Article pubs.acs.org/IECR
Gibbs Free Energy Minimization for Prediction of Solubility of Acid Gases in Water Ashwin Venkatraman,*,† Larry W. Lake,† and Russell T. Johns‡ †
Department of Petroleum and Geosystems Engineering, The University of Texas at Austin, 200 East Dean Keeton Street, Stop C0300, Austin, Texas 78712, United States ‡ Department of Energy and Mineral Engineering, Pennsylvania State University, 110 Hosler Building, University Park, Pennsylvania 16802, United States ABSTRACT: The disposal of acid gas (CO2/H2S mixtures) is a critical aspect in the production of hydrocarbons from sour gas fields. The increasing emphasis on CO2 sequestration has also renewed interest in the disposal of flue gas mixtures (primarily containing CO2 and H2S). A common strategy for safe disposal, in either case, is to inject the acid gas in aquifers close to production plants. These strategies rely on solubility calculations at different pressures and temperatures, governed by the field operating conditions. We present a comprehensive approach using Gibbs free energy minimization to calculate acid gas solubility in water at high temperatures (298−393 K) and pressures (0.1−80 MPa). The advantage of this approach is the flexibility to use different thermodynamic models for different phases. The proposed model uses the Peng−Robinson (PR) Equation of State (EOS) description for gas components while the liquid components are described using the ideal assumption for the temperature range 298−323 K and the Nonrandom Two-Liquid (NRTL) activity coefficient model at temperatures greater than 323 K. The model predictions compare well with experimental data for binary (CO2−H2O and H2S−H2O) and ternary mixtures (CO2− H2S−H2O). The model can also be easily extended to predict the solubility of any gas in water as well as brine containing ions to incorporate geochemical reactions.
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INTRODUCTION About 40% of world gas reserves have been estimated to be contaminated with the acid gases CO2 and/or H2S.1 There are vast reserves of such fields, also called sour gas fields, having more than 10% of such contaminants in Canada, North Africa, SE Asia/NW Australia, and the Middle East. New separation technologies2−5 have enabled production of hydrocarbons from these sour gas fields. This, however, brings with it the challenge to safely dispose the acid gas resulting from the separation process. An effective acid gas disposal strategy is imperative for continuous hydrocarbon production from these sour gas fields. Aquifers are commonly used for the disposal of the acid gas in sour gas operating fields.6,7 The International Energy Agency (IEA), as part of the global efforts to reduce CO2 in the atmosphere, estimates the Carbon Capture and Storage (CCS) technologies to contribute onefifth of the total target of halving current levels of emission by 2050.8 The capture and storage of flue gases from fossil power plants is part of this plan to reduce emissions. Flue gas is a mixture containing CO2, H2S, and trace amounts of CH4. The research efforts pertaining to CO2 capture and storage have also identified aquifers as efficient storage sites for storing CO2 by the solubility trapping mechanism.9 In the above applications, it is critical to accurately estimate the acid gas solubility in both water as well as brine containing ions at equilibrium. Geochemical reactions occur in the presence of ions. Hence, the solubility of gas in brine is likely to be different than that in pure water. In this research, we propose a model, using Gibbs free energy minimization, to predict the acid gas solubility in water where only phase equilibrium occurs. This unified approach can also be extended © 2014 American Chemical Society
to predict solubility in brine, where the presence of ions can cause reactions and hence model phase and chemical equilibrium. Unlike conventional approaches, which rely on experimental data to parametrize Henry’s law constant, the proposed model is predictive for the temperature range of 298−323 K and uses only the Gibbs free energy values at the standard state conditions (P0 = 101.325 kPa and T0 = 298 K). The Gibbs free energy values at the standard state conditions for different components have been tabulated in the National Institute of Standards and Technology (NIST) database.10 The predictions from the model compare well with the experimental values for binary mixtures at varying pressures (0.1−55 MPa) in this temperature range. At temperatures greater than 323 K, the ideal liquid assumption is no longer valid and an activity coefficient model is required to describe the solubility of gas in water. The Nonrandom Two-Liquid (NRTL) activity coefficient model for the aqueous phase has been proposed for predicting solubility. The interaction parameters, used in the proposed activity coefficient model, have been developed as a function of temperature for the binary systems H2S−H2O and CO2−H2O to predict solubility at temperatures greater than 323 K. The linear correlation for interaction parameters, obtained using experimental data for binary systems between 323 and 383 K, is used to predict mixture solubility for the ternary system of Received: Revised: Accepted: Published: 6157
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CO2−H2S−H2O. The ternary mixture solubility predictions also compare well with the experimental data at 393 K. A practical approach to obtain equilibrium composition, at high pressures and temperatures, by using the Gibbs free energy as a unifying function for different phase descriptions is presented in this research. In particular, the presented approach is capable of combining any Equation of State (EOS) description for gas phase components and the activity coefficient model for aqueous phase components. The Gibbs free energy function also provides the flexibility to find equilibrium compositions at different scales by adding the appropriate energy function relevant at that scale. As an example, energy due to capillarity can be appropriately added for finding equilibrium composition at microscopic scales. In addition to chemical compositions, the presented framework can also be used to compute the equilibrium state of a system under deformation by addition of the strain energy to the Gibbs free energy function. The equilibrium compositions presented in this manuscript are at the macroscopic scale where the energy due to capillarity as well as strain energy is neglected. We begin with a review of reference states and discuss how they are used to predict solubility for our system of aqueous and gas phases. This review has been drawn from some excellent texts11−13 in the subject area. In the Literature Review section, we summarize the different types of solubility prediction models previously developed for acid gases. The details of the proposed Gibbs free energy minimization model to find equilibrium compositions is presented in the Methodology section. Finally, we present a comparison between the model predictions and the experimental data for the binary systems as well as for the CO2−H2S−H2O mixture.
