Modeling Phase Equilibria for Acid Gas Mixtures using the Cubic-Plus

Article pubs.acs.org/jced

Modeling Phase Equilibria for Acid Gas Mixtures using the Cubic-Plus-Association Equation of State. 3. Applications Relevant to Liquid or Supercritical CO2 Transport Ioannis Tsivintzelis,*,†,‡ Shahid Ali,† and Georgios M. Kontogeorgis† †

Center for Energy Resources Engineering (CERE), Department of Chemical and Biochemical Engineering, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark ‡ Aristotle University of Thessaloniki, Department of Chemical Engineering, 54124, Greece

ABSTRACT: The CPA (cubic-plus-association) equation of state is applied in this work to a wide range of systems of relevance to CO2 transport. Both phase equilibria and densities over extensive temperature and pressure ranges are considered. More specifically in this study we first evaluate CPA against density data for both CO2 and CO2−water and for vapor−liquid equilibrium for mixtures of CO2 with various compounds present in transport systems. In all of these cases we consider various possibilities for modeling CO2 (inert, self-associating using two-, three-, and four sites) and the possibility of cross-association with water. Finally, we evaluate the predictive performance of CPA for multicomponent CO2 mixtures in transport systems which also include water, methane, and H2S. The results are compared to both experimental data and selected other approaches from literature. The results for the multicomponent systems are predictions using parameters solely estimated from binary data. The target of this work is two-fold: to assess the performance of the model for mixtures of practical significance but also to identify the best modeling approach so that we can arrive to an “engineering approach” for applying CPA to acid gas mixtures. The overall conclusion is that CPA performs satisfactorily; the model in most cases correlates well binary data and predicts with good accuracy multicomponent vapor−liquid equilibria. Among the various approaches investigated, the best ones are when cross association of CO2 with water is accounted for or when CO2 is considered to be a self-associating molecule (with three or four sites). The final choice on the best approach requires investigating a much larger set of mixtures including also alcohols and glycols, which will be considered in future works.

1. INTRODUCTION AND MODELING APPROACHES Besides the great importance of CO2 mixtures in petroleum engineering (e.g., injection for enhancing oil recovery1) and in supercritical form as solvent2 or medium for reactions,3,4 it is today mostly known, unfortunately, due to its well-known climate change behavior and its role as a greenhouse gas. To reduce CO2 atmospheric emissions, CO2 generated from coalbased and other power plants must be captured, for example, with amine/ammonia postcombustion type processes, and then transported and stored. None of these processes is particularly easy from a technical point of view; no final solution is available today, and all of them require knowledge of a variety of thermodynamic (phase equilibria, densities) and other thermophysical properties (e.g., speed of sound, Joule-Thomson coefficients, etc.). 5 In this work we focus on certain thermodynamic properties which are of importance during the © XXXX American Chemical Society

pipeline transportation of CO2. In these cases extreme and extended temperature and pressure conditions may exist, and together with CO2, water, hydrocarbons, and a wide range of “other” compounds in small amounts sometimes termed “impurities” like oxygen, nitrogen, and amines can be present. The exact nature and composition ranges of impurities are not always known but may range up to 10 %. The temperature/ pressure ranges of interest in CO2 transport processes are rather extended up to 300 bar and between −25 and 60 °C, which means that CO2 can be either liquid or supercritical. Special Issue: Modeling and Simulation of Real Systems Received: January 26, 2014 Accepted: April 15, 2014

A

dx.doi.org/10.1021/je500090q | J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Journal of Chemical & Engineering Data

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Table 1. Modeling Approaches Used with CPA for Modeling of CO2 and H2S with Watera cross association parametersc for the acid gas−water interaction

association schemeb for acid gases

modeling approach A B1 B2 C2 D1 D2 E1 E2

binary interaction parameter

CO2

H2S

εcross

βcross

kij

inert 1 negative site 1 negative site 2B 3B 3B 4C 4C

inert 2 positive sites 2 positive sites 2B 3B 3B 4C 4C

mCR-1 rule expt value expt value CR-1 rule expt value CR-1 rule expt value

adjustable adjustable adjustable CR-1 rule adjustable CR-1 rule adjustable

adjustable adjustable adjustable adjustable adjustable adjustable adjustable adjustable

The experimental values for the CO2−water and H2S−water cross-interactions and all the CPA binary parameters are given by Tsivintzelis et al.6,7 (Table 5 of ref 6 and Table 4 of ref 7). bFor the definition of the various association schemes see ref 11. cFor the definition of the CR-1 and the modified CR-1 combining rules see ref 4. a

Table 2. Association Schemes11 Used for Carbon Dioxide and Hydrogen Sulfide association Schemes

association sites on each molecule

inert 2B 3B

no sites one positive−one negative sites one positive−two negative sites for CO2 two positive−one negative sites for H2S two positive−two negative sites

4C

It is the purpose of this study to investigate various modeling approaches for CO2 and H2S and their interactions with water. The various modeling approaches are presented in Table 1. In multicomponent mixtures with both CO2 and H2S with water, combined modeling approaches were used. For example, the combined approach where the H2S and CO2 are treated as in the D2 and E2 approach of Table 1, respectively, is mentioned as the D2−E2 approach; that is, the fist part denotes the H2S scheme and the second one denotes the CO2 scheme. For all multicomponent systems, binary parameters were adopted from the corresponding binary systems. In cases A, D1, and E1 there is one adjustable parameter for each binary system with water. In all other cases, CO2 and/or H2S mixtures with water were modeled using two adjustable parameters, the binary interaction parameter (kij) and the cross-association volume (βcross). Mixtures of the two acid gases, when considered as associating molecules, are described with the CR-1 rule. The number of adjustable parameters should be considered together with the other details of the modeling approach when validating the results. On the basis of the results from our previous studies,6,7 we mainly consider CO2 with 3 and 4 sites (3B, 4C) and H2S with only three sites (3B) in the detailed calculations for mixtures. The association scheme terminology mentioned above is the one introduced by Huang and Radosz for the SAFT model.11 Table 2 summarizes the association schemes used in this study for carbon dioxide and hydrogen disulfide. For all fluids, except nitrogen and oxygen (see the next sections), the CPA pure compound parameters used in this work are taken from recent literature.6−8

In this work we continue our recent investigations with CPA for some H2S6 and CO27,8 systems, and we apply CPA to a much wider range of mixtures, properties (phase equilibria and density), and conditions of importance to CO2 transport. For the description of CPA and details on the model, we refer to these articles or a recent monograph.9 We will also compare the results to other approaches, especially the SRK−Huron Vidal (SKR/HV) employed by Austegard et al.10 for CO2−water− methane. There are not many more literature studies which consider such a type of multicomponent mixtures. Most importantly, in an attempt to provide a more complete investigation, we will consider a variety of modeling cases which will permit us to arrive at a single engineering approach, which may be useful for a wide range of acid gas mixtures. All pure fluid parameters are adopted from our previous studies,6,7 except those otherwise mentioned. In all cases, binary interaction parameters will be estimated from binary phase equilibrium data, and the same parameters will be used without changes for predicting densities and multicomponent multiphase behavior for mixtures containing CO2 with a wide range of relevant to transport processes compounds (water, alkanes, N2, O2, and H2S).

