Simple Particle Relaxation Modeling in One-Dimensional Oscillating Flows

Round 1

Reviewer 1 Report

The relaxation of a rigid particle suspended in a one-dimensional oscillating flow is calculated according to different drag models and the results are compared.

Thematically the work is interesting for the researchers and professionals and the proposed manuscript is relevant to the scope of the journal.

I found it appropriate for publication in the Processes journal, but only after some modifications and clarification from the Authors.
The title is a clear representation of the manuscript's content.

The abstract is too short, it should be extended with a few lines more!

The overall organization and structure of the manuscript are appropriate. The paper is well written and the topic is appropriate for the journal.
The aim of the paper is well described and the discussion was well approached, its results and discussion are correlated to the cited literature data.

The literature review is comprehensive and properly done.

The novelty of the work must be more clearly demonstrated.

The significance of the Work: Given the large number of analyzed data, this is an interesting study with a possible significant impact in this area.

The verification of the model should be performed. Model validation is possibly the most important step in the model building sequence. 

The developed model was compared to other models and not compared to experimental data, which could be one of the main drawbacks of this paper.

Other Specific Comments: The work is properly presented in terms of the language. The work presented here is very interesting and well done, it is presented in a compact manner.

In general, there are no doubtful or controversial arguments in the manuscript. The methodology applied in the research is presented in clear manner, so that it is repeatable by other authors.

The results are presented in a logical sequence and the discussion and analysis of the results are properly elaborated. 

The appendix section is very informative, giving the important data for better understanding the study. 

Author Response

Dear Reviewer,

we thank you for taking the time to review our manuscript. We will address your comments in the following.

The abstract is too short, it should be extended with a few lines more!
The novelty of the work must be more clearly demonstrated.

• The abstract & the conclusion were extended by emphasizing the novelty of the work.

The verification of the model should be performed. Model validation is possibly the most important step in the model building sequence. The developed model was compared to other models and not compared to experimental data, which could be one of the main drawbacks of this paper.

• Each considered model is well established in literature and received already a lot of experimental validation (and validation via DNS). In this work the models are set in relation to each other and are compared to each other within their stated ranges of validity on a purely mathematical basis (a comment addressing this issue was added in line 83).

The novelty of this work comes from the introduction of the ε-Re plane and the discussion on how to calculate the slip velocity between fluid and particles at each point of the ε-Re plane.

All changes in the manuscript have been highlighted.

Best regards,

Stefan Heidinger

Reviewer 2 Report

The authors arranged the manuscript in well manner.  However, I suggested the following:

1. There is no reference mentioned for Equation A1 to A21. The author's should include the references of all equations.

2. The author's should incorporate the error analysis.

3.  The author's should include some more recent references.   

Author Response

Dear Reviewer,

we thank you for taking the time to review our manuscript. We will address your comments in the following.

1. There is no reference mentioned for Equation A1 to A21. The author's should include the references of all equations.

• In case referencing refers to the mention of the equations in the text, this is done for almost all of the equations A1 to A21. In case referencing refers to citing the source of the equations, all equations are derived by the authors or are so basic (e.g. definition of Reynolds number) that this would add no useful information or clarification.

2. The author's should incorporate the error analysis.

• Each considered model is well established in literature and received a lot of experimental validation (and validation via DNS). An error analysis for each model with the experimental data can be found there. In this work, drag models are set in relation to each other and are compared to each other within their stated ranges of validity on a purely mathematical basis. (a comment addressing this issue was added in line 83). The error of applying Stokes outside of its limits of validity is discussed.

3. The author's should include some more recent references. 

• Several sources from the last three years have been added.

All changes in the manuscript have been highlighted.

