Vibrational Model of Heat Conduction in a Fluid of Hard Spheres
Round 1
Reviewer 1 Report
the article addresses a very interesting topic and combines it with various models or methods of analyzing these properties. Although it is not my area of specialty, I consider that the article is publishable from the point of view of the writing, the congruence in the ideas that they explore and in the amount of theoretical information that it contains that facilitates the understanding of the research. However, I consider it important that the results of the investigation be evaluated by a specialist in the team.
Author Response
I would like to thank the Reviewer 1 for the examination of this work and its positive evaluation. This Reviewer has not requested changes or made recommendations for revisions:
The article addresses a very interesting topic and combines it with various models or methods of analyzing these properties. Although it is not my area of specialty, I consider that the article is publishable from the point of view of the writing, the congruence in the ideas that they explore and in the amount of theoretical information that it contains that facilitates the understanding of the research. However, I consider it important that the results of the investigation be evaluated by a specialist in the team.
Reviewer 2 Report
Vibrational mechanism of heat conduction in a fluid of hard
spheres
(1) Abstract is not convincing. It is quite difficult to understand the novelty of current work. Modify abstract with motivations, research gap and main findings.
(2) Justify equation (1).
(3) Improve the discussion section. More physical applications needed.
(4) How integral (17) is valid for excess entropy?
(5) Improve the introduction section with following research
(a)A proposed unsteady bioconvection model for transient thin film flow of rate-type nanoparticles configured by rotating disk, Journal of Thermal Analysis and Calorimetry, Volume 144, pages 1639–1654 (2021).
(b)Thermal and boundary layer flow analysis for MWCNT-SiO2 hybrid nanoparticles: An experimental thermal model, Modern Physics Letter B, Vol. 35, No. 18, 2150303 (2021).
Author Response
I would like to thank the Reviewer 2 for the examination of this work and useful suggestions. This Reviewer has raised the following points:
(1) Abstract is not convincing. It is quite difficult to understand the novelty of current work. Modify abstract with motivations, research gap and main findings.
RESPONSE: I have rewritten the abstract, putting more emphasis on the novelty of this work. The main novel aspects of this work are: (i) application of the vibrational model of heat transfer to a new system – the fluid of hard spheres (HS) and the finding that although the model is not particularly suited for this highly anharmonic system, the results are nevertheless in fair agreement with MD simulation results at moderate density; (ii) the finding that Bridgman’s expression is applicable for HS fluids and the determination of the numerical coefficient involved; (iii) The performed analysis of different theoretical models has become possible only after publication of the accurate MD simulation results by Pieprzyk et al. (2020); (iv) A new expression for the dependence of the reduced thermal conductivity coefficient on the excess entropy in HS fluids.
(2) Justify equation (1).
RESPONSE: Eq. (1) is just a mathematical definition of the HS interaction potential.
(3) Improve the discussion section. More physical applications needed.
RESPONSE: This paper is focused on the application of the vibrational model of heat transfer to the fluid made of hard spheres. The vibrational picture should not be expected to be suitable for hard spheres. Nevertheless, formal application of the theoretical results demonstrates that the model predictions are not too far from the reality, in particular at moderate densities. This is a useful lesson that can help to estimate the thermal conductivity coefficient in other fluids with steep pairwise interactions. I added a related sentence to the Abstract.
(4) How integral (17) is valid for excess entropy?
RESPONSE: Eq. (17) represents a well-known thermodynamic identity combined with Carnahan-Starling (CS) equation of state for the HS fluid.
(5) Improve the introduction section with following research (a)A proposed unsteady bioconvection model for transient thin film flow of rate-type nanoparticles configured by rotating disk, Journal of Thermal Analysis and Calorimetry, Volume 144, pages 1639–1654 (2021). (b)Thermal and boundary layer flow analysis for MWCNT-SiO2 hybrid nanoparticles: An experimental thermal model, Modern Physics Letter B, Vol. 35, No. 18, 2150303 (2021).
RESPONSE: The two papers mentioned by Reviewer 2 are indeed interesting and I have read these carefully. However, I have difficulties to relate them with the content of the present manuscript. In view of the Editors request “Please check that all references are relevant to the contents of the
manuscript” I do not feel it would be appropriate to include these papers into the reference list.
Reviewer 3 Report
What is the novelty of the work? Several similar and more detailed works can be found in the literature.
A schematic presenting the studied configuration is to be presented for a better understanding
What do you mean by ‘’ liquid regime’’?
The theory and calculation are briefly presented.
How are the calculations performed? Is it a direct calculation?
The used assumptions and approximations are to be justified.
The presented results are not sufficient and not provide any novelty.
Author Response
I would like to thank the Reviewer 3 for the examination of this work. This Reviewer has raised the following points:
What is the novelty of the work? Several similar and more detailed works can be found in the literature.
