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Acta Materialia 58 (2010) 2864–2875

Modeling fluid flow in three-dimensional single crystaldendritic structures

J. Madison a,*, J. Spowart b, D. Rowenhorst c, L.K. Aagesen d, K. Thornton a, T.M. Pollock a

a Department of Materials Science & Engineering, University of Michigan, Ann Arbor, MI 48109, USAb Air Force Research Laboratory/RXLMD, Wright Patterson AFB, OH 45433, USA

c Naval Research Laboratory, Washington, DC 20375, USAd Department of Materials Science & Engineering, Northwestern University, Evanston, IL 60208, USA

Received 30 September 2009; received in revised form 8 January 2010; accepted 11 January 2010Available online 13 February 2010

Abstract

Convection during directional solidification can cause defects such as freckles and misoriented grains. To gain a better understandingof conditions associated with the onset of convective instabilities, flow was investigated using three-dimensional (3D) computational fluiddynamics simulations in an experimentally obtained dendritic network. A serial-sectioned, 3D data set of directionally solidified nickel-base superalloy measuring 2.3 � 2.3 � 1.5 mm was used to determine the permeability for flow parallel and normal to the solidificationdirection as a function of solid fraction (fS). Anisotropy of permeability varies significantly from 0.4 < fS < 0.6. High flow velocity chan-nels exhibit spacings commensurate with primary dendrite arms at the base of the mushy zone but rapidly increase by a factor of three tofour towards dendrite tips. Permeability is strongly dependent on interfacial surface area, which reaches a maximum at fS = 0.65. Resultsfrom the 3D simulation are also compared with empirical permeability models, and the microstructural origins of departures from thesemodels are discussed.� 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Nickel alloys; Dendritic growth; Directional solidification; Modeling; Permeability

1. Introduction

Convective instabilities induced by alloying or process-ing conditions during solidification of nickel-base singlecrystals are a major challenge in the production of turbineairfoils [1–3]. Predictions for the formation of defects dur-ing solidification of single crystal and columnar grainsuperalloys have primarily used the Rayleigh criterion asan indicator of thermal and solutal conditions at whichconvective instabilities will initiate [4–9]. As the ratiobetween the buoyant and frictional forces, the Rayleighnumber can be described as:

Ra ¼ Dqq0

� �gKhav

ð1Þ

1359-6454/$36.00 � 2010 Acta Materialia Inc. Published by Elsevier Ltd. All

doi:10.1016/j.actamat.2010.01.014

* Corresponding author. Tel.: +1 734 6155163; fax: +1 734 6155168.E-mail address: jonnymad@umich.edu (J. Madison).

where Dqq0

� �refers to the density gradient in the melt, g is

the acceleration due to gravity, and av is the product ofthe thermal diffusivity and kinematic viscosity. The termK is the permeability, a property which is anisotropic fordirectional structures such as those produced in single crys-tal dendritic solidification. Permeability is influenced by thethermal conditions present during solidification as well asby material properties that influence the morphology andfraction of dendrites present during solidification.

In porous media, permeability varies strongly with vol-ume fraction. As such, a number of empirical equationsrelating volume fraction to permeability have been devel-oped. Most common are the Kozeny–Carmen and Blake–Kozeny descriptions [10–13]:

KKC ¼ð1� fSÞ3

5S2V

ð2Þ

rights reserved.

J. Madison et al. / Acta Materialia 58 (2010) 2864–2875 2865

KBK ¼ C2

d21f 3

L

ð1� fLÞ2ð3Þ

where fS and fL (where fL = 1�fS), correspond to solid andliquid volume fractions, respectively. SV is the solid–liquidinterfacial area per unit volume, d1 is the primary dendritearm spacing, and C2 is an empirically determined constant.Applying these relations to a directionally solidified struc-ture across all ranges of solid volume fraction is challeng-ing owing to the inherent anisotropy of the dendrites andthe associated permeability tensor. In anisotropic materi-als, permeability is fully described by a 3 � 3 tensor withdiagonal values corresponding to each of the three princi-pal directions in an orthogonal coordinate system. Giventhe symmetry of the dendritic structure, the tensor caneffectively be described by two components, a vertical(KY) and a cross or normal component (KX). Poirier mod-ified the Blake–Kozeny relation to describe permeability inflow parallel and transverse to the solidification direction(KBKy and KBKx , respectively) for fS greater than 0.6 [13].

