A Helmholtz Free Energy Equation of State of CO2-CH4-N2 Fluid Mixtures (ZMS EOS) and Its Applications
Round 1
Reviewer 1 Report
The authors proposed a Helmholtz free energy EOS of CO2-CH4-N2 fluid mixtures. It can be used to calculate all thermodynamic properties, excess enthalpies and the isochores of CO2-CH4-N2 system, predict the solubility, quantitatively estimate the impact of the impurities on the CO2 storage capacity. The quality of this paper is generally high.
Minor problems or suggestions list as following:
(1) Whether the calculation and pictures of all figures in the text are processed manually or by software needs to be clearly/detailed explained. This is an important indicator of the repeatability/reproducibility of the content of the article. All the data involved in the pictures should be listed, which is convenient for review and subsequent verification.
(2) The words "this EOS" appear many times in the text. These two words represent an EOS of CO2-CH4-N2 fluid mixtures developed by the authors. In order to better spread and understand this EOS, the author is suggested to use a proper noun to express it (such as ZMS EOS, and so on).
(3) L127: "experiment data", Do your own experiments? Otherwise, please add references.
(4) Page 5/Figure 1: It is recommended to use color legend to identify these data points.
(5) Page 5/Figure 2: The reference of the experimental data (Jaeschke and Hinze (1991)) is missing at the end of the article. Please add it to the "References" section. In addition, why is the reference data not quoted in Figure 1?
(6) Page 6/Figure 3: What about calculated and experimental CH4? Please also give the CH4 vapor-liquid phase equilibrium curves.
(7) Line 170: Please add references in the end of the sentence.
(8) Page 7/Figure 4: The experimental data should also be obtained at different temperatures. Therefore, just as the calculated lines with different colors, the experimental data points in Figure 4 need to be marked with different colors. Similar problems also appear in Figure 8 in Page 10. Please check and correct if necessary.
(9) Line206-207: The font of these words "i" and "xi" should be consistent with those in the formula.
(10) Line 338-341: delete the space between "A" and "/ACO2".
(11) Line 342: add a space before "MN2"
Author Response
We appreciate you for your previous time in reviewing our manuscript and providing valuable comments and suggestions. Below we provide the point-by-point responses.
(1) Whether the calculation and pictures of all figures in the text are processed manually or by software needs to be clearly/detailed explained. This is an important indicator of the repeatability/reproducibility of the content of the article. All the data involved in the pictures should be listed, which is convenient for review and subsequent verification.
Response 1: The calculations are got by our Fortran program. Then Origin is used to draw the graphs according to the calculation results generated by the program. “All Fortran computer codes and Origin drawing projects in this paper can be obtained freely from the corresponding author. “This sentence is also added to the abstract in the revised manuscript. All the data involved in the pictures can be got from Origin drawing projects. All Origin drawing projects are provided by the supplementary material.
(2) The words "this EOS" appear many times in the text. These two words represent an EOS of CO2-CH4-N2fluid mixtures developed by the authors. In order to better spread and understand this EOS, the author is suggested to use a proper noun to express it (such as ZMS EOS, and so on).
Response 2: ZMS EOS is used to express the EOS of CO2-CH4-N2 fluid mixtures as suggested in the revised manuscript.
(3)L127: "experiment data", Do your own experiments? Otherwise, please add references.
Response 3: It doesn’t. The reference of "experiment data" is added in the revised manuscript.
(4)Page 5/Figure 1: It is recommended to use color legend to identify these data points.
Response 4: It is recommended to use color legend to identify these data points.
(5) Page 5/Figure 2: The reference of the experimental data (Jaeschke and Hinze (1991)) is missing at the end of the article. Please add it to the "References" section. In addition, why is the reference data not quoted in Figure 1?
Response 5: The reference of the experimental data (Jaeschke and Hinze (1991)) is added to the "References" section. Because the pressure range of the reference data is small and there is already experimental data in the corresponding pressure range, the reference data is not picked in Figure 1.
(6) Page 6/Figure 3: What about calculated and experimental CH4? Please also give the CH4vapor-liquid phase equilibrium curves.
Response 6: The CH4 vapor-liquid phase equilibrium curves are given in the revised manuscript.
(7)Line 170: Please add references in the end of the sentence.
Response 7: The references are added in the end of the sentence.
(8)Page 7/Figure 4: The experimental data should also be obtained at different temperatures. Therefore, just as the calculated lines with different colors, the experimental data points in Figure 4 need to be marked with different colors. Similar problems also appear in Figure 8 in Page 10. Please check and correct if necessary.
