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An Innovative Approach to Predict the Diffusion Rate of Reactant’s Effects on the Performance of the Polymer Electrolyte Membrane Fuel Cell

Mathematics 2023, 11(19), 4094; https://doi.org/10.3390/math11194094

by Nima Ahmadi^1,* and Sajad Rezazadeh²

Reviewer 1:

Salawu Sulyman

Reviewer 2: Anonymous

Reviewer 3: Anonymous

Mathematics 2023, 11(19), 4094; https://doi.org/10.3390/math11194094

Submission received: 6 September 2023 / Revised: 20 September 2023 / Accepted: 25 September 2023 / Published: 27 September 2023

(This article belongs to the Section Engineering Mathematics)

Round 1

Reviewer 1 Report (New Reviewer)

The report on an innovative approach to predict the diffusion rate of reactant's effects on the performance of the PEMFC. The abstract and the introduction are developed clearly in terms, and relevant and related articles are cited. Information on the cited articles is given in the introduction section. Mathematical governing equations are developed and solved. The conclusion of the study is appropriately highlighted. Thus, the manuscript is improved and well structured. Following all these observations, it is therefore recommended for publication in your journal.

Author Response

Dear Editor and reviewers,

We would like to thank you for your appropriate and valuable comments. We attempted to answer all the comments raised by the referees. Your kind recommendations and scientific hints were very useful for us.

Best Regards

Nima Ahmadi

Department of Mechanical Engineering, Technical and Vocational University (TVU), Tehran, Iran

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Comments of Reviewer #1:

Answer: Dear reviewer, thank you for your kind consideration of our paper.

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Finally, we should thank the reviewer comments again which help us to enhance the quality of our research.

Author Response File: Author Response.docx

Reviewer 2 Report (New Reviewer)

This manuscript proposed an innovative analytical approach based on the perturbation method. This method utilizes the governing equations, which consist of the continuity, momentum, species, and energy equations, are solved analytically by using the regular perturbation method. The perturbation parameter is the function of the penetration (diffusion) velocity. The topic discussed in the submitted article is very interesting and valuable. However, I have found a few mistakes in the manuscript in English description and pointed out a few places for improvement. There are some obscure equations in the proposed control design process while the stability analysis should be proved in more detail. Besides, some statements in this study should be modified to match the suitable content with the other studies. Especially, the contribution of the article is not enough. Therefore, this manuscript cannot be published with this version.

Minor revision:

(1) Many formulas need to be arranged more logically for reader convenience

Major revision:

(2) The simulation results should be compared with previous studies ?

(3) The proposed method should consider species diffusion rate in different humidity conditions ?

Minor editing of English language required

Author Response

Dear Editor and reviewers,

Best Regards

Nima Ahmadi

Department of Mechanical Engineering, Technical and Vocational University (TVU), Tehran, Iran

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Comments of Reviewer #2:

Question 1: Many formulas need to be arranged more logically for reader convenience

Answer: Thank you for your valuable comment. It is tried to arrange the formulas. For this reason, we structured an Appendix into the paper.

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Question 2: The simulation results should be compared with previous studies ?

Answer: Thank you for your very scientific comment. As we mentioned in the work, the objective of the present work is to introduce an analytical method to model the transport phenomena more accurately and simply, and a numerical simulation is developed just to verify the accuracy of our result. However, you are right, and each numerical procedure needs verification. Due to this fact, we examined and validated our 3D numerical code in our previously published works for the conventional model of the PEMFC. The numerical procedure is verified in the following and many other published works:

Ahmadi, N., Rezazadeh, S., Mirzaee, I., & Pourmahmoud, N. (2012). Three-dimensional computational fluid dynamic analysis of the conventional PEM fuel cell and investigation of prominent gas diffusion layers effect. Journal of mechanical science and technology, 26, 2247-2257.
Ahmadi, Nima, Abdolrahman Dadvand, Sajad Rezazadeh, and Iraj Mirzaee. "Analysis of the operating pressure and GDL geometrical configuration effect on PEM fuel cell performance." Journal of the Brazilian Society of Mechanical Sciences and Engineering 38 (2016): 2311-2325.
Ahmadi, N., & Rostami, S. (2019). Enhancing the performance of polymer electrolyte membrane fuel cell by optimizing the operating parameter. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 41(5), 220.

