# Mathematical Model of Small-Volume Air Vessel Based on Real Gas Equation

*Reviewer 1:*Anonymous

*Reviewer 2:*Anonymous

**Round 1**

*Reviewer 1 Report*

The article presented for review: Mathematical model of small volume air vessel based on real gas equation is a corrected version of the previously reviewed article with number water-547023: Mathematical model of small volume air vessel based on real gas equation under isothermal process.

In the current article, the authors have improved most of the issues raised in the review of the previous article, which improves the overall reception of the article.

Minor errors indicated when presenting the calculation model have been corrected. Part of the article in this regard has been completely rebuilt.

A mathematical model of the tank was presented, which definitely improved the shape of the whole article. In this section, I have minor comments about the description of this chapter. In figure 1 junction P may be marked with a different letter, letter P may be recognized as pressure. Another minor comment - the water level is marked as Z and the "water depth" appears in the rest part article - it should be standardized.

Also, the air height can be better described as, for example, the air level in the tank. In line 165 the value of H is indicated as the atmospheric pressure with the unit m. If we already indicate the pressure with the unit of the length of the liquid column, it is necessary to indicate what liquid it is. Probably it is about the height of the water column corresponding to atmospheric pressure, but it is advisable to indicate this fact in article text. In addition to the comments made, the article is clearly written, the results of the research are clearly presented and the conclusions are related to the obtained results.

Additional language correction is required, mainly minor language or typing errors.

After making these corrections, I suggest that the article should be approved for publication.

*Author Response*

**Response to Reviewer 1 Comments**

**Point 1: **A mathematical model of the tank was presented, which definitely improved the shape of the whole article. In this section, I have minor comments about the description of this chapter. In figure 1 junction P may be marked with a different letter, letter P may be recognized as pressure. Another minor comment - the water level is marked as Z and the "water depth" appears in the rest part article - it should be standardized.

**Response 1: **Thank you for your suggestion. In figure 1, junction P is marked with a different letter as junction B. *H _{S}* is the water level of air vessel and Z is the water depth, the relationship between

*H*and Z is shown in Figure 1.

_{S}

**Point 2: **Also, the air height can be better described as, for example, the air level in the tank. In line 165 the value of H is indicated as the atmospheric pressure with the unit m. If we already indicate the pressure with the unit of the length of the liquid column, it is necessary to indicate what liquid it is. Probably it is about the height of the water column corresponding to atmospheric pressure, but it is advisable to indicate this fact in article text.

**Response 2: **The air height is replaced with the air level in the paper. is the atmospheric pressure head, m (line 167).

*Reviewer 2 Report*

I have been refereeing the paper for the third time now. It is substantially improved: The authors have included the section 'Mathematical model of the air vessel' in the main text, which provides the needed modeling equations. They have also extended their figures which are now more informative.

This version still needs a major revision, to address the following issues.

1. Real-gas equations

Last time, the pressure labels were incorrect. This time the volume labels are incorrect.

The ideal gas equation is of course

PV = mRT (0).

The real gas (VdW) equation is

(P+a m^2/V^2)(V-mb)=mRT, (1)

where P is the pressure in VdW approximation, labeled as P_vdw in the study. The volume (V-mb) is the effective volume in the VdW approximation, diminished due to the finite size of the molecules. Hence, it is incorrect to label V as V_vdw. The effective volume V_eff=V-mb is the V_vdw volume!

The reason for the confusion is that authors seem unaware of the derivation of the VdW equation. The equation stems from

P V_eff=N_eff kT, (2)

where the volume of molecules is accounted by the effective volume, and the intermolecular attractions by the effective (smaller) number of molecules

V_eff=V-mb (3)

N_eff=N(1-Cm/TV) (4)

where m is number of moles (as in the above equations) and C is a constant.

Putting equations (3) and (4) into (2), after some algebra and rescaling of the second-order pressure terms, one obtains the VdW equation (1).

Furthermore, authors are insisting that the constants C1, C2 and C3 in their equations (3), (10) and (20) are different. But the limit to ideal gas state (a->0, b->0) should recover the ideal gas polytropic equation, see (1) and (0) above. Yet, their C2 and C3 do not go to C1 in this limit, since they are constants, independent of a and b. Thus, the problem with authors' claim.

