference numerical model for the evolution of an equiaxed dendrite and the determination of the variation in permeability as the dendrite evolved.
2. The numerical model
The model attempts to predict the variation in the permeability of an equiaxed dendritic structure as the structure evolves. It assumes that a uniform distribution of static nuclei solidifies to become an array of cube-shaped grains of equal size. Though somewhat unrealistic in the sense that grains are not cubic in shape and, at the onset of freezing, the grains would not remain static, the model is thought to be reasonably representative once a rigid interlocking arrangement of grains has formed. In the model, flow through a cubic domain, the size of a single solidified grain, is assumed to be representative of the bulk material behaviour. The model therefore comprises two stages, first the numerical evolution of a 3D dendrite (in this instance an Al3Cu3Si alloy was arbitrarily selected since it freezes with a large fraction of primary solid solution phase) and second the numerical determination of the permeability of the domain as a function of fraction solid.
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2.1. Stage one: a cellular automaton-finite difference (CAFD) model of an evolving 3D dendrite Various 2D/3D cellular automaton-finite difference (CAFD) models for the simulation of columnar/equiaxed growth and the investigation of microsegregation in binary/ternary alloys have been reported in recent years [21,22]. The present model details, relating to solute redistribution during freezing (including back diffusion) and to growth, are described in detail elsewhere [22]. Initially a nucleus is placed at the centre of a cubic domain consisting of 100×100×100 cubic elements. The edge length of the domain, which corresponds to an appropriate grain size for an ascast aluminium alloy, is 300 µm. The domain is assumed to be of a uniform temperature and cooling at a rate of 50 K s⫺1, again a reasonably typical value. Latent heat is not emitted during the simulation. During cooling, the model simulates the evolution of an equiaxed dendrite and the accompanying redistribution of the solutes. This involves modelling solute diffusion (using a fully implicit finite difference scheme and employing temperature-dependent diffusion coefficients for Cu and Si in the liquid and solid phases) and predicting the solidification of elements from a thermodynamically predicted phase diagram generated by ThermoCalc [23] with data from ThermoTech [24]. 3D snapshots of the evolving solid dendrite are retained for use in the permeability studies. The software is able to distinguish between liquid sites that have become completely isolated by solid and those that are still able to contribute to the flow. Isolated liquid sites are considered to be effectively solid sites (since they can no longer take part in flow) and to behave as such in the calculation of the ‘effective’ values for fraction solid, SV and S0. 2.2. Stage two: a CFD model for determining permeability An in-house developed commercial CFD code MAVIS-FLOW [25] was used to determine 3D permeability as a function of the effective solid volume fraction. The code employs a full Navier– Stokes treatment of flow through the domain
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(dimensions as above) for a wide range of dendrite solid fractions/geometries obtained from the dendrite evolution model. Having modelled the growth of the dendrite within the cubic domain, it is assumed that liquid enters the domain at one face and leaves at the opposite face. To facilitate the determination of the permeability of the dendrite, the domain is extended by adding 10 layers of liquid elements to the inlet and outlet faces. The liquid enters and leaves the extended domain at a fixed uniform velocity of 1×10⫺3 m s⫺1, no-slip boundary conditions operate during flow, and a constant kinematic viscosity of 4×10⫺6 m2 s⫺1 is assumed. The other four faces of the extended domain are assumed to be solid. For each effective dendrite fraction selected, the permeability of the domain is calculated using Darcy’s law which states, K ⫽ vhL / ⌬P
(3)
where v is the rate of flow, h is the absolute viscosity (kinematic viscosity×liquid density) and L is the length (in this case 300 µm) over which the pressure drop ⌬P is calculated. ⌬P is determined for each simulation, once a steady state flow has been achieved, by obtaining the average pressure for all liquid sites in each of the two all-liquid layers immediately adjacent to the original cubic domain used for constructing the dendrite.
3. Observations Fig. 1 illustrates the evolution of the dendrite for different values of the ‘effective’ solid volume fraction, gS. There is a slight asymmetry in the evolving solid dendrite shape which may have an influence on the determined permeability vis-a`-vis a symmetrical dendrite. However, this paper reports the findings of the preliminary study of the performance of the model and investigations are currently under way to determine the influences of major changes in dendrite geometry due to alloy composition, grain size and cooling rate. It can be seen that in the early stages of freezing, prior to the partially solidified dendritic grain having developed a cube-shaped geometry, as discussed by Wang et al. [16], flow occurs both
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Fig. 1. Numerical evolution of a dendrite for an Al3Cu3Si alloy at effective solid fractions of (a) 0.1%, (b) 1%, (c) 6% and (d) 55%.
through the dendrite and around the dendrite. At high volume fractions of solid, liquid pools become isolated in the dendrite as shown for a 2D slice through the domain for 95% fraction solid, see Fig. 2. This figure can be considered as equivalent to a 2D micrograph that might be obtained from an actual alloy sample. However, from a 2D micrograph, it is impossible to distinguish between liquid areas isolated by solid and those that are open and contributing to liquid flow through the dendrite. Fig. 3 illustrates in 3D, again for 95% solid, the liquid channels that retain connectivity between the inlet and outlet faces of the cubic domain. Many of the channels are themselves also interconnected. It is recognized that a limitation of the present modelling approach is that, as the local solid fraction increases, flow channels may dimensionally be reduced to the size of one element. Also shown are the additional liquid layers, introduced in the extended domain for the permeability simulations, at the inlet and outlet faces. Fig. 4 shows a pressure map over the domain height for a halfsection of the domain, indicating the pressure drop that occurs as the liquid flows through the domain,
Fig. 2. 2D section through a dendrite (grey) for an effective solid fraction of 95%, illustrating isolated liquid regions (black).
and the pressure variation across the exit face (top of figure). Fig. 5 indicates how the 3D numerically determined KS02 term varies with the volume fraction solid gS. For the plotted numerical results, both gS and S0 are ‘effective’ values. Also shown are equivalent variations, predicted using the Kozeny–Carman expression for KC constant values of 1 and 5. As explained in the introduction, this form of presentation of permeability data is often used when comparing experimental results, or data from numerical simulations, with the predictions of the KC equation. K, S02 and gS are all determined from the numerical model. It would seem illogical to attempt to compare the values of KS02 derived from the KC expression with those obtained from the model at low solid fractions, prior to the partially solidified dendrite having acquired a cubic shape. At low solid fractions (prior to the dendrite attaining a cubic shape) there would be wholly liquid regions within the domain and liquid would
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Fig. 3. 3D illustration of the dendrite in Fig. 2 showing the liquid channels that retain connectivity between the inlet (bottom) and outlet (top) faces of the domain.