constant, both at the reference state and at the state of interest, while the pressure may vary. The chemical potential can also be defined in terms of activity (aij) of the component.12 Here, the reference state is implicit in the definition of activity and given as Gij̅ = Gi̅ ref + RT ln aij
The reference state could be a pure (single) component property or the property of the component in an ideal mixture at a particular temperature and pressure. The convention for the reference state for components varies and depends on the nature of the component (solute or solvent) as well as the thermodynamic description (activity coefficient model or EOS description) of the phase in which the component is present. The NIST database10 lists the Gibbs free energy reference state values for different components in the Lewis−Randall convention. Table 1 describes the reference states for components present in our system based on the Lewis− Randall convention. Table 1. Reference States for Components As Reported in the Lewis−Randall Convention in the NIST Database10,a component type gas phase component aqueous solute water a
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EQUILIBRIUM AND GIBBS FREE ENERGY: A REVIEW The components in a system are distributed among phases so that the total Gibbs free energy function attains its global minimum at equilibrium. As a consequence, the partial molar Gibbs free energies G̅ (also called the chemical potential) of components are equal in each of the phases they are distributed. This equality of chemical potentials G̅ is a necessary, but not a sufficient condition. As the total Gibbs free energy function can have multiple minima, equilibrium compositions cannot be always found by equating the chemical potentials, especially in the presence of hydrocarbons.14 Most solubility models, which describe experimental measurements of phase equilibrium between gas and aqueous phase components, are constructed by equating the chemical potential of components in the two phases. The general expression for chemical potential of any component i in a nonideal phase mixture is measured from a reference state11 and is given as Gij̅ =
Gi̅ ref
⎛ f̅ ⎞ ij + RT ln⎜⎜ ref ⎟⎟ ⎝ f ij̅ ⎠
(2)
reference state hypothetical ideal gas at standard conditions hypothetical ideal solution of unit molality at standard conditions pure liquid at standard conditions
The standard conditions are T0 = 298 K and P0 = 101.325 kPa.
The components of our system (gas, aqueous solute, and water) have different reference states. The chemical potentials G̅ ij are equal at equilibrium (a necessary condition for equilibrium) even though different reference states might be used for the same component in different phases. Because one can relate the two different reference states using eq 1, the component fugacities in these phases11 are also equal. However, the activities of the components are equal at equilibrium only if the same reference states are used for both the phases.
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LITERATURE REVIEW The acid gas solubility models, previously developed, can be classified into two broad types depending on the thermodynamic description of phases. In models of the first type (Type I), aqueous phase components are assumed to be ideal while the gas phase components are described by an EOS model. In models of the other type (Type II), both the aqueous phase and the gas phase are described using a modified EOS. The developments in both these types of solubility models are discussed in this section. The starting equation for most solubility models is equal fugacities of components in different phases. In our system, the solvent is H2O (i = 1) while the solute is CO2 or H2S (i = 2). ̂ = x γf f ̅ = f ̅ ⇒ y ϕP i = 1, 2 i
(1)
Here, G̅ ij is the chemical potential of component i in phase j at the pressure and temperature conditions of interest, fij̅ is the fugacity of component i in phase j at the state of interest, G̅ ref i is the chemical potential of the component at the reference state, and fij̅ ref is the fugacity at that reference state. Lewis and Randall15 defined fugacity using eq 1 for consistency with the corresponding equation for an ideal gas. The temperature is
ig
il
i i
ii
(3)
Here, xi and yi are the mole fractions of the component in the gas and aqueous phases, respectively, ϕ̂ i is the fugacity coefficient of component i in the gas phase calculated using an EOS at the system pressure P, and f i is the fugacity of the component in the liquid phase at that pressure. The fugacity of 6158
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systems and water at low pressures. We extend this for higher pressures and temperatures. Numerous algorithms using this approach have been developed to compute equilibrium compositions for not just phase equilibrium27 but also for coupled phase and chemical equilibrium.28−33 We use the RAND algorithm proposed by White et al.33 to find the equilibrium composition for our system.
