Structural properties measurement: A Morphological tool for transport properties determination

Full text

Structural Properties Measurement: A Morphological Analysis Tool for Transport
Properties Determination
Jérome Vicente, Frédéric Topin, Lounès Tadrist
Ecole Polytechnique Universitaire de Marseille - Laboratoire I.U.S.T.I - CNRS- UMR 6595 Université de Provence
Technopôle de Château-Gombert - 5, Rue Enrico Fermi 13453 Marseille Cedex 13 – France
The aim of this work is to develop morphology analysis tools to study the impact of foams structure on physical transport properties. The reconstruction
of the solid-pore interface allows the visualization of the 3D data and determination of specific surface and porosity. We present an original method to
measure the geometrical tortuosity of a porous media for the two phases (solid and pore). This technique is based on numerical fast marching
implementation and calculates the geodesics in the medium. A centerline extraction method structures allows us to model the solid matrix as a network of
linear connected segments. Results obtained on a set of nickel foams covering a wide range of pore size are discussed.
Keywords: metallic foam, morphology, transport properties
1 Introduction
The control of the texture of porous materials used for the
optimization of compact and multipurpose heat exchangers
(boiler, vapo-reformer...) represents a significant
technological stake. Indeed, the choice of foam optimized
for a given application requires correlating the microscopic
structure to the transport properties. A first study showed up
the feasibility of the 3D reconstruction and basics
measurements on a X-ray tomography of a 10 PPI copper
foam 1).
To analyze geometry of foams different methods of
visualization, segmentation and morphometry are needed.
We develop tools to characterize both pore space and solid
matrix as these two phases may have different geometric
characteristics that impact on various properties (e.g. heat
conductivity is linked mainly to matrix structure, flow laws
are governed by pore shape). It provides geometrical
measurements (e.g. specific area) and segmentation as an
idealized network which gives access to structural
properties. Segmentation of pores in individualized cells
gives access to both porosimetry and morphometry. A
centerline extraction method allows us to model the solid
matrix as a network of linear connected segments. As
physical transport phenomena are directly linked to the path
line notion, we calculate geodesics in the medium using a
technique based on numerical fast marching implementation
to calculate geodesics in the medium. We then determine
geometrical tortuosity of each phase.
This approach will enable us to proceed to the
morphology analysis in correlation with the physical
transport properties obtained via numerical simulations or
on experimental data using the tomographied samples. We
discuss here the used methods and present several results
obtained on a set of Nickel foams (Table 1)
2 Polygonal model
Two options are usually available for viewing the scalar
volume datasets, direct volume rendering 2,3) and volume
segmentation combined with conventional surface
rendering4). The direct volume rendering only supply
images of the data whiles the volume segmentation open
access to measurements.
Fig. 1 3D rendering: Solid matrix and segmented pores (Sample Ni27-33).
We use the classic "Marching cubes" algorithm for
extracting interface between the phases 5). This technique
creates a polygonal model that approximates the iso-surface
embedded in a scalar volume dataset for a particular iso-
value. The surface represents all the points within the
volume that have the same scalar value.
The reconstruction of the dividing surface between solid
and pore allows the visualization of the 3D data (Fig.1).
The polygonal surface is created by examining each cube of
eight voxels and defining a set of triangles that
approximates the piece of the iso-surface within the space
bounded by the eight points. The efficiency of the algorithm
is due to the limited number of cases (256) for which a
surface cuts a cube. This allows their tabulation and reduces
greatly the calculations. Due to the variation of level
variations of X-Ray reconstructed images, an optimal
threshold (iso-density) based on the density histogram was
calculated for each images.

Table 1 Geometrical characterisation results.
Thus later are then re-normalized such as a unique level
corresponds to a physical density value in the entire volume.
The reconstructed surface is made of regular meshes, of
which we check connectivity of edges, and which does not
present holes. We can then carry out the direct calculation
of surfaces and specific volumes of each phase of the foam.
We also export the surface meshes of the solid matrix (or
poral space) as toward research or commercial CFD codes
to simulate the heat and mass transfers in these mediums.
2.1 Results
We compare our evaluation of both specific area and
porosity in function of mean pore diameter (see below) with
nominal values for several foam samples (Fig. 2). Flow
laws depend on open porosity. These later are determining
by filling the hollow strut and measuring the remaining
porosity. Even the high resolution of X-ray images shows
up holes between the macro pore and struts cavities. These
holes are either real or created when segmenting the thin
walls of struts. To measure both total and open porosity, we
apply a 3d closure morphological operation by dilating and
eroding the 3d binary images. We measure the respective
volumes of both pore kinds.