Table 3. CPA Pure Fluid Parameters for CO27 and Deviations between Literature and Calculated Vapor Pressures, Liquid and Vapor Densitiesa a0 assoc. scheme n.a. 2B 3B 4C

b

L bar mol 3.5079 2.6911 3.0558 3.1404

ε

b −2

2

−1

L mol

0.0272 0.0273 0.0281 0.0284

c1 0.7602 0.5560 0.6703 0.6914

%AAD −1

bar L mol 78.12 51.68 39.23

β 0.0568 0.0411 0.0297

c

sat

P

0.3 (0.3) 0.5 (0.5) 0.3 (0.4) 0.4 (0.5)

%AAD c

ρ

liq

1.6 (1.6) 1.0 (0.9) 0.9 (0.9) 0.9 (0.9)

%AAD c

ρvap

6.3 (6.1) 5.6 (5.5) 7.4 (7.2) 7.3 (7.1)

a

The temperature range is 218 K to 303 K and CPA calculations are compared to the correlations from Angus et al.12 The CPA parameters had been regressed based only on vapor pressures and liquid densities in the temperature range 216.6 k to 274 K.7 In parentheses are given the deviations exp exp between CPA and the Span-Wagner13 EoS results. bn.a.: non self-associating. c % AAD = (1/n)∑i|((Xcal i − Xi )/(Xi ) × 100)| where X stands for sat liq vap P , ρ or ρ and n is the number of experimental data points. B



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Figure 3. Comparison of the CPA and Span-Wagner EoS13 for pure CO2 density at 400 bar.

Figure 1. Densities of supercritical CO2. Literature data12 (points) and CPA calculations (lines).

Table 4. CPA Parameters Together with the %AAD between Literature and Calculated Vapor Pressures or Liquid Densities (both Fluids Are Modeled as Inert Compounds)

fluid O2 N2

temp range

a0

b

% AAD

% AAD

K

L2 bar mol−2

L mol−1

c1

Psat

ρliq

1.3900

0.0216

0.4754

0.3

1.9

1.3733

0.0260

0.4985

0.5

2.1

77.3 to 139.2 63.1 to 113.6

Table 5. CPA Binary Interaction Parameters and Deviations from Experimental Data22,23 for CO2−O2 and CO2−N2 Systems temp range association scheme of CO2

Figure 2. Densities of liquid CO2. Experimental data (points) and CPA calculations (lines) (the hydrostatic pressure at one ocean depth is reported, dash line).

0 2B 3B 4C

2. MODELING OF PURE CO2 AND SIMPLE CO2-CONTAINING SYSTEMS 2.1. Pure CO2. Table 3 summarizes the recently presented7 CPA parameters for CO2 where the percentage deviation in vapor densities (property not included in the parameter estimation) is also shown. The deviations for the vapor pressure and liquid density are higher than those presented earlier7 because a much more extended temperature range is used. Nevertheless, the performance is still satisfactory. The error in the vapor density, however, is rather high and it can be reduced if all three properties are included in the parameter estimation. However, this happens at the significant cost of increasing the error of both vapor pressures and liquid densities. Thus, we maintain in the current investigation the parameters shown in Table 3. In addition, it is well-known that such equation-of-state models overestimate the critical point of the fluids. Consequently, their use at conditions close to the critical point should be considered carefully, having in mind this drawback. It is important that even the “self-associating” CO2 parameters used in the corresponding modeling approaches should be

K CO2−O2 223.15 to 283.15

% AAD

% AAD

xCO2

yCO2

0.0897 0.0403 0.0297 0.0219

1.2 1.7 1.6 1.4

6.9 9.7 8.7 7.8

−0.0578 −0.1217 −0.1313 −0.1426

1.2 1.6 1.6 1.4

5. 7 6. 8 6.9 6.4

kij

CO2−N2 0 2B 3B 4C

270.00

reasonably low, since any CO2 self-associating interactions are expected to be very weak. The association energy parameters from Table 3 are (in K) 939.6 (2B), 621.6 (3B), and 471.5 (4C). They compare well to other values from literature, for example, for CO2-4C with PR-CPA of Perakis et al.14 (436.6 K) but also the recent values for PC-SAFT by Diamantonis and Economou15 which are 512.9 K, 371.2 K, and 307.4 K, respectively for 2B, 3B, and 4C CO2. While there are no “experimental” association energies for CO2, these values are “acceptably” low and comparable to the reported association energies for H2S and of course much lower than the association energies of water, alcohols, and other strongly hydrogen-bonding compounds. For comparison, the H2S association energies proposed for CPA are 973 K, 654 K, and 448 K (for 2B, 3B, and 4C, respectively)6 and 455 K (for 3B as suggested by Ruffine et al.16). Similarly, for PC-SAFT and H2S C



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Figure 5. CO2−H2S VLE. Experimental data24 (points) and CPA calculations (lines) considering that H2S and CO2 are inert compounds (solid lines) or that H2S has (3B) three association sites and CO2 (4C) has four (dotted lines).

Figure 4. CO2−O2 (a) and CO2−N2 (b) VLE. Experimental data22,23 (points) and CPA calculations (lines) assuming CO2 as a nonassociating fluid.

Table 6. CPA Correlation Results for H2S−CO2 VLE Using Different Association Schemes for H2S and CO2 in the Temperature Range of 254.1 K to 349.3 K association scheme of H2S

association scheme of CO2

kij

0 0 0 0 2B 2B 2B 2B 3B 3B 3B 3B 4C 4C 4C 4C

0 2B 3B 4C 0 2B 3B 4C 0 2B 3B 4C 0 2B 3B 4C

0.117 93 0.015 92 0.049 74 0.054 44 0.035 60 0.158 93 0.163 46 0.155 86 0.051 89 0.156 37 0.186 81 0.151 57 0.054 26 0.143 33 0.145 43 0.132 19

εcross/βcross

CR-1 CR-1 CR-1 CR-1 CR-1 CR-1 CR-1 CR-1 CR-1

ΔTa

Δya

1.09 1.07 1.04 1.02 1.32 1.15 1.16 1.16 1.15 1.09 1.21 1.15 1.10 1.04 1.12 1.10

0.0092 0.0079 0.0081 0.0079 0.0149 0.0095 0.0099 0.0098 0.0118 0.0103 0.0113 0.0108 0.0102 0.0106 0.0106 0.0104

Figure 6. CO2−water VLE (water content in the vapor phase). Experimental data25−27 and CPA correlations using methods A (inert), B1 (solvation 1st approach), and B2 (solvation 2nd approach).

the following association energies have been reported: 537 K to 781 K (2B) and 566 K (4C) by Tang and Gross,17 and 273 K from Diamantonis et al.18 Finally, for SAFT Diamantonis et al. used the value of 804 K.18 The parameters of Table 3 represent satisfactorily the densities of CO2 at other conditions as well. Using the n.a. set of CO2 parameters, the % AAD in liquid density is 2.3 and in vapor density 7.1, in the range 218 K to 304.21 K. For supercritical CO2, in the range 310 K to 430 K and 0 bar to 350 bar, the %AAD in density is 2.8 (a typical plot is shown in Figure 1). Figure 2 shows the densities of liquid CO2 over an extensive pressure range (200 bar to 400 bar) in a plot inspired by the presentation shown by Angelo Lucia19 and Brewer et al.20 Using the data from the latter manuscript20 (Figure 2) and in the range 273.15 K to 283.15 K and 202 bar to 406 bar, the % AAD

exp ΔX = (1/n)∑i|(Xcal i − Xi ) where X stands for T or y and n is the number of experimental data points.

a

D



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Figure 7. CO2−water VLE at 298 K. Experimental data49−58 (points) and CPA calculations (lines) using one adjustable binary parameter (kij) with the n.a. CO2 parameters of Table 3 (a,b) and the experimental values of Tc, Pc, and ω as CO2 pure fluid parameters (c,d).