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flow both around and through the dendrite. Fig. 5 illustrates, for solid fractions above 苲20%, that there is reasonable correlation between the simulated values of KS02 and those anticipated from the KC expression for a k value of unity. This implies that the modelled dendritic structure, for given values of fraction solid and S02, has a higher permeability than would have been anticipated from the Kozeny–Carman expression had a k value of 5 been assumed. Also shown in Fig. 5, by open circles, are numerically predicted values of KS02 versus fraction solid determined from 2D sections through the cubic domain for the same set of partially solidified 3D dendrites, i.e. equivalent to 2D metallographic sections through real dendrites. For each 3D partially solidified dendrite examined, 100×2D slices were taken through the domain. gS and S0 values were obtained for each slice and average values calculated for the 100 slices. These average values represent the ‘true’ fraction solid and S0 values for the structure, since, as when examining real 2D metallographic images, it is impossible to identify
Fig. 4. Pressure map (b) for flow through the dendrite, (a) for an effective solid fraction of 6%. As the colour darkens pressure decreases.
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values for a dendrite structure having the same ‘effective’ volume fraction solid. In the one case, for the edge length of 300 µm, the BCC structure has a higher permeability than the dendrite, whereas the opposite is true for the second case. Another value of KS02 for a BCC structure is given in Table 1 and plotted in Fig. 5. This value was calculated from the drag coefficient, kd, of 163 reported by Zick and Homsy for a BCC structure (gS=0.68) [15]. As mentioned briefly in the introduction, this coefficient was calculated from a theoretical expression derived by the authors for Stokes flow through periodic arrays of spheres which was then applied to flow through simple cubic, BCC and FCC arrays. The relationship between permeability and drag coefficient is given by K ⫽ 2r2 / 9gSkd 2
Fig. 5. Determined values of KS0 versus solid fraction compared with values expected from the Kozeny–Carman expression. Filled circles, 3D numerically predicted values for the evolved dendrite. Open circles, 2D numerically predicted values for the evolved dendrite. Crosses, numerically predicted values for a BCC structure for two different sphere sizes. Open square, calculated value for a BCC structure from a numerically predicted drag coefficient reported by Zick and Homsy [15].
isolated liquid pools. As can be seen, the difference between the 3D predictions (isolated liquid pools identified) and those obtained from 2D analysis of the 3D dendrites (isolated pools not identified) becomes significant at high solid fractions when liquid sites become isolated. To determine the influence of liquid channel geometry on the predictions of the numerical model, permeabilities were also calculated for a regularly packed BCC structure of spheres. The same domain of 100×100×100 elements was used. However, two different domain sizes were examined with edge lengths of 300 µm (i.e. the same as for the dendrite domain) and 30 µm. These lengths also corresponded to the lattice parameters of the BCC unit cell. The numerically predicted values of K and S0 for the two different edge lengths are given in Table 1. It can be seen that the KS02 values for the two are almost identical and lie on a Kozeny–Carman curve for a KC constant of 5. Also shown in the table are the equivalent predicted
(4)
where r is the radius of the spheres. The value of KSO2 agrees very well with those determined from the present numerical model and again is in close agreement with the value determined from the Kozeny –Carman expression for a KC constant of 5. In Fig. 6, numerically determined K values for the 3D dendrite data shown in Fig. 5 are compared with the reported experimental data of Nielsen et al. [14] for Al–Cu alloys. The fraction solid values in both cases are ‘true’ values, i.e. they do not account for the possible isolation of liquid volumes.The experimental data cover a variety of grain sizes ranging from 60 to 800 µm, obtained by adding different amounts of the grain refiner Al5Ti1B. The experimental permeabilities for grain sizes in the range 200–250 µm (the closest to the present modelled data for 300 µm) increased from 0.36×10⫺12 to 4.2×10⫺12 m2 as the fraction solid decreased from 0.91 to 0.73. In the case of the predicted values, the permeabilities increased from 0.39×10⫺12 to 2.8×10⫺12 m2 as the fraction solid decreased from 0.91 to 0.74. It can therefore be seen that there is excellent correlation between the two sets of data. Out of interest, to determine the influence of the use of ‘no-slip’ or ‘free-slip’ boundary conditions in the model on the predicted permeability values, predictions were compared for two high solid frac-
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Table 1 Summary of data in Fig. 5 for a solid fraction of 苲0.68 K (m2) 3D dendrite Domain size 300 µm (gs=0.686) BCC Domain size 300 µm (gs=0.685) BCC Domain size 30 µm (gs=0.685) BCC Zomsky and Homsy [15] Domain size 300 µm (gs=0.68)
S0 (m⫺1)
KS02
3.53×10⫺12
1.61×105
9.15×10⫺2
8.14×10⫺12
3.71×104
1.12×10⫺2
8.28×10⫺12
3.71×105
1.14×10⫺2
3.4×10⫺11
2.31×104
1.81×10⫺2
Fig. 6. Numerically determined K values for the 3D dendrite data in Fig. 5 compared with reported experimental data for Al–Cu alloys [14].
tions (in the range where interdendritic flow is important to various metallurgical phenomena). For ‘effective’ solid fractions of 0.74 and 0.96, the use of free-slip boundary conditions, as opposed to no-slip conditions, produced increases in the permeability of 4.6% and 1.5%, respectively.