Fig. 2 Specific surface versus pore size. Comparison with nominal
Recemat values.
The specific surface is calculated from the interfacial
meshes generated for each sample. The number of triangles
constituting the samples meshes is varying from 2M for the
Ni1010 sample up to 5M for the Ni3743 sample. We found
that specific surface varies like the inverse of the pore
diameter, fitting our data lead to:
(1)
This ind
specific surface data and considering the hypothesis that the
icates that the tested foams are homothetic. Using
p
pD
S91.2
=
struts are equilateral prisms of side a, their size could be
valuated. Surface and volume of a length l of struts are:
e
given by:
laValS bb
2
3
3== (2)
4
The specific surface is express by the following expression:
()
ε
−= 1
b
pV
S
S(3)
b
Substituting S and Vb by their expression we evaluate the
ize (equivalen eter) of the struts as:
b
s t diam
()
ε
−= 1
34
S
a(4)
p
s expected, the st linearly with Dp
diameter for
Th f a porous medium was for the first time
i an cited in 6) and given, in a particular
direction, li
di
0
50
100
150
200
250
300
350
400
01000 2000 3000 4000 5000
Strut Diameter, Ds/µm
0
1000
2000
3000
4000
5000
6000
7000
0 0,5 1 1,5 22,5 3 3,5 44,5 5
Recemat data
Present Work
Specific Area, Sp/m-1
Fig. 3 Evaluation of struts diameter versus pore diameter.
Pore Diameter, Dp/mm
Pore Diameter, Dp/mm A ruts diameter varies
(Fig.3); the strut thickness is about 7% of pore
samples studied here. Note that the hypothesis on the shape
(prismatic) of the struts as to be confirmed through 3D
image analysis.
3. Tortuosity
e tortuosity o
def ned by Carm
ke the square of the ratio of the average effective
stance traversed by the fluid at the Euclidean distance

between 2 sections. As we reconstruct the shape of the solid
matrix it is interesting to define a geometrical tortuosity for
each phase. This one is defined, for a couple of points
contained in the same phase and connected according to:
() ()
2
21
21min
21
,
,⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−
=pp
ppL
pp
g
τ
(5)
with Lmin(p1,p2), the length of the shortest path in the ph
joining p1 to p2.
t method is a numerical technique for
acking moving interfaces. The related FMM, which is
for monotonically advancing
fro
ase
3.1 Fast marching method
The level se
tr
computationally attractive
nts 7,8). FMM are mainly used for the construction of
geodesic on surfaces, or calculation of the optimal ways
circumventing of the obstacles 9). The key advantages of
these methods are that they rely on a fixed grid (adapted to
discrete 3D images), handle topological changes in the
interface naturally.
A moving interface Γ(t) can be formulated as the zero
level curve of a scalar-valued function Ψ : R3xRÆ R,
where
{
}
0),(:)( 3=Ψℜ∈=Γ txxt (6)
Ψ is the crossing time map, a function that gives the ti
when a moving front crosse he point x. The crossing t
unique if the front is monotonically advancing. Thus, Ψ-
me
s t ime
is
1(0) is the initial position of the front and at any later time t,
the front is given by Ψ-1(t). The crossing time map is
constructed by solving an equation of the form
)(
1
)( xF
x=Ψ∇
r
(7)
where F(x) is the front speed at the point x.
If F(x) =Cte, en the solution Ψ(x) = ϕ(x) gives th
istance from x to the zero contour Ψ-1(0) = ϕ-1(0).
t replacing the
gr
twards
fro
at allows us to determine:
To c cu media
from e
to the fro ints Γ(0) ={x | x ∈δΩ}.
ase we calculate
eodesic joining a given point p0 to any point p. a 4th order
seek the parametric curve
C
th e
d
The FMM solves this equation by firs
adient by suitable upwind operators, and then
systematically advancing the front by marching ou
m the boundary data in an upwind fashion. For N nodes,
the method has a total operation count of O(N log N).
