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

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This paper presents a numerical study on improving the design of a Halbach cylinder–based magnetic assembly for a rotating magnetic refrigerator. The authors propose a four-pole concentric Halbach configuration and introduce twelve improvement schemes involving partial substitution of permanent magnet regions with soft magnetic, air, or Teflon materials. Using numerical simulations (ANSYS Maxwell), they aim to enhance the difference between high and low magnetic flux density regions and reduce the total amount of permanent magnet material.

While the topic is relevant and the simulation work shows considerable effort, the manuscript requires substantial revision before being suitable for publication. It should be rewritten and supplemented with deeper analysis and clearer quantitative evaluation prior to submission to MDPI Magnetism or similar journals.

1. The idea of substituting permanent magnet material with soft magnetic or non-magnetic regions has been explored in previous studies. The manuscript does not clearly explain what is unique about the proposed configuration or why this particular design is significant in the introduction part.
2. The paper does not provide detailed metrics such as field uniformity, flux density difference, energy efficiency, or material utilization ratio.
3. No quantitative comparison is made between the proposed design and existing models reported in the literature.
4. The results rely entirely on simulation without any experimental verification or at least discussion of the design’s practical feasibility.
5. Several references appear as “[?]” and the numbering is out of order. The reference list needs careful revision.
6. The discussion only states that flux differences were improved and magnet usage reduced, without detailed justification or quantitative validation to support these claims.

In summary, the work demonstrates some engineering effort but lacks the clarity, and novelty required for a journal publication. A thorough rewrite with stronger quantitative analysis, comparison to prior designs, corrected language and references, and—ideally—some experimental verification is strongly recommended before resubmission.

Author Response

We would like to sincerely thank the reviewers for their invaluable and constructive comments, which have significantly improved the scientific quality, clarity, and rigor of the manuscript entitled:

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

The comments received have been carefully reviewed and incorporated into the new version of the manuscript.

We have:

• we have completely rewritten the introduction in order to better situate our contribution within the existing literature, in accordance with the journal's recommendations.

Di Gerlando, A.; Negri, S.; Ricca, C. A Novel Analytical Formulation of the Magnetic Field Generated by Halbach Permanent Magnet Arrays. Magnetism 2023, 3, 280-296. doi: 10.3390/magnetism3040022Di Gerlando, A.; Negri, S.; Ricca, C. A Novel Analytical Formulation of the Magnetic Field Generated by Halbach Permanent Magnet Arrays. Magnetism 2023, 3, 280-296. doi: 10.3390/magnetism3040022

Lin, S.; Chang, L.; Su, P.; Li, Y.; Hua, W.; Shen, Y. Research on High-Torque-Density Design for Axial Modular Flux-Reversal Permanent Magnet Machine. Energies 2023, 16, 1691. doi: 10.3390/en16041691

Li, B.; Zhang, J.; Zhao, X.; Liu, B.; Dong, H. Research on Air Gap Magnetic Field Characteristics of Trapezoidal Halbach Permanent Magnet Linear Synchronous Motor Based on Improved Equivalent Surface Current Method. Energies 2023, 16, 793. doi: 10.3390/en16020793

Kresse, T.; Martinek, G.; Schneider, G.; Goll, D. The Field-Dependent Magnetic Viscosity of FeNdB Permanent Magnets. Materials 2024, 17, 243. doi: 10.3390/ma17010243

• clarification of the numerical methodology (ANSYS Maxwell 3D FEM) with simulation parameters.
• linguistic correction and updating of references.
• and the addition of new technical elements concerning materials and experimental feasibility.

Response to Reviewer 1 Comments

We sincerely thank Reviewer 1 for the constructive and detailed feedback.
The comments helped us substantially improve the structure, clarity, and scientific depth of the paper. All points have been carefully addressed in the revised version, Detailed responses to each comment are provided below.

Please find below the detailed responses, including the corresponding revisions in green and the English-language corrections in red in the re-submitted files.

This paper presents a numerical study on improving the design of a Halbach cylinder–based magnetic assembly for a rotating magnetic refrigerator. The authors propose a four-pole concentric Halbach configuration and introduce twelve improvement schemes involving partial substitution of permanent magnet regions with soft magnetic, air, or Teflon materials. Using numerical simulations (ANSYS Maxwell), they aim to enhance the difference between high and low magnetic flux density regions and reduce the total amount of permanent magnet material.

While the topic is relevant and the simulation work shows considerable effort, the manuscript requires substantial revision before being suitable for publication. It should be rewritten and supplemented with deeper analysis and clearer quantitative evaluation prior to submission to MDPI Magnetism or similar journals.

1. The idea of substituting permanent magnet material with soft magnetic or non-magnetic regions has been explored in previous studies. The manuscript does not clearly explain what is unique about the proposed configuration or why this particular design is significant in the introduction part.