Response 8: Now the experimental data points in Figure 4 and Figure 8 are marked with different colors in the revised manuscript.
(9)Line206-207: The font of these words "i" and "xi" should be consistent with those in the formula.
Response 9: The font of these words "i" and "xi" are in consistent with those in the formula in the revised manuscript.
(10)Line 338-341: delete the space between "A" and "/ACO2".
Response 10: The space between "A" and "/ACO2" has been deleted in the revised manuscript.
(11) Line 342: add a space before "MN2"
Response 11: A space has been added before "MN2" in the revised manuscript.
Reviewer 2 Report
In manuscript applsci-2202730 entitled "A Helmholtz free energy equation of state of CO2-CH4-N2 fluid mixtures and its applications," authors fitted interaction parameters for the system of CH4-N2 based on 703 out of 1988 data points that the author gathered from literature by applying LJ-1999 EoS. However, some amendments are required.
1. What were the criteria for selecting those 703 points?
2. What was the method of fitting?
3. What was the objective function of the fitting?
4. Authors just rely on graphical representation; why is there no statistical evaluation (just Table 4)?
5. The temperature range in Figure 1 should be cleared.
6. In Figure 2, the Error of each EoS also needs to be plotted.
7. In Figure 3, you should put more isotherm, and critical points must be shown.
8. The superiority of this fit over REFPROP and TREND should be illustrated.
9. In Figure 5, why the model output is not smooth?
10. While in Fig 6, BIC has a good effect on PR, but it can not see in Fig 7; why?
11. Fig 10 needs to be plotted with more points for the model; there is weird behavior in b and d.
Author Response
We appreciate you for your previous time in reviewing our manuscript and providing valuable comments and suggestions. Below we provide the point-by-point responses.
（1）What were the criteria for selecting those 703 points?
Response 1: Line 114-117 give the criteria for selecting those 703 points: “Since experimental PVTx data available are much more than experimental VLE data, only part of new and accurate experimental PVTx data are selected to fit the parameters to make the PVTx and VLE data keep similar weights in the fitting”.
（2）What was the method of fitting?
Response 2: The method of fitting is the Levenberg-Marquardt algorithm of the nonlinear least squares method. “In this work, the Levenberg-Marquardt algorithm of nonlinear least squares method is used for fitting, which is an efficient and widely used mathematical optimization technique.” This sentence is added in the revised manuscript.
（3）What was the objective function of the fitting?
Response 3: The objective function of the fitting is to minimize the weighted sum of squares of the calculation errors of the equation on the selected experimental PVTx and VLE data. “The fitting condition is to minimize the weighted sum of squares of the calculation errors of the equation on the selected experimental PVTx and VLE data.” This sentence is added in the revised manuscript.This sentence is added in therevised manuscript.
（4）Authors just rely on graphical representation; why is there no statistical evaluation (just Table 4)?
Response 4: There is another statistical evaluation (Table 3). In other places, because there are not many data and the images can be more intuitive to show whether these EOSs and the experimental data are consistent, graphical representation is used. “All Fortran computer codes and Origin drawing projects in this paper can be obtained freely from the corresponding author. “This sentence is also added to the abstract in the revised manuscript. All the data involved in the pictures can be got from Origin drawing projects.
（5）The temperature range in Figure 1 should be cleared.
Response 5: The temperature range is given in Figure 1 in the revised manuscript.
（6）In Figure 2, the Error of each EoS also needs to be plotted.
Response 6: The errors of three EOSs are plotted in Figure 2 in the revised manuscript.
(7) In Figure 3, you should put more isotherm, and critical points must be shown.
Response 7: More isotherms are put in Figure 3 and critical points are shown in the revised manuscript.
(8) The superiority of this fit over REFPROP and TREND should be illustrated.
Response 8: Compared the Newton's method (e.g. The Newton-Raphson method used in REFPROP[44]), it combines the advantages of two numerical minimization algorithms: the gradient descent method and the Gauss-Newton method, and it have good convergence and robustness. EXCEL TREND function is a linear regression, this fit is a nonlinear regression. A linear system is an approximation of some simple nonlinear systems under certain conditions.
The superiority of this fit is illustrated in the revised manuscript. “Compared the Newton's method (e.g. The Newton-Raphson method used in REFPROP[44]), it combines the advantages of two numerical minimization algorithms: the gradient descent method and the Gauss-Newton method, and it have good convergence and robustness.” This sentence is added in the revised manuscript.