Marching algorithm makes use of an upwind finite
differences scheme to compute the value u at a given point
xi,j of the grid
We use this method to compute efficiently the distance
map, and the minimal path between any pair of point in a
given phase. Th
- the geometrical tortuosities of each phases
- An accurate morphological criterion to segment the
pore.
al late the distance map of every point in the
th bound we fix the initial position of the front Ψ-1(0)
ntier po
3.2 Minimal path extraction
Using the Ψ(x) map over one ph
g
runge-Kutta method is used to
(t) solution of retropropagation equation.
pCwithtx
dt
tdC =Ψ∇−= )0(),(
)(
r
(8)
Using a constant front propagation velocity for the FMM
calculation, we obtain the true ge esic path. While a fr
ve city proportional to the distance of the other phase leads
to
od ont
lo
create an “averaged” path similar to the Carman
definition (Fig. 4).
Fig. 4 Geodesic distance map and minimal between 2 points in the pore.
Left constant velocity propagation, right distance to solid driven velocity.
etween two parallel planes situated at the two extremity of
This evaluation is made for two orthogonal
di
an
text pores. On the other hand, the tortuosity
di
3.3 Results
For each phase, we measure the geometrical tortuosity
b
the sample.
rection named horizontal and vertical. We don’t report
results on the third direction as where is not enough pores in
the thickness of the sample to valuable results. The figure 5
shows the distribution of these tortuosities for each phase
and direction.
Fig. 5 Tortuosity distribution – left pore, right solid matrix (sample Ni
27-33 ).
The tortuosity distribution of poral space is very narrow
d centered on a low values that indicates a very open
ure of
stribution of the solid phase is rather large, with an
average value of 1.2 very different from the pore tortuosity
30
0
5
10
15
20
25
1,00 1,05 1,10 1,15 1,20 1,25 1,30
Geometrical Tortuosity, Τ
Pores
Solid matrix
Cross : Vertical direction
Diamond : Horizontal direction
Frequency (%)

value. Moreover, the results show clearly that solid
tortuosity depends on direction and the solid structure is
slightly anisotropic as the effective thermal conductivity (§
4.2.2).
Figure 6 shows influence of pore size on tortuosities
(average on the two directions). We do not compare
tortuosity on each direction because vertical and horizontal
di
4.
4.1. Aperture diameter
to determine the
meter Ouv(P)
of an object as the diameter of the
la
rections are arbitrary for each sample. Both pore and solid
values seems to decrease slightly with Dp. more work is
needed to clarify this point.
Fig. 6 Tortuosity of both solid and pore phases versus pore size.
Segmentation of pore and solid phase
One way to characterize the pore size is
perture diameters map. The local aperture dia
a
is defined in any point P
rgest ball included in the object and containing the point
P, as proposed by Delerue 10).
()
⎭
⎬
⎫
⎩
⎨
⎧⎟
⎞
⎜
⎛
∈⊂
⎟
⎠
⎞
⎜
⎝
⎛
=,,
2
,max d
CBPO
d
CBdPOuv (9)
⎠⎝ 2
Fig. 7 Aperture map in a cross section of sample Ni 27-33.
The aperture map is then defined as an 3D image of local
ap figure
. The histogram of this map gives access to the pore size
di
ata.
is necessary to characterize
co pletely the cells shape (fig. 1). We use the watershed
ker distance function 11). First we
co
c
vo f
struts and cells, as well as presence ore
th
erture; each voxel value being Ouv(P) as shown on
7
stribution (Fig. 8)
Nevertheless, this technique is limited to the evaluation
of pore size, no information on cells shape could be
deduced from these d
4.2. Cell morphology
3D segmentation of the cells
m
transform of the mar
mpute the distance map whose values are interpreted as a
topographic surface (fig. 9).
Fig. 8 Pore aperture size diameter distribution (sample Ni 27-33 ).
Then Markers, defined as the minima of the topographi
o
lume, are identified. Because of shape irregularities
of constrictions, m
an one minimum exist in each cell. But, the method needs
one unique marker for each cell. The key point of this
technique is to eliminate irrelevant markers to keep only the
one corresponding to the center of the cell. A first class of
“false” markers is easy to eliminate using topographic
conditions (markers belonging to centerline of oblong
objects, markers near the solid that would belong to too
small cells).
Fig. 9 Distance map in a cross section of sample Ni 27-33.
On the other hand, markers belonging to cells throat are
no ey mark a
al geometric structure that could have a characteristic size
cl
t so easily eliminated. That was expected as th
re
ose to the one of the cells. We develop an elimination
method based on analysis of the neighborhood of the
markers. We study the distribution of minimal distance to
solid for points located on a sphere centered on the marker.
