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Phase equilibria modeling of polar systems with Cubic-Plus-Polar (CPP) equation of state
Journal Pre-proof Phase equilibria modeling of polar systems with Cubic-Plus-Polar (CPP) equation of state Hossein Jalaei Salmani, Mohammad Nader Lotfollahi, Seyed Hossein Mazloumi PII:
S0167-7322(19)32525-5
DOI:
https://doi.org/10.1016/j.molliq.2019.111879
Reference:
MOLLIQ 111879
To appear in:
Journal of Molecular Liquids
Received Date: 4 May 2019 Revised Date:
1 September 2019
Accepted Date: 4 October 2019
Please cite this article as: H.J. Salmani, M.N. Lotfollahi, S.H. Mazloumi, Phase equilibria modeling of polar systems with Cubic-Plus-Polar (CPP) equation of state, Journal of Molecular Liquids (2019), doi: https://doi.org/10.1016/j.molliq.2019.111879. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.
Phase equilibria modeling of polar systems with Cubic-Plus-Polar (CPP) equation of state Hossein Jalaei Salmani1, Mohammad Nader Lotfollahi1*, Seyed Hossein Mazloumi2 1 2
Faculty of Chemical, Petroleum and Gas Engineering, Semnan University, Semnan 35131-19111, Iran
Chemical Engineering Department, Faculty of Engineering, Ferdowsi University of Mashhad, Mashhad, Iran
*
Corresponding author Tel.: +98 23 33654120; E-mail address: [email protected]
1
Abstract In this study, an equation of state (EoS), Cubic-Plus-Polar (CPP), has been constructed to model phase equilibria of polar and practical systems in a wide range of temperatures and pressures. In the proposed model, Soave-Redlich-Kwong (SRK) EoS takes into account dispersive forces while a set of polar terms was employed to account for dipolar and quadrupolar interactions. By applying these terms, dipole-dipole, dipole-quadrupole, and quadrupole-quadrupole interactions were directly taken into consideration. The CPP EoS with four pure-compound parameters could accurately correlate both vapor pressure and saturated liquid molar volume of studied pure compounds (water, methanol, propanol, 2-propanol, 2-butanol, acetone, N-methyl-2-pyrrolidone (NMP), CO2, and H2S). By utilizing two temperature-dependent binary interaction parameters (BIPs), the phase equilibria of binary mixtures including water-CO2, water-H2S, CO2-H2S, methanol-CO2, ethanol-CO2, propanol-CO2, acetone-CO2, water-acetone, ethanol-acetone, waterNMP, 2-propanol-NMP, and 2-butanol-NMP were satisfactorily calculated by the model. Moreover, the proposed model is sufficiently efficient which is capable of representing minimum water mole fraction in the nonaqueous phase of water-CO2 and water-H2S mixtures. Additionally, the CPP EoS accurately predicted the saturated and single phase mixtures densities of water-CO2 mixture. Finally, the solubility values of CO2 and H2S mixtures in water were reasonably predicted by the model.
Keywords: Polar compounds; Dipolar and quadrupolar interactions; Phase equilibria; Modeling; EoS.
2
1. Introduction It has been proved that applying an efficient and simple thermodynamic model to correlate and/or predict thermophysical properties of pure compounds and phase equilibria of mixtures are crucial to model, simulate, and design chemical engineering processes [1, 2]. Among numerous available thermodynamic models, cubic equations of state (EoSs) are still popular due to their simplicity, fast calculations and capabilities over an extensive range of pressures and temperatures. They can truly describe dispersive forces but they have significant weakness in considering polar interactions. Therefore, it is not recommended to apply cubic EoSs to thermodynamically describe compounds with a high polarity where numerous chemical engineering processes are associated with such significant systems [3, 4]. For example, water is a main polar compound which its significance is obviously evident for all engineers and scientists. Thermodynamic modeling of binary mixtures of water and a quadrupolar or dipolar gas (such as CO2 and H2S) is still attractive in oil and gas industries. Nowadays, thermodynamic study of polar solvents with other compounds has been become a very important subject [3]. Other examples, water and CO2 individually and also their mixture play an important role in different researches and industrial areas [5, 6]. Among industrial applications of water-CO2 mixture, the CO2 capture by chemical absorption techniques can be mentioned, which is very significant due to a considerable increase in CO2 emission [6-8]. CO2 is a common acid gas in natural gas streams, which can be eliminated through amines solution. Supercritical extraction is another practical example, which is involved with water and CO2 [6, 9]. In addition to CO2, H2S is also another important acid gas which knowledge on its solubility in aqueous solutions is vital to design sour gas sweetening processes where investigating the phase equilibria of water-H2S binary mixture is primarily required [10-12]. Binary mixtures of CO2 with alcohols have also 3
great of interest from scientific and practical points of view [9]. Acetone is a polar solvent which its binary mixtures with water and alcohols have several applications [13]. N-methyl-2pyrrolidone (NMP) is a high polar compound. It is a low volatile, low toxic and highly effective solvent, which contributes to various industrial applications such as fuel cells, coatings, pharmaceuticals, and membranes [14-18]. Additionally, it is a solvent with a great potential for extractive distillation process due to its high selectivity [19, 20]. Because of this, vapor-liquid equilibrium (VLE) modeling of NMP with water or an organic solvent becomes extremely interesting from scientific and practical viewpoints. Moreover, a considerable amount of moisture can be easily absorbed by NMP from the surrounding [15]. Hence, water can be its common contaminant, which is usually separated by distillation. Therefore it is strongly required to model the VLE of the NMP-water binary system. The aforementioned systems, in addition to their extensive applications in industry, have their own complexity from a thermodynamic point of view, and this study aims to model phase equilibria of such polar and non-ideal systems by an efficient model with an acceptable accuracy and feasible simplicity. Therefore, in this research, a set of polar terms was added to SoaveRedlich-Kwong (SRK) EoS [21] to take explicitly into account dipolar and quadrupolar interactions for phase equilibria modeling of water-CO2, water-H2S, CO2-H2S, methanol-CO2, ethanol-CO2, propanol-CO2, acetone-CO2, water-acetone, ethanol-acetone, water-NMP, 2propanol-NMP, and 2-butanol-NMP systems. There are several successful polar models that are mostly used in various types of SAFT EoS such as the models proposed by Jog and Chapman [22], Gross [23], Gross and Vrabec [24], Vrabec and Gross [25] and Larsen et al. [26]. Polar terms employed in this study are those
4
presented by Larsen et al. [26] and also applied by Shahriari et al. [27] in perturbed hard sphere chain (PHSC) EoS. In summary, in this study, a Cubic-Plus-Polar (CPP) EoS is introduced. In the first step, the model is applied to correlate the liquid molar volume and saturated vapor pressure of pure compounds by which the model parameters for the studied components (water, methanol, ethanol, propanol, 2-propanol, 2-butanol, NMP, acetone, CO2, and H2S) are obtained. In what follows, the phase equilibria calculation of binary mixtures including water-CO2, water-H2S, CO2-H2S, methanol-CO2, ethanol-CO2, propanol-CO2, acetone-CO2, water-acetone, ethanolacetone, water-NMP, 2-propanol-NMP, and 2-butanol-NMP are investigated. Finally, the density of water-CO2 mixture and the solubility values of CO2 and H2S gas mixture in water are predicted by the proposed model. 2. Modeling 2.1. General In the proposed model, Cubic-Plus-Polar (CPP) EoS, a set of polar terms was added to SRK EoS. The Helmholtz energy (A) is a key thermodynamic property from which other thermodynamic properties such as pressure, compressibility factor and chemical potential can be obtained. Therefore, the CPP EoS in terms of residual Helmholtz free energy (AR) is defined as follows:
A R = ACPP = A SRK + A Polar
(1)
ASRK, the Helmholtz energy for the SRK EoS, is obtained from Eq. (2) [28]:
5
v an v + A SRK = n R T ln ln b v − b v + b
(2)
where n is the number of total moles, R, T, v, b, and a are the universal gas constant (= 8.314 J.mol-1.K-1), temperature (K), molar volume (m3.mol-1), the co-volume parameter (m3.mol-1), and attractive parameter (Pa.m6.mol-2), respectively. a is calculated by Eq. (3):
a = a0
1 + c1 1 −
T Tc
2
(3)
where Tc is the critical temperature while b, a0, and c1 are the constants. APolar is written in the following form [27]:
APolar = Add + Aqq + Adq
(4)
where Add, Aqq, and Adq are dipole-dipole, quadrupole-quadrupole, and dipole-quadrupole contributions to the Helmholtz free energy, respectively. Polar terms can be obtained by the Padé approximation [27]:
A
Polar
A2Polar = 1 − A3Polar / A2Polar
(5)
where A2Polar and A3Polar are the second-order and third-order perturbation polar terms, respectively. It is worth noting that while the term A3Polar is extremely complicated, it has a slight influence on improving the results. This issue is further discussed in “Results and Discussion” section for pure compounds. For this reason, only the term A2Polar was used and the term A3Polar was eliminated which significantly simplified the proposed model. Furthermore, for convenience, the expressions related to the term A3Polar were not presented, but they are available 6
in the work of Shahriari [27]. Therefore, it can be concluded that:
APolar = A2Polar
(6)
and
A2dd ρ = − 2 N kB T 6 ( kB T ) ( 4 π ε 0 A2dq ρ = − 2 N kB T 2 ( kB T ) ( 4 π ε 0
x ) ∑∑
i
2
i
ri rj
j
x ) ∑∑
i
2
i
A2qq 7ρ = − 2 N kB T 10 ( k B T ) ( 4 π ε 0
n p i n p j µi µ j
xj
xj
n p i nq j µi2 Q2j
j
x ) ∑∑
i
2
i
d
ri r j
xj
j
d
5 ij
d
(7)
( )
(8)
I 8HS ρ ∗
nq i nq j Qi2 Q 2j ri r j
( )
I 6HS ρ ∗
3 ij
7 ij
( )
I10HS ρ ∗
(9)
where x, np, nq, r, µ, and Q are mole fraction, number of molecules with dipolar moments, number of molecules with quadrupolar moments, number of segments, experimental dipole moment, and experimental quadrupole moment, respectively. The Eqs. (7-9) have been used in SAFT-type models [27]. They consequently require a few changes in order to be implemented in the proposed model (CPP EoS). Thus, in Eqs. (7-9), r which indicates the number of segments was set to one in the CPP EoS. N (= n NA) refers to the total number of molecules and kB stands for the Boltzmann constant (=1.38065 × 10−23 J.K-1). NkBT was replaced by nRT with some simple mathematical operations. Therefore: A2dd = − n RT 6
ρ N A2
( RT ) ( 4 π ε0 2
x ) ∑∑ 2
i
i
x j np i n pj
j
7
µi µ j d i3j
( )
I 6HS ρ ∗
(10)
ρ N A2 A2dq = − 2 n RT 2 ( RT ) ( 4 π ε 0
x ) ∑∑ i
7 ρ N A2 A2qq = − 2 n RT 10 ( RT ) ( 4 π ε 0
x j n p i nq j
i
2
µi2 Q 2j d
j
x ) ∑∑
i
2
i
x j nq i nq j
5 ij
Qi2 Q2j
j
d
7 ij
( )
I 8HS ρ ∗
( )
I10HS ρ ∗
(11)
(12)
where NA, is Avogadro’s number (= 6.02205 × 10−23 mol-1). The number density, ρ, is defined as follow: NA n NA N = = V V v
ρ =
(13)
where v and V are molar volume (m3.mol-1) and total volume (m3), respectively. As proposed by Villiers et al. [29], the segment diameter (d) in SAFT can be replaced by co-volume parameter (b) in the cubic-plus-association (CPA) EoS as follows:
d =
3
3b 2π N A
(14)
And also the reduced density, ρ*, is calculated as:
ρ* = ρ d3 =
3b
(15)
2π v
IkHS(ρ*) in Eqs. (10-12) is approximated by the following series [27]:
( ) ∑J
I kHS ρ * =
5
i =0
i,k
ρ*
i
(k
= 6, 8, 10
)
(16)
The coefficients in Eq. (16) have been given by Larsen et al. [26] and are listed in Table 1.
