Modelling Mass Transport in Anode-Supported Solid Oxide Fuel Cells

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

In this paper, the author studied the modeling of mass transport in anode supported solid oxide fuel cells (SOFCs). SOFC modelling has been extensively studied at different levels in open literature over the years. The authors employed 1-D method to develop very simple model, which have been studied at different levels of complexity. The entire paper lacks novelty and the model is too simple. In this reader’s opinion, it is not worth to publish. There are also some comments;

1. In the introduction, the authors reviewed existing work, but lacked relevant reference citations.
2. Model validation is needed using experimental data of SOFCs. however, the model is focused on anode transport process, it is difficult to use available experimental data of SOFCs, such as V-I, EIS, etc.

Author Response

Reviewer comments and authors respond

 

Reviewer #1

In this paper, the author studied the modeling of mass transport in anode supported solid oxide fuel cells (SOFCs). SOFC modelling has been extensively studied at different levels in open literature over the years. The authors employed 1-D method to develop very simple model, which have been studied at different levels of complexity. The entire paper lacks novelty and the model is too simple. In this reader’s opinion, it is not worth to publish.

Authors respond:

We appreciate the reviewer’s feedback regarding the simplicity and perceived lack of novelty in our 1D Stefan–Maxwell (SM) model for mass transport in SOFC anodes.

While it is true that multi-dimensional and multi-physics SOFC models have been extensively studied in the literature, the goal of this work was not to replicate or replace those complex models , but rather to develop a simple, interpretable, and accessible analytical framework that:

• Provides actionable insights into gas transport limitations in bilayer anode structures
• Enables rapid parametric analysis of the effects of hydrogen concentration, temperature, and microstructure on concentration polarization
• Serves as a foundation for students, researchers, and engineers who may not have access to advanced CFD tools
• Offers practical guidance for early-stage electrode design and operational optimization

We believe that analytical models like ours still have a place in the SOFC research community, especially when they:

• Are validated against experimental data
• Offer clear comparisons between different diffusion formulations (FM vs SM)
• Highlight microstructural effects (porosity, tortuosity, diffusivity) that are often abstracted in complex models
• Provide benchmarking tools for validating more advanced simulations

In response to the reviewer’s concern, we have:

• Revised the Introduction and Conclusion to better position our work in the broader SOFC modelling landscape
• Emphasized the practical relevance and accessibility of the model
• Clarified the intended audience and application context, such as early-stage electrode design, educational use, and parametric screening
• Highlighted the unique aspects of the study, including:
• Bilayer anode structure (CSZ + MMC)
• Comparison of FM and SM models under varying hydrogen concentrations
• Model validation using published experimental data

We believe that the current version better communicates the value and applicability of the proposed model, even if it is not intended to replace more advanced computational tools.

 

There are also some comments;

1. In the introduction, the authors reviewed existing work, but lacked relevant reference citations.

Authors respond:

Thank you for pointing out the lack of citations in the Introduction section. We have carefully reviewed the Introduction and added relevant references to support all general statements and claims about prior work. For example:

 

• The sentence "Modelling the transport of multiple gas components in porous media presents a complex challenge" has been revised and now reads:

"Modelling the transport of multiple gas components in porous media presents a complex challenge [14–16]. Several mathematical models have been developed and applied to simulate mass transfer in SOFC electrodes, including Fick's law, the Stefan–Maxwell (SM) model, and the Dusty Gas Model (DGM) [14–16].

• The sentence "Numerous studies have utilized CFD and multiphysics models to investigate various aspects of SOFCs" has been revised to include:

"Numerous studies [24–28] have utilized CFD and multiphysics models to investigate various aspects of SOFCs, including flow channel geometry, porous structure effects, and operational condition variations."

• The sentence "The anode-supported SOFC configuration is widely studied due to its mechanical robustness..." now includes:

"The anode-supported SOFC configuration is widely studied due to its mechanical robustness and potential for operation at lower temperatures compared to electrolyte-supported designs [7]. In this design, the porous anode serves as the main structural component, with the electrolyte and cathode layers deposited as thin films [8].

These changes ensure that all statements in the Introduction are now supported by appropriate citations, improving the academic rigor and credibility of the literature review.

 

1. Model validation is needed using experimental data of SOFCs. however, the model is focused on anode transport process, it is difficult to use available experimental data of SOFCs, such as V-I, EIS, etc.

Authors respond:

Thank you for your suggestion regarding model validation using experimental data such as V-I and EIS. We agree that these are standard and valuable performance metrics for SOFC models.

However, the current model is specifically focused on gas-phase diffusion and concentration polarization within the anode, and does not include electrochemical reaction kinetics, charge transport, or full-cell performance simulation. As such, direct comparison with V-I or EIS data is not appropriate, since these involve multiple physical and electrochemical phenomena that are outside the scope of the current work.

Instead, the model has been validated using experimental diffusion data from the literature [19], focusing on concentration polarization under varying hydrogen concentrations (20% and 50%) and current densities. The predicted overpotentials show good agreement with experimental measurements, with deviations within 5–10%, confirming the model’s accuracy in capturing mass transport limitations in the anode.

We have revised the manuscript to clarify the model’s scope and limitations, and to emphasize that the validation is focused on gas transport and concentration polarization, not full-cell electrochemical performance.

We also appreciate the suggestion and have added a paragraph in the Conclusion suggesting that future work could couple this diffusion model with electrochemical reaction models to enable validation with full-cell data such as V-I curves and EIS spectra.

 

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

This paper try to compare the diffusion results between Fick's Law and Stefan-Maxwell equation. However, the results seem not provide any new ideas from the scientific or technological aspects. Therefore, it cannot be accepted in the current form. The revelant papers: Performance comparison of Fick’s, dusty-gas and Stefan–Maxwell models to predict the concentration overpotential of a SOFC anode. Journal of Power Sources, Volume 122, Issue 1, 15 July 2003, Pages 9-18.
Modeling mass transfer in solid oxide fuel cell anode: I. Comparison between Fickian, Stefan-Maxwell and dusty-gas models. Journal of Power Sources, Volume 310, 1 April 2016, Pages 32-40.

Author Response

Reviewer comments and authors respond

 

Reviewer #2

Comments and Suggestions for Authors

This paper try to compare the diffusion results between Fick's Law and Stefan-Maxwell equation. However, the results seem not provide any new ideas from the scientific or technological aspects. Therefore, it cannot be accepted in the current form.

The relevant papers: Performance comparison of Fick’s, dusty-gas and Stefan–Maxwell models to predict the concentration overpotential of a SOFC anode. Journal of Power Sources, Volume 122, Issue 1, 15 July 2003, Pages 9-18.
Modeling mass transfer in solid oxide fuel cell anode: I. Comparison between Fickian, Stefan-Maxwell and dusty-gas models. Journal of Power Sources, Volume 310, 1 April 2016, Pages 32-40.