Once the cells are identified, their diameter Dp is taken as
the diameter of the equivalent volume sphere.
10 500 1000 1500 2000 2500 3000
1
5
5
Pore Tortuosity,
Τ
Aperture diameter, DA /µm
1,05
1,1
1,15
1,2
1,25 1,025
6
7
5
4
1,00
1,01
1,01
1,02
solid
pore
0
1
2
3
60 179 298 418 537 656 776 895 1015 1134 1253
Pore Diameter, Dp/mm
uosity
10E6), N Frequency (x
,
Τ
Solid matrix Tort

Fig. 10 Cross section of segmented cells (sample Ni 27 .
The 3D inertia matrix of the equivalent ellipsoid is
de
-33)
termined for each individualized cell. The 3 principal
axes of the ellipsoid are denoted by c< b<a. An example of
distribution of lengths values is presented on figure 11. The
principal axis values are clearly differentiated. All the
analyzed foams show similar distributions.
0
50
100
150
200
250
300
200 250 300 350 400 450 500 550 600 650
c axis
b axis
a axis
axis length (µm)
Frequency
Fig. 11 Half axis length ellipsoid tistic
able 2 Number of segmented cells used to calculate the principal axis
10 Ni1116 Ni1723 Ni2733 Ni3743
of equivalent distribution. Sa
made from 2000 cells (sample Ni 27-33).
T
values distribution.
samples Ni10
cells 122 604 1224 1187 2305
he elongation calculated from mean principal axis length
.3. Solid network
nd the automatic recognition of the
str
rix enables us
T
is presented in Table 1.
4
The extraction a
ucturing elements constitute the base of the geometrical
description. Indeed, the identification and the three-
dimensional localization of nodes (branching detection),
branches and the connectivity table allow us to access the
geometrical characteristics. We focus especially the
determination of the connectivity of the solid matrix and of
the poral space. The network reconstruction open access to
statistic treatment of : Segment length, orientation, as well
as highlighting preferentially directed planes.
The detection of the junctions of the solid mat
to cut out it in structuring elements (segment, nodes) from
which we build an idealized network of linear segments.
4.3.1. Distance ordered homotopic thinning
Skeletons are compact representations that allow
mathematical analysis of objects. It must be homotopic, thin
and medial in relation to the object it represents. The
obtained skeleton is connected, topologically equivalent to
the object, centered and thin. We are interested in discrete
methods, generally fast and easy to use.
We choose the Distance Ordered Homotopic Thinning 12)
(DOHT), which uses a homotopic thinning, this means an
iterative deletion of simple points but in the increasing
distance map order leading to a centered skeleton.
The skeleton is computed by iteratively peeling off the
boundary of the object, layer-by-layer. A point is said
simple if its deletion preserves the object topology 13). If all
simple points are removed iteratively the result object is
topologically equivalent to the original one, but far too
simple: a connected component without hole or cavity will
be shrunk to a single point. In order to better preserve
rotation invariance, we add to this method a directional
strategy as proposed by 12). For each distance, we
systematically consider border points, in the following
order: east, bottom, west, south then top of the object. We
eliminate all the false branches induced by struts
irregularities. Such branches are short and connected from a
single point.
Fig. 12 Idealized solid network (sample Ni 27-33).
Oversegmentation often occurs at struts intersection.
Morphological operations are then used to remove these
small bad struts. This operation take into account different
scale of analysis to cleanup these regions.
Fig. 13 Ligament length distribution. (sample Ni 27-33).
0
500
1000
1500
2000
2500
3000
3500
0 100 200 300 400 500 600 700
Ligament length, L/µm
Len
g
th,
l
/
µm
Fre
q
uenc
y
,
N
Fre
q
uenc
y
,
N

On the cleaned skeleton (Fig.12) we measure the struts
length distribution (Fig 13). This idealized network is used
to determine effective thermal conductivity.
4.3.2. Effective thermal conductivity
Effective thermal conductivity on each direction
x,y,z is determined. A cube is cut into the sample RLS and
nodes constituted by intersection of each cube face and
segments are identified. We impose different temperature
on two opposite faces. We calculate the total heat flux
Φ
eff
i
K
i
across the network using the nodal temperature deduced
from simple one-dimensional conduction transfer on each
segment. On each node p of the network, energy balance is
given by
()
0
∑∑ ∈∈
=−−=Φ
ptoconnected
Nodesj
pjpj
pj
solid
ptoconnected
Nodesj
pj TTS
l
k (10)
Where
φ
ij , lij and Sij are resp. the heat flux and the length
and the cross section of the segment. ksolid is the solid
thermal conductivity, and Ti, Tj are the nodal temperatures.
