**2.2 Thermodynamic approach**

The absolute entropy, which is a function of temperature and induction in the magnetocaloric material is a combination of the magnetic entropy, the entropy of the lattice and the entropy of the conduction electrons (assumed negligible). It is given by the following equation:

$$S\left(T, B\right) = S\_m + S\_l \tag{1}$$

where S (J K-1) is the entropy (subscripts m and l are respectively for magnetic and lattice entropies), T (K) is the temperature and B (T) is the magnetic filed induction.

In magnetocaloric materials, a significant variation of the entropy can be observed by the application or removal of an external magnetic field. For a given material, MCE depends only on its initial temperature and the magnetic field. The MCE can be interpreted as the isothermal entropy change or the adiabatic temperature change.

The separation of entropy into three terms given in (1) is valid only for second order phase transition materials characterized by a smooth variation of the magnetization as a function of temperature. For first order transitions (abrupt change of magnetization around the transition temperature), this separation is not accurate (Kitanovski, 2005). For most applications, it is sufficient to work with the total entropy which - in its differential form can be given as:

$$dS\left(T, B\right) = \left(\frac{\partial S}{\partial T}\right)\_B dT + \left(\frac{\partial S}{\partial B}\right)\_T dB \tag{2}$$

The specific heat capacity CB (J m-3 K) of the material is given as:

$$C\_B = \left(\frac{\partial S}{\partial T}\right)\_B T \tag{3}$$

This gives:

226 Trends in Electromagnetism – From Fundamentals to Applications

electricity to operate the compressor, magnetic refrigeration can efficiently (and economically) replace compressor-based refrigeration technology. Some potential advantages of the magnetic refrigeration technology over the compressor-based refrigeration are: [1] green technology (no toxic or antagonistic gas emission); [2] noiseless technology (no compressor); [3] higher energy efficiency; [4] simple design of machines; [5] low maintenance cost; and [6] low (atmospheric) pressure (this is an advantage in certain

This chapter is concerned with the magnetic refrigeration technology form the material-level to the system-level. It provides a detailed review of the magnetic refrigeration prototypes available until now. The operational principle of this technology is explained in depth by making analogy between this technology and the conventional one. The chapter also investigates the study of the magnetocaloric materials using the molecular field theory. The thermal and magnetic study of the magnetic refrigeration process using the finite difference

The chapter is organized as follows. Section 2 introduces the magnetocaloric effect and its application to produce cold. It also introduces active magnetic regenerative refrigeration. Section 3 reviews ten various magnetic refrigeration systems and highlights their pros and cons. In Section 4 and 5, the thermal and magnetic study of the magnetic refrigeration process using the finite difference method are explained and the results from the thermal study are

The magnetocaloric effect (MCE) is an intrinsic property of magnetic materials; it consists of absorbing or emitting heat by the action of an external magnetic field (Tishin, 1999). This

 Fig. 1. Magnetocaloric effect (the arrows symbolize the direction of the magnetic moments).

The absolute entropy, which is a function of temperature and induction in the magnetocaloric material is a combination of the magnetic entropy, the entropy of the lattice and the entropy of

*STB S S* ( ) , = + *m l* (1)

the conduction electrons (assumed negligible). It is given by the following equation:

also presented and discussed in detail. Finally, the conclusions are drawn in Section 6.

results in warming or cooling (both reversible) the material as shown in Fig. 1.

applications such as in air-conditioning and refrigeration units in automobiles).

method (FDM) is also explained and are presented and discussed in detail.

**2. The magnetocaloric effect** 

**2.2 Thermodynamic approach** 

**2.1 Definition** 

$$
\left(\frac{\partial S}{\partial T}\right)\_{\mathcal{B}} = \frac{C\_B}{T} \tag{4}
$$

From (2) and (4) we can write:

$$dS\left(T, B\right) = \frac{C\_{\rm B}}{T}dT + \left(\frac{\partial S}{\partial B}\right)\_{T}dB\tag{5}$$

In the case of an adiabatic process (no entropy change 0 Δ = *S* ) the temperature variation can be written as:

$$dT = -\frac{T}{C\_B} \left(\frac{\partial S}{\partial B}\right)\_T dB \tag{6}$$

Using the Maxwell relation given as:

$$
\left(\frac{\partial S}{\partial B}\right)\_T = \left(\frac{\partial M}{\partial T}\right)\_B \tag{7}
$$

where *M* (A m-1) is the magnetization.

