Laminar flow characterization in ceramic porous foams

Andre Tomas Silva Conceicao Caladoandre.calado@ist.utl.pt

Instituto Superior Tecnico, Universidade de Lisboa, Lisboa, Portugal

November 2016

Abstract

Open-cell foams such as ceramic foams have desirable characteristics such as high porosity, specificarea and inter-connectivity between cells in different directions. This facilitates applications for heat ormass transfer with a low pressure drop. These types of media have been studied using representative cells(such as the Kelvin cell) although some intrinsic features are lost as the real structure is non-periodic,therefore real geometries should be regarded in order to understand the underlying physics with sufficientdetail. This is accomplished by the use of CT scan reconstructions which are introduced in a commercialCFD software. A grid independence and validation study are performed for a representative volume ofporous sample. The internal flow is characterized in terms of probability density functions for differentvariables, under Darcy and Forchheimer regimes. Statistical evaluation of flow tortuosity and residencetime are also included in this work. The anisotropy of the foam is also quantified for different flow axisand resulting pressure drop. Finally, the analysis is extended for a conical diffuser geometry comparingpore-scale simulation to an equivalent porous model.Keywords: ceramic foam, porous media, CFD, single-phase, laminar flow

1. Introduction

The current study is focused on ceramic porousfoams which have numerous applications. Some ex-amples include: liquid metal filtering, first intro-duced in the 1970’s and now a common practice infoundries, furnace thermal insulators, or in trans-ports as gas filters although not as common as dieselparticle traps or catalyser substrates due to theirfragility; even though the performance is better to-wards NOx reduction and reduced pressure dropcompared to typical honeycomb structures [1]. An-other current and interesting application are porousburners in which premixed air/fuel enters and burnswithin the medium thus transferring energy fromthe combustion products to the fresh mixture allow-ing for higher flame speeds and better flammability(leaner mixtures are possible) [2]. The flame size iscompacted and reduced emissions are achieved.

Beyond the interesting applications of ceramicfoams, the focus on flow characterization withinsuch foams has not yet been subject of compre-hensive study. The use of Computerized Tomog-raphy (CT) scans or Magnetic Resonance Imaging(MRI) to reconstruct real porous foams are quiterecent in the academic world. These provide highfidelity representations of the geometry as input forstructural and fluid dynamics calculations. Withthe interest in engineering applications, the shifttowards the phenomena underlying the pore level

flow is mandatory for a comprehensive understand-ing. The application of numerical methods to modeltransport in such foams is a proven technique tounderstand the physics of fluid flow which excelsover certain limitations of experimental techniques.The bridge between experimental results and CFDanalysis is key to explore the physics of the pro-cess. With the resolution under 100 µm/voxel thereconstructed surfaces result in a high quality repre-sentation of the real foams which may then be usedin fluid flow simulations. This serves as a big moti-vation for in-depth research in the topic. In Ref. [3]the real foam geometry was reconstructed throughmagnetic resonance imaging (MRI)and a random-ized Kelvin structure was used as a model with ad-justable parameters. Matching porosity and specificsurface alone were insufficient to describe the pres-sure drop in the real structure, however by closing40% of the artificial structure’s pore windows a bet-ter match was achievable. The statistic distributionof tortuosity and residence time was also analyzedthrough particle tracking. Ref. [4] used CT scansto simulate pore-level flow examining pressure dropand tortuosity proposing new Ergun type correla-tion based on the studied foams by adapting theviscous and inertial coefficients. It is noticed thatthe inertial contribution is related to flow tortuos-ity.

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2. BackgroundModeling of transport in porous media can be stud-ied by means of the volume averaging theory. Con-sidering a saturated porous medium with a singlefluid, volumetric averages can be defined as phaseaverage (over the whole porous region), intrinsicphase average (fluid region only) and are given inequations 1a and 1b respectively. These averagesshould be taken over a Representative ElemetaryVolume (REV) in order to obtain meaningful val-ues.

