Active Magnetic Regeneration - Magnetocaloric Energy Conversion: From Theory to Applications - page 124

4.3.4.1 The Demagnetization Field

It was shown that the internal magnetic eld in the magnetic material (and sub-

sequently in the AMR) is, in general, always lower than the applied magnetic eld

in the empty air gap of the magnetic eld source [ 93 , 94 ]. This is due to the

demagnetizing eld, which is a consequence of the magnetization of the magnetic

body and opposes the magnetization inside the body, thus reducing the resulting
internal

eld. In order to apply the correct values for the magnetocaloric effect into
the numerical model of the AMR, the internal magnetic eld in the magnetocaloric

material must be used instead of the applied one. It can be written as:

H in ¼ H appl H dem ¼ H appl N ð x Þ M ð T ; H Þ ð 4 : 24 Þ

where H appl , H dem , N and M are the applied magnetic eld, the demagnetization

eld, the demagnetization tensor and the magnetization, respectively. Equa-
tion ( 4.24 ) is usually expressed as a scalar equation and the average demagneti-
zation factor is used instead of the demagnetization tensor [ 95 ].
The internal demagnetization

eld is, in general, spatially dependent over the
magnetized body. It depends on the geometry of the body, its temperature and the
internal magnetic eld. In the case of an AMR, which is a non-uniformly mag-

netized body, the demagnetization eld therefore strongly depends on its geometry,

the temperature span at which it operates, the magnetocaloric material(s) applied
and the applied magnetic

eld. Its fully correct application is therefore non-trivial
and cannot be solved analytically (analytical solutions are only available for an
uniformly magnetized body, like ellipsoidal bodies or an in nite sheet and cylinder

[ 95 ]). Recently, a 3-D magnetostatic numerical model for the calculation of the
demagnetization tensor and subsequently the demagnetization

eld was presented
for a non-uniformly magnetized rectangular prism [ 46 ] and a stack of rectangular
prisms [ 96 ] (e.g., a parallel-plate AMR). Bj rk and Bahl [ 97 ] numerically analysed

the demagnetization factor of non-uniformly magnetized, randomly packed,
spherical particles (e.g. a packed-bed AMR). Furthermore, a coupled AMR model
with the demagnetization model can be found in [ 92 , 98 , 99 ].
Various studies showed that the impact of the demagnetization

ø

eld on the

internal eld in the AMR can be signi cant. In practice, it depends mostly on the

'

AMR s outer dimensions and the distribution of the material inside (for a particular

magnetic eld) [ 93 , 94 , 96 ]. In the case of a typical AMR with a porosity of about

30 % subjected to a magnetic

eld of 1 T the internal magnetic eld, in the two

extreme cases (parallel-plate AMR with the plate s distribution parallel to the

applied magnetic eld and perpendicular to it) can be reduced from 10 % (parallel-
distribution) and up to 70 % (perpendicular distribution) [ 96 , 100 ]. It was also
shown numerically as well as experimentally that a parallel-plate AMR with the
plate

'

cantly higher
cooling characteristics compared to the perpendicular distribution [ 92 , 99 , 101 ].

' s distribution parallel to the magnetic eld can generate signi