These are serious omissions that need to be corrected. As for the format, the section 2 is now shorter and much easier to follow.

2. Figures

The figures are good, but the lines in Fig. 5 are difficult to follow. Please annotate the lines directly i.e. write (P_id, D=2m) etc. next to each of the corresponding lines and not in legends. In addition, in Fig. 5d the line p_vdw/p_id is plotted as p_id/p_vdw, and the two lines for D=3m are annotated with the same symbol (dash dot). The authors got confused for the same reason, namely that it is hard to follow the current labels.

3. Wordiness

After a nice presentation of the theory and then of the results in Tables and Figures, the authors use too much space for explaining the obvious and trivial. For example, in the lines 234-240, they write:

'The above results show that the minimum water depths and air pressures from the actual model are lower than that from the ideal gas model, and the maximum air heights from the actual model are higher than that from the ideal gas model. For instance, the difference of the minimum water depth between VDW model and ideal model is 0.49 m, and the difference of the minimum water depth between RK gas model and ideal gas model is 0.32 m, which greatly reduce the minimum water depth, that is, greatly reduces the safe margin of water-depth required in the air vessel.'

I rewrote:

'The results in Table 2 show that the real gas models predict the water depth to be critically low, reducing the safe margin required in the air vessel (p_vdw-p_id=0.49 m, p_rk-p_id=0.32m). Hence the necessity for the inclusion of the real gas approximations.'

Another example, lines 257-260:

'As can be seen in Table 2, the minimum water depths calculated by ideal gas model, VDW gas model and R-K gas model are 0.48 m, 0.43 m and 0.45 m, respectively. The maximum air heights for the three models are 2.82 m, 2.87 m and 2.85 m, respectively. And the minimum air pressures for the three models are 1.61 MPa, 1.58 MPa and 1.59 MPa, respectively.'

My take:

'As seen from Table 2, the pressure and height differences are now much smaller compared to the ideal gas (~1%), as a consequence of the negligible influence of the interatomic attractions in the high pressure regime (cohesion pressure in Eqs. (4) and (8) tends to zero).'

There are other examples (see below).

The equations and tables are there to be used as reference points - no need to re-state what is already there. One needs to summarize main points not repeat! This will save you much of the space as well as writing effort. The authors are welcome to incorporate my written pieces above (and suggestions below) as they are.

4. Sec.4 'Discussion'

It is clear that section 4, 'Discussion', does not add anything new to the text: main points were already said in relations to the figures (Sec. 3), as well as in the 'Conclusion'. The authors are advised to completely remove the section (which in addition is poorly written).

As said in the previous point, using the equations is the best way to point to trends, and this can be done once, early on, to prevent the repetition of the same (self-evident) points over and over.

I warn the authors that I will be sending the paper back until it reaches the satisfactory format of succinct, kernel messages, and only the kernel messages.

5. Minor

The text has some additional spelling/wordiness errors which are here highlighted. The above points are here repeated for the sake of completeness.

line 63:

Therefore, -> Thus,

line 87:

...this work showed... W

Which work? The authors' paper or the reference [31]?

line 111:

dipole-dipole -> induced dipole-induced dipole

line 139:

Add at the end: 'We see that the polytropic exponents of VdW and for the given approximation of RK do not differ from the ideal gas case.'

line 157:...air follow ->... air follows

line 170, Eq 29:

rewrite CP, CM BP and BM with underscripts Cp, Cm, ...

what do subscripts 1 and 2 refer to?

line 191: double check the claim a=1000 m/s

line 194: ...are both... -> ...are all...

line 196:

Remove: ', and vice versa.'

line 197: focus -> focuses

line 225:

Remove one instance of: 'in order to'

lines 234-240: 'The above results...in the air vessel' ->

The results in Table 2 show that the reals gas models predict the water depth to be critically low, reducing the safe margin required in the air vessel (p_vdw-p_id=0.49 m, p_rk-p_.id=0.32m). Hence the necessity for the inclusion of real gas approximations.

lines 257-260: As can be seen in Table...respectively. -> As seen from Table 2, the pressure and height differences are now smaller as a consequence of the negligible influence of the interatomic attractions in the high pressure regime (cohesion pressure in Eq tends to zero).

