Loss of Methanol and Monoethylene Glycol in VLE and LLE

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Cite This: J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Loss of Methanol and Monoethylene Glycol in VLE and LLE: Prediction of Hydrate Inhibitor Partition Felipe C. Jacomel, Thales H. Sirino, Moiseś A. Marcelino Neto,* Dalton Bertoldi, and Rigoberto E. M. Morales

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Multiphase Flow Research Center (NUEM), Post-Graduate Program in Mechanical and Materials Engineering (PPGEM), Federal University of TechnologyParaná (UTFPR), 80230-901 Curitiba, Paraná, Brazil ABSTRACT: Phase equilibria of water−alcohol−hydrocarbons are important when it comes to flow assurance issues in the petroleum industry. Thermodynamic inhibitors [usually alcohols and glycols, such as methanol and monoethylene glycol (MEG)] change the thermodynamic equilibrium, thus avoiding the hydrate zone. Accurate results for the loss of the volatile inhibitor in the gas or condensate phase (partition) are of extreme importance for the oil and gas industry. In this work, a flash algorithm using the cubic-plus-association (CPA) equation of state was developed to estimate the partition of each component (water, hydrate-forming gas, and inhibitor) in any phase over a wide range of temperatures and pressures in vapor−liquid equilibrium and liquid−liquid equilibrium. Different temperaturedependent functions were optimized and evaluated for the CPA binary interaction parameters. The flash algorithm was applied to several systems with water, methane, ethane, propane, carbon dioxide, methanol, and monoethylene glycol (MEG). The results were then compared with experimental data available in the literature. The loss of methanol to the gas and/or condensate phases was satisfactorily predicted. Yet, the CPA underestimated the loss of MEG to the gas phase in a gas mixture containing carbon dioxide. The average absolute deviation for the predicted loss of methanol and monoethylene glycol ranged between 3 and 45%.

1. INTRODUCTION The formation of gas hydrates is one of the main problems faced by the flow assurance professionals in the oil and gas industry. Natural gas hydrates are crystalline compounds formed by water and gas molecules. Under high pressures and low temperatures, these water molecules rearrange themselves to form a solid structure that encapsulates a guest molecule, mainly light hydrocarbons such as methane and ethane. These crystals can agglomerate and form plugs inside the pipeline, damaging equipment, impairing the production or even stopping it completely, and consequently causing economic and safety issues. As the world demand for energy is in constant growth, the oil industry continuously seeks new horizons. The offshore exploration in ultradeep waters has been expanding over the last years and, in this scenario, the formation of hydrates became a major concern since production lines can often operate within the envelope of hydrate formation.1 The most common method used by the industry to avoid the hydrate formation is the injection of additives known as thermodynamic inhibitors, usually alcohols, glycols, and salts. Methanol and monoethylene glycol are the most widely used in the oil and gas industry. These inhibitors shift the hydrate formation envelope to regions of higher pressures and lower temperatures, thus preventing their formation in the production lines. Methanol and MEG are often injected at rates higher than the actually necessary ones due to © XXXX American Chemical Society

uncertainties in the required dosage. Methanol is relatively expensive and usually not recoverable; moreover, a large quantity of inhibitor may be required to suppress the hydrate formation temperature, thus making inhibition operations costly. MEG is used in large amounts, it is also expensive, and its regeneration is required. These facts increase the operational and capital costs as well as the demand for land or deck surface area, especially on offshore installations. Thus, optimization of the amount of inhibitor injected, as well as keeping it to a minimum, decreases environmental and operational issues, reducing costs as well.2 In petroleum production, the injection rate of a thermodynamic inhibitor is directly related to the phase equilibrium of the produced mixture (usually hydrocarbons, water, and hydrate inhibitors). When added to a mixture, an inhibitor distributes among the present phases. Hence, to determine the correct inhibition rate required, it is necessary to accurately know the fraction of the inhibitor that will be wasted into the gas or the condensate phase, since only the fraction that is effectively into the aqueous phase will contribute to the inhibition of the system. According to Tybjerg et al.,3 a difference of 10% in the predicted inhibitor distribution may cause an increase of 50% in the inhibitor quantity injected. Received: April 10, 2019 Accepted: August 6, 2019

A

DOI: 10.1021/acs.jced.9b00312 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Journal of Chemical & Engineering Data

Article

Therefore, it is understandable that reliable and precise phase equilibrium data are in constant demand in the oil and gas industry. Despite their visible importance, accurate experimental measurements related to the partition of an inhibitor into the gas or condensate phases are difficult to obtain, as they are expensive and time-consuming. Besides, experimental data available in the literature are scarce and their precision is questionable at best. This scarcity of data can be justified by the fact that the solubility of an inhibitor in the gas phase is on the order of a few parts per million (ppm), close to the limit of a gas chromatography analysis.4 Recently, some works with the aim of providing more reliable phase equilibrium data have been published. Folas et al.4 and Kruger et al.5 provided measurements for the vapor−liquid equilibrium (VLE) of a ternary system composed of MEG, water, and methane. Frost et al.6 analyzed the VLE of a system containing methanol, water, and methane. In all of these works, a thermodynamic modeling based on the cubic-plus-association (CPA) equation of state (EoS) was performed and the results were compared with experimental data. In this context, where pinpointing the amount of inhibitor required becomes important, the necessity of applying a reliable thermodynamic model capable of accurately predicting the phase behavior of systems containing hydrate-forming gases, water, and inhibitors becomes evident. Due to the importance of methanol−water−hydrocarbon systems, equilibrium data have been measured and are available in the literature.7−21 However, a few glycol−water−hydrocarbon equilibrium data are accessible.5,6,16,20 To model alcohol− water−hydrocarbons systems, since water, methanol, and glycols are hydrogen-bonded, associative models seem to be a usual choice. Several research studies using statistical association fluid theory (SAFT)-type models22−32 have been reported. Nevertheless, a smaller number of studies investigated the use of cubic-plus-association (CPA) in this application.3−5,33−35 Motivated by the scarcity of studies addressing this subject, a flash algorithm was developed and implemented in this work. The model utilizes the CPA equation of state (EoS) and the isofugacity condition to estimate both vapor−liquid equilibrium (VLE) and liquid−liquid equilibrium (LLE) of ternary and quaternary systems containing hydrate-forming gases, water, and thermodynamic inhibitors (methanol and MEG). In addition to the most common hydrocarbons present in natural gases (methane, ethane, and propane), systems composed by carbon dioxide, an acid gas capable of forming hydrates and commonly found in natural gas reservoirs, were also implemented. Additionally, different kinds of functions for the CPA binary interaction parameters were evaluated and optimized.

