A Numerical Analysis of an Active Magnetic Multilayer Regenerative Cycle

Dipartimento di Ingegneria Meccanica, Università di Salerno, Salerno, Italia

1. Introduction

Magnetic refrigeration is an emerging technology based on the magneto-caloric effect in solid-state refrigerants [1,2]. In the case of ferromagnetic materials the magnetocaloric effect (MCE) is a warming as the magnetic moments of the atom are aligned by the application of a magnetic field, and the corresponding cooling upon removal of the magnetic field.

Compared to conventional vapour compression systems, magnetic refrigeration can be an efficient and environmentally friendly technology. The high efficiency arises because the analogues to the compression and expansion parts of the vapour compression cycle are accomplished by the magnetization and demagnetization of a magnetic material. Furthermore, the magnetic refrigerant is a solid and has essentially zero vapour pressure and therefore is ecologically sound with no Ozone Depletion Potential (ODP) and zero direct Global Warming Potential (GWP) [3].

The magnetic field of magnetic refrigeration can be supplied by electromagnet, superconductor or permanent magnet, which has no need for compressors with movable components, large rotational speed, mechanical vibration noise, bad stability and short longevity.

Recently, the research for magnetic refrigeration working materials has been greatly expanded. In this paper attention is focused in the near room temperature MCE.

A good material for the refrigeration at room temperature is gadolinium, which is a member of the lanthanide group of elements. At the Curie temperature TC of 294 K, Gd undergoes a second order paramagnetic – ferromagnetic phase transition. There is no detectable magnetic hysteresis in single gadolinium crystals. A variety of Gd – R alloys, where R is another lanthanide metal have been prepared in an attempt to improve the MCE in Gd. Gd can be alloyed with terbium (Tb) [4], dysprosium (Dy) [5], or erbium (Er) [6] to lower the Curie temperature in order to construct a layered regenerator.

The Curie point is significant because at this temperature the magnetic material exhibits its greatest MCE. Therefore, only over a narrow temperature range near the Curie point the magnetic material show a large MCE. Figure 1 shows the adiabatic temperature variation, starting by the temperature value reported in abscissa, that has been obtained during the process of magnetization/demagnetization of the gadolinium considering a magnetic field of 1.5 T and of 0T, respectively. As a result, there is only a small temperature range where an active magnetic regenerator composed of a single magnetic material can maintain its otherwise potentially high performance. One method to span the temperature range is to choose several second order transition alloys and arrange them in the regenerator from the cold end to the hot end in a multi-layer arrangement (AMMR cycle). The alloys have been chosen in such a way that their Curie point follows the regenerator temperature profile. In this paper Gdx Dy1-x alloys are chosen as constituent material for the regenerator bed in the temperature range 260 – 280 K. These materials have a very convenient property to produce layered beds, namely that the Curie temperature changes with the fraction of change of two components. Using alloys it is possible to fabricate a layered bed composed of several magnetic alloys, each placed at the location in the regenerator where the average temperature is near its Curie temperature.

2. Magnetocaloric regeneration

The Active Magnetic Regenerative (AMR) refrigeration cycle is a special kind of regenerator for the magnetic refrigerator, in which the magnetic material matrix works both as a refrigerating medium and as a heat regenerating medium, while the fluid flowing in the porous matrix works as a heat transfer medium. An AMR cycle consists of the four following processes: (1) bed magnetization; (2) iso-field cooling; (3) bed demagnetization; (4) iso-field heating.

The initial and the boundary conditions of each process connect each step of the four sequential processes to allow a cyclical operation of the AMR system.

3. The model of magnetization and demagnetization processes
A homogeneous ferromagnetic material model has been used to characterize the thermal and magnetic behaviours. The basic thermodynamics of the MCE is well known [7]. An entropy balance for the magnetocaloric solid refrigerant and the entrapped fluid in the porous matrix has been performed [8, 9]:

To study the transient behaviour, ignoring the mass of the entrapped fluid compared to the mass of the magnetic material, the temperature variation is valuable integrating the following differential equation:

The derivative of the magnetization in respect to the temperature at constant magnetic field, is obtainable numerically by the function M(B,T) provided by the ferromagnetic homogeneous model with the simplified molecular theory based on the Brillouin function. The thermomagnetic properties of Gdx Dy1-x varying the alloy composition have been calculated using the molecular field theory and the Debye approximation, with the De Gennes factors [10, 11]:

4 The model of the regenerative warm and cold blow processes
The secondary fluids in the simulation are: water-monoethylenglycol mixture (50% by weight), water- monoethylenglycol mixture (34% by weight) and water-1,2 propylenglycol mixture (38% by weight). The analysis and equations in this section are based on the following simplifying assumptions:
1. The temperature of the secondary fluid entering at each end of the refrigerant bed is constant.
2. The axial conduction in the magnetic bed is assumed negligible.
3. The bed is assumed adiabatic towards the environment.
4. The properties of the magnetic material are assumed constant throughout the bed.
5. The secondary fluid is considered incompressible.
6. The secondary fluid velocity is constant during the period of flow blowing.
7. The fluid flow through the bed is parallel and uniform throughout any cross section.
9. The regenerator surface area is evenly distributed throughout its volume.
Based on the above assumptions, an energy balance for the secondary fluid and for the magnetic material can be performed, which results in two partial differential equations:

5 Numerical solution

There is not analytical solution to solve for the equations presented previously. The Runge-Kutta explicit method has been used to solve the equations system. The refrigeration energy and the energy supplied to the environment are calculated according to the following equations:

Using these equations the Coefficient of performance is evaluated:

The presented model don’t take into account the work of the pump.
The Ergun equation [12] reported below allows the evaluation of the pressure drop in secondary fluid flow:

Integrating Equation (7) along the magnetic bed with regard to the time, the pressure drop is evaluated. The work of the pump can be expressed as:

Using these equations the Coefficient of Performance is valuable:

6 Results and discussion

By means of the simulation with the previous equations, integrated with the boundary and the initial conditions, the refrigeration power, the Coefficient of Performance and the temperature profile of the magnetic bed have been obtained. The numerical program simulates layered regenerators made of Gdx Dy1-x alloys. The Curie temperature of the alloy varied with the change of the fraction of the two components. Therefore it is possible to make the bed of different numbers of layer each working at its optimal point selecting the composition of the alloy. An iterative procedure has been adopted in order to determine the optimal composition of each layer of the bed.

The parameters reported in Table 1 are used to carry out the simulation to investigate the effect on cycle performance of layering bed with Gd-Dy alloys. In the simulation the cooling capacity was held constant selecting the appropriate regenerating fluid mass flow rate. In Figure 2 is reported as an example the GdxDy1-x bed for a 6 layers bed varying the composition of the alloy.

Figure 3 shows the COPwpd values varying the number of the bed’s layer. In the graph the zero layer is referred to a bed make of pure Gd. The COP pertaining to a bed made of Gd is low because the Curie temperature is out of the temperature range. In the graph is reported the maximum COP referred to the Carnot cycle and the COP of a vapour compression plant working with the same operating conditions.

The COPwpd of the AMMR cycle is an increasing function of the layer’s number. Indeed, increasing the layers of Gd-Dy alloys, placing each layer at the location where the average temperature is near its Curie temperature, a larger magnetocaloric effect can be obtained. Beyond 6 layers the COPwpd increases slightly. The non layered bed significantly out performs the layered bed and the COP is similar to that of a vapour compression plant. The COPwpd values, for each number of layers, are similar between the three different secondary fluids used in the simulation. The water-1,2 propylenglycol mixture (38% by weight) shows COPwpd values slightly better than the water- monoethylenglycol mixture (34% by weight) (by a mean factor of +10%) and the water-monoethylenglycol mixture (50% by weight) (by a mean factor of + 19%). Indeed, the greater values of the specific heat and of the thermal conductivity of this fluid allow a better heat exchange between the magnetic material and the regenerating fluid.

Comparing the value of the best COPwpd of an 8 layers AMMR cycle with that pertinent to a classical vapour compression plant, the AMMR shows an energetic performance greater than 63%.

In Figure 4 is reported the adiabatic temperature variations of the 1 layer bed along the abscissa x during the magnetization and demagnetization. Along the magnetic bed in each phase a maximum variation of the adiabatic temperature is evidenced corresponding to the Curie temperature of the alloy.

In Figure 5 is reported the adiabatic temperature variations of the 6 layer bed along the abscissa x during the magnetization and demagnetization. Along the magnetic bed a maximum variation of the adiabatic temperature is evidenced corresponding to Curie temperature of the alloy placed in each layer. In this case the bed works with greater adiabatic temperature variations and therefore with a greater magnetocaloric effect producing an increase of the energetic performance of the cycle.

In figure 6 are reported the COP values taking into account the work of the pump. In the porous bed the pressure drops are very strong, therefore the work of the pump is significant and the COP values decreases significantly taking into account the latter contribution. The viscosity of the water-monoethylenglycol mixture (50% by weight) is greater than that pertaining to the other secondary fluids tested in this analysis. Therefore with a greater work of the pump, the COPpd values are much lower. Indeed, the COP of the AMRR cycle working with this mixture as secondary fluid is higher than that of a vapour compression plant only with a regenerator made of a number of layers greater than 4. The best secondary fluid is water- monoethylenglycol mixture (34% by weight). With this fluid the AMMR cycle always over performs a vapour compression cycle (from a minimum of + 4 to a maximum of +59 %).



A heat transfer surface, m2
B magnetic induction, T
c specific heat, J/kgK
COP Coefficient of Performance
D diameter of the regenerator section, m
dp diameter of the particles, m
h heat transfer coefficient, W/m2K
L length of the regenerator, m
M magnetization , A/m
m mass, kg
Q thermal energy, J
S entropy [J/K]
s specific entropy, J/kgK
T temperature, K
t time, s
v specific volume, m3/kg
W work, J
w local velocity, m/s
x space, m
x mass fraction of Gd

Greek symbols

Δ finite difference [-]
ε porosity [-]
η isentropic efficiency [-]
μ viscosity [Pa s]


ad adiabatic
b bed
C Curie
c cold
CF cold water flow
D demagnetization phase
f fluid
h hot
inf undisturbed flow
HF hot water flow
M magnetization phase
M.C. Carnot machine
p particle
ref refrigeration
rej reject


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Table 1 – Model parameters


Figure 1. The gadolinium adiabatic temperature variation versus initial process temperature.

Fig. 2 – Arrangement of the GdxDy1-x alloys in a 6 layers AMMR bed.

Fig. 3 – COPwpd as a function of the layer’s number with GdxDy1-x alloys.

Fig.4 – Adiabatic temperature variation in a 1 layer Gd- Dy bed .

Fig.5 – Adiabatic temperature variation in a 6 layers Gd- Dy bed .

Fig. 6 – COPpd as a function of the layer’s number with GdxDy1-x alloys.