Fluid Phase Equilibria 349 (2013) 71–82
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Assessment of clathrate hydrate phase equilibrium data for CO2 + CH4 /N2 + water system Ali Eslamimanesh a , Saeedeh Babaee b , Farhad Gharagheizi b,c , Jafar Javanmardi d , Amir H. Mohammadi b,e,∗ , Dominique Richon b,f a
Department of Chemical & Biomolecular Engineering, Clarkson University, Potsdam, NY 13699-5705, USA Thermodynamics Research Unit, School of Engineering, University of KwaZulu-Natal, Howard College Campus, King George V Avenue, Durban 4041, South Africa c Department of Chemical Engineering, Buinzahra Branch, Islamic Azad University, Buinzahra, Iran d Department of Chemical Engineering, Shiraz University of Technology, Shiraz 71555-313, Iran e Institut de Recherche en Génie Chimique et Pétrolier (IRGCP), Paris, Cedex, France f Department of Biotechnology and Chemical Technology, School of Science and Technology, Aalto University, Aalto, Finland b
a r t i c l e
i n f o
Article history: Received 25 January 2013 Accepted 19 March 2013 Available online 27 March 2013 Keywords: Data evaluation Gas hydrate Greenhouse gas Phase equilibria Thermodynamics
a b s t r a c t Outlier diagnostic in phase equilibrium data of binary clathrate hydrates containing CO2 is the main aim of the present work. The treated experimental data are concerning the clathrate hydrates of CO2 + CH4 /N2 in the presence of water. The utilized algorithm applies the basis of a mathematical approach, in which the statistical Hat matrix, Williams plot, and the residuals of two models results bring about the probable outliers detection. The range of applicability of the applied models and quality of the existing experimental data are also investigated. The van der Waals and Platteeuw (vdW–P) solid solution theory is used to model the hydrate phase, and the Valderrama–Patel–Teja equation of state (VPT-EoS) along with the non-density dependent (NDD) mixing rules is applied to deal with the fluid phases in the first model. The compositions of the vapor phase in equilibrium with gas hydrate and liquid water as well as the equilibrium pressures are predicted through the mentioned model. The second model includes a correlation proposed by Adisasmito et al., which is utilized to represent the hydrate dissociation pressures for three-phase equilibrium conditions (liquid water–vapor–hydrate). It is interpreted from the obtained results that the applied models for calculation/estimation of the phase behavior of the investigated binary clathrate hydrate systems have wide ranges of applicability. Consequently, we may, with high confidence level, say that among all data treated, one experimental equilibrium pressure value and four experimental hydrate dissociation values are probable doubtful ones. © 2013 Elsevier B.V. All rights reserved.
1. Introduction Adsorption of a molecule (or molecules) of a compound in a structure formed by the molecules of another compound normally generates clathrate structures or inclusion compounds [1,2]. The unique characteristics of the water molecules result in formation of hydrogen-bonded three-dimensional networks able to encage particular kinds of molecules [3]. The final compound is called the “hydrate”, which has been the subject of many studies from the 19th century [1–3]. At relatively high pressures and low temperatures, water molecules form various crystalline structures generally depending on the size and shape of the guest molecule(s) [1].
∗ Corresponding author at: Institut de Recherche en Génie Chimique et Pétrolier (IRGCP), Paris Cedex, France. E-mail addresses:
[email protected], amir h
[email protected] (A.H. Mohammadi). 0378-3812/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.fluid.2013.03.015
Structures I (sI), II (sII), and H (sH) are known to form as the three common structures of clathrate hydrates. The first industrial importance of the gas hydrate formation has been attributed to the blockage of gas/oil transportation pipe lines since this structure can form from association of water with natural gas constituents in petroleum industry (or found abundantly in nature) [1,2]. More rigorously, gas hydrate formation may occur during the steps of natural gas production and processing when traceable amounts of associated water exist (even in the form of very low water content). Besides these negative effects of hydrate formation there are indeed positive effects. Novel technologies utilizing gas hydrates have attracted much attention in the recent years (positive applications) [1,3,5]. For instance, clathrate structures may be used as media for the storage and transportation of natural gas and hydrogen [1,3,5]. A very promising industrial application of gas hydrates is carbon dioxide (CO2 ) capture from industrial and flue gases [3]. Ternary systems of water and carbon dioxide with either methane
72
A. Eslamimanesh et al. / Fluid Phase Equilibria 349 (2013) 71–82
or nitrogen can be pointed out as major systems in CO2 capture processes [4]. With respect to the aforementioned points, accurate and reliable phase equilibrium data are required to design efficient processes (either to prevent gas hydrate formation or to use it as a promising technique for other applications) and to adjust the parameters of the developed models (thermodynamic/numerical ones) for the corresponding phase equilibrium calculation/prediction [5]. Experimental hydrate dissociation data for the binary clathrate hydrate systems of CO2 + CH4 /N2 have been extensively reported in the literature (the readers can refer to the book of Sloan and Koh [1] for seeing a thorough summary of these data). Different experimental techniques such as phase equilibrium measurements using high pressure cell, Raman spectroscopy, nuclear magnetic resonance (NMR) spectroscopy, X-ray and neutron diffraction and so forth have been thus far applied for obtaining these kinds of data [1,3]. However, the experimental data of vapor phase composition of carbon dioxide in equilibrium with gas hydrate + liquid water in gaseous systems containing carbon dioxide (such as ternary systems of water and carbon dioxide with either methane or nitrogen) seem to be scarce. Therefore, checking the reliability of the described hydrate dissociation and vapor phase composition experimental data are of great attraction. In a previous work, we performed consistency tests for the first time on compositional data of vapor phase in equilibrium with gas hydrate and liquid water for ternary systems of water and carbon dioxide with either methane or nitrogen [4]. However, the preceding method is useful only in the case of applicability of molar compositions of equilibrium phases. It is worth pointing out that the studied phase equilibrium data include the molar compositions of the phases generally in three phase equilibrium conditions which are not necessarily the hydrate dissociation conditions. For instance, at each temperature step of the heating curve in isochoric pressure search method, fluid samples can be taken from the gas phase (or the liquid phase) to determine the corresponding composition at three phase equilibria. However, only one of these data points represents the hydrate dissociation point. Thus, it would be motivating to propose a statistical method for simultaneous detection of the doubtful data and quality evaluations along with checking the domain of applicability of a correlation and a thermodynamic model for representation/prediction of the dissociation data of binary clathrate hydrates and compositional data of vapor phase. In the present study, we use the Leverage approach [5–10] for this purpose.
