Improving Designs of Halbach Cylinder-Based Magnetic Assembly with High- and Low-Field Regions for a Rotating Magnetic Refrigerator

Abstract

The present paper investigates the generation of the alternating almost zero and strong homogeneous magnetic fields for rotary magnetic refrigeration. In order to achieve an alternating magnetic field with eight regions, a soft magnetic rod is inserted in the bore. Four high-flux-density regions (FDRs) for magnetization and four low-flux-density regions for demagnetization of magnetocaloric materials are obtained by the proposed design. The design procedure for the four-pole structure and its implementation using 3D finite-element simulation are presented. To meet the predefined requirements, some magnet segments are replaced with high-permeability soft magnetic material. The proposed magnetic design for the rotary refrigerator allows good field distribution in the air gap, a high ratio of high-field-to-permanent-magnet volume, a minimized low-field volume, reduced magnet usage to the permanent-magnet volume, reduction of the amount of magnet material used, and increased flux density between the low- and high-field regions.

1. Introduction

Magnetic refrigeration (MR) technology is becoming increasingly suitable for mobile applications due to its advantages, such as avoiding the use of hazardous, ozone-depleting chemicals and greenhouse gases [1,2]. This technology offers a high energy efficiency while respecting the environment aspects [3,4]. MR uses the magnetocaloric effect (MCE) [5,6], during which the temperature increases in some parts of a material when exposed to a magnetic field and decreases when the magnetic field is removed [2,4]. This temperature change is called the adiabatic temperature change (ΔTad). This temperature change depends on the magnetic field, which is greatest near the Curie temperature (Tc). For different magnetocaloric materials, the Tc is different [2]. Currently, the most efficient magnetocaloric materials, such as gadolinium and its alloys, exhibit a maximum (ΔTad) of approximately 4 K under a 1 T magnetic field near their Curie temperature [7].

Therefore, to produce a sufficiently large temperature span in order to satisfy the refrigeration purposes, a cascaded system including different materials should be implemented. This used system is called active magnetic regeneration (AMR) [6,7,8,9,10].

Very recently, many research works have reported on the design of refrigerators based on MCE with different temperature ranges. They are designed to provide cooling and an adequate temperature span.

The authors of [11] described a magnetic refrigerator using an adiabatic magnetic refrigerator. This device used gadolinium as a magnetic material in the experimental test. The presented experiment demonstrated the efficiency of this device. To improve the design of a concentric Halbach cylinder magnet, the authors in [12] proposed an original method by applying two general schemes. The designed prototype produces a maximum of 1.24 T in the high-field region and 0.08 T in the low-field region. In [13], the design of a five-permanent-magnet structure, which consists of optimizing the magnetic-flux density, was analyzed using numerical simulations. The proposed design provided a higher magnetic-flux density using the least amount of magnet material. Another study in [14] proposed a rotary magnetic refrigerator, which generated 1.2 T of value in the air gap. Similarly, the work presented in [15] dealt with the design of a rotary magnetic refrigerator. Some important designs for permanent-magnet magnetic refrigerator were carried out in [16], and are characterized by compactness, simple sealing, ease of use, and high operating frequency.

A comprehensive paper on the design of a magnetic refrigerator is reported in [17]. The authors studied the removal of unnecessary magnet material in the original design to increase the difference in magnetic-flux density between a low and high magnetic field region. The final design reduced 42% of the magnetic material. Moreover, the authors in [18] presented a continuously rotating active magnetic regenerator using elongated plates of the perovskite La0.67Ca0.33xSrxMn1.05O3 material.

On the other hand, a six-pole concentric Halbach cylinder for rotary magnetic refrigerators was designed in [19] and tested using numerical simulation in COMSOL Multiphysics software. In the developed design, magnet material profiles are optimized and adjusted to improve their performance. A total of 20.2% of magnet material was replaced by soft iron to optimize the magnet material used in this devise. In order to calculate the optimal direction of the remanent flux density and optimal shape for segmentation of a two-dimensional magnetic field, an optimization algorithm approach was proposed in [20]. Therefore, in [21], a novel two-pole device including a stator and rotor system was designed. The designed magnetic refrigerator produced 1 T in a high-flux-density region. The study developed in [22] presented an optimization method to obtain the optimal magnetization direction and optimal shape for a rotary magnetic refrigerator. The obtained results showed a strong performance in terms of the globally optimal shape of 3D permanent-magnet assemblies.

