Environmental Engineering Reference
In-Depth Information
q
MCE
¼q
T
s
o
M
0
o
Þ
o
l
0
H
o
t
¼q
T
s
o
s
m
Þ
o
l
0
H
o
_
ð
T
s
; l
0
H
o
l
0
H
T
s
; l
0
H
ð
ð
4
:
14
Þ
T
t
where M
0
is the speci
c magnetization (per mass of magnetocaloric material), s
m
is
the magnetic entropy and
is the density. The term
o
l
0
H
o
t
ˁ
describes the magnetic
eld
pro
ned by the magnet assembly and the operating conditions. Equa-
tion (
4.14
) applies the well-known Maxwell relation
le de
s
m
=
o
l
0
H, nor-
mally used to calculate the magnetic entropy change from the measured values of
the speci
o
M
0
=
o
T
¼
o
c magnetization.
An example of a derivative of the speci
c magnetization over temperature for
gadolinium is shown in Fig.
4.12
b. Such an inclusion of the released energy in the
magnetocaloric material during the (de)magnetization process over a period of time
is more appropriate as a more realistic magnetic eld prole and time-dependent
(de)magnetization with simultaneous heat transfer can be applied. Various models,
e.g. [
41
,
42
,
51
] include the magnetocaloric effect in such a way. However, this
method requires a detailed data set (or input functions) of the magnetization or
magnetic entropy and the speci
c heat at different temperatures and magnetic
elds,
which may not always be available. It should also be noted that the speci
c heat of
magnetocaloric materials around the transition temperatures strongly depends on
the temperature and magnetic
eld and cannot be taken as a constant value, as
shown for gadolinium in Fig.
4.12
c.
The developed AMR numerical models, in general, rely on the experimentally or
theoretically obtained magnetocaloric properties of a particular magnetocaloric
material. Especially in the case of gadolinium, which is a kind of reference mag-
netocaloric material for the AMR models, the Mean Field Theory became a well-
established tool for estimating its magnetocaloric and thermal properties. As a
result, it was applied by various authors of AMR models, e.g. [
16
,
49
,
51
]. The
Mean Field Theory (together with the applied Maxwell relation) is able to generate
a data set for the required magnetocaloric properties that can be included in the
AMR model using both above-presented methods (the direct or built-in method),
even though due to the applied assumptions it over predicts the magnetocaloric
effect. However, the experimentally obtained data are usually measured for an
insuf
elds, to be
correctly and consistently used via the built-in method. An example of suf
cient number of different temperatures, and especially magnetic
ciently
detailed, measured magnetocaloric properties for gadolinium and La
Si
magnetocaloric materials, which allows its implementation into a built-in AMR
model, was presented by Bj
Fe
Co
-
-
-
rk et al. [
32
].
As was also noted by some authors of AMR models [
79
], it is extremely
important to obtain all the magnetocaloric properties used in the model (adiabatic
temperature change or speci
ø
c heat) from the same
source. So, all should be calculated using the Mean Field Theory or experimentally
measured (the latter can also suffer from a signi
c magnetization and speci
cant measuring uncertainty). Even
a minimal mismatch between the applied magnetocaloric data can lead to a potential
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