Best regards,

Stefan Heidinger

Reviewer 3 Report

 

This paper shows a theoretical investigation on the movement of a particle within an oscillating flow. To this purpose, the authors introduce the concept of the epsilon-Re diagram, which allows mapping the possible different regimes in which this oscillating fluid and particle may be found and eventually calculate the slip velocity in each case. The concepts introduced are really interesting, however, the results are presented at some parts in a confusing manner. Therefore, I consider the decision on publishing this paper must be postponed after a deep revision on the text.

First of all, this paper lacks of an introduction, so there is no any information about the motivation and goals of this investigation. This lack of context makes really difficult to evaluate the quality of this work as well as understanding the adequateness of the methodology and the relevance and significance of this work. For the sake of clarity, please, do include an introduction section to give some context.

Figure 1 present the following problems making difficult to understand it.

-        *Please, specify in an explicit manner how this graph was made: was it calculated or was it plotted just by taking the limits shown in previous works?

-         *The plot shows the Stokes model is used for Re<1 while it is crossed by the line Wo^2=1. On the other hand, the table shows that this model must be used for Wo^2<<1. Please, clarify.

-         *The table in the right side says NS limit happens for Re<1000, however, in the plot seems that this model begins in Re=1, please, check it.

-        * For a certain set of values of Re and epsilon, can be used two different models indistinctly (e.g. Basset & Stokes or Basset & Landau Lifshitz). However, using solid colors for each model may result confusing and it is difficult to understand in which parts two different models can co-exist. Please, consider using a clearer way of representing (e.g. instead of filling the different regions with solid colors, fill them with patterns, so the superposition of two or more models can be appreaciated).

 

Lines 40 & 41: you say that to simplify the problem, instead of calculating the whole velocity field, you reduce the problem to calculate the sum of the forces in a particle, then, are you considering a punctual particle problem? If so, please do specify it. Also do consider the possibility of including the ODE of motion with a generic dragging force for the sake of clarity.

 

Lines 59 to 63: Here you say that for Re>1 you use the SN model. However, in figure 1 you say the limit of this model is for Re>1000.  How is this possible?

 

Lines 95 & 96 and figure 3: You introduce the density ratio and show the slip velocity as a function of the epsilon-Re plane for two different values of this ratio. However, why is the importance of this ratio, i.e. what makes the difference between Fig. 3a and Fig. 3b? Please, include a brief discussion on the difference between this two cases.

 

You say also that the chosen values of the density ratio correspond to iron particles in hot air and water. Please, do include a reference for this data.

 

Lines 101 & 102 and Figure 4: Here you introduce the concept of critical density ratio but its definition its not clear. Please, give a more specific definition on what is this critical density ratio and how did you calculate it. Moreover, you don’t define the cut-off criterion you introduce. Why do you use three different criteria in Fig. 4?.

 

Conclusions: The section “conclusions” is just an enumeration of the tasks did in this work but I miss a discussion on the results obtained: what is the relevance of the obtained results? Is one drag model preferable to the rest? If yes, what should we do in a real experiment to as to work in that regime? Please, do include a discussion on that things.

Author Response

Dear Reviewer,

we thank you for taking the time to review our manuscript. We will address your comments in the following.

1. First of all, this paper lacks of an introduction, so there is no any information about the motivation and goals of this investigation.

• The point of origin of this research is particle behavior in pulsation reactors. However, it was decided to publish this work detached from the pulsation reactor. This work captivates with its general applicability and considerations of large parameter ranges, which are only partly reflected by the pulsation reactor (e.g. the applicability of this work on solid-liquid systems).

Figure 1 present the following problems making difficult to understand it.

2. Please, specify in an explicit manner how this graph was made: was it calculated or was it plotted just by taking the limits shown in previous works?

• This graph was created by plotting the validity ranges of the models in the ε-Re plane. A clarifying comment has now been added; now line 51.

3. The plot shows the Stokes model is used for Re<1 while it is crossed by the line Wo^2=1. On the other hand, the table shows that this model must be used for Wo^2<<1. Please, clarify.