between CPA and experimental density values, assuming that CO2 is a nonassociating compound (n.a.), is 1.5. In addition, we have carried out a comparison of CPA with the Span and Wagner EoS13 for pure CO2, which is often used by the industry for practical applications. This is a multiparameter equation of state that can be used only for pure CO2 properties, such as vapor pressures, densities, speed of sound, heat capacities, Joule-Thomson coefficients, etc. It uses numerous adjustable parameters regressed using data for CO2 properties in a very wide temperature and pressure range and it is valid up to 1100 K and pressures up to 800 MPa. It cannot be used for CO2 mixtures, but according to the authors, the Span and Wagner EoS is able to represent even the most accurate data within experimental uncertainty. The uncertainty of predictions in pure CO2 densities ranges between 0.03 % to 0.05 % (up to 523 K and 30 MPa). Thus, for such temperature and pressure ranges, the correlations of the Span and Wagner EoS can be considered almost as accurate as the experimental data. The deviations between CPA and Span-Wagner results are shown in parentheses in Table 3

and, as expected, they are not that different from the deviations between CPA and the data of Angus et al.12 We have then compared CPA to the Span-Wagner EoS13 for the representation of single phase densities (vapor, liquid, and supercritical) for isobaric data (at (1, 10, 30, 50, 70, 150, 200, 300, 400) bar) over an extensive temperature range (230 K to 900 K). The average deviations ( %AAD) are quite low and similar for the various ways of modeling CO2 (1.7 %, 1.3 %, 1.2 %, and 1.3 % for n.a., 2B, 3B, and 4C, respectively). The deviations (difference between CPA and Span-Wagner) are lower than 1 % under 70 bar but they are around 2.5 % in the worst case scenario. The highest deviations were calculated when CO2 was treated as a 2B molecule, for example, 2.5 % at 400 bar. However, overall the association schemes (2B, 3B, and 4C) show almost similar results. When looking at the individual phases in the aforementioned temperature and pressure range, the % deviations (CPA vs Span-Wagner) in vapor, liquid, and supercritical densities are equal to 2.2, 1.0, 1.0, respectively, with CO2-3B and 2.2, 0.8, 1.1, respectively, with CO2-4C. Clearly, the performance is very similar in both modeling cases, best for liquid E



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Table 7. Percentage Average Absolute Deviations for Densities of CO2−Water Saturated Phases data points

temp (K)

waterrich phase

Teng et al. (1997)33 278.0 6 283.0 6 288.0 6 293.0 6 King et al. (1992)34 288.15 11 293.15 10 298.15 6 Li et al. (2004)35 332.15 6 Hebach et al. 36 (2004) 283.0 4 284.4 to 284.9 17 287.6 to 287.8 5 292.8 to 292.9 6 302.6 to 303.2 5 Chiquet et al. (2007)37 307.4 to 309.6 9 322.8 to 324.4 9 342.1 to 343.7 9 362.3 to 364.2 8 382.9 to 384.2 8 Kvamme et al. (2007)38 322.8 11 Zappe et al. (2000)39 298.0 313.0 323.0 Tabasinejad et al. (2010)40 468.470 10 423.060 11 382.410 10 422.980 445.740 461.620 478.350 Overall

CO2rich phase

B1 approach

B2 approach

% AAD in aqueous phase density

% AAD in aqueous phase density

% AAD in CO2 phase density

1.1 0.9 0.7 0.6

1.2 1.0 0.7 0.6

0.2 0.2 0.1

0.3 0.2 0.1

0.7

0.8

0.9 0.6 0.6 0.4 0.3

0.8 0.7 0.6 0.4 0.2

% AAD in CO2 phase density

9 9 9 8 8

0.9 0.2 0.3 0.6 0.6

3.7 2.2 1.6 4.3 3.3

0.8 0.2 0.4 0.8 0.8

3.3 2.5 1.5 4.3 3.8

11

0.3

2.2

0.3

2.2

6 7 10

1.7 2.5 2.0 1.0 0.9 0.6

10 10 9 9

1.7 2.7 2.4 0.7 0.6 0.4

4.8 5.5 5.4 5.8 0.5

Figure 8. Density of CO2−water saturated phases. Experimental data37,40 for water rich phase (solid symbols), CO2 rich phase (open symbols), and CPA predictions (lines).

3.6

used directly in the regression. Equal weights were given to vapor pressures and liquid densities. Alternatively, such pure fluid parameters for oxygen and nitrogen could be estimated through the values of the critical properties (critical temperature, pressure, and acentric factor), since the CPA reduces to SRK for nonassociating fluids. However, in this study we have used the same approach for all fluids (both associating and nonassociating), that is, to adjust the parameters to vapor pressure and liquid density data. The followed approach for nonassociating fluids results in more accurate description of volumetric properties at the expense of an overestimation of the critical point. The CO2−oxygen and CO2−nitrogen vapor−liquid (VLE) equilibrium was calculated and the results are presented in Table 5, while characteristic calculations are illustrated in Figure 4. The lowest deviations from experimental data are obtained if CO2 is treated as non-self-associating fluid. Acceptable results are obtained if CO2 is treated as self-associating fluid, especially via the 4C scheme (four sites). Furthermore, as shown in Figure 4, the model overestimates the mixture’s critical point. Such behavior is very common in all equation-of-state models, even in models with much more sophisticated physical terms than CPA, such as the SAFT-type models. Consequently, at conditions close to the critical point, the use of such models for

5.4 6.2 6.1 6.7 0.6

3.8

densities and worst for the vapor densities. A typical example is shown in Figure 3. 2.2. Simple Binary CO2-Containing Mixtures. The binary interaction parameters for mixtures of CO2 with light gases (oxygen and nitrogen) and H2S are estimated using vapor− liquid experimental data for the corresponding binary systems. The latter system has been studied before6 but here we present a more extensive investigation where both acid gases are also considered as self-associating fluids. The CPA pure fluid parameters (a0, b, and c1), for oxygen and nitrogen, which are presented in Table 4, were estimated in this study by fitting the predictions of the model to vapor pressure and liquid density data from DIPPR21 correlations, which are based on experimental data. Thus, no experimental data were F



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Table 8. Percentage Average Absolute Deviations for Densities of CO2−Water Single Phase Mixtures water content

temperature

pressure

B1 method

B2 method

mol fraction

K

bar

% AAD

% AAD

ref of expt data

0.0612 0.1470 0.2101 0.2870 0.3670 0.4349 0.5317 0.6017 0.6934 0.7523 0.7913 0.0030 0.1210 0.1210 0.2173 0.2173 0.2173 0.5008 0.5008 0.5008 0.5008 0.7032 0.7032 0.7032 0.9347

415.4 to 697.7 455.2 to 697.9 469.5 to 698.1 644.8 to 698.4 518.6 to 693.1 544.2 to 693.1 563.0 to 697.8 580.0to 599.2 616.6 to 698.6 625.4 to 299.2 634.7 to 699.3 308.2 373.2 398.2 373.2 398.2 423.2 398.2 423.2 448.2 473.2 423.2 448.2 473.2 448.2

58.8 to 108.3 70.7 to 117.6 78.3 to 127.1 127.5 to 140.0 105.0 to 154.1 121.7 to 170.0 143.1 to 198.3 165.8 to 227.1 214.0 to 273.5 242.6 to 313.5 267.9 to 345.8 79.1 to 124.8 5.7 to 8.6 6.1 to 19.9 2.5 to 5.1 2.8 to 11.0 3.0 to 18.3 2.1 to 3.4 2.3 to 5.8 2.1 to 16.0 2.8 to 33.5 3.9 to 6.7 4.1 to 13.1 4.6 to 22.6 9.3 to 16.5

1.1 1.2 1.2 0.8 0.7 0.6 1.5 1.1 1.6 2.5 3.2 8.0 0.6 0.8 0.6 0.7 0.6 1.3 0.8 1.0 1.5 1.1 1.3 1.1 5.7

1.4 1.8 2.0 1.7 2.1 2.2 3.4 3.2 4.1 5.4 6.4 6.5 0.7 0.9 0.6 0.8 0.7 1.3 0.8 1.1 1.8 1.0 1.3 1.0 5.6

41 41 41 41 41 41 41 41 41 41 41 42 43 43 43 43 43 43 43 43 43 43 43 43 43

1.5

2.2

overall

Figure 9. Density of CO2−water one-phase mixtures. Experimental data41 (symbols) and CPA predictions (lines).