We can write:

$$dT = -\frac{T}{C\_B} \left(\frac{\partial \mathcal{M}}{\partial T}\right)\_B d\mathcal{B} \tag{8}$$

Magnetic Refrigeration Technology at Room Temperature 229

( ) 21 21 1 1 coth coth 2 2 22 *<sup>J</sup> J J B x <sup>x</sup> <sup>x</sup>*

where: *J* (N m s) is the total angular momentum, *n* (mol−1) is the Avogadro number, *<sup>J</sup> g* is

( ) ( ) 21 1 ln sinh sinh 2 2 *m J <sup>J</sup> Sx R <sup>x</sup> x xB x*

The lattice contribution can be obtained using the Debye model of phonons (Allab, 2008). It

3ln 1 12

μ

Table 1. Parameters used for applying MFT to the gadolinium.

2 (f) shows the magnetocaloric effect calculated by the MFT.

*D*

*<sup>T</sup> <sup>r</sup> <sup>y</sup> <sup>D</sup> <sup>T</sup> <sup>y</sup> SR e dy <sup>T</sup> <sup>e</sup>*

*J J*

*T T*

<sup>−</sup>

=− − + <sup>−</sup>

where: *TD* (K) is the Temperature of Debye and *R* (J K−<sup>1</sup> mol−1) is the universal gas constant.

In this section the theoretical study based on the MFT developed in the previous section is applied to the gadolinium. Table 1 gives the parameters used to calculate the magnetocaloric

*<sup>B</sup> Bk*

3.5 6.023 1023 2 9.2740154 10-24 1.380662 10-23 4π 10-7 293 184

The numerical solution of equations (14), (15) and (16) allows getting the isotherms of magnetization and its evolution as a function of temperature calculated by the method of Weiss as shown in Fig. 2 (a) and Fig. 2 (b). Fig. 2 (c) represents the total heat capacity calculated from the equation (3) for different levels of induction. The magnetic entropy and its variation with temperature are shown respectively in Fig. 2 (d) and Fig. 2 (e). Finally, Fig.

the Landé factor,

properties.

*J n <sup>J</sup> g*

μ

λ

The magnetic entropy is given by the relationship of Smart (Allab, 2008):

*B x <sup>J</sup>* ( ) is the Brillouin function,

is given by the following equation:

**2.3.1 Application of MFT to gadolinium (Gd)** 

the Permeability of vacuum.

*J J JJ*

+ + = − (16)

μ

<sup>0</sup> (T m A-1) is

*<sup>B</sup>* (J T−1) is the Bohr magnetron, *Bk* (J K-1) is the Boltzmann constant,

<sup>+</sup> = − (17)

3 3

*D*

*T*

1

μ

<sup>0</sup> *TC TD*

(18)

0

is the Weiss molecular field coefficient and

The magnetocaloric effect (the adiabatic variation in temperature) can then be expressed as follows:

$$
\Delta T\_{ad} = -\int\_{B\_i}^{B\_f} \frac{T}{C\_B} \left(\frac{\partial M}{\partial T}\right)\_B dB = MCE \tag{9}
$$

In the case of an isothermal process, the temperature does not change during the magnetization and we can express the entropy as:

$$dS\left(T, B\right) = \left(\frac{\partial S}{\partial B}\right)\_T dB\tag{10}$$

Using the Maxwell relation given by (7), the magnetic entropy change can be expressed as:

$$
\Delta S = \int\_{B\_i}^{B\_f} \left(\frac{\partial M}{\partial T}\right)\_B dB \tag{11}
$$

and the heat saved in this way is transferred to the lattice thermal motion.