4. Discussion It is assumed that the numerically evolved dendrite provides a reasonable representation of
dendrite/liquid channel geometry. In previously reported papers using the CAFD technique, dendrite arm spacings have been shown to correlate well with experimentally reported values [21]. The dendrite model clearly demonstrates that for ‘effective’ solid fractions in excess of 苲80%, significant volumes of liquid become completely isolated by solid and can no longer contribute to flow/permeability. This cut-off occurs in the solid fraction range where interdendritic flow to compensate for shrinkage and convective flow due to liquid density differences take place. Therefore, the influence of cut-off on overall flow behaviour needs to be appreciated. In contrast to some recently reported experimental data [9,13], where there appeared to be a reasonable correlation with the Kozeny–Carman expression for a KC constant of 5, in the present work there was a much better correlation with a constant of unity. Some doubt must exist regarding the experimental data, as detailed in the introduction, because of the problems of maintaining rigid dendritic structures with geometries that do not change during the course of the permeability test. Also, when quantifying the microstructures in experimental studies, it is impossible to detect and account for the isolation of liquid pools in the latter stages of freezing. For given values of S0 and fraction solid, lower values of the KC constant imply a higher permeability. Although the Kozeny–Carman equation appears to be valid over the whole fraction solid range for unconsolidated randomly
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packed beds of particles, as mentioned in the introduction it is not valid for flow along a bundle of capillaries of widely varying radius or for flow along parallel-oriented fibres [5]. It appears that in the two latter cases the Kozeny equation breaks down because local values of the hydraulic radius differ significantly from the mean hydraulic radius, (1⫺gS)/SV. Considering the former case, the greater rates of flow in the larger capillaries more than compensate for the reduction associated with the smaller capillaries. The greater flow rates/permeabilities predicted from the present numerical model for interdendritic flow, than would be expected from the KC equation for a KC constant of 5, might be expected due to the fact that the solid is consolidated and the liquid channels vary in geometry from one to another and along their lengths. The finite-difference numerical predictions of flow through BCC type structures, in comparison with interdendritic flow for the same solid fraction, are very interesting. In contrast to the random packing of spheres, where the fractional area of liquid can be considered to be constant, when flow occurs in regular structures the fractional area constantly varies. Furthermore, the form of the variation will depend on orientation, i.e. permeability will differ with orientation. One of the conditions of the Kozeny equation is that the fractional area of liquid remains constant during flow. A number of numerical studies and experimental investigations of flow through periodic porous media have been made [15,26,27]. Specifically relevant to flow through BCC structures, i.e. for gS=0.68, is the numerical investigation of Zick and Homsy [15] and the experimental investigation of Susskind and Becker [27]. The experimentally determined value of 170 for the drag coefficient [27] agreed extremely well with the numerically calculated value of 163 [15]. As reported in the results section, these give KSO2 values that agree extremely well with the present numerical results for BCC structures and also with the Kozeny–Carman expression for a KC constant of 5. The excellent correlation between the various sets of data for BCC structures would appear to indicate that the flow behaviour through BCC structures (expressed
in terms of KSO2) is similar to that through randomly packed unconsolidated beds of particles. In the present investigation of BCC structures, the determined permeability was greater for the larger sphere size than for the smaller sphere size. This reflects the higher value of S0 in the latter case. Direct comparison of these permeabilities with that of the dendrite structure is impossible because of the completely different geometries of the liquid channels for the dendrite and BCC structures. Good agreement between the predicted variation of permeability of the dendritic Al3Cu3Si alloy with fraction solid, for large solid fractions, and experimental data for other Al alloys would tend to confirm the general validity of the dendrite evolution model and that numerical modelling is a viable approach to predicting permeability of dendritic structures.
5. Conclusions (a) A numerical model has been developed for the simulation of 3D flow through equiaxed dendrites of an Al3Cu3Si alloy and the determination of the variation in permeability of the structure as solidification progresses. The model involves the evolution of an equiaxed dendrite and the application of an in-house CFD program MAVIS-FLOW to calculate permeability from Darcy’s law. (b) A particular advantage of the approach is that it also facilitates a more accurate quantitative analysis of the evolving dendrite microstructure than is possible from experimental studies on actual alloys. In particular, the model is able to identify liquid sites that become surrounded by solid sites in the later stages of freezing and are therefore isolated from the flow. (c) In contrast to recent experimental studies of dendritic structures that have reported a reasonable correlation between measured permeabilities and those calculated from the Kozeny–Carman (KC) expression for a KC constant of 5, in the present numerical study a better correlation was found for a constant of unity. The deviation of the KC constant from a value of 5,
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which is normally associated with unconsolidated randomly packed beds of particles, is thought to be due to the fact that the channels through the consolidated dendrite have different cross-sections with respect to each other and along their flow paths. This research would not therefore support the use of a KC constant of 5 for flow through equiaxed dendrites in numerical models which rely on the KC expression for evaluation of permeability. (d) Experimental studies are obviously confined to high solid fractions. There was excellent correlation at these high solid fractions between permeabilities predicted by the model and experimental data for equiaxed structures. (e) Numerical models offer an alternative method to experimental studies for investigating permeability during microstructural evolution. (f) Finite-difference numerically calculated KSO2 values for BCC structures over a range of sphere sizes showed good agreement with the value calculated from the Kozeny–Carman expression for a KC constant of 5 and with other numerically calculated and experimentally obtained data that have been reported.
[4] [5] [6] [7] [8] [9] [10] [11] [12]
[13] [14] [15] [16] [17] [18] [19]
[20] [21]
References [1] Schneider MC, Beckermann C, Lipinski DM, Schaefer W. In: Thomas BG, Beckermann C, editors. Modeling of casting, welding and advanced solidification processes VIII. Warrendale, USA: The Minerals, Metals and Materials Society; 1998. p. 25-7. [2] Pequet Ch, Rappaz M. In: Sahm PR, Hansen PN, Conley JG, editors. Modeling of casting, welding and advanced solidification processes IX. Aachen, Germany: Schaker Verlag; 2000. p. 7-1. [3] Carman PC. Flow of gases through porous media. London: Butterworths Scientific Publications, 1956.