the component in the liquid phase is further simplified in Type I models to be ⎡ ν̲ (P − P0) ⎤ f1 = P1sat exp⎢ 1 ⎥; ⎣ ⎦ RT
⎡ v ∞(P − P0) ⎤ f2 = / exp⎢ 2̅ ⎥ ⎣ ⎦ RT (4)
Psat 1
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Here, is the saturation pressure of water at the temperature of interest, P0 = 101.325 kPa, and / is the Henry’s law constant for the binary system. The exponential terms represent corrections to the fugacities at the total system pressure P and ν̲1 is the molar volume of water while v∞ 2̅ is the partial molar volume of aqueous solute CO2 or H2S at infinite dilution. Carroll and Mather16 used this relationship (eq 4) to predict solubility of acid gas H2S at low pressures in the aqueous phase. The fugacity coefficient was calculated using a modification to the PR EOS suggested by Stryjek and Vera17 and a correlation for the Henry’s law constant / was developed using experimental values at different temperatures. Enick and Klara18 used PR EOS to develop similar correlations for Henry’s law constant. Li and Nghiem19 have also used the same approach for predicting the solubility of CO2 and light hydrocarbon gases in pure and saline water. In Type II models, the equation of state representations by themselves are not capable of describing the aqueous phase. Several modifications to the EOS17,20 have been proposed to incorporate the aqueous phase for better prediction of vapor liquid equilibrium and vapor pressure data. These modifications include either using separate sets of binary interaction parameters for the components in the aqueous and nonaqueous phases or using different mixing rules for polar asymmetric mixtures.21 More specifically for acid gases, more accurate EOS for the binary system of both H2S−H2O and CO2−H2O using virial expansions have been proposed to predict the solubilities.22,23 These EOS models regress on experimental data to find the coefficients in the virial expansion EOS. The Henry’s law approach, used conventionally, works well at low pressures. However, the acid gas disposal for hydrocarbon processing as well as carbon capture and sequestration occurs at high pressures (200−600 bar). Duan et al.22 have also shown that the Henry’s law approach does not predict accurate solubilities for H2S at high pressures. The number of coefficients in their proposed virial expansion EOS makes it cumbersome for use in simulators that perform phase equilibrium calculations of mixtures. The other disadvantage of both these models (types I and II) is its inability to incorporate geochemical reactions as these models are based only on phase equilibrium computations. In this research, we use the Gibbs free energy minimization approach to predict solubility. This approach is adapatable to diffferent brine compositions and provides the flexibility to incorporate ion concentrations in the brine for solubility calculations and, hence, incorporate geochemical reactions. The Gibbs free energy minimization approach to find equilibrium composition has been extensively used for process engineering applications as well as for geochemical analysis to predict mineral solubilities in water.24,25 The Gibbs free energy minimization has been used to predict compositions for individual cases of either phase equilibrium or geochemical reactions, but not together. Luckas et al.26 have used a similar approach to predict phase and chemical equilibrium for flue gas
METHODOLOGY As explained in the previous section, the chemical potential at any desired temperatures and pressure are measured from the reference state and is different for the aqueous and gas phase components (Table 1). In this section, we explain how chemical potentials for the components are obtained from the reference state values and formulate the minimization problem. The reference state for components in a gas phase described using an EOS is the pure component ideal gas property (fref i̅ = ref IG IG f IG ; G = G̲ ). The pure component property (G̲ ) is a ̅ i i i i measure when the component is an ideal gas (hypothetical), at standard conditions (fIG i̅ = P0 = 101.325 kPa and temperature T0 = 298 K). Using eq 1, we have for gas phase components (j = 1), Gi̅ 1(T0 , P) = G̲ iIG(T0 , P0) + RT ln
̂ yi ϕiP P0
i = 1, 2 (5)