We would like to thank the reviewer for this valuable comment. We agree that several previous studies have explored replacing parts of the permanent magnetic material with soft magnetic or non-magnetic regions to improve flux distribution and reduce material usage. However, the novelty of our work lies in the specific four-pole rotating Halbach configuration, which combines 3D optimization of magnetic segment removal with Teflon substitution to minimize flux leakage and maintain high magnetic contrast.

Our contribution differs from previous studies (e.g., Bjørk et al. 2011; Lorenz et al. 2017) in three key aspects:

• Four-pole 3D concentric Halbach configuration optimized specifically for rotary magnetic refrigeration, whereas earlier works mainly focused on two- or six-pole 2D arrangements.
• Systematic optimization across twelve improvement schemes, combining low-carbon steel, air, and Teflon substitutions to balance cost, weight, and flux performance—no prior study investigated such hybrid replacements simultaneously.
• Quantitative reduction of magnetic material (≈ 45 %) while maintaining the same flux contrast (1.6 T / ≈ 0 T).

To provide clarity to this contribution, we have revised the introduction (page 3, lines 84-89) to explicitly highlight these unique aspects and emphasize the motivation and significance of the proposed design relative to existing work.

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.

2. The paper does not provide detailed metrics such as field uniformity, flux density difference, energy efficiency, or material utilization ratio.

We thank the reviewer for this observation.

In the revised manuscript, we explicitly specify that the magnets used are Neodymium–Iron–Boron (NdFeB or FeNdB) permanent magnets, grade N52, with remanence = 1.44 T, coercivity = 836 kA/m, and relative permeability = 1.04457 (see Table 1, page 8, Lines 215). This information ensures reproducibility and consistency with typical Halbach-based systems.

3. No quantitative comparison is made between the proposed design and existing models reported in the literature.

We have added a Quantitative performance comparison (page 11, lines 271-275):

The improvement in the magnetic field achieved in the proposed designs is achieved through two complementary mechanisms:

1. Flux concentration in high-field regions by the low-carbon-steel inserts, which locally reduce magnetic reluctance and guide the magnetic lines.
2. Flux blocking in low-field regions due to non-magnetic Teflon, which eliminates leakage paths and maintains a quasi-zero flux density.

These effects generate higher contrast and better uniformity in active areas.

Compared to previous works 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.

4. The results rely entirely on simulation without any experimental verification or at least discussion of the design’s practical feasibility.

We acknowledge the importance of this comment.

We have added a discussion of experimental feasibility (page 10, line 237-245):

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.

5. Several references appear as “[?]” and the numbering is out of order. The reference list needs careful revision.

All references have been reviewed, renumbered, and verified.

Errors such as “[?]” have been corrected, and citations are now numbered from [1] to [36].

Each reference has been reformatted according to MDPI style.

6. The discussion only states that flux differences were improved and magnet usage reduced, without detailed justification or quantitative validation to support these claims.

We thank the reviewer for this essential observation.

Respond to comment 4 (experimental feasibility).

We have added text to the end of the conclusion (pages 12-13, line 298-302):

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.

In summary, the work demonstrates some engineering effort but lacks clarity, and novelty required for a journal publication. A thorough rewrite with stronger quantitative analysis, comparison to prior designs, corrected language and references, and—ideally—some experimental verification is strongly recommended before resubmission.

We sincerely thank the reviewer for emphasizing the need for stronger quantitative analysis and clear novelty.

The revised paper now provides:

• A clear statement of originality and engineering significance,
• Comprehensive quantitative metrics and comparisons,
• Corrected references and figures, and
• A discussion of experimental feasibility.

 

We believe that these improvements make the manuscript substantially stronger and suitable for consideration by MDPI Magnetism or similar journals.

 

 

 

 

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

​magnetism-3978785

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

The authors presented interesting results on the design of Halbach cylinders for use in magnetic refrigeration. Halbach cylinders offer clear advantages, as they can generate magnetic fields in the range of 1–1.5 T, which is often sufficient to realize the magnetocaloric effect, and they minimize eddy current losses. Several examples of refrigerators based on Halbach-cylinder magnetic s​ystems can be found in the literature. The concept itself is not new (e.g., https://doi.org/10.1016/j.ijrefrig.2006.07.025, https://doi.org/10.3390/en14154662, https://doi.org/10.1016/j.ijrefrig.2024.04.022), and the approach has already been implemented. A closely related article, “Designing of Halbach Cylinder Based Magnetic Assembly for a Rotating Magnetic Refrigerator. Part 1: Designing Procedure”, was published in the International Journal of Refrigeration, 73 (2017), 246–256. However, the authors’ proposal to use low-carbon steel concentrators with a modified design is potentially interesting from an engineering perspective.

