Theoretical Analysis of Mass Transfer Behavior in Fixed-Bed Electrochemical Reactors: Akbari-Ganji’s Method
Round 1
Reviewer 1 Report
The significance of your work has not been adressed well. Please reorganize your language and logics to let the significance of your work more clear.
Author Response
We gratefully acknowledge the reviewer's helpful and insightful comments, which were extremely helpful in revising the article. Accordingly, the revised manuscript has been systematically improved with new information and interpretations. Our responses to the referee's comments has been attached.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
The authors developed a simple approximate analytical expression for concentration and effectiveness factors, current for all possible reaction/diffusion parameter values for the slab, cylinder and spherical geometries. The numerical simulation results is consistent with theoretical predictions. These analytical results will be useful to analyze the effect of the behavior of porous catalyst particles subject to both internal mass concentration gradients and temperature gradients in endothermic or exothermic reactions. The results are well presented and support the conclusion. I recommend to accept this manuscript as the current form.
Author Response
We gratefully acknowledge the reviewer's helpful and insightful comments, which were extremely helpful in revising the article. Our responses to the referee's comments has been attached.
Author Response File: Author Response.pdf
Reviewer 3 Report
This paper presents a theoretical analysis of the mass transfer behavior in electrochemical reactors. The Authors use Akbari-Ganji’s method for solving the reaction-diffusion equation that describes the concentration profile of a reactant in the catalyst pellets. The topic of the paper is interesting, however the text is often not clear and the following major issues need to be address for considering the manuscript for publication.
1. The novelty of the paper does not emerge clearly from the manuscript. The Authors should show how the use of Akbari-Ganji’s method for solving the model brings novelty in the field. The numeric solution of Eq. (1) can apparently be achieved fast (my everyday laptop takes ~0.1 s for running the code in Appendix C). Can the Authors demonstrate that it is convenient to use their approach instead than the standard numeric approach with which they benchmark it? The computational time needed for achieving the solution with Akbari-Ganji’s method (also including the solution of Equation 7) should be compared with the computational time of the standard MATLAB numerical method. If the Akbari-Ganji’s method is not significantly faster, the standard MATLAB numerical method seems more appealing, considering the error of the proposed AGM that emerges from Figure 1.
2. The Authors identify as current issue that they aim to tackle the absence of a “rigorous analytical expression for the concentration of a reactant at the catalyst's surface of general geometries has been reported.” (lines 54-58, p. 2). However, the Authors do not obtain such rigorous analytical solution, but a “a simple approximate analytical expression” (line 56, p. 2). Which are the approximations that make the “approximate” solution obtained by the Authors not “rigorous”?
3. It should be specified in the manuscript how Equation (7) is solved. If a numeric approach is used, the methodology proposed by the Authors is numeric, and it cannot be considered “analytical”, as often referred to in the text.
4. Equations 5 and 9: I cannot find the definition of flux ?. The variables and the surface that define flux ? should be specified. Why is this flux important within the discussion?
Minor comments
5. A thorough proofreading is needed. For example, please use consistency when naming the methodology presented in the paper (at different times called “analytical solution”, “approximate analytical solution”, and “proposed AGM”). Also, the expression “effect of the parameter” in the titles of the subsections of the Discussion appears vague
6. Line 10, p.1: do the Authors derive the model for “any shape” or only for slab, cylinder, and sphere geometries? Please amend if the model is not for any shape
7. Figure A1: the x-axis should be defined also for cylindric and spheric geometries
8. Please considering (briefly) expanding the background on Akbari-Ganji’s method in Appendix B
Author Response
We gratefully acknowledge the reviewer's helpful and insightful comments, which were extremely helpful in revising the article. Our responses to the referee's comments has been attached.
Author Response File: Author Response.pdf
Round 2
Reviewer 3 Report
The Authors significantly improved the manuscript and clarified their claim, which consists in the derivation of an approximate analytical expression that correlates effectiveness factor, current, and concentration profiles to physicochemical parameters in electrochemical reactor systems. The topic is interesting, and the manuscript writing has also been improved. However, the following issue should be fixed before publication:
1. The Authors claim in the abstract and in the conclusions that they obtain an analytical expression for the current in the catalyst pellets. I don’t explicitly find it in the paper. Can the Authors clarify? It is maybe related to the flux ψ, but I cannot find the definition of variables i and FD reported in Equation 5 neither in the main text nor in the List of symbols at the end of the manuscript. Please clarify these aspects in the manuscript: this is a key contribution of the paper, since theoretical mass transfer in chemical reactors not involving electrochemical phenomena is already widely discussed in the literature.
Secondary comments:
2. Please define AGM in l. 60 at p.2 instead than at p.3 (acronyms should be defined at the first occurrence).
3. Several typos remain in the manuscript, please carefully amend (e.g., “anelectrochemical” l. 42 in p. 1, and “dimensionlessheat ofreation” p. 2 l.72-73)
4. p.7 l. 185: in this case it should be "exact" analytical solution, if I understand correctly
5. p. 9 l. 207: please clarify within the wording that the obtained solution is not analytical
6. The expression “effect of the parameter” in the titles of the subsections of the Discussion still appears vague
7. The Authors might consider removing the last sentence of the abstract, or rephrasing it to highlight that the accordance with results of a reliable numerical method shows the good accuracy that their approximate solution yields.
Author Response
We gratefully acknowledge the reviewer's helpful and insightful comments, which were extremely helpful in revising the article. Accordingly, the revised manuscript has been systematically improved with new information and interpretations. Our responses to the referee's comments are given below.
Author Response File: Author Response.pdf
Round 3
Reviewer 3 Report
The Authors clarified in their cover letter that their manuscript involves the analysis of "theoretical mass transfer in chemical reactors not involving electrochemical phenomena" (response to comment #1). Hence, this article does not discuss electrochemical reactors, as the title suggests, and as the Authors claimed upon submission of the first version of the manuscript and in the previous revisions.
The novelty of the work does not appear clearly. There is already a wide literature of theoretical mass transfer analysis in chemical reactors not involving electrochemical phenomena, that the Authors do not discuss nor compare against their approach. See, for instance, the following selection of references, that also contain several (exact) analytical solutions for effectiveness factor and concentration for non-spherical catalyst pellets in nonisothermal reactors:
[1] Bird, R., Stewart, W. and Lightfoot, E. (2002) Transport Phenomena. 2nd Edition, John Wiley and Sons, New York.
[2] Treybal, R.E., 1980. Mass transfer operations. New York, 466.
[3] Tavera, E.M., 2005. Analytical expression for the non-isothermal effectiveness factor: the nth-order reaction in a slab geometry. Chemical engineering science, 60(4), pp.907-916.
[4] Paterson, W.R. and Cresswell, D.L., 1971. A simple method for the calculation of effectiveness factors. Chemical Engineering Science, 26(5), pp.605-616.
Author Response
We gratefully acknowledge the reviewer's helpful and insightful comments, which were extremely helpful in revising the article. Accordingly, the revised manuscript has been systematically improved with new information and interpretations.
Author Response File: Author Response.pdf