KBKy ¼ ð4:53� 10�4 þ 4:02� 10�6 ðfL þ 0:1Þ�5Þ d21f 3

L

ð1� fLÞð4Þ

KBKx ¼ 1:73� 10�3ðd1=d2Þ1:09� � d2

2f 3L

ð1� fLÞ0:749ð5Þ

However, the Blake–Kozeny relation and these subse-quent modifications do not intrinsically consider theinfluence of high surface areas characteristic of dendriticstructures. To address this, Heinrich and Poirier latersuggested a three-regime description for both vertical(KY) and cross flow permeability (KX) in directionalstructures derived from regression analysis on numericalsimulation results where no prior experimental dataexisted [14]:

Kx ¼ Kcross ¼

1:09� 10�3f 3:32L d2

1 fL 6 0:65

4:04� 10�6 fL1�fL

h i6:7336

d21 0:65 6 fL 6 0:75

�6:49� 10�2 þ 5:43� 10�2 fL1�fL

h i0:25� �

d21 0:75 6 fL 6 1:0

8>>>>><>>>>>:

ð6Þ

Ky ¼ Kvertical ¼

3:75� 10�4f 2L d2

1 fL 6 0:65

2:05� 10�7 fL1�fL

h i10:739

d21 0:65 6 fL 6 0:75

0:074 ln ð1� fLÞ�1 � 1:49þ 2ð1� fLÞ � 0:5ð1� fLÞ2h i

d21 0:75 6 fL 6 1:0

8>>><>>>:

ð7Þ

In Eqs. (4)–(7), d1 and d2 refer to the spacing of the pri-mary and secondary dendrite arms, respectively. Irrespectiveof the model adopted, the predicted variation in permeabilitywith volume fraction is three to four orders of magnitude,

producing a relative change in the Rayleigh number of thesame order, assuming all other factors remain relatively con-stant. Since these large variations in Rayleigh number canstrongly influence the onset of convection, a more detailedinvestigation of flow through three-dimensional (3D) den-dritic structures is needed to understand the role of dendriticstructure in the defect formation process.

While it is apparent that permeability is influenced bythe fraction of liquid and solid (Eqs. (4)–(7)), the influenceof dendrite morphology is less clear. There have been sev-eral attempts to consider explicitly the influence of den-dritic structure on fluid flow. Ganesan et al. [15] usedmicrographs transverse to the primary growth directionin Pb–Sn alloys at discrete heights in the mushy zone tostudy flow parallel to the solidification direction with aboundary element method. Permeabilities were then calcu-lated using Darcy’s law. This approach worked well forhigh liquid fraction (fL) regions, but overestimated perme-abilities experimentally observed for fL < 0.61 [13,16,17].While one of the first studies to consider flow explicitly indendritic structure, it was assumed that the dendrite mor-phology remained uniform along the flow direction, thusneglecting the details of the secondary dendrite arms.Nonetheless, Ganesan and co-workers were among the firstto suggest a transition in dependence of permeability onliquid fraction at or near fL = 0.65 [15].

In an effort to calculate permeability normal to the pri-mary growth direction at high liquid fraction in direction-ally solidified alloys, Bhat and co-workers performednumerical simulations on both artificially created andexperimentally derived dendritic cross sections in Pb–Snalloys with liquid fractions fL ranging from 0.5 to 0.9. Thiswork offered some of the first calculations of KX above 0.6fL in experimentally obtained structures and demonstratedagreement with lower fraction liquid results. Based on this,an empirical relationship between non-dimensional perme-

ability and volume fraction was developed. This model alsodemonstrated that changes in flow orientation laterallyacross dendritic arrays have little to no impact on perme-ability normal to the primary growth direction [18].

Fig. 1. Exposed dendritic networks obtained via decant casting.

2866 J. Madison et al. / Acta Materialia 58 (2010) 2864–2875

Later, Nielsen, Bernard and co-workers [12,19] usedmicrotomography to quantify permeability directly in equi-axed dendritic structures of Al–Cu alloys and comparethem with experimental measurements. This techniquehad an upper limit in reconstructed volume on the orderof 1 � 1 � 0.6 mm and an average resolution for all fea-tures near 1 lm [19], with likely smaller volume limits forhigher density materials. Multiple 3D tomograms of simi-larly quenched, directionally solidified Al–Cu samples wereproduced by X-ray microtomography, and flow modelingin each of the primary orthogonal directions was per-formed [20]. Permeabilities normal and parallel to the pri-mary growth direction were determined using thepreviously developed Bernard model [19]. Results werecompared with solutions obtained with the Stokes equa-tion, the Kozeny–Carmen relation, their previous experi-mental measurements [12] and a host of otherexperimental data [21–24]. It was suggested that volumeelements with an edge length measuring at least twice thecharacteristic length (i.e., twice the secondary dendritearm spacing) would yield a reasonable permeability assess-ment. Anisotropic parallel and cross flow permeabilitiesthrough the columnar dendritic network were observed,with a general trend towards lower permeabilities for crossflow when compared with vertical-flow for solid fraction< 0.47.

