Compositional Analysis and Hydrate ... - ACS Publications

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Compositional Analysis and Hydrate Dissociation Conditions Measurements for Carbon Dioxide þ Methane þ Water System§ Veronica Belandria,† Ali Eslamimanesh,† Amir H. Mohammadi,*,‡ Pascal Theveneau,† Herve Legendre,† and Dominique Richon† † ‡

nergetique et Procedes, 35 Rue Saint Honore, 77305 Fontainebleau, France MINES ParisTech, CEP/TEP—Centre E Thermodynamics Research Unit, School of Chemical Engineering, University of KwaZulu-Natal, Howard College Campus, 6 King George V Avenue, Durban 4041, South Africa ABSTRACT: A detailed description of an experimental setup based on the “static-analytic” technique with gas phase capillary sampling designed, built, and improved “in-house” to measure phase equilibria (pressure, temperature, and compositions) under gas hydrate formation conditions is presented in this work. The apparatus is suitable for measurements at temperatures ranging from 233 to 373 K and pressures up to 60 MPa. It was used to study phase equilibria in the carbon dioxide þ methane þ water system under hydrate formation conditions. An isochoric pressure-search method was used to measure hydrate dissociation conditions. The experimental data have been compared successfully with the literature data. The compositions of the gas phase in equilibrium with the hydrate and aqueous phases were measured using a gas chromatography technique and compared successfully with the literature data. The compositions of the hydrate and aqueous phases were determined by applying material balance equations. The experimental data on the compositions of the hydrate have been compared successfully with the literature data. To solve the latter equations, the Newton’s numerical method coupled with the differential evolution optimization strategy was employed. All the aforementioned experimental data (hydrate dissociation conditions þ composition analyses) have been compared with the predictions of two thermodynamic models, namely CSMGem and HWHYD. A discussion is made on the reliability of the predictions of the latter models.

1. INTRODUCTION Gas hydrate technology is a reversible approach in which pressurized gas and water combine to form a solid, called gas hydrate or clathrate hydrate.1 In gas hydrates, the gas molecules are trapped in water cavities that are composed of hydrogenbonded water molecules.1 Considerable research has been devoted in the last decades to examine potential industrial applications of gas hydrate technology.1-4 Examples are natural gas processing, storage and transportation, carbon dioxide (CO2) capture from industrial/ flue gases, CO2 sequestration, steam reforming processes, hydrogen (H2) storage, water desalination, etc. Thermodynamic models based on accurate experimental equilibrium data are needed to reliably predict hydrate thermodynamic properties for potential industrial applications. As most of the existing models have been developed for hydrocarbon systems, model parameters must be reconsidered for hydrates containing carbon dioxide using reliable phase equilibrium data.1-5 In addition, any deviation in the measurement of hydrate phase equilibrium properties will lead to significant errors in predictions of the model. Consequently, measuring accurate experimental data on the phase behavior of mixed clathrate hydrates containing CO2 is of importance. A number of experimental devices and methods for measuring hydrate phase equilibrium of various systems reported in the literature have been reviewed by Sloan and Koh.1 Traditionally, the experimental methods involve measuring hydrate phase boundaries.1 Experimental determination of the compositions of the existing phases in equilibrium with gas hydrate by the pVT method is generally not easy. Some of the possible technical difficulties are long metastable periods, ineffective agitation, difficulties to sample the phases without disturbing the r 2011 American Chemical Society

thermodynamic equilibrium, lack of complete visibility inside the equilibrium cell, plugging of sampling valves, etc. All of the above can be considered in the design of an experimental apparatus for hydrate phase equilibrium measurements. Although a significant amount of research has been carried out for mixed hydrates of carbon dioxide and methane,3-10 the compositions of the gas, hydrate, and fluid phases reported in the literature are still limited. A summary of the experimental conditions at which literature data have been reported is given in Tables 1 and 2. Recently, we have been exploring an alternative and possibly more efficient way to estimate the compositions of the existing phases in equilibrium with gas hydrates. The approach is based on the “static-analytical” method12 with capillary gas phase sampling. For this purpose, we have designed and constructed an experimental setup, suitable for measurements in the temperature range of 233-373 K and compatible with corrosive gases and materials. The core of the apparatus is an equilibrium cell entirely designed, built, and improved in-house that can withstand pressures up to 60 MPa. A major advantage of this apparatus is the combination of precise and accurate measuring devices and the visual observation of the gas hydrate formation and phase behavior. An isochoric pressure-search method13-15 is used to determine the hydrate temperature and pressure dissociation conditions. For composition measurement of the gas phase under hydrate formation Received: September 23, 2010 Accepted: February 14, 2011 Revised: February 5, 2011 Published: March 24, 2011 5783

