Environmental Engineering Reference
In-Depth Information
The terms on the left-hand side of both equations describe the temperature
variations of the
fl
uid and solid with time. The
fl
uid equation also contains the
convection
ow
direction (x-direction), this term is reduced to the one-dimensional form, where the
fl
fl
uid transport term. Since the 1-D model applies only to the
fl
uid
fl
uid velocity (as a scalar) is input data. It also usually assumes that the
fl
uid
fl
ow
has a uniform temperature at each cross section of the
fl
uid channel and the velocity
pro
le is uniform along the entire length of the AMR (fully developed
fl
ow). In the
2-D and 3-D models, a velocity pro
le (vector) is calculated using the well-known
Navier
ed into an analytical expression, as
showed by Nielsen et al. [
50
] for a parallel-plate AMR or solved numerically at the
same time as the governing energy equations (Eqs.
4.5
and
4.6
)[
49
]. Since the
Stokes equations, which can be simpli
-
fl
uid
fl
ow in the AMR, especially at high operating frequencies, is oscillating and thus
not fully developed, the correct application of this effect can play an important role,
mostly with respect to the heat transfer characteristics.
The
rst terms on the right-hand side of Eqs. (
4.7
) and (
4.8
) describe the thermal
conductivity through the borders of the differential control volume. In the 1-D
model the thermal conductivity is applied only in the
ow direction (x-direction). In
order to simplify the 1-D model and increase the computational ef
fl
ciency, some
models, e.g. [
16
,
41
,
42
] apply the effective thermal conductivity in the
ow
direction. In doing so, they combine the thermal conductivity of the solid and the
fluid into the effective thermal conductivity applied only in the solid equation. This
is de
fl
ned as [
65
]:
k
eff
¼ k
stat
þ k
f
D
d
ð
4
:
7
Þ
The effective thermal conductivity can be divided into the static thermal con-
ductivity, with no
fl
uid
fl
ow
k
stat
, and the thermal conductivity due to the
fl
uid
dispersion (D
d
) of the
fl
uid
fl
ow. The static thermal conductivity depends on the
solid and
uid thermal conductivity and the porosity of the regenerator. The cor-
relations for the static thermal conductivity can be found in the literature, e.g.
[
65
fl
69
] and can be, in general, separated into the ordered structures (parallel-plate)
and the packed-bed structures (spheres, cylinders, irregular particles). The thermal
dispersion of the
-
fl
uid
fl
ow re
fl
ects the thermal conduction due to the hydrodynamic
mixing in the
fl
uid
fl
ow through the porous structure. It occurs due to the velocity
fl
uctuations in the
fl
uid
fl
ow and the separation and reuni
cation of the
fl
uid along
its path [
70
]. In the literature, e.g. [
65
,
71
s
thermal dispersion can be found. Again, one can distinguish between the correla-
tions for the ordered and the packed-bed structures.
On the other hand, the 2-D model usually applies the thermal conductivity (of
73
] various correlations of the
fl
uid
'
-
the
ow direction as well as perpendicular to it (x- and y-
directions). This is, of course, physically more correct, but computationally less
ef
fl
uid and the solid) in the
fl
cient.
The term
q
HT
applies to the heat transfer between the magnetocaloric material
(solid) and the heat-transfer
_
fl
uid. In the 1-D model the heat transfer term is de
ned
using the well-known Newton
'
s law of cooling:
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