〈ψ〉 =1

V

∫Vf

ψd∨ (1a)

〈ψ〉f =1

Vf

∫Vf

ψd∨ (1b)

〈ψ〉 = ε〈ψ〉f (1c)

with ε defined as the volumetric porosity.Having defined the phase average over a REV of

a quantity ψ the relation between the local valueinside the volume and the average value is given bythe following equation:

ψ = 〈ψ〉f + ψ (2)

where ψ is the deviation from intrinsic phase aver-

age of such REV and by definition 〈ψ〉f

= 0.For further analysis of the balance equations we

introduce the theorem for the volume average of adivergence:

〈∇ · ψ〉 = ∇ · 〈ψ〉+1

V

∫Asf

ψ · ndA (3)

where n is the unit vector perpendicular to the areaAsf of the solid-fluid interface.

The Volume-Averaged Navier-Stokes equationsfor incompressible flow are summarized below ([5]):

∇ · 〈v〉 = 0 (4)

ρ

[∂

∂t(ε〈v〉f +∇ · [ε〈v〉f 〈v〉f ] +∇ · [ε〈vv〉f ]

]=

−∇(ε〈p〉f ) + µ∇2(ε〈v〉f ) + ερg + R

(5)

R =µ

V

∫Asf

n · (∇v)dS − 1

V

∫Asf

npdS (6)

R is the drag term due to the interaction betweenthe fluid and the solid boundaries. For creeping flowin a porous medium the pressure gradient is propor-tional to the velocity, however when inertial effects

become relevant this relation is proportional to thesquared velocity. The non-linear relation betweenpressure gradient and superficial velocity associatedto the works of Forchheimer and proposed by [6]

∇p = − µK

v− CFK1/2

ρ|v|v (7)

where K is the medium’s permeability, with unitsof area and CF is the form-drag coefficient whichlike the permeability depends on the nature of theporous medium but also on the Reynolds number.These two parameters are usually obtained exper-imentally, however pore-scale simulations can alsobe used to compute such values. These parame-ters and other relevant flow quantities are calcu-lated and presented in the following sections.

3. Implementation3.1. Foam reconstruction

The first step towards simulating flow within thereal geometry involves segmentation of the CT scan.The process consists in taking a stack of 2D images,segmenting the pixels in either void region (air) orsolid. This segmentation is based on the differentgrey scale intensities which are a result of the differ-ent absorption of air and ceramic of the incomingX-rays. By stitching the region of interest into a 3Dsurface mesh (STL format), it can then be importedinto the CFD program. The chosen software for thissegmentation was freeware ITK-SNAP. It is impor-tant to mention that the segmentation process isalso prone to some degree of freedom and dependingon threshold parameters the resulting geometry willbe slightly different. The ’clustering’ mode withinthe software was chosen, and with few iterationsbased on the different grey scale intensity values,two Gaussian curves are fitted for the void and solidpixels. By taking different reconstructions based onthese iterations, three different foams were achievedwith 76, 78 and 83% porosity. The parameters forthese test segmentations are explicit in table 1. Theused segmentation was the one with highest poros-ity, since it was closest to the nominal porosity.

Porosity Mean Std. Dev.0.76 99 630.78 108 590.83 129 49

Table 1: Segmentation parameters (pixel intensityis relative to solid fraction)

A simple image processing algorithm was used tofill the inner porosities of the foam, easing the seg-mentation process and avoiding troublesome sur-faces. In fig. 1 we can see in the raw and processedimages.

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Figure 1: Image processing depiction: on the leftthe raw image and on the right the processed image

A summarized schematic of the steps from initialCT scan to CFD process is given in Fig. 2.

Figure 2: Schematic of the segmentation to CFDprocess

Two different ceramic foams are studied in thepresent work: pure alumina (Al2O3) and silicon in-filtrated silicon carbide (SiSiC) both with 10PPI.The first one for the basis of the present work andthe second for a priori tests. The alumina foamhas a volumetric porosity ε = 0.86, a specific sur-face area σ = 643m−1 and average pore diameterof 4.8mm.

