Evaporation of Small Sessile Drop Deposited on a Horizontal Solid Surface: New Exact Solutions and Approximations

Round 1

Reviewer 1 Report

The formulas for calculating the slow evaporation of an axisymmetric drop of capillary dimensions deposited on a flat solid surface is reviewed. Such characteristics as vapor density, evaporation flux density, and total evaporation rate are considered. Exact solutions for the drop the size of which is much smaller than the capillary constant (Bond number, Bo<<1) obtained in the framework of the Maxwellian model, in which the evaporation process of the drop is limited by vapor diffusion from the drop surface to the surrounding air, are presented. The new formulas, concerning exact solutions for several specific contact angles are presented. For the first time new approximate solutions are presented for total evaporation rate and mass loss per unit surface area per unit time in the whole range of contact angles q Î[0, p).

The manuscript is clear and presented in a well-structured manner. Presented research relevant for the field. The cited references are mostly recent publications and do not include an excessive number of self-citations. The manuscript’s results reproducible based on the given details. The data interpreted appropriately and consistently throughout the manuscript and they easy to understand.

But from my point of view this paper have room for improvement.

1- Figures (graphs) resolution can be updated to the more appropriate level.

2- In line 196 “Table 1 shows the first two solutions corresponding to j=1, 2, 3.” , and in line 198 “First three solution of the expression (22) with j=1,2,3 are placed into Table 1.”  It is not understandable what authors mean here?

As one can see Table 1 represents three solutions not two.

3- In line 224 approximate expression (30) presented. Why this simple formula working? May be possible to give some physical explanation of using such approximation?

Author Response

Thanks for the comments. In the following, I will respond point by point to the objections of the review.

“1- Figures (graphs) resolution can be updated to the more appropriate level”.

We will try to improve the resolution of the pictures.

“2- In line 196 “Table 1 shows the first two solutions corresponding to j=1, 2, 3.” , and in line 198 “First three solution of the expression (22) with j=1,2,3 are placed into Table 1.”  It is not understandable what authors mean here?

As one can see Table 1 represents three solutions not two”.

You're right. This sentence has to be removed.

“3- In line 224 approximate expression (30) presented. Why this simple formula working? May be possible to give some physical explanation of using such approximation?”

This approximate expression was proposed based on an analysis of the graph of the exact solution. Obviously, it can be found by simplifying (coarsening) the exact solution, and it is a certain mathematical fact. Consequently, the physical meaning of this approximation is based purely on the same model that leads to the exact solution.

Reviewer 2 Report

The present work will be better if compared to existing experimental (or analytical) published paper. See for instance DOI: 10.1039/C5SM00878F  

Link is below and underline the thermal coupling with substrate and the dimensionless SB number justifying your work  (you ll find the sefiane reference in the same given DOI)

www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=&cad=rja&uact=8&ved=2ahUKEwi_nNmPgbSBAxXbWEEAHSYnAtw4FBAWegQIEhAB&url=https%3A%2F%2Fpubs.rsc.org%2Fen%2Fcontent%2Fgetauthorversionpdf%2FC5SM00878F&usg=AOvVaw15DYzr-hUIyroNdKUtMp1a&opi=89978449

Author Response

Thanks for the comments.

The introduction to the manuscript states that we are not proposing a new physical model in this article, but are solving the same Dirichlet problem for the Laplace equation, which is solved in the papers of Picknett and Bexon, as well as Popov et al. Thus, the exact solutions we propose are mathematically equivalent to the solutions proposed in the above papers. Since the solutions proposed by Picknett and Bexon, as well as by Popov, have been compared with experiment in numerous other papers (for example, Dash, S., Garimella, S.V. Langmuir 2013 29, 10785). The conclusions in this regard are quite well known, and, we can say that all these conclusions are equally applicable to our work. The novelty of our work is only that we propose new mathematically equivalent (simpler and more convenient) solutions to the same problem that was previously solved by Popov et al. Our solutions numerically give the same result as the formulas of Popov et al. Therefore, the previously performed verification of Popov’s formulas also applies to our new formulas. We shouldn't do this again.

So, when we talk about approximate solutions, taking into account the above, it is enough to compare them with the exact solution, and not with experiment, since the exact solution, as said above, has already been verified before us.