In the aqueous phase, the solutes are described using a reference state G̅ *1 of unit molality m of the solute (CO2 or H2S, i = 1) at P0 and T0.a The solvent, H2O (i = 2), on the other hand, is described using a reference state, G̲ 02, which is the pure component water property at P0 and T0. Using these reference states, the Gibbs free energy expressions for the solute and solvent in the aqueous phase (j = 2) are G̅12(T0 , P) = G̅1*(T0 , P0) + RT ln = G̅10(T0 , P0) + RT ln G̅22(T0 , P) = G̲ 20(T0 , P0) + RT ln
m1γ1f1 f0 x1γ1f1 f0 x 2γ2f2 f0
(6)
The Henry’s law constant / and the saturation pressure of b water Psat 1 (eq 4) are also related to the reference states. It is desirable to choose a reference state at the system P and T conditions so that the second term in eqs 5 and 6 is a measure of deviation from ideality (ϕ̂ i or γi). We then have the following equations, Gi̅ 1(T , P) = G̲ iIG(T , P) + RT ln yi ϕî
i = 1, 2
G̅12(T , P) = G̅10(T , P) + RT ln x1γ1 G̅22(T , P) = G̲ 20(T , P) + RT ln x 2γ2 (7)
These new reference state values are related to the tabulated values at P0 and T0 by the molar volumes (v̲) and enthalpies (H̲ ) of the components (partial molar volume, ν,̅ and partial molar enthalpy, H̅ , for solute in the aqueous phase) by the following relations13 6159
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Industrial & Engineering Chemistry Research ⎛P⎞ IG Gi̅ IG 1 (T , P) = G̲ i1 (T0 , P0) + RT ln⎜ ⎟ + T ⎝ P0 ⎠ i = 1, 2 G̅120(T , P) = G̅10(T0 , P0) +
∫P
P
0
0 G̅22 (T , P) = G̲ 20(T0 , P0) +
∫T
∫P
( v1̅ )T0 dP + T
0
P
0
⎛ − Hi1 ⎞ ⎜ ⎟ dT ⎝ T 2 ⎠P
T
∫T
( v̲ 2)T0 dP + T
Article
T
0
∫T
⎛ − H11 ̅ ⎞ ⎜ 2 ⎟ dT ⎝ T ⎠P
T
0
and temperature conditions, H2O may or may not be present in the gas phase. We perform equilibrium computations for both cases and choose the equilibrium composition that gives the lowest total Gibbs free energy (GTotal). If the objective function, GTotal, is convex, the global minimum can be obtained irrespective of initial guesses. While GTotal for ideal gas mixture and ideal liquid assumption is a convex function,33 the use of the Wilson activity coefficient model for aqueous phase components also results in a convex objective function37 so that the global minimum may be obtained. In this manuscript, the experimental data presented for comparison have been drawn from different sources where only two phases were observed during experiments. The initial guess values to the optimization problem were varied to check for the presence of multiple solutions. Multiple solutions did not occur at the temperature and pressure conditions of the solubility models presented in this manuscript. The RAND algorithm33 has been used to obtain the solution to the optimization problem. The RAND formulation uses Lagrangian multipliers and the steepest descent method.38 We use this approach to find the equilibrium composition and, hence, solubility for the binary systems of CO2−H2O and H2S−H2O as well as for the ternary system of CO2−H2S−H2O. The conventional approach to calculate phase equilibrium composition is by using the Rachford−Rice algorithm39 using successive substitution where components in both phases are described using EOS (Type II models as defined in the Literature Review section) with an initial guess provided by Wilson’s correlation. A test system was designed to compute equilibrium composition for the CO2−H2O system with two phases using the conventional approach and the Gibbs free energy minimization method. As the phases were predefined, no phase stability analysis was performed in either approach. For the test case, the Gibbs free energy minimization approach using the RAND algorithm was found to be 30% faster than the conventional approach. In addition to computational speed, the Gibbs free energy minimization algorithm also provides the advantage of combining different phase descriptions of components (EOS and activity coefficient model). This approach can be extended to include ions present in brine as additional components in the aqueous phase and, hence, incoporate any geochemical reactions that may occur.40,41 The specific parameter values for such ions (molar volumes and partial molar enthalpies at T0 and P0) are available in the literature.36
⎛ − H̲ 22 ⎞ ⎜ ⎟ ⎝ T 2 ⎠P
dT (8) 34−36