3.2. Numerical modelSTAR-CCM+ R© was the used software to run thenumerical analysis as it comprises of several pack-ages such as 3D CAD and physics modelling whichserve their purpose for engineering flow, heat trans-fer and stress studies in a commercial tool. Af-ter the initial import of the CT scanned surfacemesh, the finite volume mesh is built on the flowdomain and the user may choose from different vol-ume mesh models. The finite volume discretizationwas based on polyhedral cells. The segregated flowmodel with SIMPLE (semi-implicit method pres-sure linked equations) as a widely used iterativesolver is used to solve the Navier-Stokes equationsin the flow field. Regarding the physical model, thefollowing assumptions were taken into account: in-compressible flow, laminar and isothermal. Fluid

properties for all simulations were: ρ = 1.18kg/m3

and µ = 1.86 × 10−5Pa.s. For the a priori testsdescribed in the following subsection the foam wasinserted in a pipe flow with 25mm diameter, withan extruded mesh for the inlet and outlet bound-aries of 1 and 3 pipe diameters respectively. Theboundary conditions are of constant velocity inletand a pressure outlet at the pipe ends. The cylinderwall and foam surface are no-slip walls.

3.3. Verification and validationAs part of the verification process a grid indepen-dence study was carried out for the SiSiC foam with90% porosity. Six different meshes were used toevaluate grid independence. Starting from an av-erage (base) size of 0.5mm for each cell in the firstmesh, the grid was refined from there with the re-spective base sizes displayed in table 2. The meshinside the porous region is set to be more or lessuniform, and in the rest of the domain (entranceand exit) the mesh is gradually stretched to allowfor less cells (since these regions are not particularlyrelevant to study and can have larger cells). For thefinal mesh (no. 6) the base size of the polyhedralcells is the same as the previous mesh however aprism layer is added to check the influence of thewall layer refinement.

Mesh No. Base size [mm] No. cells in

porous region

1 0.5 4.15E+05

2 0.3 1.87E+06

3 0.2 6.26E+06

4 0.15 1.48E+07

5 0.11 2.68E+07

6 0.11 3.98E+07

Table 2: Mesh sizes and cell count

The convergence of the numerical scheme usedwas judged not only on the macroscopic pressuredrop, but also on the probability density functionof axial velocity within the porous region, as seenin Fig. 3. Looking at the plots, beyond Mesh 4there appears to be no significant gain in refiningthe mesh size for accurate values, hence the choiceof the respective base size (0.15mm) for the follow-ing simulations but with the addition of a prismlayer.

The validity of the model can be proven bycomparing the present work results against liter-ature values for similar foams in terms of pres-sure drop. From the numerical results for differ-ent superficial velocities, the pressure gradient isplotted and by curve-fitting the viscous and iner-tial coefficients are extracted. In table 3 the re-sults from the present work and literature refer-

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(a) Red = 13 (b) Red = 266

Figure 3: PDF for axial velocity for different meshsizes

ences for similar foams are displayed. The valuesof permeability and inertial coefficient are withinthe same order of magnitude, and some variabil-ity is expected since foams with the same poros-ity and material depend on the manufacturing pro-cess to which they are submitted. The results of[4] are numerical, and those of [7] were obtainedexperimentally for a foam with 3.7mm cell diame-ter. Both authors use higher superficial velocities intheir computations/measurements where turbulentflow is expected.

Since the agreement is satisfactory, the model as-sumptions hold and the discretisation errors are lowenough for the intended analysis.

Author ε(−) KF × 107(m2) CF (−)Present 0.90 3.65 0.08[7] 0.91 3.42 0.2[4] 0.85 2.37 0.103

Table 3: Pressure drop coefficients for SiSiC foam

4. Results4.1. REV

As introduced in section 2, for a proper study usinga ”macroscopic continuum” approach, the choiceof the averaging volume is important. From a cu-bic shape, 3 centroid points were chosen within themedium and the porosity was calculated for differ-ent cube sizes. The cube length l is varied between0.15-17mm. The porosity for these REVs are plot-ted against a non-dimensional length: l/d in figure4. The dashed line in the figure is the computedporosity by taking almost the entire foam for thecomputation (a cylindrical REV with a length of40mm). The fluctuations of porosity tend to de-crease with the increase in the size of the REV,as expected. For cube sizes over 3 pore diame-ters (2986mm3) the fluctuations in porosity are verylow, resulting in an average porosity of 83%.