The simulation results are also noteworthy, serving as an example of solving this problem in ANSYS Maxwell. Nevertheless, the manuscript, in its current form, cannot be accepted due to several critical issues:

1.     The authors refer to a “regular permanent magnet” without specification. It is unclear which type of magnet is meant. Typically, NdFeB or SmCo magnets are used, with the magnetization vector rotating circumferentially. Coercivity, remanence, and other essential parameters are not provided.​

2.     The choice of magnetocaloric material for this configuration is not justified. Typically, gadolinium is used due to its Curie temperature near room temperature, though alternatives exist.

3.     Literature citations are inaccurate or missing. Many references are replaced with question marks (e.g., [?] on pp. 46, 58, 70, 141, 143), making verification impossible. Furthermore, the introduction begins with references 3 and 5, skipping 1 and 2.

4.     Some statements are ambiguous, e.g., “the best magnetocaloric materials exhibit a maximum temperature change of 4 K in a 1 T magnetic field” (p. 25). It is unclear which materials are meant and according to what criteria.

5.     The manuscript suggests that “the magnetocaloric material should be porous” (p. 95), but the material type is unspecified. How could such a structure be realized, thin films, microchannels, sintered particles? What porosity level is implied, and what fabrication method would be used?

6.     Certain design recommendations appear impractical, such as: “The ratio between the volume of magnetocaloric material and the volume of magnetic material used in the concentric Halbach cylinder should be maximized for reasons of cost and weight” (pp. 126–127, 129, 205).

7.     On p. 193, data is presented for only one material, and units are not specified (e.g., S/m, A/m), making interpretation difficult.

8.     Several figures contain errors and inconsistencies. For example, Figure 11(b) contains typos, the coordinate system is unclear, and Figure 11(a) appears to repeat Figure 10. Figure 11 shows a symmetrical profile with a maximum of 1.65 T, whereas elsewhere the maximum is listed as 1.6 T. Statements such as “magnetic flux density in the low field region is about 0.178689 T and in the high field region is only 1.66632 T” (p. 213) or “the magnetic flux density in the high field region is only 1.581 T and in the low field region is about 0.02 T” (p. 221) are not substantiated. Similar inconsistencies appear in Figures 14–16; overlapping lines and unclear deviation from the color scale prevent quantitative evaluation. Only the difference from the original design is apparent (Fig 17). Inclusion of graphs or histograms would strengthen the conclusions.

9.     In conclusion, it is unclear where the statement “the magnet material consumption is used as the evaluation criterion” (p. 249) originates. The claim that “for the proposed improvement scheme, the magnetic material used is reduced by 45%” appears for the first time in the conclusions. It remains unclear how reducing the amount of magnet material while increasing the field difference would affect field uniformity and cooling efficiency in this specific case.

Author Response

We would like to sincerely thank the reviewers for their invaluable and constructive comments, which have significantly improved the scientific quality, clarity, and rigor of the manuscript entitled:

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

The comments received have been carefully reviewed and incorporated into the new version of the manuscript.

We have:

• we have completely rewritten the introduction in order to better situate our contribution within the existing literature, in accordance with the journal's recommendations.

Di Gerlando, A.; Negri, S.; Ricca, C. A Novel Analytical Formulation of the Magnetic Field Generated by Halbach Permanent Magnet Arrays. Magnetism 2023, 3, 280-296. doi: 10.3390/magnetism3040022Di Gerlando, A.; Negri, S.; Ricca, C. A Novel Analytical Formulation of the Magnetic Field Generated by Halbach Permanent Magnet Arrays. Magnetism 2023, 3, 280-296. doi: 10.3390/magnetism3040022

Lin, S.; Chang, L.; Su, P.; Li, Y.; Hua, W.; Shen, Y. Research on High-Torque-Density Design for Axial Modular Flux-Reversal Permanent Magnet Machine. Energies 2023, 16, 1691. doi: 10.3390/en16041691

Li, B.; Zhang, J.; Zhao, X.; Liu, B.; Dong, H. Research on Air Gap Magnetic Field Characteristics of Trapezoidal Halbach Permanent Magnet Linear Synchronous Motor Based on Improved Equivalent Surface Current Method. Energies 2023, 16, 793. doi: 10.3390/en16020793

Kresse, T.; Martinek, G.; Schneider, G.; Goll, D. The Field-Dependent Magnetic Viscosity of FeNdB Permanent Magnets. Materials 2024, 17, 243. doi: 10.3390/ma17010243

• clarification of the numerical methodology (ANSYS Maxwell 3D FEM) with simulation parameters.
• linguistic correction and updating of references.
• and the addition of new technical elements concerning materials and experimental feasibility.

