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Pressure drop in flow across ceramic foams—A numerical and experimental
study
Article in Chemical Engineering Science · July 2015
DOI: 10.1016/j.ces.2015.06.043
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Chemical Engineering Science 137 (2015) 320–337
Contents lists available at ScienceDirect
Chemical Engineering Science
journal homepage: www.elsevier.com/locate/ces
Pressure drop in flow across ceramic foams—A numerical
and experimental study
W. Regulski a,n, J. Szumbarski a, Ł. Łaniewski-Wołłk a, K. Gumowski a, J. Skibiński b,
M. Wichrowski c, T. Wejrzanowski b
a
Institute of Aeronautics and Applied Mechanics, Warsaw University of Technology, Poland
Faculty of Materials Engineering, Warsaw University of Technology, Poland
c
Institute of Fundamental Technological Research (IPPT), Polish Academy of Sciences, Poland
b
H I G H L I G H T S
We investigate pressure drop of foam filters experimentally and numerically.
10 ppi (pore-per-inch), 20 ppi and 30 ppi foams of porosity 75–79% are used.
Experiments are done in water channel, simulations use D3Q19 MRT-LBM.
Excellent agreement between simulations and experiment is reported.
Comparison to external data and correlations yields varying outcomes.
art ic l e i nf o
a b s t r a c t
Article history:
Received 19 February 2015
Received in revised form
11 June 2015
Accepted 13 June 2015
Available online 30 June 2015
The unique properties of ceramic foams make them well suited to a range of applications in science and
engineering such as heat transfer, reaction catalysis, flow stabilization, and filtration. Consequently, a
detailed understanding of the transport properties (i.e. permeability, pressure drop) of these foams is
essential. This paper presents the results of both numerical and experimental investigations of the
morphology and pressure drop in 10 ppi (pores per inch), 20 ppi and 30 ppi ceramic foam specimens
with porosity in the range of 75–79%. The numerical simulations were carried out using a GPU
implementation of the three-dimensional, multiple-relaxation-time lattice Boltzmann method (MRTLBM) on geometries of up to 360 million nodes in size. The experiments were undertaken using a water
channel. Foam morphology (porosity and specific surface area) was studied on post-processed,
computed tomography (CT) images, and the sensitivity of these results to CT image thresholding was
also investigated. Comparison of the numerical and experimental data for pressure drop exhibited very
good agreement. Additionally, the results of this study were verified against other researchers' data and
correlations, with varying outcomes.
& 2015 Elsevier Ltd. All rights reserved.
Keywords:
Ceramic foam
Pressure drop
Lattice Boltzmann method
Darcy–Forchheimer equation
Specific surface area
Pore-scale simulation
1. Introduction
The industrial importance of materials with open porosity
structures in the form of ceramic or metallic foams has grown in
recent years. These materials exhibit specific properties such as
high specific surface area, high porosity, low density, favourable
mechanical, thermal and corrosion resistance. Thus they are well
suited to serve as compact heat exchangers, reaction catalyst
support, flow stabilizers or filters (Twigg and Richardson, 2007).
n
Corresponding author.
E-mail address: [email protected] (W. Regulski).
http://dx.doi.org/10.1016/j.ces.2015.06.043
0009-2509/& 2015 Elsevier Ltd. All rights reserved.
This results in an increased need for a priori knowledge of their
hydrodynamic properties. Thus, much effort has been devoted to
link the foam structural parameters to their pressure drop.
An extensive review of pressure drop correlations for foams
was provided by Edouard et al. (2008). All presented formulae are
based on the Darcy–Forchheimer relation
Δp μ
ρ
¼ U þ U2;
k2
ΔL k 1
ð1Þ
where Δp=ΔL is the average pressure gradient, U is the mean flow
velocity, ρ is the fluid density, μ is the fluid dynamic viscosity, and
the coefficients k1 and k2 are called viscous and inertial
W. Regulski et al. / Chemical Engineering Science 137 (2015) 320–337
permeabilities. These permeabilities should be functions of the
foam's geometry. The key geometrical parameters of the foam are
its porosity, ε ¼ V void =V total , which is the ratio of the void space,
Vvoid, to total volume, Vtotal, occupied by the foam, foam strut
diameter, dstrut, mean pore diameter, dp, pore per linear inch
number, ppi, and specific surface area, ac ¼ S=V total , ratio of the
surface in contact with the flow, S, to the whole volume occupied
by the specimen. Obviously, some of these parameters are
correlated.
The primary purpose of this study is to present results of
morphology assessment and a series of simulations performed for
typical ceramic foam structures. Three foams of nominal pore
densities 10 ppi, 20 ppi and 30 ppi and porosities in the range
ε ¼ 75–79% are investigated. The obtained results (porosity, specific surface area and pressure drop) are compared to data
available in the literature. Pressure drops are compared to those
of similar specimens reported in other works. Additionally, the
appropriateness of pressure drop correlations proposed by other
authors is explored. Special attention is paid to the postprocessing of the computer tomography (CT) images and its
influence on the obtained parameters.
1.1. Experimental data and relations for pressure drop
The Darcy–Forchheimer equation is a general relation pressure
drop in porous media. Although obtained empirically by
Forchheimer (1901), it can be shown that it emerges from a proper
averaging of the Navier–Stokes equations (Whitaker, 1996) and it
can actually be applied to various kinds of porous structures like
packed beds of spheres (van der Sman, 2002), rings, or virtually
any other structure (Rautenbach, 2009). In particular, for the
porous bed composed of uniformly spaced spheres, the specific
Darcy–Forchheimer-type formula was proposed by Ergun (1952):
Δp
ð1 εÞ2
1ε
¼ 150
μU þ 1:75 3 ρU 2 ;
2
3
ΔL
ε ds
ε ds
Δp
ð1 εÞ2 a2c
ð1 εÞac 2
¼α
μU þ β
ρU ;
ΔL
ε3
ε3
and 45 ppi. The work explored pressure drop dependence on
morphological parameters. The authors compared their data to
previously existing correlations and proposed their own formula.
Additionally, the tortuosity of the medium was investigated.
Incera Garrido et al. (2008) investigated experimentally a series
of 10 ppi, 20 ppi, 30 ppi and 45 ppi ceramic foams with porosity in
the range ε ¼ 75–85%. The morphology of the foams, pressure
drop as well as heat transfer coefficients were assessed. Additionally, another correlation for pressure drop was developed. The
foam data presented in their work will serve as a benchmark to
validate our results and will be reviewed in detail in Sections 3.1.2
and 3.3.2. Inayat et al. (2011b) focused on 10 ppi, 20 ppi and 30 ppi
foams of porosity about ε ¼ 85% and drew their morphology
correlations for α and β from Eq. (3).
In their review, Edouard et al. (2008) remarked that it is always
possible to come up with a set of coefficients to fit one's experimental data and that many previously proposed correlations have
only limited application. In their other works (Lacroix et al., 2007;
Huu et al., 2009) they proposed a relation that would be universal,
namely the Ergun equation with the original numerical coefficients but a properly chosen equivalent particle diameter, dh,
instead of the sphere diameter, ds. The equivalent particle diameter for a particular foam was obtained from the assumption that
the equivalent packed bed of spheres and the foam have equal
specific surface areas and equal porosities. The total volume of the
representative unit cell containing the sphere can be found from
the sphere volume and the porosity of the packing:
V total ¼
ds ¼ 6
ð3Þ
where the characteristic dimension is the reciprocal of the specific
surface area, ac. This formulation was used by Richardson et al.
(2000), where a series of high-porosity ðε 4 90%Þ alumina foams
of ppi number in range from 10 to 65 were investigated. They
reviewed a few specific relations between ac and the mean pore
diameter, dp, and chose the one proposed by Kozeny (1927) for
packed beds of particles. The pore diameter was found from the
image analysis of slices of the foam specimens. The coefficients α
and β were fitted to the experimental data, and they became
functions of porosity and mean pore diameter. Additionally, the
influence of the surface roughness of the foam was investigated
and some correlations for this effect were drawn as well. The
relative surface roughness of a specially prepared 30 ppi foam was
investigated by means of the nitrogen sorption. The roughness
factor was defined as the ratio of this measured area versus ac of a
“smooth” foam geometry.