Also, this code is used in references 1, 4, and 6 which have been cited in the paper. As we didn’t include the electrochemical reaction in this paper, we didn’t present the verification of the code in this work. However, we will refer verification of references 1 and 4 in the revised paper. Also, the validation of the 3D numerical procedure which included the electrochemical reaction is as follows for the polarization curve which has been compared to Wang et al.[*] (Voltage versus Current density curve) which have good accordance:

[*] L. Wang, A. Husar, T. Zhou, and H. Liu, A parameteric study of PEM fuel cell performances, J. Hydrog. Energy, 28 (2003) 1263-1272.

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Question 3: The proposed method should consider species diffusion rate in different humidity conditions ?

Answer: Nice question. In this work, we assumed the fully humid condition of the inlet reactant gases. Because, in our previously published work (Ahmadi et al. [**]), we focused on the PEMFC performance analysis in different inlet gas humidity. So, the repetition of the analysis would not be useful in the present work, and for more information, the reader can refer to our previous work which has been mentioned below. Also, As can be seen, the governing equations are solved in the gas channel in this study. As it is clear, in the reactant gas channel there are no electrochemical reactions and the humidity is related to the electrochemical reactions. However, the reactant gas channel has a great impact on the performance of the PEMFC because of its role in transferring the reactant to the reaction area. In the present study, we tried to solve the non-linear governing equation of the reactant flow and study the effect of the diffusion rate of the reactant which is the function of the back pressure and GDL porosity. The solution of the non-linear governing equation of the PEMFC is a phenomenon that rarely has been addressed before except in a few works. But, it is a very important point that determines the performance and efficiency of the PEMFC, However, in future works, we have planned to solve the governing equation in the reaction areas. In this region, the governing equations have source terms that are related to the electrochemical reactions.

[**]Ahmadi, Nima, Sajad Rezazadeh, Mirkazem Yekani, Alireza Fakouri, and Iraj Mirzaee. "NUMERICAL INVESTIGATION OF THE EFFECT OF INLET GASES HUMIDITY ON POLYMER EXCHANGE MEMBRANE FUEL CELL (PEMFC) PERFORMANCE." Transactions of the Canadian Society for Mechanical Engineering 37, no. 1 (2013): 1-20.

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Comments on the Quality of English Language: Minor editing of English language required

Answer: Dear reviewer, we rechecked the paper and edited the English of the paper again.

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Finally, we should thank the reviewer comments again which help us to enhance the quality of our research.

Author Response File: Author Response.docx

Reviewer 3 Report (New Reviewer)

1. Please avoid the use of abbreviations in your title (EMFC).

2. The abbreviation GDL is not introduced in the abstract when first used.

3. I believe the first three lines of the abstract are unnecessary. Please streamline your abstract by presenting the essence of your topic, the methodology used, and numerical results obtained from your analytical resolution of the Navier-Stokes equations coupled with the species model.

4. Kindly reduce the use of the phrase "in recent years" on page 2.

5. The Reynolds number must be calculated correctly. Please perform the calculation and include it in the section on your assumptions. Otherwise, you will need to provide a reference to justify the value Re=200 that you mentioned. (You should present the characteristic length used correctly to facilitate the reproducibility of your study.)

6. Please address your assumptions point by point to facilitate the reproducibility of your study.

7. Please include the units for the dimensions in your Figure 1.

8. All established equations must be mentioned in the body of your text.

9. Please correct Equation 2, as there are Chinese characters appearing in the line presenting it.

10. You need to justify the numbering of your equations correctly!

11. References should be included next to the equations you present (continuity, momentum, energy, species).

12. Please provide a clear diagram on which you outline the boundary conditions applied to your fluid domain.

13. You will need to enlarge the caption for Figure 3. It is not visible!

14. Please standardize the writing style of all your equations. A significant difference is noticeable each time (page 10, 9 compared to page 5).

15. The details of mathematical calculations (pages 10-16) should be presented in an appendix section to lighten your article.

16. Kindly format and label all your figures appropriately (Fig. 4 – Fig. 23). The captions are not visible. Consider using notation (a), (b), and (c) ... for multiple sub-figures to aid in their identification in the text when discussing them.

17. Numerical results must be included in both your conclusion and abstract.

18. Please deepen the discussion of your results using recent references to discuss the advantages and disadvantages of your analytical resolution compared to 2D or 3D CFD modeling.

Moderate editing of the English language required

Author Response

Dear Editor and reviewers,

Best Regards

Nima Ahmadi

Department of Mechanical Engineering, Technical and Vocational University (TVU), Tehran, Iran

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Comments of Reviewer #3:

Question 1: Please avoid the use of abbreviations in your title (EMFC).