2. Theory 2.1. Leverage method Detection of the outliers is to diagnose individual datum (or a dataset) that may differ from the bulk of the data (a database) [5,9,10]. The proposed methods for this purpose normally consist of simultaneous numerical and graphical algorithms [5,9,10]. The Leverage method [5–10] (employed here) utilizes the values of the residuals (i.e. the deviations of a model results from the corresponding experimental data) and a matrix (Hat matrix) composed of the experimental data and the represented/predicted values obtained from the model [5,9,10]. Hence, a suitable mathematical model is also required to pursue the calculation steps of the algorithm [5,9,10]. The Leverage or Hat indices are determined as a Hat matrix (H) with the following definition [5–10]: H = X(X t X)
−1
Xt
(1)
where X is a two-dimensional matrix composed of n data (rows) and k model parameters (columns) and t stands for the transpose matrix. The Hat values in the feasible region of the problem are the diagonal elements of the H matrix. The Williams plot is later sketched for graphical identification of the suspended data or outliers on the basis of the calculated H values through Eq. (1). This plot shows the correlation of Hat indices and standardized cross-validated residuals (R), which are defined as the difference between the represented/predicted values and the implemented data [5,9,10]. A warning Leverage (H* ) is generally fixed at the value equal to 3p/n, where n is number of training points and p is the number of model (or correlation) input parameters plus one [5,9,10]. The leverage of 3 is normally considered as a “cut-off” value to accept the points within ±3 range (two horizontal red lines) standard deviations from the mean (to cover 99% normally distributed data) [5,9,10]. Existence of the majority of data points in the ranges 0 ≤ H ≤ H* and −3 ≤ R ≤ 3 reveals that the representations/predictions of the model (or correlation) are done in its applicability domain. “Good High Leverage” points are located in domain of H∗ ≤ H and −3 ≤ R ≤ 3. The Good High Leverage can be designated as the ones, which are outside of applicability domain of the applied model [5,9,10]. In other words, the model (or correlation) is not able to represent/predict the following data at all. The points located in the range of R < −3 or 3 < R (whether they are larger or smaller than the H* value) are designated as outliers of the model or “Bad High Leverage” points. These erroneous representations/predictions can be attributed to the doubtful data [5,9,10]. 2.2. Thermodynamic model The gas hydrate thermodynamic model [4,11–13], applied in our previous work [4] has been used in this study. The general phase equilibrium criterion, which is the equality of fugacities of each component throughout all phases is considered to model the phase behavior as follows [4,11–13]: fiV = fiL fwV
=
fwL
(2) =
fwH
(3)
where f is the fugacity, i refers to the ith component in the mixture, subscript w stands for water, and superscripts V, L, and H denote the vapor, liquid, and hydrate phases, respectively. The Valderrama modification of the Patel and Teja equation of state (VPT-EoS) [14] with the non-density-dependent (NDD) mixing rules [15] was used to calculate the equilibrium pressures and the mole fractions of components in the liquid and vapor phases and the solid solution theory of van der Waals–Platteeuw [16] was applied to determine the fugacity of water in the hydrate phase. 2.3. Correlation We have herein used the following correlation recommended by Adisasmito et al. [17] to represent/predict the binary mixed hydrate dissociation data of CO2 + CH4 /N2 system: ln(p/MPa) = A + B(T/K)
−1
+ Cy + D(T/K)
−2
+ Ey(T/K)
−1
+ Fy2
(4)
where y is the mole percent of CO2 in the feed gas, A, B, C, D, E, and F are adjustable parameters. Adisasmito et al. [17], have reported the values of these parameters to determine the phase behavior of the clathrate hydrates of CO2 + CH4 . In addition, the optimal values of the corresponding parameters for the CO2 + N2 clathrate hydrates have been herein evaluated using the experimental hydrate dissociation data [23,36,41]. Both sets of the parameters are reported in Table 1.
A. Eslamimanesh et al. / Fluid Phase Equilibria 349 (2013) 71–82
73
Table 1 The parameters of Eq. (4). Parametersa
System
CO2 + CH4 b CO2 + N2 c a b c
A
B
C
175.3 415.2
−89,009 −219,543
0.07392 0.28337
D 11,307,000 29,198,870
E
F
−23.392 −1330.392
0.000039566 1.969164551
The numbers of the digits of the parameters have been determined by sensitivity analysis of the results to the values of these parameters. The parameters have been taken from Adisasmito et al. [17]. The parameters have been evaluated assuming no structural change of clathrate hydrates.
3. Experimental data Selected experimental gas hydrate dissociation data [17–20,23,34–36,41] for the systems containing binary hydrate formers (CO2 + CH4 /N2 ) in liquid water–hydrate–vapor (Lw–H–V) region together with vapor phase compositional data in equilibrium with gas hydrates [18,21–26] for the similar systems have been treated in this work. Tables 2 and 3 report the ranges of the experimental data along with their sources. 4. Results and discussion The details of the thermodynamic model [4,11–13] along with its applied parameters are presented as appendix. The absolute relative deviations of the thermodynamic model results as well as those of the Adisasmito et al. correlation [17] from the experimental values [17–26,34–36,41] are presented in Tables 4 and 5. As can be seen, the deviations of the representations (by the correlation)/predictions (by the thermodynamic model) from the corresponding experimental data [17–26] are generally acceptable to be used for the Leverage statistical approach [5–10]. To pursue our objectives, the H values have been calculated through Eq. (1) and the Williams plots have been sketched in Figures 1 to 38 shown in the supplementary material. The calculated
H and R values are presented in Tables 4 and 5. The warning Leverages (H*) have been fixed at 3p/n for the entire data. In addition, the recommended cut-off value of 3 has been applied [5,9,10]. The following results are interpreted from application of the aforementioned methodology: (1) Accumulation of the data points [17–26,34–36,41] in the ranges 0 ≤ H ≤ H* and −3 ≤ R ≤ 3 for each datasets for clathrate hydrates of the investigated binary hydrate formers reveals that the applied model and correlation are statistically valid for representation/prediction of the treated experimental values [17–26,34–36,41]. Around 98% of the data (219 data points) from the evaluated data sets are groped in this category. (2) All of the hydrate dissociation data points [17–20,23,34–36,41] (224 data points) as well as the compositional data of vapor phase in equilibrium with gas hydrate and liquid water [18,21–26] for the CO2 + CH4 /N2 systems can be declared to within the applicability domain of the thermodynamic model [4,11–13] and the applied correlation [17]. Furthermore, good high leverage points are accumulated in the domains of H* ≤ H and −3 ≤ R ≤ 3. These points may be declared to be outside of applicability domain of the applied correlation though cannot be assigned as doubtful experimental data. It should be noted
Table 2 The range of experimental vapor phase compositional data [18,21–26] treated in this work. System
Ta (K)
Nb
1 2 3 4 5 6
280.3 273.6 275.2 276.1 278.1 280.2
30 8 8 8 7 6
7 8 9 10 11 12 13 14 15
274 277 280 272.1 275.3 273.6 275.2 276.1 278.1
9 8 9 9 7 5 7 5 5
Set no.
Ref.c
Range of data p ranged (MPa)
CO2 + CH4
CO2 + N2
a b c d e
y2 rangee
3.77–4.36 1.51–2.44 1.792–2.766 1.985–3.027 2.450–3.802 3.139–4.655
0.143–0.384 0.081–0.630 0.086–0.657 0.096–0.669 0.103–0.694 0.108–0.620
[18] [21] [21] [21] [21] [21]
1.394–17.926 1.953–24.041 2.801–32.308 3.2–14.5 1.6–3.5 2.032–11.943 2.29–12.745 2.5–8.58 2.974–14.260
0–1 0–1 0–1 0.012–0.847 0.436–1 0.171–0.617 0.16–0.729 0.196–0.682 0.127–0.729
[22,23] [22,23] [22,23] [24] [25] [26] [26] [26] [26]
Temperature. Reference of experimental data. Number of experimental data points. Pressure. Mole fraction of CO2 in vapor phase.