Having ensured a sufficient computational do-main for the intended analysis, the relevant physical

Figure 4: Porosity vs. non-dimensional cube length

quantities can be computed from the resolved flowfield.

4.2. Pressure dropThe transition from Darcy to Forchheimer regimecan be understood by looking at the pressure dropand superficial velocity relation. By modifying theForchheimer equation as in Fig. 5, the transition ismarked from the departure of the constant valueµ/K. This transition happens at Red ∼ 10, orbased on the square root of permeability ReK ∼ 1.

Figure 5: Pressure gradient normalized by velocityvs. Re

The reconstructed foam is known to have somedegree of anisotropy, as given in the reports of [8]where through image analysis the mean pore diam-eters for the three orthogonal directions are given,and one of the directions has a slightly higher di-ameter (in the referenced work it is X, which cor-responds to the Y axis in the present work). Thisgeometrical feature is expected to evoke a distinct

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pressure drop depending on the main flow direction.With this in mind, flow simulations in all three di-rections were carried out using a cube sample ofrepresentative volume size (25mm length).

These simulations lead to the conclusion that infact the direction where mean pores are greater inlength has substantially less pressure drop whencompared to the other 2 directions orthogonal axesas seen in Fig. 6.

Figure 6: Pressure drop for the 3 normal directions

The PDF for Z and Y directions in fig. 7 re-flects the anisotropy as the local velocity fields arespecific to each simulation. In the Y axis the max-imum velocity is around 4 times the superficial ve-locity, reducing the tail size of the distribution. Thevelocities are more uniform, and can be associatedto a diminished effect of the expansion/contractionthrough pores linked to the pressure drop. Since thepermeability itself is higher, there is less resistancefor the flow passage which is due to more flow areor the pore configuration itself.

The viscous and form-drag terms can also becompared between each other at varying Reynoldsin order to judge the relative weight of each con-tribution. In table 4 the second column lists thecalculated total drag in Newton for the axial direc-tion. The next column shows the ratio of viscousand form drag computed from the surface integrals,and the last gives the percentage of drag due to thefriction on the tube walls. Increase in the Reynoldsis translated into an increase of the form/viscousdrag ratio. This is caused by increase of separa-tion/recirculation zones, inducing low pressures insuch areas. The formation of such wakes is whatmarks the transition into the Forchheimer regime.

A different scope to look into the pressure dropis by quantifying the energy loss in the fluid.With the isothermal assumption, the fluid losesenergy through viscous dissipation. The viscous-

Sim-Z: Red=0.37Sim-Y: Red=0.37Sim-Z: Red=37Sim-Y: Red=37Sim-Z: Red=185Sim-Y: Red=185

Prob

abili

ty d

ensi

ty

0

0.25

0.5

1

1.25

Uaxial/Uin

−2 0 2 4 6 8

Figure 7: PDF for axial velocity for Z and Y direc-tions

Red Total Form % Tubedrag [N] Viscous wall drag

0.037 4.13E-07 3.9 3.53.7 4.23E-05 4.0 3.57.4 8.88E-05 4.2 3.418.5 2.65E-04 5.1 3.137 6.90E-04 6.6 2.7185 1.01E-02 15.4 1.7

Table 4: Pressure loss integral calculations and rel-ative contributions

dissipation function Φ, can be divided into the sumof two terms - elongation and shear, and are givenby Eq. 8a and Eq. 8b, respectively.

Φe = 2µ

(∂ui∂xi

)2

(8a)

Φs = µ

(∂ui∂xj

+∂uj∂xi

)2

(8b)

The integral of the viscous dissipation over thewhole porous fluid region was computed and thevalues are presented in Table 5.