Moreover, the substitution of the magnet material, such as neodymium–iron–boron, by soft magnetic low-carbon steels in [23] gave 2 T as the maximum flux density in the air gap of the rotary refrigerator magnetic field. An analytical method to optimize a two-pole system for a magnetic refrigerator was proposed in [24]. Bjørk et al. in [25] presented a topology optimization for a Halbach cylinder with iron, and as a result, this optimized method increased 15% in magnetic efficiency. Another paper dealt with the multi-material topology optimization of iron and magnet segments [26]. On the other hand, the study in [27] described an effective design for a magnetic refrigeration device, which used an octagonal orientation like a Halbach cylinder. In this system, the soft magnetic material is FeVCo, which increases the average flux. The work presented in [28] investigates the Halbach optimization for achieving field homogeneity by varying the remanence of its magnetic materials. Di Gerlando et al. [29] proposed an innovative closed-form analytical solution for the magnetic field of a permanent magnet with arbitrary magnetization, providing a fast and accurate alternative to FEM simulations for modeling the air-gap field in Halbach-type structures. Based on flux modulation theory, Lin et al. proposed an axial modular flux reversal permanent-magnet machine (AM-FRPM) to improve torque density and reduce torque ripple. Two magnetization modes were analyzed and experimentally validated to confirm the effectiveness of the proposed design.

The present paper proposes a new four-pole design configuration for a rotary magnetic refrigeration system. Compared to previous designs [2,17,19,21] that replaced only part of the permanent-magnet material with soft magnetic areas, the new design uses a 3D-optimized four-pole rotating Halbach structure to minimize flux losses and magnetic material consumption. The strategy of substituting non-magnetic Teflon further improves both efficiency and compactness, providing a new balance between magnetic performance and structural simplicity. The design procedure steps are presented to apply any requirements of the used materials in this design. The proposed devises are applied in a 3D Halbach cylinder, which is optimized by removing the unnecessary magnetic segmentations. The proposed device generates the maximum flux density of 1.6 T in four high regions in the air gap. In addition, this scheme guarantees a value of almost zero for the flux density in the four lower regions. In order to further optimize the magnetic material used in this device, an important part is removed and replaced with Teflon material. The obtained results using ANSYS Maxwell 16.0 software confirm the performance of the suggested rotary magnetic refrigeration. The paper is structured as follows: Section 2 presents the prototype design. Section 3 introduces the physical model of the improvement scheme. Section 4 derives the requirements on magnet design. The optimization procedure and its implementation are presented in Section 5. Section 6 gives the simulation procedure. The simulation results are presented and discussed in Section 7. Finally, the last section is the conclusion of this paper.

2. Prototype Design

The prototype design consists of a rotating cylinder of magnetocaloric material FeNdB. Two concentric multipole magnet assemblies are arranged around and inside the cylinder. Therefore, as the cylinder rotates, each part of it successively experiences alternating high- and low-magnetic-field zones (see Figure 1). To ensure the circulation of the heat-transfer fluid, the regenerator is composed of thin plates of magnetocaloric material mounted radially, thus creating controlled flow channels. This geometry minimizes pressure loss through the regenerator, while allowing the regenerator to be assembled with selected spacing and plate thickness [30,31,32].

Gadolinium may appear as a reference material in the literature, but in our prototype simulations, the permanent-magnet assembly is exclusively composed of FeNdB. The mention of Gd is solely contextual and does not reflect the materials used in the simulations. In this prototype, a parallel table support is essential to maintain the equipment’s alignment and ensure proper operation. Mechanical bearings are employed to guide the shaft’s rotation (see sectional view Figure 1).

3. Physical Model

The concentric Halbach cylinder design proposed in this work is based on an ideal four-pole configuration. This device consists of two cylinders; the first is the inner magnet (internal cylinder), which is expressed by two radiuses $r_{in}$ and $r_{ex}$; the second is the outer magnet (external cylinder), which is expressed by radius $R_{in}$ and $R_{ex}$ radiuses. The current concentric Halbach cylinder shown in Figure 2 consists of a cylindrical magnet with an air gap between the internal cylinder and the external cylinders. The internal and external cylinders are magnetized such that they have uniform magnitudes of remanent flux density in polar coordinates [33].