Table 3 The range of experimental hydrate dissociation data [17–20,23,34–36,41] treated in this work. System
Equilibrium region
T range (K)
p range (MPa)
Ref.*
CO2 + CH4 CO2 + N2
Lw + H + V Lw + H + V
272.66–287.60 272.85–284.25
1.45–10.95 1.22–22.23
[17–20,34,35] [23,36,41]
*
Reference of experimental data.
74
Table 4 The results of the statistical approach [5–10] for checking the quality of experimental vapor phase compositional data [18,21–26] treated in this work. Set no.
Sysa
Texp b (K)
1
CO2 + CH4
280.3
273.6
3
275.2
4
276.1
3.04j 3.24 3.38 3.60 3.64 3.67 3.71 3.77 3.86 3.98 4.00 4.01 4.06 4.07 4.15 4.20 4.22 4.31 4.32 4.34 4.37 4.37 4.44 4.50 4.57 4.58 4.63 4.75 4.85 4.99 2.234 2.416 2.440 1.844 1.941 2.048 1.510 1.607 2.583 2.712 2.766 2.123 2.220 2.400 1.792 1.865 2.813 3.025 3.027 2.318 2.503 2.690 1.985 2.174
ppred d (MPa) 2.67 3.13 3.31 3.52 3.61 3.62 3.66 3.78 3.86 4.02 4 3.99 4.06 4.05 4.13 4.21 4.22 4.32 4.31 4.36 4.36 4.44 4.46 4.5 4.6 4.62 4.61 4.79 4.86 4.98 2.167 2.254 2.303 1.8 1.894 1.973 1.535 1.586 2.546 2.65 2.706 2.131 2.247 2.343 1.825 1.892 2.787 2.901 2.962 2.343 2.474 2.579 2.012 2.089
ARD %k , ppred 12 3.4 2 2.2 0.8 1.5 1.3 0.3 0.1 1 0.1 0.4 0.1 0.5 0.5 0.2 0.1 0.2 0.2 0.5 0.2 1.6 0.4 0 0.7 0.8 0.4 0.8 0.1 0.1 3 6.7 5.6 2.4 2.4 3.7 1.6 1.3 1.4 2.3 2.2 0.4 1.2 2.4 1.8 1.4 0.9 4.1 2.2 1.1 1.2 4.1 1.4 3.9
He
SRf 0.287 0.150 0.111 0.075 0.063 0.062 0.058 0.046 0.041 0.034 0.035 0.035 0.034 0.034 0.033 0.035 0.035 0.039 0.039 0.042 0.042 0.048 0.049 0.053 0.064 0.067 0.065 0.092 0.105 0.129 0.213 0.292 0.348 0.158 0.128 0.127 0.400 0.335 0.208 0.290 0.346 0.159 0.12768 0.128 0.408 0.333 0.206 0.289 0.346 0.159 0.127 0.129 0.412 0.332
4.523 −0.324 −0.739 −0.052 −0.867 −0.439 −0.349 −1.289 −0.910 −1.357 −0.602 −0.224 −0.470 −0.093 0.082 −0.343 −0.121 −0.103 0.276 −0.216 0.386 −1.046 0.003 0.495 0.111 −0.048 1.148 0.334 1.118 1.830 −1.870 1.672 0.080 0.191 −0.437 0.106 −0.708 1.067 −0.362 0.337 −0.126 −0.238 −1.824 1.784 0.306 0.197 −0.903 0.992 −0.420 −1.200 −0.316 1.140 −0.857 1.585
exp g
y2
(mole fraction)
1 0.683 0.585 0.488 0.45 0.448 0.429 0.384 0.357 0.302 0.31 0.311 0.288 0.293 0.268 0.245 0.241 0.215 0.217 0.203 0.203 0.183 0.179 0.169 0.144 0.141 0.143 0.104 0.09 0.065 0.141 0.125 0.081 0.345 0.288 0.22 0.63 0.545 0.166 0.129 0.086 0.384 0.302 0.228 0.657 0.565 0.179 0.134 0.096 0.405 0.315 0.232 0.669 0.579
pred h
y2
1 0.679 0.581 0.484 0.446 0.444 0.425 0.381 0.354 0.299 0.307 0.308 0.285 0.29 0.265 0.243 0.239 0.213 0.215 0.201 0.201 0.181 0.177 0.167 0.142 0.139 0.141 0.103 0.089 0.064 0.199 0.149 0.122 0.466 0.388 0.328 0.736 0.677 0.197 0.144 0.117 0.463 0.378 0.315 0.733 0.667 0.196 0.142 0.115 0.461 0.373 0.309 0.732 0.661
(mole fraction)
pred
ARD % k, y2 0 0.5 0.7 0.8 0.8 0.8 0.8 0.9 0.9 1 0.9 0.9 1 1 1 1 1 1 1 1 1 1 1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 41 19 50 35 35 49 17 24 19 12 37 20 25 38 12 18 9.7 5.8 19 14 18 33 9.4 14
H
R 0.469 0.160 0.103 0.063 0.052 0.051 0.047 0.039 0.036 0.033 0.033 0.033 0.034 0.033 0.035 0.036 0.037 0.040 0.040 0.042 0.042 0.046 0.047 0.050 0.056 0.057 0.057 0.069 0.074 0.084 0.215 0.270 0.306 0.143 0.125 0.133 0.455 0.354 0.211 0.269 0.304 0.145 0.125 0.135 0.462 0.349 0.209 0.268 0.303 0.145 0.125 0.136 0.468 0.345
−4.940 0.959 1.094 1.232 1.288 1.291 1.319 0.376 0.419 0.509 0.496 0.494 0.532 0.524 0.565 −0.408 −0.401 −0.359 −0.362 −0.340 −0.340 −0.307 −0.301 −0.285 −0.244 −0.239 −0.243 −1.207 −1.187 −1.152 0.097 1.418 0.406 −1.038 −0.584 −1.331 1.727 −0.205 0.628 1.232 0.160 −0.391 −0.713 −1.650 1.656 −0.492 0.768 1.025 0.215 −0.003 −0.579 −1.971 1.344 −0.426
Ref.i [18]
[21]
[21]
[21]
A. Eslamimanesh et al. / Fluid Phase Equilibria 349 (2013) 71–82
2
pexp c (MPa)
Table 4 (Continued ) Set no.