In addition, in the manuscript, we compared the approximate solutions obtained for the droplet evaporation times in the two main modes (CCA and CCR) with a well-designed experiment described in the Dash and Garimella’s paper. We think that this is sufficient for the main purpose of the work.

Following the reviewer's remarks, we have added references to works that contributed to the substantiation of the diffusion evaporation model:

Yilin Wang, Liran Ma, Xuefeng Xu, Jianbin Luo. Combined effects of underlying substrate and evaporative cooling on the evaporation of sessile liquid droplets Soft Matter, 2015,11, 5632-5640,

Sefiane, K.; Wilson, S. K.; David, S.; Dunn, G. J.; Duffy, B. R. (2009). On the effect of the atmosphere on the evaporation of sessile droplets of water. Physics of Fluids, 21(6), 062101.

Reviewer 3 Report

This paper presented the new approximate solutions of sessile droplet. The total evaporation rate and mass loss per unit surface area per unit time in the whole range of contact angles can be calculated through elementary functions, without needing integral. This paper provided an alternative method for the calculating the evaporation rate of sessile droplet in the whole range of contact angles. It is thus interesting and helpful for the readers in this fields. This paper can be published as its current form. Only the follow mirror is necessary to revise.

1. In Eq. (4), the definition of symbol beta is not given in the paper.  

Author Response

Thanks for the comments.

The symbol beta appears only in the integrand of Eq.(4), being the variable of integration between the two limits of the definite integral. Therefore, it is not an input parameter on which the discussed physical quantities (W and J) depend. Thus, beta is an internal parameter that has the meaning of an integration variable running from phi to theta. This is simply a mathematical quantity that does not have an independent physical meaning: it does not need to be specified in order to obtain the evaporation rate or evaporation flux density.

Reviewer 4 Report

Comments about the manuscript entitled “Evaporation of small sessile drop deposited on horizontal solid surface: new exact solutions and approximations” with manuscript ID: colloids-2583319. The comments about the manuscript are the following:

1.       On lines 2-3: In the title, “Evaporation of small sessile drop deposited on horizontal solid surface: new exact solutions and approximations”, this reviewer recommends a change to “Evaporation of small sessile drop deposited on a horizontal solid surface: new exact solutions and approximations”.

2.       About the title of the work, “Evaporation of small sessile drop deposited on horizontal solid surface: new exact solutions and approximations”. This reviewer is concerned about the originality of the present manuscript in section 2, "2. New exact solutions for some values of contact angles” since the “new exact solutions” have already been reported by reference [26]. How do the authors justify this in the title of the work? So, the real contribution of this manuscript is section 3, "3. New approximate solutions".

3.       On lines 51-52: When the authors say, “The smallness of the convective Stefan flux also has to be satisfied. Stephan showed 51 for the first time [12] that near the surface…”; Is the name "Stephan" spelled correctly? or is it Stefan? Review this.

4.       On line 58: It is recommended that the equations between paragraphs be edited linearly.

5.       On line 72: Include the mathematical definition of the Bond number for this work and that it must be much less than one.

6.       All nomenclature used in the manuscript must be defined systematically and immediately after it appears for the first time. Different variables such as \tao (on page 2), \varsigma (on page 5), z_0 (on page 3), \beta (on page 3), among others, are never defined in the manuscript.

7.       Review the nomenclature format used; authors sometimes use the variables \pi, \beta, and \theta as italics and, on other occasions, as normal.

8.       On line 85: The equation in this line should be removed from the paragraph and numbered below as another equation.

9.       On lines 113-115: When the authors say, “It requires significant computational resources. To accelerate the calculations of the evaporation flux density, it is reasonable to use simplified approximate expressions.” How could these significant computational resources be measured? In time, infrastructure? As? In those terms, how much would these approximate expressions save? This reviewer believes that explaining this situation in more detail would help to justify the work.

10.   On page 4: Improve the editing of Figure 2, the reader should differentiate the results of equations (4) and (10). In the current state, they cannot be differentiated.

11.   On page 4: Improve the position of Figure 2; it should be placed preferably after where it is first mentioned.

12.   On line 134: The equation in this line should be removed from the paragraph and numbered below as another equation.

13.   On line 137: The equation in this line should be removed from the paragraph and numbered below as another equation.