Answer: Thank you for your valuable comment. It is performed and the title is changed according to your comment.

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Question 2: The abbreviation GDL is not introduced in the abstract when first used.

Answer: It is corrected in the abstract.

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Question 3: I believe the first three lines of the abstract are unnecessary. Please streamline your abstract by presenting the essence of your topic, the methodology used, and numerical results obtained from your analytical resolution of the Navier-Stokes equations coupled with the species model.

Answer: Very nice comment. It is applied in the abstract and the abstract is shortened.

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Question 4: Kindly reduce the use of the phrase "in recent years" on page 2.

Answer: It is performed on page 2.

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Question 5: The Reynolds number must be calculated correctly. Please perform the calculation and include it in the section on your assumptions. Otherwise, you will need to provide a reference to justify the value Re=200 that you mentioned. (You should present the characteristic length used correctly to facilitate the reproducibility of your study.)

Answer: Very scientific comment. As mentioned in the many previously published works, the flow in the PEMFC is the internal and laminar flow and it is mentioned in references 35 and 38 of the paper. However, in many works the Reynolds is assumed to be less or about 200. For this, it can be referred to the Sung work on page 7 (Which pointed to the Reynolds number equal to 200) and Ahmadi et al.

Sung, Y. (2006). An approximate analytical solution to channel flow in a fuel cell with a draft angle. Journal of power sources, 159(2), 1051-1060.
Ahmadi, Nima, and Mihkel Kõrgesaar. "Analytical approach to investigate the effect of gas channel draft angle on the performance of PEMFC and species distribution." International Journal of Heat and Mass Transfer 152 (2020): 119529.
Cox, S. M., & King, A. C. (1997). On the asymptotic solution of a high–order nonlinear ordinary differential equation. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 453(1959), 711-728.

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Question 6: Please address your assumptions point by point to facilitate the reproducibility of your study.

Answer: It is applied in the paper.

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Question 7: Please include the units for the dimensions in your Figure 1.

Answer: Dear reviewer, thank you for your valuable comment. All of the parameters of the paper are nondimensionalized and the explanation of the nondimensionalizing channel dimension is added to section 2.2.

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Question 8: All established equations must be mentioned in the body of your text.

Answer: It is adjusted in the paper.

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Question 9: Please correct Equation 2, as there are Chinese characters appearing in the line presenting it.

Answer: Dear reviewer, we cannot find any Chinese character in equation 2, as can be seen in the picture presented below it is correct. It might be due to your system setting or other system problems. However, we rechecked it.

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Question 10: You need to justify the numbering of your equations correctly!

Answer: it is applied in the text.

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Question 11: References should be included next to the equations you present (continuity, momentum, energy, species).

Answer: Dear reviewer, the reference is added next to the equations.

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Question 12: Please provide a clear diagram on which you outline the boundary conditions applied to your fluid domain.

Answer: Dear reviewer, the boundary conditions are presented as the mathematical relation for each equation the function of the channel dimension. As it is indicated for all of the equations which have been solved the mathematical boundary conditions are presented. For instance, equations 2, 35…Also, it should be noticed that in the analytical works the boundary conditions are presented in the form of the mathematical relations. However, for more details can refer to our previously published works like references 1 and 2 of the paper.

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Question 13: You will need to enlarge the caption for Figure 3. It is not visible!

Answer: it is adjusted.

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Question 14: Please standardize the writing style of all your equations. A significant difference is noticeable each time (page 10, 9 compared to page 5).

Answer: Dear reviewer, all of the equations are typed using MathType version 7 in the same style and size. However, due to your valuable comment, we rechecked and fixed it. All of the numbers and characters are now in Italic style.

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Question 15: The details of mathematical calculations (pages 10-16) should be presented in an appendix section to lighten your article.

Answer: Thank you for your suggestion. It is performed.

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Question 16: Kindly format and label all your figures appropriately (Fig. 4 – Fig. 23). The captions are not visible. Consider using notation (a), (b), and (c) ... for multiple sub-figures to aid in their identification in the text when discussing them.

Answer: It is adjusted.

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Question 17: Numerical results must be included in both your conclusion and abstract.

Answer: Thank you for your comment. The abstract and conclusion are revised according to your comment.

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Question 18: Please deepen the discussion of your results using recent references to discuss the advantages and disadvantages of your analytical resolution compared to 2D or 3D CFD modeling.