Sysa
Texp b (K)
5
278.1
6
280.2
CO2 + N2
274
8
277
9
280
10
272.1
11
275.3
ppred d (MPa)
ARD %k , ppred
He
SRf
exp g
y2
(mole fraction)
pred h
y2
(mole fraction)
pred
ARD % k, y2
H
R
Ref.i
3.416 3.631 3.802 3.037 3.319 2.450 2.580 4.486 4.655 3.541 4.109 3.139 3.481 5.767 7.190
3.408 3.545 3.616 3.064 3.193 2.506 2.612 4.375 4.456 3.839 3.994 3.318 3.493 5.438 6.645
0.2 2.4 4.9 0.9 3.8 2.3 1.2 2.5 4.3 8.4 2.8 5.7 0.3 5.7 7.6
0.208 0.289 0.344 0.147 0.146 0.486 0.380 0.126 0.125 0.168 0.149 0.273 0.231 0.240 0.689
−1.389 −0.424 1.285 −0.976 1.282 0.114 0.238 0.201 0.770 −2.217 0.914 −0.316 0.804 0.105 −0.459
0.202 0.139 0.103 0.323 0.233 0.694 0.609 0.147 0.108 0.344 0.235 0.62 0.49 0.114 0.115
0.194 0.137 0.109 0.361 0.293 0.728 0.647 0.132 0.104 0.348 0.278 0.63 0.525 0.099 0.095
4 1.7 5.9 12 26 4.9 6.2 10 3.9 1.1 18 1.7 7.2 13 17
0.212 0.271 0.307 0.143 0.153 0.532 0.382 0.193 0.222 0.142 0.125 0.532 0.326 0.228 0.232
1.113 0.681 0.192 −0.654 −1.892 0.729 0.151 0.537 −0.199 0.319 −2.066 1.467 −0.781 0.414 0.689
[21]
1.394 1.769 2.354 2.835 3.560 7.235 11.200 14.928 1.953 2.600 3.377 5.233 11.980 15.500 19.174 2.801 3.600 4.233 5.068 8.275 14.974 20.753 26.690 14.5 13.0 10.5 7.7 5.0 4.1 3.5 3.2 1.6 2.0 2.2 3.4 3.4 3.4 3.5
1.496 1.832 2.502 2.956 3.68 6.423 9.306 13.036 2.106 2.511 3.693 5.52 10.587 13.514 16.894 3.052 3.824 4.615 5.564 8.535 12.639 16.403 21.934 13.355 11.695 8.069 5.53 3.45 2.395 1.821 1.435 1.73 2.292 2.488 3.922 3.881 3.905 3.982
7.3 3.6 6.3 4.3 3.4 11 17 13 7.8 3.4 9.4 5.5 12 13 12 9 6.2 9 9.8 3.1 16 21 18 7.9 10 23 28 31 42 48 55 8.1 15 13 15 14 15 14
0.237 0.217 0.184 0.165 0.143 0.138 0.269 0.645 0.302 0.281 0.226 0.169 0.180 0.300 0.542 0.255 0.226 0.200 0.174 0.128 0.154 0.268 0.594 0.492 0.345 0.155 0.126 0.168 0.211 0.241 0.263 0.515 0.281 0.227 0.244 0.233 0.239 0.261
0.642 0.512 −0.351 −0.604 −1.154 0.141 2.034 −1.588 0.651 1.185 −0.820 −1.732 0.514 0.721 −0.471 0.923 0.565 −0.076 −0.705 −1.727 0.211 1.571 −0.800 −1.094 −0.657 1.999 1.100 −0.648 −0.347 −0.475 −0.281 1.708 −1.620 −0.260 −0.834 0.508 −0.275 1.083
1 0.8205 0.5999 0.5048 0.3994 0.2057 0.1159 0.0498 1 0.8491 0.5867 0.3899 0.1761 0.1159 0.0663 1 0.825 0.6999 0.5917 0.3924 0.251 0.1709 0.0905 0.012 0.033 0.099 0.184 0.333 0.499 0.665 0.847 1 0.763 0.704 0.443 0.448 0.445 0.436
0.999 0.817 0.594 0.498 0.393 0.199 0.111 0.047 1 0.846 0.58 0.382 0.169 0.111 0.063 1 0.821 0.693 0.584 0.383 0.243 0.164 0.086 0.011 0.031 0.095 0.178 0.327 0.493 0.66 0.844 0.999 0.759 0.699 0.436 0.441 0.438 0.429
0.1 0.4 0.9 1.3 1.7 3.1 4.2 5.2 0 0.4 1.2 2.1 3.8 4.5 5.2 0 0.5 0.9 1.4 2.4 3.4 4.1 5 5.8 5.4 4.2 3.1 1.9 1.2 0.7 0.3 0.1 0.6 0.7 1.6 1.6 1.6 1.7
0.491 0.286 0.148 0.127 0.130 0.208 0.274 0.335 0.510 0.333 0.163 0.148 0.239 0.282 0.325 0.458 0.263 0.176 0.135 0.142 0.210 0.270 0.347 0.277 0.258 0.207 0.159 0.125 0.165 0.288 0.520 0.670 0.226 0.175 0.232 0.227 0.230 0.240
−1.569 −0.175 0.785 1.109 0.763 0.688 −0.427 −1.782 −1.372 0.003 1.074 1.229 0.552 −0.554 −1.425 −1.709 −0.112 0.826 0.903 1.178 0.326 −0.327 −1.654 −1.449 −0.874 0.203 1.194 1.005 0.841 0.113 −1.581 −1.971 0.634 1.907 −0.330 −0.155 −0.260 −0.577
[22,23]
[21]
[22,23]
[22,23]
A. Eslamimanesh et al. / Fluid Phase Equilibria 349 (2013) 71–82
7
pexp c (MPa)
[24]
[25]
75
76
Table 4 (Continued ) Set no.
Sysa
Texp b (K) 273.6
13
275.2
14
276.1
15
278.1
a b c d e f g h i j
k
2.032 8.149 11.943 2.962 3.761 2.290 2.643 3.256 4.045 7.450 8.246 12.745 2.500 2.865 3.703 4.401 8.580 2.974 3.411 4.194 9.146 14.260
ppred d (MPa) 1.937 5.342 6.441 2.393 3.026 2.332 2.415 3.008 3.681 4.824 6.528 7.959 2.594 2.691 3.444 4.123 7.328 3.311 3.451 4.753 9.558 11.777
ARD %k , ppred 4.7 34 46 19 19 1.8 8.6 7.6 9 35 21 38 3.8 6.1 7 6.3 15 11 1.2 13 4.5 17
System. Experimental equilibrium temperature. Experimental equilibrium pressure. Predicted equilibrium pressure. Hat value. Standardized residuals. Experimental molar composition of CO2 in vapor phase. Predicted molar composition of CO2 in vapor phase. Reference of experimental data. Probable doubtful data. Absolute relative deviation: ARD % = 100 ×
(y2 /p)iexp . −(y2 /p)pred. i exp . i
(y2 /p)
.