To date, studies on directionally solidified structureswith sufficiently large simulation cells are still lacking. Thisstudy combines serial sectioning with numerical fluid flowanalyses to examine the permeability in a directionallysolidified structure. The approach described in this paperuses a full, 3D reconstruction of a dendritic networkobtained from directionally solidified commercial nickel-base superalloy, Rene N4. The full data set of the dendriticvolume and its subsets are used to generate meshes for flowsimulations. Each subset contains, at minimum, cross-sec-tional dimensions greater than twice the secondary den-drite arm spacing and up to 18 primary dendrite cores inthe most densely populated cross sections. Permeabilitiesare determined via a Darcy’s law approach. The inhomoge-neity of the flow process, its relation to the dendritic micro-structure present and the interfacial surface area (ISA) areconsidered in detail.

2. Experimental method

The commercial nickel-base superalloy Rene N4 withnominal composition 4.2Al–0.05C–7.5Co–9.8Cr–0.15Hf–1.5Mo–0.5Nb–4.8Ta–3.5Ti–6.0 W–bal. Ni (wt.%) wassolidified directionally in an ALD Vacuum TechnologiesBridgman Furnace with a withdrawal rate of 2.5 mm min�1

(0.042 mm s�1) and a thermal gradient of �40 �C cm�1

[25]. During withdrawal, a controlled fracture was initiatedin the ceramic mold to allow molten liquid to flow out ofthe mold, leaving the solidified dendritic structuresexposed. A micrograph illustrating dendrites obtainedusing this technique is shown in Fig. 1. The decanting event

occurs in a matter of seconds and avoids coarsening and“globularization” typical in permeameter experiments[26]. With a DTliquidus–solidus range of 45 �C, and a thermalgradient of 40 �C cm�1, a mushy zone height up to 11 mmcould be expected with a linear thermal gradient. However,an ability to decant fully from high fraction solid regionslower in the mushy zone and a local non-axial gradientat the mold wall are most likely responsible for a shortenedmushy zone height. Nonetheless, microstructural investiga-tions reveal good correlation of both primary and second-ary dendrite arm spacings with spacings expected fromthermal gradient and withdrawal rate models [27–30]. Inaddition, isolated porosity was identified with a volumefraction in the range typical of single crystal nickel-basecastings [31–33]. Additional details of the reconstructionand initial characterization of resulting structures havebeen presented elsewhere [34–36]. Here, the focus will beon examination and description of dendritic morphologyand its effect on local fluid flow.

Samples of the dendritic structure spanning from the den-drite tips to fully solidified material measuring�1 cm3 in sizewere removed from the surface of the casting, mounted andserial-sectioned parallel to the solidification direction at anaverage interval of 2.2 lm per slice, using the automatedRoboMET.3De serial sectioning system [37,38].

The reconstructed solid and liquid domains are shown inFig. 2, where the primary growth directions are indicatedby arrows. Fluid domains are generated by assuming thatall non-solid features in the volume are liquid. By invertingthe volume of interest, a 3D representation of liquid can beobtained, as shown in Fig. 2b. The solid inverse of the 3Ddendritic structure can then be used as the simulationdomain for local flow models.

3. Simulation approach

Using the 3D rendering and meshing software, 3-Matice by Materialise�, a surface mesh containing morethan 2.9 � 105 elements (Fig. 3a) was generated then con-verted into a volume mesh of more than 1 � 106 elements(Fig. 3b) using the Gambite FEM pre-processor. Afterfunctional volume meshes had been produced, mesh refine-ment was continued iteratively until numerical flow solu-tions converged. All flow simulations were performedusing FLUENTe by ANSYS�.

Fig. 2. 3D reconstruction of dendritic network (a) solid and (b) liquid in Rene N4. Arrows indicate direction of solidification.

Fig. 3. (a) Surface mesh and (b) volume mesh of 3D dendritic network for vertical-flow.

J. Madison et al. / Acta Materialia 58 (2010) 2864–2875 2867

For additional simulations, smaller subsets of the vol-ume mesh shown above were taken and iteratively refinedfor full Navier–Stokes simulations. These cells were thencategorized by variation in volume fraction. The effect ofgrid element quantity was also monitored to ascertain thegrid independence of the flow results. Decreases in elementcount by up to 30% within individual simulations producedno variation in velocity or pressure drop, verifying the gridquality and repeatability of simulation results. Addition-ally, in an effort to understand better the effects of theanisotropy of dendritic structures on flow channel tortuos-ity, two principal orthogonal flow directions were investi-gated. For the remainder of this paper, these primaryflow directions will be referred to as cases of either “verti-cal” (denoted by the subscript Y) or “cross” (denoted bythe subscript X) flow and are parallel and normal to the

Fig. 4. (a) Surface mesh and (b) volume mesh

primary solidification direction, respectively. A subset ofthe above mesh is shown in Fig. 4 as an example of themesh type used for the cross flow studies.