$$\begin{array}{c}\hfill B_{rem,r}=B_{rem}cos\left(h\mathsf{\Psi }\right),B_{rem,\varphi }=B_{rem}sin\left(h\mathsf{\Psi }\right)\end{array}$$

(1)

The notation $B_{rem,r}$ denotes the remanent flux-density magnitude, h is an integer wave number, and $\mathsf{\Psi }$ is the peripheral angle. A negative value of h generates a field that is directed outwards from the cylinder, whereas a positive value of h produces a field that is directed inward toward the device bore. The current Halbach cylinder has four low- and four high-flux-density regions. These regions can be created by having an external and internal cylinder, whose wave numbers equal $h=2$ and $h=-2$, respectively. The dimensions of the Halbach cylinder proposed in this work are $R_{in}=150$ mm, $R_{ex}=220$ mm, $r_{ex}=120$ mm, $r_{in}=10$ mm, which are illustrated in Figure 2.

In this work, we define:

$$\Delta B={\overline{B}}_{\mathrm{opt}}-{\overline{B}}_{\mathrm{ref}}$$

(2)

where ${\overline{B}}_{\mathrm{opt}}$ and ${\overline{B}}_{\mathrm{ref}}$ denote the volume-averaged magnetic-flux density within the regenerator region for the optimized and the reference magnet configurations, respectively.

4. Requirements on Magnet Design

In this subsection, the requirements of the concentric Halbach cylinder magnet segmentations are formulated to show clearly the magnetocaloric effect in the proposed scheme. The Halbach magnet array must produce the low- and high-flux density over a large volume in the magnet device. For mobile applications such as magnetic refrigeration, and for reasons of weight and cost, it is essential to optimize the available space and ensure continuous use of both high- and low-field regions [34,35]. The ratio between the volume of magnetocaloric material and the volume of magnetic material used in the concentric Halbach cylinder should be maximized for the reasons of cost and weight. In addition, in the device design shown in Figure 3, a geometrical simplicity of soft and hard magnet segmentations is required. Additionally, some parts of the design can be replaced by alternative materials characterized by low cost and low weight compared to the soft and hard magnetic components. The requirements of magnet design can be formulated as follows:

• High magnetic flux density over two large regions
• Very low-magnetic-flux density over two large regions
• Homogeneous field distribution within two high- and two low-field regions
• Air-gap volume maximized ratio to the volume of the magnet
• Magnetocaloric material continuous use
• Replacement of some hard magnet components with another low-cost material

5. Optimization Procedure and Its Implementation

To optimize the magnetic flux within a specific region of the system, the field properties can be exploited [17]. For geometrical simplicity, the design optimization shown in Figure 3 is based on regular permanent magnets. Furthermore, each segment in the design system depicts the direction of magnetization. In [17], the authors show that the removal of the $A_z$ equipotential encompasses a low flux density in a given region. Thus, to minimize flux density in certain regions of the design, the magnet material can be replaced with soft magnetic material, air, or Teflon material. Also, the cost and weight of the system can be enhanced using this method. Figure 4 shows the flow diagram of the design procedure used in the paper. It consists of four configurations: the concentration of magnetic-flux density; reduction of magnetic-flux density in low regions; optimization of magnet material with reduction of the magnetic-flux density in low regions; and optimization of the magnet operating point by including another low-cost material and reduction of leakage flux.

5.1. Concentration of Magnetic-Flux Density

Concentrating the magnetic-flux density in specific parts of the design is essential for improving flux distribution within the air gap. For this purpose, two near-zero field regions and two high-field regions were created. The improvement designs shown in Figure 5, called, respectively, Imp. Schemes 1, 2, and 3, are achieved by replacing some special segmentations with important soft magnet materials. The soft magnet material reduces the magnet resistance [19]. Three modifications to the original design were implemented, as shown in Figure 5. To increase the magnetic field in the high regions, certain parts of the magnet design were replaced with high-permeability soft magnetic materials. Also, a low-magnetic-flux density in a low field is achieved.

5.2. Reduction of Magnetic-Flux Density in Low Regions

Achieving near-zero field regions presents a challenge in magnetic refrigeration. In this section, two improvement schemes (Figure 6) are proposed. Specific portions of the magnet material are removed and replaced by air to achieve near-zero magnetic-flux density in selected regions of the system. This improvement not only gives the low-magnetic-flux density but also allows the reduction of the magnetic material used in the design.

