In foundry practice, as-cast microstructure is a key concern. It is well accepted that gravitydriven transport processes have an influence on the final microstructure of shape cast components. A predictive model to simulate equiaxed solidification, which treats dendrite transport in the solidifying alloy melt, at low computational cost, is presented in this article. The non-equilibrium solidification model considers nucleation from industrial inoculants, growth and transport of the equiaxed dendrites. The motion of free dendrites is computed as the combined effects of sedimentation settling and transport of free dendrites by the liquid flow due to natural convection. When free dendrites become coherent at the latter stage of solidification, the equiaxed mush of coherent dendrites is assumed to demonstrate characteristics of a porous medium, until it becomes completely solid. Simulations are used to establish the sensitivity of
the as-cast structure of Al-Si shape castings to the magnitude of mould/alloy heat transfer coefficients. It was found that the effects of dendrite sedimentation are higher for low values of this heat transfer coefficient.Other funderEuropean Space Agency (ESA) via
PRODEX funding (contract number 90267).Deposited by bulk importkpw18/10/1

Browne, David J.

Mirihanage, Wajira U.

Research Repository UCD

Title As-cast grain size distribution prediction for grain refined castings via simulating free equiaxed dendrite transport during solidificationAuthors(s) Mirihanage, Wajira U., Browne, David J.Publication date 2011Publication information Mirihanage, Wajira U., and David J. Browne. “As-Cast Grain Size Distribution Prediction for Grain Refined Castings via Simulating Free Equiaxed Dendrite Transport during Solidification” 63, no. 1 (2011).Publisher GiessereiverlagItem record/more informationhttp://hdl.handle.net/10197/4788Downloaded 2023-10-31T04:02:18ZThe UCD community has made this article openly available. Please share how this accessbenefits you. Your story matters! (@ucd_oa)© Some rights reserved. For more information1  As-cast grain size distribution prediction for grain refined castings via simulating free equiaxed dendrite transport during solidification.   Wajira U. Mirihanage, David J. Browne*  School of Electrical, Electronic and Mechanical Engineering,  University College Dublin, Belfield, Dublin 4, Ireland.  Keywords:  Equiaxed Solidification, Grain transport, Dendrite sedimentation, Modelling   Abstract In foundry practice, as-cast microstructure is a key concern. It is well accepted that gravity- driven transport processes have an influence on the final microstructure of shape cast components. A predictive model to simulate equiaxed solidification, which treats dendrite transport in the solidifying alloy melt, at low computational cost, is presented in this article. The non-equilibrium solidification model considers nucleation from industrial inoculants, growth and transport of the equiaxed dendrites. The motion of free dendrites is computed as the combined effects of sedimentation settling and transport of free dendrites by the liquid flow due to natural convection. When free dendrites become coherent at the latter stage of solidification, the equiaxed mush of coherent dendrites is assumed to demonstrate characteristics of a porous medium, until it becomes completely solid. Simulations are used to establish the sensitivity of the as-cast structure of Al-Si shape castings to the magnitude of mould/alloy heat transfer coefficients. It was found that the effects of dendrite sedimentation are higher for low values of this heat transfer coefficient. (*  email: david.browne@ucd.ie) 2   Introduction Solidification of alloy melts can lead to various grain structures such as columnar, equiaxed or mixed. When grain refiners are used, usually solidification results in a fully equiaxed macrostructure. For many structural load-bearing applications, such equiaxed macrostructures are preferable. Prediction of the structure of castings has long been of interest to engineers, scientists and others involved with the foundry industry. During the last three decades, various analytical and numerical models [1-8] have been proposed to predict formation and growth of equiaxed grains. Some of the earlier models [1-5,7] were simplified by ignoring movement of equiaxed dendrites during solidification. Only a few multiphase models [6,8] considered such dendrite transport, but they came with a requirement for high computational power and time, and so were not suited for use at industrial scales.   