8
Table 1. The Constants of series in Eq. (16) K 6 8 10
J0,k 4.1888 2.5133 1.7952
J1,k 2.8287 2.1795 1.7551
J2,k 0.8331 1.0423 1.0376
J3,k 0.0317 0.2596 0.3890
J4,k 0.0858 0.1097 0.1561
J5,k - 0.0846 - 0.0573 - 0.0082
2.2. Mixing and combining rules The mixing and combining rules are necessary for successful extension of an EoS to mixtures. The simple van der Waals one fluid (vdW1f) mixing rules were applied in the cubic part:
a=
∑∑ x
i
x j ai j
(17)
∑∑ x
x j bi j
(18)
i
b=
i
j
i
j
with the following combining rules:
ai j =
bi j =
ai a j (1 − k i j 1 ( bi + b j 2
)
(19)
) (1 − l )
(20)
i j
The polar terms do not need the mixing rules and only a simple combining rule was used to calculate dij:
di j = 0.5 ( di + d j
)
(21)
9
2.3. Phase equilibria calculations When two phases, for example α and β, are in equilibrium, the fugacity of all components in the phases are equal:
fiα = fi β
(22)
This is the basis of phase equilibria calculations. The fugacity coefficients of a pure component (φ) and a compound in the mixture (φi) were obtained by the following equations at a certain temperature (T) and pressure (P) [30]:
A f ln ϕ = ln = + Z − 1 − ln Z n RT P
(23)
f µ 1 ln ϕi = ln i = i − ln Z = RT xi P R T
(24)
∂A − ln Z ∂ ni T , V , n j
where, Z, f, fi and µ i are the compressibility factor, fugacity of the pure compound, fugacity and chemical potential of a component in the mixture, respectively and:
Z=
∂A 1 ρ n R T ∂ ρ T , n
(25)
while:
Z = Z CCP = Z SRK + Z Polar =
∂ A SRK 1 ρ n RT ∂ ρ
∂ A Polar + T , n ∂ ρ
10
T , n
(26)
The final expressions for the compressibility factor and chemical potential of SRK and polar terms were given by Danesh [31] and Shahriari et al. [27], respectively. 3. Results and discussion 3.1. Pure component calculations The critical and physical properties of pure components are given in Table 2 [32-34]: Table 2. Physical properties of pure compounds [32-34] Compound
water methanol ethanol propanol 2-propanol 2-butanol acetone NMP hydrogen sulfide carbon dioxide
Critical temperature Tc/K 647.13 512.64 514.00 536.78 508.30 536.05 508.10 727.10 373.40 304.13
Acentric factor ω 0.344 0.565 0.644 0.629 0.665 0.574 0.307 0.358 0.090 0.225
Critical pressure Pc/bar 220.6 80.9 61.4 51.7 47.6 41.8 47.0 45.2 89.6 73.8
Dipole moment µ × 2.99788 × 1029/C.m 1.84 1.7 1.7 1.68 1.66 1.66 2.88 4.09 0.9 0
Quadrupole moment Q × 1040/C.m2 0 0 0 0 0 0 0 0 0 -13.4
The SRK EoS parameters are determined by the expressions defined in Soave’s work [21]: R 2 Tc2 a 0 = 0.42747 Pc RT b = 0.08664 c Pc
(27)
(28)
c = 0.480+1.574ωi − 0.176ωi2
(29)
11
while the proposed model parameters were determined by fitting to the experimental saturated liquid molar volumes and vapor pressures data. The average absolute relative deviation percent (AARD%) together with the average absolute deviation percent (AAD%) were employed to indicate the model deviation from experimental data, which were determined by Eqs. (30) and (31):
A A RD % =
AAD% =
100 NP
∑ i
q icalc − q iexp q iexp
(30)
100 qicalc − qiexp ∑ NP i
(31)
where qiexp and qicalc are the experimental and model-calculated values, respectively, while NP is the number of data points. SRK parameters together with its AARDs% in calculating saturated liquid molar volumes and vapor pressures are given in Table 3. According to AARDs% in Table 3, the results for liquid molar volume by SRK EoS are, as expected, unsatisfactory. It should be mentioned that the experimental saturated liquid molar volumes data of NMP reported in literature are scattered and do not cover the studied range of temperature. Hence, the experimental vapor pressure data were only utilized in this study for this component.
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Table 3. SRK parameters together with its AARDs% for saturated liquid molar volumes and vapor pressures Compound water methanol ethanol propanol 2-propanol 2-butanol acetone NMP hydrogen sulfide carbon dioxide
T/K 323-583 255-460 255-463 280-483 279-459 281-488 253-456 330-522 190-337 217-275
a0 /Pa.(m3)2.mol-2 0.5609 0.9599 1.2714 1.6468 1.6039 2.0318 1.6231 3.4056 0.4597 0.3705
b(105)/m3.mol-1 2.1130 4.5645 6.0301 7.4788 7.6920 9.2398 7.7872 1.1504 3.0009 2.9695
c AARD%Psat AARD%vliq 1.0006 4.77 38.27 1.3131 8.86 34.22 1.4207 4.43 22.21 1.4004 2.93 16.61 4.00 18.78 1.4489 1.3259 12.18 16.89 2.97 29.00 0.9471 4.40 1.0205 0.6202 1.56 5.72 0.8252 0.31 11.06
overall
4.64
21.42
The results of the CPP EoS, for water and CO2, with and without the third-order perturbation polar term, A3Polar, are given in Table 4. It can be seen from Table 4 that the third-order perturbation polar term did not improve the results and consequently this term was eliminated. This elimination made the calculations significantly simpler, and, thus, CPP EoS was applied with only the second-order perturbation polar term, A2Polar, in this study. Accordingly, the CPP parameters together with calculated results for all studied compounds are shown in Table 5. Table 4. The parameters and the AARDs% of the CPP EoS for saturated liquid molar volumes and vapor pressures of water and carbon dioxide with and without A3Polar term Compound Water Water carbon dioxide carbon dioxide
CPP EoS with A3 without A3 with A3 without A3
a0 /Pa.(m3)2.mol-2 0.4667 0.4663 0.3211 0.3132
b(105) /m3.mol-1 1.5719 1.5724 2.9308 2.9205
13
c
np
nq
0.7817 0.0298 0.7805 0.0364 0.5799 0.4291 0.5823 0.4289
AARD%Psat AARD%vliq 0.77 0.70 0.13 0.14
0.35 0.36 0.11 0.08
Table 5. CPP parameters together with its AARDs% for saturated liquid molar volumes and vapor pressures for all studied compounds Compound
T/K
water methanol ethanol propanol 2-propanol 2-butanol acetone NMP hydrogen sulfide carbon dioxide
323-583 255-460 255-463 280-483 279-459 281-488 253-456 330-522 190-337 217-275
a0 b(105) 3 2 -2 /Pa.(m ) .mol /m3.mol-1 0.4663 1.5724 0.8237 3.4735 1.1363 5.0026 1.4623 6.4748 1.4040 6.5537 1.3571 8.4743 1.4000 6.1784 3.5013 14.6090 0.4285 2.9238 0.3132 2.9205
c 0.7805 1.0133 1.2351 1.3654 1.4285 1.2926 0.7915 0.8047 0.5001 0.5823
np
nq
AARD%Psat AARD%vliq
0.0364 0.0521 0.1131 0.2818 0.2693 2.8907 0.0400 0.8904 1.0016 0.4289
overall
0.70 2.58 2.13 0.90 0.95 1.57 0.20 4.07 0.53 0.14
0.36 0.26 0.88 1.26 1.03 1.31 0.72 0.72 0.08
1.38
0.74
A comparison between Table 3 and Table 5 demonstrates that CPP EoS significantly improved the calculation values of both saturated vapor pressure and liquid molar volume. 3.2. Mixture calculations 3.2.1. Phase equilibria correlation Since the SRK EoS in calculating pure compound properties shows unsatisfactory results and the goal of this study is to achieve a suitable model for both pure and mixture fluids, the calculations of CPP EoS for mixtures were only presented. According to Eqs. (19) and (20), the CPP EoS has two binary interaction parameters (BIPs) which were determined by fitting to experimental binary VLE data. Table 6 shows the calculated BIPs for each binary mixture.