• The line Wo^2=1 would equal the axis Re_S = Re ε, which does not cross the validity range of the Stokes model.
• The origin in Figure 1 (the point of intersection of the four lines Re, ε,Wo^2, Re_S) does not have the coordinates Re = 0, ε =0, but rather Re = 1, ε =1. This is a crucial information that was missing and is now stated in the manuscript.

4. The table in the right side says NS limit happens for Re<1000, however, in the plot seems that this model begins in Re=1, please, check it.

• Entire areas of validity of drag models in Figure 1 are indicated by solid colors. In case two or three models are valid in the same area, the area was in the color of the preferable model (mostly simpler model), while now the validity of the other models in this area is indicated by a striped pattern. This explanation is stated in the body of the text (now line 52) and in the caption of Figure 1.

Since the Stokes model is preferable to the SN model, the area for Re<1 & Wo^2<<1 is in the color of the Stokes model (red). The area for the SN model is solid blue for 1<Re<1000 & Wo^2<<1 (where only the SN model is valid) and the blue stripe pattern indicates that the SN model is also valid in the area of the Stokes model.

5. For a certain set of values of Re and epsilon, can be used two different models indistinctly (e.g. Basset & Stokes or Basset & Landau Lifshitz). However, using solid colors for each model may result confusing and it is difficult to understand in which parts two different models can co-exist. Please, consider using a clearer way of representing (e.g. instead of filling the different regions with solid colors, fill them with patterns, so the superposition of two or more models can be appreciated).

 

• Patterns have been added for better readability of areas with multiple valid models.

Lines 40 & 41: you say that to simplify the problem, instead of calculating the whole velocity field, you reduce the problem to calculate the sum of the forces in a particle, then, are you considering a punctual particle problem? If so, please do specify it. Also do consider the possibility of including the ODE of motion with a generic dragging force for the sake of clarity.

• A remark was added (now line 41) in order to clarify the mass point assumption.

Lines 59 to 63: Here you say that for Re>1 you use the SN model. However, in figure 1 you say the limit of this model is for Re>1000.  How is this possible?

• At no point in the paper is it stated that the SN model is valid for Re > 1000, which would be false.

Lines 95 & 96 and figure 3: You introduce the density ratio and show the slip velocity as a function of the epsilon-Re plane for two different values of this ratio. However, why is the importance of this ratio, i.e. what makes the difference between Fig. 3a and Fig. 3b? Please, include a brief discussion on the difference between this two cases.

• A reference was added in what is now line 105 and a more substantial discussion was added in what is now line 129.

You say also that the chosen values of the density ratio correspond to iron particles in hot air and water. Please, do include a reference for this data.

• The chosen values of the density ratios are somewhat arbitrary; they just needed to represent high and low density ratio cases in order to highlight the influence of the density ratio on the problem. The density ratios of iron in hot gas or water were mentioned to give the reader a better understanding of the values and how they can tie into an applicable case.
• The densities of iron, hot air and water are easily accessible via numerous sources. It is not necessary to include a reference here.

 Lines 101 & 102 and Figure 4: Here you introduce the concept of critical density ratio but its definition is not clear. Please, give a more specific definition on what is this critical density ratio and how did you calculate it. Moreover, you do not define the cut-off criterion you introduce. Why do you use three different criteria in Fig. 4?.

• Some more explanations were added (in what is now line 109) in order to address this point.

Conclusions: The section “conclusions” is just an enumeration of the tasks did in this work but I miss a discussion on the results obtained: what is the relevance of the obtained results? Is one drag model preferable to the rest? If yes, what should we do in a real experiment to as to work in that regime? Please, do include a discussion on that things.

• The conclusion heading now reads “Summary & Conclusion” and was extended by some more concluding remarks.

All changes in the manuscript have been highlighted.

Best regards,

Stefan Heidinger

Round 2

Reviewer 3 Report

After the changes introduced I consider this work can be published