ARUP; ISBN: 978-0-9562121-5-3 - CMBBE 2012 - Cardiff University

1
u dV
V u. (Eq. 2)
V
The permeability K is then determined by the equation 1. In our case, the size of the
spherical pores in our bioceramics (radius about 300m) leads to neglect the influence
of the gravitation force 11-12 .
There are many empirical models to predict the permeability K but only for packed beds
or granular porous media. To use theses models the interconnected spherical pores of
our bioceramics porous media must become the “spaces” of the packed bed or of the
granular porous media.
3. Validation of the numerical model.
To validate the numerical model, an academic 3D case of a porous medium was tested
(similar to Fig. 2). The variation of the distance between the uniform cubes (edge size 1
mm) allows fixing a porosity ranging from 0.2 to 0.8. This porous medium is isotropic
and its structure is periodic. So, we are able to simplify the simulation using the UC
(Fig.2b) instead of the REV. The procedure above mentioned (for a structured grid and
a pressure gradient equal to 20 Pa.m -1 imposed in one couple of periodic boundary
condition) leads to the fields of velocity and of pressure shown in the figure 4.
(a) (b)
Fig.4. Fields of velocity (a) and fields of pressure (b) in UC.
The permeability K of this porous medium was then calculated and compared with the
results issued from classical models. Note that the second order tensor K becomes a
scalar in this case; we need only to solve the Navier-Stokes equations by applying a
pressure gradient in one direction. The table 1 shows the permeability determined
through three empirical models: Ergun 13-14 , Carman-Kozeny 15 , Rumpf-Guptes 16 , an
analytical model: Du Plessis 17 and our numerical simulation method.
Porosity Ergun’model
3
K
2
2
d 150 1
Carman-
Kozeny
3
K
2
2
d 1801
Rumpf-
Gupte
5.5
K
2
d 5.6
Du Plessis
1 2
3 3
K
11 11
2
4
d
3 411
Numerical
simulation
0.4 1.185E-003 9.877E-004 1.156E-003 7.930E-004 1.022E-003
0.5 3.333E-003 2.778E-003 3.946E-003 2.190E-003 2.749E-003
0.6 9.000E-003 7.500E-003 1.076E-002 5.742E-003 6.955E-003
0.7 2.541E-002 2.117E-002 2.511E-002 1.542E-002 1.761E-002
Tab.1. Dimensionless permeability according to the porosity and to the model (d mean
particle diameter of the spherical granular medium).
For an isotropic porous medium , the numerical simulation is close to the Duplessis's
model as shown by the results (tab.1).
5