14
Table 6. Calculated BIPs by fitting to binary VLE data System
T/K
Water-CO2 Water-CO2 Water-CO2 Water-CO2 Water-H2S Water-H2S H2S-CO2 Methanol-CO2 Ethanol-CO2 Popanol-CO2 Acetone-CO2 Water-Acetone Ethanol-Acetone Water-NMP 2-Propanol-NMP 2-Butanol-NMP
288.26-373.15 373.15-383.15 383.15-473.15 473.15-573.15 310.00-344.26 344.26-444.26 258.41-348.15 304.20-313.14 291.15-333.4 313.40-333.40 313.15-333.15 333.15-433.15 305.15-321.15 343.15-363.15 353.15-373.15 373.15
kij = A + BT+ CT2 A B/K-1 C/K-2 0.57990 -0.0012545 0 11.31044 -0.0300111 0 -0.63631 0.0011692 0 -0.01758 -0.0001385 0 2.16537 -0.0064519 0 -0.19330 0.0003995 0 -3.89127 0.0271527 -4.6339324 0.26229 -0.0009165 0 -0.02826 0.0001669 0 0.14567 -0.0003139 0 -0.15209 0.0002842 0 0.22035 0.0002717 0 0.22712 -0.0005686 0 1.13441 0.0007285 0 1.41900 -0.0043857 0 0.06300 0 0
lij = A + BT + CT2 A B/K-1 C/K-2 0.68297 -0.0012534 0 6.30208 -0.016312 0 -0.01962 0.0001873 0 0.14706 -0.0001650 0 1.66928 -0.0046616 0 0.11743 -0.0001538 0 -5.11292 0.0358433 -6.2484813 1.10965 -0.0003412 0 0.23617 -0.0006879 0 0.44974 -0.0013370 0 0.02661 0 0 0 0.44526 -0.0003526 0.20764 0.0005851 0 0.06789 -0.0001595 0 1.46425 -0.0040128 0 0.09875 0 0
Phase equilibria investigation of water-CO2 and water-H2S binary mixtures has a significant role in thermodynamics and numerous applications in industry. Because at some specified temperatures and pressures nonaqueous phase is converted to a liquid phase, its water content is suddenly increased and water mole fraction in the nonaqueous phase has a minimum value. It should be noted that most the thermodynamic models are unable to represent this minimum. Modern thermodynamic models such as statistical associating fluid theory (SAFT) and CPA EoSs considered CO2 and H2S as a self-associating or solvating component to correctly compute water content of nonaqueous phase of these binary systems [9, 35, 36]. However, CO2 and H2S are not real associating compounds. Moreover, interactions between CO2 and water originate from quadrupole-dipole and dispersive forces [9]. In this study, the dipole-quadrupole interactions between H2O and CO2 molecules and dipole-dipole interactions between H2O and H2S molecules were directly taken into consideration. 15
Pappa et al. [35] studied mixture of water-CO2 in a wide range of pressure and temperature. They employed the Peng-Robinson (PR) EoS, universal mixing rule with PR-EoS (UMR-PR) (The universal mixing rule refers to a class of EoS/GE models), and PR-based CPA EoS. Their results showed that the results of CPA with two BIPs were in better agreement with experimental data in which both water and CO2 were modeled as an associating component with four association sites (4C). Table 7 shows our results obtained by CPP EoS and those reported by Pappa et al. [35]. Accordingly, it can be observed that the CPP could generate better results than the results provided by CPA EoS. Additionally, the CPP has a higher level of accuracy compared to PR EoS. Table 7. Water-CO2 VLE calculation results (experimental data from [37-42]) Number of data points
P/bar
T/K
8 4 9 9 7 15 15 15 15 13 10
5.0-202.7 101.3-202.7 40.5-141.1 40.5-131 3.3-23.1 100.0-1500.0 100.0-1500.0 100.0-1500.0 100.0-1500.0 200.0-1230.0 100.0-550.0
288.26 313.15 333.15 353.15 373.15 383.15 423.15 473.15 523.15 543.15 573.15
overall
CPP CPA-PR PR This work [35] [35] AARD% AARD% AARD% AARD% AARD% AARD% xCO2 yH2O xCO2 yH2O xCO2 yH2O 3.55 6.31 8.8 8.9 3.1 31.6 3.88 2.85 3.1 4.7 2.7 4.0 1.93 5.64 1.8 8.2 4.0 4.6 2.18 2.57 2.3 2.3 8.8 2.5 9.62 11.16 8.0 9.6 14.1 10.6 1.76 17.62 6.0 20.2 11.3 69.8 8.40 8.45 5.3 10.3 9.5 60.3 5.80 3.63 2.5 12.2 9.0 40.4 13.75 8.43 8.4 12.4 3.7 29.1 4.84 3.11 10.8 6.0 19.4 28.2 6.11 8.31 6.6 2.0 9.5 10.2 5.62
7.10
5.78
8.8
8.65
26.48
Water-H2S calculation results provided by CPP EoS are also given in Table 8. Moreover, some typical results for water-CO2 and water-H2S mixtures are demonstrated in Figs. 1 to 4, which indicate that the CPP EoS could correctly determine both water rich and nonaqueous phases
16
compositions. An accurate representation of minimum water mole fraction in CO2 rich phase and H2S rich phase is evident in Figs. 1 and 3, respectively, which is another noteworthy point of the proposed model.
Table 8. Water-H2S VLE calculation results (experimental data from [32, 43]) Number of data points
P/bar
T/K
9 9 6 6 6
3.447-206.84 6.8948-48.263 6.8948-48.263 6.8948-48.263 6.8948-48.263
310.00 344.26 377.59 410.93 444.26
overall
CPP (This work) AARD% xH2S AARD% yH2O 4.27 11.50 0.72 7.64 2.70 4.97 5.21 1.15 7.40 2.51 4.06
5.55
Fig. 1. Water mole fraction in CO2 rich phase at 288.26 K. CPP calculation (solid line) and experimental data (points) [38, 41]
17
Fig. 2. CO2 mole fraction in water rich phase at 288.26 K. CPP calculation (solid line) and experimental data (points) [38, 41]
Fig. 3. Water mole fraction in H2S rich phase at 310.00 K. CPP calculation (solid line) and experimental data (points) [32]
18
Fig. 4. H2S mole fraction in water rich phase at 310.00 K. CPP calculation (solid line) and experimental data (points) [32]
The VLE calculation results of CO2-H2S obtained by the proposed model are presented in Table 9. It can be seen that CPP EoS by BIPs with a quadratic temperature dependency could accurately calculate the VLE data of this binary mixture. The graphical representation of CPP calculation for CO2-H2S system at some arbitrary temperatures is available in Fig. 5. Table 9. CO2-H2S VLE calculation results (experimental data from [44, 45]) Number of data points 6 6 9 4 3 3
P/bar
T/K
10.02-19.23 13.75-28.99 20.49-52.64 34.34-54.75 55.1581-82.7371 60.795-81.06
258.41 273.15 293.47 313.02 338.00 348.15
overall
CPP (This work) AARD%P AAD%y 0.80 1.14 1.49 1.13 0.72 1.31 0.47 1.52 1.02 0.38 1.54 0.61 1.01
19
1.02
Fig. 5. VLE plot for CO2-H2S system. CPP calculation (solid line) and experimental data (points) [45]
The calculated results for methanol-CO2, ethanol-CO2, and propanol-CO2 provided by CPP EoS are available in Tables 10 to 12. The BIPs of these binary mixtures were obtained by fitting to only bubble point pressure, and the vapor phase compositions of these binary systems were left to be as a predictive property. The deviation between model-calculated results and experimental data for all three systems are satisfactory while the results of propanol-CO2 are more precise.