Figure 10. Solubility of H2O in CO2 with 5.31 mol % CH4. Experimental data45 (points) and CPA calculations (lines).

process design should be done carefully, having in mind this drawback. Otherwise, the so-called “crossover” approach should be used, which results in better description of the critical point at the expense of more adjustable parameters. Mixtures with H2S and CO2 were studied in our previous work for H2S systems.6 However, this previous study was focused on H2S and, consequently, in all the studied modeling approaches, CO2 was treated as a non-self-associating fluid (as inert or solvating compound). Subsequently, modeling approaches in which both H2S and CO2 were treated as self-associating

compounds were not presented. In the next sections, new calculations are presented for multicomponent mixtures, in which both H2S and CO2 were treated as self-associating compounds. To perform such calculations for multicomponent mixtures, the binary interaction parameters for the H2S−CO2 mixture should be estimated. In Table 6, results are presented for all possible combinations of modeling approaches for the acid gases. Some characteristic results are presented in Figure 5. Almost all combinations result in a good description of the VLE of this system and similar deviations from the experimental data. Thus, an appropriate G



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Table 9. Binary parameters Water−CO2−Methane Using Various Approaches from Table 1 εcross/(bar L mol−1)

βcross

0.1145 0.0098 0.0882

mCR1:83.275

0.1836

0.1141 0.0098 0.0882

expt: 142.0

0.0162

0.0261 0.0098 0.0349

expt: 142.0

0.0087

0.0173 0.0098 0.0339

expt: 142.0

0.0040

0.0300 0.0098 0.0292

expt: 142.0

0.0030

kij Case A H2O−CO2 methane−H2O methane−CO2 Case B1 H2O−CO2 methane−H2O methane−CO2 Case B2 H2O−CO2 methane−H2O methane−CO2 Case C2 H2O−CO2 methane−H2O methane−CO2 Case D2 H2O−CO2 methane−H2O methane−CO2 Case E2 H2O−CO2 methane−H2O methane−CO2

additional calculations are presented here against recent measurements. New data for the water content of the CO2− water vapor phase have been recently published.25 The data cover a very wide pressure range (1200 bar, Figure 6). The CPA equation of state was applied using the binary parameters obtained based on previously published data (only up to moderate pressures, mostly up to 250 bar). Such binary parameters were published in our previous work7 (binary parameters for all the CO2−water modeling approaches of Table 1 were presented in Table 4 of ref 7). The three choices which have been tested here correspond to methods A, B1, and B2 of Table 1. Recall that with method A we have only one interaction parameter, while the other two methods have two adjustable parameters. As it is shown in Figure 6, CPA accurately describes the water content at low and moderate pressures. However, the deviations are increased at higher pressures (higher than 500 bar). We will return to the CO2−water phase behavior later in sections 4 and 5, where we will consider also multicomponent CO2−water−methane mixtures and comparison to literature models. The industrially very important CO2−water mixture has been further studied in several recent studies which shed light on what is possible when modeling this mixture with various association models. In two separate works14,28 Perakis et al. have shown with the PR-CPA EoS that CO2−water can be modeled almost equally successfully over an extensive pressure range using either the 4C scheme for CO2 or treating it as non-self-associating, but solvating with water. Kim et al.29 have used CPA (with the same parameters as in ref 7) and they correlated very well their own data for the water solubility in liquid CO2 using a temperature dependent kij. They used the experimental value for the cross association. Tang and Gross17 presented also rather good results for CO2− water with a polar PC-SAFT for both the liquid and vapor phase using a temperature dependent kij in the region 293 K to 473 K.

−0.0232 0.0098 0.0882

selection of the best modeling approach cannot be based on binary data but we should consider multicomponent mixtures.

3. PHASE BEHAVIOR AND DENSITIES OF CO2−WATER MIXTURES OVER EXTENSIVE CONDITIONS 3.1. Phase Behavior. The phase behavior of this system has been studied extensively in a recent publication,7 but a few

Table 10. Compositions of the Studied Mixtures (1−7) for the Ternary Water−Methane−CO2 System mole fraction CO2/methane = 20/80 methane CO2 water

CO2/methane = 70/30

1

2

3

4

5

6

7

0.799 73 0.199 93 0.000 34

0.799 66 0.199 91 0.000 43

0.799 52 0.199 88 0.000 60

0.798 68 0.199 67 0.001 65

0.299 84 0.699 62 0.000 55

0.299 75 0.699 41 0.000 84

0.299 58 0.699 02 0.001 40

Table 11. Average Deviations from Experimental Data for the Ternary System Water−CO2−Methane (Dew Point Calculations) %AAD in P mixture 1 Case A 18.62 Case B1 29.6 Case B2 31.47 Case C2 32.8 Case D2 27.6 Case E2 27.7

mixture 2

mixture 3

mixture 4

mixture 5

mixture 6

mixture 7

overall

13.1

13.3

9.1

30.7

13.3

9.3

15.5

20.9

17.6

9.9

56.2

26.6

15.3

25.4

22.0

18.3

9.9

62.3

28.6

16.1

27.2

22.9

18.8

10.0

65.3

29.6

16.5

28.2

19.6

17.0

9.7

51.2

24.2

14.3

23.6

19.7

17.0

9.7

52.0

24.5

14.4

23.8

H



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Table 12. Binary Interaction Parameters6,7 Used in Modeling H2S−CO2−CH4

CO2 parameters (Table 3) and using the experimental critical properties and acentric factor. The first set of CPA parameters (see Table 3) provides a good description of the vapor pressures and liquid densities, but overestimates the critical point of the pure compound and, consequently, results in nonaccurate correlations in this region. This is a common behavior of all classical equation of state models. To overcome such a drawback, in a second approach, the experimental Tc, Pc, and ω were used as pure fluid parameters (for CO2: Tc = 304.21, Pc = 73.84, and ω = 0.2236), since the CPA equation of state reduces to the SRK EoS for nonassociating fluids. In this way the model does not overpredict the critical point, however, at a cost of lack in accuracy for densities. Using this later approach, we have used a temperature independent kij (optimized at the 278 K to 478 K temperature range) for modeling the CO2 − water phase behavior. First, we carried out calculations with a single a kij optimized using data for both phases that are in equilibrium (water rich and CO2 rich phases). As it is shown in Figure 7, in this case and with both approaches, CPA can accurately describe the liquid (water-rich) phase, but not the CO2-rich phase(s). As it is expected, it predicts the appearance of the three phase VLLE equilibrium, but does not accurately predict the minimum in the solubility, which exists at the VLLE pressure. According to the model predictions, the second liquid phase and the vapor phase that are in equilibrium have very similar compositions in contrast to the experimental data, according to which the second liquid phase contains much more water than the vapor phase. Then, we performed calculations using kij optimized only using data for the CO2-rich phase. As it is shown in Figure 7, in this way the model predicts a minimum in the solubility of the water, however the overall description is not satisfactory, especially for the water-rich phase. As it is clear from these calculations, CPA can describe the correct phase behavior for CO2−water mixture (VLE, VLLE, and LLE); however, if no cross-association is accounted for, the composition of the CO2-rich phase is not accurately correlated either using the CPA parameters, or using the experimental Tc, Pc, and ω for CO2. For these reasons, we will therefore continue in this and subsequent articles to model the CO2−water systems via the solvating or the self-associating approaches. Finally, we should mention that the description of the complex CO2−water phase behavior (both liquid and vapor phases) is also possible with empirical cubic equations of state, but only at the cost of many and different adjustable parameters in the aqueous and nonaqueous phases which are strongly temperature and composition dependent, an approach that is thermodynamically inconsistent.32 Satyro and van der Lee32 mention “it is recognized that this formulation is not entirely consistent from a thermodynamic point of view...but for gas processing modeling

kij methane CO2 CO2 (3B) CO2 (4C)

methane

H2S

H2S (3B)