For both vertical (Y) and cross flow (X), boundary con-ditions of zero pressure at the outlet as well as no-slip on allinternal boundaries and side walls were imposed. Since rel-atively thin slices were used for cross flow investigation, azero-shear condition was set on the top and bottom sur-faces to alleviate any drag or boundary layer effects causedby the artificially imposed walls associated with these flowcells. For vertical-flow cases, the nominal flow velocity wasprescribed to be in the range of the withdrawal rate. Forcross flow, inlet velocities were selected to yield velocityranges equivalent to 0.1 of those in the vertical-flow cases.Any changes resulting from variation in inlet velocity werebalanced by commensurate changes in the volumetric flow

of 3D dendritic network for cross flow.

Table 1Summary of flow simulations by case.

Case Inletvolumefractionliquid

Cell volume (lm3) Inletvelocity(lm/s)

Velocitymin–

max (lm/s)Pressuredifference(Pa)

Vertical-flow

Y1 0.08 2180 � 1635 � 980 6.0 53.2–1060 7.73Y2 0.10 2180 � 1650 � 840 5.0 52.7–1050 5.40Y3 0.22 2180 � 1640 � 570 4.5 53.7–1070 5.15Y4 0.32 2180 � 1650 � 480 4.0 53.6–1070 3.79Y5 0.43 2180 � 1650 � 400 65.0 53.8–1080 5.45Y6 0.60 2180 � 1650 � 350 110 53.5–1070 3.77

Case Volumefractionliquid

Cell volume (lm3) Inletvelocity(lm/s)

Velocitymin–

max (lm/s)Pressuredifference(Pa)

Cross flow

X1 0.42 2160 � 1620 � 20 1.95 5.09–102 1.62X2 0.56 2160 � 1620 � 20 5.5 5.20–104 0.633X3 0.73 2160 � 1620 � 20 7.8 5.12–102 0.214X4 0.89 2160 � 1620 � 20 18.6 5.02–100 0.079

2868 J. Madison et al. / Acta Materialia 58 (2010) 2864–2875

rate and pressure, leaving the final results unaltered for allconverged results. Details of the inlet boundary conditionsare summarized in Table 1 and listed by volume fractioncase. The physical properties for the molten nickel alloywere approximated as 6980 kg m�3 for density (q) [39]and 0.0045 kg m s�1 for viscosity (l) [40] in all cases. Itshould be emphasized that no temperature or density gra-dients have been imposed in these simulations, as the focushere is not to model solidification, but rather understandthe nature of fluid flow through the heterogeneous environ-ment of an actual dendritic network.

4. Results

4.1. Fluid flow

As mentioned previously, flow simulations were focusedon the two primary orthogonal directions. Flow was stud-ied in a total of 10 cells, with six vertical-flow and fourcross flow cases. A summary of the specific boundary con-ditions, resultant velocity ranges and pressure differencesexperienced across the cells is given in Table 1. The inletvelocities for each case are listed and were adjusted via scal-ing by the area fraction available in the vicinity of the inletand optimizing from this value to arrive at a minimumvelocity equal to 0.05 mm s�1. This allowed unrestrictedand heavily restricted channels to experience flow fieldswith velocities comparable with the withdrawal velocity.This can be noted in velocity ranges across all vertical-flowcases (denoted by Y), and similarly among all cross flowcases (denoted by X) as listed in Table 1.

Color-coded velocity vector images, where blue corre-sponds to minimum velocities and red corresponds to max-ima, detail the flow behavior through the dendriticnetwork, and are shown in Figs. 5 and 6 for cases Y1and X2, respectively.

4.2. Interfacial surface area

The ISA as a function of height in the reconstructionwas analyzed. All individual elements of the reconstructionwere compiled and indexed according to their height. Bybinning all surface elements within a pre-determined heightrange, measures of ISA as a function of height wereobtained. By changing the bin size, measures as coarse oras fine as desired were obtained. Ultimately, the bin sizeequal to the vertical resolution in the data set(4.16 lm pixel�1) was selected to maintain equal spatialcorrelation between height, volume fraction and ISA. AnISA normalized by the associated volume was also calcu-lated to obtain the surface area per unit volume (SV), whichhas a unit of length and is a measure of the characteristiclength scale. The relative change in SV with volume frac-tion is shown in Fig. 7. Interestingly, SV reaches a maxi-mum at fS = 0.65. Undulations in SV correspond directlywith local fluctuations in the solid volume fraction causedby collections of secondary dendrite arms. The undulatingpeak widths range from 80 to 200 lm and indicate that SV

is influenced by individual and concurrent sets of second-ary dendrite arms.