Table 10. Methanol-CO2 VLE calculation results (experimental data from [46, 47]) Number of data points 10 6
P/bar
T/K
37.5-69.5 6.83-69.54
304.2 313.4
CPP (This work)
overall
20
AARD%P 3.10 2.72
AAD%y 0.21 0.18
2.91
0.20
Table 11. Ethanol-CO2 VLE calculation results (experimental data from [46, 48]) Number of data points 9 8 7
P/bar 20.94-54.00 5.14-71.02 5.44-60.94
T/K
CPP (This work)
291.15 313.40 333.40
overall
AARD%P 2.02 2.84 1.23
AAD%y 0.45 0.12 0.08
2.03
0.22
Table 12. Propanol-CO2 VLE calculation results (experimental data from [46]) Number of data points 7 5
P/bar
T/K
5.18-69.79 6.68-60.81
313.40 333.40
CPP (This work)
overall
AARD%P 1.34 0.37
AAD%y 0.11 0.09
0.86
0.10
Fig. 6. VLE plot for ethanol-CO2 system at T=291.15 K. CPP calculation (solid line) and experimental data (points) [48]
21
The AARDs% for bubble point pressure of the acetone-CO2 system are presented in Table 13 where an overall AARD% of 0.55% indicates a high level of accuracy achieved by the CPP EoS. Table 13. Acetone-CO2 VLE calculation results (experimental data from [49]) Number of data points 9 7
P/bar 10.01-69.93 9.87-70.23
T/K
CPP (This work)
313.15 333.15
overall
AARD%P 0.59 0.50
AAD%y -
0.55
-
In the cases of water-acetone and ethanol-acetone, according to Tables 14 and 15, the CPP EoS shows a good performance so that the representation of azeotropic points is evident in Fig. 7.
Table 14. Water-acetone calculation results (experimental data from [32]) Number of data points 10 10
P/bar
T/K
CPP (This work)
0.739-1.149 10.418-14.135
333.15 433.15
overall
AARD%P 1.63 1.14
AAD%y 0.78 1.67
1.37
1.23
Table 15. Ethanol-acetone calculation results (experimental data from [13]) Number of data points 16 16 16
P/bar
T/K
0.1369-0.39717 0.20292-0.54582 0.29771-0.73807
305.15 313.15 321.15
CPP (This work) AARD%P AAD%y 0.58 0.42 1.40 0.25 1.29 0.33
overall
1.09
22
0.33
Fig. 7. P-x-y plot for water-acetone system at T=333.15 K. CPP calculation (solid line) and experimental data (points) [32]
Fig. 8. P-x-y plot for water-acetone system at T=433.15 K. CPP calculation (solid line) and experimental data (points) [32]
23
Fig. 9. P-x-y plot for ethanol-acetone system. CPP calculation (solid line) and experimental data (points) [13]
VLE modeling of important mixed solvents, water-NMP, 2-propanol-NMP, and 2butanol-NMP, was evaluated by CPP EoS. Tables 16 to 18 provide the AARDs% of these systems obtained by the proposed model. It can be seen that the proposed model works properly while the results of 2-propanol-NMP are more accurate. These findings are also obtained based on Figs. 9 to 11. Table 16. Water-NMP calculation results (experimental data from [50]) Number of data points 10 10
P/bar
T/K
0.0079-0.3100 0.0217-0.7014
343.15 363.15
CPP (This work) AARD%P AAD%y 2.67 0.26 4.61 0.39
overall
3.64
24
0.33
Table 17. 2-Propanol-NMP calculation results (experimental data from [51]) Number of data points 25 18
P/bar 0.013-0.927 0.036-1.977
T/K AARD%P 1.33 1.03
353.15 373.15
overall
CPP (This work) AAD%y 0.07 0.54
1.18
0.31
Table 18. 2-Butanol-NMP calculation results (experimental data from [51]) Number of data points 15
P/bar 0.036-1.070
T/K AARD%P 2.38
373.15
CPP (This work) AAD%y 0.36
Fig. 10. P-x-y plot for water-NMP system. CPP calculation (solid line) and experimental data (points) [50]
25
Fig. 11. P-x-y plot for 2-propanol-NMP system. CPP calculation (solid line) and experimental data (points) [51]
Fig. 12. P-x-y plot for 2-butanol-NMP system. CPP calculation (solid line) and experimental data (points) [51]
26
3.2.2. Density prediction The BIPs in the CPP model, kij and lij, were set to zero to represent a fully predictive model for calculation of water-CO2 mixture density. The AARDs% of the model for mixture densities of equilibrated and single phases are given in Tables 19 and 20, respectively. The CPP results of water-CO2 mixture density were compared with those obtained by CPA EoS reported in the work of Tsivintzelis et al. [52] in which CO2 was considered as a solvating molecule. The obtained results indicated that the predicted densities by the proposed CPP-EoS model were in a good agreement with the experimental data. Table 19. Density prediction results of water-CO2 saturated mixtures T/K
283 288 293 298.15 322.8 298 313 323 382.41 423.06 468.47 422.98 445.74 461.62 478.35 overall
Number of data points water rich CO2 rich phase phase
6 6 6 6 11 10 11 11 -
11 6 6 10 10 10 10 10
CPP This study AARD% AARD% water rich CO2 rich phase phase
CPA CO2 with solvation [52] AARD% AARD% water rich phase CO2 rich phase
0.7 0.6 0.5 0.1 0.5 0.9 1.4 1.5 -
2.3 0.6 2.5 1.4 2.8 3.5 3.5 3.6
0.9 0.7 0.6 0.1 0.3 0.6 0.9 1.0 -
2.2 1.7 2.5 2.0 4.8 5.5 5.4 5.8
0.65
2.52
0.64
3.74
27
Reference
[53] [53] [53] [38] [54] [55] [55] [55] [56] [56] [56] [56] [56] [56] [56]
Table 20. Density prediction results of water-CO2 single phase mixtures T/K
P/bar
Number of data points
Water mole fraction
415.36-697.66 455.23-697.94 469.55-698.08 497.21-698.40 518.59-693.10 544.21-693.07 562.96-697.77 580.3-699.18 616.59-698.57 625.44-699.19 634.73-699.30
58.84-108.28 70.76-117.62 78.32-127.12 91.20-139.98 104.99-154.14 121.69-170.00 143.08-198.32 165.85-227.11 214.03-273.47 242.26-313.52 267.89-345.77
22 19 18 16 14 12 11 12 10 9 8
0.0612 0.1407 0.2101 0.2870 0.3670 0.4349 0.5317 0.6017 0.6934 0.7523 0.7913
overall
CPP This study
Reference
AARD%
CPA CO2 with solvation [52] AARD%
0.2 0.1 0.2 0.4 0.5 0.5 1.1 1.1 2.2 3.9 5.2
1.1 1.2 1.2 0.8 0.7 0.6 1.5 1.1 1.6 2.5 3.2
[57] [57] [57] [57] [57] [57] [57] [57] [57] [57] [57]
1.40
1.41
3.2.3. Solubility prediction Savary et al. [58] presented a set of solubility data of H2S and CO2 mixtures in water at temperature 393.15 K and pressures up to 350 bar. Their data were utilized for our predictions where Table 21 gives deviations between model-predicted results and experimental data. The computed BIPs of water-H2S and water-CO2 systems reported in Table 6 were applied for prediction calculations but the BIP of CO2-H2S system was set to zero, because the temperature 393.15 K was higher than the studied temperature range of CO2-H2S system (258.41-348.15 K). Table 21. Prediction results of solubility of CO2+H2S mixture in water (experimental data from [58]) Number of data points 23
P/bar
T/K
39-350
393.15
CPP (This work) AAD% xCO2 AAD% xH2S 0.29 0.90
28
Conclusion In this study, Cubic-Plus-Polar (CPP) equation of state (EoS), which is based on a combination of Soave-Redlich-Kwong (SRK) EoS and a set of polar terms, was developed to model phase equilibria of polar systems over a wide range of pressures and temperatures. The proposed model was significantly simplified by eliminating the third-order perturbation polar term (A3Polar) owing to slight influence of this term on improving the results. The CPP EoS was able to obtain satisfactory results for pure water, methanol, propanol, 2-propanol, 2-butanol, acetone, Nmethyl-2-pyrrolidone (NMP), CO2, and H2S as well as their mixtures including water-CO2, water-H2S, CO2-H2S, methanol-CO2, ethanol-CO2, propanol-CO2, water-acetone, acetone-CO2, ethanol-acetone, water-NMP, 2-propanol-NMP, and 2-butanol-NMP with four pure-compound parameters and two temperature-dependent binary interaction parameters. Moreover, the proposed model is capable of correctly representing minimum water mole fraction in the nonaqueous phase of water-CO2 and water-H2S mixtures. Additionally, the CPP EoS could accurately predict the saturated and single-phase densities of water-CO2 mixtures. The solubility values of CO2 and H2S mixture in water was also reasonably predicted by the proposed model. The aim of introducing such EoS was to propose an efficient and simple model, which is compatible with the physics of the problem through directly taking into account dipole-dipole, quadrupole-quadrupole and dipole-quadrupole interactions in pure polar components and their mixtures.
29
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31
Highlights Cubic-Plus-Polar (CPP) EoS has been constructed to model phase equilibria of practical systems with a high polarity. The proposed model takes into account dipolar and quadrupolar interactions to be more compatible with physics of the problem The Phase equilibria of studied systems were properly modeled in a wide range of temperatures and pressures. The proposed model represents minimum water mole fraction in the nonaqueous phase of water-CO2 and water-H2S mixtures. The CPP EoS accurately predicted water-CO2 mixture density and the solubility values of CO2 and H2S mixture in water.