The main motivation for the work described here is to develop a computationally efficient equiaxed solidification model which can be used to predict the Columnar to Equiaxed Transition (CET) via combination with a Front Tracking (FT) model of columnar solidification [9]. The current work bridges macro and micro phenomena in equiaxed solidification. At a microscopic scale it considers nucleation and growth, and then average values are taken into the macroscopic level representation, based on macro-scale conservation equations and evolution of flow characteristics. The model computes the nucleation, growth, movement and impingement of equiaxed dendrites during solidification from the liquid state.  A microscopic detail of each dendrite is not computed, thus reducing the computational overhead considerably. Macroscopic conservation is treated via a single phase formulation and that cuts down the requirement for 3  complex supplementary relations between two phases and hence for costly computer resources.   This model coupled with the FT columnar model was recently used for CET prediction, ignoring equiaxed dendrite motion via sedimentation settling but including their advection due to natural thermal convection [10]. The simulated CET predictions found good agreement with experimental results under reduced effects of gravity [11].   This article demonstrates the capabilities of the proposed model and provides a preliminary analysis of sedimentation/settling of dendrites leading to the final as-cast microstructure. The main contribution here comes from integrating a model of equiaxed dendrite  sedimentation [12] into a previous single phase equiaxed solidification model [13], via superimposition.     The Model The descriptive mathematical details of the first version of this equiaxed solidification model, which only considered dendrite transport in the liquid flow due to natural thermal convection, can be found in ref. [13]. The analytical and numerical equations (numbed 1 through 23) used in that model are listed here (table 1). The development of the mathematical expression to compute dendrite sedimentation velocity is outlined elsewhere [12]. Only the relevant dendrite growth kinetics, solid fraction evolution using the Scheil approximation instead of a basic linear one, self-determination of dendrite coherency, and integration of dendrite sedimentation into the dendrite transport model, are explained briefly here.   4   Nomenclature Cp  - specific heat(Jkg-1K-1) E  - source term account for latent heat K  - permeability tensor (m-1) KO  - morphological constant (m-1) L  - latent heat of fusion (Jm-3) N  - number of grains  Sx  - source term to account for momentum modifications in horizontal direction  Sy - source term to account for momentum modifications in vertical direction T   -  temperature (K) V  - volume of grains (m3) oldV   - average envelope volume at the previous time step(m3) exV   - volume  of previously nucleated (existing) equiaxed grains (m3) newV  - envelope volume of newly nucleated grains (m3)  Z  - an increment of radius of the equiaxed mushy envelope (m)  d  - grain diameter (m) fs - solid volume fraction g  - gravitational acceleration (ms-2) gs  - solid fraction k  - thermal conductivity (Wm-1K-1) ls  - solutal diffusion length (m) nv   - number of seeds nucleated p  - pressure (kgm-1s-2) t  - time (s) u  - horizontal velocity (ms-1) v - vertical velocity (ms-1) vt -dendrite tip velocity at time (ms-1)  β  - expansion coefficient (K-1) Ø  - net equiaxed volume fraction of total grown envelopes ΔSv  - volumetric entropy of fusion (Jm-3K-1) μ  - apparent viscosity (kgm-1s-1) μ0   - dynamic viscosity (kgm-1s-1) ρ  - density (kgm-3) γ  - solid-liquid interfacial energy (Jm-2) ηTi  - fraction of  seeds initiated at undercooling ΔTi  σ  - standard deviation  φ   - geometric mean ω  - seed density (m-2) 5  )7()(1 dFiTConservation of energy, momentum and mass Nucleation Envelop Evolution, Averaging and Impingement )1(2222EyTxTCkyvTxuTtTp)2(122222xxref SgTTyuxuxpyuvxutu)3(122222PSgTTyvxvypyvxuvtvyyref)4(0yvxu)5(4dSTvi)6(0ddssfdF)8(. jiTv in)9(2lnlnexp21)(22sssf)10(tvZ tt)11(2toldex ZVV)12(2tnew ZV)13()( newoldvvexoldvTotal VnnVnV)14(vTotal