[22] [23]
[24] [25] [26] [27]
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Kozeny JSB. Akad Wiss Wien, Abt IIa 1927;136:271. Sullivan RR. J Appl Phys 1942;13:725. Piwonka TS, Flemings MC. Trans AIME 1966;236:1157. Apelian D, Flemings MC, Mehrabian R. Metall Trans 1974;5:2533. Streat N, Weinberg F. Metall Trans B 1976;7B:417. Murakami K, Okamoto T. Acta Metall 1984;32:1741. Poirier DR, Ganesan S. Mater Sci Eng A 1992;A157:113. Poirier DR, Ocansey P. Mater Sci Eng A 1993;A171:231. Paradies CJ, Arnberg L, Thevik HJ, Mo A. In: Cross M, Campbell J, editors. Modeling of casting, welding and advanced solidification processes VII. Warrendale, USA: The Minerals, Metals and Materials Society; 1995. p. 60-9. Duncan AJ, Han Q, Viswanathan S. Metall Mater Trans B 1999;30B:745. Nielsen O, Arnberg L, Mo A, Thevik H. Metall Mater Trans A 1999;30A:2455. Zick AA, Homsy GM. J Fluid Mech 1982;115:13. Wang CY, Ahuja S, Beckermann C, de Groh III HC. Metall Mater Trans B 1995;26B:111. McCarthy JF. Acta Metall Mater 1994;42:1573. Bhat MS, Poirier DR, Heinrich JC, Nagelhout D. Scr Metall Mater 1994;31:339. Goyeau B, Benihaddadene T, Gobin D, Quintard M. In: Thomas BG, Beckermann C, editors. Modeling of casting, welding and advanced solidification processes VIII. Warrendale, USA: The Minerals, Metals and Materials Society; 1998. p. 35-3. Diepers H-J, Beckermann C, Steinbach I. Acta Mater 1999;47:3663. Brown SGR, Spittle JA. In: Thomas BG, Beckermann C, editors. Modeling of casting, welding and advanced solidification processes VIII. Warrendale, USA: The Minerals, Metals and Materials Society; 1998. p. 17-9. Jarvis DJ, Brown SGR, Spittle JA. Mater Sci Technol 2000;16:1420. ThermoCalc. Department of Materials Science and Engineering, Royal Institute of Technology, Stockholm, Sweden. ThermoTech Ltd, Surrey Technology Centre, Guildford, UK. Spittle J, Brown SGR. Diecast World 2001;175:7. Martin JJ, McCabe WL, Monrad CC. Chem Eng Prog 1951;47:91. Susskind H, Becker W. AIChE J 1967;13:1155.
Numerical determination of liquid flow permeabilities for equiaxed dendritic structures S.G.R. Brown a, J.A. Spittle a,*, D.J. Jarvis b, R. Walden-Bevan c a
Department of Materials Engineering, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, UK b ESTEC-MSM-GA-P, Keplerlaan 1, PO Box 299, 2200 AG Noordwijk ZH, The Netherlands c Enviros QRA, Culham Science Park, Abingdon OX14 3DB, UK Received 31 July 2001; received in revised form 18 December 2001; accepted 19 December 2001
Abstract Darcy’s law has been applied to the 3D finite difference numerical determination of the influence of solid fraction and geometry on the permeability of equiaxed dendritic structures. The micro-model computes the permeability for flow through a domain equivalent to the volume ultimately occupied by a single solid solution dendritic grain in an Al3Cu3Si alloy. Evolution of the dendrite shape during solidification was simulated using a novel cellular automatonfinite difference technique. Numerically determined permeabilities compare well with reported experimental data for aluminium alloys. For solid fractions in excess of 苲20%, there is also reasonable correlation with the Kozeny–Carman (KC) expression for a KC constant of unity. A significant feature of the micro-model is that it is able to account for the isolation of interdendritic liquid pools in calculating the effective values of the solid–liquid interfacial area and of the fraction liquid. 2002 Published by Elsevier Science Ltd on behalf of Acta Materialia Inc. Zusammenfassung Das Darcysche Gesetz ist bei den 3D numerischen Finite-Differenz Berechnungen angewendet worden, um den Einfluß des Festko¨rperanteils und der Geometrie auf die Permeabilita¨t von dendritischen Strukturen zu bestimmen. Das mikroskopische Modell berechnet die Permeabilita¨t durch ein repra¨sentatives Volumen wa¨hrend des Fließens. Die Gro¨ße dieses Volumens entspricht der Endgro¨ße eines alleinstehenden prima¨rphasigen dendritischen Kornes einer Al3Cu3Si Legierung. Die Entwicklung der Dendritenmorphologie wa¨hrend der Erstarrung ist mittels einer neuartigen ZellularAutomaten-Finite-Differenz Methode simuliert worden. Die numerisch bestimmten Werte von der Permeabilita¨t stimmen mit vero¨ffentlichten Versuchsergebnissen fu¨r Aluminiumlegierungen gut u¨berein. Daru¨ber hinaus gibt es eine angemessene Korrelation mit der Kozeny–Carman (KC) Gleichung, wenn die KC Konstante der Eins gleicht und der Festko¨rperanteil gro¨ßer als 苲20% ist. Eine wichtige Besonderheit des mikroskopischen Modelles liegt darin, daß es beim Berechnen der effektiven Werte von der Grenzfla¨che Festko¨rper/Schmelze und dem Schmelzanteil, auch die Isolierung von interdendritischen Schmelztro¨pfchen beru¨cksichtigen kann. 2002 Published by Elsevier Science Ltd on behalf of Acta Materialia Inc. Re´sume´ La loi Darcy a e´te´ applique´e a` la methode de differences finies 3D afin de determiner l’influence de la fraction solide et la ge´ometrie sur la permeabilite´ des structures dendritiques e´quiaxe´es. Le micro-mode`le simule la permeabilite´ de 1359-6454/02/$22.00 2002 Published by Elsevier Science Ltd on behalf of Acta Materialia Inc. PII: S 1 3 5 9 - 6 4 5 4 ( 0 2 ) 0 0 0 1 4 - 9
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l’e´ coulement a` travers un domaine equivalent au volume occupe´ par un seul grain de solution solide dendritique dans un alliage Al3Cu3Si. Pendant la solidification, l’e´ volution de la forme dendritique a e´ te´ simule´ e a` l’aide d’une nouvelle technique cellulaire automaton couple´ e avec les diffe´ rences finies. Les permeabilite´ s determine´ es numericalement sont comparables aux donne´ es experimentales pour les alliages d’aluminium. Pour les fractions solides en exce`s de 20%, une bonne correlation existe egalement avec l’expression de Kozeny–Carman (KC) pour une unite´ de KC constante. Une caracte´ ristique importante du mode`le est sa capacite´ a` prendre en conside´ ration l’isolation de poches de liquide interdendritiques en calculant les valeurs correctes de la region interfaciale solide–liquide et de la fraction solide. 2002 Published by Elsevier Science Ltd on behalf of Acta Materialia Inc. Keywords: Modelling; Transport; Microstructure; Permeability; Aluminium
1. Introduction Modelling of various solidification phenomena, related to fluid flow in the interdendritic liquid channels of shaped castings and continuously cast alloys, requires a knowledge of the way in which permeability changes with local solidification conditions and the associated geometry of the solid dendrite. Such phenomena include macrosegregation (due either to convection or to liquid flow to compensate for shrinkage) [1] and microporosity formation [2]. In the absence of experimental permeability data, many modellers [1,2] resort to using the Kozeny–Carman (KC) expression for permeability, K. This expression arises from a consideration of viscous flow through a bundle of parallel capillaries [3] where K ⫽ (1⫺gS)3 / kS2V
(1)
gS is the volume fraction of solid and SV is the solid–liquid interfacial area per unit volume. This is based on an earlier expression due to Kozeny [4], which can also be written as in Eq. (1), whence k is the Kozeny or KC constant. Alternatively, K ⫽ (1⫺gS)3 / kS20g2S
(2)
where S0 is the solid/liquid interfacial area per unit volume of solid. Equations (1) and (2) assume that there is a uniform distribution of liquid in the solid, that the
* Corresponding author. Fax: +44-1792-295244. E-mail address: [email protected] (J.A. Spittle).