S0167-7322(19)32525-5
DOI:
https://doi.org/10.1016/j.molliq.2019.111879
Reference:
MOLLIQ 111879
To appear in:
Journal of Molecular Liquids
Received Date: 4 May 2019 Revised Date:
1 September 2019
Accepted Date: 4 October 2019
Please cite this article as: H.J. Salmani, M.N. Lotfollahi, S.H. Mazloumi, Phase equilibria modeling of polar systems with Cubic-Plus-Polar (CPP) equation of state, Journal of Molecular Liquids (2019), doi: https://doi.org/10.1016/j.molliq.2019.111879. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.
Phase equilibria modeling of polar systems with Cubic-Plus-Polar (CPP) equation of state Hossein Jalaei Salmani1, Mohammad Nader Lotfollahi1*, Seyed Hossein Mazloumi2 1 2
Faculty of Chemical, Petroleum and Gas Engineering, Semnan University, Semnan 35131-19111, Iran
Chemical Engineering Department, Faculty of Engineering, Ferdowsi University of Mashhad, Mashhad, Iran
*
Corresponding author Tel.: +98 23 33654120; E-mail address: [email protected]
1
Abstract In this study, an equation of state (EoS), Cubic-Plus-Polar (CPP), has been constructed to model phase equilibria of polar and practical systems in a wide range of temperatures and pressures. In the proposed model, Soave-Redlich-Kwong (SRK) EoS takes into account dispersive forces while a set of polar terms was employed to account for dipolar and quadrupolar interactions. By applying these terms, dipole-dipole, dipole-quadrupole, and quadrupole-quadrupole interactions were directly taken into consideration. The CPP EoS with four pure-compound parameters could accurately correlate both vapor pressure and saturated liquid molar volume of studied pure compounds (water, methanol, propanol, 2-propanol, 2-butanol, acetone, N-methyl-2-pyrrolidone (NMP), CO2, and H2S). By utilizing two temperature-dependent binary interaction parameters (BIPs), the phase equilibria of binary mixtures including water-CO2, water-H2S, CO2-H2S, methanol-CO2, ethanol-CO2, propanol-CO2, acetone-CO2, water-acetone, ethanol-acetone, waterNMP, 2-propanol-NMP, and 2-butanol-NMP were satisfactorily calculated by the model. Moreover, the proposed model is sufficiently efficient which is capable of representing minimum water mole fraction in the nonaqueous phase of water-CO2 and water-H2S mixtures. Additionally, the CPP EoS accurately predicted the saturated and single phase mixtures densities of water-CO2 mixture. Finally, the solubility values of CO2 and H2S mixtures in water were reasonably predicted by the model.
Keywords: Polar compounds; Dipolar and quadrupolar interactions; Phase equilibria; Modeling; EoS.
2
1. Introduction It has been proved that applying an efficient and simple thermodynamic model to correlate and/or predict thermophysical properties of pure compounds and phase equilibria of mixtures are crucial to model, simulate, and design chemical engineering processes [1, 2]. Among numerous available thermodynamic models, cubic equations of state (EoSs) are still popular due to their simplicity, fast calculations and capabilities over an extensive range of pressures and temperatures. They can truly describe dispersive forces but they have significant weakness in considering polar interactions. Therefore, it is not recommended to apply cubic EoSs to thermodynamically describe compounds with a high polarity where numerous chemical engineering processes are associated with such significant systems [3, 4]. For example, water is a main polar compound which its significance is obviously evident for all engineers and scientists. Thermodynamic modeling of binary mixtures of water and a quadrupolar or dipolar gas (such as CO2 and H2S) is still attractive in oil and gas industries. Nowadays, thermodynamic study of polar solvents with other compounds has been become a very important subject [3]. Other examples, water and CO2 individually and also their mixture play an important role in different researches and industrial areas [5, 6]. Among industrial applications of water-CO2 mixture, the CO2 capture by chemical absorption techniques can be mentioned, which is very significant due to a considerable increase in CO2 emission [6-8]. CO2 is a common acid gas in natural gas streams, which can be eliminated through amines solution. Supercritical extraction is another practical example, which is involved with water and CO2 [6, 9]. In addition to CO2, H2S is also another important acid gas which knowledge on its solubility in aqueous solutions is vital to design sour gas sweetening processes where investigating the phase equilibria of water-H2S binary mixture is primarily required [10-12]. Binary mixtures of CO2 with alcohols have also 3
great of interest from scientific and practical points of view [9]. Acetone is a polar solvent which its binary mixtures with water and alcohols have several applications [13]. N-methyl-2pyrrolidone (NMP) is a high polar compound. It is a low volatile, low toxic and highly effective solvent, which contributes to various industrial applications such as fuel cells, coatings, pharmaceuticals, and membranes [14-18]. Additionally, it is a solvent with a great potential for extractive distillation process due to its high selectivity [19, 20]. Because of this, vapor-liquid equilibrium (VLE) modeling of NMP with water or an organic solvent becomes extremely interesting from scientific and practical viewpoints. Moreover, a considerable amount of moisture can be easily absorbed by NMP from the surrounding [15]. Hence, water can be its common contaminant, which is usually separated by distillation. Therefore it is strongly required to model the VLE of the NMP-water binary system. The aforementioned systems, in addition to their extensive applications in industry, have their own complexity from a thermodynamic point of view, and this study aims to model phase equilibria of such polar and non-ideal systems by an efficient model with an acceptable accuracy and feasible simplicity. Therefore, in this research, a set of polar terms was added to SoaveRedlich-Kwong (SRK) EoS [21] to take explicitly into account dipolar and quadrupolar interactions for phase equilibria modeling of water-CO2, water-H2S, CO2-H2S, methanol-CO2, ethanol-CO2, propanol-CO2, acetone-CO2, water-acetone, ethanol-acetone, water-NMP, 2propanol-NMP, and 2-butanol-NMP systems. There are several successful polar models that are mostly used in various types of SAFT EoS such as the models proposed by Jog and Chapman [22], Gross [23], Gross and Vrabec [24], Vrabec and Gross [25] and Larsen et al. [26]. Polar terms employed in this study are those
4
presented by Larsen et al. [26] and also applied by Shahriari et al. [27] in perturbed hard sphere chain (PHSC) EoS. In summary, in this study, a Cubic-Plus-Polar (CPP) EoS is introduced. In the first step, the model is applied to correlate the liquid molar volume and saturated vapor pressure of pure compounds by which the model parameters for the studied components (water, methanol, ethanol, propanol, 2-propanol, 2-butanol, NMP, acetone, CO2, and H2S) are obtained. In what follows, the phase equilibria calculation of binary mixtures including water-CO2, water-H2S, CO2-H2S, methanol-CO2, ethanol-CO2, propanol-CO2, acetone-CO2, water-acetone, ethanolacetone, water-NMP, 2-propanol-NMP, and 2-butanol-NMP are investigated. Finally, the density of water-CO2 mixture and the solubility values of CO2 and H2S gas mixture in water are predicted by the proposed model. 2. Modeling 2.1. General In the proposed model, Cubic-Plus-Polar (CPP) EoS, a set of polar terms was added to SRK EoS. The Helmholtz energy (A) is a key thermodynamic property from which other thermodynamic properties such as pressure, compressibility factor and chemical potential can be obtained. Therefore, the CPP EoS in terms of residual Helmholtz free energy (AR) is defined as follows:
A R = ACPP = A SRK + A Polar
(1)
ASRK, the Helmholtz energy for the SRK EoS, is obtained from Eq. (2) [28]:
5
v an v + A SRK = n R T ln ln b v − b v + b
(2)
where n is the number of total moles, R, T, v, b, and a are the universal gas constant (= 8.314 J.mol-1.K-1), temperature (K), molar volume (m3.mol-1), the co-volume parameter (m3.mol-1), and attractive parameter (Pa.m6.mol-2), respectively. a is calculated by Eq. (3):
a = a0
1 + c1 1 −
T Tc
2
(3)
where Tc is the critical temperature while b, a0, and c1 are the constants. APolar is written in the following form [27]:
APolar = Add + Aqq + Adq
(4)
where Add, Aqq, and Adq are dipole-dipole, quadrupole-quadrupole, and dipole-quadrupole contributions to the Helmholtz free energy, respectively. Polar terms can be obtained by the Padé approximation [27]:
A
Polar
A2Polar = 1 − A3Polar / A2Polar
(5)
where A2Polar and A3Polar are the second-order and third-order perturbation polar terms, respectively. It is worth noting that while the term A3Polar is extremely complicated, it has a slight influence on improving the results. This issue is further discussed in “Results and Discussion” section for pure compounds. For this reason, only the term A2Polar was used and the term A3Polar was eliminated which significantly simplified the proposed model. Furthermore, for convenience, the expressions related to the term A3Polar were not presented, but they are available 6
in the work of Shahriari [27]. Therefore, it can be concluded that:
APolar = A2Polar
(6)
and
A2dd ρ = − 2 N kB T 6 ( kB T ) ( 4 π ε 0 A2dq ρ = − 2 N kB T 2 ( kB T ) ( 4 π ε 0
x ) ∑∑
i
2
i
ri rj
j
x ) ∑∑
i
2
i
A2qq 7ρ = − 2 N kB T 10 ( k B T ) ( 4 π ε 0
n p i n p j µi µ j
xj
xj
n p i nq j µi2 Q2j
j
x ) ∑∑
i
2
i
d
ri r j
xj
j
d
5 ij
d
(7)
( )
(8)
I 8HS ρ ∗
nq i nq j Qi2 Q 2j ri r j
( )
I 6HS ρ ∗
3 ij
7 ij
( )
I10HS ρ ∗
(9)
where x, np, nq, r, µ, and Q are mole fraction, number of molecules with dipolar moments, number of molecules with quadrupolar moments, number of segments, experimental dipole moment, and experimental quadrupole moment, respectively. The Eqs. (7-9) have been used in SAFT-type models [27]. They consequently require a few changes in order to be implemented in the proposed model (CPP EoS). Thus, in Eqs. (7-9), r which indicates the number of segments was set to one in the CPP EoS. N (= n NA) refers to the total number of molecules and kB stands for the Boltzmann constant (=1.38065 × 10−23 J.K-1). NkBT was replaced by nRT with some simple mathematical operations. Therefore: A2dd = − n RT 6
ρ N A2
( RT ) ( 4 π ε0 2
x ) ∑∑ 2
i
i
x j np i n pj
j
7
µi µ j d i3j
( )
I 6HS ρ ∗
(10)
ρ N A2 A2dq = − 2 n RT 2 ( RT ) ( 4 π ε 0
x ) ∑∑ i
7 ρ N A2 A2qq = − 2 n RT 10 ( RT ) ( 4 π ε 0
x j n p i nq j
i
2
µi2 Q 2j d
j
x ) ∑∑
i
2
i
x j nq i nq j
5 ij
Qi2 Q2j
j
d
7 ij
( )
I 8HS ρ ∗
( )
I10HS ρ ∗
(11)
(12)
where NA, is Avogadro’s number (= 6.02205 × 10−23 mol-1). The number density, ρ, is defined as follow: NA n NA N = = V V v
ρ =
(13)
where v and V are molar volume (m3.mol-1) and total volume (m3), respectively. As proposed by Villiers et al. [29], the segment diameter (d) in SAFT can be replaced by co-volume parameter (b) in the cubic-plus-association (CPA) EoS as follows:
d =
3
3b 2π N A
(14)
And also the reduced density, ρ*, is calculated as:
ρ* = ρ d3 =
3b
(15)
2π v
IkHS(ρ*) in Eqs. (10-12) is approximated by the following series [27]:
( ) ∑J
I kHS ρ * =
5
i =0
i,k
ρ*
i
(k
= 6, 8, 10
)
(16)
The coefficients in Eq. (16) have been given by Larsen et al. [26] and are listed in Table 1.