μiα (P , T , w1 , w2 , ..., wn) = μi β (P , T , x1 , x 2 , ..., xn) = μi γ (P , T , y1 , y2 , ..., yn )

(1)

where P is the pressure, T is the temperature, and μ is the chemical potential. The variables wi, xi, and yi are the mole fractions of component i in each of the coexisting phases. However, in certain situations, it is more convenient to use the definition of fugacity instead of the chemical potential. For an isothermal system, it is possible to ensure the chemical equilibrium through an equality of fugacities of each component (fî ) in the present phases.36,37 In this case, eq 1 can be rewritten as fi ̂

α

β γ = fi ̂ = fi ̂

(2)

The isofugacity criterion was implemented in the modeling presented in this work. The fugacities of the components were calculated using the CPA EoS. 2.1. Cubic-Plus-Association Equation of State (CPA EoS). The CPA EoS was developed by Kontogeorgis et al.38 with the aim of improving the prediction efficacy of cubic EoS in systems containing polar/hydrogen-bonding compounds. This was done by adding an association term based on Wertheim’s first-order perturbation theory. The modeling of multicomponent systems containing associative components is a difficult task and, moreover, a cubic EoS is often unable to present satisfactory results. Complex associating systems are important in many practical cases, and several of them are of interest to the oil and gas industry, especially systems composed of methanol and glycols, mostly because of their extensive use as hydrate inhibitors.39 The CPA EoS can then be expressed in terms of pressure as P=

RT a 1 RT ijj 1 ∂ln(g ) yzz − − jjj1 + z vm − b vm(vm + b) 2 vm k vm ∂(1/vm) zz{

∑ xi ∑ (1 − X A ) i

i

(3)

Ai

where a and b are two SRK parameters, the temperaturedependent energy parameter and the covolume, respectively. T is the absolute temperature, R is the universal constant of the gases, vm is the molar volume, g is the radial distribution function, XAi is the mole fraction of molecules i that are not bonded through the active site A, and xi is the mole fraction of the component i. The temperature-dependent parameter (a) derived from the SRK EoS is given by eq 4 a = a0[1 + c1(1 −

T /Tc )]2

(4)

Here, a0 and c1 are pure component parameters, T is the absolute temperature, and Tc is the critical temperature. However, to extend the CPA EoS to mixtures containing multiple components, it is necessary to apply mixing and combining rules. To achieve this, the traditional van der Waals mixing rules given by eqs 5 and 6 were used

2. THERMODYNAMIC MODELING For a system in equilibrium, from a thermodynamic perspective, the chemical potential of each component throughout the system must be uniform.36,37 In a system containing n components that are distributed into three distinct phases (α, β, γ), the equality of the chemical potential can be represented according to eq 1

a=

∑ ∑ xixjaij i

B

j

(5) DOI: 10.1021/acs.jced.9b00312 J. Chem. Eng. Data XXXX, XXX, XXX−XXX


Article

Table 1. CPA Parameters for Pure Fluids and Used Association Schemea,b fluid

association scheme

a0 (L2 bar/mol2)

b (L/mol)

c1

ε (bar L/mol)

β

H2O CH3OH C2H6O2 CH4 C2H6 CO2c

4C 2B 4C n.a. n.a. n.a.

1.2277 4.0531 10.8190 2.3204 5.5093 3.5079

0.0145 0.0310 0.0514 0.0291 0.0429 0.0272

0.6736 0.4310 0.6744 0.4472 0.5846 0.7602

166.55 245.91 197.52

0.0692 0.0161 0.0141

a

SRK pure component parameters (a0, b, c1), association energy (ε), association volume (β). bSource: Kontogeorgis and Folas.2 cSource: Tsivintzelis et al.42

b=

∑ bixi

where Bj denotes summation over all sites. The ΔAiBj term is the association strength between site A on molecule i and site B on molecule j, and it is estimated by eq 10 ÅÄÅ i A iBj y ÑÉÑ Å jε z Ñ zz − 1ÑÑÑbijβ A iBj ΔA iBj = g (vm)ÅÅÅÅexpjjj ÑÑ ÅÅ k RT z{ ÑÑÖ (10) ÅÇ

(6)

i

where xi and xj are the mole fractions of the components i and j, respectively, aij is the energy parameter of the mixture, and bi is the covolume of component i. Moreover, the classical combining rules, given by eqs 7 and 8, were also applied aij =

∑

ajai xixj(1 − kij) (7)

i,j

bij =

Here, b is the covolume parameter from the cubic part of the model, g(vm) is the radial distribution function, εAiBj and βAiBj are the association energy (well depth) and association volume (well width) between site A from the molecule i and site B from molecule j, respectively, T is the absolute temperature, and R is the universal constant of the gases. In systems composed of molecules that are able to undergo cross-association between themselves, it is necessary to implement combining rules for the association energy and volume parameters. In several studies of phase equilibria, various combining rules have been analyzed. Among the existing rules, Derawi et al.43 demonstrated that the CR-1 and the ECR (Elliot rule) combining rules have physical meaning. The geometric mean of the cross-association volume (βAiBj) is associated with the cross-entropy of the hydrogen bonding, while the arithmetic mean of the cross-association energy (εAiBj) is proportional to the enthalpy of the hydrogen bonding. Kontogeorgis et al.44 recommend the CR-1 for VLE calculations in MEG−water systems, whereas the ECR is recommended for VLE of methanol−water systems. However, both combining rules give similar and satisfactory results. In the case of heavy alcohol−water systems, the use of CR-1 is highly recommended. The latter combing rule was used in this work and is represented by eqs 11 and 12.