The work of Moreira and Coury (2004) concerned a series of
high-porosity foams of nominal linear pore density of 8 ppi, 20 ppi
Vs
:
1ε
ð4Þ
On the other hand, the specific surface area of the packing is
ac ¼ Ss =V total . Thus it is possible to link porosity, specific surface
area and the diameter:
ð2Þ
where ds is the sphere's diameter. Foam-like structures are viewed
as an inverse of bed-like structures where solid spheres are
substituted by pores. Therefore, many researchers use the Ergun
formula to fit their own experimental data. They usually replace
sphere diameter, ds, with the equivalent hydraulic diameter, dh,
that should somehow be linked to the foam geometry. Additionally, varying numerical constants preceding the geometrydependent coefficients are often used. This kind of general
correlation was proposed by Gibson and Ashby (1999):
321
1ε
:
ac
ð5Þ
Since the foam has the same porosity and specific surface area,
the relation in (5) holds for it as well. It also defines the foam's
equivalent hydraulic diameter, dh. The authors went one step
further and postulated the specific geometrical shape of a foam
cell. First, a cell composed of three cylindrical struts was utilized
(Lacroix et al., 2007), and this was later modified to a decahedron
(Huu et al., 2009). Based on assumptions about the shape of the
strut and detailed geometrical calculations, the equivalent hydraulic diameter, dh, and the foam strut diameter, dstrut, were linked in
those works. In this work, however, we will exploit the presented
formulation that involves only Eq. (5), without postulating any
specific shape of the foam. By combining Eq. (2) with Eq. (5) the
following expression for the pressure gradient can be written:
Δp
a2
ac
¼ 6:25 3c μU þ 0:29 3 ρU 2 :
ΔL
ε
ε
ð6Þ
Another approach that aims to be universal was proposed by
Dietrich (2012). He provided an extensive review of over 100
previously available pressure drop data sets and proposed his
pressure-flow velocity relation in a non-dimensional form
Hg ¼ 110Re þ1:45Re2 ;
ð7Þ
with non-dimensional pressure gradient (the Hagen number),
Hg ¼ ρðΔp=ΔLÞðdh =μ2 Þ, as a function of the Reynolds number,
3
Re ¼ ρUdh =εμ. Eq. (7) translates to the following dimensional
322
W. Regulski et al. / Chemical Engineering Science 137 (2015) 320–337
formula:
Δp
μU
ρU
¼ 110 2 þ 1:45 2
ΔL
ε dh
εdh
2
ð8Þ
In previous works, Dietrich et al. (2009) investigated a few
available formulations for dh. They were based on semi-empirical
correlations using three distinct approaches. First involved pore
strut diameter and pore window size, next used only the specific
surface area, while the third one assumed regular geometrical
models for foams. In our case, due to the lack of pore window size
and strut diameters, we will only use the second idea and connect
dh to specific surface area:
ε
dh ¼ 4 :
ac
ð9Þ
The relation in Eq. (9) comes from the so-called pipe-model
(Richardson et al., 2000; Große et al., 2009). This model postulates
that the foam has the same porosity and surface area as the set of
hollow circular channels of diameter, dh, traversing the medium.
Substitution of Eq. (9) into Eq. (8) yields the following expression:
Δp
a2
ac
¼ 6:88 3c μU þ 0:36 3 ρU 2 :
ΔL
ε
ε
ð10Þ
As one can observe, the choice of the appropriate hydraulic
diameter is somewhat arbitrary and different models were used. It
should be noted that the relations for dh given by Eqs. (5) and (9)
exhibit completely different behaviour. The former increases with
increasing ε while the latter decreases. This issue was highlighted
by Edouard et al. (2008) in their review, and it has a straightforward explanation. In the first model, the characteristic dimension
refers to the object that is solid (the sphere) so its diameter must
decrease with the increase of ε while in the second it refers to size
of the void space itself (the pipe diameter). Eventually both
models yield relations for pressure that differ only in terms of
numerical constants.
It is also important to emphasize that, quite often, the pressure
drop in porous structures is investigated by using idealized tesselations (based on the so-called representative unit cell, RUC) or
random packings in the form of the Voronoi cells (Wejrzanowski
et al., 2013), rather than real geometries. The RUC may have a shape
of a cube (Fourie and du Plessis, 2002), Kelvin tetrakaidecahedron
(Inayat et al., 2011a,c) or the Weaire–Phelan cell (Große et al., 2009).
In particular, the Weaire–Phelan structure seemed to be very
promising when representing the real foam-like structures. The
average number of pore-to-pore connections in this structure is 13.5
(in the Kelvin cell it is 12) while real foams exhibit 13.7 connections
according to Matzke (1946) who studied soap bubbles. Detailed
comparison of the Weaire–Phelan conglomerate to real structures
revealed, however, that some of their properties significantly
differ from the ones of real foams (Lautensack and Sych, 2008).
For this reason, structural models based on Kelvin and Weaire–
Phelan tesselations were also reported to be inaccurate (Große et al.,
2009; Incera Garrido et al., 2008).
1.2. Flow simulations in real foam structures
The literature on the simulation of real, foam-like, highporosity structures is not very rich. The analysis of real porous
structures requires reconstruction of the geometry obtained from
computer tomography scans, which can be intractable. This
reconstruction results in a voxelized image with each cube
belonging to a fluid or solid space. Thus the reconstructed fluid–
solid interface appears as a “staircase” geometry. Moreover, in the
case of the classical computational fluid dynamics (CFD) tools such
as the finite element method (FEM) or the finite volume method
(FVM), additional preprocessing of the surface is required in order
to generate the computational mesh.
Some simulations performed using FVM were reported by
Petrasch et al. (2008) for 10 ppi ceramic foam for solar receiver
applications. The pressure drop in linear and inertial flow regimes
for a 12.5-million-cell tetrahedral mesh was investigated. FVM from
commercial code ANSYS CFX was used. The Nusselt number
variation with flow velocity and foam morphological characteristics
was studied as well. Habisreuther et al. (2008) investigated real CTobtained and artificial porous structures by means of a commercial
FVM code. Skibiński et al. (2012) investigated pressure drop data of
a 15 mm 15 mm 15 mm sample cut from a 10 ppi foam geometry, using FVM. Additionally, he compared this data with
pressure drops of a set of randomly generated Laguere–Voronoi
tesselations of apparently similar characteristics to the original
foam, and he reported significant discrepancies. Another case
concerned research aimed at an electromagnetic improvement of
the alloy filtration (Kennedy et al., 2013), using the COMSOL module
based on FEM. Only 2D flow simulations were carried out, however.
Ranut et al. (2014) performed a full 3D simulation of combined heat
and fluid flow across 10 ppi, 20 ppi and 30 ppi metallic foams.
Again, the simulations were carried out using a commercial FEM
ANSYS software with extensive preprocessing of the geometry and
mesh generation in the ANSYS ICEM CFD package. Lastly, simulations by de Carvalho et al. (2014) concerned flow in a metallic foam
for aero-engine separators. The CT-reconstructed real geometry and
artificial Weaire–Phelan structures were studied. Simulations were
performed on a tetrahedral mesh with use of the ANSYS Fluent
package. The realizable k–ε turbulence model with enhanced wall
treatment (EWT) was utilized.
Alternatives to conventional CFD are available for the simulation of foams. The lattice Boltzmann method (LBM), which has
received increasing attention over the past two decades and is
used in this work, is particularly attractive (Succi, 2001). It is a
method based on the mesoscopic kinetic description of fluid flow.
The algorithm consists of two steps, namely streaming and
collision. These are performed on particle populations moving on
a Cartesian grid. The LBM can be formulated from a range of
collision models (BGK, MRT, ELBM) and particle velocity sets
(D2Q9, D3Q19, D3Q27), these issues are covered in detail in
Section 2.2.1. LBM is straightforward to implement and parallelize.
Because of the use of the Cartesian grid, there is no need for
geometry-fitted mesh generation. The geometry, after designation
of solid and liquid voxels, can be implemented directly in the
“staircase” form. LBM has been shown to reproduce detailed flow
characteristics of confined flows (Regulski and Szumbarski, 2012)
and it has received significant attention both in single- and multicomponent flow across porous media, see an extensive review by
Liu et al. (2015). However, few studies refer to simulations of flows
in real foam-like structures.
The first D3Q19 BGK-LBM simulation of flow across a SiC foam
was presented by Bernsdorf et al. (1999). The size of the simulation domain was Lx Ly Lz ¼ 100 149 149 and the flow
regime was linear with a coarse specimen resolution of
δx ¼ 0:5 mm. Still, the pore-scale flow was resolved – pore throats
had the size of at least several voxels. Graf von den Schulenburg
et al. (2007) investigated the flow in a compressed polyurethane
foam with different levels of compression using a D3Q19 BGK-LBM
model on a 128 128 47 node geometry. They verified it against
Magnetic Resonance Imaging (MRI) measurements of flow field in
the experiment and a good agreement was found. Work by
Magnico (2009) concerned flows across a CT-obtained Ni–Cr foam.