#### **2.3 Theoretical approach of MCE: molecular field theory**

The theoretical calculation of the MCE is based on the model of Weiss (MFT: Molecular Field Theory) and the thermodynamic relations (Huang, 2004). To interpret quantitatively the ferromagnetism, Weiss proposed a phenomenological model in which the action of the applied magnetic field **B** was increased from that of an additional magnetic field proportional to the volume magnetization density *Bv* as:

$$B\_{\upsilon} = \mathcal{A}\mu\_0 M \tag{12}$$

The energy of a magnetic moment is then:

$$E = -\mu \left( B + B\_{\upsilon} \right) \tag{13}$$

The magnetic moments will tend to move in the direction of this new field. Adapting the classical Weiss-Langevin classical calculations to a system of quantum magnetic moments, one finds:

$$M(\mathbf{x}) = \eta \mathbf{g}\_{\parallel} \mu\_{\oplus} B\_{\parallel}(\mathbf{x}) \tag{14}$$

where:

$$\alpha = \frac{\lg\_{\text{J}} \mu\_{\text{B}} \left( \text{B} + \text{\AA} \mu\_{\text{0}} M(\text{x}) \right)}{k\_{\text{B}} T} \tag{15}$$

and

228 Trends in Electromagnetism – From Fundamentals to Applications

The magnetocaloric effect (the adiabatic variation in temperature) can then be expressed as

*B B B T M <sup>T</sup> dB MCE C T* <sup>∂</sup> Δ =− <sup>=</sup>

In the case of an isothermal process, the temperature does not change during the

*<sup>S</sup> dS T B dB B* <sup>∂</sup> <sup>=</sup>

Using the Maxwell relation given by (7), the magnetic entropy change can be expressed as:

*f*

*B*

*i*

and the heat saved in this way is transferred to the lattice thermal motion.

**2.3 Theoretical approach of MCE: molecular field theory** 

proportional to the volume magnetization density *Bv* as:

The energy of a magnetic moment is then:

one finds:

where:

and

*B B <sup>M</sup> <sup>S</sup> dB T* <sup>∂</sup> Δ =

The theoretical calculation of the MCE is based on the model of Weiss (MFT: Molecular Field Theory) and the thermodynamic relations (Huang, 2004). To interpret quantitatively the ferromagnetism, Weiss proposed a phenomenological model in which the action of the applied magnetic field **B** was increased from that of an additional magnetic field

> *B M <sup>v</sup>* = λμ

*E BB* ( ) *<sup>v</sup>* =− + μ

The magnetic moments will tend to move in the direction of this new field. Adapting the classical Weiss-Langevin classical calculations to a system of quantum magnetic moments,

> *M*( ) *x n* = *gJ BJ* μ

> > μ

*x*

*J B* ( ) <sup>0</sup> ( ) *B Jg B Mx*

*k T*

 λμ *T*

<sup>∂</sup> (9)

<sup>∂</sup> (10)

<sup>∂</sup> (11)

0 (12)

*B x*( ) (14)

<sup>+</sup> <sup>=</sup> (15)

(13)

*f*

*B*

*ad*

magnetization and we can express the entropy as:

*i*

( ) ,

follows:

$$B\_{I}(\mathbf{x}) = \frac{2J+1}{2J} \coth\left(\frac{2J+1}{2J}\mathbf{x}\right) - \frac{1}{2J} \coth\left(\frac{1}{2J}\mathbf{x}\right) \tag{16}$$

where: *J* (N m s) is the total angular momentum, *n* (mol−1) is the Avogadro number, *<sup>J</sup> g* is the Landé factor, μ *<sup>B</sup>* (J T−1) is the Bohr magnetron, *Bk* (J K-1) is the Boltzmann constant, *B x <sup>J</sup>* ( ) is the Brillouin function, λ is the Weiss molecular field coefficient and μ<sup>0</sup> (T m A-1) is the Permeability of vacuum.

The magnetic entropy is given by the relationship of Smart (Allab, 2008):

$$S\_m(\mathbf{x}) = R \left( \ln \left( \sinh \left( \frac{2J+1}{2J} \mathbf{x} \right) \bigg/ \sinh \left( \frac{1}{2J} \mathbf{x} \right) \right) - \mathbf{x} \mathbf{B}\_I(\mathbf{x}) \right) \tag{17}$$

The lattice contribution can be obtained using the Debye model of phonons (Allab, 2008). It is given by the following equation:

$$S\_r = R\left(-3\ln\left(1 - e^{\frac{-T\_D}{T}}\right) + 12\left(\frac{T}{T\_D}\right)^3 \int\_0^{\frac{T\_D}{T}} \frac{y^3}{e^y - 1} dy\right) \tag{18}$$

where: *TD* (K) is the Temperature of Debye and *R* (J K−<sup>1</sup> mol−1) is the universal gas constant.
