Influence of Rheological, Ionic–Electrostatic, and Van Der Waals Forces on the Flow Structure of Water–Coal Fuel in Pipeline

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The manuscript presents a theoretical development that describes the flow of structured suspensions, such as water-coal fuel (WCF), through circular pipes. It describes a mathematical model that combines conventional rheological forces and interparticle forces of an ionic-electrostatic and van der Waals nature within the framework of DLVO theory. The manuscript addresses the issue of determining the radius of the undeformed flow core, which is a critical parameter in transporting these non-Newtonian fluids through pipes. The methodology involves deriving a cubic equation to describe the balance of forces on the suspension structure, followed by analyzing its solutions using Cardano's method. Key findings reveal that physically possible solutions depend critically on the sign of the "lyophobicity parameter" (which encapsulates colloidal forces) and the radius of the undeformed flow core's magnitude relative to the flow parameters.

1. The summary describes the problem, approach, and main results sequentially. However, it lacks quantitative results and is purely descriptive.
2. In the methods section, the description of the mathematical derivation is detailed. However, replicability is compromised by simplifications. For example, when moving from inequality (10) to (11), the authors state that the second term and the hydraulic particle size component b can be neglected based on “the analysis of orders of magnitude.” This crucial analysis is not presented, which weakens the validity of the final equation (11), which is the cornerstone of the entire study.  Thus, it is recommended that you demonstrate, using scaling analysis, why the neglected terms are several orders of magnitude smaller than the terms that are retained under typical WCF flow conditions.
3. The figures are not entirely self-explanatory, which is important for readers. For example, the physical meaning of the “proportionality coefficient k” is not defined, which makes the x-axes of almost all figures abstract. It is introduced as a mathematical convenience in equation (19), but its connection to the physical properties of the system (solid concentration, zeta potential, etc.) is not discussed.
4. The comparison of results with the literature is very limited. The study does not revisit the experimental discrepancy mentioned in the introduction to show how the new model resolves it.
5. It is important to explicitly discuss the limitations of the study.
6. The mathematical analysis is interesting and sound. However, from a physical perspective, it can be ambiguous due to the lack of connection between parameter k and measurable properties.
7. Revise: The conclusion states that "the quadratic term considers the influence of the distance between two particles," yet this term was previously disregarded. While this refers to the original equation (10), it could cause confusion. More precise wording is recommended.

Author Response

Commets 1: The summary describes the problem, approach, and main results sequentially. However, it lacks quantitative results and is purely descriptive.
Response 1: We agree. We added explicit quantitative statements and updated the abstract

Commets 2: In the methods section, the description of the mathematical derivation is detailed. However, replicability is compromised by simplifications. For example, when moving from inequality (10) to (11), the authors state that the second term and the hydraulic particle size component b can be neglected based on “the analysis of orders of magnitude.” This crucial analysis is not presented, which weakens the validity of the final equation (11), which is the cornerstone of the entire study.  Thus, it is recommended that you demonstrate, using scaling analysis, why the neglected terms are several orders of magnitude smaller than the terms that are retained under typical WCF flow conditions.
Response 2: Thank you for this comment. We agree with it. We have also added a brief justification for this possibility before equation (11).

Commets 3: The figures are not entirely self-explanatory, which is important for readers. For example, the physical meaning of the “proportionality coefficient k” is not defined, which makes the x-axes of almost all figures abstract. It is introduced as a mathematical convenience in equation (19), but its connection to the physical properties of the system (solid concentration, zeta potential, etc.) is not discussed.
Response 3: Thank you for your comment. We agree with it. We have added appropriate changes to explain the nature of k, and have made changes to the captions to the figures accordingly.

Commets 4: The comparison of results with the literature is very limited. The study does not revisit the experimental discrepancy mentioned in the introduction to show how the new model resolves it.
Response 4: We will add a comparison subsection that explicitly revisits the small and near-unity core-radius anomalies and shows how the cubic model resolves or bounds them, with parameterized examples linked to reported ranges in [15, 22].

Commets 5: It is important to explicitly discuss the limitations of the study.
Response 5: Agreed. We will add a Limitations subsection detailing modeling assumptions, regime validity, parameter sensitivity, and calibration needs.

Commets 6: The mathematical analysis is interesting and sound. However, from a physical perspective, it can be ambiguous due to the lack of connection between parameter k and measurable properties.
Response 6: Agreed. We will add a practical estimation pathway for k using experimentally accessible quantities (ζ-potential, ionic strength → Debye parameter, A via material pairing, τw via hydraulic calculations).

Commets 7: Revise: The conclusion states that "the quadratic term considers the influence of the distance between two particles," yet this term was previously disregarded. While this refers to the original equation (10), it could cause confusion. More precise wording is recommended.
Response 7: Agreed. We will clarify that the quadratic term appears in the original cubic polynomial (10) but is neglected in the reduced form (11) under justified scaling, avoiding confusion

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The manuscript presents an interesting theoretical attempt to unify rheological and interparticle interaction models within a cubic analytical framework. However, the presentation would benefit from clearer justification of the simplifying assumptions, improved English, and updated international references. Adding experimental comparison or parametric validation would significantly enhance its scientific impact. 

Comments for author File: Comments.pdf

Comments on the Quality of English Language

The English is understandable but cumbersome and often based on Slavic structures. A review by a native English speaker is necessary.

Author Response

Thank you for the careful and constructive review. Your comments helped us to substantially improve the clarity, physical grounding and practical relevance of the manuscript. We have revised the text throughout to address the concerns you raised and to make the model and its implications easier to interpret and apply.

We clarified and made explicit the physical assumptions underlying the analysis and the range of flow regimes for which the adopted velocity profile and simplifications are intended.

To remove ambiguity in notation and parameter estimation, we introduced a consistent nomenclature and provided practical guidance on how the key model parameters (including the lyophobicity parameter and the proportionality coefficient) can be estimated from measurable quantities such as zeta potential, Debye length, Hamaker constant, particle size and hydraulic conditions. The discussion of DLVO applicability has been expanded to acknowledge common non‑DLVO contributions and to indicate how such effects may be incorporated when they are significant.

Interpretative text was added after major derivations to explain the physical meaning of the mathematical results for pipeline operation, including implications for structure breakdown, deposition risk and restart behaviour.

Figures and captions were updated to include units and to make the graphical results directly interpretable; notation and equation formatting were standardised across the manuscript. The English text has been professionally edited for clarity and concision, and the bibliography was expanded with recent peer‑reviewed studies that support the expanded discussion.

We appreciate the reviewer’s thoughtful suggestions, which materially improved the manuscript’s transparency and usefulness. We believe the revised version addresses the major concerns and presents a clearer, better‑justified analytical framework for understanding how rheological and interparticle forces jointly determine the undeformed core in slurry pipeline flow.

Sincerely,

E. Semenenko, O. Krut’ and A. Zaporozhets

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The recommended changes and/or clarifications have been made.  It is recommended that the manuscript be accepted for publication. 

Reviewer 2 Report

Comments and Suggestions for Authors

The revised manuscript has addressed the major scientific and editorial concerns raised during the review process.

While the work remains theoretical, the discussion now  positions the model with respect to experimental observations and practical pipeline operation, making its scope and limitations transparent.

Comments on the Quality of English Language

The English is understandable but cumbersome and often based on Slavic structures. A review by a native English speaker is necessary.