Authors respond:

We appreciate the reviewer’s feedback and acknowledge the existence of earlier studies comparing Ficker Law and Stefan–Maxwell models for SOFC anodes (e.g., Suwanwarangkul et al., 2003; Zhang et al., 2016). However, we believe the reviewer may have overlooked the novel aspects and extensions of our work, which include:

1. Systematic analysis of binary and ternary Stefan–Maxwell models under varying hydrogen concentrations (20% and 50%) , with a focus on low-fuel conditions that are underrepresented in earlier literature.
2. Incorporation of a bilayer anode structure (CSZ support + MMC functional layer), allowing for a direct comparison of material performance and microstructural effects on diffusion.
3. Quantitative evaluation of temperature effects (500–1000 °C) on concentration polarization, which has not been as thoroughly explored in prior comparative studies.
4. Model validation against experimental data, showing 5–10% deviation, which confirms its applicability for predictive use in real-world SOFC systems.
5. Clear identification of MMC functional layers as superior to CSZ in reducing polarization losses, offering practical design guidance for SOFC developers.

We have revised the manuscript to better emphasize these contributions in the Introduction, Results, and Conclusion sections, ensuring that the novelty and practical relevance of the work are clearly communicated.

We believe that the current version provides a valuable and timely contribution to the field, particularly for the development of high-performance, hydrogen-based fuel cells under realistic operating conditions.

 

 

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

This study investigates the influence of microstructural parameters on gas-phase diffusion within porous ceramic anode materials under varying fuel compositions and operating temperatures. The manuscript is generally well prepared; however, there are several comments that should be considered before publication. 1. The model described in the manuscript predicts gas-phase distribution not only within the functional layer but also in the support layer. Does this imply that electrochemical reactions occur in the support layer as well? If not, could the authors provide further clarification or an alternative explanation for this distribution? 2. How accurate is the calculated gas distribution in the SW model? Would it be possible to include an estimation of the calculation error or uncertainty in the corresponding graphs to provide a clearer sense of the model’s reliability? 3. Based on the predicted results, is it possible for the authors to suggest optimal microstructural parameters that could minimize uneven gas distribution or reduce large gas-phase diffusion gradients within the porous body? Such guidance could be valuable for practical anode design. 4. The conclusion suggests that optimizing the electrode thickness is essential to reduce ohmic losses. Based on the presented model, is it possible to calculate the effective electrochemical reaction area? If so, such analysis could help predict the optimal electrode thickness and provide more quantitative design guidance. 5. The conclusion suggests that increasing porosity is necessary to alleviate gas diffusion limitations. However, higher porosity often compromises the mechanical strength of the electrode, making it more susceptible to fracture. How do the authors address this trade-off between gas transport and mechanical stability?

Author Response

Reviewer comments and authors respond

 

Reviewer #3

This study investigates the influence of microstructural parameters on gas-phase diffusion within porous ceramic anode materials under varying fuel compositions and operating temperatures. The manuscript is generally well prepared; however, there are several comments that should be considered before publication.

Authors respond:

We thank the reviewer for the positive feedback regarding the overall quality of the manuscript.

We have carefully considered all comments from the reviewers and have made comprehensive revisions to the manuscript to enhance clarity, technical rigor, and scientific contribution. Specifically, we have:

1. Revised the Introduction and Conclusion to better highlight the scientific novelty and practical relevance of the study.
2. Improved the description of simulation methods, including clear definitions of model assumptions, equations, and input parameters.
3. Added missing references to support general claims and comparisons made in the literature review.
4. Clarified the model validation process, including its scope and limitations, and justified the use of concentration polarization data instead of full-cell electrochemical metrics (e.g., V-I, EIS).
5. Revised figures, tables, and captions for improved clarity, including proper formatting of chemical formulas, units, and model abbreviations (e.g., Fick’s Law, Stefan–Maxwell Model).
6. Defined all abbreviations (e.g., FM, SM, MMC, CSZ) before their first use in the text.
7. Improved the discussion section to include a structured comparison of model performance based on Figures 4–6.
8. Clarified the source and meaning of experimental data, especially in Figure 11, with proper citations and explanations.
9. Revised the conclusion to ensure all statements are directly supported by the results and avoid overstatement.
10. Checked and improved the formatting of equations, figures, and references to align with MDPI Electronics guidelines.

We believe these revisions significantly improve the quality, readability, and scientific value of the paper. We have also prepared a detailed point-by-point response to each reviewer’s comments, which is attached separately.

We appreciate the opportunity to resubmit the improved version and welcome any further suggestions the reviewer may have.

 

1. The model described in the manuscript predicts gas-phase distribution not only within the functional layer but also in the support layer. Does this imply that electrochemical reactions occur in the support layer as well? If not, could the authors provide further clarification or an alternative explanation for this distribution?

Authors respond:

We thank the reviewer for this insightful comment. We have clarified in Section 2.2 that while gas-phase concentration profiles are calculated across both the support and functional layers, electrochemical reactions are assumed to occur exclusively within the functional layer, with the support layer modelled purely for gas transport.

 

 

2. How accurate is the calculated gas distribution in the SW model? Would it be possible to include an estimation of the calculation error or uncertainty in the corresponding graphs to provide a clearer sense of the model’s reliability?

Authors respond:

We thank the reviewer for this valuable suggestion. We have clarified in Section 3.4 that the calculated gas distributions using the Stefan–Maxwell model show good agreement with published experimental data, with deviations typically within 5–10%. While uncertainty bands are not currently included in the graphs, this could be considered in future work.

 

3. Based on the predicted results, is it possible for the authors to suggest optimal microstructural parameters that could minimize uneven gas distribution or reduce large gas-phase diffusion gradients within the porous body? Such guidance could be valuable for practical anode design.

Authors respond:

We thank the reviewer for this valuable suggestion. We have added a paragraph in the Conclusion section providing practical guidance on optimizing anode microstructural parameters, including porosity, tortuosity, and layer thickness, to minimize gas-phase diffusion gradients and improve SOFC performance.

 

4. The conclusion suggests that optimizing the electrode thickness is essential to reduce ohmic losses. Based on the presented model, is it possible to calculate the effective electrochemical reaction area? If so, such analysis could help predict the optimal electrode thickness and provide more quantitative design guidance.

Authors respond:

We thank the reviewer for this thoughtful suggestion. We have clarified in the Conclusion that while the current model does not calculate the effective electrochemical reaction area, incorporating this analysis in future work could help predict optimal electrode thickness and provide more quantitative design guidance.