liquid channels have the same cross-section, that the channels are of uniform cross-section and that k remains constant as gS changes. For simplicity, it is often assumed in numerical models of interdendritic liquid flow that k has a value of 5 and that S0 remains constant as gS increases. These assumptions may not be valid for interdendritic fluid flow for the following reasons. First, the value of 5 is typical, for example, of fluid flow through a randomly packed bed of spheres. However, for other solid geometries, k will be governed by the channel cross-sectional shape and the uniformity of the dimensions of the channels. Experimental observations on flow along paralleloriented fibres for porosity fractions ranging from 0.984 to 0.093 [5] have shown that k decreases continually to very low values with increasing fraction solid instead of approaching a constant value, i.e. the Kozeny equation fails. Similarly, the equation does not hold for flow through a bundle of capillaries of widely different cross-sections. Second, S0 will not remain constant as solidification proceeds but varies continuously with gS. Finally, as solidification progresses, i.e. gS increases, some interdendritic liquid regions will become enclosed by solid and hence isolated from flow channels. Since these regions do not contribute to the permeability, the effective values of gL (the volume fraction of liquid), SV and S0 are all lowered. A number of investigators, using a variety of permeameter designs, have attempted to determine experimentally the permeability of semi- solid dendritic materials [6–14]. Significant problems with experimentation include preparing test samples that initially exhibit uniformity of structure, main-
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taining a uniform temperature throughout a sample for the duration of a test, and ensuring that the solid dendritic grains remain interconnected and static during the test. An additional major source of error is the natural tendency of the microstructure to change during the test period due to coarsening and back diffusion in the solid. Most investigators have also attempted to relate their experimentally observed permeabilities to values calculated from expressions based on structural geometry, such as Eqs. (1) and (2). Often this has been achieved by comparing plots of KSV2 versus gL. Many of the experimental studies referred to in the paragraph above have been performed on structures having equiaxed primary grains in an attempt to study isotropic structures. Murakami and Okamoto [9] measured the permeabilities of mushy zones in borneol–paraffin mixtures for gL values in the range 0.27–0.48. The permeabilities were measured just above the eutectic temperature which allowed the SV values to be measured at room temperature on structures consisting of equiaxed borneol grains in a eutectic matrix. At a fraction liquid of 0.523 the solid in the mushy zone moved with the liquid, resulting in a higher permeability than expected. The investigators found fairly good agreement between measured permeabilities and values calculated from Eq. (1) for k=5. Several investigators have measured the permeabilities of equiaxed structures, for alloys containing eutectic, at temperatures just above the eutectic temperature. By using a metallostatic head of molten eutectic alloy, the liquid flowing into the structures has the same composition as the interdendritic liquid. This approach was first used by Streat and Weinberg for Pb–Sn alloys and demonstrated that the permeabilities of equiaxed structures are less than those for columnar structures [8]. A number of recent investigations have concentrated on Al–Cu alloys [10,13,14]. Poirier and Ganesan [10], from measurement of permeabilities for an Al–15.6 wt% Cu alloy with equiaxed structures, investigated the influence of the morphology of the dendritic solid for samples whose microstructures exhibited a range of values of grain size, of specific surface area S0 and of gL. They recognized that K represented an average measured value and that the microstructure and hence per-
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meability changed continuously during a test. They identified a time tm for each test during which the permeability data were assumed to apply. Depending on length of test, the morphologies changed from dendritic to dendritic-globular to globular with time. They also recognized that liquid inclusions could be present and that these do not participate in fluid flow. The microstructures corresponding to the measured K values for particular times tm were characterized from either quenched test specimens or samples that were separately isothermally coarsened. They observed for gL values in the range 0.166–0.434 that permeability in equiaxed structures is very structure sensitive. In addition, they found that the measured dimensionless permeability data KS02 compared reasonably well with equivalent theoretical data for flow through arrays of uniform spheres using a theoretical expression due to Zick and Homsy [15]. The experimental S0 values were corrected to account for the liquid inclusions. Duncan et al. [13] examined the permeabilities of Al–15.42 and Al–8.68 wt% Cu alloys at just above the eutectic temperature. Recognizing that Poirier and Ganesan had determined an average value for K over a period of time, they used a modified technique for calculating experimental permeability as a function of time. They demonstrated that permeability initially increases continuously to a maximum value as coarsening proceeds and then falls due to blockages at the sample inlet surface. From a comparison of microstructures from isothermal coarsening experiments and from quenched samples during permeability tests, they also found that samples coarsen significantly faster (in the presence of fluid flow) in the permeability tests. They also observed from both sets of samples that gL decreases with time to the equilibrium value due to back diffusion. Using this more accurate method of calculating K and the more accurate values for SV and gL determined from quenched permeability test samples, they found from a plot of KSV2 versus gL for k=5 that the Kozeny–Carman equation gave an accurate representation of permeability. No mention was made, however, of allowing for entrapment of liquid pools in the calculation of SV. Nielsen et al. [14] measured the permeability of