Fig 5a-5d: Legends are hard to follow; it is better to annotate the lines directly i.e. write (P_id, D=2m) etc. next to each of the corresponding lines.

5d: Careful!: line p_vdw/p_id is plotted as p_id/p_vdw. Also, two lines for D=3m are annotated with the same symbol (dash dot). That's why it is important to label the curves next to them.

line 270: ...are more violently... -> ...are more violent

lines 272-274: In addition, inasmuch as the influence of vessel diameter, the fluctuations of water depth and air pressure for varying diameter under the same gas model present hysteretic nature. ->

Note the asymmetry in the upward and downward oscillations (including the sudden changes in the water depth and air pressure, Fig. 5a and 5c), akin to hysteresis.

line 274: The smaller the diameters...the shorter...are. -> The smaller the diameters, the shorter are the periods of the fluctuations of water depth and air pressure.

Discussion 4 -> Completely remove the section! It has already been mentioed and is unnecessary. I will reject the article.

(?? line 285: ...remain positive values. -> ...is positive.)

line 322: ...used water hammer -> used for water hammer

line 325: ...is not suitable -> ...is less suitable

lines 328-329: Hence,...vessel diameters. -> Hence, VDW and R-K gas equations are numerically compared with ideal gas equation for the cases of different vessel diameters.

line 333: ...are both derived -> are all derived

lines 334-336: The cohesion...gas model. -> The air pressure in the vessel calculated by the real gas models is always smaller compared to the ideal gas model on the account of non-zero intermolecular attractions i.e. the positive cohesion pressure. Likewise, the corresponding water levels are lower than those of the ideal gas.

line 339: ...is significant... -> ...is more significant...

lines 341-346: For small volume...downstream system. -> For small volume vessel, the real gas models predict critically low water levels (~0.5m), so that an adequate safety margin can be provided in the operation of pump-air vessel downstream systems.

*Author Response*

**Response to Reviewer 2 Comments**

**Point 1: **Last time, the pressure labels were incorrect. This time the volume labels of VDW gas model are incorrect.

**Response 1: **Thank you for your suggestion. I have modified the wrong labels of the VDW volume.

**Point 2: **Furthermore, authors are insisting that the constants C1, C2 and C3 in their equations (3), (10) and (20) are different. But the limit to ideal gas state (a->0, b->0) should recover the ideal gas polytropic equation, see (1) and (0) above. Yet, their C2 and C3 do not go to C1 in this limit, since they are constants, independent of a and b. Thus, the problem with authors' claim.

**Response 2: **I have completely accepted your opinion and deleted the claim.

**Point 3: **the lines in Fig. 5 are difficult to follow. Please annotate the lines directly i.e. write (*P*_{Ideal}, D=2m) etc. next to each of the corresponding lines and not in legends. In addition, in Fig. 5d the line *P*_{VDW}/* P*_{Ideal} is plotted as *P*_{Ideal} /* P*_{VDW}, and the two lines for D=3m are annotated with the same symbol (dash dot). The authors got confused for the same reason, namely that it is hard to follow the current labels.

**Response 3:** Thank you for your suggestion. I have modified the mistakes.

**Point 4:** It is clear that section 4, 'Discussion', does not add anything new to the text: main points were already said in relations to the figures (Sec. 3), as well as in the 'Conclusion'. The authors are advised to completely remove the section (which in addition is poorly written). As said in the previous point, using the equations is the best way to point to trends, and this can be done once, early on, to prevent the repetition of the same (self-evident) points over and over. I warn the authors that I will be sending the paper back until it reaches the satisfactory format of succinct, kernel messages, and only the kernel messages.

**Response 4: **Thank you for your suggestion. I have removed the previous section (Discussion) and rewritten the conclusion.

**Point 5: **The text has some additional spelling/wordiness errors which are here highlighted.

**Response 5: **Thank you for your suggestion. I have modified the mistake.

Author Response File: Author Response.docx

**Round 2**

*Reviewer 2 Report*

The volume labels are now corrected but not in Eqs. 11 and 12.

I realized that Figures 4d and 5d have air levels and air pressure ratios in the same figure which is rather confusing. The correct way is shown in Fig. 3d that has air pressures and air-pressure ratios. Authors are asked to make Fig. 4d and 5d consistent with Fig. 3c.