0.0882 0.0339 0.0292

0.0760 0.1179 0.0497 0.0544

0.0221 0.0519 0.1868 0.1516

They do not report percentage deviations, only plots from where it can be seen that there are significant deviations in the vapor phase above 373 K and at high pressures. Diamantonis and Economou15 carried out recently a systematic study on the performance of various SAFT models and modeling approaches for CO2−water and compared with literature models. The best results, which are comparable to CPA,7 are obtained with PC-SAFT assuming solvation between CO2 and water and using two fitted adjustable parameters (kij and cross association volume). This approach is essentially identical to the one we recommend for CPA7 and moreover the deviations reported in the 298 K to 533 K range (11 % in the liquid phase and 22 % in the vapor phase) are similar to CPA7 results with the solvating approach (around 12 % to 13 % for both solubilities in the 298 K to 478 K range). It is clear from the above analysis that, in almost all investigations, the most successful approaches for modeling CO2−water with association models include either accounting explicitly for the solvation and/or considering CO2 to be a self-associating molecule. However, Galindo and co-workers30,31 have successfully modeled CO2−water with SAFT-VR including (qualitatively correct) the minimum of the water content in the vapor phase without such considerations. The minimum in the water content of the CO2-rich phase of the water−CO2 mixture is related to the three phase equilibrium (VLLE), which exists at temperatures lower than the upper critical end point (UCEP) of the binary mixture (which is very close to the critical point of pure CO2). Thus, at temperatures lower than the UCEP, VLLE can be observed at a specific pressure. However, the minimum in the water content of the vapor phase is observed up to high temperatures at least up to 373 K as indicated by experimental studies. We have investigated with CPA an approach for modeling CO2−water based on the approach of dos Ramos et al.,30,31 who showed the SAFT-VR equation of state can capture (at least qualitatively) the minimum in the water content of the CO2-rich phase for the water−CO2 mixture, if the pure CO2 parameters are rescaled to capture the critical properties of this compound. We have performed calculations with CPA using the n.a.

Table 13. % AAD of CPA Predictions from Experimental Data46 for H2S (1)−CH4 (2)−CO2 (3) VLE at 222.15 K to 238.75 K and 20.7 bar to 48.3 bar association schemea for CO2−H2S

% AAD in y1

% AAD in y2

% AAD in y3

% AAD in P

n.a.−n.a. n.a.−3B 3B−n.a. 3B (1d−2a)−3B (2d−1a)b 4C−n.a. 4C−3Bb

1.3 1.4 1.2 1.3 1.2 1.3

2.0 2.0 2.2 2.1 2.0 2.0

2.0 1.9 2.2 2.0 2.1 2.0

7.8 9.9 7.2 10.3 7.5 10.1

a

Abbreviations: n.a., nonassociating (inert); d, proton donor (positive site); a, proton acceptor (negative site). bParameters for the cross interactions were calculated using the CR-1 rule. I



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this formulation works remarkably well and provides a simple, flexible, and reliable modeling platform”. However, our opinion is that thermodynamically consistent models should be used and indeed, as shown, association models with suitable parametrization can represent well CO2−water over extensive temperature and pressure ranges with few adjustable parameters. 3.2. Densities. In our previous study7 we presented a few results for CO2−water saturated densities at two temperatures (308 K and 323 K). There are, however, many more data for the densities of CO2−water saturated phases from various other sources,33−40 which cover a wide temperature and pressure range (278 K to 480 K and 0 bar to 1300 bar, respectively). CPA was applied to predict the densities over the whole range of pressures and temperatures for which experimental data exist using two representative modeling approaches. In both of them CO2 was modeled assuming that each molecule has one site, which is only able to cross associate with water. In the first one the cross association energy was estimated using the modified CR-1 combining rule (B1 approach of Table 1), while the experimental value was used in the second one (B2 approach of Table 1) The results for the densities of saturated phases are summarized in Table 7, where it can be seen that CPA represents the aqueous phase density very well ( % AAD less than 1 %) and with deviations around 3.6−3.8 % for the CO2-rich phase density. The results are very similar for both modeling methods used. Typical examples are shown in Figure 8. Experimental data are also available for the densities of CO2−water single phase mixtures,41−43 covering the range of 308 K to 700 K and 0 bar to 350 bar. The results are shown in Table 8, while some of them are illustrated in Figure 9. Very good agreement is obtained with the experimental data in all cases except for the data42 at 308 K.

4. MULTICOMPONENT SYSTEMS 4.1. Ternary Systems with Water, H2S, and Methane. a. Water−CO2−Methane. First we consider the industrially important system water−CO2−methane which has been also studied by Austegard et al.10 with CPA and SRK/HV. This system is also of interest because there are two types of data, the dew points measured by Jarne et al.44 for different mixture compositions and water content data of a CO2 gas mixture containing 5.31 % methane, measured by Song and Kobayashi.45 In all calculations the parameters used are shown in Table 9. In all cases, such binary parameters were estimated7 using solely data for the corresponding binary mixtures and, consequently, the model calculations for the ternary mixture can be characterized as predictions. Results for cases A and B2 were also shown in our previous work8 (also for B1, however in ref 8 CO2 was modeled with two negative sites (for solvation), instead of one used in this work), but in this work we consider a much broad range of cases which also include self-associating CO2 (using the 2B, 3B, or 4C association schemes). Initially, CPA was applied to predict the dew points of seven mixtures with compositions shown in Table 10, and the results were compared with the experimental data of Jarne et al.44 The results are shown in Table 11. Subsequently, the model was applied to predict the vapor phase of the ternary mixtures, and characteristic calculations for the water content are shown in Figure 10. It is interesting to see that the two types of calculations suggest different modeling approaches. For the dew point calculations, the best results are obtained with the simplest case A

Figure 11. H2S−CH4−CO2 VLE with CPA and experimental data46 at 238.75 K and 20.68 bar (a), 34.47 bar (b), 48.26 bar (c).

(no solvation, no self-association, only one kij per binary system). For all mixtures, cases B2 and C2 yield similar results. Cases D2 and E2 perform better than cases B1 and B2. On the other hand, for the water content calculations as a function of pressure, case A fails to represent the minimum observed in the water content of the CO2−CH4 rich phase. All other cases perform qualitatively correct; that is, both the solvating and self-associating CO2 J



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Table 14. Binary Parameters for Water−H2S−CO2−Methane System kij

εcross (bar L mol−1)

βcross

kij

εcross (bar L mol−1)

βcross

CH4−H2O 0.0098 CH4−H2S 0.0221 H2S−CO2 0.0519 CH4−CO2 0.0882 Case D1−D1: H2S 3B, CO2 3B (CR1) H2O−H2S 0.2601 CR-1:110.47 CR-1:0.0635 H2O−CO2 0.2531 CR-1:109.11 CR-1:0.0533 CH4−H2O 0.0098 CH4−H2S 0.0221 H2S−CO2 0.1868 CR1:53.04 CR1:0.0490 CH4−CO2 0.0339 Case D2−D2: H2S 3B, CO2 3B, (expt εcross for binaries with water) H2O−H2S 0.0991 Exp: 108.78 0.0299 H2O−CO2 0.0173 Exp: 142.00 0.0040 CH4−H2O 0.0098 CH4−H2S 0.0221 H2S−CO2 0.1868 CR1:46.81 CR1:0.0416 CH4−CO2 0.0339 Case D1−E1: H2S 3B, CO2 4C (CR1) H2O−H2S 0.2601 CR-1:110.47 CR-1:0.0635 H2O−CO2 0.2673 CR-1:102.89 CR-1:0.0453 CH4−H2O 0.0098 CH4−H2S 0.0221 H2S−CO2 0.1516 CR1: CR1: CH4−CO2 0.0292 Case D2−E2: H2S 3B, CO2 4C, (expt εcross for binaries with water) H2O−H2S 0.0991 Exp: 108.78 0.0299 H2O−CO2 0.0300 Exp: 142.00 0.0030 CH4−H2O 0.0098 CH4−H2S 0.0221 H2S−CO2 0.1516 CR1: CR1: CH4−CO2 0.0292