nVV)15(yxVTotalext)16(exp1 extTable 1. Conservation, Nucleation and Growth relations  6  Table 2. Source terms and transport relations   The empirical relationship between undercooling ∆T and dendrite tip velocity given in ref. [14] is used to compute the growth of equiaxed dendrite envelopes, which are assumed to be spherical in shape. Dendrite tip velocity vt is given by  )24(nt TCv  )17(...........1,,1,1,,,tyuNyuNxvNxvNyxNNjiwjiejisjinjinewji)18(........1,,1,1,,,tyuVyuVxvVxvVyxVVjiwjiejisjinjinewjiViscosity and Permeability  Source Terms Dendrite Number and Volume Transport   )19(exp05.105.21 20 sss fBAff)20(132ssOffKK)22()21(00vKSuKSyx)23(tVgCLEsp7    for Al-7%wt.Si  ;  C = 2.9x10-6 and n = 2.7; and tip velocity vt is then in unit of  ms-1.   With the assumption of zero solute diffusion in the solid phase and complete mixing in the liquid phase between dendrite arms, the Scheil approximation is used to calculate internal solid fraction gs of an equiaxed envelope.  )25(111klmmsTTTTg   Where Tl , Tm and k are equilibrium liquidus temperature, melting temperature of pure solvent material and partition coefficient of the alloy, respectively.  In computations, the dendrite transport regime changes from a model of slurry to a porous medium when the solid volume fraction reaches coherency. Unlike the initial model [13] the current development does not use a fixed experimental value for coherency fraction and the model itself computes the coherency fraction in 2D [15]. The coherency point is defined as the point where equiaxed envelopes fill the local space (control volume). Mathematically this is the instance when equiaxed envelope volume fraction reaches unity.   Due to the density difference existing between solid dendrites and liquid, the former are subjected to motion in the vertical direction. Generally, solid is denser than the liquid of similar composition and so solidifying dendrites start to sediment. The pure settling velocity w at any new time step, for a single dendrite can be written [12] as  8  )26(163161)(34111320210031101000rgReCmttggrwmmmwwsslss where, w0 is the settling velocity at previous time step. The average dendrite sedimentation settling speed, w is calculated using averaged characteristic dendrite parameters. Here the superscript “1” denotes a new time-step value and the superscript “0” denotes previous time-step value. The source term, P of the vertical momentum equation (equation (3)) incorporates dendrite settling where the complete system is treated as single phase without inter-phase closure relations.  )27(dtdwfP s   For net dendrite velocity, sedimentation only affects the dendrite transport in the vertical direction. The average dendrite transport velocity across a horizontal control volume (CV) face in this vertical direction is modified to  )28(2,,,,jsjnjijiwwvv      where, wn,j and ws,j are the average settling speed in the CV to the north (above) and the CV to the south (below) of the face, respectively.   9  The model presented here comes with a few simplifications and/or assumptions. The computations do not consider impacts between free dendrites on each other or with coherent solid, during their movement, nor the resultant momentum changes. Also, the complex momentum relationships between liquid and solid are not included in the model, for computational efficiency, and the net effect is instead accounted for as a modified viscosity of the slurry, which is a function of the local solid fraction – see equation (19). In reality, dendrites may deviate from a representative spherical envelope shape, as all primary dendrite arms may not be equal in length. In such instances dendrite rotation can happen as a result of Magnus forces. This model considers only an average representation of a large number of dendrites. It is assumed that most of these interactive forces may cancel each other out and so have very minimal net effect on overall developments.    Results and Discussion Model simulations were used to study the sensitivity of the as-cast structure of Al-7%wt.Si shape castings to the magnitude of mould/alloy heat transfer coefficients (h). A rectangular mould casting (height-‘y’ direction and width-‘x’ direction) was taken with the assumption that the front and back faces are fully insulated and hence adiabatic. This brings zero heat flux along the perpendicular direction (‘z’); which converts the system to 2D. The mould temperature was set to a constant 473K.  