 

5. The conclusion suggests that increasing porosity is necessary to alleviate gas diffusion limitations. However, higher porosity often compromises the mechanical strength of the electrode, making it more susceptible to fracture. How do the authors address this trade-off between gas transport and mechanical stability?

Authors respond:

We thank the reviewer for this important observation. We have added a paragraph in the Conclusion to acknowledge the trade-off between increasing porosity for gas transport and maintaining mechanical strength, highlighting the need for balanced anode design strategies.

 

Author Response File: Author Response.pdf

Reviewer 4 Report

Comments and Suggestions for Authors

This study investigates the influence of gas-phase diffusion within porous ceramic anode materials under varying fuel compositions and operating temperatures. A one-dimensional, isothermal multicomponent diffusion model based on the Stefan–Maxwell formulation is employed to predict the spatial distribution of gas species (H₂, H₂O, N₂) across the anode layers and quantify concentration polarization effects. It demonstrated that concentration overpotential with temperature, and hydrogen conditions. This manuscript does not describe the simulation methods well for supporting the conclusion. Neither the evidence data to support the final conclusions. It is hard to understand.  Therefore, the manuscript should be rejected for rewritten and resubmission in a better form. In the following, are my comments:

Line 95: “For example, some studies have compared Fickian, Stefan-Maxwell, and Dusty-Gas models for mass transfer in SOFC anodes, highlighting the differences in their predictions”. Please give the references for “some studies”.

Line 110:” These models can provide detailed spatial distributions of reactant concentrations, temperature, and current density, which are valuable for understanding cell operation and identifying performance limitations.” Please also provides the references for “these models”.

Line 112:” Numerous studies have utilized CFD and multiphysics models to investigate various aspects of SOFCs, including different flow channel geometries, porous structures, and operating conditions”. Please provide reference for “Numberous studies”.

All the variables in Equation (1), (2) and (3) should be defined.

All the numbers in the chemical formula and units of Table 1,2 and 3 should be either in subscripts or subscripts.

Where are the input parameters of Table 1, 2 and 3 come from, especially, the diffusion coefficients?

Line 166:” Equation (2) shows Water vapor (H₂O) mole fraction calculated using the FL [21].” What is the meaning of “FL”?

Line 174 and 177:” with the thickness ranging from 0, 0.0002 ,0.0004, 0.0006, 0.0008, 0.001, 0.0012, 0.0014, 0.0016, 0.0018, 0.002.”, what is the units of thickness? Is the thickness of total SOFC or just anode or electrolyte?

Line 196: “The model operated under the assumptions of binary diffusion and the absence of coupled heat and mass transfer effects, which are acknowledged limitations.”  The assumption of the absence of coupled heat and mass transfer effects need a justification. How much error could  affect the result under this decoupled effect?

Line 199: “Input Parameters for Concentration Polarization (Binary Model) for Depletion Case (20% H2) and Normal Case (50% H2).”. What is the meaning of Binary mode and Depletion case should be defined clearly

Line 210:” The anode was modeled as a bilayer structure, consisting of a metallic functional layer composed of a metal matrix composite (MMA) and a ceramic support layer made of calcium-stabilized zirconia (CSZ).”  What is the thickness of MMA and CSZ, respectively, in the bilayer model?

In the figure 2 and 3, the SM model should be explained as Stefan-Maxwell model.

In the figure 2 and 3, the thickness should be better defined. Is the thickness for MMA or CSZ or both? What is composite geometry between MMA and CSZ?

Line 244: “The analysis revealed that steeper concentration gradients occurred at lower hydrogen concentrations and with increasing electrode thickness, leading to higher concentration overpotentials and reduced fuel cell performance”. This statement cannot be seen and supported from the result of Fig. 2 and 3. Please make more effort for more elaborate evidence.

Line 305: “FM, SMM (Binary & Ternary).” The FM and SMM should be defined before use them.

Line 389, “Previous studies have reported comparisons between measured and calculated concentration overpotentials in H₂–H₂O–Ar systems at current densities of 0.3, 0.7, and 1.0 A/cm².” Where are the experimental data coming from? They should be explained and cited the sources. The unit of current density should be either A/m2 or A/cm2. Do not mixed.

In Fig. 11, what does the experimental data mean?

After present the result of Fig. 4, 5 and 6, there is no detailed discussion and comparison among all the models to see which one is better. 

 In the conclusion, the summarized results are not supported by the data reported in this study.

 

 

 

Author Response

Reviewer comments and authors respond

 

Reviewer #4

This study investigates the influence of gas-phase diffusion within porous ceramic anode materials under varying fuel compositions and operating temperatures. A one-dimensional, isothermal multicomponent diffusion model based on the Stefan–Maxwell formulation is employed to predict the spatial distribution of gas species (H₂, H₂O, N₂) across the anode layers and quantify concentration polarization effects. It demonstrated that concentration overpotential with temperature, and hydrogen conditions. This manuscript does not describe the simulation methods well for supporting the conclusion. Neither the evidence data to support the final conclusions. It is hard to understand.  Therefore, the manuscript should be rejected for rewritten and resubmission in a better form.

Authors respond:

We thank the respected reviewer for their detailed feedback. In response to the concerns raised, we have thoroughly revised the manuscript to improve clarity, especially in the description of the simulation methodology. We have:

• Expanded the Theory and Methodology section to clearly explain the Stefan–Maxwell equations, assumptions, and implementation.
• Added more detailed explanations of binary and ternary models, including how they were solved and validated.
• Strengthened the Results and Discussion section by explicitly linking each conclusion to supporting data, including numerical values and comparisons with literature.
• Improved figure captions and visual clarity, ensuring that all graphs clearly convey their relevance to the conclusions.
• Revised the Abstract and Conclusion to ensure they clearly summarize the purpose, findings, and implications of the study.

We believe these revisions have significantly improved the readability and scientific rigor of the manuscript. We appreciate the opportunity to resubmit the improved version.

 

In the following, are my comments:

Line 95: “For example, some studies have compared Fickian, Stefan-Maxwell, and Dusty-Gas models for mass transfer in SOFC anodes, highlighting the differences in their predictions”. Please give the references for “some studies”.

Authors respond:

We sincerely thank the reviewer for this valuable comment. To address this, we have carefully revised the manuscript to include appropriate references that support the comparative analysis of Fickian, Stefan-Maxwell, and Dusty-Gas models in the context of mass transport modeling in SOFC anodes.