Helgeson and co-workers have developed EOS models for the partial molal volumes ν̲ as well as expressions for partial molal enthalpies H̲ for aqueous solutes (CO2 and H2S as well as ionic solutes) to evaluate the reference state values at high temperatures and high pressures (up to 1273 K and 500 MPa). The typical pressure range for acid gas injection is between 20 and 80 MPa. The molar volumes and enthalpies at T0 and P0 (tabulated in the NIST database10 and also listed in Table 5) are assumed to be constants over the pressure and the temperature range of acid gas injection. We can then simplify eq 8 to obtain ⎛P⎞ ⎛ T⎞ IG Gi̅ IG ⎟ 1 (T , P) = G̲ i1 (T0 , P0) + RT ln⎜ ⎟ + [H̲ i1]P0 , T0 ⎜1 − T0 ⎠ ⎝ P0 ⎠ ⎝ i = 1, 2 ⎛ T⎞ G̅120(T , P) = G̅10(T0 , P0) + ( v1̅ ∞)T0 (P − P0) + [H11 ̅ ]P0 , T0 ⎜1 − ⎟ T0 ⎠ ⎝ ⎛ T⎞ 0 G̅22 (T , P) = G̲ 20(T0 , P0) + ( v̲ 2)T0 (P − P0) + [H̲ 22]P0 , T0 ⎜1 − ⎟ T0 ⎠ ⎝
(9)
The total Gibbs free energy function of the entire system,c which includes all components in all phases, is minimized to obtain the equilibrium compositions. The balance among elements (C or S, O, and H) forms the equality constraint for this minimization. For a system with two phases (j = 1,2) and two components (i = 1,2), the equilibrium composition is the solution to the constrained nonlinear optimization problem below, 2
Minimize GTotal =
2
∑ ∑ nijGij̅ (T , P) j=1 i=1
Subject to AN = E
and
nij ≥ 0
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(10)
RESULTS AND DISCUSSION The reference state values for the Gibbs free energy at different temperatures and pressures are evaluated using the expressions in eq 9 and the thermodynamic properties of components tabulated in Appendix A (Table 5). The fugacity coefficient for components at different pressures is calculated using the standard expressions available for the PR EOS. The critical properties and the binary interaction parameters (BIPs) Kij for PR EOS are listed in Table 6. The choice of BIPs has an influence on gas phase mole fractions (see Appendix A) but does not affect the aqueous phase mole fractions (solubility), the focus in this manuscript. The BIPs are assumed to be constants for all computations at different pressures and temperatures. A comparison of the model predictions, assuming ideal liquid solution, with experimental data for varying pressures at different temperatures is presented in Figures 1, 2 and 3. The
Here, A is the elemental matrix representing the number of specific elements in each component and N is the matrix comprising moles of each component nij in each phase (unknowns), while E is formed by the total number of moles of each element. As an example, the matrices for the binary system comprising of H2S (i = 1) and H2O (i = 2) when both components are present in the gas phase (j = 1) and the aqueous phase (j = 2) are ⎡1 0 1 0⎤ ⎢ ⎥ A = ⎢ 0 1 0 1 ⎥; ⎣2 2 2 2⎦
⎡ n11 ⎤ ⎢n ⎥ 12 N = ⎢ ⎥; ⎢ n21 ⎥ ⎢n ⎥ ⎣ 22 ⎦
⎡ eS ⎤ ⎢ ⎥ E = ⎢ e0 ⎥ ⎢⎣ eH ⎥⎦
(11)
The total moles of each element e (S, O and H) is obtained from the initial number of moles. Depending on the pressure 6160
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Figure 1. Comparison between experimental data and model prediction at moderate temperatures for H2S−H2O (a and b) and CO2−H2O (c and d).
Figure 2. Comparison of experimental data with ideal solution prediction and NRTL model with tuned interaction parameter τ0 to fit experimental data for the H2S−H2O binary system.
break-over trend in the figures corresponds to the critical pressure of the mixtures at the corresponding temperature. It can be inferred that the ideal liquid assumption is only valid in the temperature range of 298−323 K for both the binary systems and not at higher temperatures. The proposed solubility model has been classified into two temperature rangesthe moderate temperature range (between 298 and 323 K) where the liquid phase is ideal and the high temperature range (temperatures greater than 323 K) where the liquid phase is described using an activity coefficient model.
Ideal Aqueous Model. The Gibbs free energy minimization model predictions assuming ideal aqueous solution have been compared with experimental values at the moderate tempeature range of 298−323 K for both the binary systems. The model predictions match well with the experimental values (Figure 1). This is reflected in the low average deviation values between the model prediction and the experimental values as well as large R2 values (coefficients of determinationd) at these temperatures (Table 2). The only exception is H2S−H2O at 323 K where predictions are still reasonable compared to the 6161
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Figure 3. Comparison of experimental data with ideal solution prediction and with NRTL tuned interaction parameter τ0 to fit experimental data for the CO2−H2O binary system.