The total number of inoculant particles in each melt was set to 1 x 106 particles per m2 (for 2D) and the initial melt temperature was set at 898K for all cases. Height and width of the mould are both 0.2m.   10  Simulations predicted a gradient of average grain diameters along the vertical centre line of the casting. Relatively large diameter grains were predicted at locations close to the top of the mould and grains of lower diameter were predicted in the bottom half of the mould. A simple qualititative comparison between the simulated results and the average grain diameter distribution in the central part of a relatively small experimental Al-7%Si casting is shown in figure 1. A reasonable qualititative similarity between grain size variation predicted by the model and the experimental findings can be found.    Figure 1. Comparison of (a) predicted and (b) experimental, average grain diameter variation along centre line of the rectangular casting (distance measured from bottom) h=3,000 Wm-2K-1  Further simulations were performed with two different mould heat transfer coefficients (h=600 Wm-2K-1 and h=15,000 Wm-2K-1). During solidification, the predicted convection currents vs. dendrite movements are vector mapped in figure 2. The left portion shows convection current vectors while the right portion shows the dendrite movement vectors, which is the combined 11  effect of convection and settling. The dendrite movement (where free dendrites actually exist) is very similar to the liquid flow, as seen from figure 2. Numerically, these values are close but the percentage difference of dendrite velocity to the liquid flow velocity is higher when the mould heat transfer coefficient is low. It could be inferred that natural convection has a greater influence on dendrite transport than sedimentation (for the case of the binary alloy system that we discuss here) when cooling rates are high. In the central part of the mould, after 30s, there is no dendrite movement as it contains superheated liquid. Similarly, no dendrite movements and only convective fluid flow are present in the coherent region (lightly shaded). Figure 3 shows predicted average as-cast grain size maps for the three different cooling conditions.    Figure 2. Velocity vectors: liquid flow (left half) and dendrite movement (right half) after 30 seconds. Temperature (left half) and solid volume fraction (right half) also scaled.  For (a) h=600 Wm-2K-1    (b) h=3,000 Wm-2K-1    (c) h=15,000 Wm-2K-1  According to the simulations, the lower the heat transfer coefficient, the greater the difference between average grain size on top and bottom of the casting. The effect of sedimentation settling of dendrites seems to be higher at the lower cooling rate. At low mould heat transfer coefficient, 12  a relatively low thermal gradient is present in the melt as the temperature difference between centre and wall is small. Higher temperature differences cause a higher magnitude of natural convection currents, and convective flow dominates sedimentation settling (for the system we discuss here and similar), regarding movements of dendrites. As a result for high h, the effects of  dendrite settling are small, as is the overall macrostructure difference between top and bottom portions of the mould. It should also be noted that free dendrites have more time to descend in the melt via sedimentation when the cooling rate is low.   Figure 3. Average grain size map (a) h=600 Wm-2K-1 (b) h=3,000 Wm-2K-1 (c) h=15,000Wm-2K-1  It is important to note that the results yielded here could be significantly different from results for an alloy system which reject solute with higher density than the bulk liquid alloy (eg. Al-Cu). In such a case, the sedimentation settling action may be the reverse to that predicted here. Generally, solid metals have higher density than liquid melts of the same composition. But, a higher density of rejected solute can overturn this general situation by increasing bulk melt density in comparison to solute-poor dendrites. At initial stages of solidification, such solute poor (and hence less dense) dendrites may even rise (float) due to the increased density of bulk 13  liquid as a result of the rejected denser solute. Such a scenario has been observed experimentally via in-situ X-ray studies of solidification of Al-Cu alloys [16].  