4.3. Permeability

By measuring the pressure gradient across a given flowdirection in these simulations, Darcy’s law:

Q ¼ KADPlL

ð8Þ

can be used to calculate a global permeability for the sim-ulation cell, where Q is the volumetric flow rate, A is theinlet cross-sectional area, L is the length associated withthe pressure drop DP, and l is the fluid viscosity. Localpressures are assessed through planar averages over the en-tire cross section of the simulation normal to the principalflow direction. In vertical-flow cells, average pressure dropssignificantly in the vicinity of the inlet and decreases to aplateau at increasing heights. In cross flow, pressure dropoccurs less rapidly across the domain. As a result, in verti-cal-flow simulations, tangents of the maximum pressuredifferential were assessed from the pressure profile for iden-tification of the pressure change, DP. L is the length overwhich the maximum and minimum pressures of the tangentline are observed. In cross flow, planes offset from the inletand outlet free of potential boundary effects are used, andDP is identified and assessed as the difference between theiraverage planar pressures. L is the length of separation be-tween the two identified planes. Fig. 8 illustrates the selec-tion of DP and L through illustration of the pressureprofiles in cases Y2 and X2 respectively. Since permeabilityis anisotropic, KY for parallel-flow and KX for cross flowwere independently calculated, and are listed as a functionof volume fraction inlet for vertical-flow cases and totalvolume fraction for cross flow cases as listed in Table 2and shown graphically in Fig. 11.

Fig. 6. (a) Velocity vectors for case X2 with (b) an exploded view highlighting flow behavior across the dendritic network in the plane of secondarydendrite arms. Velocities in units of mm s�1.

Fig. 7. Variation in measured ISA per unit volume (SV) with solid volumefraction.

Fig. 5. (a) Velocity vectors for case Y1 for vertical-flow with (b) an exploded view detailing the flow behavior in the vicinity of multiple inlets. Flowvelocities are shown in units of m/s.

J. Madison et al. / Acta Materialia 58 (2010) 2864–2875 2869

4.4. Anisotropy of permeability

Although there is little overlap in volume fractions forcalculated KY and KX in this study, there is a discernabledifference in the dependence of KY and KX on solid volumefraction. To evaluate the difference, the dependence ofdimensionless permeability (KSVS

2) on ð1� fSÞ3, for KY

and KX was calculated by combining the simulation resultswith measured ISA per unit volume solid (SVS) in the 3D

data set at the corresponding volume fractions (Fig. 9).Comparison with ð1� fSÞ3 is depicted, as it is the funda-mental quantity relating to solid fraction in both Eqs. (4)and (5). Vertical-flow permeabilities KY show �16%increase in dimensionless permeability (KSVS

2) per unitincrease in ð1� fSÞ3, while cross flow simulations exhibita much higher sensitivity to the solid fraction. These differ-ences are likely to result from variation in flow path tortu-osity. For this reason differences in flow-path lengths forvertical- and horizontal-flow cases were also investigated.

4.5. High-velocity-flow channels

As discussed previously, convective flow can erode, frag-ment and transport portions of solidified material resultingin the formation of freckle chains containing high angleboundaries. While this process is driven by density gradi-ents in the liquid, dendritic channels that permit markedlyfaster flow may locally increase the likelihood of fragmen-tation and migration of solid material. Therefore, channelswith continuous flow velocities on the order of ten timesthe withdrawal rate and higher were examined for physicallocation and relative spacing. By thresholding each simula-tion cell to identify regions with velocities no less than tentimes the withdrawal rate and noting their occurrencesthroughout the vertical-flow cells, a measure of the spacing

Fig. 8. DP and L assessed for permeability calculation for cases Y2 and X2.

Table 2Calculated permeabilities from flow simulations.

Case Inletvolumefractionliquid

Pressuredifferential(Pa)

Length overpressuredrop(�10�4 m)

Volumetricflow rate(�10�12 m3/s)

Permeability(�10�11 m2)

Vertical-flow

Y1 0.08 7.7 2.4 19 0.11Y2 0.10 5.4 1.5 19 0.09Y3 0.22 5.2 1.5 170 1.3Y4 0.32 3.8 0.74 160 1.4Y5 0.43 5.5 0.55 240 1.8Y6 0.60 3.8 0.51 390 3.7

Case Volumefractionliquid

Pressuredifferential(Pa)

Length overpressuredrop(�10�4 m)

Volumetricflow rate(�10�12 m3/s)

Permeability(�10�11 m2)

Cross flow

X1 0.42 1.6 20 0.07 4.5X2 0.56 0.63 20 0.18 21X3 0.73 0.21 20 0.26 183X4 0.89 0.079 20 0.59 659

2870 J. Madison et al. / Acta Materialia 58 (2010) 2864–2875

of high-velocity channels was obtained (Fig. 10). The spac-ing of high-velocity channels increases from �400–700 lmnear the inlets at fL= 8%, 10%, 22%, 32%, 43% and 60%, to

