Civil Engineering Reference
In-Depth Information
Lafarge model to pressure diffusion, in a first step, the dynamic thermal permeability
with
α
given by Equation (5.35) can be rewritten
q
0
jω
2
ω/ω
t
1
/
2
1
+
j
M
t
q
(ω)
=
ω
t
+
(5.111)
8
q
0
2
φ
M
t
=
(5.112)
ωφδ
2
2
q
0
ω
t
=
(5.113)
A function
D
(
ω
), similar to
q
(
ω
), can be defined for the acoustic pressure by
D(
0
)
jω
2
ω/ω
d
1
/
2
1
+
j
M
d
D(ω)
=
ω
d
+
(5.114)
8
D(
0
)
d
(
1
M
d
=
(5.115)
−
φ
p
)
(
1
−
φ
p
)P
0
q
0
m
φ
m
ηD(
0
)
ω
d
=
(5.116)
The parameters
D
(0) and
d
are geometrical factors which are defined similar to
q
0
and
. The contact surface
sp
is the same for both problems, but for the thermal
conduction problem the volume is the volume of the pores
fp
and for the pressure
diffusion it is the volume
sp
out of the pores. Therefore
d
=
2
sp
(
1
−
φ
p
)
φ
p
p
sp
=
(5.117)
where
p
=
2
fp
/
sp
is the thermal characteristic length of the pore network.
In a second step, the relation
q
(ω)jωp/(φκ)
which relates the average tem-
perature to the pressure for a simple porosity medium is reinterpreted. The temperature
field
τ
can be considered as the sum of two fields
τ
1
and
τ
2
,τ
1
=
τ
=
ν
p/κ
is the spatially
constant field created for a specific mass capacity of the frame equal to 0 and a temper-
ature of the boundary equal to
τ
1
. In order to satisfy the boundary condition
τ
0on
the second field
τ
2
is the diffused temperature field related to the boundary condition
τ
=
ν
p/κ
on
. The average temperature
τ
2
is given by
=−
τ
2
=
τ
−
τ
1
1
−
q
(ω)jω
q
0
ω
t
ν
p
κ
=
−
(5.118)
1
−
(
q
(ω)jω
q
0
ω
t
=
−
τ
1
)
This expression can be transposed to express the average diffused pressure field in
the air in the microporous medium
sp
1
p
p
j
ω
ω
d
D(ω)
D(
0
)
p
m
=
−
(5.119)
