Civil Engineering Reference
In-Depth Information
volume where diffusion occurs is the microporous medium and
φ
must be replaced by
1
−
φ
p
. The skin depth is
δ
d
and
D
(
ω
)isgivenby
1
−
tanh
j
1
/
2
(b
√
2
/δ
d
)
2
j
1
/
2
(b
√
2
/δ
d
)
2
D(ω)
=−
j(
1
−
φ
p
)
δ
d
√
2
(5.123)
φ
p
)b
2
/
3and
d
=
2
b
. The pressure field in the microporous domain is represented in Figure 5.12. When
the diffusion skin depth is much smaller than
b
, the pressure is negligible compared
with
p
p
except in a small volume close to the pores. Volume variations are restricted
because the pressure is not transmitted in the whole volume of the microporous medium.
Large variations of the macroscopic bulk modulus are created by the variations of
F
.
Moreover, in the transition range where
F
varies rapidly from 1 to 0, the loss angle
of the macroscopic bulk modulus can be larger than for a single porosity medium (see
Figure 9 in Olny and Boutin 2003). The semi-phenomenological models for the thermal
exchanges and the bulk modulus described in Sections 5.1 - 5.5 are not valid for double
porosity media with a high permeability contrast.
The semi-phenomenological model could be used with
D(
0
)
=
(
1
−
5.10.6 Practical considerations
Double porosity materials may exist either in a natural state (fractured material with
a porous frame) or result from a manufacturing process (recycled materials, perfo-
rated porous materials). In particular, Olny (1999) and Olny and Boutin (2003) showed
both theoretically and experimentally that the absorption coefficient of highly resistive
porous materials could be significantly increased in a wide frequency band by perform-
ing properly designed mesoperforations in the materials. Atalla
et al
. (2001) presented a
finite-element-based numerical model accounting naturally for the assembling of air cavi-
ties and multiple porous materials, thereby alleviating the limitations of Olny's HSP-based
model and extended his model to three-dimensional configurations. In particular, they
confirmed Olny's results and showed the influence of several design parameters (size of
holes, meso-perforation rate and distribution of holes) on the absorption coefficient. An
example illustrating the increased absorption of properly selected double porous materials
is given in Section 13.9.5 with a comparison between the analytical and the numerical
models.
The review by Sgard
et al
. (2005) gives practical design rules to develop optimized
noise control solutions based on the concept of perforating a properly selected porous
material. The acoustic behaviour of these perforated materials is governed by three impor-
tant parameters: the size of the perforation, the mesoporosity and a shape factor which
depends on the perforation shape and on the mesopores distribution. The enhancement of
the absorption coefficient is intimately linked to the position of the viscous characteristic
frequency of the microporous domain
ω
vm
and the diffusion frequency defined in Equation
(5.116):
ω
d
φ
p
)P
0
q
0
m
(φ
m
ηD(
0
))
. Two conditions are necessary for this poten-
tial enhancement. The first condition is that the flow in the microporous domain be viscous
that is:
ω
=
(
1
−
ηφ
m
(ρ
0
α
∞
m
q
om
)
. The second condition imposes that the
diffusion frequency
ω
d
be much smaller than the viscous characteristic frequency of the
microporous domain
ω
vm
,thatis:
ω
d
ω
vm
where
ω
vm
=
ω
vm
. This condition means that the wavelength
in the microporous domain is of the same order as the size of the pore, so that pressure