14.   On pages 5-8: Section 2 of this manuscript is the same as sections 2 and 3 of reference [26]. How is the originality of this manuscript and the appropriate way of citing already published works supported, as well as a similarity analysis of the information presented?

15.   On line 155: For Equation (11), this reviewer believes it is unnecessary to place the "\times" symbol twice; it is enough to put it on the second line of the equation. See the journal's author guide for long equations.

16.   On line 155: For Equation (11), the end point of the equation must be placed on the second line of this equation.

17.   On line 157: This reviewer recommends listing equations with consecutive numbers only. That is, Eq. (11), Eq. (12), Eq. (13), and so on...Do not use Eq. (11) and then Eq. (11a).

18.   On line 216: When the authors say, “It is easy to verify by direct calculation that formula (4) gives the same curves…”. First, this sentence is the same than reference [26]. Second, please obtain them from Eq. (4) and overlap with the results shown in Figure 3 in the present manuscript.

19.   On line 256: Where was the time variable "t" in Eq. (36)?

20.   On lines 264-265: When the authors say, “substituting it into (36), and performing trigonometric transformations, we get the evaporating time…”, What trigonometric transformations? Can the authors please detail better how to obtain Eq. (38).

21.   On line 276: Number all the equations in the manuscript, including the one on this line.

22.   On page 11: Please place the evaporation time in seconds on the vertical scale, as reference [27] does.

23.   On line 283: Improve the edition of physical units “kg/m3”. The number 3 is superscript.

24.   On page 11: The graphical results of the equation. (40a) and Eq. (41) in Figure 5, can it be compared with the results (theoretical and experimental) of Figure 5 in reference [27]?

25.   On lines 296-298: When the authors say, "At the same time, as indicated in the paper [26], the experiment that was made for such a drop drying in the CCA mode gives the evaporation time of 1009 s.”. This evaporation time of 1009 s was never found in reference [26]. Review and correct. In general, review the cross-references of the manuscript.

26.   On lines 363-369: This paragraph had already been reported on page 7 of the reference [26].

Author Response

Thanks for the comments.

In the attached pdf file, we respond point by point to the objections of the review.

Author Response File: Author Response.pdf

Reviewer 5 Report

colloids-2583319

Evaporation of small sessile drop deposited on horizontal solid surface: new exact solutions and approximations

Peter Lebedev-Stepanov, Olga Savenko

This manuscript reviewed the arsenal of formulas for calculating the slow evaporation of an axisymmetric drop of capillary dimensions deposited on a flat solid surface. New exact solutions and approximations are also presented. Although detailed derivation process is given, but the following comments should be addressed.

1.      The authors should be careful when using “for the first time” in the Abstract.

2.      The authors reviewed the formulas for calculating the slow evaporation of an axisymmetric drop of capillary dimensions. This work has been done in many previously published papers. So, what’s novelty of this work?

3.      Why don’t the authors compare the results by the new solutions with the evaporation experiments?

4.      Why don’t the authors compare the results between different evaporation formulas in different published papers?

5.      In Figs. 5 and 6, the variable of vertical axis is missing.

6.      In Fig. 6, φ and ϑ are repetitive.

7.      The quality of all figures should be improved.

N/A

Author Response

Thanks for the comments.

In the attached pdf file, we respond point by point to the objections of the review.

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

1- There no marangoni effect because The authors consider the case of isothermal evaporation and such behavior is possible under the validity of SB dimensionless number , so compleet by the adequate condition based on the small SB number (Journal of Fluid Mechanics , Volume 667 , 25 January 2011 , pp. 260 - 271 DOI: https://doi.org/10.1017/S0022112010005446 ).

2- improve the figures quality.

3- the analytical results must be compared to experimental to demonstrate the utility of such work. Results are avialable on net  (not the best but you ll find some in this paper Paper No: HT-14-1690 https://doi.org/10.1115/1.4032954

Some long sentences but not complicate to read.

Author Response

Thanks for the comments.

To view our responses, please view the attached pdf-file.

Author Response File: Author Response.pdf

Reviewer 4 Report

This reviewer has no further comments and accepts the manuscript in its present form.

Author Response

Thank you for your response.

Reviewer 5 Report

N.A.

Author Response

Thank you for your response.