Two cases were analyzed with FVM and LBM. The first case was
the realistic geometry and second one was the same geometry
W. Regulski et al. / Chemical Engineering Science 137 (2015) 320–337
after a transformation that introduced anisotropy. The tensorial
form of the Darcy–Forchheimer relation was proposed there. In
the work of Gerbaux et al. (2010) D3Q19 MRT-LBM and FVM
simulations as well as experiments were used to investigate three
specimens of metallic foams. In this work, the domain size for LBM
simulations was of order 250 120 120 but the flow regime was
only linear. Very good agreement between both simulation methods were reported. Experiment and simulations agreed in two
cases, whereas they significantly diverged in the third one, with
this issue attributed to improper geometry reconstruction (large
difference of porosities between the real structure and the CTreconstructed image was reported). Beugre et al. (2010) performed
a complete simulation of the flow across a highly-porous
(ε ¼ 0:955) nickel foam using D3Q19 MRT-LBM formulation. The
size of the computational domain was 4003 nodes. They provided
experimental results as well and reported good agreement. Twodimensional flow simulations using slices of the geometry
obtained from 3D imaging were also reported. A ceramic foam
filter was investigated by Laé et al. (2013) for metal filtration
applications with D2Q9 model. Finally, the LBM simulation of fluid
flow with heat transfer was performed by Prasianakis et al. (2013)
in the porous gas diffusion layer of a fuel cell. The authors analysed
the permeability and the relative effective diffusivity using the
state-of-the-art D3Q27 thermal LBM model. The size of the
computational domain was 200 100 180 nodes. Yet another
recent example (Ettrich et al., 2014) presented an LBM simulation
of a ceramic foam used as catalyst support, with the specific
feature of this work being the use of a special diffusive boundary
condition for solid–liquid interface.
2. Materials and methods
2.1. Ceramic foams
In this work, three foam specimens were investigated experimentally and numerically. These were Al2 O3 VUKOPOR A ceramic
foam filters with 10 ppi, 20 ppi and 30 ppi nominal pore densities.
The frontal images of the filters are given in Fig. 1. Each specimen
was a cube of 50 mm 50 mm 50 mm size. The nominal porosities were not provided by the supplier. Inspection of the filters
confirmed that these structures were polyurethane-matrix based
(manufactured by the so-called ”replica-technique”) with hollow
spaces in the struts left after burning out the polymer template
(Richardson et al., 2000). Polyurethane template based filters are
very common. Filters of this kind were also investigated by other
groups (Incera Garrido et al., 2008; Große et al., 2009; Inayat et al.,
323
2011a). The hollow strut spaces will be referred to as the “internal
porosity” or “internal void space”, see Fig. 1 for details. The
presence of internal porosity significantly influences some of the
foam's characteristics (see Section 2.1.3).
The porosity of foams was calculated on the post-processed
image after removal of the internal void space in the struts.
Obviously, only this porosity has any practical meaning when
one wants to draw the correlations for pressure drop. The
measurements of the real structures’ morphological characteristics
were not performed. Therefore neither total porosity nor mean
strut diameter and window diameter are available.
2.1.1. Reconstruction of CT-images
Image acquisition of the foams was conducted using the high
resolution SkyScan X-Radia XCT-400 tomograph. The samples
were radiated with a directional microfocus GEj phoenixj x
ray xsj 240D tube at an acceleration voltage of 150 kV and a
current of 50 A. The radiography images were reconstructed using
the XCT-Reconstruction software. This process resulted in a set of
256-level greyscale bitmap-format tomograms. In order to recover
the actual geometry of the specimen, this data set had to be
filtered and binarized i.e. each pixel had to be prescribed with a
value 0 or 1 (black or white) denoting void or solid phase. Both
processes were performed using SkyScan cTAN software. First, a
median filter was used to remove the noise. This filtering procedure has an additional advantage of smoothing the solid–liquid
boundaries. Then, an appropriate threshold level for binarization
was chosen. Finally, the internal porosity was closed. The last
process, however, was only partially successful. There existed
regions that should be regarded as internal porosity, but they still
had contact with external void space. This resulted either from
imperfections in the reconstruction process or the existence of
some narrow connections with the void space in reality (see Fig. 2
for details). An additional algorithm that would completely close
those spaces had to be used, this issue is addressed in the next
section. The presence of the internal porosity as well as the effect
of using a median filter is clearly visible in Fig. 3.
It is important to note that there exists no standard procedure
for the appropriate choice of the threshold for the grey-scale CTimage. Große et al. (2008) postulated that the threshold should be
taken at the level when the rate of change of the specific surface
area with respect to the change of threshold value, dac =dt r , reaches
minimum. They, however, did not support this claim by any
evidence. Therefore, in our case, the thresholding was performed
at the level that seemed to be the most appropriate representation
of the structure based on the ’naked-eye’ comparison with the
Fig. 1. Frontal images of Al2 O3 filters used in the investigation. Each specimen has dimensions 50 mm 50 mm 50 mm. (a) 10 ppi. (b) 20 ppi. (c)30 ppi.
324
W. Regulski et al. / Chemical Engineering Science 137 (2015) 320–337
greyscale image. Nevertheless, the influence of the threshold level
on the structure morphology and pressure drop is discussed in
Sections 3.1.3 and 3.2.1.
2.1.2. Removing internal porosity in the CT-scans
The foam structure contains closed porosity in the foam struts
that has to be removed in order to extract hydrodynamically
relevant foam characteristics such as the surface area in contact
with the flow. An initial idea for the removal was to additionally
threshold the foams after performing numerical flow simulations.
The dead ends of strut porosity would be erased based on the
criterion of very small velocity there. This approach, however, had
two drawbacks. First, it was very memory-consuming because the
new geometry had to be obtained after post-processing of simulation data. Furthermore, it incorrectly included areas in the separation zones on the external surface of the struts. Therefore this
approach was rejected.
A different kind of algorithm was then implemented. This
procedure relied only on two-dimensional images without any
need of flow simulations at all. The consecutive steps of the
correction algorithm effects are demonstrated in Fig. 4. Fig. 4a
shows a fragment of a sample slice of the 20 ppi foam after postprocessing of a greyscale CT-scan. One can clearly see that there
exist internal areas with no contact with the surrounding fluid as
well as many regions that seem to belong to the internal porosity
(one can deduce that from a preprocessed greyscale CT-image) but
Fig. 2. Classification of porosity present in the post-processed image. (1) “open
porosity” or “external void space” – void region that is filled with fluid and where
the flow takes place, only this region is relevant from the hydrodynamic point of
view, (2) “internal/closed porosity” – void space in the foam's struts that has no
connection with the external void space, (3) strut porosity that should be regarded
as internal porosity (practically no flow occurs here) but which topologically
belongs to external void space either due to imperfect CT-scan reconstruction or by
being connected by narrow channels in the real structure. The last kind of porosity
is most difficult to remove from the post-processed image.
due to the imperfect post-processing of the scan remain unclosed.
Yet, some of these areas can be in actual contact with the external
area, but this issue is insignificant due to the fact that the
connecting throat is very narrow and there is little flow in there.
This procedure requires the adjustment of an appropriate
thickening level and the maximal size of the removed spot. These
parameters are chosen by method of trial and error. The effects of
corrections with various thickening levels are shown in Fig. 5. The
procedure is performed along three perpendicular directions of
the foam specimen. It must be noted that this approach is totally
heuristic. On one hand, it is very efficient because it operates on
2D images. On the other hand, it has a few drawbacks. It results in
closing the ‘tips’ of large pores that emerge as small entities within
the solid regions. Additionally, it generates some artefacts on the
solid surface and sometimes creates false connections (like the top
left corner in Fig. 5d). The latter is removed by using an additional
procedure not described here. It is worth underlining, however,
that it is quite easy to detect those artefacts during the inspection
of the three-dimensional geometry. Eventually, however, they
have to be removed manually.
2.1.3. Measurement of the specific surface area
The specific surface area of the post-processed images had to
be measured. Two distinct approaches were used. First, the
Cauchy–Crofton theorem was explored (Petrasch et al., 2008;
Große et al., 2009; Liu et al., 2010). It states that the surface area
of the object can be calculated by generating a random set of lines
that cross the object and counting the number of intersections.
More specifically, the following ratio is defined:
R
SΓ
L nΓ ϕ; ψ ; h dϕ dψ dh
;
ð11Þ
¼R Ω SΩ
LΩ nΩ ϕ; ψ ; h dϕ dψ dh
in which SΓ and SΩ are the areas of two objects. The first, Γ, is the
targeted geometry while the second, Ω, is the encapsulating ball.
LΩ represents the set of all lines crossing
Ω. These lines are
parametrized with the set of variables ϕ; ψ ; h . The number of
line-geometry cross sections nΩ (with the ball) and nΓ (with the
foam) are taken for each line. Obviously, for the ball nΩ 2 almost
always.