Fig. 9. Current study: (a) vertical-flow permeability, KY; (b) cross flow

nearly 1900 lm after flow permeates through the dendriticnetwork. The primary dendrite arm spacing (PDAS) forthe experimental material studied here was measured tobe in the range 450–600 lm and is consistent with the rangepredicted from the thermal gradient and withdrawal rate[35]. While the spacings of the flow cells at low liquid frac-tions are indicative of the PDAS, high-velocity spacingsexpand to three to four times the PDAS after flow throughthe entire mushy zone. The larger spacing at higher loca-tions within the mushy zone is a result of a decreasing num-ber of high-velocity-flow channels at increasing heightsowing to dendrite morphology.

Additionally, these upward, high-velocity-flow pathsthrough the dendritic array can be examined by tracingvelocity profile peaks associated with them. In this manner,identification of high-velocity channels and their relationto localized features of dendrite morphology is possible.Fig. 11c–f illustrates velocity profiles for select paths takenfrom within the global-flow domain of a vertical-flow cellas shown in Fig. 11a and depicted as functions of path lengthtraveled. Path length is defined as the distance traveled by aparticle through the simulation domain. Representative flowpaths corresponding to these profiles are highlighted inFig. 11b for reference. While not all interdendritic channelsexhibit these long paths of higher velocity, those that do tend

permeability Kx detailing difference in volume fraction dependence.

Fig. 10. High-velocity channel spacing for vertical-flow cases as functionsof height in the mushy zone.

Fig. 11. Velocity profiles for selected pa

J. Madison et al. / Acta Materialia 58 (2010) 2864–2875 2871

to exhibit velocity peaks on the order of 0.5–1 mm s�1, andhigh-velocity-flow regions develop and subside over veryshort length intervals. This indicates that large velocity fluc-tuations can occur over relatively small paths of travel. Thepredominant flow direction in all vertical-flow simulations isupward. As such, increases in path length correspond toincreased height and consequently, higher liquid volumefraction (fL) levels in the meshed domain. Differences in loca-tion, profile and magnitude of peak velocities with respect topath length in Fig. 11c–f indicate an influence of dendritemorphology and not merely volume fraction solid. Localvariations in dendrite arm spacing may impact these varia-tions and give rise to rapid flow in localized regions. Withinthe reconstructed data set, an average PDAS variability of±62 lm was determined by investigating subsets of the

th lines as functions of path length.

2872 J. Madison et al. / Acta Materialia 58 (2010) 2864–2875

reconstruction. The spatial heterogeneity of high velocitiesand overall flow behavior suggest that local dendrite mor-phology affects both flow speed and path. Future work willfocus on relating local spacing variation and morphologyto flow behavior and the resultant potential for flowinstabilities.

4.6. Flow paths and tortuosity

The lengths of paths traveled (Fig. 11a and b) also pro-vide an indication of the tortuosity associated with flowthrough each cell. Tortuosity is defined as the ratio betweenpath lengths associated with flow through the dendritic net-work and the specific length associated with the pressuredrop used for calculation of the permeability. Average tor-tuosity (Tavg) and the local ISA per unit volume (SV) foreach case are given in Table 3. Tavg is determined frompaths distributed uniformly over the inlet surface. In crossflow cases, the normalized flow-path lengths are only some-what greater than one. However, for vertical-flow cases,average tortuosity ranges from 1.5 to 2.6. Increased tortu-osity in flow would be expected with increased ISA and

Table 3Flow path ratios, total ISA per unit volume & permeability by case.

Case Inlet volumefraction liquid

Tavg

{±Var.}Local Sv

(�104 m�1)Permeability(�10�11 m2)

Vertical-flow

Y1 0.08 2.29 ± .83 218 0.11Y2 0.10 2.56 ± 1.4 225 0.09Y3 0.22 1.61 ± .33 128 1.3Y4 0.32 1.57 ± .63 86.1 1.4Y5 0.43 1.67 ± .63 42.8 1.8Y6 0.60 1.51 ± .32 24.6 3.7

Case Volume fractionliquid

Tavg

{±Var.}Local Sv

(�104 m�1)Permeability(�10�11 m2)