below 323 K. To accurately represent the aqueous phase at high temperatures, we use the NRTL activity coefficient model50 for components in the aqueous phase. This activity coefficient model has been used extensively to model acid gas solubility in amine solutions.51 In the NRTL model, the activity coefficient is a function of the randomness parameter αij and the interaction parameter τij. The expressions for a binary system (mole fractions x1 and x2) are
Table 2. Experimental Data Source for Binary Systems and Comparison with Model Prediction binary system
temperature (K)
experimental data source
H2S−H2O
310
H2S−H2O
323
Selleck et al.42 Koschel et al.43 Lee and Mather44 Dodds et al.45 King et al.46 Wiebe47 Bamberger et al.48 Tödheide and Franck49 Wiebe47
CO2−H2O
CO2−H2O
298
323
coefficient of determination (R2)
average deviation (percent)
0.98
4.45%
0.78
0.93
0.98
14.3%
⎤ ⎡ ⎛ ⎞2 τ12G12 G21 ⎥ ln γ1 = x 2 2⎢τ21⎜ ⎟ + ⎢⎣ ⎝ x1 + x 2G21 ⎠ (x 2 + x1G12)2 ⎥⎦
13.52%
⎡ ⎛ ⎤ ⎞2 τ21G21 G12 2⎢ ⎥; γ = τ + ln 2 x1 12⎜ ⎟ ⎢⎣ ⎝ x 2 + x1G12 ⎠ (x1 + x 2G21)2 ⎥⎦
2.78%
ln G12 = −α12τ12
ln G21 = −α21τ21
(12)
We assume τ12 = τ21 and α12 = α21 = 0.2 (value for most systems50) which reduces the activity coefficient expression (eq 12) to just one variablethe interaction parameter τ. A linear correlation for τ is proposed as a function of temperature specific to each binary system (Table 3). This correlation is developed using available experimental values in the temperature range 323−383 K (see Appendix B). The complete
experimental values. Also, the H2S−H2O system pressure range is low because, at high pressures, H2S forms a separate liquid phase. We do not perform regression with the experimental data in this temperature range. This makes the model predictive in this temperature range as only the thermodynamic properties at T0 and P0 (listed in Table 5) have been used to obtain the solubility values using the minimization algorithm. NRTL Aqueous Model. At higher temperatures, the predictions using the ideal liquid assumption do not match the experimental data at those temperatures for both binary systems (Figures 2 and 3). This implies that the ideal assumption for the liquid phase is valid only at temperatures
Table 3. Constants A and B in the Interaction Parameter Correlation (τ = A + (B/T)) for Binary Systems Obtained by Linear Regression over Experimental Data (See Appendix B)
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binary system
A
B (K)
H2S−H2O CO2−H2O
−2.78 −1.88
883 613.1
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aqueous phase NRTL model (given below) along with the thermodynamic properties at T0 and P0 can be used to estimate the solubility at temperatures greater than 323 K.
Table 4. Experimental Data Source for Binary Systems at 393 K and Comparison with Model Prediction (Figure 4)a
⎡ ⎛ ⎤ ⎞2 τG G ⎥ ln γ1 = x 22⎢τ ⎜ ⎟ + ⎢⎣ ⎝ x1 + x 2G ⎠ (x 2 + x1G)2 ⎥⎦ ln γ2 =
binary system H2S−H2O
⎤ ⎞2 τG G ⎥ ⎟ + 2 ⎢⎣ ⎝ x 2 + x1G ⎠ (x1 + x 2G) ⎥⎦ ⎡ ⎛
x12⎢τ ⎜
ln G = −0.2τ ;
τ=A+
B T
CO2−H2O
(13)
experimental data source Lee and Mather44 Savary et al.52 Koschel et al.43 Prutton and Savage53 Savary et al.52
τ
coefficient of determination (R2)
−0.53
0.94
−0.32
0.97
average deviation (percent) 19%
6.7%
a τ is calculated from the correlation in Table 3 for the corresponding binary system.
We validate this proposed NRTL model, applicable at high temperatures, by comparing with the experimental values at 393 K for the binary systems. This temperature is higher than the range originally used to obtain the interaction parameter correlation and helps validate the correlation. The interaction coefficient τ at 393 K, along with the thermodynamic properties listed in Table 5, are used to find the solubility values using the Gibbs free energy minimization method. Figure 4 shows a good match between the model prediction and the experimental values and hence validates the correlation for τ. The results are summarized in Table 4.
393 K (Figure 5a,c). The percentage deviation between the model prediction and the experimental value at every data point has been shown in Figure 5b,d. The errors in the experimental measurements are about 10−20% for the mixture solubilities.52 It can be seen that the percentage deviation of the majority of points for both H2S as well as CO2 lie in this experimental error range. Experimental data for the ternary mixture is not available at other temperatures, currently. As more data becomes available, a comparison between the experimental data and model prediction, using the interaction coefficients from the binary mixture data, can further help establish temperature ranges where this model is applicable. In summary, the PR EOS description for gas phase components together with the ideal aqueous solution (between 298 and 323 K) and the NRTL activity coefficient model with a temperature dependent interaction parameter τ (for temperatures greater than 323 K) can be used to predict the solubility of acid gases in water for the binary systems of H2S−H2O and CO2−H2O. This approach can also be used to obtain estimates for solubility in the complete mixture (H2S−CO2−H2O) at high temperatures and in the absence of any experimental data.
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CONCLUSION An acid gas solubility model has been developed using the Gibbs free energy minimization method for binary systems of H2S−H2O and CO2−H2O at high temperatures (298−393 K) and varying pressures (1−80 MPa). This model uses the tabulated10 thermodynamic properties at standard conditions of temperature and pressure. While any algorithm can be used to obtain equilibrium composition using the Gibbs free energy minimization approach, the RAND algorithm was observed to be 30% faster than the conventional successive substitution method for a test case of the two phase CO2−H2O system at standard conditions. The model predictions in the temperature range of 298−323 K, assuming ideal liquid phase and PR EOS for gas phase components, compare well with experimental values. The NRTL activity coefficient model with temperature dependent interaction parameters (τ) has been proposed to predict solubility at temperatures greater than 323 K. The interaction parameters for the binary systems can be further used as an estimate for CO2 and H2S solubility in the ternary system of H2S−CO2−H2O. The average deviation for the ternary model prediction with the experimental values at 393 K is 14.4% for H2S and 21.1% for CO2 in the pressure range 1−40 MPa. However, ternary mixture experimental data at different
Figure 4. (a) Comparison of experimental and model prediction for H2S solubility in the aqueous phase for the binary system H2S−H2O at 393 K. (b) Comparison of experimental and model prediction for CO2 solubility in the aqueous phase for the binary system CO2−H2O at 393 K.