(9) In Figure 5, why the model output is not smooth?
Response 9: This is because the method proposed in this paper to calculate the phase equilibrium properties of the mixture has a higher accuracy, which can be seen in three binary systems. But the requirement for the initial value is high and it needs to go through repeated iterations, this method for ternary system is not smooth.
(10) While in Fig 6, BIC has a good effect on PR, but it cannot see in Fig 7; why?
Response 10:The Peng-Robinson (PR) EOS, either optimized (optimal kij) or not (kij = 0) are from the reference (S. Lasala, P. Chiesa, R. Privat, J.-N. Jaubert, Measurement and prediction of multi-property data of CO2-N2-O2-CH4 mixtures with the Peng-Robinson + residual Helmholtz Energy- Based”model, Fluid Phase Equilibria (2017), doi: 10.1016/j.fluid.2017.01.016.)
As for the reason I think it might be that the PR(kij = 0) EOS deviates greatly from the experimental value for the CH4-CO2 mixture, so PR (optimal kij) EOS can have a good effect on PR. But the error of the PR (kij = 0) EOS for the CO2-N2 mixture is already small.
（11）Fig 10 needs to be plotted with more points for the model; there is weird behavior in b and d.
Response 11: More points for the model are plotted in the revised manuscript.
Reviewer 3 Report
The authors established a new equation of state for calculating phase equilibria in the system CO2-CH4-N2. In addition, the solubility of these gas mixtures in aqueous solutions has been calculated. The results of L-V immiscibility in the system CO2-CH4-N2 were compared with data from the literature (at least for Al-Sanhaf, 1990). The calculations well reproduce published experimental data. This work is an important contribution on the physical properties of the system CO2-CH4-N2. The EOS has applications to the storage capacity in the CO2 capture and storage (CCS) as well as to isochore calculations in fluid inclusion studies.
The paper is well written (English flawless) and can be published almost as it stands. Some points of critics /recommendation are given below.
(1) The software developed and described in this paper is not available to the reader (at least I cannot find any). It would be advantageous for any application that the reader has access to the software.
(2) It is somewhat strange that several papers with very similar intention are not cited here. The system CO2-CH4-N2 has been studied in much detail by several authors during the last 50 years, a.o. in relation to the study of fluid inclusions and Raman spectrometry. It is clear that this paper is not a review paper on the system CO2-CH4-N2, but for my feeling, some more basic papers on the topic can be mentioned.
A selection of references is given below:
The work of R. Thiéry:
Thiéry R., Vidal J, Dubessy J (1994) Phase equilibria modelling applied to fluid inclusions: Liquid-vapor equilibria and calculation of the molar volume in the CO2-CH4-N2 system. Geochim. Cosmochim. Acta 58-3, 1073-1082.
Thiéry R, Van den Kerkhof AM, Dubessy J (1994) vX properties of CH4-CO2 and CO2-N2 fluid inclusions: modelling for T<31°C and P<400 bars. Eur. J. Mineral. 6, 753-771.
Some older literature may be mentioned as well, like the work of G. Heyen (“Heyen EOS”) for the binary systems CO2-CH4 and CO2-N2:
Heyen G. (1981) An equation of state with extended range of application. Proc. 2nd World Congress of Chemical Engineering, Montreal, Paper 8.2.6.
Heyen G, Ramboz C, Dubessy (1982) CR Acad Sc. Paris 294, 203-206.
Darimont & Heyen (1988) … CO2-N2… Bull. Mineral 111, 179-182.
The work of Van-Hoan Le :
PhD thesis 2020 (Université de Lorraine), published in Analytical Chemistry and Chemical Geology. Even when these papers mainly deal with Raman shifts in pressurized gas mixtures, a comparison of these and present results would be interesting.
(3) The work of Bakker (compilation of several EOS) applied to fluid inclusions (isochore calculations). The much lower pressure calculations compared to other EOS for gas mixtures may be shortly discussed (Bakker, 2009, Package FLUIDS. Part 3: correlations between equations of state, thermodynamics and fluid inclusions. Geofluids 9, 63-74, and references in here).
Testing the EOS for pressure calculations: Comparing the calculations obtained from this software package (freeware from the author /homepage) gives following results for the example of a gas mixture with composition CO2=0.70; CH4=0.15; N2 =0.15 (like in Fig. 13), molar volume = 35 cm3/mol and T=600K (in order of decreasing calculated pressures):
Redlich-Kwong (1949) P=12005 bar
Holloway (1977) P=12273 bar
Bakker (1999) P= 12386 bar
Duan etal (1992, 1996) P= 9148 bar
Peng & Robinson (1976) P=7890 bar
Present EOSà Zhang etal (2023) P= ca. 6700 bar (estimated from Fig. 13)
From these examples, it is clear that the calculations presented in the present paper result in by far the lowest pressures. The discrepancies between this and other EOS may be shortly discussed. Do any experimental data support these results?