Eq. (11) is valid for the complete set LΩ which is infinite.
Therefore a Monte-Carlo approximation is used and a random set
of lines that cross that ball and internal geometry is generated,
implying that the algorithm requires an appropriate amount of
lines. We used a set of one and two million lines for each geometry
Fig. 3. CT-image of 10 ppi foam specimen at various stages of reconstruction: (a) 256-level greyscale image, (b) binarized image, (c) binarized image with median filter (used
before the binarization). Internal void spaces are clearly visible. They are artefacts of binarization because the threshold value is low so that some volume that is definitely
solid is not captured in the resulting picture, causing an exaggeration of the internal void space.
W. Regulski et al. / Chemical Engineering Science 137 (2015) 320–337
325
Fig. 4. Correction procedure shown on a fragment of slice of the 20 ppi foam. (a) Base scan, (b) Morphological thickening of the solid part – the solid part starts to grow a
specific number of pixels (shown in red), some of the open regions become closed, (c) reverse liquid and solid pixels with each other – obtain a negative image, (d) remove
new ’hanging’ solid spots below specified size (those are in fact former inner porosities), (e) thicken new solid material – that effectively goes back to initial shape, (f) reverse
solid with liquid to come back to positive image which is the final state. (For interpretation of the references to colour in this figure caption, the reader is referred to the web
version of this paper.)
and the resulting specific surface areas were very similar (discrepancy less than 0.1%).
Another approach to determine the specific surface area was
also used. Lindblad (2005) demonstrated the method of the socalled local weighted configurations and proved its accuracy. The
key idea of this technique was to come up with a proper surface
for each possible configuration of 2 2 2 voxel cube consisting
of either solid or void pixels.
Results from both methods differ no more than by 71% but
only the results of the Cauchy–Crofton algorithm are presented,
and they are listed in Table 1.
the number of velocity vectors). In our implementation, the set
D3Q19 was used. The evolution equation, known as the lattice
Boltzmann equation (LBE), relates the populations at neighbouring
grid nodes
f i ðt þ Δt; xj þ ci ΔtÞ f i ðt; xj Þ ¼ Ωi ðf Þ;
where xj represents the point on the grid, c i is the lattice velocity
and Ωðf Þ is the collision operator.
The macroscopic flow variables, namely the density and velocity, are the consecutive moments of the discrete probability
density functions:
2.2. Simulations
2.2.1. Lattice Boltzmann method
The simulations were performed using the lattice Boltzmann
method (LBM), which is an approach to simulate fluid flow by
repetitive streaming and collision of a certain set of discrete
probability density functions (often called ‘populations’) f i t; xj
placed on a Cartesian grid (Chen and Doolen, 1998). The populations move (stream) with prescribed velocities. The velocity set is
constructed in a way that this dynamic system possesses enough
degrees of freedom to recover the governing equations of hydrodynamics. It usually consists of vectors that connect the neighbouring nodes. The standard choice for two-dimensional problems
is the D2Q9 lattice while for three-dimensional problems D3Q15,
D3Q19 and D3Q27 sets are used (”D” stands for dimension, ”Q” for
ð12Þ
ρ¼
N
1
X
i¼0
f i;
u¼
1
1 NX
ρi¼0
ci f i :
ð13Þ
The LBM has received ever more recognition in the CFD
community due to its straightforward implementation and ease
of handling complex geometries via the so-called bounce-back
boundary condition (Pan et al., 2006).
Originally, the LBM was derived from the lattice gas cellular
automata (LGCA) which were formulated to recover the behaviour of gas systems (Higuera and Jiménez, 1989). Nowadays,
LBM is viewed as a certain discrete form of the Boltzmann
transport equation from the kinetic theory of gases. Using the
Chapman–Enskog analysis it can be proved (Dellar, 2003; Viggen,
326
W. Regulski et al. / Chemical Engineering Science 137 (2015) 320–337
Fig. 5. Correction of closed porosity in the 20 ppi foam slices, showing 336 400 pixel images. Blue colour represents the fluid pixels, green – solid, red – solid after
correction. Thickening levels: (a) input image, (b) 4 pixels, (c) 6 pixels, (d) 8 pixels. (For interpretation of the references to colour in this figure caption, the reader is referred
to the web version of this paper.)
Table 1
Parameters of foam images before and after post-processing. Values of specific surface area ac are based on the Cauchy–Crofton method with two million generated lines. The
discrepancy between measurements with Lindblad and Cauchy–Crofton methods are approx. 1%. Similar discrepancies as well as absolute accuracy of 1% are reported in Liu
et al. (2010). Therefore the error of the measurement is at the level of 2%. The number of significant digits for presented numbers (both for ϵ and ac) follows the convention
used in other works (e.g. Incera Garrido et al., 2008).
No.
ppi
Lx Ly Lz (–)
resolution ðμmÞ
Lx Ly Lz (mm)
εpre ð–Þ
εpost ð–Þ
apre ðm2 =m3 Þ
apost ðm2 =m3 Þ
1.
2.
3.
10
20
30
672 672 800
672 672 800
672 672 800
45.252
45.252
25
30.41 30.41 36.20
30.41x30.41x36.20
16.8 16.8 20
0.805
0.768
0.797
0.777
0.751
0.787
1039.7
1337.9
1666.9
721.5
1075.0
1361.7
2009) that the LBE recovers the pseudo-incompressible Navier–
Stokes equations in the limit of vanishing Mach number.
The LBM algorithm has other advantages, namely there is no
need to solve the Poisson equation for pressure and the computational operations are local. However, the lack of a pressure solver
translates to the lack of the incompressibility constraint. For this
reason, simulations must be run with limited velocities and
pressure in the fluid is recovered from the equation of state,
p ¼ ρc2s . The speed of sound on the standard lattice is cs ¼ p1ffiffi3 and
the practical limit for flow velocity is jujo 0:1 (that translates to
Ma o0:17).
Regarding the collision operator, Ωðf Þ, its oldest and simplest
form is the BGK collision model (Bhatnagar et al., 1954) of the form
1
;
Ωi ðf Þ ¼ f i f eq
i
τ
ð14Þ
represents the equilibrium distribution
where the population feq
i
given by the truncated Maxwell–Boltzmann form
!
u c ðu c i Þ2 u2
eq
f i ¼ ρw i 1 þ 2 i þ
;
ð15Þ
cs
2c4s
2c2s
and
τ is the collision relaxation time that is strictly connected to
W. Regulski et al. / Chemical Engineering Science 137 (2015) 320–337
the kinematic viscosity of the fluid
1
νLB ¼ c2s τ ;
2
ð16Þ
and wi are the weights associated with specific lattice velocity
vectors. It must be noted that the velocities and viscosity in the
LBM simulation are non-dimensional and the physical values of
velocity and viscosity follow from an appropriate rescaling. This
issue is addressed in Section 2.2.1.
MRT collision model: It can be shown that the BGK collision
model used together with the bounce-back boundary condition
results
in
spurious
permeability–viscosity
dependence
(d'Humieres and Ginzburg, 2009). Additionally, the BGK collision
operator is not the most stable collision operator – the stability
drops with increasing τ (i.e. decreasing viscosity). In order to
alleviate both effects, the Multiple-Relaxation-Time model was
proposed by d’Humieres et al. (2002). The key idea of the MRT
model is to transform the set of populations, fi, into a set of
moments using the transformation matrix, M (see Appendix A of
d’Humieres et al. (2002) for the exact form of the matrix). The
constructed moments have various hydrodynamic and nonhydrodynamic interpretations: some are density, velocity and
energy, while others lack relevant interpretation – they are the
so-called ghost modes. Then the system is relaxed towards its
equilibrium (also defined in the moment
space) using a set of
relaxation values, S ¼ diag s0 ; s1; …; s18 . After that, the distribution functions are recovered via the inverse transformation, M 1 .
The MRT-lattice Boltzmann equation takes then the form
eq f i ðt þ Δt; xj þ c i ΔtÞ f i ðt; xj Þ ¼ ðM 1 SMÞi f f :
ð17Þ
In our case, we use the so-called two-relaxation time model,
where the relaxation rates are si ¼ ω; i ¼ 0; …; 18 except
s4 ¼ s6 ¼ s8 ¼ 8ð2 ωÞ=ð8 ωÞ . This formulation results in the fixed
value
of
the so-called
magic
number
3
Λ ¼ 12 1=ω1 12 1=ω2 ¼ 12 1=ω 12 ð8 ωÞ=8ð2 ωÞ ¼ 16
. This
keeps the error of simulation results completely viscosityindependent in the linear flow regime and decreases spurious
dependence in other cases (Pan et al., 2006; d'Humieres and
Ginzburg, 2009). It must be noted that there exist other advanced
collision models such as the Cascaded model (Geier et al., 2007) or
the so-called Entropic LBM (Karlin et al., 2014). Both exhibit
advantageous behaviour especially for high Reynolds numbers
flows. Nevertheless, the MRT model performed satisfactorily in
our case. Additionally, the mentioned advanced models have not
been thoroughly analysed yet in combination with the bounceback boundary condition. More complex boundary conditions
dedicated to Entropic LBM have been proposed as well
(Chikatamarla and Karlin, 2013).
