Active Magnetic Regeneration - Magnetocaloric Energy Conversion: From Theory to Applications - page 109

(Eqs. 4.5 and 4.6 ). Regarding the number of addressed dimensions in the energy
equation, we can distinguish between one-dimensional (1-D), two-dimensional
(2-D) and three-dimensional (3-D) time-dependent AMR models. The 1-D
approach is the most widely applied, mostly due to its higher computational ef

-

ciency. In the 1-D models, it is assumed that the ow and the thermal con-

ductivity (if included) only occur in the direction of the

fl uid fl

fl uid fl ow. A crucial

parameter for the 1-D models is the heat transfer coef cient (e.g. the Nusselt

number), which de nes the heat transfer rate between the magnetocaloric material

and the heat-transfer uid. The accuracy of the 1-D models, therefore, often very

much depends on the suitability and the accuracy of the correlation of the heat
transfer coef

fl

cient being used. In recent years, several 1-D time-dependent models
were developed. They can be found in, e.g. [ 16 , 38 48 ]. Recently, also a few 2-D

-

models have been applied [ 49 52 ]. The 2-D model generally assumes a two-

-

dimensional le and includes the longitudinal and transversal

thermal conductivity of the magnetocaloric material and the heat-transfer

fl uid velocity pro

fl uid

(parallel and perpendicular to the direction of the ow). The governing

equations for the 2-D model are not directly coupled through the heat transfer
coef

fl uid fl

cient, as in the case of 1-D models, but through the additional boundary
condition, which makes them physically more consistent. However, its application
is, in general, limited by the ordered AMR geometries (e.g. parallel-plates, wire-like
and perforated structures or similar). Random geometries, like packed-bed struc-
tures, cannot be fully addressed in a two-dimensional space. The 1-D models, in
contrast, do not have this limitation, as the impact of the geometry is considered
through the use of suitable thermohydraulic correlations. However, Li et al. [ 53 ]
developed a partial 2-D model of a packed-bed AMR with spheres, where the
thermal conduction of the spheres is considered in 2-D, while the uid equation is

applied in 1-D and the required thermohydraulic properties are included through the
appropriate correlations. A similar approach was presented for a honeycomb AMR
by

fl

Š arlah et al. [ 54 ].

4.3.2 Mathematical (Physical) Model of an AMR (Basic
Energy Balance Equations)

The mathematical model of an AMR is based on the well-established passive
regenerator model. The major difference between passive regenerator models and
AMR models is the implementation of the magnetocaloric effect and the timing
between the magnetic eld ' s pro le and the fl uid fl ow ' spro le. The basic model of

the passive heat regenerator was rst developed by Anzelius [ 55 ]. A few years later,

Nusselt [ 56 ] and Hausen [ 57 ] separately described the general operation of the heat
regenerator and its mathematical model. The applications and research on heat
regenerators were greatly expanded in later years due to the increasing need for heat