Cross flow

X1 0.42 1.26 ± .22 8.75 4.5X2 0.56 1.39 ± .01 8.51 21X3 0.73 1.21 ± .04 1.06 183X4 0.89 1.13 ± .003 3.50 659

Fig. 12. (a) Tortuosity and (b) permeability as fun

result in a decrease in permeability [41]. Fig. 12a and bshows the dependence of tortuosity and permeability onSV, with error bars in Fig. 12a illustrating the variance inTavg, which is also listed in Table 3. For permeability, ver-tical and cross flow scale inversely with SV by power lawwith an exponent of 1.6 or 1.7 (Fig. 12b). While tortuosityis not significantly affected by SV across a large range ofvolume fraction for cross flow, Tavg in vertical-flow wasstrongly affected by SV. Apparently, cross flow is stronglyaffected by the presence of primary arms, while vertical-flow is more strongly influenced by the arrangement ofthe secondary arms. This is understandable, as primaryarms are oriented parallel to vertical-flow and do not hin-der flow as much as secondary arms, which are orientedperpendicular to the vertical-flow direction. Alternatively,primary arms influence cross flow more, as these are theprimary microstructural feature impeding flow laterallyacross the dendritic network.

4.7. Comparisons with Blake–Kozeny and Kozeny–Carmen

relations

Comparison of the results of this study with commonempirical relations for permeability can give insight intothe structural origins of changes in permeability. Fig. 13compares the calculated permeabilities with both theKozeny–Carmen approximation (Eq. (2)) [10,12] and theHeinrich–Poirier-modified Blake–Kozeny relationshipsfor flow parallel and normal to the primary dendriticgrowth direction (Eqs. (6) and (7), respectively) [11,13].Error bars denote the total variation in local solid fractionover the assessed pressure drop (Eq. (8)), within each sim-ulation cell. KY error bars are one-sided, as the critical andmaximum solid fraction is the solid fraction located at theinlet.

Vertical-flow permeabilities (KY) agree well with the pre-dictions of the Heinrich–Poirier-modified Blake–Kozenydescription for all simulation results. Conversely, crossflow permeabilities (KX) are higher by a factor of 3–5 inthe range 0.4 < fS < 0.6, with better agreement in the lowsolid fraction regime, fS < 0.35. With regard to the

ctions of measured ISA per unit volume (SV).

Fig. 13. Calculated permeabilities plotted as a function of solid volumefraction with modified Blake–Kozeny and Kozeny–Carmen relations.

J. Madison et al. / Acta Materialia 58 (2010) 2864–2875 2873

Kozeny–Carmen approximation, permeabilities in thisstudy are in good agreement with KY at high solid fraction,fS > 0.7. For fS < 0.6, when SV is measured as opposed toapproximated by the inverse of the secondary dendritearm spacing as previously suggested [42], the Kozeny–Car-men approximation (Eq. (2)), for KX agrees reasonably wellwith the simulation results. Interestingly, the maximumdivergence between measured and approximated Kozeny–Carmen predictions arise at fS < 0.3, which occurs nearthe dendrite tips, where SV decays rapidly. It is worth not-ing that, while the Kozeny–Carmen relation is not formu-lated to describe anisotropic flow, it is very effective atcapturing the influence of ISA on permeability, providedSV can be well quantified.

For volume fractions in the range 0.4 < fL < 0.6, the cur-rent study shows a greater resistance to vertical-flow com-pared with cross flow. This trend is not consistent withearlier Blake–Kozeny models (Eqs. (4) and (5)), as higherKY across all volume fractions are predicted. However,Heinrich and Poirier’s Blake–Kozeny formulation (Eqs.(6) and (7)) does predict higher cross flow permeabilitieswithin the range 0.3 < fS < 0.55. The simulations of thecurrent study confirm this assertion, albeit with greater dif-ference in the magnitudes of KY and KX.

5. Discussion

The availability of a 3D data set containing 18 primarydendrite trunks permits direct assessment of the influenceof dendritic structure on fluid flow and permeability. Verti-cal- and cross-flow permeabilities, KY and KX, respectively,are sensitive to the ISA per unit volume as:

Kx;y ¼/Sn

V

where n is slightly less than two, and / is a material con-stant related to the flow path and the solid fraction. Thedependence of permeability on SV is somewhat less thansuggested by the Kozeny–Carmen relation and suggestsan important role for flow path tortuosity.

Tortuosity was investigated by evaluation of flow-pathlengths associated with vertical and cross flow. In this data

set, vertical-flow occurs with a more tortuous path com-pared with cross flow. Conversely, cross flow is more sensi-tive to solid fraction. As a result of flow tortuosity,channels for high-velocity vertical-flow from the bottomto the top of the mushy zone initially occur at the spacingof the primary dendrite arms, but assume a spacing three tofour times the PDAS at the dendrite tips.