Implementation of body force: There exist several methods to
introduce the body-force term (e.g. gravity) into LBM simulations.
An extensive review can be found in Huang et al. (2011). It is stated
there that in the case of single-phase flows all methods give
practically the same results. In this work, the formulation of
Kupershtokh et al. (2009) known as the exact difference method
(EDM) was used. The body-force is introduced by adding appropriate difference of the equilibrium distribution functions. The LBE
with force source term takes the form
eq
f i ðt þ Δt; xj þ c i ΔtÞ f i ðt; xj Þ ¼ ðM 1 SMÞi f ðuÞ f ðuÞ
eq
eq
ð18Þ
þf i u þ Δu f i ðuÞ:
Consecutive equilibrium distributions need to be calculated
either at current velocity u or the one increased by Δu ¼ F Δt=ρ.
Additionally, the actual flow velocity is obtained after introduction
of a correction term: ureal ¼ u þ 12Δu.
327
Scaling from non-dimensional to physical units: Simulations are
carried out in non-dimensional, or lattice, units. Nondimensionality results from the normalization of the populations
and setting lattice cell size to unity. Thus there is a need to rescale
the quantities from physical units to non-dimensional and the
other way round. The rescaling is performed based on the dynamic
similarity of dimensional and non-dimensional flows. Flows need
to have the same Reynolds number:
Re ¼
UL
ν
¼
U LB N
νLB
:
ð19Þ
In Eq. (19), U and ν denote velocity and kinematic viscosity in
physical units, respectively, L is the physical size of some characteristic dimension of the system (e.g. size of the foam) while ULB
and νLB are respective quantities in the lattice (i.e. non-dimensional) units. N is the characteristic dimensions given in voxels (it
can be imposed as the scan resolution in the case of this work).
The expression for physical size of the LBM cell is straightforward:
L
N
δx ¼ :
ð20Þ
Lattice resolution, δx, together with the physical duration of
one simulation step, δt, link the non-dimensional and physical
velocities:
U ¼ U LB
δx
:
δt
ð21Þ
Using Eqs. (19)–(21) one can show that the simulation time step is
given by
δt ¼
νLB
ðδxÞ2 :
ν
ð22Þ
The physical time step is fixed to the lattice resolution. This
results from the fact that the postulated velocity set on the lattice
is scaled together with the lattice resolution (populations must
travel from node to node in one time step). The resulting physical
velocity of the flow is thus inversely proportional to the image
resolution and lattice viscosity:
U ¼ U LB
δx
ν
¼ U LB
:
δx νLB
δt
ð23Þ
From this relation one sees the main limitation of LBM, namely
the fact that the physical velocity is tied to the parameters of the
simulation. In order to obtain bigger physical velocity in the
simulation, we must either increase lattice velocity, or increase
the resolution of specimen (decrease δx) or decrease the lattice
viscosity. The first is limited by the incompressibility constraint,
the second necessitates significant memory consumption, whilst
the third results in the loss of stability of the computation at some
level of νLB.
2.2.2. Solver
On the contemporary computational hardware the LBM would
not be efficient without proper parallelization. Since this computational method relies on local operations and the computational
data is very well-structured (the foam geometry resides in a
prism), the parallelization on a GPU-type architecture seems
natural. A GPU processor can be considered as a vectorized unit
that is capable of performing numerical operations faster than a
CPU, provided that those operations are alike and that memory
access patterns are regular. The code was written using NVIDIACUDA language that provides extensions to the C language which
enable the user to create functions (called kernels) that are
executed on a GPU. Due to limited operating memory resources
of a GPU (up to 6 GB RAM on NVIDIA Tesla M2090 in our case) and
328
W. Regulski et al. / Chemical Engineering Science 137 (2015) 320–337
a huge memory demand of LBM (compared to FEM of FVM), the
parallelization had to also cover the multi-GPU communication
level. This was performed using the Message Passing Interface
library. An efficient GPU implementation has to consider issues
such as hierarchy of memory types in the GPU and memory access
latency. In order to consider the above remarks, a special implementation technique was utilized. About 70% of the code was
generated in an automatic way using a specially developed tools
based on the R-language (Łaniewski-Wołłk, 2014).
The calculations were performed on up to 24 NVIDIA Tesla M2090
GPUs divided into 3 racks. Code tests revealed very good scalability
even for up to 48 GPUs (Łaniewski-Wołłk and Rokicki, 2015).
2.2.3. Simulation setup
Pressure drop measurements can be carried out in various
ways. Some approaches were analyzed by Narváez Salazar and
Harting (2010). The simplest idea is to postulate a velocity field at
the inlet and pressure value at the outlet. Then, in the course of
the LBM simulation, the pressure at the inlet adjusts itself to the
proper value. However, this configuration raises a few problems.
First, it requires a certain free space upstream and downstream of
the specimen because, in the case of a porous structure, the
velocity field at its inlet and pressure field at the outlet are not
uniform. This results in an increased memory demand. Second,
this configuration also introduces an additional error into the
simulation because of the compressible nature of LBM – the
pressure in front of the specimen is bigger, thus the density is
bigger as well. While flowing across the porous structure, the fluid
expands and increases its velocity (obviously mass flow rate is
kept constant). Therefore the question arises: which velocity –
inlet or outlet one is the “real” flow velocity? This effect becomes
more pronounced when the pressure drop increases.
The approach used in this work employed periodic boundary
conditions on the inlet and outlet together with no-slip (bounceback) boundary conditions at the sides of the sample (see Fig. 6).
Thus, a channel of a rectangular cross-section was created. The
driving force was aligned with the channel. It is important to
highlight that there is no free space at the inlet and the outlet.
Obviously in the case of a real foam geometry, the foam shape at
the inlet is non-conformal with the outlet shape, thus the velocity
profile in this area is disturbed. Our tests revealed that the
difference between the pressure drop measured in such a configuration and one with velocity inlet-pressure outlet is insignificant.
A similar configuration was used for example by Prasianakis et al.
(2013). Actually, we noticed that the relative difference in both
Fig. 6. Boundary conditions used in the flow simulation: red – periodic inlet/outlet,
grey – the channel walls (visualized only at the bottom and the back of the sample).
Driving force was directed along the horizontal direction. (For interpretation of the
references to colour in this figure caption, the reader is referred to the web version
of this paper.)
configurations dropped as we increased the average velocity of
flows. Just to illustrate, we analyzed two times compressed 20 ppi
foam geometry. For the average flow velocity of U in ¼ 0:002 m/s
the relative difference reached in pressure drop was 5% while for
U in ¼ 0:02 m/s discrepancy dropped to less than 1%. The smaller
discrepancy for flows with higher velocity is attributed to the
increasing contribution of the inertial drag occurring within the
structure.
The influence of the presence of walls is investigated in Section
3.2.2, because it has more significant effect on the pressure drop.
2.3. Experimental setup
2.3.1. Construction of the experimental stand
In order to investigate the flow resistance of the porous
structures experimentally, a dedicated apparatus was constructed.
It is presented in Fig. 7. The measurements were taken automatically – the control software adjusted the forcing (in the form of
the power of the pump) and the response was measured (pressure
drop and volumetric flow rate). The measurement output consisted of the set of points in the pressure–velocity space as well as
the coefficients of the least-square fitted parabola representing the
Darcy–Forchheimer curve.
The channel had a square cross-section of side length
L ¼ 50 mm. The flow was in horizontal direction. The schematic
view of the installation is shown in Fig. 8.
Siemens Sitrans P DSII differential manometer was used. It
permitted measurements with up to Δp ¼ 6000 Pa, with measurement error less than 0.075%. It was equipped with a display to
check the parameters. Data acquisition was realized with use of
the current of intensity of 4–20 mA. The flowmeter model was
Kobold DOM A20. It was capable of measuring the flow rate in the
range of 1–40 l=min with error not greater than 0.5%. This
translated to a velocity range of 0:0066–0:2667 m=s. In our case,
the velocity magnitude was limited to 0:2 m=s.
The main channel was made from transparent plexiglas. The
total length of the channel was 30L ¼ 1:5 m. A honeycomb flow
straightener was placed 8L downstream from the inflow. After the
next 8L the place for the investigated specimen was prepared. The
measurement space was designed so that the channel crosssection was kept almost constant. The specimen was fixed in place
by an additional elastic board (see Fig. 9).