bi + bj (8)

2

where ai and aj are the energy parameters of the components i and j, respectively; bi and bj are the covolumes of the components i and j, respectively; kij is the binary interaction parameter between components i and j; and bij is the covolume of the mixture. On the right-hand side of eq 3, the third term is the association term of the CPA, which is similar to the one presented in the statistical association fluid theory (SAFT) developed by Chapman et al.40 All of the assumptions taken into account during the development of the SAFT EoS apply to the CPA. In the case of nonassociating compounds, the association term becomes zero and the CPA EoS reduces to the SRK one. The fundamental feature of the association term is the fraction of molecules not bonded at site A (XAi). During the calculation of this fraction, it is necessary to adopt an associative scheme in agreement with the distribution of charges and the geometry of the involved molecules.39 This scheme is used to determine the quantity and the type of the bonding sites in each molecule. Following the terminology presented by Huang and Radosz,41 a four-site (4C, two electron donors + two electron acceptors) scheme for MEG and water and a two-site (2B, one electron donor + one electron acceptor) scheme for the methanol have been adopted in this work. Carbon dioxide was modeled with two electronacceptor sites and was capable of cross-associating with selfassociating compounds (i.e., glycols, water, and alcohols) but unable to perform self-association.42 All of the other gases implemented in this work are nonassociative compounds. Once the association scheme is defined, it is possible to calculate XAi by a set of implicit equations given by X Ai =

1 vm

∑j xj ∑j X BjΔA iBj

ε A iBi + ε A jBj 2

(11)

β A iBj =

β A iBi β A jBj

(12)

where εAiBi and εAjBj are the association energies of the components i and j, respectively, and βAiBi and βAjBj are the association volumes of the components i and j, respectively. For carbon dioxide, a non-self-associating compound, a modified CR-1 combining rule was applied. This rule considers the cross-association between the gas and the other associative compounds εassociating ε A iBj = (13) 2 where εassociating is the value of the associating compound. Kontogeorgis et al.45 presented a simplified equation for the radial distribution function (g). In this new approach, a simplified hard-sphere radial distribution function form was

1 1+

ε A iBj =

(9) C



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used as proposed by Elliott et al.46 This simpler expression was adopted in this work, and it is given by eq 14 g=

1 B

1 − 1.9 4v

(14)

m

In the case of associative compounds, five pure component parameters for the implementation of the CPA EoS are necessary: three from the physical part (a0, c1, b) and two from the association part (βAiBi, εAiBi). These parameters are usually obtained by fitting the model to vapor pressure and liquid density data.39 Table 1 provides the parameters for pure fluids, and Table 2, the cross-association parameters between carbon dioxide and the other associative compounds. Table 2. Cross-Association Parameters between Carbon Dioxide and Other Associative Compoundsa gas

liquid

εcross(bar L/mol)

βcross

CO2

H2O CH3OH C2H6O2

83.28 122.96 98.76

0.0911 0.0108 0.0034

a

Source: Tsivintzelis et al.42

The purpose of utilizing the CPA EoS to perform the flash calculations presented in this work relies upon the fact that the systems studied herein are composed by molecules capable of associating between each other. In fact, it is the ability to associate with water molecules that makes alcohols and glycols hydrate inhibitors. The presence of the hydroxyl group in these compounds leads them to perform hydrogen bonding with water molecules, reducing their availability and competing directly with the hydrate formation.1 The formation of hydrogen bonds among molecules of the same kind is named self-association (e.g., in pure water, pure ethanol, pure monoethylene glycol), while the hydrogen bonding between two different molecules is called cross-association (e.g., water−methanol, water−monoethylene glycol).2 These molecular interactions affect the fugacity of the compounds directly, thus influencing the distribution and composition of the phases present in the system (partition).

3. CALCULATION METHOD A robust flash calculation algorithm, as shown in Figure 1, was developed and implemented to predict the equilibrium conditions of systems containing associative compounds. The calculations were based on the φ−φ approach and used the isofugacity concept as the criterion of convergence. The CPA EoS was utilized to determine the fugacity of each compound. At the end, the existing phases and the distribution of the components among them were known. In Figure 1, the input data of the program are temperature, pressure, and global composition of the system. After reading the initial data, the program proceeds by estimating the Ki’s, which are the phase equilibrium constants. In the first iteration, the estimation of these constants can be performed by considering an ideal behavior of the mixtures applying Raoult’s law. Once the phase equilibrium constants are determined, it is possible to calculate the fraction and the composition (xi) of each phase using the Rachford−Rice equation. It must be emphasized that either the Rachford−Rice equation or a modified version of it is derived from a mass balance and can

Figure 1. Computational algorithm’s flow chart.

be applied to several types of equilibrium (e.g., vapor−liquid, liquid−liquid, and liquid−liquid−vapor). The total volume is required to estimate the fugacity coefficient of a component in a mixture. During the calculation of the volume using the CPA EoS, it is necessary to estimate the mole fraction of molecules that are not bonded through the active site A, XAi, and, as can be observed in eq 9, this fraction depends on the molar volume of the mixture. As an initial guess, the volume is calculated utilizing the SRK EoS, and with this volume, the XAican be estimated. Using the volume calculated by the SRK EoS and the estimated values of XAi, a D