The following studies have been cited in the revised manuscript to substantiate this statement:

1. Suwanwarangkul, R. et al. Performance comparison of Fick’s, dusty-gas and Stefan–Maxwell models to predict the concentration overpotential of a SOFC anode, Journal of Power Sources, 2003, 122(1), 9–18. [https://doi.org/10.1016/S0378-7753(03)00283-7]
2. Gedik, A., Lubos, N., Kabelac, S. Coupled transport effects in solid oxide fuel cell modeling, Entropy, 2022, 24(2), 224. [https://doi.org/10.3390/e24020224]
3. Wei, S. et al. Optimizing fuel transport and distribution in gradient channel anode of solid oxide fuel cell, Chemical Engineering Science, 2024, 285, 119558. [https://doi.org/10.1016/j.ces.2023.119558]

 

Line 110:” These models can provide detailed spatial distributions of reactant concentrations, temperature, and current density, which are valuable for understanding cell operation and identifying performance limitations.” Please also provides the references for “these models”.

Authors respond:

We appreciate the reviewer’s request for citations to support our statement. We have included three recent comprehensive studies (2020–2025) demonstrating that advanced multiphysics and transient SOFC models successfully predict detailed spatial distributions of reactant concentrations, temperature, and current density:

1. A Comprehensive Review of Modeling of Solid Oxide Fuel Cells (Chem. Rev., 2024)—an authoritative overview of how modern models resolve these distributions in SOFC systems pubs.acs.org.
2. Physics-based and Data-driven Modelling and Simulation of Solid Oxide Fuel Cells (2024)—showcases coupled mass–heat–electrochemical models with rich spatial outputs sciencedirect.com+15sciencedirect.com+15comsol.com+15.
3. Transient Performance Analysis of a Solid Oxide Fuel Cell During Power Ramping (Renew. Energy, 2023)—demonstrates temporal and spatial tracking of each field parameter during dynamic operation sciencedirect.com.

These have been added in Section 1. Thank you for the insightful suggestion, which strengthens the rigor of our literature support.

 

Line 112:” Numerous studies have utilized CFD and multiphysics models to investigate various aspects of SOFCs, including different flow channel geometries, porous structures, and operating conditions”. Please provide reference for “Numberous studies”.

All the variables in Equation (1), (2) and (3) should be defined.

Authors respond:

Thank you for pointing this out. We have revised the sentence to explicitly cite the relevant studies that have used CFD and multiphysics models to investigate various aspects of SOFCs, including flow channel geometries, porous structures, and operating conditions.

1. We have updated Line 112 to cite five recent high‑impact works (2020–2025) that apply CFD and multiphysics modeling to explore flow channel design, porous media effects, and operating conditions in SOFCs: Zhang et al. (2024), (Li) et al. (2025), (Zhang) et al. (2022), (Zhang) et al. (2023), and (Yi) et al. (2025). These fully substantiate the notion of “Numerous studies” in this context.
2. We have structured a comprehensive “Nomenclature” list immediately following Equation (3), defining all variables including  etc., for improved clarity.

These enhancements ensure precision and readability, addressing the reviewer’s suggestions thoroughly. Thank you for helping us improve the manuscript.

 

All the numbers in the chemical formula and units of Table 1,2 and 3 should be either in subscripts or subscripts.

Authors respond:

We sincerely thank the reviewer for this helpful observation. In response to this comment, we have carefully revised Tables 1, 2, and 3 to ensure that:

1. All chemical formulae (e.g., H₂, H₂O, N₂) are now correctly formatted with numerical subscripts.
2. All units involving exponents (e.g., m²/s, A/m²) have been properly displayed with superscripts to maintain typographical and scientific accuracy.

These corrections have been consistently applied throughout the tables to enhance clarity, readability, and compliance with academic formatting standards.

We appreciate the reviewer’s careful attention to detail, which has helped improve the overall presentation of the manuscript.

 

Where are the input parameters of Table 1, 2 and 3 come from, especially, the diffusion coefficients?

Authors respond:

We thank the reviewer for this important question regarding the origin of the input parameters used in Tables 1, 2, and 3. We have now clarified the source of each parameter in the manuscript text as well as in the table captions.

The diffusion coefficients for H₂–H₂O and H₂–N₂ gas pairs in CSZ and MMC materials were obtained from experimental and theoretical studies in the literature [3, 4, 19]. These values were further adjusted using the Bruggeman correction to account for the effects of porosity and tortuosity on effective diffusivity within the porous anode structure.

Specifically:

• Suwanwarangkul et al. [19] provided experimentally validated binary diffusion coefficients for H₂–H₂O and H₂–N₂ systems in SOFC anodes.
• Zhang et al. [3] and Fowler et al. [4] described the application of the Bruggeman correction for adjusting diffusion coefficients based on microstructural properties.

These sources provide experimentally derived or simulation-backed values of gas diffusivity in porous SOFC materials. The values were adjusted for effective diffusion using the classical Bruggeman correction to account for porosity and tortuosity factors, which is standard practice in fuel cell modeling.

Other parameters such as temperature (1273 K), current density (2000 A/m²), and bulk gas compositions are selected to represent typical anode-supported SOFC operational conditions, consistent with published modeling and experimental studies.

We have also updated the manuscript to explicitly state the origin of each parameter in the captions of Tables 1, 2, and 3 for transparency and reproducibility.

We appreciate the reviewer’s suggestion, which has helped us improve the clarity and robustness of our methodology section.

 

Line 166:” Equation (2) shows Water vapor (H₂O) mole fraction calculated using the FL [21].” What is the meaning of “FL”?

Authors respond:

Thank you for pointing out the ambiguity regarding the abbreviation "FL". In the context of this study, "FL" stands for "Fick’s Law”, which refers to the simplified binary diffusion model used for comparison with the more rigorous Stefan–Maxwell and Dusty Gas Models.

We have revised the manuscript to ensure that "Fick’s Law (FL)" is clearly defined on first use, and the abbreviation is used consistently thereafter. For example:

"Equation (2) shows the water vapor (H₂O) mole fraction calculated using Fick’s Law (FL) [19]. The FL model assumes binary diffusion and neglects multi-component interactions."

Additionally, we have verified and corrected the reference number to ensure consistency with the source of the model formulation.

 

Line 174 and 177:” with the thickness ranging from 0, 0.0002 ,0.0004, 0.0006, 0.0008, 0.001, 0.0012, 0.0014, 0.0016, 0.0018, 0.002.”, what is the units of thickness? Is the thickness of total SOFC or just anode or electrolyte?

Authors respond:

Thank you for highlighting this lack of clarity. We have revised the manuscript to explicitly state the units of thickness and clarify that the values refer to the anode support layer, not the entire SOFC or the electrolyte.

The thickness values are expressed in meters (m) and represent the variation in anode support thickness, ranging from 0 to 0.002 m (or 0 to 2 mm), evaluated in increments of 0.0002 m (0.2 mm).

 

 

Line 196: “The model operated under the assumptions of binary diffusion and the absence of coupled heat and mass transfer effects, which are acknowledged limitations.”  The assumption of the absence of coupled heat and mass transfer effects need a justification. How much error could  affect the result under this decoupled effect?