2.3.2. Measurement procedure
The control and data acquisition was realized in the LabView
software. The created software enabled the control of the pumps
and registration of the volumetric flow rate and pressure difference between the front and the back of the specimen. The
measurement of pressure drop lasted one second. The impulse
number (representing the flow rate) was gathered at this time.
Fig. 7. Experimental apparatus consisting of a plexiglass channel, piping and
auxiliary equipment. Everything is mounted on an aluminum frame.
W. Regulski et al. / Chemical Engineering Science 137 (2015) 320–337
This procedure was then repeated and the average of two such
measurements was taken as the representative value. Then, the
329
control software changed the pump power in order to change the
flow rate.
3. Results and discussion
3.1. Foam morphological parameters
Fig. 8. Scheme of the experimental setup. The elements of the set-up: (1) Pumps.
Maximum power P max ¼ 90 W each with three gears available, (2) water filter to
protect the flowmeter, (3) flowmeter with the impulse output, (4) gauge system,
(5) differential manometer Kobold DOM A20 with analog output, (6) water
container, (7) honeycomb flow straightener with eyes of diameter 5 mm and a
net at its inlet with holes of size 0:75 mm, (8) space for investigated structure,
(9) auxiliary pneumatic containers compensating for rapid pressure variations, (10)
main flow channel, (11) analog-digital converter National Instruments USB6009,
(12) impulse counter, (13) computer, (14) inverter. The connections are marked
with different colours: black – the plumbing, red – electric measurement circuit,
brown – circuit powering the pumps, blue – auxiliary water cables, green – USB
cables. (For interpretation of the references to colour in this figure caption, the
reader is referred to the web version of this paper.).
Fig. 9. Measurement space with a specimen. Tightening screws and plate are
visible.
3.1.1. Parameters of reconstructed images
Complete foam data is given in Table 1. Data presented there
does not refer to the entire foam because the scanning process
could not cover the entire foam specimen. The reasons are the
following: Firstly, serious disturbances occurred on the borders of
the measured specimen. Secondly, the total scanned volume was
related to the scanning resolution. We were able to read about
1000 pixels in each direction. After removing external layers full of
disturbances and cropping the geometry to multiplicity of 32 (due
to GPU memory alignment constraints), we arrived at final specimen sizes of 672 672 800 voxels. 10 ppi and 20 ppi foam scans
were performed with resolution of about 45 μm while the 30 ppi
foam was scanned at 25 μm. The last choice was dictated by the
fact that we did not want to risk the quality of the image, as the
scanner may miss the thinnest struts and the foam topology
would be distorted. One can observe that the corrections of void
space did not affect the porosity much, but had a tremendous
impact on the specific surface area (20–30% reduction). Rendered
foam specimens are shown in Fig. 10.
3.1.2. Comparison with other data
Similar foam specimens were investigated by other groups.
Each of the following cases concerns Al2 O3 ceramic sponges and
the selected specimen data is presented in Table 2. Große et al.
(2008) provided extensive experimental data for a series of 20 ppi
foams of nominal porosity (i.e. given by the supplier) of
εnom ¼ 80%. Data was obtained by two teams after postprocessing of CT-images. Discrepancies are attributed to different
preprocessing of the specimen and insufficient resolution. They
are considerable in the case of porosity. In their other paper, Große
et al. (2009), the morphological data of series of 10 ppi, 20 ppi and
30 ppi foams were given. In that case, Hg-intrusion porosimetry
was used to obtain both open and total porosity (penetration
abilities of mercury are dependent on external pressure applied).
Additionally, the gravimetric method was utilized, which uses the
weight of the sample, its external dimensions and material
density. The latter was obtained by helium pycnometry. Their
third work, Dietrich et al. (2009), focused on investigation of the
series of foams of the same kind as before. It is not clear if it
concerned exactly the same specimens or ones very similar to
them. Some data presented there are exactly the same (ac Þ, while
Fig. 10. Rendered foam specimens used in the simulations. Note that 30 ppi was scanned with higher resolution. (a) 10 ppi. (b) 20 ppi. (c) 30 ppi.
330
W. Regulski et al. / Chemical Engineering Science 137 (2015) 320–337
Table 2
Parameters of similar foams found in the literature. Both open and closed porosity were determined by Hg-intrusion except Große et al. (2008) where CT-analysis was used.
Specific surface area was determined using the Cauchy–Crofton formula (Eq. (11)) after reconstruction of the CT-images.
No.
ppi
εnom
εtot
εopen
ac m2 =m3
Porosity measurement
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
10
20
30
30
10
20
30
20
20
10
30
30
0.8
0.8
0.8
0.85
0.8
0.8
0.8
0.8
0.8
0.8
0.8
0.85
0.818
0.804
0.816
0.852
0.808
0.802
0.806
0.741–0.781
0.721–0.775
0.794–0.815
0.816–0.808
0.870–0.842
0.752
0.751
0.766
0.807
0.765
0.748
0.752
0.736–0.757
0.718–0.745
0.773
0.758
0.793
675.4
1187
1437.8
1422
664
1204
1402
1187–1244
1213–1268
664
1474
1520
Hg-intrusion
Hg-intrusion
Hg-intrusion
Hg-intrusion
Hg-intrusion
Hg-intrusion
Hg-intrusion
CT-post-processing
CT-post-processing
Gravimetry, Hg-intrusion
Gravimetry, Hg-intrusion
Gravimetry, Hg-intrusion
other parameters are very close to each other. Incera Garrido et al.
(2008) investigated the same ppi series. Foams of various porosities were taken into account. Two last works also provided
extensive pressure drop data that will be compared against our
results in Section 3.3.2.
One can immediately conclude that foam specimen nos. 2, 10
and 12 match porosities of our foam closely ð 7 0:5%Þ, but the
10 ppi foam has lower ac by 100 m2 =m3 while the 20 ppi and
30 ppi specimens have bigger ac. On the other hand, presented
results also demonstrate that the range of discrepancies when
using experimental methods can be significant (porosity errors of
up to 5% for specimen no. 9).
Source
Incera Garrido et al. (2008)
Dietrich et al. (2009)
Große et al. (2008)
Große et al. (2009)
Table 3
Parameters of the 10 ppi foam obtained at different thresholding levels.
No.
Threshold case
ε
ac m2 =m3
1.
2.
3.
Low
Medium
High
0.7914
0.7769
0.7658
696.4
721.5
746.3
3.1.3. Influence of threshold value on foam parameters
The influence of the thresholding level on the foam parameters
was investigated in the case of the 10 ppi foam. Thresholding was
performed after appropriate filtering processes and closing of
internal void spaces. Threshold levels are referred to as “low”,
“medium” and “high”. Obtained data is shown in Table 3. Porosity
varies relatively about 3.3% while the specific surface area ac about
7%. Obviously it affects the predicted pressure drop when typical
Darcy–Forchheimer-type correlations are used and it also distorts
input geometries for the numerical simulations. The scale of the
potential error in the latter case will be investigated in Section 3.2.1.
3.2. Simulation results
3.2.1. Influence of the threshold on the pressure drop
Fig. 11 shows the Darcy–Forchheimer curves obtained from
simulations of the 10 ppi specimen with different threshold levels.
From Section 3.1.3 we know that the relative differences in
porosity and specific surface area are 3.3% and 7% respectively.
The simulations showed that these differences translate to the
relative error between lowest- and highest-porosity foam at a
level of more than 20%. This error was measured for the following
velocities: U inlet ¼ 0:001 m/s, 0.1 m/s, 0.2 m/s. The obtained results
emphasize the significance of careful image analysis and
reconstruction.
3.2.2. Influence of channel walls
The measurement of pressure drop for the foam specimen
confined in the channel is obviously affected by the presence of
channel walls. These generate additional drag in two ways. Firstly,
they introduce another surface in contact with the liquid, and
secondly, their presence creates an additional boundary layer so
that more fluid passes through central part of the foam rather than
the region closer to the sides of the channel. The narrower the
Fig. 11. Pressure drop for 10 ppi foam obtained from LBM simulations with the
same base CT-image with different thresholding levels. DF-fit denotes Darcy–
Forchheimer curves that were least-square fitted to the data from the LBM
simulations.
channel is, the greater the relative influence on the obtained
pressure drop. As discussed earlier, one of the key differences
between the experimental and simulation foam configuration was
the fact that the CT-obtained geometries for simulations did not
encompass the entire physical specimen. Thus, we would expect
that the presence of channel walls introduces relatively more drag
to the specimen in the simulated case than in the experimental
one.