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equiaxed Al–Cu alloys for gL values in the range 0.09 to 0.32 to more rigorously examine the match between experimental data and the Kozeny–Carman equation. Whereas the latter equation is based on a single length scale (SV)⫺1 (which equates to the arm spacing), Wang et al. [16] developed a relationship for permeability that also takes into account the equiaxed grains, arguing that liquid flow is partitioned between flow through the dendrite grains and flow around the grains. Permeabilities were therefore related to five parameters defining the solid morphology [14]. On completion of each permeability test (lasting about 50 s) the sample was quenched and the microstructures analysed for the five features and an average value for K calculated for the 50-s period. From plots of KS02 versus gS it was concluded that the measurements constitute experimental validation of the model of Wang et al. [16]. However, any microstructural coarsening and any liquid pool entrapment occurring during a test were ignored. Also, as noted by the authors [14] and illustrated in several representative micrographs, in a number of the tests individual grains had been completely surrounded by liquid. This would allow solid movement as well as fluid flow. It is certainly unclear at the present time whether either the Kozeny–Carman [3], Zick and Homsy [15] or Wang et al. [16] relationships are applicable to the equiaxed solidification of alloys over a wide range of values of gL. Even if one or more of them are, their implementation for the accurate prediction of interdendritic flow would require a quantitative knowledge of the geometry of the solid as a function of gL. Experimental determination of permeability requires not only precise measurement of the permeability but also specification of the microstructure corresponding to the permeability measured. Since microstructure changes continuously during a permeameter test (and these microstructures can only be subsequently examined two dimensionally) and tests can only be performed if there is a rigid interconnected solid, the accuracy of experimentation is questionable. The only alternative to the above approaches for obtaining permeability values is to employ a numerical method to model the permeability of 3D
microstructures. In this way an attempt can be made to overcome the problems relating to maintaining microstructural rigidity and stability experienced during experimentation and to evaluating SV, S0, gL or any other required geometrical parameters for 3D structures. To date, a variety of numerical methods have been applied to determining the permeability of dendritic structures, using either simplified dendrite-like shapes or images of actual dendrites [17– 19]. A numerical simulation has also been performed of the influence of Ostwald ripening of an Al– 4% Cu solid–liquid mixture on permeability [20]. All of these studies [17–20] are limited by the fact that they were performed for 2D structures and for large liquid fractions. The purpose of the present research was to develop a comprehensive 3D finite difference numerical model for the evolution of an equiaxed dendrite and the determination of the variation in permeability as the dendrite evolved.
2. The numerical model
The model attempts to predict the variation in the permeability of an equiaxed dendritic structure as the structure evolves. It assumes that a uniform distribution of static nuclei solidifies to become an array of cube-shaped grains of equal size. Though somewhat unrealistic in the sense that grains are not cubic in shape and, at the onset of freezing, the grains would not remain static, the model is thought to be reasonably representative once a rigid interlocking arrangement of grains has formed. In the model, flow through a cubic domain, the size of a single solidified grain, is assumed to be representative of the bulk material behaviour. The model therefore comprises two stages, first the numerical evolution of a 3D dendrite (in this instance an Al3Cu3Si alloy was arbitrarily selected since it freezes with a large fraction of primary solid solution phase) and second the numerical determination of the permeability of the domain as a function of fraction solid.
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2.1. Stage one: a cellular automaton-finite difference (CAFD) model of an evolving 3D dendrite Various 2D/3D cellular automaton-finite difference (CAFD) models for the simulation of columnar/equiaxed growth and the investigation of microsegregation in binary/ternary alloys have been reported in recent years [21,22]. The present model details, relating to solute redistribution during freezing (including back diffusion) and to growth, are described in detail elsewhere [22]. Initially a nucleus is placed at the centre of a cubic domain consisting of 100×100×100 cubic elements. The edge length of the domain, which corresponds to an appropriate grain size for an ascast aluminium alloy, is 300 µm. The domain is assumed to be of a uniform temperature and cooling at a rate of 50 K s⫺1, again a reasonably typical value. Latent heat is not emitted during the simulation. During cooling, the model simulates the evolution of an equiaxed dendrite and the accompanying redistribution of the solutes. This involves modelling solute diffusion (using a fully implicit finite difference scheme and employing temperature-dependent diffusion coefficients for Cu and Si in the liquid and solid phases) and predicting the solidification of elements from a thermodynamically predicted phase diagram generated by ThermoCalc [23] with data from ThermoTech [24]. 3D snapshots of the evolving solid dendrite are retained for use in the permeability studies. The software is able to distinguish between liquid sites that have become completely isolated by solid and those that are still able to contribute to the flow. Isolated liquid sites are considered to be effectively solid sites (since they can no longer take part in flow) and to behave as such in the calculation of the ‘effective’ values for fraction solid, SV and S0. 2.2. Stage two: a CFD model for determining permeability An in-house developed commercial CFD code MAVIS-FLOW [25] was used to determine 3D permeability as a function of the effective solid volume fraction. The code employs a full Navier– Stokes treatment of flow through the domain
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(dimensions as above) for a wide range of dendrite solid fractions/geometries obtained from the dendrite evolution model. Having modelled the growth of the dendrite within the cubic domain, it is assumed that liquid enters the domain at one face and leaves at the opposite face. To facilitate the determination of the permeability of the dendrite, the domain is extended by adding 10 layers of liquid elements to the inlet and outlet faces. The liquid enters and leaves the extended domain at a fixed uniform velocity of 1×10⫺3 m s⫺1, no-slip boundary conditions operate during flow, and a constant kinematic viscosity of 4×10⫺6 m2 s⫺1 is assumed. The other four faces of the extended domain are assumed to be solid. For each effective dendrite fraction selected, the permeability of the domain is calculated using Darcy’s law which states, K ⫽ vhL / ⌬P
(3)
where v is the rate of flow, h is the absolute viscosity (kinematic viscosity×liquid density) and L is the length (in this case 300 µm) over which the pressure drop ⌬P is calculated. ⌬P is determined for each simulation, once a steady state flow has been achieved, by obtaining the average pressure for all liquid sites in each of the two all-liquid layers immediately adjacent to the original cubic domain used for constructing the dendrite.
