Environmental Engineering Reference
In-Depth Information
When the system is isotropic and the physical properties of the two phases are con-
stant, Eqs. (6.271) and (6.272) reduce to
γ
β
∂
T
β
β
∂
T
σ
σ
−
T
β
β
k
β
Δ
T
β
β
T
σ
σ
+
=
+
k
βσ
Δ
ha
υ
,
(6.273)
t
and
T
σ
σ
−
T
β
β
T
σ
σ
∂
k
σβ
Δ
T
β
β
γ
σ
∂
T
σ
σ
+
=
k
σ
Δ
+
ha
υ
,
(6.274)
t
where
γ
β
=
ϕ
(
ρ
c
)
β
and
γ
σ
=(
1
−
ϕ
)(
ρ
c
)
σ
are the
β
-phase and
σ
-phase effective
thermal capacities, respectively,
ϕ
is the porosity,
k
β
and
k
σ
are the effective ther-
mal conductivities of the
β
-and
σ
-phases, respectively, and
k
βσ
=
k
σβ
is the cross
effective thermal conductivity of the two phases.
The one-equation model is valid whenever the two temperatures
T
β
β
and
T
σ
σ
are sufficiently close to each other so that
T
β
β
T
σ
σ
=
=
T
.
(6.275)
This
local thermal equilibrium
is valid when any one of the following three con-
ditions occurs (Quintard and Whitaker 1993, Whitaker 1999): (1) either
∈
β
∈
σ
or
tends to zero, (2) the difference in the
β
-phase and
σ
-phase physical properties
2
tends to zero (e.g.
tends to zero, (3) the square of the ratio of length scales
(
l
βσ
/
L
)
βσ
=
∈
β
∈
σ
∈
β
k
σ
+
∈
σ
k
β
for steady, one-dimensional heat conduction). Here
l
2
/
(
ha
υ
)
,and
L
=
L
T
L
T
1
with
L
T
and
L
T
1
as the characteristic lengths of
∇
T
and
∇∇
T
,
respectively,
such
that
∇
T
=
O
(
Δ
T
/
L
T
)
and
∇∇
T
=
O
.
When the local thermal equilibrium is valid, Quintard and Whitaker (1993) add
Eqs. (6.271) and (6.272) to obtain a one-equation model
(
Δ
T
/
L
T
1
L
T
)
K
ef f
C
∂
T
ρ
=
∇
·
·
∇
T
.
(6.276)
∂
t
Here
ρ
is the spatial average density defined by
ρ
=
∈
β
ρ
β
+
∈
σ
ρ
σ
,
(6.277)
and
C
is the mass-fraction-weighted thermal capacity given by
=
∈
β
(
ρ
c
)
β
+
ε
σ
(
ρ
c
)
σ
C
.
(6.278)
∈
β
ρ
β
+
∈
σ
ρ
σ
The effective thermal conductivity tenor is
K
ef f
=
K
ββ
+
2
K
βσ
+
K
σσ
.
(6.279)
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