The degree to which the foam confinement increased the
pressure drop was investigated for each foam separately for
multiple values of driving force. The reference case was the
configuration where no side walls were used. Instead, periodic
boundary conditions on the sides were implemented. Since the
flow was driven by the same external force field for both configurations, the difference is observed in the average velocity. The
W. Regulski et al. / Chemical Engineering Science 137 (2015) 320–337
discrepancy is defined as
ϵ¼
wall
U open
avg U avg
U wall
avg
;
ð24Þ
open
in which Uwall
avg and Uavg are the mean flow velocities with and
without the channel walls, respectively. The error values are
presented in Table 4. The relative difference in each case is at
the level of several percent. It is the biggest in the case of the
30 ppi foam probably because it is the smallest physical specimen.
Interestingly, this difference is almost constant across all values of
driving force. Nevertheless, we observe that it is not as significant
as the error introduced by improper image preprocessing.
Table 4
Mean velocity difference (given by Eq. (24)) between unconfined and confined
foam configuration. For the whole range of driving force, in each case the difference
is practically constant. The simulations were performed for non-dimensional force
values of g ¼ 4 10 7 ; 1 10 6 ; 3 10 6 ; 5 10 6 for all specimens and additional
g ¼ 7 10 6 in the case of 20 ppi foam. That translates to all but the smallest and
biggest pressure drops reported in Fig. 12 for the 10 ppi and 30 ppi foams and all
but the smallest one for the 20 ppi foam.
No.
ppi
Avg. rel. difference ϵ (%)
Rel. difference standard deviation σ ϵ (%)
1.
2.
3.
10
20
30
4.41
7.00
8.89
0.40
0.13
0.17
331
3.2.3. Simulation results for all three foams
All three foams were investigated with use of the LBM solver
and all three perpendicular flow directions were taken into account.
The simulation results together with the Darcy–Forchheimer
curves obtained for them are presented in Fig. 12. The Darcy–
Forchheimer curves were obtained by the least-square fit. In Section
3.3.1, the pressure drops is compared against experimental values.
The latter were obtained for velocities up to U ¼ 0:2 m/s. In the
simulations, however, such speeds were impossible to achieve due
to the speed and stability limitations of the LBM discussed in
Section 2.2.1. Flow across the 30 ppi foam, due to a better scan
resolution, resulted in the highest physical velocity.
One can clearly observe the anisotropy of the foam structure.
This is a common issue that was previously reported in other
works (Große et al., 2008, 2009) and can be explained by the
features of the formation process of the foam specimen (Scheffler
and Colombo, 2006). In contrast, in the case of metallic foams it
was reported that the pressure drop is lower in one direction
(Ranut et al., 2014).
The coefficients k1 and k2 are provided in Table 5.
3.3. Comparison of simulation results to other data
3.3.1. Comparison to experimental results
Data shown in Fig. 13 shows very good agreement between
results obtained from numerical simulations and experiments for
10 ppi and 20 ppi foams. The relative discrepancy is defined in the
Fig. 12. Simulation results of pressure gradients for 10 ppi, 20 ppi and 30 ppi foams. Sim. represent values obtained in the LBM simulations, DF-fit stands for the Darcy–
Forchheimer curves that were fitted to this data. (a) 10 ppi. (b) 20 ppi. (c) 30 ppi
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W. Regulski et al. / Chemical Engineering Science 137 (2015) 320–337
Table 5
Darcy–Forchheimer coefficients for the foams based on the LBM simulations. Water parameters at 20 1C: μ ¼ 1:002 10 3 Pa s; ρ ¼ 998:3 kg=m3 . The average value of the
coefficient is the harmonic average because the coefficients appear in the denominator of the DF-formulae (Eq. (1)). The accuracy of the presented coefficients follows the
approach presented in other works on this subject (Incera Garrido et al., 2008).
Direction
x
y
z
Average
10 ppi
20 ppi
30 ppi
k1 ðm2 Þ 10 9
k2 ðmÞ 10 3
k1 ðm2 Þ 10 9
k2 ðmÞ 10 3
k1 ðm2 Þ 10 9
k2 ðmÞ 10 3
122.86
171.83
165.16
149.90
1.73
2.51
2.81
2.25
81.37
77.64
71.48
76.61
1.41
1.20
0.95
1.16
45.28
49.60
48.14
47.61
0.81
0.93
0.83
0.85
Fig. 13. Comparison between numerical and experimental data for 10 ppi, 20 ppi and 30 ppi foams. Exp. stands for experimental results, Sim. denotes the Darcy–
Forchheimer curves fitted to the results from the LBM simulations. (a) 10 ppi. (b) 20 ppi. (c) 30 ppi.
following way:
ϵ¼
ΔpSIM ΔpEXP
:
ΔpEXP
ð25Þ
In the case of velocities U avg 40:025 m=s for 10 ppi foam, the
difference did not exceed 10% for x-direction and 15% in two other
directions. For these latter directions there was a tendency for the
discrepancy to drop with the increase of the flow velocity (ϵ o10%
for U avg 4 0:1 m/s). In the case of 20 ppi foam, the agreement was
even better. For the x-direction the discrepancy grew from ϵ ¼ 3%
for 0:05 m=s 4 U avg 4 0:02 m=s to ϵ ¼ 8% at U avg ¼ 0:18 m/s. The yand z-directions also showed the deviation well below ϵ ¼ 10%
with strong tendency for convergence for higher velocities (ϵ ¼ 4%
for U avg ¼ 0:1 m=s and ϵ ¼ 1% for U avg ¼ 0:18 m/s). Velocities
smaller than U avg ¼ 0:025 m/s were neglected in the estimation
because the experimental results showed very high oscillation.
These oscillations occurred because the pump driving the flow in
the experiment was working with minimal power and its output
was very unstable in that regime.
It should be noted that in the case of 10 ppi and 20 ppi foams a
very good agreement was observed despite three important
differences between the experimental and numerical configuration. Firstly, numerical data for 10 ppi and 20 ppi foams were
obtained for flow velocities only up to 50% of the maximal flow
speed in the experimental case. Furthermore, the boundary conditions were different (as discussed in Section 2.2.3). Finally, the
foam geometries used in the simulations did not represent the
complete volume of the real foams used in the experiment. Thus,
these results confirm that the inlet/outlet effects are negligible.
W. Regulski et al. / Chemical Engineering Science 137 (2015) 320–337
Additionally, the specimen size turned out to be big enough to
diminish the influence of the presence of the channel walls and
can be treated as a representative volume of the whole foam.
In the case of the 30 ppi foam, the agreement was not as good
as in the previous cases, with results for only one flow direction
agreeing quite well. For the x-direction the discrepancy was in the
range of ϵ ¼ 10% while for the two other directions deviation
exceeded ϵ ¼ 20%. These results suggest that the sample size was
not representative for the whole specimen. Another difference
between the simulation and the experiment that is more significant than in the case of 10 ppi and 20 ppi specimen was the
influence of the channel walls, because a relatively smaller portion
of the specimen is confined in the channel in the case of the 30 ppi
foam. Additionally, inspection of the geometry of this foam
revealed significant heterogeneity within the porosity distribution
of the structure. This heterogeneity can be attributed to the
phenomena occurring during the forming process and it was also
reported by Große et al. (2009). Maximal velocities obtained in
333
this simulation reached 60% of the maximal values reached in the
experimental case.
3.3.2. Comparison to other results
In this section data from the LBM simulations is compared with
the experimental data of pressure drop from other research groups
(Incera Garrido et al., 2008; Dietrich et al., 2009). Darcy–Forchheimer curves for foams most similar to the foams used in our
research are plotted in Fig. 14. The Darcy–Forchheimer curves
corresponding to lowest and highest permeability from our
simulations are also presented. The values of the coefficients used
for plotting are given in Table 6.
In the case of the 10 ppi foam shown in Fig. 14a, one can
observe that porosity values are very close to each other but the
specific surface areas of the experimental specimens are smaller.
Yet, the experimental curves fit well into the range spanned by our
data for higher velocity range. However, due to the smaller specific
Fig. 14. Comparison of the LBM simulation results with the experimental results of other groups estimated for water parameters at 20 1C:
μ ¼ 1:002 10 3 Pa s; ρ ¼ 998:3 kg=m3 . The numbers in the brackets refer to consecutive specimens from Table 2, abbreviations: TW – this work, D'09 – Dietrich et al.
(2009), G'08 – Incera Garrido et al. (2008). (a) 10 ppi. (b) 20 ppi. (c) 30 ppi.
Table 6
Darcy–Forchheimer coefficients for the foams from other groups.