Red Φe[W ] Φs[W ] Φe/Φs[%]0.037 1.51E-11 2.33E-11 65.13.7 1.54E-07 2.37E-07 64.77.4 6.36E-07 9.93E-07 64.118.5 4.55E-06 7.29E-06 62.437 2.24E-05 3.68E-05 60.9185 1.28E-03 2.16E-03 59.3

Table 5: Elongation and shear dissipation values fordifferent Reynolds

The elongation / shear ratio does not changemuch with Reynolds number, and the computed

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values are close to the 60% reported by [9], com-puted at Re ≈ 0.1. This may suggest that the ratiois a function of the type of porous medium, andcan measure how far it is from the capillary model,where elongation strain is zero.

The hydraulic tortuosity, which influences the in-ertial term in the pressure drop, can have differ-ent definitions. Perhaps the most accepted is theratio between an equivalent streamline length be-tween the boundaries of the medium and the systemlength:

τ =LeL

(9)

However the treatment of this equivalent length isnot straight-forward. Ref. [3] calculated the tortu-osity by tracing particle trajectories within the flowfield and integrating the length traveled for eachparticle, defining the equivalent length as the arith-metic mean of 30000 particle trajectories. Althoughthe particles had mass, their size and density waschosen as so to minimize the slip velocity and onlydrag force was considered. Ref. [10] arrived at asimple relation for arbitrary porous medium of theform based on the ratio of volume-averaged veloc-ity magnitude and volume-averaged axial velocity.For the present work, both approaches describedabove were implemented. For the particle trackingapproach, close to 30000 massless particles were in-jected at the inlet section, and the tracks were post-processed to compute the distance traveled insidethe matrix. Only paths which enter and exit the do-main considered, filtering out ”dead particles” withexcessive residence times. Since the particles arevirtually massless, the track results in fluid stream-lines (zero slip velocity).

Red 0.037 18.5 37 185τ 1.216 1.222 1.251 1.415στ 0.044 0.053 0.078 0.155τM 1.233 1.239 1.272 1.461t∗ 0.765 0.757 0.770 0.766σt∗ 0.305 0.337 0.369 0.2831/cos(θ) 1.257 1.332 1.378 1.683

Table 6: Computed tortuosity values for bothmethodologies

[3] pointed that since mollecular diffusion is notconsidered in the model the wake regions becomeimpenetrable and this shifts the mean value of nor-malized residence time t∗ to lower values as com-pared to ε. A separate although translatable statis-tic to measure flow tortuosity (and recirculationzones) is the probability density function for theflow angle defined between the local velocity vectorand the axial velocity. The inverse cosine of the av-erage flow angle, θ is listed in Table 6. The values

slightly overestimate the computed tortuosity, butfollow the same trend as expected.

By intuition, the longer the path the particle hasto travel, the longer the residence time (assuminga constant velocity). To understand if there is anycorrelation between residence time and tortuosity,these values were plotted in a scatter fashion to un-derstand if there is any correlation between the two- Fig. 8. The correlation coefficient is also com-puted for each Reynolds number. For Red = 0.037the correlation is weak, showing a wide spread inresidence times. With the increase in Reynoldsthe correlation becomes stronger. This evidencepoints towards a more random process in the Darcyflow, where some streamlines capture low velocityregions. It can be deduced that the particle velocityalong the streamlines is not uniform and betweenstreamlines there is a significant variance in veloc-ity.

(a) Red = 0.037 (b) Red = 18.5

(c) Red = 37 (d) Red = 185

Figure 8: Scatter plots for tortuosity and normal-ized residence time

4.3. Conical porous diffuser

A separate scenario investigated was of a conicalporous diffuser geometry. Such diffusers can be usedfor thermal partial oxidation processes as an energyconversion system for fuel cells.

The dimensions of the diffuser were chosen as thesame as found in [11], with a reduced cylindricalsection downstream of the diffuser: 18mm for theupstream diameter, 100mm length of the conicalsection and 100mm diameter of the downstream sec-tion (divergence angle of 22o). The interest is in the

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conical section and how the velocity field changes inthe axial and radial direction, therefore the cylin-drical section of porous media after the diffuser iscut down to 20mm in length. This increase in vol-ume size limits the number of cells to be used, hencea larger base size is chosen (0.3mm).