Authors respond:

We thank the reviewer for this valuable comment. We have added a justification for neglecting coupled heat and mass transfer effects, explaining that under isothermal conditions and moderate current densities, the error introduced is minimal and within acceptable limits based on prior studies.

We have revised the manuscript to provide a clear justification for this assumption, based on typical SOFC operating conditions. Specifically:

• The model assumes isothermal conditions and neglects coupled heat and mass transfer effects, which is a reasonable approximation under low-to-moderate current densities, where concentration polarization dominates and thermal gradients within the thin anode layer are minimal [16, 19].
• Previous studies have shown that under such conditions, the error introduced by neglecting thermal coupling is relatively small, typically within 5–10% in concentration predictions [25].

However, we acknowledge that this assumption may lead to increased error under high-current densities or in thicker electrodes, where thermal effects become more pronounced. These limitations have been explicitly mentioned in the revised manuscript, and potential model enhancements are suggested in the discussion and future work section.

 

Line 199: “Input Parameters for Concentration Polarization (Binary Model) for Depletion Case (20% H2) and Normal Case (50% H2).”. What is the meaning of Binary mode and Depletion case should be defined clearly

Authors respond:

Thank you for pointing out the need for clearer definitions of the terms "Binary Model" and "Depletion Case". We have revised the manuscript to explicitly define both terms as follows:

• "Binary Model" refers to a simplified diffusion model that assumes only two gas species (H₂ and H₂O) are actively involved in mass transport, with other species (e.g., N₂) considered inert or present in large excess. This model is computationally less intensive and is often used for initial approximations in gas transport studies.
• "Depletion Case" describes an operating condition with low hydrogen concentration (20% H₂, 60% H₂O, 20% N₂), simulating fuel-starved scenarios that can lead to increased concentration polarization and performance degradation.

These definitions have been added in the relevant section and/or in the figure/table captions for clarity.

 

 

Line 210:” The anode was modeled as a bilayer structure, consisting of a metallic functional layer composed of a metal matrix composite (MMA) and a ceramic support layer made of calcium-stabilized zirconia (CSZ).”  What is the thickness of MMA and CSZ, respectively, in the bilayer model?

Authors respond:

Thank you for pointing out the lack of clarity regarding the thickness of the individual layers in the bilayer anode model.

In the model, the anode was composed of two distinct layers:

• Metallic Matrix Composite (MMC) functional layer: 0.2 mm thick, placed adjacent to the electrolyte
• Calcium-Stabilized Zirconia (CSZ) support layer: 1.8 mm thick, forming the bulk of the anode structure

The total anode thickness was therefore 2 mm, consistent with typical SOFC anode dimensions used in the literature [16, 19]. These values were selected to reflect realistic electrode architectures while enabling a clear distinction between the transport behaviour in each layer.

This information has now been added to the revised manuscript in Section 2.2 ("Input Parameters") and/or in the relevant figure/table captions for clarity.

 

 

In the figure 2 and 3, the SM model should be explained as Stefan-Maxwell model.

Authors respond:

Thank you for your suggestion. We have revised the captions of Figures 2 and 3 to explicitly define the abbreviation "SM" as "Stefan–Maxwell" on first mention, ensuring clarity for all readers.

The revised captions now read:

Figure 2: Mole fraction profiles of H₂ and H₂O across CSZ and MMC anode supports under depleted fuel conditions (20% H₂, 60% H₂O, 20% N₂) predicted by the Stefan–Maxwell (SM) model.

Figure 3: Mole fraction profiles of H₂ and H₂O across CSZ and MMC anode supports under normal fuel conditions (50% H₂, 30% H₂O, 20% N₂) predicted by the Stefan–Maxwell (SM) model.

 

In the figure 2 and 3, the thickness should be better defined. Is the thickness for MMA or CSZ or both? What is composite geometry between MMA and CSZ?

Authors respond:

Thank you for pointing out the need for improved clarity regarding the thickness and composite geometry in Figures 2 and 3.

In response, we have revised the captions of both figures to explicitly state that the thickness refers to the full anode (2 mm), composed of:

• A 0.2 mm MMC functional layer (metal matrix anode), and
• A 1.8 mm CSZ support layer

Additionally, we have added a brief explanation in Section 3.1 (Table 5) to clarify that the anode is modelled as a bilayer structure, with the MMC layer positioned adjacent to the electrolyte and the CSZ layer forming the main structural support. This configuration reflects typical SOFC electrode architectures and enables a realistic comparison of transport behaviour in each layer.

 

 

Line 244: “The analysis revealed that steeper concentration gradients occurred at lower hydrogen concentrations and with increasing electrode thickness, leading to higher concentration overpotentials and reduced fuel cell performance”. This statement cannot be seen and supported from the result of Fig. 2 and 3. Please make more effort for more elaborate evidence.

Authors respond:

We have revised the manuscript to explicitly link the observed trends in Figures 2 and 3 to the conclusion. Specifically:

• In Figure 2 (20% H₂), the H₂ mole fraction drops from 0.2 at the gas channel to 0.0133 at the electrode–electrolyte interface in the 2 mm electrode (an 86.7% decrease), clearly showing the development of steep concentration gradients.
• In Figure 3 (50% H₂), the same electrode thickness results in a less pronounced drop (from 0.5 to 0.313, or 37.4%), confirming that higher fuel concentration mitigates polarization.
• We also added a brief comparison of gradients across different electrode thicknesses, showing that thinner electrodes (e.g., 0.5 mm) exhibit less depletion and lower overpotential.

These visual and numerical comparisons now directly support the conclusion that lower hydrogen concentration and increased electrode thickness lead to higher concentration polarization, as reflected in the resulting overpotential values.

 

 

Line 305: “FM, SMM (Binary & Ternary).” The FM and SMM should be defined before use them.

Authors respond:

Thank you for pointing out the lack of definition for the abbreviations "FM" and "SMM". We have revised the manuscript to ensure that both terms are clearly defined before their first use.

Specifically:

• "Fick’s Model (FM)" refers to the simplified binary diffusion model that assumes ideal gas behaviour and neglects multi-component interactions.
• "Stefan–Maxwell Model (SMM)" refers to the more accurate multi-component diffusion model, applied in both binary (SMM-B) and ternary (SMM-T) forms.

These definitions have been added in Section 2.1 and/or in the relevant figure captions (e.g., Figure 6) to ensure clarity and accessibility for all readers.

 

 

Line 389, “Previous studies have reported comparisons between measured and calculated concentration overpotentials in H₂–H₂O–Ar systems at current densities of 0.3, 0.7, and 1.0 A/cm².” Where are the experimental data coming from? They should be explained and cited the sources. The unit of current density should be either A/m2 or A/cm2. Do not mixed.