Mixture Solubility. We use the interaction parameters in the NRTL activity coefficient model developed for the binary system to investigate whether it can predict solubility for the ternary mixture of CO2−H2S−H2O. Several batch experiments to measure solubility for this ternary system were reported by Savary et al.52 at 393 K. These batch experiments were performed with different initial compositions of the mixture. The predictions from the Gibbs free energy minimization model are compared with the experimental results available at 6163
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Figure 5. (a) Comparison of experimental52 and model prediction for H2S solubility in the aqueous phase for the mixture H2S−CO2−H2O at 393 K. (b) Comparison of experimental and model prediction for CO2 solubility in the aqueous phase for the ternary mixture H2S−CO2−H2O at 393 K. A solubility measurement at 39 MPa was neglected as authors report lack of confidence in the low pressure measurement.
Table 5. Thermodynamic Properties54 at Standard Conditions (T0 = 298 K and P0 = 101.325 kPa) and Used for Equilibrium Computation
temperatures are required to establish when interaction parameters, obtained from binary mixture data, can be used for predicting the ternary mixture solubility. The parameters of the solubility model presented in this manuscript can be used as input in flowsheet simulators. The solubility estimates obtained using this model can aid in designing acid gas injection schemes that are critical to producing hydrocarbons from these sour gas fields. These estimates can also be used to evaluate the storage capacity of potential aquifers for flue gases. The Gibbs free energy minimization is a unified approach and can be further extended to incorporate specific ions in the brine and model the impact of geochemical reactions on solubility.
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component H2S
CO2
APPENDIX A
In this appendix, the constants used for equilibrium computations are presented. The thermodynamic properties used for computations are presented in Table 5. The critical properties of components used to calculate fugacities of gas phase components are presented in Table 6. The binary constant parameters for CO2−H2O,55 H2S−H2O, and CO2−H2S56 were assumed constant for all equilibrium computations at different pressures and temperatures. A BIP value of 0.087 was assumed for the H2S−H2O system. This assumption was tested by choosing a different value for BIP and obtaining equilibrium compositions. While similar results were obtained, values lower than 0.03 had convergence problems at high pressures using this approach. Figure 6 shows the impact of varying BIPs on model prediction for the CO2−H2O system. As can be seen, the choice of BIP has an influence on equilibrium computations of gas phase compositions but not on the aqueous phase composition. As the values of BIPs are increased, the match
H2O
thermodynamic property Gibbs free energy of an ideal gas, G̲ IG 11 (kJ/mol) molar enthalpy in gas phase, H̲ 11 (kJ/mol) partial molar Gibbs free energy of aqueous solute, G̅ 01 (kJ/mol) partial molar enthalpy of solute in aqueous phase, H̲ 11 (kJ/mol) 3 partial volume at infinite dilution, ν̅∞ 1 (cm ) IG Gibbs free energy of an ideal gas, G̲ 11 (kJ/mol) molar enthalpy in gas phase, H̲ 11 (kJ/mol) partial molar Gibbs free energy of aqueous solute, G̅ 01 (kJ/mol) partial molar enthalpy of solute in aqueous phase, H̅ 11 (kJ/mol) 3 partial volume at infinite dilution, ν̅∞ 1 (cm ) Gibbs free energy of an ideal gas, G̲ IG (kJ/mol) 21 molar enthalpy in gas phase, H̲ 21 (kJ/mol) molar Gibbs free energy, G̅ 02 (kJ/mol) molar enthalpy of H2O in aqueous phase, H̲ 22 (kJ/mol) molar volume of H2O, ν̲2 (cm3)
−33.02 −20.15 −27.36 −39.33 34.92 −394.38 −393.51 −386.23 −412.92 32.8 −228.57 −241.81 −237.19 −285.84 19
with gas phase mole fractions is better. In this manuscript, we focus on aqueous phase composition and, hence, choose a set of BIPs listed in Table 6. The same procedure described in the manuscript can be employed to obtain NRTL parameters with a different choice of BIP for either binary system.