Since the area is no longer constant in the ax-ial direction, the pressure gradient is a function oflocal superficial velocity, as the cross-sectional areaincreases, the average velocity decreases. One of theobjectives of this test case is to understand if theequivalent porous medium is accurate enough to de-scribe the pressure drop compared to the pore levelsimulation of the scanned geometry. This is accom-plished using the porous region feature in STAR-CCM+ R©.

The software allows for the definition of a porousregion when given the viscous and inertial resistancecoefficients (α[kg/m3.s], β[kg/m4]). With these co-efficients known from simulation, the medium is as-sumed isotropic using the permeability values of theconical axis, where the bulk flow is aligned is alsoimplemented.

The inlet conditions are of ReD = 570 and whichis equal to a maximum pore Reynolds number ofRed = 185. The flow field in these conditions withthe porous medium is quite different than for a’clean’ diffuser, as one would infer.

The validity of the Forchheimer type equation forthe pressure gradient is investigated, now consider-ing the change of superficial velocity with the axialdistance; and since Vs ∝ 1/x2, integration of pres-sure in x leads to a pressure variation of the form:

p(x) =a

x+

b

x3+ c (10)

where the the constants a, b are function of the flowrate and the pressure drop coefficients α and β,respectively. Figure 10 shows the average surfacepressure as a function of the axial distance. Boththe pore-scale and equivalent model show the sametrend in pressure decay, even though there is somenumerical difference in the values, possibly due tosome uncertainty in the pressure loss coefficients orfoam anisotropy effects. Equation 10 is fitted to theobtained results from the detailed with good fit. Toaccurately represent flow in a conical diffuser, whereboth axial and radial velocity components are of in-terest, the concept of a REV should be taken withcertain care. Perhaps the most meaningful REVto consider would be an annular disk for which theouter and inner radii may vary to compute locallyaveraged values. However, there is a trade-off be-tween what should be considered a representativevolume size and discretization limits: a large REVshould rule out local fluctuations, but on the otherhand if the sample is too big the values cease to

be relevant for a local analysis, and must be withinphysical bounds.

With these concepts in mind, the annular REVis used to find local volume averages of axial andradial velocity throughout the diffuser. Symmetryis assumed, hence tangential velocity is assumed tobe negligible, and the radial component of velocityis zero in the centerline.

To set out the process of local volume averag-ing, two approaches were followed: the first oneusing disks with constant area (dr decreases withr), and the second using disks with a constant drinterval, which leaves the area of the disks unbal-anced but differentiates the points being averagedwith a constant spacing. A sensitivity study fornumber of disks and disk thickness (dx) was per-formed and can be seen in Fig. 9 for a cross-sectionat x = 20mm. It is evident that for the axial ve-locity component, following the constant area ap-proach, the thickness of the slice has little influenceon the averaged values, hence 7 disks were usedwith a thickness h = 1mm as default. Followingthe second approach of constant spacing, the axialvelocity showed little influence of h but the radialcomponent is more sensitive (as seen in Fig. 9 (b)).Thus the default value for h using this approachwas 5mm. For the final section of x = 100mm anextra disk was used to increase resolution near thewall for the isospacing approach.

(a) Iso area - axial profile (b) Iso spacing - radial profile

Figure 9: Sensitivity analysis of REV size(X=20mm)

Figures 11 and 12 shows the locally averaged val-ues for axial and radial velocity profiles. The isoareaaverages are plotted in blue, the isospacing onesin red and in black is the average velocity profileobtained from the equivalent porous medium cor-rected by the porosity factor. Overall it can besaid that the averaged values compare well to theequivalent model, but as area increases the averagestend to be more significant. The iso area approachappears to give more consistent values around themean curve since the volume is more or less thesame for each point. As stated before, using aconstant spacing will result in smaller volumes for

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smaller radii, therefore explaining some of the dis-crepancy in those scenarios.

Figure 10: Pressure variation for porous diffuser

Figure 11: Axial velocity profile

Probability density functions for the axial andradial velocities inside the conical region are shownin Figs. 13 and 14. The mean values and standarddeviation is well captured by the simpler model, butnaturally the deviation velocities are non-existant,such as local negative axial and radial velocities.