5.3. Augmentation of Magnetic-Flux Density in High Regions with Concentration of the Magnetic-Flux Density

The design improvements presented in Figure 7 aim to achieve a significant enhancement in high-magnetic-flux-density regions. Also, some magnet materials are replaced by soft materials due to the concentration of the distribution of the magnetic-flux density.

5.4. Optimization of Magnet Material with Reduction of the Magnetic-Flux Density in Low Regions and Augmentation of Magnetic-Flux Density in High Regions

To achieve both high and low-magnetic-flux densities while reducing overall weight and cost, the volume of magnetocaloric and magnetic material should be minimized. For this, some important parts of the design are removed and replaced with Teflon.

Figure 8 shows the improvement designs with Teflon.

5.5. Optimization of Magnet Operating Point by Including Another Low-Cost Material and Reduction of Leakage Flux

Parts (a) to (d) of Figure 9 show the proposed designs for a magnetic refrigerator by improving the distribution of the magnetic-flux density in the low regions. Some parts of the soft or hard magnet material are removed and replaced with air.

6. Simulation Procedure

The commercial software ANSYS Electromagnetics Suite (version 16.1.0), incorporating ANSYS Maxwell 2015.1.0, was adopted to validate the proposed designs. This software was selected for its straightforward integration with the system’s 3D CAD model. The ANSYS software has been used for many applications, such as electric motor development. Therefore, there exists a wide experience to evaluate the use of this software. ANSYS Maxwell has capabilities in 2D and 3D electromagnetic field simulation due to its finite-element method. The different designs proposed in this paper are created in Maxwell. The characteristics of the material used in the simulations are illustrated in

Table 1. Materials used in finite-element simulations.

Parameter	FeNdB	AISI 1010
Remanence [T]	1.44	–
Coercivity [kA/m]	836	–
Relative permeability [-]	1.04457	2000–5000
Conductivity [S/m]	265,000	6,000,000
Density [kg/m3]	7500	7870

. The solver error is monitored by the percentage errors and the absolute energy.

• Nature of the permanent magnet: The device uses Neodymium–Iron–Boron (FeNdB) permanent magnets, grade N52, due to their high remanence and energy product.
• Temperature dependence: The remanent flux density of FeNdB magnets decreases linearly with temperature, approximately −0.11% per °C, consistent with data from Kresse et al. [36].
• Temperature-related risk: FeNdB magnets may experience reversible demagnetization when the combined temperature and external field exceed the intrinsic coercivity, which was checked to remain within the safe operating range in all simulated cases.

In this work, we used the finite-element method (FEM) (Figure 10) as the numerical approach for solving the governing equations. The device was meshed using tetrahedral (tetra-type) elements, ensuring adequate spatial resolution in regions with high field gradients. The mesh parameters were selected based on a convergence study to balance computational efficiency and accuracy; the maximum element size was set to 2 mm, and the convergence criterion was 0.5% on magnetic energy. To ensure a physically realistic simulation, a field decay condition was imposed far from the device, such that the field intensity tends toward zero, following a Neumann-type decay consistent with the physical boundary of the problem.

7. Simulation Results and Discussion

In this section, the numerical simulation results are presented to validate the proposed designs for rotary magnetic refrigeration. Moreover, the replacement of inner and outer magnet parts with low-carbon steel, air, or Teflon was performed. In the permanent magnets with high poles, both low-field and high-field regions are required to demagnetize and magnetize magnetocaloric materials repeatedly in the air gap. In addition, the problem of magnetic-flux leakages in the system is addressed. The original design shows that the average B is equal to 1.66632 T and 0.178689 T for the low region and high region, respectively. Therefore, this section will address the problem of minimizing magnetic-flux leakages. On another hand, the magnet material used in the original design will be minimized and replaced by a low-cost material due to reasons of weight and the cost of the system.

The proposed prototype is designed to be made from FeNdB segments and AISI 1010 steel inserts [36]. These materials are commercially available and can be machined at low cost. The configuration can be reproduced for a rotor with a radius of 150 mm, with Hall probe instrumentation. This experimental phase is currently being planned. The device is designed to be manufactured in a laboratory using standard NdFeB N52 segments and AISI 1010 mild steel concentrators. The dimensions chosen (Rex = 220 mm, Rin = 150 mm) allow for simple mechanical assembly. Experimental field measurements using Hall sensors and a Lake Shore 475 3D magnetometer are planned for future validation. The concept remains compatible with real-world applications in compact rotary refrigerators.