Response to Reviewer 2 Comments

We sincerely thank Reviewer 2 for the constructive and detailed feedback.
The comments helped us substantially improve the structure, clarity, and scientific depth of the paper. All points have been carefully addressed in the revised version, Detailed responses to each comment are provided below.

Please find below the detailed responses, including the corresponding revisions in green and the English-language corrections in red in the re-submitted files.

The authors presented interesting results on the design of Halbach cylinders for use in magnetic refrigeration. Halbach cylinders offer clear advantages, as they can generate magnetic fields in the range of 1–1.5 T, which is often sufficient to realize the magnetocaloric effect, and they minimize eddy current losses. Several examples of refrigerators based on Halbach-cylinder magnetic systems can be found in the literature. The concept itself is not new e.g.

• https://doi.org/10.1016/j.ijrefrig.2006.07.025,
• https://doi.org/10.3390/en14154662,
• https://doi.org/10.1016/j.ijrefrig.2024.04.022

and te approach has already been implemented. A closely related article, “Designing of Halbach Cylinder Based Magnetic Assembly for a Rotating Magnetic Refrigerator. Part 1: Designing Procedure”, was published in the International Journal of Refrigeration, 73 (2017), 246–256. However, the authors’ proposal to use low-carbon steel concentrators with a modified design is potentially interesting from an engineering perspective. The simulation results are also noteworthy, serving as an example of solving this problem in ANSYS Maxwell. Nevertheless, the manuscript, in its current form, cannot be accepted due to several critical issues.

1.     The authors refer to a “regular permanent magnet” without specification. It is unclear which type of magnet is meant. Typically, NdFeB or SmCo magnets are used, with the magnetization vector rotating circumferentially. Coercivity, remanence, and other essential parameters are not provided.​

In the revised manuscript, we explicitly specify that the magnets used are Neodymium–Iron–Boron (NdFeB or FeNdB) permanent magnets, grade N52, with remanence = 1.44 T, coercivity = 836 kA/m, and relative permeability = 1.04457 (see Table 1, page 8, Lines 215). This infor8mation ensures reproducibility and consistency with typical Halbach-based systems.

2. The choice of magnetocaloric material for this configuration is not justified. Typically, gadolinium is used due to its Curie temperature near room temperature, though alternatives exist.

In the revised manuscript, we explicitly specify that the magnets used are Neodymium–Iron–Boron (NdFeB or FeNdB) permanent magnets with remanence = 1.44 T, coercivity = 836 kA/m, and relative permeability = 1.04457 (see Table 1, page 8, Line 215). This information ensures reproducibility and consistency with typical Halbach-based systems.

3. Literature citations are inaccurate or missing. Many references are replaced with question marks (e.g., [?] on pp. 46, 58, 70, 141, 143), making verification impossible. Furthermore, the introduction begins with references 3 and 5, skipping 1 and 2.

All references have been reviewed, renumbered, and verified.

Errors such as “[?]” have been corrected, and citations are now numbered from [1] to [36].

Each reference has been reformatted according to MDPI style.

4. Some statements are ambiguous, e.g., “the best magnetocaloric materials exhibit a maximum temperature change of 4 K in a 1 T magnetic field” (p. 25). It is unclear which materials are meant and according to what criteria.

The sentence has been revised to read (page 1, line 25-27):

“Currently, the most efficient magnetocaloric materials, such as gadolinium and its alloys, exhibit a maximum adiabatic temperature change of approximately 4 K under a 1 T magnetic field near their Curie temperature [7].”

This clarification specifies the materials and conditions for that value.

5. The manuscript suggests that “the magnetocaloric material should be porous” (p. 95), but the material type is unspecified. How could such a structure be realized, thin films, microchannels, sintered particles? What porosity level is implied, and what fabrication method would be used?

We expanded Section 1 to explain that the term “porous” refers to a regenerator geometry composed of radially mounted thin plates of magnetocaloric material, ensuring fluid flow with minimal pressure drop. This structure can be manufactured by sintering or stacking thin Gd plates not by microchannels or films.

6.     Certain design recommendations appear impractical, such as: “The ratio between the volume of magnetocaloric material and the volume of magnetic material used in the concentric Halbach cylinder should be maximized for reasons of cost and weight” (pp. 126–127, 129, 205).

The statement on maximizing the ratio between magnetocaloric and magnetic materials has been revised for clarity:

“The ratio between the volumes of magnetocaloric and magnetic materials should be optimized to balance performance, cost, and weight.”
We emphasize that this is a design objective, not a strict rule, and that our optimization process explicitly considers this trade-off.

7. On p. 193, data is presented for only one material, and units are not specified (e.g., S/m, A/m), making interpretation difficult.

