**4. Thermal study**

240 Trends in Electromagnetism – From Fundamentals to Applications

Fig. 14. The second prototype of the G2Elab, Components of the prototype (left) and the

An interesting prototype of a rotary magnetic refrigerator (Tusek et al., 2009) has been built on the basis of permanent magnets at the University of Ljubljana in Slovenia. Their rotary magnetic refrigerator consisted of a rotating drum (cylinder) that rotated around an internally positioned stationary soft iron core and externally positioned stationary permanent magnets. As shown in Fig. 15, the magnetic structure was composed of four NdFeB permanent magnets and low carbon 1010 steel used as a soft ferromagnetic material, and two magnetic circuits existed to allow the rotary movement of the AMR's. After optimization of the magnet structure geometry, a range of magnetic field intensities from 0.05 T to 0.98 T was obtained in the air gaps. There were 34 AMR's in the rotary drum and each AMR had the dimensions 10 mm × 10 mm × 50 mm. Gd plates, with a thickness of 0.3 mm, were filled in the AMR's and the total mass of Gd was approximately 600 g. The prototype could operate up to a frequency of 4 Hz. This reference mainly focused on the experience in development of such a rotary magnetic refrigeration prototype and no experimental results were reported. However, first predictions according to the researchers

are that approximately a 7 K temperature difference will be achieved (Yu, 2010).

Fig. 15. The rotary magnetic refrigerator developed at the University of Ljubljana in Slovenia

Prototype in its actual environment (right) (Bouchekara, 2008).

Magnet

MCE material

(Tusek et al., 2009).

**3.10 The Slovenian system** 

Heat exchanges play an important role in magnetic refrigeration systems, both in the cold production cycles, and in the interaction with external environments, including the substance to be cooled. Thus, a thermal study is needed to determine the performance of a magnetic refrigeration system and optimize it. The aim of this section is to focus on the thermal modeling of magnetic AMRR systems.

Most of heat exchanges operating in the magnetic refrigeration are via convection. The convection represents transfer processes performed by the motion of fluids (Bianchi, 2004). In a solid (index 's') in contact, with a fluid (index 'f'), the flow through the wall (index 'w') can be written as:

$$\mathcal{A}\_s \left(\frac{\partial T}{\partial n}\right)\_{\text{us}} = \mathcal{A}\_f \left(\frac{\partial T}{\partial n}\right)\_f = \varphi\_p \tag{19}$$

where : n is the normal to the wall and λ*W mK* ( ) is the thermal conductivity

whereas, the continuity of temperatures can be given by:

$$\left(\left(T\_s\right)\_{w\mathcal{M}} = \left(T\_f\right)\_{w\mathcal{M}}\tag{20}$$

where : ( )*<sup>s</sup> wM <sup>T</sup>* is the temperature of the solid at a point 'M' of the wall and ( )*<sup>f</sup> wM T* represents the temperature of the fluid at this point.

According to Newton, there is a linear relationship between the density of heat flow ϕ and temperature difference Δ= − *TT T <sup>s</sup> <sup>f</sup>* between the solid (*Ts* ) and the fluid (*Tf* ):

$$
\varphi = h \Delta T = h \left( T\_s - T\_f \right) \tag{21}
$$

where: <sup>2</sup> *h W Km* represents the coefficient of heat transfer by convection or simply the convection coefficient.

Using the first law of thermodynamics, by subtracting the mechanical energy, we get the balance of internal energy that gives us the heat equation governing the temperature field at any point in the domain (Janna, 2000)

$$
\rho \mathcal{C}\_p \left( \frac{\partial T}{\partial t} + \mathbf{V} .\mathbf{grad} T \right) = \beta \Gamma \left( \frac{\partial p}{\partial T} + \mathbf{V} .\mathbf{grad} T \right) + P + \Phi + \mathcal{A} \operatorname{div} \left( \mathbf{grad} T \right) \tag{22}
$$

where: <sup>3</sup> ρ *kg m* is he volume density, *Cp J kg K* ( ) is the specific heat, V [*m s*] is the velocity of the fluid, *p* [ ] *Pa* is the pressure, β [1 *K*] is the coefficient of dilatation, Φ [*W* ] is the dissipation function and *P* [*W* ] is the local thermal power produced or absorbed.

