Environmental Engineering Reference
In-Depth Information
The viscous losses term
ð _
q
vis
Þ
in Eq. (
4.6
) on the macroscopic scale represents
the pressure drop of the
ow in the AMR. Since the hydraulic diameters of
AMRs are usually very small (in the micro-heat exchanger range) the viscous losses
are mostly affected by the viscous dissipation caused by friction in the core of the
AMR (the entry and exit effects are thus usually neglected), which further causes a
degradation of the mechanical energy into heat. Viscous losses can play an
important role, especially in the packed-bed AMRs, which suffer from order higher
pressure drops compared to an ordered structure, e.g. parallel-plate AMRs [
76
].
Most 1-D models include a viscous losses term in the governing energy equations,
while most 2-D models, which are in general limited to the ordered-structure
AMRs, do not [
33
]. In the 1-D models, the viscous losses term is usually applied as
a pressure drop along the AMR (which is assumed to be constant) calculated
through the friction factor
fl
uid
fl
Reynolds number correlations (see the Section Heat
transfer and Fanning friction factor correlations for details). It can be written as:
—
v
3
q
f
d
h
p
o
x
¼
v
D
p
L
¼
v
o
q
vis
¼
2f
F
ð
4
:
12
Þ
where v,
ʔ
p, L,
ˁ
f
,d
h
,f
F
are the average velocity of the
fl
uid, pressure drop, AMR
length, density of the
uid, hydraulic diameter and Fanning friction factor. Besides
the impact on the AMR
fl
-
ciency of the process, since high viscous losses require a larger input work to pump
the
'
s energy state, the viscous losses strongly affect the ef
uid (see Eq.
4.17
).
The
fl
q
MCE
term in Eq. (
4.5
) applies the magnetocaloric effect. In general, AMR
models implement the magnetocaloric effect using different approaches. A more
straightforward and simple way to include the magnetocaloric effect is to apply the
adiabatic temperature change of the magnetocaloric material during the magneti-
zation and demagnetization processes, as was done by, e.g. [
43
,
49
,
52
]. In doing
so, the term
_
_
q
MCE
is not included in Eq. (
4.5
) directly, but the following equations
are applied (solely) during the magnetization and demagnetization processes
instead:
T
s
;
fi
¼
T
s
;
i
þ D
T
ad
;
mag
T
s
;
i
; l
0
H
fi
; l
0
H
i
ð
4
:
13a
Þ
; l
0
H
fi
; l
0
H
i
T
s
;
fi
¼
T
s
;
i
D
T
ad
;
dem
T
s
;
i
þ D
T
ad
;
mag
ð
4
:
13b
Þ
where
and i denote the initial and
nal temperatures (T) and magnetic
elds
ðl
0
H
Þ
, respectively, and mag relates to the magnetization, while dem relates to the
demagnetization. The adiabatic temperature change
ðD
T
ad
Þ
depends on the mate-
rial
eld change. An example of an adiabatic
temperature change for gadolinium (as a function of the magnetic
'
s temperature and the magnetic
eld change and
temperature) calculated using the Mean Field Theory and the well-known Maxwell
relation
is shown in Fig.
4.12
a for both the magneti-
zation and demagnetization processes. The Mean Field Theory model
ð
ð
o
s
m
=
o
l
0
H
Þ¼ð
o
M
0
=
o
T
Þ
Þ
is a
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