Lines 281-285:

Especially, the values of pressure ratios exceed 1 after the fluctuation of air level curve along with longer oscillation period reaches the minimum level inasmuch as the influence of lag. Consequently, the values of pressure ratios are always lower than 1. That means the pressure calculated by real gas model is lower than the value calculated by ideal gas model whether it is obvious or not during the transient. ->

The ratios exceed 1 after the crossover of the curves (around 116s, Fig. 5d). However, the corresponding points of the pressure curves corrected for the lag are always lower for the real gases, Fig. 5c.

Remove the numbering in 'Conclusions':

lines 287-288: The following...1. A simplified... -> In this study, a simplified...

line 290: 2. The effect... -> The effect...

line 296: 3. This study... -> The...

*Author Response*

Thank you for your suggestions. Please see the attachment

Author Response File: Author Response.docx

*This manuscript is a resubmission of an earlier submission. The following is a list of the peer review reports and author responses from that submission.*

**Round 1**

*Reviewer 1 Report*

Some minor comments and corrections to the text below:

1) In equation 2, the gas constant R used should be marked as the individual gas constant for air, the units of constant R and the specific heat Cp and Cv should be consistent. In the rest of the text the values are calculated for 1 kg of air.

2) In line 111, it is probably incorrectly indicated that the calculations are for 1 kilomole of air, while in the rest of the article indicates for 1 kg.

3) In line 124, the unit of the aRK coefficient is incorrect, the current unit should be multiplied by K^0.5, because in the formula on the aRK coefficient there is Tcr^2.5.

4) In line 248, units should be added to the initial data shown in this line.

5) In line 303 instead of "for gas" should be "for air", this should be corrected in all cases in the article, because the article refers specifically to air.

*Author Response*

Please see the attachment.

Author Response File: Author Response.docx

*Reviewer 2 Report*

The manuscript has been improved substantially, and is now much easier to read and follow. The authors have made an effort to include the earlier objections. Importantly, the article is now shorter and to the point.

There are still issues which became more transparent in the trimmed down version. I addressed them below.

1. Three-line derivation of the RK's polytropic isothermic exponent

One of the main 'innovative features' of the article is '(1) derivation of the R-K gas polytropic equation of air vessel during isothermal process;'. It is spread over two pages (Sec 2.3). I now derive this in a few lines.

On the one hand, it is valid (Eqs. 3 and 6)

P_j V_j=mRT, (*)

where subscript j={RK, vdW}.

On the other hand, one defines n by

equations analogous to (1)

P_j V_j^n=C_j (**)

where C_j is a characteristic constant.

For T=const., Eq. (*) becomes

P_j V_j = const.,

which together with Eq. (**) yields n=1 and C_j = const. The solution is exact. Hence authors' C2 and C3 are the same. This does not mean that the pressures in vdW and RK cases are the same (see the next point).

Whenever a very simple result appears in multiple circumstances, there is likely a very simple identity at work.

2. Implying that p_vdW(=p + m^2*a/v^2) and p_RK are pressures of vdW and RK gases - they are not.

Eqs. 3 and 5 are formally correct. One is also allowed to label quantities in the first parentheses as p_vdW or p_RK, but the labeling is misleading as these are NOT the actual pressures predicted by the vdW and RK equations! The actual pressures are always labeled p. In the case of RK equation, p is calculated correctly by equation (5) (analogously for vdW). From these it can be seen that the pressure of real gases are lowered from those of ideal gas (by the amount of 'internal pressure'), and not increased as the equation p_vdW=p + m^2*a/v^2 would suggest. The 'internal pressure', is properly called the pressure of cohesion and it is always lowering the pressure p because of attractive interactions (both, the frequency and the strength of collisions to the walls is lowered). Thus the minus sign in Eq. 5. This might seem a basic comment, but the authors are making conflicting conclusions about the vdW and RK contributions with respect to Figs. 2 (see below).

I remark that the attractive interactions are atomic dipole-dipole interactions, i.e. electrical in nature, not gravitational as the gravity is about 40 orders of magnitude weaker at that scale (line 113).