Table 1 now includes proper SI units (e.g., T, S/m). Table 1 lists the properties of both NdFeB and AISI 1010 steel. This resolves the ambiguity regarding material parameters and units (page 9, line 207-208)

8. Several figures contain errors and inconsistencies. For example, Figure 11(b) contains typos, the coordinate system is unclear, and Figure 11(a) appears to repeat Figure 10. Figure 11 shows a symmetrical profile with a maximum of 1.65 T, whereas elsewhere the maximum is listed as 1.6 T. Statements such as “magnetic flux density in the low field region is about 0.178689 T and in the high field region is only 1.66632 T” (p. 213) or “the magnetic flux density in the high field region is only 1.581 T and in the low field region is about 0.02 T” (p. 221) are not substantiated. Similar inconsistencies appear in Figures 14–16; overlapping lines and unclear deviation from the color scale prevent quantitative evaluation. Only the difference from the original design is apparent (Fig 17). Inclusion of graphs or histograms would strengthen the conclusions.

All figures have been carefully reviewed and corrected:

• Typos and labeling errors in Fig. 11 have been fixed.
• The coordinate system and color scales are now clearly indicated.
• Additional plots (Fig. 17) and histograms comparing all 12 configurations have been included to provide quantitative analysis and visual clarity

9. In conclusion, it is unclear where the statement “the magnet material consumption is used as the evaluation criterion” (p. 249) originates. The claim that “for the proposed improvement scheme, the magnetic material used is reduced by 45%” appears for the first time in the conclusions. It remains unclear how reducing the amount of magnet material while increasing the field difference would affect field uniformity and cooling efficiency in this specific case.

We expanded the conclusions to explicitly explain that “magnet material consumption” refers to the ratio of magnet mass to total system mass used as a quantitative performance indicator.
The claim of 45% reduction in magnet material is now supported by numerical comparison between the original and optimized configurations. This optimization was achieved by replacing part of the NdFeB segments with low-carbon steel (AISI 1010) and non-magnetic Teflon, maintaining the same high/low field contrast (1.6 T / ~0 T) while reducing weight and cost.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

The paper presents an improvement of a device made of permanent magnets associated to generate high and low magnetic flux zones. This device is used in a magnetic refrigerator.

1/The materials are not clearly presented:

*units are missing in table 1

*what is the nature of the permanent magnet?

*how is its remnant magnetic flux sensitive to temperature?

*how is its remnant magnetic flux sensitive to combination of temperature and external field?

…

2/The calculation method is not described: some details should be given.

*what is the numerical method?

*if the device is meshed, what is the mesh parameter?

*what is the field decrease condition far from the device?

…

Comments on the Quality of English Language

3/There are many errors and typos :

*in the introduction “is bawled” should be replaced by “is called”

*in the introduction “which is greatest” should be replaced by “and is greatest”

*in the introduction “4k” should be replaced by “4K”

*in the introduction “in order to satisfied” should be replaced by “in order to satisfy”

*in line 46, reference is missing

*in line 56, “devise” should be replaced by “device”

*in line 58, reference is missing

*in line 70, reference is missing

*in line 77, “devises” should be replaced by “devices”

*in line 78, “devise” should be replaced by “device”

*in line 107, “consists a cylindrical magnet” should be changed into “consists of a cylindrical magnet”

*in line 109, “an uniform norms” has to be changed into “a uniform norm”

*in line 112, “”genets a field is directed” → “generates a field that is directed”

*in line 115, “which its” → “with”

*in line 141, reference is missing

*in line 143, reference is missing

…

 

Author Response

We would like to sincerely thank the reviewers for their invaluable and constructive comments, which have significantly improved the scientific quality, clarity, and rigor of the manuscript entitled:

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

The comments received have been carefully reviewed and incorporated into the new version of the manuscript.

We have:

• we have completely rewritten the introduction to better situate our contribution within the existing literature, in accordance with the journal's recommendations.

Di Gerlando, A.; Negri, S.; Ricca, C. A Novel Analytical Formulation of the Magnetic Field Generated by Halbach Permanent Magnet Arrays. Magnetism 2023, 3, 280-296. doi: 10.3390/magnetism3040022Di Gerlando, A.; Negri, S.; Ricca, C. A Novel Analytical Formulation of the Magnetic Field Generated by Halbach Permanent Magnet Arrays. Magnetism 2023, 3, 280-296. doi: 10.3390/magnetism3040022

Lin, S.; Chang, L.; Su, P.; Li, Y.; Hua, W.; Shen, Y. Research on High-Torque-Density Design for Axial Modular Flux-Reversal Permanent Magnet Machine. Energies 2023, 16, 1691. doi: 10.3390/en16041691

Li, B.; Zhang, J.; Zhao, X.; Liu, B.; Dong, H. Research on Air Gap Magnetic Field Characteristics of Trapezoidal Halbach Permanent Magnet Linear Synchronous Motor Based on Improved Equivalent Surface Current Method. Energies 2023, 16, 793. doi: 10.3390/en16020793

Kresse, T.; Martinek, G.; Schneider, G.; Goll, D. The Field-Dependent Magnetic Viscosity of FeNdB Permanent Magnets. Materials 2024, 17, 243. doi: 10.3390/ma17010243

• clarification of the numerical methodology (ANSYS Maxwell 3D FEM) with simulation parameters.
• linguistic correction and updating of references.
• and the addition of new technical elements concerning materials and experimental feasibility.