He
SRf 0.432 0.349 0.643 0.334 0.242 0.294 0.282 0.211 0.161 0.150 0.305 0.596 0.338 0.320 0.223 0.201 0.918 0.378 0.363 0.255 0.350 0.654
0.494 −1.519 1.692 0.399 −0.554 0.110 0.410 −0.191 −0.700 1.176 −1.898 1.366 −0.937 1.427 0.344 −1.056 0.792 0.240 0.555 −0.403 −1.592 1.699
exp g
y2
(mole fraction)
0.617 0.171 0.179 0.429 0.32 0.656 0.729 0.449 0.357 0.174 0.176 0.16 0.682 0.731 0.488 0.396 0.196 0.729 0.734 0.521 0.229 0.127
pred h
y2
0.727 0.233 0.181 0.584 0.454 0.725 0.7 0.559 0.451 0.334 0.232 0.177 0.723 0.697 0.543 0.449 0.231 0.719 0.691 0.504 0.23 0.171
(mole fraction)
pred
ARD % k, y2 18 36 1.1 36 42 10 4 24 26 92 32 10 6 4.6 11 13 18 1.3 5.8 3.3 0.3 34
H
R 0.598 0.393 0.505 0.303 0.202 0.400 0.355 0.181 0.143 0.193 0.316 0.412 0.434 0.376 0.201 0.239 0.749 0.453 0.401 0.206 0.410 0.530
1.546 −0.338 1.212 −0.858 −0.955 −0.264 1.590 −0.773 −0.415 −1.433 0.409 1.251 −0.852 1.684 −0.747 −0.508 0.840 −1.302 0.892 0.407 1.054 −1.312
Ref.i [26]
[26]
[26]
[26]
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12
pexp c (MPa)
A. Eslamimanesh et al. / Fluid Phase Equilibria 349 (2013) 71–82
77
Table 5 The results of the statistical approach [5–10] for checking the quality of experimental hydrate dissociation data [17–20,23,34–36,41] treated in this work. System
Texp a (K)
pexp b (MPa)
Mole fraction of CO2 in the feed
CO2 + CH4
273.56 274.76 273.56 275.86 277.16 273.16 275.36 277.96 279.16 276.16 278.06 280.16 281.46 279.6 281.46 283.26 272.66 273.56 274.36 274.76 273.56 274.36 275.86 276.56 277.16 273.16 275.36 276.76 277.96 278.26 279.16 276.16 278.06 279.26 280.16 280.76 281.46 279.6 281.46 282.56 283.26 283.56 273.5 274.2 275.2 275.6 275.7 276.8 278.4 278.7 279.5 280.1 281.8 283 283.1 280.3 280.3 280.3 280.3 280.3 280.3 280.3 280.3 280.3 280.3 280.3 280.3 280.3 280.3 280.3 280.3 280.3 280.3 280.3 280.3
1.5 1.5 2 2 2 2.6 2.6 2.6 2.6 3.5 3.5 3.5 3.5 5 5 5 1.5 1.5 1.5 1.5 2 2 2 2 2 2.6 2.6 2.6 2.6 2.6 2.6 3.5 3.5 3.5 3.5 3.5 3.5 5 5 5 5 5 1.78 1.83 2.05 2.12 2.2 2.4 2.825 2.851 3.301 3.37 4.41 5.001 5.07 3.04 3.24 3.38 3.6 3.64 3.67 3.71 3.77 3.86 3.98 4 4.01 4.06 4.07 4.15 4.2 4.22 4.31 4.32 4.34
0.6169 1 0.2634 0.5648 1 0 0.1854 0.6195 1 0 0.2009 0.6087 1 0 0.1971 0.5989 0.4067 0.6169 0.9041 1 0.2634 0.3375 0.5648 0.7954 1 0 0.1854 0.3972 0.6195 0.7843 1 0 0.2009 0.4265 0.6087 0.7617 1 0 0.1971 0.4089 0.5989 0.8052 0.4653 0.466 0.4697 0.4701 0.4729 0.4801 0.4861 0.496 0.5002 0.504 0.5042 0.5058 0.5162 1 0.683 0.585 0.488 0.45 0.448 0.429 0.384 0.357 0.302 0.31 0.311 0.288 0.293 0.268 0.245 0.241 0.215 0.217 0.203
pcal c (MPa) 1.58 1.49 2.1 2.09 2 2.68 2.68 2.57 2.6 3.49 3.47 3.36 3.55 4.91 5.07 5.02 1.69 1.58 1.47 1.49 2.1 2.12 2.09 2 2 2.68 2.68 2.58 2.57 2.46 2.6 3.49 3.47 3.35 3.36 3.41 3.55 4.91 5.07 5.03 5.02 4.91 1.75 1.88 2.07 2.16 2.18 2.45 2.92 3 3.29 3.53 4.34 5.05 5.1 3.02 3.31 3.46 3.64 3.73 3.73 3.77 3.88 3.95 4.11 4.08 4.08 4.15 4.13 4.21 4.28 4.3 4.38 4.38 4.42
ARD %d
He
Rf
Ref.g
5.1 0.8 5 4.3 0.1 3.1 3.1 1.3 0.2 0.2 0.8 3.9 1.3 1.8 1.4 0.4 12.9 5.1 1.9 0.8 5 5.9 4.3 0.2 0.1 3.1 3.1 0.6 1.3 5.2 0.2 0.2 0.8 4.4 3.9 2.5 1.3 1.8 1.4 0.5 0.4 1.8 1.7 2.5 1.2 2 0.8 1.9 3.3 5.3 0.4 4.6 1.5 1.1 0.5 0.5 2.2 2.4 1.2 2.4 1.7 1.7 3 2.4 3.2 2.1 1.7 2.2 1.5 1.4 2 1.8 1.7 1.3 1.9
0.019 0.02 0.015 0.015 0.015 0.011 0.011 0.012 0.011 0.008 0.008 0.009 0.008 0.01 0.011 0.01 0.018 0.019 0.02 0.02 0.015 0.015 0.015 0.015 0.015 0.011 0.011 0.012 0.012 0.012 0.011 0.008 0.008 0.009 0.009 0.008 0.008 0.01 0.011 0.01 0.01 0.01 0.018 0.016 0.015 0.014 0.014 0.012 0.01 0.01 0.009 0.008 0.008 0.01 0.011 0.01 0.009 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008
−0.728 0.435 −0.985 −0.857 0.304 −0.728 −0.728 0.688 0.302 0.428 0.685 2.098 −0.343 1.453 −0.604 0.039 −2.148 −0.728 0.693 0.435 −0.985 −1.243 −0.857 0.304 0.304 −0.728 −0.728 0.559 0.688 2.104 0.302 0.428 0.685 2.227 2.098 1.456 −0.343 1.453 −0.604 −0.09 0.039 1.453 0.692 −0.341 0.046 −0.212 0.561 −0.342 −0.921 −1.615 0.441 −1.756 1.196 −0.334 −0.09 0.558 −0.6 −0.728 −0.215 −0.857 −0.472 −0.472 −1.114 −0.858 −1.372 −0.73 −0.601 −0.858 −0.473 −0.473 −0.73 −0.73 −0.602 −0.474 −0.731
[34]
[35]
[20]
[18]
78
A. Eslamimanesh et al. / Fluid Phase Equilibria 349 (2013) 71–82
Table 5 (Continued ) System
CO2 + N2
Texp a (K)
pexp b (MPa)
Mole fraction of CO2 in the feed
pcal c (MPa)
ARD %d
He
Rf