Eventually, the flux
Φ
i is identified with macroscopic
conductive heat flux across a homogeneous medium placed
in the same conditions and
T
e
Ki
eff
iΔ
Φ
= . Note that we
supposed here, that the fluid phase doesn’t contribute
significantly to the effective conductivity and that radiative
transfer between solid surfaces is negligible. The segment
section is taken as its mean value deduced from total solid
volume and length of the network. The results show a
slightly anisotropy of the foam (Table 3).
Table 3 Effective thermal conductivity. Copper foam 10 ppi grade.
Metal foam
Sample descritpion
Thermal Conductivity,
K/W.m-1.K-1
dx/m 3.43E-03 Keffx3.31
dy/m 3.44E-03 Keffy4.05
dz/m 6.57E-03 Keffz3.38
strut section, S/m² 1.65E-07 Keffmean 3.58
4 Conclusions
We develop a morphological analysis tool that gives
access to quantification of the main structural parameters of
metallic foams. It provides the functions of geometrical
measurements (specific area, pore size distribution,..) and of
schematization of idealized network which gives access to
structural properties.
The reconstruction of the dividing surface between solid
and pore (made of regular mesh) allows the visualization of
the 3D data. The interface tessellation allows us to measure
porosity, specific surface, and evaluate struts diameter.
Dependence of these later with pore size has been
established.
An efficient centerline extraction method gives the
skelletization of the solid phase. Identification of nodes,
segments, and connectivity of the idealized network
modelling the solid matrix has been carried out. This
network allows us to calculate the effective thermal
conductivity.
An original method based on numerical fast marching
implementation has been developed to measure the
geometrical tortuosity of the two phases (solid and pore).
Tortuosity of poral space is very low compare to solid
matrix value. A slight anisotropy of solid tortuosity is
observed. The measurements carried out on our set of
samples shows no clear influence of the pore size on the
tortuosity.
A systematic study of transport properties dependence on
parameters such as tortuosities and porosity is undergoing
using other samples of different textures. We also export the
surface mesh as toward research or commercial computer
codes to simulate Heat and fluid flows into these medium.
Acknowledgments
The authors wish to thank Recemat Company for
providing the samples, the ID19 beam team for helpful
assistance at ESRF synchrotron facility, and the French
government in the frameworks of a PACo programme and
CNRS Energy program: PR Specimousse.
REFERENCES
1) S. Catillon, C. Louis, F. Topin, J. Vicente and R. Rouget: Journal of
Hydrogen Energy. (2005) to appear
2) R.A. Drebin, L. Carpenter and P. Hanrahan: Computer Graphics. 22
(1988) 65-74.
3) S. Parker, M. Parker, Y. Livnat, P. Sloan, C. Hansen and P. Shirley:
IEEE Trans. on Viz. and Comp. Graph. 5 (1999) pp.238-250.
4) A. Singh, D. Goldgof and D. Terzopoulos (eds.): Deformable Models in
Medical Image Analysis, IEEE Computer Society Press Los Alamitos (1998)
5) W.E. Lorensen, H.E. Cline: Computer Graphics. 21 (1987) 163-169
6) Pierre M. Adler: Media, geometry and transport, series in chemical
engineering, Butterworth-Heinemann (1992).
7) J.A. Sethian: Level Set Methods and Fast Marching Methods,
Cambridge University Press, Cambridge UK (1999)
8) J.A. Sethian, R. Kimmel: Proc. of Nat. Acad. Sci, 95 (1998) pp.8431-8435.
9) T. Deschamps, L.D. Cohen: Medical Image Analysis 5 (2001) 281-299.
10) J.F. Delerue, E. Perrier, Z. Yu, B. Velde: Physics and Chem. of the
earth. 24 (1999) 639-644.
11) T. Dillard & al.: Metfoam (2003) pp.301-307.
12) C.J. Pudney: Proc. of CVIU, 72 (1998) pp.404-413.
13) C. Lohou, G. Bertrand: Proc. of DGCI , 23 ( 2002) pp.102-113.