Response to Reviewer 3 Comments

We sincerely thank Reviewer 3 for their valuable and detailed comments, which have helped us to improve the clarity, completeness, and linguistic accuracy of the manuscript. Below, we provide a point-by-point response to each issue raised.

Please find below the detailed responses, including the corresponding revisions in green and the English-language corrections in red in the re-submitted files.

1/The materials are not clearly presented:

We thank the reviewer for these observations. The materials section has been completely revised and clarified (see Table 1, page 8, Lines 207-215).

• Nature of the permanent magnet: The device uses Neodymium–Iron–Boron (NdFeB or FeNdB) permanent magnets, grade N52, due to their high remanence and energy product.
• Temperature dependence: the remanent flux density of NdFeB magnets decreases linearly with temperature, approximately −0.11 % per °C, which is consistent with data from Kresse et al. [36].
• Temperature–field interaction: NdFeB 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.

These clarifications ensure that the materials and their properties are now fully defined and physically justified.

*units are missing in table 1

*what is the nature of the permanent magnet?

*how is its remnant magnetic flux sensitive to temperature?

*how is its remnant magnetic flux sensitive to combination of temperature and external field?

In the revised manuscript, we explicitly specify that the magnets used are Neodymium–Iron–Boron (NdFeB or FeNdB) permanent magnets with remanence = 1.44 T, coercivity = 836 kA/m, and relative permeability = 1.04457 (see Table 1, page 8, Line 215). This information ensures reproducibility and consistency with typical Halbach-based systems.

2/The calculation method is not described: some details should be given.

*what is the numerical method?

*if the device is meshed, what is the mesh parameter?

*what is the field decrease condition far from the device?

We agree and have significantly expanded the “Simulation Procedure” section to include the requested details (page 1, line 25-27):

The calculation method is now described in more detail. In this work, we used the Finite Element Method (FEM) 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 to zero follows a Neumann-type decay consistent with the physical boundary of the problem.

These additions make the simulation process reproducible and transparent.

3/There are many errors and typos :

*in the introduction “is bawled” should be replaced by “is called”

*in the introduction “which is greatest” should be replaced by “and is greatest”

*in the introduction “4k” should be replaced by “4K”

*in the introduction “in order to satisfied” should be replaced by “in order to satisfy”

*in line 46, reference is missing

*in line 56, “devise” should be replaced by “device”

*in line 58, reference is missing

*in line 70, reference is missing

*in line 77, “devises” should be replaced by “devices”

*in line 78, “devise” should be replaced by “device”

*in line 107, “consists a cylindrical magnet” should be changed into “consists of a cylindrical magnet”

*in line 109, “an uniform norms” has to be changed into “a uniform norm”

*in line 112, “”genets a field is directed” → “generates a field that is directed”

*in line 115, “which its” → “with”

*in line 141, reference is missing

*in line 143, reference is missing

All typographical and grammatical errors listed by the reviewer have been carefully corrected throughout the revised manuscript. Specifically:

• “is bawled” → “is called”
• “which is greatest” → “and is greatest”
• “4k” → “4 K”
• “in order to satisfied” → “in order to satisfy”
• “devise/devises” → “device/devices” (all occurrences corrected)
• “consists a cylindrical magnet” → “consists of a cylindrical magnet”
• “an uniform norms” → “a uniform norm”
• “genets a field is directed” → “generates a field that is directed”
• “which its” → “with”
• All missing references at lines 46, 56, 58, 70, 141, and 143 have been verified and inserted correctly.

In addition, the entire text was rechecked for consistency in English style and units, particularly the use of “T”, “A/m”, and “K”.

The entire manuscript has been proofread and corrected in red:

lexical errors (e.g., “devise” → “device,” “bawled” → “called,” “in order to satisfied” → “in order to satisfy”),

standardization of units (K, mm, µ₀),

rewording of ambiguous sentences in the introduction and conclusion.