280.3 280.3 280.3 280.3 280.3 280.3 280.3 280.3 280.3 280.3 280.3 273.7 275.8 277.8 280.2 283.2 285.1 287.2 274.6 276.9 279.1 281.6 284 286.1 287.4 273.8 279.4 283.4 285.2 287.6 273.7 276.9 280.7 283.1 285.1 287.4h 275.6 278.5 280.9 281.8 285.1 287.4h 274.6 276.4 278.2 280.2 282 273.7 275.9 277.8 279.6 281.6 282.7
4.37 4.37 4.44 4.5 4.57 4.58 4.63 4.75 4.85 4.99 5.46 2.52 3.1 3.83 4.91 6.8 8.4 10.76 2.59 3.24 4.18 5.38 7.17 9.24 10.95 2.12 3.96 6.23 7.75 10.44 1.81 2.63 4.03 5.43 6.94 9.78 1.99 2.98 4.14 4.47 6.84 9.59 1.66 2.08 2.58 3.28 4.12 1.45 1.88 2.37 2.97 3.79 4.37
0.203 0.183 0.179 0.169 0.144 0.141 0.143 0.104 0.09 0.065 0 0.1 0.09 0.08 0.08 0.08 0.08 0.09 0.14 0.13 0.13 0.13 0.13 0.12 0.13 0.25 0.22 0.22 0.21 0.25 0.44 0.42 0.4 0.39 0.39 0.39 0.5 0.47 0.4 0.41 0.44 0.45 0.73 0.7 0.68 0.68 0.67 0.79 0.78 0.76 0.75 0.74 0.85
4.42 4.5 4.51 4.55 4.64 4.65 4.65 4.8 4.86 4.97 5.28 2.51 3.07 3.78 4.85 6.8 8.53 10.99 2.61 3.29 4.13 5.43 7.19 9.37 10.98 2.17 3.95 6.27 7.88 10.49 1.82 2.58 4.03 5.44 7.02 9.54 2.12 2.98 4.13 4.57 6.85 9.3 1.65 2.05 2.56 3.28 4.14 1.44 1.86 2.35 2.95 3.84 4.32
1.2 2.9 1.6 1 1.6 1.6 0.4 1.1 0.3 0.3 3.3 0.4 0.9 1.4 1.1 0 1.5 2.1 0.8 1.5 1.3 1 0.3 1.4 0.3 2.6 0.3 0.7 1.7 0.5 0.7 1.9 0 0.1 1.1 2.5 6.4 0 0.3 2.3 0.2 3 0.9 1.5 0.8 0.1 0.5 0.5 1.1 0.7 0.6 1.2 1.2
0.008 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.01 0.01 0.012 0.012 0.009 0.008 0.01 0.025 0.051 0.11 0.011 0.009 0.008 0.013 0.03 0.069 0.11 0.014 0.008 0.019 0.04 0.096 0.017 0.012 0.008 0.013 0.028 0.073 0.015 0.01 0.008 0.009 0.025 0.067 0.018 0.015 0.012 0.009 0.008 0.02 0.017 0.013 0.01 0.008 0.008
−0.345 −1.373 −0.602 −0.345 −0.603 −0.603 0.04 −0.346 0.168 0.553 2.612 0.431 0.686 0.941 1.068 0.294 −1.413 −2.82 0.044 −0.343 0.94 −0.348 0.035 −1.428 −0.108 −0.341 0.427 −0.222 −1.403 −0.376 0.175 0.945 0.298 0.167 −0.743 3.483 −1.372 0.301 0.426 −0.988 0.165 4.136 0.434 0.69 0.559 0.3 0.041 0.435 0.562 0.56 0.558 −0.344 0.94
274.95 277.45 280.25 282.55 283.55 274 276.15 280.65 283.45 284.25 273.75 276 279 281 282 272.85 274.05 277.45 278.65 280.55h 274.25 275.65 277.6
1.565 2.06 2.9 4 5.115 2 2.6 4.225 6.45 7.445 3.195 4.257 5.867 7.449 8.975 7.24 8.12 10.65 11.748 14.22 11.02 13.87 18.1
0.9659 0.9659 0.9659 0.9659 0.9659 0.778 0.778 0.778 0.778 0.778 0.4815 0.4815 0.4815 0.4815 0.4815 0.1761 0.1761 0.1761 0.1761 0.1761 0.1159 0.1159 0.1159
1.51 2.06 3.1 4.56 5.47 1.69 2.14 4.01 6.45 7.47 3.07 3.87 5.65 7.6 8.93 7.78 8.59 12.29 14.29 18.64 11.13 12.74 15.86
3.4 0.1 6.9 14.1 6.9 15.4 17.7 5.2 0 0.4 3.8 9.1 3.7 2.1 0.5 7.4 5.8 15.4 21.7 31.1 1 8.2 12.4
0.044 0.039 0.031 0.024 0.022 0.042 0.038 0.026 0.021 0.022 0.031 0.027 0.021 0.023 0.028 0.023 0.026 0.057 0.085 0.175 0.044 0.063 0.113
0.087 0.072 −0.048 −0.279 −0.058 0.313 0.463 0.35 0.287 0.314 0.223 0.487 0.428 0.217 0.453 −0.098 −0.004 −0.809 −1.508 −3.06 0.435 1.583 2.772
Ref.g
[17,19]
[23]
A. Eslamimanesh et al. / Fluid Phase Equilibria 349 (2013) 71–82
79
Table 5 (Continued ) System
Texp a (K) 278.95h 273.1 274.6 278.3 279.4 280.2 273.4 274.1 276.7 279.1 274.4 276 278.1 280.2 281.1 274.1 277.5 279.7 281.3 281.9 273.4 274 275.1 276.3 277.2
a b c
d e f g h
pexp b (MPa) 22.23 1.22 1.54 2.42 2.89 2.95 1.37 1.53 1.89 3.09 3.074 3.753 4.822 6.561 7.266 1.986 2.803 3.777 4.852 5.453 6.243 6.51 7.324 8.458 9.55
Mole fraction of CO2 in the feed
pcal c (MPa)
ARD %d
He
Rf
0.1159 0.9652 0.9652 0.9652 0.9652 0.9652 0.9099 0.9099 0.9099 0.9099 0.4358 0.4473 0.46 0.469 0.4748 0.655 0.6859 0.7132 0.7189 0.7122 0.162 0.165 0.172 0.179 0.1826
18.82 1.25 1.46 2.31 2.72 3.08 1.35 1.45 1.95 2.72 3.71 4.24 5.3 6.95 7.86 2.12 2.95 3.82 4.86 5.43 8.59 8.94 9.64 10.64 11.63
15.3 2.5 5.5 4.4 5.9 4.3 1.3 5.3 3.3 12.1 20.6 13.1 9.8 5.9 8.1 6.7 5.1 1.2 0.2 0.5 37.6 37.3 31.6 25.8 21.8
0.179 0.047 0.045 0.037 0.034 0.031 0.046 0.045 0.04 0.034 0.028 0.025 0.022 0.021 0.023 0.039 0.032 0.027 0.023 0.022 0.026 0.028 0.032 0.04 0.05
4.108 0.003 0.11 0.172 0.246 0.012 0.049 0.108 0.011 0.418 −0.381 −0.235 −0.167 −0.012 −0.138 −0.039 −0.006 0.12 0.2 0.258 −1.579 −1.628 −1.504 −1.345 −1.219
Ref.g [41]
[36]
Experimental hydrate dissociation temperature. Experimental hydrate dissociation pressure. Calculated hydrate dissociation pressure. Absolute relative deviation: ARD % = 100 × Hat value. Standardized residuals. Reference of experimental data. Probable doubtful data.