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Table 3. Fitted Binary Interaction Parameters (kij)a binary system CH4 + H2O

temperature range (K)

kij (T)

number of points 120

C2H6 + H2O

274.26−444.26 141

C3H8+ H2O

277.62−368.16 67

CO2 + H2O

274.14−473.15 151

CH3OH + H2O

308.15−373.15 111

C2H6O2 + H2O

343.10−363.20 38

CH4 + CH3OH

273.15−330.00 84

CH4 + C2H6O2

273.25−423.15 120

C2H6 + CH3OH

240.00−303.20 51

CO2 + CH3OH

230.00−477.60 343

C3H8 + C2H6O2

CO2 + C2H6O2

298.15−398.15

35

288.15−398.15

138

overall

references

−4.1957 × 10−08 × T2 -4.2012 × 10−08 × T −1.2600 × 10−05 +2.3478 × 10−06 × T2 -2.9213 × 10−04 × T −6.4975 × 10−02 +6.4929 × 10−07 × T2 +6.4830 × 10−07 × T +1.9456 × 10−04 +1.3035 × 10−06 × T2 +1.8971 × 10−06 × T +4.9454 × 10−04 −9.7970 × 10−07 × T2 -5.1062 × 10−07 × T −4.4259 × 10−05 −4.6943 × 10−07 × T2 -4.6935 × 10−07 × T −1.4081 × 10−04 +4.5160 × 10−07 × T2 +4.5196 × 10−07 × T +1.3555 × 10−04 −8.1066 × 10−07 × T2 +4.6025 × 10−04 × T +1.0074 × 10−01 +5.3861 × 10−07 × T2 +5.3888 × 10−07 × T +1.6165 × 10−04 +4.7546 × 10−07 × T2 +4.7601 × 10−07 × T +1.4275 × 10−04 −2.6428 × 10−07 × T2 +2.4140 × 10−04 × T + 5.1903 × 10−02 −4.5008 × 10−07 × T2 +2.6647 × 10−04 × T +5.7070 × 10−02

274.19−323.56

6, 50−57

50, 58−64

65, 66

67−72

73−75

76

50, 77, 78

4, 50, 79−81

50, 82−85

86−96

97

79, 98, 99

1399

a

Binary systems optimized, temperature range, number of experimental points, and references of the experimental data used.

new volume utilizing the CPA EoS is calculated. With this new volume, it is possible to recalculate the values of XAi and subsequently another volume using the CPA EoS. This iterative process continues until a convergence criterion, based on the difference of the calculated volumes, is satisfied. Once the convergence criterion based on the volume is satisfied, the fugacity coefficients of every component in each phase are calculated with the CPA EoS following the methodology presented by Michelsen and Hendriks.47 Finally, the fugacities are estimated and the isofugacity criterion can be tested. If the criterion is not satisfied, new values of Ki’s are calculated using the fugacity coefficients and the program returns to the third step. Once the isofugacity criterion is attained, the mole fraction of each component in each phase is printed. The flash algorithm presented in Figure 1 was applied to various systems containing hydrate-forming gases, water, and thermodynamic inhibitors over a large range of temperatures and pressures. The results obtained were compared to experimental data available in the literature, and the absolute average deviations (AAD) were calculated according to eq 15

1 AAD = n

ij ψ cal − ψ exp yz ∑ jjjjj i exp i zzzzz ψi i k { n

(15)

where n is the number of experimental points and Ψ stands for the molar fraction of component i in the respective phase. Another important parameter to evaluate the model used in this work is the number of occurrences where it fails completely %UE =

N%AAD > 20% Ndata

(16)

where %UE stands for the percentage of unreliable estimates, N%AAD>20% is the number of absolute average deviations (AAD) higher than 20%, and Ndata is the quantity of data estimated by the model.

4. MODEL PARAMETERS’ ESTIMATION To adequately describe the properties of a mixture or predict the phase equilibrium of a complex multicomponent system utilizing an equation of state, it is usually necessary to use E



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Table 4. Summary of the Phase Equilibrium Conditions and Absolute Average Deviations (AAD) for the Systems Investigated in This Worka %AAD T-range (K)

P-range (MPa)

x

y

%UEx

%UEy

0.03 37.72 0.89 0.20 21.03 0.20 5.91 21.35 6.02 0.35 33.03 10.66 1.85 0.26 48.54 41.54 0.31

0 100 0 0 60 0 5 55 5 0 67 0 0 0 100 100 0

6

24.14 0.02 3.42 9.27 0.72 16.52 31.66 0.04 16.19 43.43 5.45 26.82 19.42 48.88 1.88 16.71 11.78 25.16

100 0 0 15 0 36 67 0 33 100 0 67 33 100 0 0 0 50

6

14.31 29.07

system

components

region

I

H2O CH4 CH3OH H2O CH4 C2H6O2 H2O CO2 CH3OH H2O C2H6 CO2 CH3OH H2O CH4 CO2 CH3OH H2O CH4 C3H8 C2H6O2 H2O CH4 CO2 C2H6O2

VLE

280.25−313.45

5.14−13.12

VLE

278.15−323.19

5−20

20

VLE and LLE

243.15−298.15

0.16−9.25

22

VLE

243.15−298.15

0.16−9.25

3

VLE

284.15

6.89

1

VLE

283.15−310.95

6.81−20.70

VLE

283.15−310.95

6.81−20.70

II

III

IV

V

VI

VII

number of points 9

overall

67

10.28

45.02 13.00

33 67

23.31

references 6

4, 5

100, 101

102

16

16

16

67 29.84

a

Types of equilibria found in a range of pressure and temperature, number of occurrences where CPA fails completely (%UE).