In this study, overall higher permeabilities and greatervariation in KX compared with KY were observed in compar-ison with prior investigations [13,16,20]. Higher permeabili-ties are observed for cross flow in the range 0.4 < fS < 0.6,which is a behavior predicted in the Heinrich–PoirierBlake–Kozeny approximations for the range0.3 < fS < 0.55 [14]. This study suggests an even greateranisotropy between KY and KX in the vicinity of 0.5fS. Thesefindings are indicative of lower restriction across the den-dritic channels at locations with approximately equalsolid–liquid ratio. Simulations show more uniform pathsof lateral flow across the array compared with the upwarddirection through the mushy zone over the same range.

In the total data set investigated (2180 � 1635 �980 lm3) paths for vertical-flow with inlets stationed atlocations < 0.4fS do not yield well-defined channels forvertical-flow through the dendritic network owing to theabrupt decrease in solid fraction over the upper 500 lmof the mushy zone. Such cells do not contain representativedendritic structures over an appreciable upward length, butrather dendrite tips in pools of extradendritic liquid. There-fore vertical-flow cells with inlet volume fractions < 0.4fS

were not included in this study. Additionally, since flowpermeability becomes very low for fS > 0.6, it is likely thatthe decanting process was influenced, resulting in a shortermushy zone than existed during solidification at the pointwhere decanting was initiated. As a result, contiguousregions of cross flow across the data set were unavailablefor heights corresponding to fS > 0.6. Nevertheless, thedependence of PDAS and SDAS on cooling rate [34,35]suggested that the decanting process did not significantlyinfluence the structure in the upper half of the mushy zone.Thus the dependence of permeability on volume fractionand dendritic morphology would extend to a fully decantedstructure.

Permeability values presented by various investigatorsusing both numerical and experimental approaches are ingeneral agreement with the current study. Furthermore,in the case of cross flow, the current study has allowed fur-ther investigation in the low solid fraction regime, whichhas been largely untreated owing to difficulties associatedwith maintaining dendritic solid in the range of very highliquid fraction levels where fL approaches or is greater than60–70% [13,18]. To compare with previous analyses,Fig. 14a and b shows KY and KX, respectively, plotted asfunctions of solid fraction.

Additionally, cross flow simulations reported here pro-vide a useful, new contribution to the understanding of per-meability at low solid volume fraction. Cross flowpermeabilities determined by the physical microstructure

Fig. 14. Comparison of current study with literature values for (a) parallel-flow permeability (KY) and (b) cross flow permeability (KX). Both are shown asfunctions of solid volume fraction.

2874 J. Madison et al. / Acta Materialia 58 (2010) 2864–2875

down to 0.10fS converge with higher solid fraction trendsreported by others, and show limited correlation with SV

while being more definitively related to the local solid frac-tion. However, correlations with the Heinrich–PoirierBlake–Kozeny and Kozeny–Carmen approximations athigh liquid fraction indicate that, with accurate knowledgeof SV, these models can provide a reasonable assessment ofpermeability. In the vicinity of 0.50fS, understanding oflocal SV appears particularly important for cross flow, asthe Heinrich–Poirier Blake–Kozeny model underestimatespermeability in this range, while the Kozeny–Carmenagrees nominally. In fact, the implementation of an ISAcomponent in Blake–Kozeny models may increase theirpredictive robustness, particularly in the region of 0.50fS.While results of the current study show good agreementwith empirical models at low fS, the dependence of crossflow at low solid fraction may extend to lower permeabili-ties than previous empirical suggestions. The implicationsof the present 3D approach to assessing permeability forconvective instabilities and defect formation will be pre-sented in a future publication.

6. Conclusions

� A new approach for assessing fluid flow at the solid–liquid interface was developed to study flow permeabil-ity through an experimentally obtained 3D dendriticstructure in a directionally solidified nickel-basesuperalloy.� ISA per unit volume (SV) reaches a maximum at

fS = 0.65. SV has also been shown to vary inversely withdirectly assessed permeability with a power law relation-ship where n is slightly less than two.� Over the regimes investigated, cross flow permeability

(KX) exhibits a much greater sensitivity to volume frac-tion compared with vertical-flow permeability (KY)owing to lower surface to volume ratios and lower tor-tuosity in cross flow.� A transition in permeability anisotropy occurs in the

range of near-equal solid–liquid ratio, with cross flowpermeabilities (KX) becoming higher in this vicinity.

� In vertical-flow cells, high-velocity channels occur with aspacing on the order of the PDAS, but rapidly increaseto three to four times the PDAS at higher liquid fractionlevels, owing to the tortuosity of flow paths.� In vertical-flow, increases in tortuosity coincide with

increased dendritic surface area and correspondingdecreased permeability.

Acknowledgements

The authors acknowledge support from the AFOSRMEANS-II Program, Grant No. FA9550-05-1-0104 andthe NSF CAREER Award. The authors are also gratefulto P. Voorhees of Northwestern University for many usefuldiscussions. The primary author also recognizes the ONRHBEC-FF Fellowship for student support.

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