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A
HT
V
q
HT
_
ðÞ¼a
x
;
t
T
s
x
ðÞ
;
t
T
f
x
ðÞ
;
t
ð
4
:
8
Þ
where
cient usually obtained through Nusselt number
correlations (see Section Heat transfer and Fanning friction factor correlations for
details), A
HT
is the total heat transfer area of the AMR and V is the volume of the
solid (for Eq.
4.5
)or
ʱ
is a heat transfer coef
uid (for Eq.
4.6
). Since the 1-D model neglects the thermal
conductivity and thus the temperature distribution in the solid perpendicular to the
fl
fl
ow, many models, e.g. [
16
,
41
,
43
] apply a correction factor for the heat
transfer coef
uid
fl
cient. In this way, we can deal with the effective (lumped) heat
transfer coef
ʱ
eff
)[
74
]. Using such an approach, the effect of the non-uniform
temperature distribution in the magnetocaloric material is taken into account to a
certain extent. The effective heat transfer coef
cient (
cient is de
ned as:
a
a
eff
¼
ð
4
:
9
Þ
Bi
a
0
1
þ
ʱ
0
depends on the geometry of the magnetocaloric material in the
AMR and has a value of 3 for spheres, 4 for cylinders and 5 for plates [
74
]. The
Biot number is de
The factor
ned as:
a
d
2
k
s
Bi
¼
ð
4
:
10
Þ
where d is the sphere diameter or plate thickness. It should be noted that a well
acceptable limit is established, i.e. if the Biot number is less than 0.1, the thermal
conductivity and the temperature gradient in the material perpendicular to the
fl
uid
fl
ow can be neglected (due to there being at least ten times higher convective heat
transfer at the surface) [
74
]. Since the Biot number in the AMR is not a priori less
than 0.1, an effective heat transfer coef
cient should be considered in the 1-D
models. Furthermore, Eq. (
4.9
) is fully valid only for the steady-state heat transfer
conditions [
75
]. Since in an AMR, the heat transfer is transient (the temperature
difference between the solid and
uid is not constant at any time in the process).
Engelbrecht et al. [
75
] developed a correction factor for transient conditions that
should be included in Eq. (
4.9
).
However, the 2-D and 3-D models may apply the heat transfer term (Eq.
4.8
)as
well, but it is more consistent to apply an additional boundary condition that
describes the heat transfer and temperature gradients at the surface instead (the term
_
fl
q
HT
is thus not applied in Eqs. (
4.5
) and (
4.6
)):
y
¼
h
¼ k
f
o
T
f
y
¼
h
k
s
o
T
s
o
ð
4
:
11
Þ
y
y
o
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