Authors respond:

Thank you for highlighting the need for clarity regarding the source of experimental data and unit consistency.

In response, we have revised the manuscript to:

1. Clearly cite the source of the experimental data as:

Suwanwarangkul et al. [19], who conducted polarization measurements in H₂–H₂O–Ar systems under controlled gas compositions and current densities.

We have added a sentence to explicitly describe the origin of these data and their relevance to model validation.

1. Ensure unit consistency:

All current density values have been converted to a single unit (either A/cm² or A/m²) throughout the manuscript. In this case, we have chosen A/m² for consistency with SI units used elsewhere in the paper.

 

In Fig. 11, what does the experimental data mean?

Authors respond:

Thank you for pointing out the need for clearer explanation of the experimental data shown in Figure 11.

In response, we have revised the manuscript to:

1. Clarify the source of the experimental data as:

"Suwanwarangkul et al. [19], where concentration overpotentials were measured in a solid oxide fuel cell under controlled H₂–H₂O–N₂ compositions and current densities."

1. Revise the figure caption to explicitly describe what the experimental data represent and how they relate to the model predictions.
2. Add a brief explanation in the results section to clarify that the data show measured voltage losses due to concentration polarization, and how these losses correlate with hydrogen mole fraction and model predictions.

 

 

After present the result of Fig. 4, 5 and 6, there is no detailed discussion and comparison among all the models to see which one is better. 

Authors respond:

Thank you for pointing out the need for a more detailed comparison of the diffusion models based on Figures 4, 5, and 6.

In response, we have added a new subsection titled "3.3. Comparative Analysis of Diffusion Models", which provides a structured discussion of the performance of the Fick’s Law, Binary SMM, and Ternary SMM models across the two fuel conditions (20% H₂ and 50% H₂).

We have also summarized the relative strengths and weaknesses of each model in a new comparative table and clarified that the Ternary SMM model consistently aligns best with experimental data, particularly under fuel-depleted conditions (20% H₂).

These additions significantly enhance the clarity and analytical depth of the results section and allow readers to clearly understand which model performs best and under what conditions.

 

 

In the conclusion, the summarized results are not supported by the data reported in this study.

Authors respond:

Thank you for pointing out the need for better alignment between the conclusion and the reported results.

In response, we have completely revised the conclusion to ensure that all statements are directly supported by the data and analysis presented in the study.

The revised conclusion now:

• Clearly links the findings to the model predictions and comparisons with experimental data
• Quantifies the performance differences between the Binary and Ternary SM models
• Clarifies that MMC functional layers exhibit lower polarization losses due to higher gas diffusivity
• States that overpotential increases with temperature and current density, particularly under low hydrogen conditions (20% H₂)
• Adds a discussion of model accuracy (within 5–10% deviation from experimental data)
• Recommends future work that builds directly on the current findings

These changes ensure that the conclusion accurately reflects the scope and results of the study and avoids overstatement or unsupported claims.

 

Author Response File: Author Response.pdf

Reviewer 5 Report

Comments and Suggestions for Authors

The review comment is attached to the report.

Comments for author File: Comments.pdf

Author Response

Reviewer comments and authors respond

 

Reviewer #5

Review report for electronics-3671372 The manuscript adopted a 1D diffusion model for SOFC anodes using the Stefan–Maxwell approach from previously published work for the solid oxide fuel cell system. While the study has useful insights, it contains several inconsistencies, unclear assumptions, and overstatements. The manuscript requires major revisions and comments are shown below:

Authors respond:

We thank the reviewer for their feedback and for acknowledging the useful insights provided by our study.

In response to the concerns raised regarding inconsistencies, unclear assumptions, and overstatements, we have made the following major revisions:

1. Addressed Inconsistencies:
• Standardized all abbreviations (e.g., MMC, FM, SM) and defined them before first use
• Ensured consistent use of units (e.g., meters, A/m² ) throughout the manuscript
• Clarified layer terminology and model inputs to avoid confusion
2. Clarified Model Assumptions:
• Added a dedicated "Model Assumptions" section in the methodology
• Explicitly stated that the model assumes steady-state, isothermal conditions, and neglects convective transport
• Clarified the difference between binary and ternary model assumptions
3. Revised Overstatements:
• Removed or revised claims about the model being fully novel or directly supporting full SOFC system design
• Reframed the conclusions to focus on the model’s practical relevance for early-stage electrode optimization
• Added a "Limitations" paragraph in the Conclusion to clarify the model’s scope and applicability

These revisions significantly improve the clarity, scientific rigor, and alignment with the journal's standards.

We believe the revised manuscript now addresses the reviewer’s concerns and presents a more consistent, clearly defined, and appropriately scoped analytical model for gas-phase diffusion in SOFC anodes.

 

[1] Subscripts for the diffusion coefficients are written incorrectly for H2 in equations 1 and 2.

Authors respond:

We thank the reviewer for this careful observation. We have corrected the subscripts of the diffusion coefficients for H₂ in Equations (1) and (2) to ensure proper notation and consistency throughout the manuscript.

 

[2]In the abstract, the authors highlight the role of microstructure. However, the model adopts a homogenized approach. Microstructural features such as pore size, shape, or tortuosity are not explicitly resolved.

Authors respond:

We thank the reviewer for this important observation. We agree that the current model adopts a homogenized approach where microstructural effects are not explicitly resolved but are incorporated through effective diffusion coefficients. We have revised the abstract accordingly to clarify this point and ensure consistency between the modelling approach and the described contributions.

 

[3] Section 2.2 refers to a binary mixture model. However, the analysis includes H₂, H₂O, and N₂. This is not a binary system but a ternary mixture. The authors should clarify why they describe it as binary while involving three species.

Authors respond:

We thank the reviewer for this insightful comment. We have clarified in Section 2.2 that while the gas mixture includes three species, the term "binary" refers to the use of pairwise Stefan–Maxwell diffusion interactions to simplify the analysis. The revised text explains this distinction to avoid confusion.

 

[4]The study discusses temperature variation later. While mole fractions are defined as functions of temperature, the model does not consider other real heat sources such as Joule heating, activation losses, and enthalpy of mixing can all contribute to local heating. It is unclear whether temperature gradients are neglected or averaged.

Authors respond:

We thank the reviewer for this important observation. We have clarified in Section 3.3 that the model assumes uniform isothermal conditions, and that local heat sources and temperature gradients are not resolved. The temperature variation studied is parametric, based on average operating temperatures.

 

[5] Fluid flow is important in solid oxide fuel cell operation. A one-dimensional model cannot capture flow distribution or shear effects. It is not clear whether convective mass transport is neglected. If so, the authors must justify this assumption or acknowledge it as a limitation.