The language was checked using Grammarly Academic Edition.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors I appreciate the authors’ efforts in substantially revising the manuscript. The introduction is more structured, references have been corrected, simulation parameters are clearer, and additional quantitative comparisons have been included. Overall, the authors have made significant improvements and have addressed most of the previous concerns. Minor clarifications as noted are needed to ensure clarity and reproducibility before the paper can be accepted.    1. The introduction should be tightened by removing repeated background statements.   2. The authors added comparisons (ΔB and magnet volume reduction), which is appreciated. However, the following still needs clarification in the manuscript: • How exactly is ΔB defined? • The statement “magnet material reduced by 45%” should include the explanation of how this percentage was calculated.  

Author Response

Dear Editors,

We sincerely thank the reviewers for their careful reading of our manuscript entitled: “Improving Designs of Halbach Cylinder Based Magnetic Assembly with High and Low Field Regions for a Rotating Magnetic Refrigerator” and for their constructive and insightful comments.
We have revised the paper thoroughly, and all modifications are clearly highlighted in the revised manuscript.

Below, we provide a point-by-point response to each reviewer.
All changes requested have been implemented unless otherwise stated, and detailed explanations are provided.

Response to Reviewer 1 Comments

I appreciate the authors’ efforts in substantially revising the manuscript. The introduction is more structured, references have been corrected, simulation parameters are clearer, and additional quantitative comparisons have been included. Overall, the authors have made significant improvements and have addressed most of the previous concerns. Minor clarifications as noted are needed to ensure clarity and reproducibility before the paper can be accepted.

We sincerely thank Reviewer 1 for the positive evaluation of our revised manuscript. We are pleased that the improvements made, namely the restructuring of the Introduction, the correction of references, the clarification of the simulation parameters, and the inclusion of additional quantitative comparisons were appreciated.

We have carefully addressed the remaining minor clarifications requested to further enhance the clarity, reproducibility, and scientific accuracy of the work. All corrections are now implemented and highlighted in the revised version.

The comments helped us substantially improve the structure, clarity, and scientific depth of the paper. All points have been carefully addressed in the revised version, Detailed responses to each comment are provided below.

Please find below the detailed responses, including the corresponding revisions in green in the re-submitted files.

1. The introduction should be tightened by removing repeated background statements.

We thank the reviewer for this helpful remark. We carefully revised the Introduction to remove redundant background sentences and to streamline the narrative. We merged repeated explanations of magnetocaloric refrigeration principles and regenerator geometries, and we clarified the specific research gap our work addresses.

2. The authors added comparisons (ΔB and magnet volume reduction), which is appreciated. However, the following still needs clarification in the manuscript:

• How exactly is ΔB defined?

We appreciate the reviewer’s request for clarification. In the revised manuscript, we now explicitly define the parameter ΔB and explain how the percentage reduction in magnet material is calculated. Specifically, ΔB is defined as the difference between the volume-averaged magnetic flux densities in the regenerator region for two compared configurations. The “magnet material reduced by 45%” refers to the relative reduction of the total magnet volume in the optimized configuration with respect to the reference design. These clarifications have been added in (Page 4, lines 135–137)

In this work, we define : 

\begin{equation} \Delta B = \bar{B}_{\mathrm{opt}} - \bar{B}_{\mathrm{ref}} \end{equation}

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

• The statement “magnet material reduced by 45%” should include the explanation of how this percentage was calculated.

These clarifications have been added in (Page 12, lines 280-284)

The reduction in magnet material is quantified as : 

       \begin{equation}
           R_V = \frac{V_{\mathrm{ref}} - V_{\mathrm{opt}}}{V_{\mathrm{ref}}} \times 100\%
        \end{equation}

 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.

We sincerely thank the reviewer for emphasizing the need for stronger quantitative analysis and clear novelty.

We believe that these improvements make the manuscript substantially stronger and suitable for consideration by MDPI Magnetism.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors magnetism-3978785

Although the authors have improved the manuscript, it still cannot be accepted for publication. Several questions remain unclear:

1. The authors should avoid misleading the readers. A porous material and a regenerator geometry composed of radially mounted thin plates of magnetocaloric material are not the same thing. These terms are not synonymous and represent different concepts. A porous material implies the presence of pores and channels that can be filled with fluid or gas, whereas the described regenerator geometry is intended to optimize heat transfer.
2. The authors should clarify which magnetic material they are using — gadolinium or NdFeB. The text contains contradictions on this point.
3. In the authors’ response, it was stated that Table 1 now includes proper SI units (e.g., T, S/m) and the properties of both NdFeB and AISI 1010 steel. However, Table 1 currently contains data only for NdFeB. Why are the properties of AISI 1010 steel missing if the authors claim otherwise?
4. The authors should explain what they mean by the negative value Magnitude [T] = –837999.9999. How should such a value be physically interpreted?

Author Response

Dear Editors,

We sincerely thank the reviewers for their careful reading of our manuscript entitled: “Improving Designs of Halbach Cylinder Based Magnetic Assembly with High and Low Field Regions for a Rotating Magnetic Refrigerator” and for their constructive and insightful comments.
We have revised the paper thoroughly, and all modifications are clearly highlighted in the revised manuscript.