8
Table 1. The Constants of series in Eq. (16) K 6 8 10
J0,k 4.1888 2.5133 1.7952
J1,k 2.8287 2.1795 1.7551
J2,k 0.8331 1.0423 1.0376
J3,k 0.0317 0.2596 0.3890
J4,k 0.0858 0.1097 0.1561
J5,k - 0.0846 - 0.0573 - 0.0082
2.2. Mixing and combining rules The mixing and combining rules are necessary for successful extension of an EoS to mixtures. The simple van der Waals one fluid (vdW1f) mixing rules were applied in the cubic part:
a=
∑∑ x
i
x j ai j
(17)
∑∑ x
x j bi j
(18)
i
b=
i
j
i
j
with the following combining rules:
ai j =
bi j =
ai a j (1 − k i j 1 ( bi + b j 2
)
(19)
) (1 − l )
(20)
i j
The polar terms do not need the mixing rules and only a simple combining rule was used to calculate dij:
di j = 0.5 ( di + d j
)
(21)
9
2.3. Phase equilibria calculations When two phases, for example α and β, are in equilibrium, the fugacity of all components in the phases are equal:
fiα = fi β
(22)
This is the basis of phase equilibria calculations. The fugacity coefficients of a pure component (φ) and a compound in the mixture (φi) were obtained by the following equations at a certain temperature (T) and pressure (P) [30]:
A f ln ϕ = ln = + Z − 1 − ln Z n RT P
(23)
f µ 1 ln ϕi = ln i = i − ln Z = RT xi P R T
(24)
∂A − ln Z ∂ ni T , V , n j
where, Z, f, fi and µ i are the compressibility factor, fugacity of the pure compound, fugacity and chemical potential of a component in the mixture, respectively and:
Z=
∂A 1 ρ n R T ∂ ρ T , n
(25)
while:
Z = Z CCP = Z SRK + Z Polar =
∂ A SRK 1 ρ n RT ∂ ρ
∂ A Polar + T , n ∂ ρ
10
T , n
(26)
The final expressions for the compressibility factor and chemical potential of SRK and polar terms were given by Danesh [31] and Shahriari et al. [27], respectively. 3. Results and discussion 3.1. Pure component calculations The critical and physical properties of pure components are given in Table 2 [32-34]: Table 2. Physical properties of pure compounds [32-34] Compound
water methanol ethanol propanol 2-propanol 2-butanol acetone NMP hydrogen sulfide carbon dioxide
Critical temperature Tc/K 647.13 512.64 514.00 536.78 508.30 536.05 508.10 727.10 373.40 304.13
Acentric factor ω 0.344 0.565 0.644 0.629 0.665 0.574 0.307 0.358 0.090 0.225
Critical pressure Pc/bar 220.6 80.9 61.4 51.7 47.6 41.8 47.0 45.2 89.6 73.8
Dipole moment µ × 2.99788 × 1029/C.m 1.84 1.7 1.7 1.68 1.66 1.66 2.88 4.09 0.9 0
Quadrupole moment Q × 1040/C.m2 0 0 0 0 0 0 0 0 0 -13.4
The SRK EoS parameters are determined by the expressions defined in Soave’s work [21]: R 2 Tc2 a 0 = 0.42747 Pc RT b = 0.08664 c Pc
(27)
(28)
c = 0.480+1.574ωi − 0.176ωi2
(29)
11
while the proposed model parameters were determined by fitting to the experimental saturated liquid molar volumes and vapor pressures data. The average absolute relative deviation percent (AARD%) together with the average absolute deviation percent (AAD%) were employed to indicate the model deviation from experimental data, which were determined by Eqs. (30) and (31):
A A RD % =
AAD% =
100 NP
∑ i
q icalc − q iexp q iexp
(30)
100 qicalc − qiexp ∑ NP i
(31)
where qiexp and qicalc are the experimental and model-calculated values, respectively, while NP is the number of data points. SRK parameters together with its AARDs% in calculating saturated liquid molar volumes and vapor pressures are given in Table 3. According to AARDs% in Table 3, the results for liquid molar volume by SRK EoS are, as expected, unsatisfactory. It should be mentioned that the experimental saturated liquid molar volumes data of NMP reported in literature are scattered and do not cover the studied range of temperature. Hence, the experimental vapor pressure data were only utilized in this study for this component.
12
Table 3. SRK parameters together with its AARDs% for saturated liquid molar volumes and vapor pressures Compound water methanol ethanol propanol 2-propanol 2-butanol acetone NMP hydrogen sulfide carbon dioxide
T/K 323-583 255-460 255-463 280-483 279-459 281-488 253-456 330-522 190-337 217-275
a0 /Pa.(m3)2.mol-2 0.5609 0.9599 1.2714 1.6468 1.6039 2.0318 1.6231 3.4056 0.4597 0.3705
b(105)/m3.mol-1 2.1130 4.5645 6.0301 7.4788 7.6920 9.2398 7.7872 1.1504 3.0009 2.9695
c AARD%Psat AARD%vliq 1.0006 4.77 38.27 1.3131 8.86 34.22 1.4207 4.43 22.21 1.4004 2.93 16.61 4.00 18.78 1.4489 1.3259 12.18 16.89 2.97 29.00 0.9471 4.40 1.0205 0.6202 1.56 5.72 0.8252 0.31 11.06
overall
4.64
21.42
The results of the CPP EoS, for water and CO2, with and without the third-order perturbation polar term, A3Polar, are given in Table 4. It can be seen from Table 4 that the third-order perturbation polar term did not improve the results and consequently this term was eliminated. This elimination made the calculations significantly simpler, and, thus, CPP EoS was applied with only the second-order perturbation polar term, A2Polar, in this study. Accordingly, the CPP parameters together with calculated results for all studied compounds are shown in Table 5. Table 4. The parameters and the AARDs% of the CPP EoS for saturated liquid molar volumes and vapor pressures of water and carbon dioxide with and without A3Polar term Compound Water Water carbon dioxide carbon dioxide
CPP EoS with A3 without A3 with A3 without A3
a0 /Pa.(m3)2.mol-2 0.4667 0.4663 0.3211 0.3132
b(105) /m3.mol-1 1.5719 1.5724 2.9308 2.9205
13
c
np
nq
0.7817 0.0298 0.7805 0.0364 0.5799 0.4291 0.5823 0.4289
AARD%Psat AARD%vliq 0.77 0.70 0.13 0.14
0.35 0.36 0.11 0.08
Table 5. CPP parameters together with its AARDs% for saturated liquid molar volumes and vapor pressures for all studied compounds Compound
T/K
water methanol ethanol propanol 2-propanol 2-butanol acetone NMP hydrogen sulfide carbon dioxide
323-583 255-460 255-463 280-483 279-459 281-488 253-456 330-522 190-337 217-275
a0 b(105) 3 2 -2 /Pa.(m ) .mol /m3.mol-1 0.4663 1.5724 0.8237 3.4735 1.1363 5.0026 1.4623 6.4748 1.4040 6.5537 1.3571 8.4743 1.4000 6.1784 3.5013 14.6090 0.4285 2.9238 0.3132 2.9205
c 0.7805 1.0133 1.2351 1.3654 1.4285 1.2926 0.7915 0.8047 0.5001 0.5823
np
nq
AARD%Psat AARD%vliq
0.0364 0.0521 0.1131 0.2818 0.2693 2.8907 0.0400 0.8904 1.0016 0.4289
overall
0.70 2.58 2.13 0.90 0.95 1.57 0.20 4.07 0.53 0.14
0.36 0.26 0.88 1.26 1.03 1.31 0.72 0.72 0.08
1.38
0.74
A comparison between Table 3 and Table 5 demonstrates that CPP EoS significantly improved the calculation values of both saturated vapor pressure and liquid molar volume. 3.2. Mixture calculations 3.2.1. Phase equilibria correlation Since the SRK EoS in calculating pure compound properties shows unsatisfactory results and the goal of this study is to achieve a suitable model for both pure and mixture fluids, the calculations of CPP EoS for mixtures were only presented. According to Eqs. (19) and (20), the CPP EoS has two binary interaction parameters (BIPs) which were determined by fitting to experimental binary VLE data. Table 6 shows the calculated BIPs for each binary mixture.