Answer: Very nice comments. It is performed in the text and the conclusion. We added a comparison of the two methods in the conclusion.

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Comments on the Quality of English Language: Moderate editing of the English language required

Answer: Dear reviewer, we rechecked the paper and edited the English of the paper again.

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Finally, we should thank the reviewer comments again which help us to enhance the quality of our research.

Author Response File: Author Response.docx

Round 2

Reviewer 2 Report (New Reviewer)

Reviewer comments have been clarified. Therefore, this manuscript can be accepted for this journal.

Minor editing of English language should be uppdated !

Reviewer 3 Report (New Reviewer)

The authors have brilliantly addressed all the remarks and comments I proposed to them. For these reasons, I recommend the publication of the article referenced as 'mathematics-2625043,' titled 'An Innovative Approach to Predict the Diffusion Rate of Reactant's Effects on the Performance of the Polymer Electrolyte Membrane Fuel Cell,' in its current form in the Mathematics journal.

This manuscript is a resubmission of an earlier submission. The following is a list of the peer review reports and author responses from that submission.

Round 1

Reviewer 1 Report

An innovative method to predict the effect of reactant diffusion rate on PEMFC performance was proposed in this manuscript through the combination of established mathematical model and simulation by Fluent. It could be considered for publishment after the following concerns are addressed,

1. Lines 40~99, the second paragraph in Introduction section was too long. Please divide it to several paragraphs.

2. Please double check the format of Equation (2), and it seemed that there were some errors in it.

3. In section 2.2.3. Comparison of Numerical and analytical results, the authors stated that “The perturbation parameters are considered as 0, 0.3, and 0.7.”. However, only the results under the perturbation parameters of 0 and 0.7 have been analyzed. Why?

4. Figure 9 should be re-drawn and the panels should be numbered to clearly show the contents.

5. Significant data should be provided in Abstract and Conclusions sections.

6. The Conclusions section should be re-written.

Moderate editing of English language required

Author Response

Dear Editor and reviewers,

Best Regards

Nima Ahmadi

Department of Mechanical Engineering, Technical and Vocational University (TVU), Tehran, Iran

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Comments of Reviewer #1:

Question 1: Lines 40~99, the second paragraph in Introduction section was too long. Please divide it to several paragraphs.

Answer: Thank you for your valuable comment. It is performed.

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Question 2: Please double check the format of Equation (2), and it seemed that there were some errors in it.

Answer: Nice comment, it is corrected.

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Question 3: In section 2.2.3. Comparison of Numerical and analytical results, the authors stated that “The perturbation parameters are considered as 0, 0.3, and 0.7.”. However, only the results under the perturbation parameters of 0 and 0.7 have been analyzed. Why?

Answer: It was an unwanted mistake. The validation and comparison between the analytical and numerical results in section 2.2.3 was performed for perturbation parameters 0 and 0.7. The reason is that as we know by increasing the perturbation parameter from 0 to the maximum values, the accuracy of this method is going to be lesser. In this case, we compared the results just for the minimum value (0) and a big value of the perturbation parameter (0.7). the error of the lesser results than 0.7 are smaller than 0.7. As can be seen, the error between the results for 0.7 is about %1.5, which is a very small error. It is expected that the error for smaller perturbation parameters should be lesser. The reason that we didn’t present the comparison for the other parameters is to prevent the increasing number of images in the paper without increasing the scientific soundness of the content. However, the text in section 2.2.3 is corrected and highlighted.

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Question 4: Figure 9 should be re-drawn and the panels should be numbered to clearly show the contents.

Answer: Dear reviewer, Figure 9, is separated and marked as 9a, 9b, and 9c to clarify the results as you mentioned.

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Question 5: Significant data should be provided in Abstract and Conclusions sections.

Answer: Very scientific comment. The abstract and conclusion are modified.

--------------------------------------------------------------------------------------------------------------------Question 6: The Conclusions section should be re-written.

Answer: Thank you again. The conclusion is re-written and re-arranged.

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Author Response File: Author Response.docx

Reviewer 2 Report

This paper presents an analytical approach to solving the diffusion rate of reactants in PEMFC. However, the author should improve their English writing and present the manuscript in a more organized manner. The following concerns are listed for the authors to address before considering publication.

1. It’s not acceptable to have only two paragraphs in the introduction section. The authors should at least follow a classic structure of “background, current research status, current research limitation, knowledge gap, objective of this study” to rearrange the introduction.