Case A: CO2 and H2S inert compounds H2O−H2S −0.0098 CH4−H2O 0.0098 CH4−H2S 0.0760 H2O−CO2 −0.0232 H2S−CO2 0.1179 CH4−CO2 0.0882 Case B1: Solvation of CO2 and H2S in water (mCR-1) H2O−H2S 0.0985 mCR1:83.275 0.0654 H2O−CO2 0.1145 mCR1:83.275 0.1836 CH4−H2O 0.0098 CH4−H2S 0.0760 H2S−CO2 0.1179 CH4−CO2 0.0882 Case B2: Solvation of CO2 and H2S in water (experimental values for εcross) H2O − H2S 0.1913 Exp: 108.78 0.0624 H2O−CO2 0.1141 Exp: 142.00 0.0162 CH4−H2O 0.0098 CH4−H2S 0.0760 H2S−CO2 0.1179 CH4−CO2 0.0882 Case D1−B1: H2S 3B, solvation of CO2 in water (CR-1 and mCR-1, respectively) H2O−H2S 0.2601 CR-1:110.47 CR-1:0.0635 H2O−CO2 0.1380 mCR1:83.275 0.0911 CH4−H2O 0.0098 CH4−H2S 0.0221 H2S−CO2 0.0519 CH4−CO2 0.0882 Case D2−B2: H2S 3B, solvation of CO2 in water (experimental values for εcross) H2O−H2S 0.0991 Exp: 108.78 0.0299 H2O−CO2 0.1141 Exp: 142.00 0.0162

Table 15. Deviations from experimental data for the H2O (1)−CO2 (2)−H2S (3)−CH4 (4) Quaternary Mixture (Experimental Data from Huang et al.,47 Mixture with Overall Composition H2O 0.5, CO2 0.3, H2S 0.05, CH4 0.15 mol) x1

x2

x3

x4

Case A: CO2 and H2S inert compounds % AAD 0.4 27 9.4 69 Case B1: Solvation of CO2 and H2S (mCR-1) % AAD 0.5 35 12 75 Case B2: Solvation of CO2 and H2S (experimental values for εcross) % AAD 0.2 11 16 71 Case D1−B1: H2S 3B, solvation of CO2 in water (CR-1 and mCR-1, respectively) % AAD 0.5 40 21 78 Case D2−B2: H2S 3B, solvation of CO2 in water (experimental values for εcross) % AAD 0.2 10 8.3 70 Case D1−D1: H2S 3B, CO2 3B, (CR-1) % AAD 0.8 69 26 95 Case D2−D2: H2S 3B, CO2 3B, (CR1 for H2S-CO2 and expt εcross for binaries with H2O) % AAD 0.7 60 11 83 Case D1−E1: H2S 3B, CO2 4C, (CR-1) % AAD 0.6 42 24 79 Case D2−E2: H2S 3B, CO2 4C, (CR1 for H2S-CO2 and expt εcross for binaries with H2O) % AAD 0.2 8.3 8.8 69

approaches can reproduce the water content-pressure behavior. From a quantitatively point of view, cases B2 and C2 yield very similar results, and the best results are obtained for cases D2 and E2. These results indicate that cases D2 and E2 should be

y1

y2

y3

y4

30

0.6

2.9

1.6

1.0

2.6

1.2

14

0.7

2.9

1.2

20

1.0

2.3

1.1

11

0.7

2.5

1.2

>100

1.7

2.4

1.1

12

0.7

2.2

1.6

88

1.4

2.5

0.9

11

0.6

2.5

1.1

9.5

selected if an overall good representation of the phase behavior must be obtained for the ternary system. In our forthcoming publication we will consider a wider range of water−CO2−alkane mixtures. K



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Figure 13. Solubility of methane in water. Experimental data48 (points) and CPA calculations (lines).

Figure 12. Dew and bubble points of the H2O−CO2−H2S-CH4 mixture ((50, 5, 40, 5) % mol). Experimental data47 and CPA predictions (case B2 of Table 1).

Table 16. Number of Binary Parameters That Were Adjusted to Binary Data for All Modeling Approaches Used to Describe the H2O−CO2−H2S−CH4 VLE approach

total number of binary adjustable parameters (all six sub-binary systems)

D2−E2 D2−B2 B2 A D1−B1 B1 D2−D2 D1−E1 D1−D1

8 8 8 6 7 8 8 6 6

Figure 14. Water content in methane vapor phase. Experimental data48 (points) and CPA calculations (lines).

Table 17. Binary Interaction Parameters for Water−Methane. The Obtained kij Values Follow a Linear Dependency from Temperature: kij = 0.00245 × T/K − 0.72917 temp (K)

kij

298.15 313.15 338.15

0.0003 0.0394 0.0987

Table 18. Temperature Specific Binary Parameters for CO2−Water, Assuming One Negative Site on Every CO2 Molecule That Is Only Able to Cross-Associate with Water

b. H2S−CO2−Methane. The H2S−CO2−methane system was considered by Tsivintzelis et al.6 assuming that both acid gases are inert. Here, we compare this approach to an “associating” approach where CO2 is modeled assuming that it has one positive and two negative (3B) or two two-positive and two-negative (4C) association sites, while H2S was modeled assuming that it has two positive and one negative sites (3B). In the latter case, parameters for the cross interactions were calculated using the CR-1 combining rule.4 In all cases, the binary interaction parameters6,7 were adopted from the corresponding binary mixtures (Table 12). One interaction parameter per binary is used for all binaries and for both cases. The results are presented in Table 13 and in Figure 11 (for the highest temperature and the three pressures). In all cases, the model’s predictions are in very satisfactory agreement with the experimental data, even when the acid gases are modeled without associating sites. The difference between the two cases (inert or self-associating acid gases) is small.

temp (K)

kij

εcross/(bar L mol−1)

βcross

298.15 to 477.6 298.0 323.2

0.1141 0.0914 0.1371

expt: 142.0 expt: 142.0 expt: 142.0

0.0162 0.0140 0.0206

4.2. The CO2−H2S−Water−Methane System. The water−H2S−CO2−methane system was modeled using several different approaches. Depending on the approach, H2S was treated as an inert compound, cross-associating with water (solvating) or self-associating fluid using the 3B association scheme (this can be considered as optimum association scheme according to our previous study, Tsivintzelis et al.6). Also, CO2 was treated as an inert compound, cross-associating with water (solvating), or as a self-associating fluid using the 3B and 4C association schemes. In cases where cross association with water occurs, the corresponding association parameters were estimated using the CR-1 combining rule4 or the experimental values for the cross association energy. The relevant binary parameters are presented in Table 14 and were estimated6,7 solely from binary system data. The results are presented in Table 15, and a typical result is shown in Figure 12. L



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Figure 15. CO2−water VLE at 298 K and 323.2 K. Experimental data49−58 (points) and CPA calculations (lines).