14
Table 6. Calculated BIPs by fitting to binary VLE data System
T/K
Water-CO2 Water-CO2 Water-CO2 Water-CO2 Water-H2S Water-H2S H2S-CO2 Methanol-CO2 Ethanol-CO2 Popanol-CO2 Acetone-CO2 Water-Acetone Ethanol-Acetone Water-NMP 2-Propanol-NMP 2-Butanol-NMP
288.26-373.15 373.15-383.15 383.15-473.15 473.15-573.15 310.00-344.26 344.26-444.26 258.41-348.15 304.20-313.14 291.15-333.4 313.40-333.40 313.15-333.15 333.15-433.15 305.15-321.15 343.15-363.15 353.15-373.15 373.15
kij = A + BT+ CT2 A B/K-1 C/K-2 0.57990 -0.0012545 0 11.31044 -0.0300111 0 -0.63631 0.0011692 0 -0.01758 -0.0001385 0 2.16537 -0.0064519 0 -0.19330 0.0003995 0 -3.89127 0.0271527 -4.6339324 0.26229 -0.0009165 0 -0.02826 0.0001669 0 0.14567 -0.0003139 0 -0.15209 0.0002842 0 0.22035 0.0002717 0 0.22712 -0.0005686 0 1.13441 0.0007285 0 1.41900 -0.0043857 0 0.06300 0 0
lij = A + BT + CT2 A B/K-1 C/K-2 0.68297 -0.0012534 0 6.30208 -0.016312 0 -0.01962 0.0001873 0 0.14706 -0.0001650 0 1.66928 -0.0046616 0 0.11743 -0.0001538 0 -5.11292 0.0358433 -6.2484813 1.10965 -0.0003412 0 0.23617 -0.0006879 0 0.44974 -0.0013370 0 0.02661 0 0 0 0.44526 -0.0003526 0.20764 0.0005851 0 0.06789 -0.0001595 0 1.46425 -0.0040128 0 0.09875 0 0
Phase equilibria investigation of water-CO2 and water-H2S binary mixtures has a significant role in thermodynamics and numerous applications in industry. Because at some specified temperatures and pressures nonaqueous phase is converted to a liquid phase, its water content is suddenly increased and water mole fraction in the nonaqueous phase has a minimum value. It should be noted that most the thermodynamic models are unable to represent this minimum. Modern thermodynamic models such as statistical associating fluid theory (SAFT) and CPA EoSs considered CO2 and H2S as a self-associating or solvating component to correctly compute water content of nonaqueous phase of these binary systems [9, 35, 36]. However, CO2 and H2S are not real associating compounds. Moreover, interactions between CO2 and water originate from quadrupole-dipole and dispersive forces [9]. In this study, the dipole-quadrupole interactions between H2O and CO2 molecules and dipole-dipole interactions between H2O and H2S molecules were directly taken into consideration. 15
Pappa et al. [35] studied mixture of water-CO2 in a wide range of pressure and temperature. They employed the Peng-Robinson (PR) EoS, universal mixing rule with PR-EoS (UMR-PR) (The universal mixing rule refers to a class of EoS/GE models), and PR-based CPA EoS. Their results showed that the results of CPA with two BIPs were in better agreement with experimental data in which both water and CO2 were modeled as an associating component with four association sites (4C). Table 7 shows our results obtained by CPP EoS and those reported by Pappa et al. [35]. Accordingly, it can be observed that the CPP could generate better results than the results provided by CPA EoS. Additionally, the CPP has a higher level of accuracy compared to PR EoS. Table 7. Water-CO2 VLE calculation results (experimental data from [37-42]) Number of data points
P/bar
T/K
8 4 9 9 7 15 15 15 15 13 10
5.0-202.7 101.3-202.7 40.5-141.1 40.5-131 3.3-23.1 100.0-1500.0 100.0-1500.0 100.0-1500.0 100.0-1500.0 200.0-1230.0 100.0-550.0
288.26 313.15 333.15 353.15 373.15 383.15 423.15 473.15 523.15 543.15 573.15
overall
CPP CPA-PR PR This work [35] [35] AARD% AARD% AARD% AARD% AARD% AARD% xCO2 yH2O xCO2 yH2O xCO2 yH2O 3.55 6.31 8.8 8.9 3.1 31.6 3.88 2.85 3.1 4.7 2.7 4.0 1.93 5.64 1.8 8.2 4.0 4.6 2.18 2.57 2.3 2.3 8.8 2.5 9.62 11.16 8.0 9.6 14.1 10.6 1.76 17.62 6.0 20.2 11.3 69.8 8.40 8.45 5.3 10.3 9.5 60.3 5.80 3.63 2.5 12.2 9.0 40.4 13.75 8.43 8.4 12.4 3.7 29.1 4.84 3.11 10.8 6.0 19.4 28.2 6.11 8.31 6.6 2.0 9.5 10.2 5.62
7.10
5.78
8.8
8.65
26.48
Water-H2S calculation results provided by CPP EoS are also given in Table 8. Moreover, some typical results for water-CO2 and water-H2S mixtures are demonstrated in Figs. 1 to 4, which indicate that the CPP EoS could correctly determine both water rich and nonaqueous phases
16
compositions. An accurate representation of minimum water mole fraction in CO2 rich phase and H2S rich phase is evident in Figs. 1 and 3, respectively, which is another noteworthy point of the proposed model.
Table 8. Water-H2S VLE calculation results (experimental data from [32, 43]) Number of data points
P/bar
T/K
9 9 6 6 6
3.447-206.84 6.8948-48.263 6.8948-48.263 6.8948-48.263 6.8948-48.263
310.00 344.26 377.59 410.93 444.26
overall
CPP (This work) AARD% xH2S AARD% yH2O 4.27 11.50 0.72 7.64 2.70 4.97 5.21 1.15 7.40 2.51 4.06
5.55
Fig. 1. Water mole fraction in CO2 rich phase at 288.26 K. CPP calculation (solid line) and experimental data (points) [38, 41]
17
Fig. 2. CO2 mole fraction in water rich phase at 288.26 K. CPP calculation (solid line) and experimental data (points) [38, 41]
Fig. 3. Water mole fraction in H2S rich phase at 310.00 K. CPP calculation (solid line) and experimental data (points) [32]
18
Fig. 4. H2S mole fraction in water rich phase at 310.00 K. CPP calculation (solid line) and experimental data (points) [32]
The VLE calculation results of CO2-H2S obtained by the proposed model are presented in Table 9. It can be seen that CPP EoS by BIPs with a quadratic temperature dependency could accurately calculate the VLE data of this binary mixture. The graphical representation of CPP calculation for CO2-H2S system at some arbitrary temperatures is available in Fig. 5. Table 9. CO2-H2S VLE calculation results (experimental data from [44, 45]) Number of data points 6 6 9 4 3 3
P/bar
T/K
10.02-19.23 13.75-28.99 20.49-52.64 34.34-54.75 55.1581-82.7371 60.795-81.06
258.41 273.15 293.47 313.02 338.00 348.15
overall
CPP (This work) AARD%P AAD%y 0.80 1.14 1.49 1.13 0.72 1.31 0.47 1.52 1.02 0.38 1.54 0.61 1.01
19
1.02
Fig. 5. VLE plot for CO2-H2S system. CPP calculation (solid line) and experimental data (points) [45]
The calculated results for methanol-CO2, ethanol-CO2, and propanol-CO2 provided by CPP EoS are available in Tables 10 to 12. The BIPs of these binary mixtures were obtained by fitting to only bubble point pressure, and the vapor phase compositions of these binary systems were left to be as a predictive property. The deviation between model-calculated results and experimental data for all three systems are satisfactory while the results of propanol-CO2 are more precise.