2. Missing the procedure of how the oscillations in Fig 2a,b and 3a,b are obtained.

It is not clear to me how exactly are the main results, i.e. oscillations in Fig. 2 and 3, obtained. The actual experimental set-up is described. Did authors use a computational fluid dynamics (CFD) simulation into which they implemented a particular gas equation? If so, is water vapor - which happens to be the real gas - also taken into account or just air i.e. nitrogen and oxygen? Is Henry's law taken into account? It is important to describe this (i) since it is the main result, and (ii) to see which pressures from vdW and RK equations - p or p_vdw,p_RK - are they plotting in the figures. I repeat, p_vdW and p_RK are NOT the correct predictions of the pressures!

In Fig. 2a the trend in the expansion (downward curves) are not consistent throughout the graph: the real gases show lower water level than the ideal gas up to 47 s, but later they show the opposite. Hence authors' conclusion (line 240)

'For small volume vessel (D vessel = 2 m), the water depth arising from the volume expansion drops slowly because of the positive constant internal pressure. Hence, the water depth calculated by the real gas model is higher than that by the ideal gas model at the same instant during the first phase of the transient.'

does not hold throughout the Fig. 2a.

Further, it is not clear (or 'obvious') why the comment (line 243)

'On the contrary, when the gas is compressed, the water depth of real gas is lower than the water depth of ideal gas at the same instant.'

is true. Authors refer to the explanations as 'theoretical analysis' (line 245), whereas one would like to deduce these trends from an equation. The fact is that Fig. 2a shows that oscillations of the real gases lag behind the ideal gas, but not as much in the beginning. How to explain this? Perhaps there is an effective increase of oscillating mass, or an effective lowering of the spring stiffness (frequency ~ sqrt(k/m), where k and m are the stiffness constant and the mass of the oscillator). More importantly, the pressures in vdW and RK equations should always (if the volume change is neglected) come smaller than those of ideal gas because of the negative contribution of the cohesion pressure

P = P_id - m^2*a/V^2, (***)

where P_id=mRT/V.

Are the results in Fig. 2a and 2b compatible with this? Lower pressure means smaller effective (restoring) force i.e. smaller stiffness k, and hence longer oscillation period. Thus, the lag seems to be compatible with the lower pressure. What about the amplitudes, especially in the first 100 s? The equation

P_vdW/P_id=1/(1-m*b/V) - m*a/R*T*V (****)

gives the ratio of the vdW gas and the ideal gas pressures, and will give the amplitudes (provided that there are no other contributions e.g. Henry's law). Authors are urged to estimate Eq (****) for different volumes, e.g. 10 m^3, 1 m^3, 10^-1 m^3 etc., to see if it can account for the effects in Figs 2. This is the useful information. The pressures are primary results and only after they are determined the water levels are deduced (if I correctly understand). By focusing first on the correct pressures and then on the indirect water levels will remove some of the confusion about 'water levels of real gases'.

Also, only one set of coefficients a and b are given in the appendix. Which gas it refers to? Nitrogen? Oxygen? The air? A textbook source, e.g., CRC Handbook of Phys. and Chem., needs to be used (and referenced) for the parameter values.

One final comment on modeling is whether the isothermal approximation at all holds in actual experimental set-up. I assume that is the reason for deriving the polytropic 'n' in isothermal conditions? Large amplitude changes in two-fluid systems e.g. in slugging in oil-gas industry, is known to create large changes in temperatures as the system is adiabatically expanding/contracting. One would like to know the limitations of the authors' two-phase modeling, since in reality, as already mentioned, water will evaporate/condense under large volume changes (vapor is a real gas), and air will partition into water based on Henry's law.

3. Emphasis of the novelty

Comparison of the three models - when expanded properly along the mentioned points above - is enough of a reason to publish the findings in my opinion. There is no need to oversell the isothermal polytropic derivation, which may or may not be applicable in the real-world (experimental) circumstances. Authors should not shy to cut even further, such as putting the derivations into appendix, removing the repetitive points from 'Conclusion' if they are clarified in 'Discussion'. On the other hand, they need to reinforce Sec 3.2, the most important part of the article, emphasizing much more clearly their numerical method and the properties of the oscillations. The equations above are written to help them.

A major revision (of the Sec 2 and 3) is needed to address these questions. I encourage the authors to make the final effort as the writing style and information content is now hugely improved with respect to the first version.