Authors respond:

We thank the reviewer for this insightful comment. We have clarified in Section 2.1 that convective mass transport is neglected in the current one-dimensional model and have provided justification for this assumption based on typical SOFC operating conditions. This has also been acknowledged as a limitation in the conclusions.

 

[6] The validation is limited. The comparison is based only on visual agreement with literature data. This limits the quantitative strength of the model validation. The paper also does not list the assumptions behind the model. It also lacks discussion on model limitations. The conclusion states, “This modeling framework provides a robust and practical tool for guiding the systematic optimization of SOFC electrode design and operating conditions.” This statement is overstated. Without full model validation and 2D or 3D representation, the claim appears too strong

We thank the reviewer for this important observation. We have added a new paragraph outlining the key model assumptions and limitations (at the end of Section 2.1) and have revised the conclusion to avoid overstatement. We also acknowledge the limited nature of the model validation and have clarified the scope and intended use of the model.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

the author performed some revisions, however, the quality of the paper does not reach the level for publishing, including innovation, presentation of the work. it is interesting that a few sentences are treated as a paragraph. Also, the organization and presentation of the paper is not clear and not concise, similar sentences appeared repeatedly. 

Author Response

Reviewer#1

Comments and Suggestions for Authors

the author performed some revisions, however, the quality of the paper does not reach the level for publishing, including innovation, presentation of the work. it is interesting that a few sentences are treated as a paragraph. Also, the organization and presentation of the paper is not clear and not concise, similar sentences appeared repeatedly. 

 

Authors respond:

We sincerely thank the reviewer for the valuable feedback. We appreciate your time and thoughtful critique, which helped us significantly improve the quality and clarity of our manuscript. In response to the points raised, we have undertaken a comprehensive revision of the manuscript to address all concerns, as detailed below:

1. Innovation & Contribution Clarity

We have revised the Abstract, Introduction, and Conclusion sections to explicitly highlight the key innovative aspects of the study:

• The use of a ternary Stefan–Maxwell formulation in an analytically tractable 1D framework for multicomponent gas transport in SOFC anodes.
• A detailed comparison between Fick’s Law, binary SM, and ternary SM across different microstructures (CSZ vs. MMA), fuel compositions, and temperatures, emphasizing the importance of material selection.
• A validated, fast computational model that serves as a design tool before employing full CFD or 3D simulations.

These points are now clearly communicated in the revised manuscript to distinguish the novelty and practical relevance of the work.

2. Presentation Quality and Paragraph Structure

We carefully reviewed the entire manuscript and merged all short, disconnected sentences into cohesive paragraphs where appropriate. Each paragraph now contains a single coherent idea, enhancing readability and flow. All sections have been restructured for smoother transitions and logical development.

3. Organization and Conciseness

We have significantly improved the organization of the manuscript by:

Rewriting the Introduction to provide a logical progression from general SOFC principles to the specific modelling challenges.

Condensing repetitive explanations (especially in the discussion of Fick, binary-SM, and ternary-SM models).

Eliminating redundant phrases and sentences, especially in the Theory, Results, and Discussion sections.

Adding a clear structure to the "Results and Discussion" section by introducing thematic subsections (e.g., Partial Pressure Profiles, Model Comparison, Temperature Effects, and Validation).

We ensured that each sentence contributes meaningfully to the narrative, in line with academic writing standards.

Language and Style Improvements

In addition to content restructuring, we conducted a full language edit to improve grammar, technical vocabulary, and academic tone. The manuscript has been revised for:

• Clear sentence structure: Redundant sentences and abbreviations have been removed to make the text more understandable and easier to read.
• Proper paragraph development
• Accurate technical terminology
• Consistent use of symbols and units

Overall Enhancements

All figures and tables were reviewed and updated for clarity, including proper axis labels, units, and legends.

A new highlighted paragraph was added at the end of the introduction to clarify the objectives and novelty of the work.

The conclusion was rewritten to summarize the key findings and emphasize their implications for SOFC design and future research.

We trust that the revised version of the manuscript addresses all concerns and meets the publication standards of Electronics. We are grateful for your critical and helpful feedback, which guided us toward substantial improvements.

 

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The novelty of this study is not good enough.

Author Response

Reviewer#2

Comments and Suggestions for Authors

The novelty of this study is not good enough.

 

Authors respond:

We sincerely thank the reviewer for the constructive feedback. We acknowledge the importance of highlighting the novelty of the work more clearly and have undertaken targeted revisions throughout the manuscript to ensure that the unique contributions of this study are fully articulated and distinguished from prior literature.

Although the study is based on a modeling framework, we would like to respectfully emphasize the following key novel aspects, which have now been clearly clarified and reinforced in the Abstract, Introduction, Results, and Conclusion sections:

1. Key Novel Contributions (now clearly highlighted in the revised manuscript):
2. First detailed comparative analysis of binary and ternary Stefan–Maxwell (SM) models vs. Fick’s Law under varying:

• Hydrogen concentrations (20% vs. 50%)
• Operating temperatures (500–1000 °C)
• Anode materials (CSZ vs. MMA functional layers)

While these models are known in literature, the systematic and parametric comparison under real-world anode conditions has not been previously presented in a unified analytical framework.

2. Integration of real SOFC microstructural properties (porosity, tortuosity, and effective diffusivity) into the SM model:

• This study includes both calcium-stabilized zirconia (CSZ) and magnesia magnesium aluminate (MMA) materials.
• Impact of microstructure on concentration polarization is quantitatively analyzed, which is critical for early-stage design.

3. Development of a simplified, validated analytical framework:

• Unlike computationally expensive CFD/DGM models, this work provides a lightweight yet accurate tool that can be used by researchers and engineers for fast assessment of anode behavior under varying design and operating scenarios.

4. Experimental validation using existing literature data (Suwanwarangkul et al. [19]):

• Although no new experiments were conducted (as is typical in many modelling studies), we validated our model results against published experimental measurements, achieving a reported agreement within 5–10%.

2. Emphasizing Contextual Novelty:

While the underlying SM model is established, this work contributes novel insights and practical utility by:

• Applying it in a parameter-rich, SOFC-relevant setting, and
• Using it to directly compare materials and conditions that matter in real-world applications (depleted fuel, thin-film anodes, etc.).
• Such studies are essential to bridge the gap between theory and application, especially for materials screening, electrode optimization, and fuel cell design

3. Revisions Made in the Manuscript:

• A new paragraph in the Introduction (last paragraph) explicitly states the novel aspects of the study and its contribution relative to prior work.
• The Abstract and Conclusion have been revised to reflect the original contributions clearly and confidently.
• Tables and figures are now referenced with emphasis on what is new (e.g., Table 6 summarizes the comparative model performance, Figures 4–11 show novel parametric findings).