RESPONSE: Briefly, the main novel aspects of this work are: (i) application of the vibrational model of heat transfer to a new system – the HS fluid and the finding that although the model is not particularly suited for this highly anharmonic system, the results are nevertheless in fair agreement with MD simulation results at moderate density; (ii) the finding that Bridgman’s expression is applicable for HS fluids and the determination of the numerical coefficient involved; (iii) The performed analysis of different theoretical models has become possible only after publication of the accurate MD simulation results by Pieprzyk et al. (2020); (iv) New expression for the dependence of the reduced thermal conductivity coefficient on the excess entropy of HS fluids has been proposed.
A schematic presenting the studied configuration is to be presented for a better understanding
RESPONSE: The configuration was previously sketched in Fig. 1 of Ref. 31 and I refer to this just before Eq. (4). I do not think it would be appropriate to reproduce this figure one more time in the present paper.
What do you mean by ‘’ liquid regime’’?
RESPONSE: Throughout the paper “liquid regime” means liquid phase, “fluid” means both liquid and supercritical fluid, “dense fluid” means liquid or fluid not too far from the freezing line. In the revised version, this is pointed out in the Introduction.
The theory and calculation are briefly presented.
RESPONSE: This is correct. The vibrational model has been presented in more details in Ref. 31. The properties of the HS fluid have been the subject of many publications and all the details are not repeated to avoid redundancy.
How are the calculations performed? Is it a direct calculation?
RESPONSE: All the main steps of calculations are outlined in the text. The details can be found in other publications, which are cited appropriately. The calculation is relatively simple and direct.
The used assumptions and approximations are to be justified. The presented results are not sufficient and not provide any novelty.
RESPONSE: A number of assumptions and approximations are involved. However, the main point is that the vibrational approximation, which does not seem reasonable for extremely anharmonic HS interactions, nevertheless leads to reasonable results regarding the thermal conductivity coefficient (when a formal procedure of averaging over collective modes is applied). As also mentioned, the Stokes-Einstein relation without the hydrodynamic radius, which arises naturally within the vibrational paradigm, also holds in HS fluids. All this represents an important lesson and can be useful for rough estimates of the transport properties of simple fluids with steep interactions when more accurate experimental results are not available. This has been mentioned in the revised manuscript. The novel aspects of this work have been summarized above.
Reviewer 4 Report
the author must explain what is new in his study. The used model is well known and widely used.
What is the interest of using such theoretical model, when we can use more efficient and precise experimental techniques?
How can you explain the overestimation Near the freezing point?
The assumptions used in the vibrational model are to be justified.
The methods used for the calculations are to be detailed.
In the results part only one figure is presented. More results are to be provided and discussed.
Add a nomenclature.
Author Response
I would like to thank the Reviewer 4 for the examination of this work and useful suggestions. This Reviewer has raised the following points:
The author must explain what is new in his study. The used model is well known and widely used.
RESPONSE: Briefly, the main novel aspects of this work are: (i) application of the vibrational model of heat transfer to a new system – the HS fluid and the finding that although the model is not particularly suited for this highly anharmonic system, the results are nevertheless in fair agreement with MD simulation results at moderate density; (ii) the finding that Bridgman’s expression is applicable for HS fluids and the determination of the numerical coefficient involved; (iii) The performed analysis of different theoretical models has become possible only after publication of the accurate MD simulation results by Pieprzyk et al. (2020); (iv) New expression for the dependence of the reduced thermal conductivity coefficient on the excess entropy of the HS fluid has been proposed.
What is the interest of using such theoretical model, when we can use more efficient and precise experimental techniques?
RESPONSE: The obtained theoretical results can be useful for rough estimates of the thermal conductivity coefficient of simple fluids with steep interactions when more accurate experimental results are not available. When precise experimental results are available, they are of course superior, and can be used to validate or disprove theoretical models.
How can you explain the overestimation Near the freezing point?
RESPONSE: Vibrational model should not be expected to work accurately in the extremely anharmonic regime of HS fluids. We just observe that in the moderately dense regime it is more consistent with MD data than at higher density. The vary fact that the vibrational model can predict correctly the order of magnitude of the thermal conductivity coefficient is, however, remarkable.
The assumptions used in the vibrational model are to be justified. The methods used for the calculations are to be detailed.
RESPONSE: A number of assumptions and approximations are involved. However, the main point is that the vibrational approximation, which does not seem reasonable for extremely anharmonic HS interactions, nevertheless leads to reasonable results regarding the thermal conductivity coefficient (when a formal procedure of averaging over collective modes is applied). As also mentioned, the Stokes-Einstein relation without the hydrodynamic radius, which arises naturally within the vibrational paradigm, also holds in HS fluids. All this can be useful for rough estimates of the transport properties of simple fluids with steep interactions when more accurate experimental results are not available. This has been mentioned in the revised manuscript. Concerning the methods, the vibrational model has been presented in more details in Ref. 31. The properties of the HS fluid have been the subject of many publications and all the details are not repeated to avoid redundancy.