2. The manuscript is filled with qualitative descriptions, which should be replaced by quantitative results. For example, lines 49-51 on page 2, increase the number of blocks from what to what? What is a certain magnitude? What is the effect? Why even include these literature? It feels like a random piling.

3. The motivation of this study needs to be justified thoroughly. The limitations of the numerical method, said by the authors, are slow and high computational costs when it involves 3D simulation. However, the presented analytical approach in this study is only 2D. Then, what is the advantage of approaching this problem analytically? Since the numerical method is also employed in this study as a benchmark, what is the computing time of running it in FLUENT?

4. The electrochemical calculation is not included in the analytical model? The current density is considered as a given parameter of 1 A/cm2? Then, what is the practical value of the analytical model in this study if it is not linked to any electrochemical calculations?

The author should improve their English writing and present the manuscript in a more organized manner.

Author Response

Dear Editor and reviewers,

Best Regards

Nima Ahmadi

Department of Mechanical Engineering, Technical and Vocational University (TVU), Tehran, Iran

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Comments of Reviewer #2:

Question 1: It’s not acceptable to have only two paragraphs in the introduction section. The authors should at least follow a classic structure of “background, current research status, current research limitation, knowledge gap, objective of this study” to rearrange the introduction.

Answer: Thank you for your very scientific and valuable comment. The introduction is re-arranged to several paragraphs and modified.

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Question 2: The manuscript is filled with qualitative descriptions, which should be replaced by quantitative results. For example, lines 49-51 on page 2, increase the number of blocks from what to what? What is a certain magnitude? What is the effect? Why even include these literature? It feels like a random piling.

Answer: More explanation is added to the text to clarify the content to make it more useful. However, since, the topic of the work is the analytical solution, just we presented the main recently published work highlight and to prevent prolonging the paper just we focused on the main achieved results without giving more detail. However, due to your valuable comment, we added more detail to the works in the literature.

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Question 3: The motivation of this study needs to be justified thoroughly. The limitations of the numerical method, said by the authors, are slow and high computational costs when it involves 3D simulation. However, the presented analytical approach in this study is only 2D. Then, what is the advantage of approaching this problem analytically? Since the numerical method is also employed in this study as a benchmark, what is the computing time of running it in FLUENT?

Answer: Nice comment. The analytical model predicts the phenomena more accurately and faster than the numerical solution. In the numerical solution, we have to discretize the domain by applying the grid and the accuracy of the model depends on the size of the grid and the discretization method which is used. But, in the analytical solution, the domain remains continuous and the properties can be calculated in each point of the domain easily and rapidly. In the numerical solution, for achieving higher accuracy, it is necessary to choose the high-density grid which occupies more memory from the computer and it has a significant running cost. For example, in this study. The time for each case of the numerical solution for the present study, with a 2.4GHz quad-core computer, took about 7 hours to catch the convergency. Also. The other limitation of the numerical solution is the grid independency check which is a time-consuming process. However, in the analytical solution, the results can be achieved in less than a second and the results are more reliable than the numerical solution. In the present study, we tried to compare the numerical and analytical results and show that we can achieve accurate results in less than a second instead of 7 hours running the results.

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Question 4: The electrochemical calculation is not included in the analytical model? The current density is considered as a given parameter of 1 A/cm2? Then, what is the practical value of the analytical model in this study if it is not linked to any electrochemical calculations?

Answer: As can be seen, the governing equations are solved in the gas channel in this study. As it is clear, in the reactant gas channel there are no electrochemical reactions. However, the reactant gas channel has a great impact on the performance of the PEMFC because of its role in transferring the reactant to the reaction area. In the present study, we tried to solve the non-linear governing equation of the reactant flow and study the effect of the diffusion rate of the reactant which is the function of the back pressure and GDL porosity. The solution of the non-linear governing equation of the PEMFC is a phenomenon that rarely has been addressed before except in a few works. But, it is a very important point that determines the performance and efficiency of the PEMFC, However, in future works, we have planned to solve the governing equation in the reaction areas. In this region, the governing equations have source terms that are related to the electrochemical reactions.

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Author Response File: Author Response.docx

Round 2

Reviewer 1 Report

This paper has been thoroughly revised and all the comments have been addressed.

Moderate editing of English language required.

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An Innovative Approach to Predict the Diffusion Rate of Reactant’s Effects on the Performance of the Polymer Electrolyte Membrane Fuel Cell

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