Author Response
We appreciate you for your precious time in reviewing our paper and providing valuable comments and suggestions. Below we provide the point-by-point responses
(1)The software developed and described in this paper is not available to the reader (at least I cannot find any). It would be advantageous for any application that the reader has access to the software.
Response 1: The software developed and described in this paper can be obtained freely from the corresponding author. “All Fortran computer codes and Origin drawing projects in this paper can be obtained freely from the corresponding author. “This sentence is also added to the abstract .
(2) It is somewhat strange that several papers with very similar intention are not cited here. The system CO2-CH4-N2 has been studied in much detail by several authors during the last 50 years, a.o. in relation to the study of fluid inclusions and Raman spectrometry. It is clear that this paper is not a review paper on the system CO2-CH4-N2, but for my feeling, some more basic papers on the topic can be mentioned.
A selection of references is given below:
The work of R. Thiéry:
Thiéry R., Vidal J, Dubessy J (1994) Phase equilibria modelling applied to fluid inclusions: Liquid-vapor equilibria and calculation of the molar volume in the CO2-CH4-N2 system. Geochim. Cosmochim. Acta 58-3, 1073-1082.
Thiéry R, Van den Kerkhof AM, Dubessy J (1994) vX properties of CH4-CO2 and CO2-N2 fluid inclusions: modelling for T<31°C and P<400 bars. Eur. J. Mineral. 6, 753-771.
Some older literature may be mentioned as well, like the work of G. Heyen (“Heyen EOS”) for the binary systems CO2-CH4 and CO2-N2:
Heyen G. (1981) An equation of state with extended range of application. Proc. 2nd World Congress of Chemical Engineering, Montreal, Paper 8.2.6.
Heyen G, Ramboz C, Dubessy (1982) CR Acad Sc. Paris 294, 203-206.
Darimont & Heyen (1988) … CO2-N2… Bull. Mineral 111, 179-182.
The work of Van-Hoan Le :
PhD thesis 2020 (Université de Lorraine), published in Analytical Chemistry and Chemical Geology. Even when these papers mainly deal with Raman shifts in pressurized gas mixtures, a comparison of these and present results would be interesting.
Response 2: Some more basic papers (the work of R. Thiéry and the work of G. Heyen) on the topic are cited in the revised manuscript. The work of Van-Hoan Le was cited as [1] to indicate non-aqueous CO2–CH4–N2 mixtures are often reported in the studies of fluid inclusions, and it is also added to make a comparison with this EOS in Table 4 in the revised manuscript.
(3) The work of Bakker (compilation of several EOS) applied to fluid inclusions (isochore calculations). The much lower pressure calculations compared to other EOS for gas mixtures may be shortly discussed (Bakker, 2009, Package FLUIDS. Part 3: correlations between equations of state, thermodynamics and fluid inclusions. Geofluids 9, 63-74, and references in here).
Testing the EOS for pressure calculations: Comparing the calculations obtained from this software package (freeware from the author /homepage) gives following results for the example of a gas mixture with composition CO2=0.70; CH4=0.15; N2 =0.15 (like in Fig. 13), molar volume = 35 cm3/mol and T=600K (in order of decreasing calculated pressures):
Redlich-Kwong (1949) P=12005 bar
Holloway (1977) P=12273 bar
Bakker (1999) P= 12386 bar
Duan etal (1992, 1996) P= 9148 bar
Peng & Robinson (1976) P=7890 bar
Present EOSà Zhang etal (2023) P= ca. 6700 bar (estimated from Fig. 13)
From these examples, it is clear that the calculations presented in the present paper result in by far the lowest pressures. The discrepancies between this and other EOS may be shortly discussed. Do any experimental data support these results?
Response 3: The isochore is also calculated by the relation of PVTx. Table 4 and Figure 4 indicate that calculated values from this EOS are in good agreement with experiment data, which can verify the accuracy of this EOS up to 573.15 K and 1000 bar. Since there are relatively few high-temperature and high-pressure data at present, the discrepancies between this and other EOS in high temperature and pressure may be discussed later.
Round 2
Reviewer 2 Report
It is OK to be published.