herein are formed by hydrate-forming gases, water, and hydrate thermodynamic inhibitors. In the case of a binary system composed of a gas and a self-associating compound (e.g., water, methanol, and monoethylene glycol), the property (ψ) assumes the value of the molar fraction of the gas in the liquid phase (solubility). If the binary system is constituted by a thermodynamic inhibitor and water, the property (ψ) is the molar fraction of the inhibitor in the vapor phase. The binary interaction among gases was disregarded. Results for the three different expressions (eqs 17−19) were compared to experimental equilibrium data available in the literature. On average, during the modeling of the multicomponent systems presented in Section 5, the lowest deviations were obtained when a quadratic dependence with respect to temperature (eq 19) was considered. Thence, the results presented in this work were calculated using this quadratic approach as displayed in Table 3. As pointed out in the literature, for cubic equations of state, the binary interaction parameter decreases with carbon number for CO2−alkane systems (data available up to C44) and there is a quadratic temperature dependency.49 For light alkanes, the temperature dependence is also quadratic.

adjustable binary interaction parameters (kij) in the mixing rules, as can be observed in eq 7. In fact, an accurate prediction of the phase equilibria of mixtures is highly dependent on these parameters and on the mixing rules. A priori, the binary interaction parameters cannot be obtained theoretically; however, they can be regressed from VLE experimental data.48 In this work, kij values are temperature-dependent. Three different expressions were tested and adjusted to VLE experimental data available in the literature. The three different functions evaluated for the kij are represented in eqs 17−19.

kij = aT + b

(17)

kij = a /T + b

(18)

kij = aT 2 + bT + c

(19)

where kij is the adjustable binary interaction parameter; T is the absolute temperature; and a, b, and c are the adjustable coefficients. The objective function presented in eq 20 was minimized for the tuning of the binary interaction parameters ij ψ cal − ψ exp yz Fob j = ∑ jjjj i exp i zzzz j z ψi i k { n

2

5. RESULTS AND DISCUSSION Table 4 presents a summary of the results obtained in this work. As one can see, the systems are composed mainly of hydrocarbons (methane, ethane, and propane) and associative compounds (water, methanol, and MEG). The presence of systems containing carbon dioxide is observed. Carbon dioxide is a hydrate-forming acid gas frequently present in the

(20)

where n is the number of experimental points and the subscript i indicates the compound. Superscripts cal and exp stand for calculated and measured property ψ, respectively. Since hydrates are the focus of this work, the systems investigated F



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Figure 2. Experimental and predicted VLE for a water, methane, and methanol system at 280.25, 298.77, and 313.15 K.6

280.25, 298.77, and 313.15 K. All of these molar fractions have the same order of magnitude, and the solubility of methane in the solution of methanol and water is greater than that presented by methanol and water in the vapor phase. These magnitudes demonstrate the difficulty in accurately modeling and measuring these values experimentally. According to Frost et al.,6 it is not surprising that different research groups in similar conditions display distinct measured data for the water content due to the well-known challenges associated with measuring trace amounts of water in gases, and it is safe to assume that the same difficulties apply to the other compounds. As expected, water exhibited a lower gas molar fraction in comparison to methanol due to its lower volatility. In this case, methane solubility exhibited the highest AAD (37.72%), whereas the lowest AAD was presented by methanol (3.42%). Deviation calculated for the molar fraction of water was 24.14%. Good agreement between experimental and correlated partitions of methanol in the vapor phase was achieved. The 2B association scheme for methanol has been adopted in this work. According to Kontogeorgis and Folas,2 the two association schemes (2B and 3B) are typically used for alcohols returning similar results, though the performance is slightly better once the simpler 2B association scheme is used. Thus, no attempt has been made to change the 2B scheme for methanol in this work. To give emphasis to the precision of the model in predicting the solubility of methanol in the gas phase, Figure 3

composition of natural gases, and it can perform crossassociation with other associative compounds. This behavior makes it an interesting gas to evaluate the modeling capacity of an associative equation of state. AAD, %UE, and experimental references are also shown in Table 4. As it can be observed, most of the systems investigated in this study exhibited a vapor−liquid equilibrium (VLE) in the temperature and pressure ranges evaluated. The only exception is system III that exhibited a liquid−liquid equilibrium. In the case of a vapor−liquid equilibrium, x refers to the molar fraction in the liquid phase and y to the molar fraction in the vapor phase. For the liquid−liquid equilibrium, y was also used to represent the molar fraction in the condensed gas-rich phase. Analyzing the results, it is noted that the highest deviations were obtained for the gas solubility in the liquid phase (water + thermodynamic inhibitor) and for the molar fraction of water in the vapor phase. Some of the systems described herein will be presented in detail, focusing on the partition of the thermodynamic inhibitors in the vapor phase due to its importance and wide use in the oil and gas industry as a method of preventing the hydrate formation. System I from Table 4 is composed of 0.250 mol % methane (1), 0.463 mol % water (2), and 0.287 mol % methanol (3). Results obtained for this system are shown in Figures 2−5, with the experimental data taken from the work of Frost et al.6 Figure 2 shows the solubilities of methane in the liquid polar phase and those of the water and methanol in the gas phase at G



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Figure 3. Experimental and predicted mole fractions of methanol in the gas phase in VLE for a water, methane, and methanol system at several temperatures.6