Below, we provide a point-by-point response to each reviewer.
All changes requested have been implemented unless otherwise stated, and detailed explanations are provided.

Response to Reviewer 2 Comments

1. The authors should avoid misleading the readers. A porous material and a regenerator geometry composed of radially mounted thin plates of magnetocaloric material are not the same thing. These terms are not synonymous and represent different concepts. A porous material implies the presence of pores and channels that can be filled with fluid or gas, whereas the described regenerator geometry is intended to optimize heat transfer.

We thank the reviewer for pointing out this misleading terminology. We have revised the manuscript to clearly distinguish between a “porous material” and a “regenerator composed of radially mounted thin plates.” A new paragraph has been added (Pages 3-4, lines 107-111) explaining that the regenerator geometry is not inherently porous but instead consists of thin magnetocaloric plates separated by controlled spacing to guide the heat-transfer fluid. All occurrences of “porous structure” have been corrected.

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 [30–32].

2. The authors should clarify which magnetic material they are using — gadolinium or NdFeB. The text contains contradictions on this point.

We acknowledge this inconsistency. The manuscript has been revised to clarify that gadolinium is mentioned only as a reference magnetocaloric material from the literature. In our design, the permanent magnet assembly is made exclusively of FeNdB, while the regenerator plates are not Gd-based in the current simulation model. A paragraph has been added (Page 4, lines 112-114) to avoid any confusion between magnetocaloric function and magnetic field generation.

Gadolinium may appear as a reference material in 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.

3. In the authors’ response, it was stated that Table 1 now includes proper SI units (e.g., T, S/m) and the properties of both NdFeB and AISI 1010 steel. However, Table 1 currently contains data only for NdFeB. Why are the properties of AISI 1010 steel missing if the authors claim otherwise?

The reviewer is correct; the previous version did not include AISI 1010 steel properties despite our statement. Table 1 (Page 9) has now been fully corrected and includes all soft-magnetic properties required for reproducibility (relative permeability, conductivity, density). New table to replace in the Simulation procedure section.

 

4. The authors should explain what they mean by the negative value Magnitude [T] = –837999.9999. How should such a value be physically interpreted?

We thank the reviewer for highlighting this issue. The negative value originated from an internal ANSYS Maxwell solver parameter (scalar potential magnitude) and is not a magnetic flux density. This entry has been removed from Table 1 (Page 9).

In previous versions of the manuscript, the value “Magnitude [T] = –837999.9999” appeared in the table following a direct extraction from ANSYS Maxwell. This value does not represent a magnetic flux density, but an internal parameter related to the magnetic scalar potential used by the FEM solver (negative normalized value resulting from the initialization process). This entry therefore does not correspond to any usable physical quantity and has been removed to avoid confusion.

We sincerely thank the reviewer for emphasizing the need for stronger quantitative analysis and clear novelty.

We believe that these improvements make the manuscript substantially stronger and suitable for consideration by MDPI Magnetism.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

There still remains something weird in Table 1; what means the magnitude of - 838 000 Tesla?

838 000 A/m may mean something, not 838 000 Tesla.

 

Author Response

Dear Editors,

We sincerely thank the reviewers for their careful reading of our manuscript entitled: “Improving Designs of Halbach Cylinder Based Magnetic Assembly with High and Low Field Regions for a Rotating Magnetic Refrigerator” and for their constructive and insightful comments.
We have revised the paper thoroughly, and all modifications are clearly highlighted in the revised manuscript.

Below, we provide a point-by-point response to each reviewer.
All changes requested have been implemented unless otherwise stated, and detailed explanations are provided.

Response to Reviewer 3 Comments

There still remains something weird in Table 1; what means the magnitude of - 838 000 Tesla?

838 000 A/m may mean something, not 838 000 Tesla.

We thank the reviewer for highlighting this issue. The negative value originated from an internal ANSYS Maxwell solver parameter (scalar potential magnitude) and is not a magnetic flux density. This entry has been removed from Table 1 (Page 9).

In previous versions of the manuscript, the value “Magnitude [T] = –837999.9999” appeared in the table following a direct extraction from ANSYS Maxwell. This value does not represent a magnetic flux density, but an internal parameter related to the magnetic scalar potential used by the FEM solver (negative normalized value resulting from the initialization process). This entry therefore does not correspond to any usable physical quantity and has been removed to avoid confusion.

We sincerely thank the reviewer for emphasizing the need for stronger quantitative analysis and clear novelty.

We believe that these improvements make the manuscript substantially stronger and suitable for consideration by MDPI Magnetism.

Author Response File: Author Response.pdf