Table 10. Methanol-CO2 VLE calculation results (experimental data from [46, 47]) Number of data points 10 6
P/bar
T/K
37.5-69.5 6.83-69.54
304.2 313.4
CPP (This work)
overall
20
AARD%P 3.10 2.72
AAD%y 0.21 0.18
2.91
0.20
Table 11. Ethanol-CO2 VLE calculation results (experimental data from [46, 48]) Number of data points 9 8 7
P/bar 20.94-54.00 5.14-71.02 5.44-60.94
T/K
CPP (This work)
291.15 313.40 333.40
overall
AARD%P 2.02 2.84 1.23
AAD%y 0.45 0.12 0.08
2.03
0.22
Table 12. Propanol-CO2 VLE calculation results (experimental data from [46]) Number of data points 7 5
P/bar
T/K
5.18-69.79 6.68-60.81
313.40 333.40
CPP (This work)
overall
AARD%P 1.34 0.37
AAD%y 0.11 0.09
0.86
0.10
Fig. 6. VLE plot for ethanol-CO2 system at T=291.15 K. CPP calculation (solid line) and experimental data (points) [48]
21
The AARDs% for bubble point pressure of the acetone-CO2 system are presented in Table 13 where an overall AARD% of 0.55% indicates a high level of accuracy achieved by the CPP EoS. Table 13. Acetone-CO2 VLE calculation results (experimental data from [49]) Number of data points 9 7
P/bar 10.01-69.93 9.87-70.23
T/K
CPP (This work)
313.15 333.15
overall
AARD%P 0.59 0.50
AAD%y -
0.55
-
In the cases of water-acetone and ethanol-acetone, according to Tables 14 and 15, the CPP EoS shows a good performance so that the representation of azeotropic points is evident in Fig. 7.
Table 14. Water-acetone calculation results (experimental data from [32]) Number of data points 10 10
P/bar
T/K
CPP (This work)
0.739-1.149 10.418-14.135
333.15 433.15
overall
AARD%P 1.63 1.14
AAD%y 0.78 1.67
1.37
1.23
Table 15. Ethanol-acetone calculation results (experimental data from [13]) Number of data points 16 16 16
P/bar
T/K
0.1369-0.39717 0.20292-0.54582 0.29771-0.73807
305.15 313.15 321.15
CPP (This work) AARD%P AAD%y 0.58 0.42 1.40 0.25 1.29 0.33
overall
1.09
22
0.33
Fig. 7. P-x-y plot for water-acetone system at T=333.15 K. CPP calculation (solid line) and experimental data (points) [32]
Fig. 8. P-x-y plot for water-acetone system at T=433.15 K. CPP calculation (solid line) and experimental data (points) [32]
23
Fig. 9. P-x-y plot for ethanol-acetone system. CPP calculation (solid line) and experimental data (points) [13]
VLE modeling of important mixed solvents, water-NMP, 2-propanol-NMP, and 2butanol-NMP, was evaluated by CPP EoS. Tables 16 to 18 provide the AARDs% of these systems obtained by the proposed model. It can be seen that the proposed model works properly while the results of 2-propanol-NMP are more accurate. These findings are also obtained based on Figs. 9 to 11. Table 16. Water-NMP calculation results (experimental data from [50]) Number of data points 10 10
P/bar
T/K
0.0079-0.3100 0.0217-0.7014
343.15 363.15
CPP (This work) AARD%P AAD%y 2.67 0.26 4.61 0.39
overall
3.64
24
0.33
Table 17. 2-Propanol-NMP calculation results (experimental data from [51]) Number of data points 25 18
P/bar 0.013-0.927 0.036-1.977
T/K AARD%P 1.33 1.03
353.15 373.15
overall
CPP (This work) AAD%y 0.07 0.54
1.18
0.31
Table 18. 2-Butanol-NMP calculation results (experimental data from [51]) Number of data points 15
P/bar 0.036-1.070
T/K AARD%P 2.38
373.15
CPP (This work) AAD%y 0.36
Fig. 10. P-x-y plot for water-NMP system. CPP calculation (solid line) and experimental data (points) [50]
25
Fig. 11. P-x-y plot for 2-propanol-NMP system. CPP calculation (solid line) and experimental data (points) [51]
Fig. 12. P-x-y plot for 2-butanol-NMP system. CPP calculation (solid line) and experimental data (points) [51]
26
3.2.2. Density prediction The BIPs in the CPP model, kij and lij, were set to zero to represent a fully predictive model for calculation of water-CO2 mixture density. The AARDs% of the model for mixture densities of equilibrated and single phases are given in Tables 19 and 20, respectively. The CPP results of water-CO2 mixture density were compared with those obtained by CPA EoS reported in the work of Tsivintzelis et al. [52] in which CO2 was considered as a solvating molecule. The obtained results indicated that the predicted densities by the proposed CPP-EoS model were in a good agreement with the experimental data. Table 19. Density prediction results of water-CO2 saturated mixtures T/K
283 288 293 298.15 322.8 298 313 323 382.41 423.06 468.47 422.98 445.74 461.62 478.35 overall
Number of data points water rich CO2 rich phase phase
6 6 6 6 11 10 11 11 -
11 6 6 10 10 10 10 10
CPP This study AARD% AARD% water rich CO2 rich phase phase
CPA CO2 with solvation [52] AARD% AARD% water rich phase CO2 rich phase
0.7 0.6 0.5 0.1 0.5 0.9 1.4 1.5 -
2.3 0.6 2.5 1.4 2.8 3.5 3.5 3.6
0.9 0.7 0.6 0.1 0.3 0.6 0.9 1.0 -
2.2 1.7 2.5 2.0 4.8 5.5 5.4 5.8
0.65
2.52
0.64
3.74
27
Reference
[53] [53] [53] [38] [54] [55] [55] [55] [56] [56] [56] [56] [56] [56] [56]
Table 20. Density prediction results of water-CO2 single phase mixtures T/K
P/bar
Number of data points
Water mole fraction
415.36-697.66 455.23-697.94 469.55-698.08 497.21-698.40 518.59-693.10 544.21-693.07 562.96-697.77 580.3-699.18 616.59-698.57 625.44-699.19 634.73-699.30
58.84-108.28 70.76-117.62 78.32-127.12 91.20-139.98 104.99-154.14 121.69-170.00 143.08-198.32 165.85-227.11 214.03-273.47 242.26-313.52 267.89-345.77
22 19 18 16 14 12 11 12 10 9 8
0.0612 0.1407 0.2101 0.2870 0.3670 0.4349 0.5317 0.6017 0.6934 0.7523 0.7913
overall
CPP This study
Reference
AARD%
CPA CO2 with solvation [52] AARD%
0.2 0.1 0.2 0.4 0.5 0.5 1.1 1.1 2.2 3.9 5.2
1.1 1.2 1.2 0.8 0.7 0.6 1.5 1.1 1.6 2.5 3.2
[57] [57] [57] [57] [57] [57] [57] [57] [57] [57] [57]
1.40
1.41
3.2.3. Solubility prediction Savary et al. [58] presented a set of solubility data of H2S and CO2 mixtures in water at temperature 393.15 K and pressures up to 350 bar. Their data were utilized for our predictions where Table 21 gives deviations between model-predicted results and experimental data. The computed BIPs of water-H2S and water-CO2 systems reported in Table 6 were applied for prediction calculations but the BIP of CO2-H2S system was set to zero, because the temperature 393.15 K was higher than the studied temperature range of CO2-H2S system (258.41-348.15 K). Table 21. Prediction results of solubility of CO2+H2S mixture in water (experimental data from [58]) Number of data points 23
P/bar
T/K
39-350
393.15
CPP (This work) AAD% xCO2 AAD% xH2S 0.29 0.90
28
Conclusion In this study, Cubic-Plus-Polar (CPP) equation of state (EoS), which is based on a combination of Soave-Redlich-Kwong (SRK) EoS and a set of polar terms, was developed to model phase equilibria of polar systems over a wide range of pressures and temperatures. The proposed model was significantly simplified by eliminating the third-order perturbation polar term (A3Polar) owing to slight influence of this term on improving the results. The CPP EoS was able to obtain satisfactory results for pure water, methanol, propanol, 2-propanol, 2-butanol, acetone, Nmethyl-2-pyrrolidone (NMP), CO2, and H2S as well as their mixtures including water-CO2, water-H2S, CO2-H2S, methanol-CO2, ethanol-CO2, propanol-CO2, water-acetone, acetone-CO2, ethanol-acetone, water-NMP, 2-propanol-NMP, and 2-butanol-NMP with four pure-compound parameters and two temperature-dependent binary interaction parameters. Moreover, the proposed model is capable of correctly representing minimum water mole fraction in the nonaqueous phase of water-CO2 and water-H2S mixtures. Additionally, the CPP EoS could accurately predict the saturated and single-phase densities of water-CO2 mixtures. The solubility values of CO2 and H2S mixture in water was also reasonably predicted by the proposed model. The aim of introducing such EoS was to propose an efficient and simple model, which is compatible with the physics of the problem through directly taking into account dipole-dipole, quadrupole-quadrupole and dipole-quadrupole interactions in pure polar components and their mixtures.
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Highlights Cubic-Plus-Polar (CPP) EoS has been constructed to model phase equilibria of practical systems with a high polarity. The proposed model takes into account dipolar and quadrupolar interactions to be more compatible with physics of the problem The Phase equilibria of studied systems were properly modeled in a wide range of temperatures and pressures. The proposed model represents minimum water mole fraction in the nonaqueous phase of water-CO2 and water-H2S mixtures. The CPP EoS accurately predicted water-CO2 mixture density and the solubility values of CO2 and H2S mixture in water.