Figure 4. Experimental and predicted mole fractions of methanol in the gas phase in VLE for a water, methane, and methanol system at various temperatures using different functions for binary interaction parameters.6

demonstrates the calculated molar fraction for the three different temperatures (280.25, 298.77, and 313.15 K). One can observe that the modeling performed accurately with an average AAD of 3.41% and, as expected, showed that the molar fraction increases with the temperature. The highest (5.84%) and the lowest (1.83%) deviations were presented by the highest and the lowest temperatures, respectively. According to Kontogeorgis and Folas,2 the influence of the binary interaction parameters, kij, on the prediction of the alcohol partition for methanol−water−alkane systems revealed that the correct representation of only two binary systems, alkane−methanol and water−methanol, is crucial for a satisfactory correlation of the ternary systems. In accordance with Table 3, in this work, three binary systems were considered for system I: methane−methanol, water−methanol, and water−methane. As presented in Section 4, three different temperaturedependent functions were evaluated for the binary interaction parameters. To demonstrate the influence of these different approaches, Figures 4 and 5 show a comparison of the results obtained using each expression. In Figure 4, it is possible to analyze the influence of the binary interaction parameters on the solubility of methanol in the gas phase for system I. As it can be observed, the discrepancy among the results provided by the different functions is not high. The linear temperaturedependent binary interaction parameter provided better results for all temperatures with an AAD of 1.95%. In general, the quadratic approach presented the second-best performance with an AAD of 3.42%. The only temperature for which the quadratic approach did not perform better than the inverse approach was the highest one (313.45 K). The AAD calculated for the inverse approach was 4.07%. It is also important to emphasize that even with the linear function presenting better results for the partitioning of methanol into the vapor phase for system I, in general taking all of the systems investigated in this work (I−VII) into account, the quadratic expression performed better. For this reason, the quadratic expression was selected to provide the results exhibited in this work. Figure 5 presents the estimated values of methane solubility for system I considering the three different types of binary interaction parameters. On analyzing this plot, the influence

Figure 5. Experimental and predicted mole fractions of methane in the liquid phase in VLE for a water, methane, and methanol system at several temperatures using different functions for binary interaction parameters.6

and importance of the binary interaction parameters in the calculation of equilibrium conditions become more evident. There is a significant deviation in the results displayed by the linear and inverse functions compared to the quadratic approach, a fact not observed in Figure 4. The AAD exhibited by the quadratic binary interaction parameter was 34.2%, while the linear and inverse approaches presented ADDs of 43.04 and 43.74%, respectively. System II from Table 4 presents two different sources of experimental data. The data taken from Folas et al.4 is composed of 0.1925 mol % methane (1), 0.6220 mol % water (2), and 0.1855 mol % MEG (3). In Figure 6, the water and MEG content in the gas phase for a temperature of 278.15 K is shown. The concentration of water in the gas phase is higher than that of MEG, in contrast to that observed in Figure 2 for methanol. The fact that methanol is more volatile than MEG causes a greater loss of inhibitor to the gas phase, making it H



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Figure 8 also presents the MEG content in the vapor phase for different temperatures and pressures. Similarly to Figure 7, the model presented a less accurate performance for points with higher pressures. The CPA model presented a satisfactory qualitative prediction of the experimental data for the lower pressure (AAD = 5.83%). The absolute average deviation calculated for the molar fraction of MEG at the higher pressure was 18.69%. Although CPA can fairly represent the partition of MEG for this system, models like SRK (using van der Waals mixing rules) usually overestimate the amount of glycols in the gas phase by almost an order of magnitude. Oppositely, SRK with advanced mixing rules [Huron−Vidal (HV) types using a modified nonrandom two-liquid (NRTL) activity coefficient] can provide reasonable prediction of the methanol partition as well. However, more interaction parameters are required in NRTL to represent the temperature dependence of phase behavior.2 There are a few articles available in the literature that present a comparison between the CPA EoS and the SRK EoS with the Huron−Vidal (HV) mixing rule.4,103,104 Folas et al.4 concluded that for multicomponent systems both models provided satisfactory predictions of water and MEG solubility in the gas phase; however, the CPA EoS demonstrated a superior predictive performance even using only one binary and temperature-independent interaction parameter. As previously highlighted in Section 1, there are a number of difficulties related to the quantification of the inhibitor content in the gas phase. Therefore, precise measurements can be difficult to achieve and only a few sources present this type of data. Having reliable experimental data is essential in the validation of thermodynamic models. For this reason, it is fundamental to compare the experimental measurements available in the literature. At a first sight, this type of investigation could be done for systems II and III. However, this analysis is not possible due to the fact that the experiments were carried out under different conditions (feed, temperature, and pressure) that directly affect the thermodynamic equilibrium. Although it is not possible to perform a comparative analysis between the experimental data of different studies, some of these works provide the experimental uncertainties. For system I, Frost et al.6 reported uncertainties of 0.05 for the molar fraction of methane in the liquid phase and methanol and water in the gas phase. Regarding system II, Folas et al.4 assert that the greatest uncertainties are presented in the points of lower temperatures. The accuracy of methane solubility is 5% in the worst case. The maximum uncertainty of the water content measurements in the gas phase is 5%, while for MEG, solubility in the gas phase is 25%. Continuing with system II, Kruger et al.5 reported a maximum uncertainty of 0.000318 for the molar fraction of methane in the liquid phase. In the worst scenario, the uncertainty values of the MEG content in the gas phase are 1.01 and 59.4 ppm for water. All of the results presented so far were related to ternary systems. The following system (IV from Table 4) to be analyzed is composed by a quaternary mixture containing 0.0168 mol % ethane (1), 0.0056 mol % carbon dioxide (2), 0.8571 mol % water (3), and 0.1205 mol % methanol (4). Experimental data were taken from Ng and Chen,102 and the comparison with the thermodynamic modeling is shown in Tables 5 and 6.