combining rules or adopted from experimental or theoretical (ab initio) studies. Consequently, this decreases the number of adjustable parameters for each binary mixture, which is shown in Table 16. Having this in mind, we can make the following conclusions: (i) The first three approaches are comparable in terms of the number of parameters, and it appears that selfassociating H2S/CO2 or combinations of self-associating H2S/solvating CO2 perform rather similarly and better than only considering solvation between CO2, H2S, and water. (ii) It is very interesting to see that the fourth best method (A) is the simplest of all with only six interaction parameters (one per binary) and no solvation/self-association considerations at all. (iii) It is of interest to consider that mentioned in points (i) and (ii) and at the same time see that the other two methods (D1−D1, D1−E1) that use six adjustable parameters are between the worst approaches. Alone this leaves us to conclude that it is not entirely certain that the good performance of approaches D2−E2 and D2−B2 is due to the assumption of selfassociation/solvation of acid gases, but it may very well be that the extra parameters of cases D2−E2 and D2−B2 make the difference or, alternatively, that CR-1 combining rules perform poorly. (iv) D2−D2 is comparable to D2−E2 and D2−B2 in terms of the number of parameters, and the fact that the former results in higher deviations shows that CO2 behaves more like 4C rather than the 3B molecule, which makes sense from an intuitive point of view. (v) Finally B1 and B2 are also comparable in terms

The results for cases A, B1, B2, D1−B1, and D2−B2 were also shown in our previous work,6 but in this work we consider several more cases, also with a self-associating CO2 (3B, or 4C). From Table 15, it can be concluded that the best approaches for this particular system are the D2−E2 and D2−B2 approach, while alternative approaches that can be used are the B2 and A. It is clear that the best modeling approaches (cases D2−E2, D2−B2 and B2) use the experimental values for the association energy between acid gases and water. Also, it can be concluded that treating H2S and CO2 as self-associating fluids, using the 3B and 4C association schemes, respectively, seems to give the best results (Case D2−E2). An alternative approach could be the accounting of solvation of acid gases in water using the experimental association energy (case B2). All the results for the multicomponent systems of this study are pure predictions, since all the needed binary parameters are obtained from the corresponding binary systems and no parameter is adjusted to the multicomponent mixture data. However, in order to compare the various modeling approaches the total number of adjustable binary parameters (adjusted solely to binary data) for all six sub-binary systems involved in this quaternary mixture should be considered. The parameters are shown in Table 14 and are typically the binary interaction parameter (kij) and the cross-association volume (βcross). However, in several sub-binary systems there is no need for adjusting the cross-association parameters, since they are estimated from M



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Figure 16. Solubility of H2O in CO2 with 5.31 mol % CH4. Experimental data45 (points), CPA, and SRK/HV calculations (lines).

Figure 17. Solubility of H2O in pure CO2, in pure CH4 and in CO2 with 5.31 mol % CH4. Experimental data45 (points), CPA using temperature dependent parameters and SRK/HV calculations (lines).

of the number of parameters and this tells us that using the experimental values for the cross association should be preferred (for the solvation cases and in general) rather than using the modified CR-1 rule.

5. DISCUSSION 5.1. Further Investigation of the CO2−Water−Methane System and Comparison to the SRK/HV Equation of State. Because of its importance for practical applications, we have further studied the CO2−water−methane ternary, both by introducing more adjustable parameters (a temperature dependent kij) and by comparing the results to those obtained by Austegard et al.10 with SRK/HV. First, temperature dependent binary interaction parameters were estimated for the water−methane system and they are presented in Table 17. The results are presented in Figures 13 and 14, where they are compared to when a temperature independent kij is used. Second, temperature specific binary parameters were estimated for the CO2−water system at 298 K and 323.2 K. All calculations were performed assuming solvation and using the experimental value of the cross association energy (εcross = 14200 J/mol). The results are presented in Table 18 and Figure 15, where results using temperature independent binary parameters are also shown.

Figure 18. Solubility of H2O in CO2 with (5.31, 7.31, and 9.31) mol % CH4. Experimental data45 (points) and CPA calculations (lines).

Then, with the use of the binary parameters of Tables 17 and 18, calculations were performed for the CO2−water−methane ternary system. The results are shown in Figures 16−18. The results are compared to those obtained with CPA using N



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5.2. Comparison to Other Studies. Some studies have been published59,60 where association models are compared against each other (and experimental data) for various systems. Often, similar results are obtained for binary systems if the parameter estimation procedure for pure compounds and for mixtures and the number of adjustable parameters are the same. Another calculation verifying this conclusion, this time for CO2−water, is shown in Figure 19 in which the CPA and NRHB EoS59,60 are compared. Both models are treated in the same way, that is, using two adjustable parameters, a temperature independent kij, and the cross-association volume or entropy (for NRHB) which are fitted to the experimental data. Moreover, the experimental value is used for the cross-association energy (−14200 J/mol). The parameters of the two models are presented in Table 19. Although there are some minor Table 19. Binary Parameters for CPA and NRHB Models model

association sites in CO2

kij

εcross/(J mol−1)

cross association parameter

CPA NRHB

1 negative site 1 negative site

0.11406 0.05460

expt: 14200 expt: 14200

βcross = 0.01623 ΔScross = − 27.25 J mol−1 K−1

differences in models’ correlations at different temperatures (as shown in Figure 18 for 298 K), the overall performance in the whole temperature range (298 K to 473 K) can be considered similar. It should be noted that without accounting for crossassociation interactions, none of the models captures the minimum in the water content of the vapor phase, which corresponds to the optimal conditions of many processes in the chemical industry. Again the performance is similar. Of course a stringent test−comparison between the various approaches would be to consider their performance for multicomponent systems. Relatively few investigations have been published for multicomponent systems, and most of them consider very few mixtures. The CO2−water−methane system has been studied, as mentioned, by Austergard et al.10 (with CPA, SRK, and SRK/HV), by Miguez et al.61 and by Kontogeorgis et al.8 (with CPA using inert and solvating CO2). Finally, the CO2− water−H2S−methane mixture has been studied with CPA in our previous work6 and by Li and Firoozabadi62 with their PR-CPA and also by Knudsen et al.,63 with conventional cubic equations of state.

Figure 19. CO2−water VLE at 298 K. Experimental data49−51 (points) and CPA and NRHB calculations (lines).

temperature independent parameters7 and with the SRK/HV model as it was applied by Austegard et al.10 The effect of the methane content was investigated using the CPA temperaturedependent parameters. These results are presented in Figure 18. When temperature-dependent parameters are used, both CPA and SRK/HV have two adjustable parameters per temperature for the CO2−water system, while for water−methane CPA has only one adjustable parameter per temperature and SRK/HV has two. For the CO2−water−5.31 % CH4 system, the % AAD in the water mole fraction is for CPA 10 and 23 at 298 K and 323 K, respectively, and for SRK/HV 3.3 and 19 at 298 K and 323 K, respectively. Thus, SRK/HV performs a bit better especially at the lowest temperature (with one more adjustable parameter). We have also investigated the possibility of obtaining the kij of the CO2−water from the ternary data (with methane). With the obtained value (kij = 0.13) for the binary interaction parameter calculations were then performed for the CO2−water binary system. The results revealed that in this way the ternary data and the vapor phase of the binary system are well described, but there are significant deviations for the liquid phase compositions of the binary mixture CO2−water. Thus, this method is not further pursued.