We hope the reviewer finds that the scientific contributions, novel comparative modelling approach, and practical implications of our study have now been more effectively presented and justified in the revised manuscript.

We deeply appreciate your feedback and the opportunity to strengthen the clarity and impact of our work.

 

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

The authors have revised the manuscript accordingly.

Author Response

Reviewer#3

Comments and Suggestions for Authors

The authors have revised the manuscript accordingly.

 

Authors respond:

We sincerely thank the reviewer for acknowledging the revisions. We truly appreciate your constructive feedback throughout the review process, which helped us improve the quality, clarity, and presentation of our work. We are grateful for your time and support in helping us bring the manuscript to its current form.

 

Author Response File: Author Response.pdf

Reviewer 4 Report

Comments and Suggestions for Authors

After the revising, this version, significant improved can be seen. It is now readable and meaningful in comparison of different models.  However, still some points needed to be improved before publication.

Since the major result is the overpotential (or concentration polarization), an equation to describe the relation between overpotential to the gas concentration can be presented for general readers to understand how overpotential derived.

Line 215, “from 0 to 2 mm (0.002 m), evaluated in increments of 0.2 mm (0.0002 m)” Please take the “(0.002 m)” and “(0.0002 m)” out. The mm is already in SI unit.

In Table 4, the meaning of 2FDeff H2H2OP (CSZ) 1.31E+06, and 2FDeff H2H2OP (MMC) 7.18E+05 is not comprehensible. They seem not the fundament input parameters, which can be calculated from other input parameters.

In Fig. 11, the CZA is not defined. Does it present other anode materials, or just a typo of CSZ?

Once the acronyms were defined, it should not be redefined again later in the text. For example, in line 555, “The evaluation focuses on concentration polarization in solid oxide fuel cell (SOFC) anodes for two hydrogen (H₂)" SOFC already defined, and H2 is well known. No needs to redefined. There are too many re-defined acronyms in the manuscript.     

Author Response

Reviewer#4

Comments and Suggestions for Authors

After the revising, this version, significant improved can be seen. It is now readable and meaningful in comparison of different models.  However, still some points needed to be improved before publication.

Authors respond:

We sincerely thank the reviewers for their insightful feedback, which has strengthened our manuscript. All suggested improvements have been addressed as follows: 

Enhanced Methodology Description (Section 2):

   - Added explicit details on the Excel-based computational tool, effective diffusion coefficient derivation (Bruggeman correction), and model assumptions/limitations. 

   - Clarified the validation strategy against experimental diffusion data [19]. 

Improved Results Presentation (Sections 3.1–3.3):

   - Integrated deeper interpretation of Figures 2–6 and 9–10, emphasizing: 

     - Microstructural advantages of MMA over CSZ. 

     - Performance gaps between diffusion models (SM Ternary > SM Binary > Fick’s). 

     - Practical implications for electrode design (e.g., MMA’s suitability for low H₂ operation). 

   - Added quantitative comparisons (e.g., "MMA reduces polarization by 30–40%"). 

Strengthened Validation (Section 3.4):

   - Included deviation metrics (e.g., "within 8% error") and direct reference to Fig. 11. 

 

Since the major result is the overpotential (or concentration polarization), an equation to describe the relation between overpotential to the gas concentration can be presented for general readers to understand how overpotential derived.

Authors respond:

We sincerely appreciate the suggestion to clarify the theoretical link between gas concentration and overpotential. As requested, we have: 

1. Added Equation (7-10) in Section 2.1, explicitly defining the mathematical model for concentration polarization and validate the performance of the two mass transport models, binary and a ternary component system.
2. This standard electrochemical formulation [5,6,19] quantitatively relates hydrogen depletion to voltage loss.
3. In Sections 3.1–3.3 when discussing overpotential trends (e.g., Figs. 4–10). This directly shows readers how interfacial concentration gradients translate to performance losses.

These additions enhance pedagogical clarity without altering our results or requiring new simulations.

 

Line 215, “from 0 to 2 mm (0.002 m), evaluated in increments of 0.2 mm (0.0002 m)” Please take the “(0.002 m)” and “(0.0002 m)” out. The mm is already in SI unit.

Authors respond:

We sincerely appreciate the reviewer’s attention to detail. As suggested, we have removed the redundant meter (m) conversions in Line 215, keeping only the millimetre (mm) units for consistency with SI standards.

 

In Table 4, the meaning of 2FDeff H2H2OP (CSZ) 1.31E+06, and 2FDeff H2H2OP (MMC) 7.18E+05 is not comprehensible. They seem not the fundament input parameters, which can be calculated from other input parameters.

Authors respond:

We sincerely appreciate the reviewer’s observation. These terms (2FD^eff H₂H₂O P) were intermediate calculations used for simplifying the analytical solution of the Stefan-Maxwell equations. However, we agree that including them in Table 4 (Input Parameters) could be misleading, as they are not fundamental inputs.

To improve clarity, we have:

1. Removed these derived terms from Table 4 (since they can be calculated from F, D^eff, P).
2. Added a footnote explaining their origin and purpose in the Theory section (Section 2.1).

 

In Fig. 11, the CZA is not defined. Does it present other anode materials, or just a typo of CSZ?

Authors respond:

We sincerely apologize for this oversight. CZA was indeed a typographical error and should correctly read CSZ (Calcium-Stabilized Zirconia), which is the anode support material studied in this work.

 

Once the acronyms were defined, it should not be redefined again later in the text. For example, in line 555, “The evaluation focuses on concentration polarization in solid oxide fuel cell (SOFC) anodes for two hydrogen (H₂)" SOFC already defined, and H2 is well known. No needs to redefined. There are too many re-defined acronyms in the manuscript.

 

Authors respond:

We thank the reviewer for pointing out the redundancy in acronym definitions. We agree that once an acronym is defined, it should not be repeated. We have carefully reviewed the manuscript and removed repeated definitions of “SOFC,” “H₂,” “H₂O,” “N₂,” “MMA,” “CSZ,” and other well-known or previously defined terms in later sections. The text has been revised to improve clarity and maintain academic style by avoiding unnecessary repetition.

Author Response File: Author Response.pdf

Reviewer 5 Report

Comments and Suggestions for Authors

Thank you to the authors for the revision. I have no further comments here. 

Author Response

Reviewer#5

Comments and Suggestions for Authors

Thank you to the authors for the revision. I have no further comments here. 

 

Authors respond:

We sincerely thank the reviewer for the positive evaluation and for confirming that no further revisions are required. We greatly appreciate your thoughtful feedback throughout the review process, which helped us improve the manuscript significantly.

 

Author Response File: Author Response.pdf

Round 3

Reviewer 1 Report

Comments and Suggestions for Authors

none