It is realistic to extend this model with further developments to predict as-cast grain size distribution in grain-refined shape castings in 3D and/or complex shapes. Research is in progress to qualitatively validate this equiaxed solidification model and integrate it into a prediction model of CET in alloy castings under the effects of gravity.   Conclusion A recently developed model of equiaxed solidification has been extended to add the effect of dendrite sedimentation to dendrite movement due to convection currents. The model can simulate developments during equiaxed alloy solidification and predict final average as-cast macrostructure. Simulations show that, under certain conditions, dendrite transport by convection currents dominate sedimentation of free dendrites in the bulk undercooled liquid. A qualititative comparison of simulation results with experimental results showed good agreement. Analysis of simulation results with different mould heat transfer coefficients found a reduced effect of dendrite settling with high cooling rates. This suggests a possible higher gradient of grain diameters in the vertical direction of a casting solidifying with a lower cooling rate. Acknowledgments The authors wish to acknowledge the support of the European Space Agency (ESA) via PRODEX funding (contract number 90267). This work is part of the ESA-MAP (Microgravity Applications Promotion) project CETSOL.  References 14  [1] Maxwell I., Hellawell A., Acta Metall., 1975, Vol. 23, pp. 229-237 [2] Hunt J.D., Mater. Sci. Eng.,  1984, Vol. 65, pp. 75-83 [3] Dustin I., Kurz W., Z. Metallkd., 1986, Vol. 77, pp. 265-263 [4] Rappaz  M., Thevoz Ph., Acta Metall., 1987, Vol. 35, pp. 1487-1497 [5] Rappaz M., Gandin Ch.-A.,  Acta Metall. Et. Mater., 1993,Vol. 41, pp.345-360 [6] Wang C.Y., Beckermann C., Metal. Mater. Trans. A, 1996, Vol.27, pp. 2754-2763 [7] Greer A.L., Bunn A.M, Tronche A., Evans P.V., Bristow D.J., Acta Mater., 2000, Vol. 48, pp. 2823–2835 [8] Wu M., Ludwig A., Metal. Mater Trans.A., 2006, Vol. 37, pp1613-1631   [9] Browne D.J., Hunt J.D., Num. Heat. Trans. B, 2004, Vol. 45, pp 395-419 [10] Mirihanage W.U., McFadden S., Browne D.J., Shape Casting- 3rd International Symposium - TMS 2009, (Edi. J. Campbell et. al.), TMS, Warrendale, PA, USA, pp. 257-263 [11] Mirihanage W.U., McFadden S., Browne D.J., Mater. Sci. Forum, 2010, Vol. 649, pp. 355-360 [12] Mirihanage W.U.,  Browne D.J.,  manuscript in preparation [13] Mirihanage W.U, Browne D.J, Comp. Mater. Sci., 2009, Vol.46, pp.777-784 [14] Ch.-A. Gandin, Acta Mater., 2000, Vol.48, pp. 2483-2501 [15] Mirihanage W.U, Browne D.J, manuscript in preparation [16] Mathiesen R.H., Arnberg L., Beleuet P., Somogyi A., Metall. Mater. Trans. A., 2006, vol.37A, pp.2515-2524  

As-cast grain size distribution prediction for grain refined castings via simulating free equiaxed dendrite transport during solidification

In foundry practice, as-cast microstructure is a key concern. It is well accepted that gravitydriven transport processes have an influence on the final microstructure of shape cast components. A predictive model to simulate equiaxed solidification, which treats dendrite transport in the solidifying alloy melt, at low computational cost, is presented in this article. The non-equilibrium solidification model considers nucleation from industrial inoculants, growth and transport of the equiaxed dendrites. The motion of free dendrites is computed as the combined effects of sedimentation settling and transport of free dendrites by the liquid flow due to natural convection. When free dendrites become coherent at the latter stage of solidification, the equiaxed mush of coherent dendrites is assumed to demonstrate characteristics of a porous medium, until it becomes completely solid. Simulations are used to establish the sensitivity of
the as-cast structure of Al-Si shape castings to the magnitude of mould/alloy heat transfer coefficients. It was found that the effects of dendrite sedimentation are higher for low values of this heat transfer coefficient.Other funderEuropean Space Agency (ESA) via
PRODEX funding (contract number 90267).Deposited by bulk importkpw18/10/1

As-cast grain size distribution prediction for grain refined castings via simulating free equiaxed dendrite transport during solidification

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