6. CONCLUSIONS The CPA equation of state has been applied to phase equilibria and densities of various mixtures relevant to CO2 transport (CO2−light gases, CO2−water, CO2−water−methane, CO2− H2S−methane, CO2−water−H2S−methane,). Interaction parameters were estimated from binary phase equilibrium data, and then CPA was used to predict densities and phase behavior for the multicomponent systems. Moreover, various modeling approaches have been used with either inert or self-associating CO2 and H2S and with possibility for solvation with water molecules (again using either combining rules or experimental values for the cross-association energy). The investigations showed that very good results are obtained for the densities of supercritical (up to 350 bar) and liquid CO2, with deviations from experimental data at 2.3 % for liquid CO2 and 2.8 % for the supercritical CO2. Moreover, good results are obtained for the densities of CO−water mixtures up to 1250 bar. The deviations against new experimental data are 1 % for the aqueous phase and 3.8 % for the CO2 phase. O



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(3) Beier, M. J.; Grunwaldt, J.-D.; Tsivintzelis, I.; Jensen, A. D.; Kontogeorgis, G. M.; Baiker, A. Selective Oxidation of Benzyl Alcohol in Dense CO2: Onsight by Phase Behavior Modeling. J. Supercrit. Fluids 2012, 63, 199−207. (4) Tsivintzelis, I.; Beier, M. J.; Grunwaldt, J.-D.; Baiker, A.; Kontogeorgis, G. M. Experimental Determination and Modeling of the Phase Behavior for the Selective Oxidation of Benzyl Alcohol in Supercritical CO2. Fluid Phase Equilib. 2011, 302, 83−92. (5) Diamantonis, N. I.; Boulougouris, G. C.; Tsangaris, D. M.; El Kadi, M. J.; Saadawi, H.; Negahban, S.; Economou, I. G. Thermodynamic and Transport Property Models for Carbon Capture and Sequestration (CCS) Processes with Emphasis on CO2 Transport. Chem. Eng. Res. Des. 2013, 91, 1793−1806. (6) Tsivintzelis, I.; Kontogeorgis, G. M.; Michelsen, M. L.; Stenby, E. H. Modeling Phase Equilibria for Acid Gas Mixtures Using the CPA Equation of State. 1. Mixtures with H2S. AlChE J. 2010, 56, 2965−2982. (7) Tsivintzelis, I.; Knotogeorgis, G. M.; Michelsen, M. L.; Stenby, E.H. Modeling Phase Equilibria for Acid Gas Mixtures Using the CPA Equation of State. PartII: Binary Mixtures with CO2. Fluid Phase Equilib. 2011, 306, 38−56. (8) Kontogeorgis, G. M.; Tsivintzelis, I.; Michelsen, M. L.; Stenby, E. H. Towards Predictive Association Theories. Fluid Phase Equilib. 2011, 301, 244−256. (9) Kontogeorgis, G. M.; Folas, G. K. Thermodynamic Models for Industrial Applications. From Classical and Advanced Mixing Rules to Association Theories; John Wiley and Sons: Chichester, UK, 2010. (10) Austegard, A.; Solbraa, E.; De Koeijer, G.; Mølnvik, M. J. Thermodynamic Models for Calculating Mutual Solubilities in H2O− CO2−CH4 Mixtures. Chem. Eng. Res. Des. 2006, 84, 781−794. (11) Huang, S. H.; Radosz, M. Equation of State for Small, Large, Polydisperse and Associating Molecules. Ind. Eng. Chem. Res. 1990, 29, 2284−2294. (12) Angus, S.; Armstrong, B.; de Reuck, K. M. International Thermodynamic Tables of the Fluid State - 3 Carbon Dioxide; Pergamon: New York, 1976. (13) Span, R.; Wagner, W. A New Equation of State for Carbon Dioxide Covering the Fluid Region from the Triple-Point Temperatture to 1100 K at Pressures up to 800 MPa. J. Phys. Chem. Ref. Data 1996, 25, 1509−1596. (14) Perakis, C.; Voutsas, E.; Magoulas, K.; Tassios, D. Thermodynamic Modeling of the Vapor−Liquid Equilibrium of the Water/ Ethanol/CO2 System. Fluid Phase Equilib. 2006, 243, 142−150. (15) Diamantonis, N. I.; Economou, I. G. Modeling the Phase Equilibria of a H2O−CO2 Mixture with PC-SAFT and tPC-PSAFT Equations of State. Mol. Phys. 2012, 110, 1205−1212. (16) Ruffine, L.; Mougin, P.; Barreau, A. How To Represent Hydrogen Sulfide with the CPA Equation of State. Ind. Eng. Chem. Res. 2006, 45, 7688−7699. (17) Tang, X.; Gross, J. Modeling the Phase Equilibria of Hydrogen Sulfide and Carbon Dioxide in Mixture with Hydrocarbons and Water using the PCP-SAFT Equation of State. Fluid Phase Equilib. 2010, 293, 11−21. (18) Diamantonis, N. I.; Boulougouris, G. C.; Mansoor, E.; Tsangaris, D. M.; Economou, I. G. Evaluation of Cubic, SAFT, and PC-SAFT Equations of State for the Vapor-Liquid Equilibrium Modeling of CO2 Mixtures with Other Gases. Ind. Eng. Chem. Res. 2013, 52, 3933−3942. (19) Lucia, A.; A. Multiscale Gibbs-Helmholtz Constrained Cubic Equation of State. J. Thermodyn. 2010, article ID 238365. (20) Brewer, P. G.; Friederich, G.; Peltzer, E. T.; Orr, F. M., Jr. Direct Experiments on the Ocean Disposal of Fossil Fuel CO2. Science 1999, 284, 943−945. (21) DIPPR DIADEM, The DIPPR Information and Data Evaluation Manager for the Design Institute for Physical Properties, version 1.1.0; AIChE: New York, 2006. (22) Fredenslund, A.; Sather, G. A. Gas-Liquid Equilibrium of the Oxygen-Carbon Dioxide System. J. Chem. Eng. Data 1970, 15, 17−22. (23) Somait, F. A.; Kidnay, A. J. Liquid−Vapor Equilibria at 270.0 K for Systems Containing Nitrogen, Methane and Carbon Dioxide. J. Chem. Eng. Data 1978, 23, 301−305.

CPA was applied to mixtures of CO2 with light gases (oxygen and nitrogen). Very good VLE correlation is achieved in all cases but the best results are obtained if CO2 is treated as a non-selfassociating fluid. However, acceptable deviations from experimental data are obtained if CO2 is treated as a self-associating fluid. Prediction of CO2−H2S−methane ternary VLE is satisfactory with all modeling approaches of CPA resulting in similar deviations from experimental data. For the ternary CO2−water− methane system, the importance of accounting for cross association between CO2 and water is evident, either via a “solvating CO2” or a “self-associating CO2”. Although an “inert” CO2 approach performs well against the dew-point measurements of this gas mixture, it cannot satisfactorily predict the water content of the vapor phase as a function of pressure. Both the “solvating” and the “self-associating” CO2 provide an overall better performance. Few literature models have been applied so extensively as CPA and only few models have been applied to multicomponent systems. SRK with Huron-Vidal mixing rules has been used for CO2−H2O−CH4. The results are similar to CPA, with somewhat better results for SRK/HV which uses four binary parameters per temperature (compared to three parameters for CPA). Finally, for the water−CO2−H2S−methane quaternary mixture, the best results are obtained when the experimental values of the cross association energy between acid gases and water are used. With this restriction, treating H2S and CO2 as self-associating fluids, using the 3B and 4C association schemes, respectively, seems to give the best results (Case D2−E2). Alternative approaches could be considering H2S to be a 3B molecule and accounting for solvation of CO2 in water (Case D2−B2) or the accounting of solvation of both acid gases in water (case B2). The overall conclusion of this study is that CPA is a successful model for describing and predicting phase behavior and densities for a variety of mixtures relevant to CO2 transport. Various modeling approaches have been compared and the best overall performance is obtained using either a “self-associating” CO2 or a solvating CO2 with water. In both cases two adjustable parameters are needed for CO2−water binary mixture but the cross association energy can be obtained from experimental values. Further investigations will be carried out in order to establish whether one of these two approaches should be preferred.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected], Tel +30 2310996249. Funding

The authors wish to thank Statoil and Gassco (Norway), BP International Limited (UK), DONG Energy (Denmark), and Petrobras (Brazil) for supporting this work as part of the CHIGP project (Chemicals in Gas Processing). Notes

The authors declare no competing financial interest.

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REFERENCES

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