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APPENDIX B The activity coefficient expression for a binary system in the NRTL model has been simplified to a single variable τ, the 6164
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Table 6. Properties Used for Evaluation of Fugacity Coefficient Using PR EOS for Gas Mixturea
a
component i
critical pressure Pc (MPa)
critical temperature Tc (K)
accentric factor ω
BIP Ki,H2O
BIP Ki,CO2
H2S CO2 H2O
8.942 7.373 22.063
373.2 304.2 647.1
0.1 0.225 0.345
0.087 −0.0576 0
0.097 0 −0.0576
BIP represents binary interaction parameters.
Figure 6. Varying BIPs and the comparison between experimental data and model prediction of gas phase mole fractions (a) and (b) and aqueous phase mole fractions (c) and (d) for CO2−H2O system at 298 and 323 K.
Table 7. τ0 Values for H2S−H2O System Using Regression over Experimental Values
interaction parameter (eq 13). In this appendix, we describe how the constants A and B, which are the coefficients for the interaction parameter in Table 3, are obtained. For each binary system, we vary τ to find the particular value τ0, so that model predictions using the Gibbs free energy minimization method agree well with the available experimental data between 333 and 383 K. We use the coefficient of determination (R2) as a measure of agreement between the model prediction and the experimental values. Thus, τ0 maximizes the coefficient of determination for a data set containing experimental values of solubility for varying pressures at a particular temperature. The source of experimental data, the value of τ0 at that temperature, and the largest coefficient of determination at τ0 are given in Tables 7 and 8 for both binary systems. In order to ensure continuity, the value of τ0 at 323 K (zero because of ideal solution) has been included in the analysis. Figures 2 and 3 compare the experimental values and calculated water solubility using the ideal model prediction and the solubility predictions obtained using the τ0 values in the NRTL activity coefficient model (tuned NRTL). These τ0 values have been used to find a linear correlation with (1/T) for both the binary systems (Figure 7). Similar linear correlation models have been obtained to describe acid gas solubility in amine solutions.51 The values of the constants A and B in the linear correlation (Table 3) are obtained using the linear least-squares minimum approach.
temperature (K) 323 333 344.1 353 363 377.4
experimental data source Koschel et al.43 Lee and Mather44 Lee and Mather44 Selleck et al.42 Koschel et al.43 Lee and Mather44 Selleck et al.42
τ0
coefficient of determination (R2)
0
0.78
−0.19 −0.18 −0.36 −0.3 −0.42
0.98 0.97 0.83 0.97 0.94
Table 8. τ0 Values for CO2−H2O System Using Regression over Experimental Values temperature (K) 323
333 348 353 373 383
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experimental data source
τ0
coefficient of determination (R2)
Bamberger et al.48 Tödheide and Franck49 Wiebe47 Bamberger et al.48 Wiebe47 Bamberger et al.48 Wiebe47 Takenouchi and Kennedy57
0
0.98
−0.04 −0.1 −0.12 −0.22 −0.31
0.99 0.99 0.99 0.99 0.99
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Figure 7. (a) Regression over τ0 to get linear correlation for H2S−H2O binary system. (b) Regression over τ0 to get linear correlation for CO2−H2O binary system.
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c
AUTHOR INFORMATION
Even though different reference states are used for components CO2 or H2S depending on whether it is in the gas or aqueous phase (Table 1), the total Gibbs free energy of the system is additive because the aqueous and gas phase reference states are themselves related by
Corresponding Author
*(A.V.) E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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G̅10(T0 , T0) = G̲ 1IG + (ΔG)hydration
ACKNOWLEDGMENTS The authors acknowledge the financial support from Abu Dhabi National Oil Company (ADNOC) at The University of Texas at Austin as well as the participating companies in the Gas Flooding JIP at Penn State University. The authors also acknowledge comments from anonymous reviewers that helped improve this manuscript. L.W.L. holds the Shahid and Sharon Ullah Endowed Chair in Petroleum and Geosystems Engineering at The University of Texas at Austin. R.T.J. holds the Victor and Anna Mae Beghini Faculty Fellowship in the John and Willie Leone Family Department of Energy and Mineral Engineering and is also associated with the EMS Energy Institute at The Pennsylvania State University.
d
Coefficient of determination R2 measures the variability of a data set consisting of model predicted values (yi) and experimental values (f i) in terms of mean of the data set y ̅ and sums of squares. R2 = 1 −
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■
We can rearrange eqs 5 and 6 to get ⎛ G̅ 0 − G̲ IG ⎞ i ⎟ = fi exp⎜ i xiγi RT ⎝ ⎠
i=1
⎛ G̲ 0 − G̲ IG ⎞ i ⎟ = fi exp⎜ i RT ⎝ ⎠
i=2
̂ yi ϕiP
∑i (yi − y ̅ )2
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ADDITIONAL NOTES a For consistency, we convert reference state value at T0 and P0 from unit molality of solute in the Lewis−Randall convention (G̅ 1*) to unit molarity of the solute (G̅ 01) using the molecular weight M0 of water, 1000 G̅10 = G̅1* + RT ln M0 b
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