(p)iexp . −(p)cal. i exp . i
(p)
.
that, in the case of encountering good high leverage points, it is recommended to use/develop other models or correlations on the basis of different theoretical concepts for their calculations/estimations in order to avoid estimation through biased model calculations [5]. (3) The data points located in the range of R < −3 or 3 < R (ignoring their H values) are designated as outliers or bad high leverage points, as already explained. All the hydrate dissociation data points [17–20,23,34–36,41] and the compositional data of vapor phase in equilibrium with gas hydrate and liquid water [18,21–26] for the investigated systems are valid (not outliers) except two points in the corresponding data of carbon dioxide + methane in Lw + V + H region, two points from the carbon dioxide + nitrogen hydrate dissociation data, and one point from the carbon dioxide + methane three-phase equilibrium pressure data (totally 5 data points which consist about 2% of the data sets). These erroneous representations/predictions, that may be attributed to the doubtful experimental data, are bolded in Tables 4 and 5 and defined as red asterisks in the figures presented in the supplementary material. (4) The data points in the ranges H* ≤ H and R < −3 or 3 < R may be designated as neither within the applicability domain of the applied correlation nor valid data. In other words, these data cannot be well calculated/estimated by the correlation and meanwhile attributed as suspended data points. There is no such a point in the treated datasets in this work. (5) The quality of the treated data [17–26,34–36,41] (even different data in the same dataset) are different. The data with lower absolute R values (near R = 0 line) and lower H values may be declared as the more reliable experimental data [13].
It should be noted that we have studied selected phase equilibrium data available in open literature for the systems of interest. However, not all of the experimental data may be fully trustable from an experimental point of view. This may be due to the inaccuracy of the used experimental techniques, as already mentioned. The results also suggest that new experimental techniques may lead to obtaining more reliable compositional data. In the present work, as we were interested in defining only the data quality (and the range of applicability of the models), consequently, we have focused on assessment tests while our objective has not been a comparison between the different experimental methods. One point should not be eliminated from our discussion. In our previous work [4], we concluded that some of the experimental data of vapor phase compositions for the CO2 + CH4 /N2 systems are not thermodynamically consistent or not fully consistent data. However, this does not mean that those data are erroneous ones merely based on this fact that they could not pass the consistency test. As described elsewhere [4], the consistency test can show to what extent the experimental data can satisfy the Gibbs–Duhem equation [27–31]. These tests are inevitably modeldependent [4,32,33]. Therefore, it is possible that some data could not pass the consistency test but still be reliable ones from mathematical point of view. We recommend the users keep all the consistent data that can be declared not to be outliers using the applied statistical approach [5–10] and some of the similar not fully consistent data, for tuning the thermodynamic models as well as application in design of the processes. Later, on the basis of their experiences, some conclusions should be made on thermodynamic inconsistent data which can pass the statistical approach [5–10]. However, we strongly advice the users not to use the data that cannot fulfill both tests.
80
A. Eslamimanesh et al. / Fluid Phase Equilibria 349 (2013) 71–82
5. Conclusion
c=
A statistical method to evaluate the hydrate dissociation data for the CO2 + CH4 /N2 systems as well as vapor phase composition in equilibrium with these binary clathrate hydrates and liquid water was proposed on the basis of the Leverage statistical approach [5–10]. Selected hydrate dissociation datasets [17–20,23,34–36,41] as well as compositional data of the vapor phase [18,21–26] from the literature have been represented/predicted by the Adisasmito et al. correlation [17] and the thermodynamic model [4,11–13] to follow the calculation steps of the evaluation method [5–10]. The results show that: I. The applied thermodynamic model [4,11–13] and the correlation [17] are statistically correct; II. All of the points from the investigated experimental data [17–26,34–36,41] are found to be within the applicability domains of the employed model [4,11–13] and correlation [17]. III. Four hydrate dissociation data and one equilibrium pressure datum from the vapor phase compositional data may be designated as suspended (probable doubtful) experimental data.
˝c RTc pc
(A.7)
where the alpha function is given as: ˛(Tr ) = [1 + F(1 − Tr )]
(A.8)
where = 0.5 and the coefficient F is given by: F = 0.46286 + 3.58230(ωXc ) + 8.19417(ωZc )2
(A.9)
The subscripts c and r in the preceding equations denote the critical and reduced properties, respectively, and ω is the acentric factor. Besides, the coefficients ˝a , ˝b , ˝c are calculated by: ˝a = 0.66121 − 0.76105Zc
(A.10)
˝b = 0.02207 + 0.20868Zc
(A.11)
˝b = 0.57765 − 1.87080Zc
(A.12)
where Zc is the critical compressibility factor. Avlonitis et al. [15], relaxed the constraints on F and for water in order to improve the predicted vapor pressure and saturated volume for these compounds: F = 0.72318,
The aforementioned results accompanied by the results of the thermodynamic consistency test [4] can be further used to conclude about the quality of the data points, which are supposed to be applied in tuning the thermodynamic models to predict the phase equilibrium of clathrate hydrates of the treated hydrate formers.
2
= 0.52084
(A.13)
Later, Tohidi-Kalorazi [13] relaxed the alpha function for water, ˛w (Tr ), using experimental water vapor pressure data in the range of 258.15–374.15 K, in order to improve the predicted water fugacity: ˛w (Tr ) = 2.4968 − 3.0661Tr + 2.7048Tr2 − 1.2219Tr3
(A.14)
Appendix A. The details of the applied thermodynamic model and its parameters [4]
Nonpolar–nonpolar binary interactions in fluid mixtures are described by applying the classical mixing rules as follows:
The general phase equilibrium criteria, which is the equality of fugacities of each component throughout all phases is considered to model the phase behavior as follows [4,11–13]:
a=
fiV
fiL
(A.1)
fwV = fwL = fwH
(A.2)
=
where f is the fugacity, i refers to the ith component in the mixture, subscript w stands for water, and superscripts V, L, and H denote the vapor, liquid, and hydrate phases, respectively. The Valderrama modification of the Patel and Teja equation of state (VPT-EoS) [14] with the non-density-dependent (NDD) mixing rules [15] was used to calculate the compressibility factor, fugacity coefficients, and the mole fractions of components in the liquid and vapor phases and the solid solution theory of van der Waals–Platteeuw [16] was applied to determine the fugacity of water in the hydrate phase.
b=
yi yj aij
(A.15)
j
yi bi
(A.16)
yi ci
(A.17)
i
c=
i
aij = (1 − kij )
ai aj
(A.18)
where kij is the standard binary interaction parameter. For polar–nonpolar interaction, however, the classical mixing rules are not satisfactory and therefore more complicated mixing rules are necessary. In this work, the NDD mixing rules developed by Avlonitis et al. [15], are applied to describe mixing in the a – parameter: (A.19)
where ac
The VPT-EoS [14] is believed to be a strong tool for modeling systems containing water and polar compounds [15]. This equation of state is written as follows [4,11–13]: (A.3)
where R is the universal gas constant, T is temperature, and v is molar volume, and ˛(Tr ) a=a
i
a = ac + aA
A.1. Fluid phase model
RT a p= − v − b v(v + b) + c(v − b)
(A.4)
= a
˝a R2 Tc2 Pc
(A.5)
b=
˝b RTc pc
(A.6)
is given by the classical quadratic mixing rules (Eqs. (A.15) and (A.18)). The term aA corrects for asymmetric interaction which cannot be efficiently accounted for by the classical mixing rules [15]: aA =
yp2
p
api =
yi api lpi
(A.20)
i
ap ai
(A.21)
0 1 lpi = lpi − lpi (T − T0 )
(A.22)
where p is the index of polar components, and l represents the binary interaction parameter for the asymmetric term. Using the above EoS [14] and the associated mixing rules, the fugacity of each component in fluid phases is calculated from: fi = yi ϕi p
(A.23)
A. Eslamimanesh et al. / Fluid Phase Equilibria 349 (2013) 71–82
The fugacity coefficient of each component in all fluid phases is derived straightforwardly from the following relation: 1 RT
ln ϕi =
∞
∂p ∂ni
− T,V,nj = / i
V
RT V
dV − ln Z
(A.24)
where V is total volume.
