**4.2 Results**

242 Trends in Electromagnetism – From Fundamentals to Applications

V.grad .

ρ

*T P T aT t C*

Governing equations for AMRR system have been developed throughout the years with the objective of to analytically or numerically describe the thermal behaviour at specific time and for a given set of boundary conditions. They consist of a system of two equations, one for the fluid and the other for the solid matrix. These equations are derived from the energy balance expression for each phase. Since they are coupled they must be solved simultaneously (Bouchekara, 2008). The model of an AMRR cycle has been developed in (Bouchekara, 2008). Fig. 16 illustrates the concept of an AMRR regenerator modelled using

The system of equations given by the energy balance (explained above) for both the magnetocaloric material and the fluid by neglecting the axial conduction (this approximation can be justified for different conditions: low thermal conductivity, very thin

( ) ( )

τ

(25)

for the time. Thus, the derivatives with

(26)

( )

To solve this system we use the finite difference method. We use a grid of elements that

respect to the time are calculated using forward formulas, and those with respect to the space are calculated using backward formulas. This gives a centered discretization scheme.

1, , , 1 (,)

12 3

( ) ( ) ( )

=+ +

1, (,) (,)

*i j i j i j*

*m mm m f*

*T AT AT* + −

= +

1 2 *i j i j i j i j*

*T AT AT AT*

*f f f f f fm*

. ( ) *f f ff f m f*

∂ ∂ + =− ∂ ∂

*T T m C T d t hS T T t x*

*<sup>m</sup> m m f m*

*<sup>T</sup> m C hS T T t*

<sup>∂</sup> = −

∂

( )

+

range from 0 to L for the space and from 0 to

Thus the system (25) becomes:

<sup>∂</sup> + = +Δ

*p*

∂ (24)

is defined as the thermal diffusivity of the fluid, thus :

if *<sup>p</sup> a C* = λ ρ

**4.1 Application to AMRR** 

one dimensional (1D) approximation.

Fig. 16. Conceptual drawing of a 1D AMRR model.

plates, etc.) can be summarized by the following system of equations:

We will now apply the model developed earlier to a regenerator in the form of plates, as shown in Fig. 17(a). The equivalent cell of the whole regenerator is given in Fig. 17 (b). This cell has the same parameters as the regenerator except the width that is *eq p l Nl* = (where *eq l* represents the equivalent width, *Np* represents the number of plates and *l* is the width of one plate).

Fig. 17. (a) A regenerator in the form of plates, (b) Equivalent cell (plate + fluid).

The model parameters (for this simulation case) are shown in Table 2. The magnetocaloric material used is gadolinium, the coolant used is water and the magnetic field is generated by permanent magnets B = 1 T.


Table 2. The parameters used in the simulation.

Fig. 18 (a) shows the temperature evolution of both sides (hot and cold) of the material versus time. After a transient phase, the two curves reach their steady state. In addition, we note that the final value is greater than the initial MCE. From this curve we can extract the evolution of the temperature at the end of each cycle (Fig. 18 (b)). The small delay between the two curves of this figure is due to programming constraints, i.e. the magnetization phase has been introduced (programmed) before the demagnetization phase.

Magnetic Refrigeration Technology at Room Temperature 245

(a) Geometry of the first structure modeled by Flux 3D.

outside the cylinder). The opposite way works also; i.e. the material remains fixed and the magnet is connected to the linear motor. In this case, the magnetic behavior is the same as in

Y0 (mm)

Y0 (mm)

Fig. 19. Geometry and magnetic characteristics of the Structure A.

Fig. 20. Cylindre d'Halbach made of eight segments.

(b) Induction (T)

(c) Force (N)

the first case.

The projection of the geometry of the structure on the plane (oxy).

Y0=-50(mm)

Y0=0(mm)

(d) The distribution of the magnetic induction.

Fig. 18. Temperature profiles given by the AMRR numerical model.