Figure 6. Experimental and predicted VLE for a water, methane, and MEG system at 278.15 K.4

more difficult to recover. For that reason, in a gas-dominant system, MEG is often preferred over methanol. AADs calculated for the molar fraction of MEG were 16.74 and 3.49% for water. Once again, turning the attention to the partition of the inhibitor in the gas phase, Figure 7 shows the molar fraction of

Figure 7. Experimental and predicted mole fractions of MEG in the gas phase in VLE for a water, methane, and MEG system at various temperatures.4

MEG in the gas phase for two distinct temperatures (278.15 and 298.15 K). The model presented a less precise performance for points with higher pressures, especially for the higher temperature (AAD 40.49%). The AAD was 23.81%. In system II, the feed used by Kruger et al.5 ranged from 0.31 to 0.48 mol % methane, 0.02 to 0.18 mol % water, and 0.41 to 0.65 mol % MEG. Figure 8 evaluates the effect of temperature and pressure on water solubility in the vapor phase. It can be observed that the vapor water content increases roughly exponentially with the temperature increase. A decrease of approximately 40% in water solubility in the gas phase is observed with the increase of the pressure from 6 to 12.5 MPa. The CPA model provided a good description of the data with an AAD of 7.83%. I



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Figure 8. Experimental and predicted mole fractions of water and MEG in the gas phase in VLE for a water, methane, and MEG system at various temperatures and pressures.5

Table 5. Experimental and Predicted Molar Compositions of the Liquid Phase in VLE of a Water, Ethane, Carbon Dioxide, and Methanol System102

exp. model exp. model exp. model

temperature (K)

pressure (MPa)

ethane in liquid × 103

270.93

1.14

275.76

2.70

280.85

3.60

0.9800 1.3328 1.6300 2.3757 1.8500 2.1706

carbon dioxide in liquid × 103

%AAD 36.00

1.7700 1.5664 3.8100 3.8435 10.2100 8.2102

45.75 17.33

%AAD

methanol in liquid

11.50 0.88 19.59

%AAD

0.1299 0.1229 0.1226 0.1225 0.1217 0.1219

5.41 0.09 0.13

water in liquid 0.8673 0.8742 0.8719 0.8713 0.8661 0.8678

%AAD 0.79 0.07 0.19

Table 6. Experimental and Predicted Molar Compositions of the Gas Phase in VLE of a Water, Ethane, Carbon Dioxide, and Methanol System102

exp. model exp. model exp. model

temperature (K)

pressure (MPa)

ethane in gas

270.93

1.14

275.76

2.70

280.85

3.60

0.8748 0.9109 0.8265 0.8876 0.7708 0.8081

%AAD

carbon dioxide in gas

4.14 7.39 4.84

0.1239 0.0879 0.1729 0.1116 0.2267 0.1904

%AAD

methanol in gas × 103

29.06

%AAD

0.6200 0.7111 0.5700 0.4709 1.5600 1.1517

35.41 15.99

14.70 17.39 26.17

water in gas × 103 0.7200 0.4346 0.4100 0.2653 0.7800 0.3482

%AAD 39.63 35.29 55.35

Table 7. Experimental and Predicted Compositions of the Aqueous Solution in LLE of a Water, Carbon Dioxide, and Methanol System101

exp. model exp. model

temperature (K)

pressure (MPa)

carbon dioxide in aqueous solution

278.50

8.72

270.75

9.25

0.0315 0.0382 0.0360 0.0513

%AAD 21.30 42.42

Table 5 shows molar concentrations in the liquid phase. Ethane solubility exhibited the highest deviations, while carbon dioxide solubility presented lower deviations. As previously explained, the molecules of carbon dioxide can associate with the molecules of the solvent (water and methanol), making its solubility greater than that of a hydrocarbon. This ability was taken into account during the calculation, and both the associative scheme and the parameters utilized for this compound are presented in Section 2. The solubility of carbon dioxide in water is, in some cases, 2 orders (or more) of magnitude higher than that of the ethane. This is a physical

methanol in aqueous solution 0.0553 0.0561 0.1120 0.1168

%AAD 1.50 4.26

water in aqueous solution 0.9132 0.9057 0.8520 0.8320

%AAD 0.83 2.35

indication of the importance of solvation for such systems. Solvation implies that carbon dioxide does not self-associate but is able to solvate with water and methanol. On the other hand, Table 6 shows the gas phase. Once again, the highest deviations are related to the mole fraction of water in the vapor phase, which is even smaller than the solubility of ethane in the liquid phase. Results obtained for methanol in the gas phase presented a better agreement with experimental data. It is known that low temperatures and high pressures are usually required for hydrate formation and that in certain J



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Table 8. Experimental and Predicted Compositions of the Condensate Gas Phase in LLE of a Water, Carbon Dioxide, and Methanol System101

exp. model exp. model

temperature (K)

pressure (MPa)

carbon dioxide in condensate

278.50

8.72

270.75

9.25

0.9956 0.9950 0.9936 0.9941

%AAD 0.06 0.05

methanol in condensate × 103 2.3900 1.9978 4.7500 3.4528

%AAD 16.41 27.31

water in condensate × 103 2.0600 3.0014 1.6200 2.4055

%AAD 45.70 48.49

Figure 9. Experimental and predicted mole fractions of water and MEG in the gas phase in VLE for a water, methane, propane, and MEG system at 283.15 and 310.95 K.16

Figure 10. Experimental and predicted mole fractions of water and MEG in the gas phase in VLE for a water, methane, carbon dioxide, and MEG system at 283.15 and 310.95 K.16

the gas phase for 310.95 K. AAD calculated for the molar fraction of water was 25.16%. AAD calculated for the molar fraction of MEG was 14.31%. The good performance of CPA for systems like this, which contain some of the main type of compounds of a typical petroleum mixture (water, gases, and a hydrate inhibitor), is important for practical applications. The accurate prediction of the partitioning of MEG between the liquid and gas phases for such a mixture is very important for the oil industry. Figure 10 shows system VII composed of 0.3007 mol % methane (1), 0.0334 mol % carbon dioxide (2), 0.5161 mol % water (3), and 0.1498 mol % MEG (4). AAD calculated for the molar fraction of water was 29.07%. AAD calculated for the molar fraction of MEG was 45.02%. MEG concentration in system VII is higher than in system VI, in a likely

conditions some of the hydrate-forming gases may liquefy. The following results address exactly this scenario. Tables 7 and 8 show the liquid−liquid equilibrium (LLE) data for a system (III from Table 4) containing carbon dioxide, methanol, and water. Experimental points shown in Table 8 allow one to observe that the molar fraction of water in the condensed carbon dioxide-rich phase is smaller than the molar fraction of methanol. The solubility of water in the condensate showed the highest deviations, followed by the molar fraction of carbon dioxide in the aqueous phase. System VI is composed of 0.3007 mol % methane (1), 0.0334 mol % propane (2), 0.5161 mol % water (3), and 0.1498 mol % MEG (4). Experimental data were obtained from Ng and Chen.16 Figure 9 shows the water content in the gas phase for 283.15 and 310.95 K and the MEG content in K