The question then arises, where are the fluctuat-ing quantities relevant? By looking at the kinetic

Figure 12: Radial velocity profile

Figure 13: Axial velocity PDF

energy profile (related to the squared velocity), thedifference is clear between the two models, as seenin Figures 15 and 16. The average profiles havehigher kinetic energy, and this is due to the veloc-ity fluctuation terms which are not negligible. Thesquared of an extensive quantity can be decomposedas such: ψ2 = (〈ψ〉f + ψ)2 and the mean of thesquared fluctuation is not zero (represented by thevariance). Hence the contribution of local velocityfluctuations to the kinetic energy, which are not en-compassed by the equivalent model. The PDF of

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Figure 14: Radial velocity PDF

kinetic supports this evidence with a larger meanvalue and standard deviation for the kinetic energyin the diffuser.

Figure 15: Kinetic energy profile

In summary despite foam anisotropy and REV

Figure 16: Kinetic energy PDF for diffuser

limitations, the average profiles for velocity andpressure resemble closely the ones obtained fromthe isotropic equivalent model. This last model hasits limitations, as it does not include sub-filter mo-tions (under the REV scale), which may be impor-tant on scalar dispersion, kinetic energy profiles orheat transfer for example.

5. Conclusions

CT reconstruction of ceramic foams is proven asa useful non-destructive technique to create a 3Drepresentation of the matrix, and for the 10PPIfoams, the necessary voxel resolution should be un-der ∼ 100µm to assure sufficient detail. There issome degree of uncertainty when recreating the realgeometry, as different threshold values will influ-ence the specific surface area and porosity of theoutput foam. Image processing can be useful to re-move inner foam porosities which are undesirablewhen creating a surface mesh to import into theCFD software. The finite volume method withinthe commercial software STAR-CCM+ R© using un-structured polyhedral cells is capable of solvingpore-level flow details, with good comparison to lit-erature values of pressure drop coefficients, tortuos-ity and residence time. The concept of volume aver-aging is useful to ensure meaningful values are beingextracted from such numerical computations. Tothe author’s knowledge there are not many worksstudying open-cell foam anisotropy, which is partof this work. Flow transition from Darcy to Forch-heimer happens at ReK ∼ 1, or Red ∼ 10 whichis consistent with other studies made with simi-lar foams. Values for elongation and shear dissipa-tion are also computed, with the ratio being around

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60%, similar to the results of [9] and is weakly de-pendent on Reynolds number. Even though eval-uations of tortuosity and residence time for thesetypes of porous media have been reported by [3]and [4], the influence of Reynolds number on thesestatistical measurements are not mentioned. Byusing Lagrangian particle tracking, the tortuosityis calculated and found to increase slightly withthe Reynolds number, and the normalized residencetime is unchanged due to the fluid accelerationin the porous medium assured by conservation ofmass. Additionally the correlation between tortuos-ity (distance travelled) and residence time is made,showing very weak dependence which however in-creases with the Reynolds number. The study of aporous conical diffuser and comparisons with equiv-alent porous model concludes the Results chapter.It is proven that the Forchheimer equation is validto describe the pressure-drop in the conical sectionwhen considering superficial velocity variation (1Dapproximation). The concept of REV is used to ob-tain locally averaged values which can then be com-pared to radial and axial profiles, as well as pres-sure. On the other hand, the kinetic energy scalarprofile is not captured by the equivalent model sincethe deviating component of velocity is not present.By taking the velocity squared, the deviation termis non-negligible and yields a higher kinetic energyfor the pore-scale simulation. The probability den-sity plots are a useful statistical measure to char-acterize flow variables in such complex geometries,and it is observed that the axial/radial profiles ob-tained from pore-scale and equivalent model havecomparable values for the mean and standard devi-ation. Plotting the PDF for kinetic energy the roleof local velocity deviations supports the evidencefor the kinetic energy of the deviation flow. Futurework could involve extension to turbulent flow (byusing turbulent models), or how scalar dispersioncan be related to the equivalent porous model byuse of an effective dispersion viscosity.

AcknowledgementsThe author would like to thank PhD. MiguelMendes for supplying the CT scan of the aluminafoam used in the simulations.

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