Source
Dietrich et al. (2009)
Incera Garrido et al. (2008)
Incera Garrido et al. (2008)
10 ppi
20 ppi
30 ppi
k1 ðm2 Þ 10 9
k2 ðmÞ 10 3
k1 ðm2 Þ 10 9
k2 ðmÞ 10 3
k1 ðm2 Þ 10 9
10 3
77
28.58
–
1.87
3.13
–
54
9.17
–
1.14
1.67
–
32
7.23
11.07
0.98
1.45
1.89
334
W. Regulski et al. / Chemical Engineering Science 137 (2015) 320–337
surface area in the experimental specimens, one would expect
smaller pressure drop than in our case.
Also in the case of the 20 ppi foam (Fig. 14b) we have the
specimens with characteristics very close to our structure. Both
reference results possess higher specific surface areas. Here,
however, only data reported by Dietrich et al. (2009) fits well into
our range, while data of Incera Garrido et al. (2008) presents a
larger pressure drop in the investigated range. One can easily
notice that Darcy–Forchheimer (DF) curves from our results have a
greater growth rate and will eventually overtake both results,
despite lower specific surface area of our specimen.
The results for the 30 ppi foam are shown in Fig. 14c. The
specimen of Incera Garrido et al. (2008) that has lower porosity
and higher specific surface area (no. 3 in Table 2) exhibits a higher
flow resistance, but it can be seen, that its DF curve has slower
growth rate. The more porous structure investigated by this group
(no. 4 in Table 2) initially remains within our range, but also
exhibits smaller growth rate later on. Dietrich et al. (2009) result
practically overlaps with the lower bound of our case.
To summarize, most of the presented experimental data is in a
fairly good agreement with our results. Still, one should recognize
the limitations of the results of the previous works. First, both
groups report the anisotropy of ceramic sponges in their works.
Neither of them, however, investigates its influence in their
measurements. Our results shown in Section 3.2.3 demonstrate
that the directional differences can be significant. In addition, we
must remember that the porosity measurements in other groups
were performed on real samples. In the work of Große et al. (2008)
it was reported that variations of porosity measurements for the
same sample were on the level of 2–3% based on the preprocessing
of acquired data. Finally, Incera Garrido et al. (2008) used tomography resolutions at the level of 50–86 μm. From the authors'
experience 50 μm is an upper resolution bound for foams of this
kind. It seems that the insufficient CT-resolution can also be a
source of error, especially in case of specific surface area
estimation.
3.3.3. Comparison with existing correlations
The results obtained in our LBM simulations can be compared
with known correlations which relate pressure drop with specific
surface area. For this we used two relations. The first is given by
Eq. (10) by Dietrich (2012) and the second was based on the
formulation proposed in Lacroix et al. (2007) and Huu et al. (2009)
given by Eq. (6).
The results are presented in Fig. 15. Due to the structure
anisotropy, data for all three directions is drawn. One can notice
that in all cases the correlations of both groups fit well to our
results for 10 ppi and 20 ppi. Obviously, the correlation by Dietrich
gives a higher pressure drop estimate. In the case of the 30 ppi
foam, Dietrich's correlation fits just below the lowest DF curve
while the data of Huu, Lacroix and co-workers (designated in
Fig. 15 as ‘HLE’) does not fit at all. Although, for the reasons
discussed earlier, geometry of this specimen is not considered as
reliable as data for 10 ppi and 20 ppi foams, we can still discuss the
applicability of the pressure drop correlations in this case because
accurate parameters of the digital 30 ppi specimen are available.
It should be stressed that verification of correlations against
simulation results should in principle be prone to lower discrepancy
than their verification to experimental data. The correlations
Fig. 15. Comparison of pressure drop from LBM simulations against other groups’ correlations. Eq. (10) is based on Dietrich (2012) while Eq. (6) is based on the idea of
Lacroix et al. (2007) and Huu et al. (2009) (referred to as HLE in the legend). (a) 10ppi, (b) 20ppi, and (c) 30ppi.
W. Regulski et al. / Chemical Engineering Science 137 (2015) 320–337
involve parameters that were obtained by investigating the digital
images of foams and exactly these images were used in the
simulations. Thus the inaccuracies are only related to determination
of the structures' parameters based on their voxelized
representation.
4. Conclusions and future work
This work presents the results of numerical and experimental
investigations of pressure drop in flow across three specimens of
ceramic foams. Very good agreement for 10 ppi and 20 ppi foams
is observed. Results for the 30 ppi foam, probably due to the
smaller and heterogeneous (thus not representative) physical
volume investigated, do not show such a good agreement. Still,
they can be considered satisfactory. Results confirm the capability
of the lattice Boltzmann method to resolve the flow field in porous
media at pore-scale level.
The influence of different factors on pressure drop was investigated. Those factors included image thresholding levels and the
presence of channel walls. The importance of proper image
segmentation was demonstrated. The obtained results were also
compared against other researchers' data for similar structures as
well as to selected correlations for pressure drop. Good agreement
was found in comparison with similar structures from the other
groups. Also, the pressure drop correlations of Huu et al. (2009)
and Dietrich (2012) agreed very well with presented data in most
cases. In all cases, foam anisotropy was detected.
The obtained results imply further research areas. First, the
analysis of pressure drop with respect to the CT-scan resolution
and total specimen size (especially for the 30 ppi foam) will be
performed. The ultimate goal of our work is to define the pressuredrop correlations that will be based purely on the geometrical
characteristics of these types of foams and image analysis is a key
issue in this case. As soon as this capacity is developed, the
detailed internal characteristics will be investigated. Additionally,
the real geometry will be manipulated in order to modify the
degree of anisotropy and come up with data for more general
correlations.
Nomenclature
Latin letters
ac
specific surface area m2 =m3
ci
lattice velocity vector associated with ith distribution
function (–)
cs
lattice speed of sound (–)
dh
equivalent hydraulic diameter (m)
ds
sphere diameter (m)
dstrut foam strut diameter (m)
dp
mean pore diameter (m)
fi
particle distribution function in the LBM simulations (–)
f
vector of all distribution functions in the LBM simulations
(–)
F
force vector in the LBM simulations (–)
h
3rd parameter used in random line generation for the
Cauchy–Crofton formula (–)
Hg
Hagen number (–)
k1
viscous permeability coefficient m2
k2
inertial permeability coefficient (m)
L
physical length (m)
LΩ
set of all lines crossing the encapsulating ball Ω in the
ΔL
M
Cauchy–Crofton formula
foam length (m)
MRT transformation matrix (–)
335
Ma Mach number (–)
N
size of the object in lattice units (–)
nΓ ;Ω number of intersections of generated lines and object of
interest used in the Cauchy–Crofton formula (–)
p
lattice pressure (–)
Δp pressure drop in flows across foams (Pa)
P
power (W)
Re
Reynolds number (–)
si
relaxation rates in the MRT collision operator (–)
S
surface area m2
S
diagonal matrix of the MRT relaxation rates (–)
t
lattice time (–)
tr
threshold level (–)
physical simulation time step ðsÞ
δt
Δt lattice time step (–)
u
lattice fluid velocity vector (–)
Δu lattice fluid velocity increase due to body force (–)
U
mean flow velocity (m/s)
ULB lattice flow velocity to rescale (–)
V
volume m3
weights for the equilibrium distribution function (–)
wi
xi
position of the lattice node (–)
δx physical lattice resolution (m)
Greek letters
α
coefficient in the viscous term of the generalized Ergun
equation (–)
β
coefficient in the inertial term of the generalized Ergun
equation (–)
Γ
index of the internal geometry for the Cauchy–Crofton
formula
ϵ
relative error (–)
ε
porosity (–)
Λ
magick number linking the relaxation rates in the TRT
collision operator (–)
μ
fluid dynamic viscosity (Pa s)
ν
fluid kinematic viscosity m2 =s
νLB fluid kinematic viscosity in lattice units (–)
ρ
fluid density kg=m3
ρ
lattice fluid density (–)
σ
standard deviation
τ
lattice relaxation time (–)
ϕ
1st parameter used in random line generation for the
Cauchy–Crofton formula (–)
ψ
2nd parameter used in random line generation for the
Cauchy–Crofton formula (–)
ω
relaxation rate in the LBM collision operators (–)
Ω index of the bounding sphere for the Cauchy–Crofton
formula
Ω(f) LBM collision operator (–)
Acknowledgements
The work was financially supported by the Polish Ministry of
Science and Higher Education research Grant no. N N507 273636.
This work was also supported by the European Union in the
framework of European Social Fund through the “Didactic Development Program of the Faculty of Power and Aeronautical Engineering of the Warsaw University of Technology”. Finally, this
research was supported in part by PL-Grid Infrastructure. The
calculations were performed on ZEUS HPC unit at the Academic
Computer Centre CYFRONET in Cracow. The author would also like
336
W. Regulski et al. / Chemical Engineering Science 137 (2015) 320–337
to thank Dr. Christopher R. Leonardi for his proof-reading of the
manuscript and his comments.
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