In the results part only one figure is presented. More results are to be provided and discussed.
RESPONSE: In this paper the vibrational model of heat transfer in simple fluids has been applied to the HS fluid. The thermal conductivity coefficient as a function of HS fluid density is the main outcome.
Add a nomenclature.
RESPONSE: The nomenclature has been added in the revised manuscript.
Round 2
Reviewer 2 Report
[1]. There are some typo mistakes that must be carefully removed from whole article.
[2]. Author need to mention briefly, what is new in this model and why is it considered?
[3]. Author should ensure that all physical quantities involved here are well defined.
[4]. Have you employed any assumption? Please explain
[5]. How to choose the values of parameters? Explain
[6] explain HS interaction.
[7] Discuss physical exploration of Eqs. (2) and (3).
[8] Include more physical applications of results.
[9] Sumerize the legends of figure (1).
[10] For heat transfer phenomenon, authors are reffered to following works:
(a)Photo-catalytic pretreatment of biomass for anaerobic digestion using visible light and Nickle oxide (NiOx) Nanoparticles prepared by sol gel method, Renewable Energy, Volume 154, (2020), pp. 128-135
(b)Swimming of gyrotactic microorganisms in unsteady flow of Eyring Powell nanofluid with variable thermal features: Some bio-technology applications, International Journal of Thermophysics, 41, Article number: 159 (2020).
(b)
Author Response
I am thankful to Referee 2 for his/her efforts. This Reviewer has raised the following points:
[1]. There are some typo mistakes that must be carefully removed from whole article.
RESPONSE: I have read the manuscript carefully and removed the defects detected
[2]. Author need to mention briefly, what is new in this model and why is it considered?
RESPONSE: In the abstract and introduction of the revised manuscript its novelty is explained. Briefly, the main novel aspects of this work are: (i) application of the vibrational model of heat transfer to a new system – the HS fluid and the finding that although the model is not particularly suited for this highly anharmonic system, the results are nevertheless in fair agreement with MD simulation results at moderate density; (ii) the finding that Bridgman’s expression is applicable for HS fluids and the determination of the numerical coefficient involved; (iii) The performed analysis of different theoretical models has become possible only after publication of the accurate MD simulation results by Pieprzyk et al. (2020); (iv) New expression for the dependence of the reduced thermal conductivity coefficient on the excess entropy of HS fluids has been proposed.
[3]. Author should ensure that all physical quantities involved here are well defined.
RESPONSE: After the nomenclature has been added to the revised manuscript, I believe that all physical quantities are properly defined.
[4]. Have you employed any assumption? Please explain
RESPONSE: The assumptions behind the vibrational model of transport properties are summarized in the first paragraph of Section 3. These assumptions are not very well suited for the case of the HS fluid. However, a formal expression arrived at using the vibrational model arguments appears reasonably accurate. This is one of the important lessons from this study.
[5]. How to choose the values of parameters? Explain
RESPONSE: There are no free parameters in the expression for the thermal conductivity coefficient that is examined.
[6] explain HS interaction.
RESPONSE: The HS interaction potential ensures that the spheres cannot overlap. I have added this note just below Eq. (1)
[7] Discuss physical exploration of Eqs. (2) and (3).
RESPONSE: Eqs. (2) and (3) represent two variants of normalization of the thermal conductivity coefficients that are often used in the literature.
[8] Include more physical applications of results.
RESPONSE: I added a sentence in the Introduction, which explains that heat transfer in fluids is an important topic of contemporary research with diverse interdisciplinary applications. Some relevant references are provided.
[9] Sumerize the legends of figure (1).
RESPONSE: The legend is present in Fig. 1
[10] For heat transfer phenomenon, authors are reffered to following works:
(a)Photo-catalytic pretreatment of biomass for anaerobic digestion using visible light and Nickle oxide (NiOx) Nanoparticles prepared by sol gel method, Renewable Energy, Volume 154, (2020), pp. 128-135
(b)Swimming of gyrotactic microorganisms in unsteady flow of Eyring Powell nanofluid with variable thermal features: Some bio-technology applications, International Journal of Thermophysics, 41, Article number: 159 (2020).
RESPONSE: I have added Ref. (b) in the revised version to the point in the Introduction where diverse interdisciplinary applications are discussed.
Reviewer 3 Report
After revision, the paper can be accepted for publication
Author Response
I would like to thank Reviewer 3 for the examination and positive evaluation of this work.
Reviewer 4 Report
Accept as it is
Author Response
I would like to thank Reviewer 4 for the examination and positive evaluation of this work.
