Civil Engineering Reference
In-Depth Information
was selected by Lafarge
et al
. (1997) as the homologue of
α
(
ω
). From Equation (5.6)
the thermal permeability is related to
α
(
ω
)by
q
(ω)
=
ν
φ/(jωα
(ω))
.Forthecaseof
identical parallel cylindrical pores, it is seen from Equations (4.45) - (4.46) that
α(ω
)
can be identified with 1/
F
(
ω
)and
α
(
ω
) with 1/
F(B
2
ω
). With the simplified Lafarge
model, which gives for
K
the same high-frequency limit as Equation (5.28), the same
low-frequency limit as Equation (5.8), and satisfies the causality condition,
α
can be
written
1
1
/
2
2
q
0
φ
2
jω
ν
ν
φ
jωq
0
α
(ω)
=
+
+
1
(5.35)
An additional parameter
p
is present in the complete expression of
α
(
ω
)givenby
Lafarge (2006). This parameter can provide minor modifications of the bulk modulus in
the low- and the medium-frequency range, but does not seem necessary in the description
of the bulk modulus of plastic foams and fibrous materials. This parameter is equal to 1
in Equation (5.35).
5.5
Simpler models
5.5.1 The Johnson
et al.
model
The dynamic tortuosity in the work by Johnson
et al
. (1987) is given by
1
1
/
2
2
α
∞
q
0
φ
2
jω
ν
νφ
jωq
0
α(ω)
=
+
+
α
∞
(5.36)
The use of causality and of the asymptotic behaviour to justify the use of this expres-
sion was an important step in the description of sound propagation in porous media. The
same expression is obtained by setting
b
1 in Equation (5.32) that was carried out later
by Pride
et al
. (1993). The effective density
ρ
=
=
α(ω)ρ
0
has the right limit to first-order
approximation in 1
/
√
ω
for large
ω
given by Equation (5.26) and for small
ω
the limit
is given by
1
+
2
α
∞
q
0
2
φ
ηφ
jωq
0
ρ(ω)
=
ρ
0
α
∞
+
(5.37)
The limit of the imaginary part is given by Equation (5.7),
j
Im
ρ
=
ηφ/(jωq
0
)
=
φσ/(jω)
. As an example, for identical circular cross-sectional shaped pores, with
=
R
R
2
φ/
8, the limit of the real part is 1.25
ρ
0
. The true limit obtained in
Chapter 4 is 1.33
ρ
0
. In spite of this small difference for the limit of Re
ρ
when
ω
tends
to zero, Equation. (5.36) and the 'exact' model give similar predictions.
and
q
0
=
η/σ
=
5.5.2 The Champoux -Allard model
The direct measurement of the static thermal permeability is not easy. The simplified
Lafarge model has been used with
q
0
replaced in Equation (5.35) by the permeability