MINOR

line 101: m->m^3

line 102: remove 'remains constant during the process.'

line 113: gravitational -> no, attractive (electrical (dipole-dipole) in nature)

line 198: obviously -> remove

line 205: ...and VdW model...0.49m... -> ...and R-K model...0.32 m...

line 206: greatly reduces safety margin -> explain

line 231: ..obvious hysteretic... -> how exactly 'hysteretic'? Not obvious.

line 243: water depth of real gas

line 246: can be negligible -> is negligible...

line 249-250 v^2 -> V^2

line 254: equations show -> ...show that

line 269: to be followed -> to follow

line 270: to a certain extent, -> specify

line 280: obvious hysteretic nature. -> why obvious? see previous.

line 282: calculated by R-K -> by vdW

line 285-286 Therefore, the air vessel model with the real gas model is more

effective and accurate. -> Therefore, the air vessel model with the real gas model is more effective and accurate in the small-vessel regime.

line 287: And a higher safe mar -> , and a higher...

*Author Response*

Please see the attachment.

Author Response File: Author Response.docx

**Round 2**

*Reviewer 2 Report*

Authors still do not understand what the problem with their paper is.

Figs. 2 and 3 are the main results of the paper. These must be substantiated with the explicit mechanism - in form of the equations and their limiting cases - where a reader can for herself find out about the trends in the graphs. In addition, by using the ratios p_vdw/p_id and p_rk/p_id, which I gave in my last comment, one will be able to deduce more precisely the difference that the two real-gas approximations provide compared to the ideal gas case. *That* will be useful.

Nobody is interested in two-page long derivation of an exponent whose, to use authors' own defending words, "...effect of polytropic exponent is not taken into account in this paper." (#MS16). It is simply too much formalism for so little gain! This also means that the two key claims (lines 94-95):

"The key innovative features of this paper are as follows: (1) derivation of the R-K gas polytropic equation of air vessel; (2) acquisition of the R-K gas polytropic exponent;"

must be scrapped and replaced with others (see below). One should not fall in love with one's own (long) derivations that serve no or very little purpose. It is not your effort that counts, but usefulness of the information to the reader.

Authors have finally included the equations which they supposedly solve numerically. For some reason these are put in the appendix. If the equations are central to the paper they should go into the main body of the text! The authors should demonstrate how the (damped) oscillations seen in Figs 2 and 3 come out of these equations: the resultant differential equation for an effective oscillator (the level Z) must be explicitly written. Probably, one has to combine Eq. 32, which is the differential equation for the level Z, with Eqs. 34-36, which are functions of the real-gases and contain the change of sign in the term Qs|Qs|, and then transform them (additionally differentiate) to obtain the equation for the second derivative in Z. This will yield the oscillations and frequencies. Something along these lines. The equation for Z seems non-linear and complicated, hence the limiting cases need to be explored if the full solution cannot be found. This will then be the main novelty and will help the authors (and readers) understand the system better.

-----------------

IMPORTANT:

Section 2 must be compressed to few main and useful results: it must include (a subset of) the equations from the Appendix A.2, the second-order differential equation for the level Z, the above mentioned pressure ratios, and considerations of the relevant limits of the Z level e.g. small vs. large volumes. These are the relevant equations from which the trends in the figures will be transparent. Any discussion will then refer to the derivations and relevant parameters and will completely remove confusing and informationless statements such as (line 244-245): "In addition, when the air volume remains expansion, the water depth of real gas model is increasingly lower than the water depth of ideal gas model at the same instant." Readers want to know the mechanism behind the trends in the figures, and not hear the repetition of what the figures are already showing.

Usually, a peer review is not supposed to make unreasonable suggestions for extra work. But in this case it is important to direct the authors to substantiate the main results of Figs. 2 and 3, and divert them away from the useless (and trivial) derivation of the polytropic exponent.

The shorter yet information-rich section 2 will then be devoid of the notation clutter V_RK, P_RK etc. which now is painful to look at.

---------------

MINOR but ANNOYING

There is a careless annotation in line 330 calling gamma 'gravity' and H_bar 'gage'; proper names must be given.

Also, please do not make trivial changes in the text just for the sake of complying with specific comments: the former phrase "present obvious hysteretic nature" is changed into "present hysteretic nature" (line 232), but still without explaining why is there hysteresis at all. If no background behind a technical term is known, please restrain from using it.