3. Observations Fig. 1 illustrates the evolution of the dendrite for different values of the ‘effective’ solid volume fraction, gS. There is a slight asymmetry in the evolving solid dendrite shape which may have an influence on the determined permeability vis-a`-vis a symmetrical dendrite. However, this paper reports the findings of the preliminary study of the performance of the model and investigations are currently under way to determine the influences of major changes in dendrite geometry due to alloy composition, grain size and cooling rate. It can be seen that in the early stages of freezing, prior to the partially solidified dendritic grain having developed a cube-shaped geometry, as discussed by Wang et al. [16], flow occurs both
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Fig. 1. Numerical evolution of a dendrite for an Al3Cu3Si alloy at effective solid fractions of (a) 0.1%, (b) 1%, (c) 6% and (d) 55%.
through the dendrite and around the dendrite. At high volume fractions of solid, liquid pools become isolated in the dendrite as shown for a 2D slice through the domain for 95% fraction solid, see Fig. 2. This figure can be considered as equivalent to a 2D micrograph that might be obtained from an actual alloy sample. However, from a 2D micrograph, it is impossible to distinguish between liquid areas isolated by solid and those that are open and contributing to liquid flow through the dendrite. Fig. 3 illustrates in 3D, again for 95% solid, the liquid channels that retain connectivity between the inlet and outlet faces of the cubic domain. Many of the channels are themselves also interconnected. It is recognized that a limitation of the present modelling approach is that, as the local solid fraction increases, flow channels may dimensionally be reduced to the size of one element. Also shown are the additional liquid layers, introduced in the extended domain for the permeability simulations, at the inlet and outlet faces. Fig. 4 shows a pressure map over the domain height for a halfsection of the domain, indicating the pressure drop that occurs as the liquid flows through the domain,
Fig. 2. 2D section through a dendrite (grey) for an effective solid fraction of 95%, illustrating isolated liquid regions (black).
and the pressure variation across the exit face (top of figure). Fig. 5 indicates how the 3D numerically determined KS02 term varies with the volume fraction solid gS. For the plotted numerical results, both gS and S0 are ‘effective’ values. Also shown are equivalent variations, predicted using the Kozeny–Carman expression for KC constant values of 1 and 5. As explained in the introduction, this form of presentation of permeability data is often used when comparing experimental results, or data from numerical simulations, with the predictions of the KC equation. K, S02 and gS are all determined from the numerical model. It would seem illogical to attempt to compare the values of KS02 derived from the KC expression with those obtained from the model at low solid fractions, prior to the partially solidified dendrite having acquired a cubic shape. At low solid fractions (prior to the dendrite attaining a cubic shape) there would be wholly liquid regions within the domain and liquid would
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Fig. 3. 3D illustration of the dendrite in Fig. 2 showing the liquid channels that retain connectivity between the inlet (bottom) and outlet (top) faces of the domain.
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flow both around and through the dendrite. Fig. 5 illustrates, for solid fractions above 苲20%, that there is reasonable correlation between the simulated values of KS02 and those anticipated from the KC expression for a k value of unity. This implies that the modelled dendritic structure, for given values of fraction solid and S02, has a higher permeability than would have been anticipated from the Kozeny–Carman expression had a k value of 5 been assumed. Also shown in Fig. 5, by open circles, are numerically predicted values of KS02 versus fraction solid determined from 2D sections through the cubic domain for the same set of partially solidified 3D dendrites, i.e. equivalent to 2D metallographic sections through real dendrites. For each 3D partially solidified dendrite examined, 100×2D slices were taken through the domain. gS and S0 values were obtained for each slice and average values calculated for the 100 slices. These average values represent the ‘true’ fraction solid and S0 values for the structure, since, as when examining real 2D metallographic images, it is impossible to identify
Fig. 4. Pressure map (b) for flow through the dendrite, (a) for an effective solid fraction of 6%. As the colour darkens pressure decreases.
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values for a dendrite structure having the same ‘effective’ volume fraction solid. In the one case, for the edge length of 300 µm, the BCC structure has a higher permeability than the dendrite, whereas the opposite is true for the second case. Another value of KS02 for a BCC structure is given in Table 1 and plotted in Fig. 5. This value was calculated from the drag coefficient, kd, of 163 reported by Zick and Homsy for a BCC structure (gS=0.68) [15]. As mentioned briefly in the introduction, this coefficient was calculated from a theoretical expression derived by the authors for Stokes flow through periodic arrays of spheres which was then applied to flow through simple cubic, BCC and FCC arrays. The relationship between permeability and drag coefficient is given by K ⫽ 2r2 / 9gSkd 2
Fig. 5. Determined values of KS0 versus solid fraction compared with values expected from the Kozeny–Carman expression. Filled circles, 3D numerically predicted values for the evolved dendrite. Open circles, 2D numerically predicted values for the evolved dendrite. Crosses, numerically predicted values for a BCC structure for two different sphere sizes. Open square, calculated value for a BCC structure from a numerically predicted drag coefficient reported by Zick and Homsy [15].
isolated liquid pools. As can be seen, the difference between the 3D predictions (isolated liquid pools identified) and those obtained from 2D analysis of the 3D dendrites (isolated pools not identified) becomes significant at high solid fractions when liquid sites become isolated. To determine the influence of liquid channel geometry on the predictions of the numerical model, permeabilities were also calculated for a regularly packed BCC structure of spheres. The same domain of 100×100×100 elements was used. However, two different domain sizes were examined with edge lengths of 300 µm (i.e. the same as for the dendrite domain) and 30 µm. These lengths also corresponded to the lattice parameters of the BCC unit cell. The numerically predicted values of K and S0 for the two different edge lengths are given in Table 1. It can be seen that the KS02 values for the two are almost identical and lie on a Kozeny–Carman curve for a KC constant of 5. Also shown in the table are the equivalent predicted
(4)
where r is the radius of the spheres. The value of KSO2 agrees very well with those determined from the present numerical model and again is in close agreement with the value determined from the Kozeny –Carman expression for a KC constant of 5. In Fig. 6, numerically determined K values for the 3D dendrite data shown in Fig. 5 are compared with the reported experimental data of Nielsen et al. [14] for Al–Cu alloys. The fraction solid values in both cases are ‘true’ values, i.e. they do not account for the possible isolation of liquid volumes.The experimental data cover a variety of grain sizes ranging from 60 to 800 µm, obtained by adding different amounts of the grain refiner Al5Ti1B. The experimental permeabilities for grain sizes in the range 200–250 µm (the closest to the present modelled data for 300 µm) increased from 0.36×10⫺12 to 4.2×10⫺12 m2 as the fraction solid decreased from 0.91 to 0.73. In the case of the predicted values, the permeabilities increased from 0.39×10⫺12 to 2.8×10⫺12 m2 as the fraction solid decreased from 0.91 to 0.74. It can therefore be seen that there is excellent correlation between the two sets of data. Out of interest, to determine the influence of the use of ‘no-slip’ or ‘free-slip’ boundary conditions in the model on the predicted permeability values, predictions were compared for two high solid frac-
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Table 1 Summary of data in Fig. 5 for a solid fraction of 苲0.68 K (m2) 3D dendrite Domain size 300 µm (gs=0.686) BCC Domain size 300 µm (gs=0.685) BCC Domain size 30 µm (gs=0.685) BCC Zomsky and Homsy [15] Domain size 300 µm (gs=0.68)
S0 (m⫺1)
KS02
3.53×10⫺12
1.61×105
9.15×10⫺2
8.14×10⫺12
3.71×104
1.12×10⫺2
8.28×10⫺12
3.71×105
1.14×10⫺2
3.4×10⫺11
2.31×104
1.81×10⫺2
Fig. 6. Numerically determined K values for the 3D dendrite data in Fig. 5 compared with reported experimental data for Al–Cu alloys [14].