Table A1 Physical properties of the investigated components [4]. Compound
pc a (MPa)
Tc b (K)
Zc c
ωd
Water Methane Nitrogen Carbon dioxide
22.055 4.599 3.394 7.382
647.13 190.56 126.10 304.19
0.2294 0.2862 0.2917 0.2744
0.3449 0.0115 0.0403 0.2276
a b c
A.2. Hydrate model
d
The van der Waals and Platteeuw statistical thermodynamic model [16], based on ideal solid solution theory, is generally used to model the gas hydrate phase equilibria. The model, which is similar to the Langmuir gas adsorption theory, considers the guest molecule to move around in a spherical cavity constructed of water molecules. Each cavity contains at most one guest molecule and there is no interaction between the encaged molecules [16]. Furthermore, the presence of the guest molecule in the cavity does not distort the hydrate crystal lattice [16]. The fugacity of water in the hydrate phase is given by [12,37]:
ˇ
fwH = fw exp where
ˇ fw
ˇ−H
−
w RT
(A.25)
ˇ−H
and is obtained from the van der Waals and Platteeuw expression [12,37,38]: ˇ−H
ˇ
= w − H w = RT
⎛
vm ln ⎝1 +
m
⎞
Cjm fi ⎠
(A.26)
j
where vm is the number of cavities of type m per water molecule in the unit hydrate cell, fj is the fugacity of the gas component j. Cjm is the Langmuir constant, which accounts for the gas–water interaction in the cavity. Numerical values for the Langmuir constant can be calculated by choosing a model for the guest–host interaction [12,37,38]: 4 Cmj (T ) = kT
r/2
w(r)
exp −
kT
r 2 dr
(A.27)
0
where k is the Boltzmann’s constant. The function w(r) is the spherically symmetric cell potential in the cavity, with r measured from center, and depends on the intermolecular potential function chosen for describing the encaged gas–water interaction. In this work, the Kihara [39] potential function is applied to evaluate the Langmuir constant as follows [40]:
( ∗ )12 w(r) = 2zε R¯ 11 r where ¯
ıN =
1 N
ˇ
The fugacity of water in the empty hydrate lattice, fw is given by [30,36]:
I/L
ˇ
fw = fw exp
1−
ˇ−I/L
w RT
r ˛ − R¯ R¯
10
ı
˛ + ı11 R¯
−N¯
− 1+
( ∗ )6 − R¯ 5 r
r ˛ − R¯ R¯
˛ ı + ı5 R¯ 4
(A.30)
I/L
where fw is the fugacity of pure ice or liquid water and the quantity inside the parentheses is given by the following equation [12,38]: ˇ−I/L
I/L
ˇ
w RT
=
w (T, p) w (T, p) − RT RT
0w − = RT0
T
P
ˇ−I/L
hw RT 2
dT +
T0
is the chemical potential difference of water
ˇ between the empty hydrate lattice (fw ) and the hydrate phase (fwH ),
w
Critical pressure. Critical temperature. Critical compressibility factor. Acentric factor.
is the fugacity of water in the empty hydrate lattice. In
Eq. (A.25), w
81
ˇ−I/L
vw RT
dp
(A.31)
P0
I/L
ˇ
where w and w are the chemical potential of the empty hydrate lattice and of pure water in the ice (I) or the liquid (L) state, respectively. p is the equilibrium pressure and T0 is the absolute temperature at the ice point. 0w is the reference chemical potential difference between water in the empty hydrate lattice and pure ˇ−I/L ˇ−I/L water in the ice phase at 273.15 K [12]. hw and vw are molar enthalpy and volume differences between an empty hydrate ˇ−I/L is given by the following lattice and ice or liquid water. hw equation [12]:
T ˇ−I/L w
=
h0w
+
Cpw dT
(A.32)
T0
where h0w is the enthalpy difference between the empty hydrate lattice and ice, at the ice point and zero pressure. The heat capacity difference between the empty hydrate lattice and the pure liquid water phase is also temperature dependent and the following equation is used [12]: Cpw = −38.12 + 0.141(T − T0 )
(A.33)
Furthermore, the heat capacity difference between hydrate structures and ice is set equal to zero [12]. A.3. Model parameters
(A.28)
−N¯ (A.29)
In the two preceding equations, z is the coordination number of the cavity (the number of oxygen molecules at the periphery of each cavity), ε would be characteristic energy, ˛ is the radius of spherical ¯ is an integer molecular core, R¯ stands for the cavity radius, and N equals to 4, 5, 10 or 11. Also, * = − 2˛, where is the collision diameter [12,40].
Table A1 shows the physical properties of the components studied in this study. The binary interaction parameters between the species of the investigated systems for the VPT-EoS [14] with NDD mixing rule [15] are reported in Tables A2 and A3. Moreover, the applied values of the Kihara [39] potential function parameters are shown in Table A4. The reference parameters for hydrate model are given in [11–13]. Appendix B. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.fluid.2013.03.015.
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Table A2 Binary interaction parameters between the investigated gases and water using the VPT-EoS [14] with NDD mixing rule [15]. Gas
CH4 (j) CO2 (j) N2 (j) a b
H2 O (i) kij = kji a
lij0 b
lij1 b
0.5028 0.1965 0.4792
1.8180 0.7232 2.6575
0.0049 0.0024 0.0064
Classical binary interaction parameters. Binary interaction parameters for the asymmetric term.
Table A3 Binary interaction parameters between the investigated gases using the VPT-EoS [14] with NDD mixing rule [15]. Gas
CH4 (i) CO2 (i) N2 (i)
kij CH4 (j)
CO2 (j)
N2 (j)
0 0.092 0.035
0.092 0 −0.036
0.035 −0.036 0
Table A4 The Kihara [39] potential parameters used in this thermodynamic model [4,11–13]. Component
˛,a (Å)
*,b (Å)
ε/k,c (K)
Methane Carbon dioxide Nitrogen
0.3834 0.6805 0.3525
3.165 2.9818 3.0124
154.54 168.77 125.15
a b c
The radius of spherical molecular core. * = − 2˛, where is the collision diameter. ε is the characteristic energy and k is the Boltzmann’s constant.
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