For low viscosity fluids and isochors (Janna, 2000), the energy equation reduces to:

$$\rho \, \rho \, \mathbf{C}\_p \left( \frac{\partial T}{\partial t} + \mathbf{V} . \text{grad} \, T \right) = P + \mathcal{A} \, \operatorname{div} \left( \text{grad} \, T \right) \tag{23}$$

Magnetic Refrigeration Technology at Room Temperature 243

, .

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

(a) (b)

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

> Lm [mm]

Values 5 1 0.157 50 573 1000 4185 1.75

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

leq [mm] ρ

*Cp* [J/(kg K)]

MCE [K]

[kg/m3]

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

ef [mm]

has been introduced (programmed) before the demagnetization phase.

<sup>2</sup> ( ) *<sup>f</sup> <sup>t</sup> A dt*

*x* <sup>Δ</sup> <sup>=</sup> <sup>Δ</sup> , 3 .

*f f h S A t m C* = Δ

,

*f*

where: .

*h S A t m C* <sup>=</sup> − Δ 

1

*<sup>m</sup>* 1

**4.2 Results** 

one plate).

<sup>1</sup> 1 () *<sup>f</sup>*

.

*m m*

by permanent magnets B = 1 T.

[ml/s]

Table 2. The parameters used in the simulation.

em [mm]

Parameters D(t)

*t hS A dt <sup>t</sup>*

*f f*

. *<sup>m</sup>*

.

*m m h S A t m C* = Δ

The AMRR model has been implemented using Matlab ® commercial software.

*x mC* <sup>Δ</sup> =− + Δ <sup>Δ</sup>

and 2

if *<sup>p</sup> a C* = λ ρis defined as the thermal diffusivity of the fluid, thus :

$$\frac{\partial T}{\partial t} + \mathbf{V}.\text{grad}T = \frac{P}{\rho \mathbf{C}\_p} + a\Delta T\tag{24}$$

## **4.1 Application to AMRR**

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 one dimensional (1D) approximation.

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

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

$$\begin{cases} m\_f \mathbf{C}\_f \left( T\_f \right) \left( \frac{\partial T\_f}{\partial t} + \dot{d}(t) \frac{\partial T\_f}{\partial \mathbf{x}} \right) = hS \left( T\_m - T\_f \right) \\\\ m\_m \mathbf{C}\_m \frac{\partial T\_m}{\partial t} = hS \left( T\_f - T\_m \right) \end{cases} \tag{25}$$

To solve this system we use the finite difference method. We use a grid of elements that range from 0 to L for the space and from 0 to τ for the time. Thus, the derivatives with 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. Thus the system (25) becomes:

$$\begin{cases} T\_{f\_{(i+1,j)}} = A\_{f1} T\_{f\_{(i,j)}} + A\_{f2} T\_{f\_{(i,j-1)}} + A\_{f3} T\_{m\_{(i,j)}} \\ T\_{m\_{(i+1,j)}} = A\_{m1} T\_{m\_{(i,j)}} + A\_{m2} T\_{f\_{(i,j)}} \end{cases} \tag{26}$$

$$\begin{aligned} \text{where: } & A\_{f1} = \left( 1 - \left( d(t) \frac{\Delta t}{\Delta \mathbf{x}} + \frac{hS}{m\_f \mathbf{C}\_f} \Delta t \right) \right), & A\_{f2} = \left( \dot{d}(t) \frac{\Delta t}{\Delta \mathbf{x}} \right), & A\_{f3} = \left( \frac{hS}{m\_f \mathbf{C}\_f} \Delta t \right), \\\ A\_{m1} = \left( 1 - \frac{hS}{m\_m \mathbf{C}\_m} \Delta t \right) \text{ and } & A\_{m2} = \left( \frac{hS}{m\_m \mathbf{C}\_m} \Delta t \right) \end{aligned}$$

The AMRR model has been implemented using Matlab ® commercial software.