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adjustable parameter for mixtures. Poor results can also be obtained for some cases.105

demonstration that the existence of carbon dioxide increases the loss of MEG to the gas phase. The cross-association between carbon dioxide and MEG might be the reason for this increase of solubility of MEG in the gas phase. However, it is worth pointing out that the CPA underpredicts the MEG concentration in system VII. In general, the model described in this work showed a good performance, given the inherent difficulty in predicting traces of water content and thermodynamic inhibitors in the gas or condensate phase. Results showed a good agreement with experimental data, especially regarding the partition of methanol and MEG into the gas phase (systems I and II), which is one of the most critical and important data for the oil and gas industry due to its wide use in large amounts as a tool in prevention of hydrate formation. However, it was found that CPA underpredicted the loss of MEG to the gas phase in a gas mixture containing carbon dioxide. The cross-association between an acid gas and MEG might be the reason for this increase of the solubility of MEG in the gas phase. In most cases, the model tends to overestimate the solubility of the gas in the liquid phase. It is important to point out that the presence of systems containing carbon dioxide and hydrogen sulfide (hydrateforming acid gases frequently present in the composition of natural gases), which can perform cross-association with other associative compounds (hydrate inhibitors), should be investigated in greater detail to evaluate some possible limitations of CPA. Despite the satisfactory performance exhibited by the CPA EoS in this work, association theories present theoretical and practical limitations and there are many unanswered questions. It is possible to see a detailed discussion regarding this subject in the work reported by Kontogeorgis.105 Only a few of these problems will be highlighted in this paper. One topic that requires special attention is the modeling of systems containing acid gases, such as CO2. Carbon dioxide is in reality a quadrupolar molecule (not a self-associating molecule), and the model does not take this type of interaction into account. However, these quadrupolar interactions can be implicitly accounted for via binary interaction parameters. Due to this fact, for highly polar and quadrupolar systems (e.g., CO2 with hydrocarbons), nonzero binary interaction parameters must be always implemented to obtain satisfactory results.105 During the modeling of the vapor−liquid equilibrium of CO2−alkanes systems, Tsivintzelis et al.106 reported better predictive results (without binary interaction parameters) when considering CO2 as a self-associative compound. Although there is no experimental evidence for strong self-associating interactions between CO2 molecules, the addition of self-association effects in the modeling may indirectly and effectively account for quadrupolar interactions. Regardless of the modeling approach, very similar results were obtained when interaction parameters were used. According to Tsivintzelis et al.,106 higher values of binary interaction parameters are necessary when CO2 is treated as a nonassociating fluid. Another problem related to associative equations of state is related to the description of cross-associating systems. In numerous systems, the interactions are very complex and they are only partially understood. In certain cases, the limited understanding is indicated by the use of high values of interaction parameters to obtain accurate results. Another indication is the need of implementing more than one

6. CONCLUSIONS The objective of this work was to implement a thermodynamic model capable of predicting hydrate inhibitor partition. CPA EoS was utilized to calculate the vapor−liquid equilibrium (VLE) and liquid−liquid equilibrium (LLE) of ternary and quaternary systems containing hydrate-forming gases (methane, ethane, propane, and carbon dioxide), water, and thermodynamic inhibitors (methanol and monoethylene glycol). Different approaches for the EoS binary interaction parameter were also implemented and tested. Results were compared with experimental data available in the literature and proved to be satisfactory for the loss of methanol and MEG to gas and/or condensate phases, especially due to the difficulty associated with predicting traces of water content and thermodynamic inhibitors in the gas and condensate phases. Yet, CPA underpredicted the loss of MEG to the gas phase in a gas mixture containing carbon dioxide. The average absolute deviation for the prediction of loss of methanol and monoethylene glycol ranged between 3 and 45%. The authors also realized that more experimental data are needed to validate the thermodynamic models.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone/Fax: +55 41 3279 6521. ORCID

Moisés A. Marcelino Neto: 0000-0001-5492-6640 Rigoberto E. M. Morales: 0000-0003-3297-7361 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge the financial support from the Brazilian National Research Council CNPq, ANP, and FINEP through the Human Resources Program to Oil and Gas segment PRH-ANP (PRH 10UTFPR) and from TE/ CENPES/PETROBRAS.

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NOMENCLATURE a,SRK attractive energy parameter (Pa m6 mol−2) aw,activity of water AAD,average absolute deviation (%) b,SRK covolume parameter (m3 mol−1) ci,CPA pure component parameter f i,fugacity of species i (Pa) g,radial distribution function kij,binary interaction parameter P,pressure (Pa) R,universal constant of gases [≈8.31451] (J mol−1 K−1) T,temperature (K) Tc,critical temperature (K) vm,molar volume (m3 kmol−1) x,mole fraction in the liquid phase y,mole fraction in the gas phase

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GREEK LETTERS ΔAiBj,association strength (Pa m3 mol−1) εAiBj,association energy (Pa m3 mol−1) βAiBj,association volume

L



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μ,chemical potential (J mol−1)

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SUBSCRIPTS AND SUPERSCRIPTS A,association site A B,association site B cal,calculated exp,experimental i, j, l,component 1, 2,component

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