Figure 11 and Figure 12 show the results of the original design presented in Figure 2. The results indicate two high-flux and two low-flux regions. The objective is to further increase the difference between these regions. The results presented in these sections are the results of a 3D electrostatic simulation in the ANSYS Maxwell software. Figure 12 depicts the angular distribution of magnetic-flux density in the air gap for the original design. It can be seen that the magnetic-flux density in the low-field region is about 0.178689 T and in the high-field region is only 1.66632 T. In order to achieve the concentration of magnetic-flux density, low-carbon steel is used as a soft magnetic material in the configuration presented in Figure 5. The results plotted in Figure 13 show that the soft magnetic material enhances the magnetic-flux density in the high-field region, with almost zero in the low-field region, as demonstrated in Figure 13. From these results, we can show that the Improvement Schemes 1, 2, and 3 have the same value of magnetic-flux density but can minimize the magnetic material used in the system.

The results of the configurations presented in Figure 6 are depicted in Figure 14. It can be observed from these results that the magnetic-flux density in the high-field region is 1.581 T and in the low-field region is about 0.02 T.

The aim is to reduce the leakage of magnetic-flux density, which is obtained in these configurations. Moreover, the magnetic-flux density in some parts of the design is not homogeneous. This represents a minor contribution of these configurations to the overall magnetic field.

Figure 15 shows the improvement scheme designed in Figure 7. The scheme design enhances the field distribution in the air gap, where the magnetic-flux density in the low-field region is equal to 0.01 T and in the high-field region is equal to 1.63 T.

Figure 16a,b shows the effect of applying the improvement schemes presented in Figure 8. It can be seen from the 2D simulation that the replacement of some parts of the inner magnet with Teflon minimizes the magnet material used. Compared to previous works such as Bjørk et al. [17] with ΔB = 1.45 T, You et al. [19] with ΔB = 1.52 T, and Celik et al. [27] with ΔB = 1.50 T, the current design achieves ΔB = 1.57–1.62 T while reducing the magnet volume by up to 45%. This represents an improvement of approximately 12% in magnetic efficiency per unit of magnetic mass. The reduction in magnet material is quantified as:

$$R_V={\displaystyle \frac{V_{\mathrm{ref}}-V_{\mathrm{opt}}}{V_{\mathrm{ref}}}}\times 100\%$$

(3)

where $V_{\mathrm{ref}}$ is the total volume of permanent magnet in the reference configuration and $V_{\mathrm{opt}}$ is the total volume in the optimized design.

Figure 17a–d depicts four versions of the improvement schemes in terms of reducing leakage flux and using a low-cost material in the system. Using these configurations, the cost of the design system is reduced, and with excellent weight.

Figure 18 and Figure 19 show a comparison of the different designs in terms of magnetic-flux density. It can be seen that Improvement Schemes 6, 7, and 8 have the same value of maximum magnetic-flux density (T) and can be classified into simple magnetic circuits. For cost reasons, Improvement Scheme 7 used the minimum amount of the magnetic material with the same performance. Also, the minimum field in the annular gap of this configuration is almost zero.

8. Conclusions

This paper discussed the design challenges of a multipolar rotary magnetic refrigeration system. Several improvement schemes were presented to enhance the difference between low- and high-flux-density regions. Many scenarios are proposed to improve this device. In this study, the design system with four poles is developed and enhanced by the numerical simulation using ANSYS Maxwell software. Magnet material consumption was used as an evaluation criterion. In the proposed improvement scheme, the magnetic material was reduced by 45%. In addition, the difference between a high and a low region is increased. When replacing the magnet material (Nd–Fe–B permanent magnets) with low-carbon steel as the soft magnetic material, the difference in flux density increased. The device proposed in this work is easy to manufacture, reduces magnetic-flux density in low regions, and enhances it in high regions. Also, the important parts of the magnetic material of the original design are removed.

Future work will include experimental validation of the proposed configuration using FeNdB segments and AISI 1010 steel inserts. The prototype will be manufactured using 3D printing with a rotor radius of 150 mm and characterized using Hall-effect sensors and a 3-axis Gaussmeter. These experiments will confirm the simulated magnetic-flux distribution and the expected 45% reduction in magnetic mass.