tions (in the range where interdendritic flow is important to various metallurgical phenomena). For ‘effective’ solid fractions of 0.74 and 0.96, the use of free-slip boundary conditions, as opposed to no-slip conditions, produced increases in the permeability of 4.6% and 1.5%, respectively.
4. Discussion It is assumed that the numerically evolved dendrite provides a reasonable representation of
dendrite/liquid channel geometry. In previously reported papers using the CAFD technique, dendrite arm spacings have been shown to correlate well with experimentally reported values [21]. The dendrite model clearly demonstrates that for ‘effective’ solid fractions in excess of 苲80%, significant volumes of liquid become completely isolated by solid and can no longer contribute to flow/permeability. This cut-off occurs in the solid fraction range where interdendritic flow to compensate for shrinkage and convective flow due to liquid density differences take place. Therefore, the influence of cut-off on overall flow behaviour needs to be appreciated. In contrast to some recently reported experimental data [9,13], where there appeared to be a reasonable correlation with the Kozeny–Carman expression for a KC constant of 5, in the present work there was a much better correlation with a constant of unity. Some doubt must exist regarding the experimental data, as detailed in the introduction, because of the problems of maintaining rigid dendritic structures with geometries that do not change during the course of the permeability test. Also, when quantifying the microstructures in experimental studies, it is impossible to detect and account for the isolation of liquid pools in the latter stages of freezing. For given values of S0 and fraction solid, lower values of the KC constant imply a higher permeability. Although the Kozeny–Carman equation appears to be valid over the whole fraction solid range for unconsolidated randomly
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packed beds of particles, as mentioned in the introduction it is not valid for flow along a bundle of capillaries of widely varying radius or for flow along parallel-oriented fibres [5]. It appears that in the two latter cases the Kozeny equation breaks down because local values of the hydraulic radius differ significantly from the mean hydraulic radius, (1⫺gS)/SV. Considering the former case, the greater rates of flow in the larger capillaries more than compensate for the reduction associated with the smaller capillaries. The greater flow rates/permeabilities predicted from the present numerical model for interdendritic flow, than would be expected from the KC equation for a KC constant of 5, might be expected due to the fact that the solid is consolidated and the liquid channels vary in geometry from one to another and along their lengths. The finite-difference numerical predictions of flow through BCC type structures, in comparison with interdendritic flow for the same solid fraction, are very interesting. In contrast to the random packing of spheres, where the fractional area of liquid can be considered to be constant, when flow occurs in regular structures the fractional area constantly varies. Furthermore, the form of the variation will depend on orientation, i.e. permeability will differ with orientation. One of the conditions of the Kozeny equation is that the fractional area of liquid remains constant during flow. A number of numerical studies and experimental investigations of flow through periodic porous media have been made [15,26,27]. Specifically relevant to flow through BCC structures, i.e. for gS=0.68, is the numerical investigation of Zick and Homsy [15] and the experimental investigation of Susskind and Becker [27]. The experimentally determined value of 170 for the drag coefficient [27] agreed extremely well with the numerically calculated value of 163 [15]. As reported in the results section, these give KSO2 values that agree extremely well with the present numerical results for BCC structures and also with the Kozeny–Carman expression for a KC constant of 5. The excellent correlation between the various sets of data for BCC structures would appear to indicate that the flow behaviour through BCC structures (expressed
in terms of KSO2) is similar to that through randomly packed unconsolidated beds of particles. In the present investigation of BCC structures, the determined permeability was greater for the larger sphere size than for the smaller sphere size. This reflects the higher value of S0 in the latter case. Direct comparison of these permeabilities with that of the dendrite structure is impossible because of the completely different geometries of the liquid channels for the dendrite and BCC structures. Good agreement between the predicted variation of permeability of the dendritic Al3Cu3Si alloy with fraction solid, for large solid fractions, and experimental data for other Al alloys would tend to confirm the general validity of the dendrite evolution model and that numerical modelling is a viable approach to predicting permeability of dendritic structures.
5. Conclusions (a) A numerical model has been developed for the simulation of 3D flow through equiaxed dendrites of an Al3Cu3Si alloy and the determination of the variation in permeability of the structure as solidification progresses. The model involves the evolution of an equiaxed dendrite and the application of an in-house CFD program MAVIS-FLOW to calculate permeability from Darcy’s law. (b) A particular advantage of the approach is that it also facilitates a more accurate quantitative analysis of the evolving dendrite microstructure than is possible from experimental studies on actual alloys. In particular, the model is able to identify liquid sites that become surrounded by solid sites in the later stages of freezing and are therefore isolated from the flow. (c) In contrast to recent experimental studies of dendritic structures that have reported a reasonable correlation between measured permeabilities and those calculated from the Kozeny–Carman (KC) expression for a KC constant of 5, in the present numerical study a better correlation was found for a constant of unity. The deviation of the KC constant from a value of 5,
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which is normally associated with unconsolidated randomly packed beds of particles, is thought to be due to the fact that the channels through the consolidated dendrite have different cross-sections with respect to each other and along their flow paths. This research would not therefore support the use of a KC constant of 5 for flow through equiaxed dendrites in numerical models which rely on the KC expression for evaluation of permeability. (d) Experimental studies are obviously confined to high solid fractions. There was excellent correlation at these high solid fractions between permeabilities predicted by the model and experimental data for equiaxed structures. (e) Numerical models offer an alternative method to experimental studies for investigating permeability during microstructural evolution. (f) Finite-difference numerically calculated KSO2 values for BCC structures over a range of sphere sizes showed good agreement with the value calculated from the Kozeny–Carman expression for a KC constant of 5 and with other numerically calculated and experimentally obtained data that have been reported.
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