dx.doi.org/10.1021/ie101959t | Ind. Eng. Chem. Res. 2011, 50, 5783–5794

Industrial & Engineering Chemistry Research

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Table 1. Experimental Data for the Dissociation Conditions of the Binary Clathrate Hydrates of Carbon Dioxide þ Methane dissociation condition ranges (T and p)

authors Adisasmito et al.2

range of mole fraction in gaseous mixture

273.7-287.6 K

number of experimental data

0.08-0.85

41

0.25-0.77

3

0.264-0.730

11

2.52-10.95 MPa Hachikubo et al.4

271.0 K 1.27-2.02 MPa

Belandria et al.5

279.1-289.9 K 2.96-13.06 MPa

Unruh and Katz6

275.5-285.7 K

0.055-0.71

17

Fan and Guo7

1.99-7.0 MPa 273.5-282.3 K

0.9652

9

0-1.0

17

0.206-0.744

9

1.10-4.80 MPa Seo et al.8

272.66-283.56 K 1.5-5.0 MPa

this work

277.9-285.5 K 2.72-8.27 MPa

Table 2. Experimental Data for the p, T, and Compositions of the Gas, Hydrate, and/or Aqueous Phases under Lw-H-G Equilibrium Conditions for the Binary Clathrate Hydrates of Carbon Dioxide þ Methane authors 3

T and p ranges

measurement of mole fraction of CO2

number of experimental

in the gas (y), hydrate (z), and/or aqueous (x) phases

data

Ohgaki et al.

280.3 K

y, z

31

Seo et al.8

3.04-5.46 MPa 274.36-283.56 K

y, z

26

y

22

y

12

1.5-5.0 MPa Beltran and Servio9

275.14-285.34 K 2.36-7.47 MPa

Bruusgaard et al.10

274.02-280.05 K 1.66-4.03 MPa

Uchida et al.11

258-274.1 K and 190 K

y, z

a

this work

0.5-3 MPa and 0.1 MPa 273.6-284.2 K

y, x, z

41

1.510-7.190 MPa a

Cannot be determined from the original article since the data are reported as curves.

conditions, a ROLSI sampler16 was installed and connected online to a gas chromatograph. The compositions of the liquid and hydrate phases are calculated using the material balance approach proposed by Ohgaki et al.3 in combination with the experimental data and the volumetric properties evaluated from equations of state for gas mixtures. The proposed nonlinear material balance equations are solved using the Newton’s numerical method17 coupled with the differential evolution (DE) optimization strategy.18 In this work, we describe the characteristics of the aforementioned unique measurement system and demonstrate its effectiveness and capabilities.

2. EXPERIMENTAL SECTION 2.1. Apparatus. With the aim of developing a flexible apparatus suitable for gas hydrate phase equilibrium measurements, combinations of new and adopted features from the literature were considered in the design and construction of the apparatus presented in this work. The main characteristics of this setup are suitability for corrosive fluids (high pressures), strong agitation, precise and accurate measuring devices, compositional analysis of the gas phase, and visual observation inside the equilibrium cell during the course of the experiment. A schematic diagram of this apparatus is shown in Figure 1. The experimental setup consists of four main sections: an

equilibrium cell, a sample supplying system, a composition analyzing system, and a pressure-temperature measurement system. The core of the apparatus is a cylindrical equilibrium cell (Figures 2 and 3a). The cell was made of a superalloy grade material, type XN26TW, and it consists of two major parts: (1) a main body and (2) two-sight sapphire windows. The main body has an outer diameter of 8.3 cm and a length of 16.6 cm. The volume of the cell (including the transfer lines) of 201.0 ( 0.5 cm3 is known from calibration using a variable volume cell coupled to a displacement transducer. The two sapphire windows are located in the front and rear end sides of the cell enabling the visual observation of the gas hydrate formation and phase behavior. All pieces of the cell are tightened by sixteen bolts, and the seal between the main body and the sapphire windows is achieved with polytetrafluoroethylene (PTFE) O-rings which sit in circular channels on both parts. There are three inlet/ outlet portholes for pressure and temperature sensors drilled in the main body of the cell. A grid support was installed inside the cell to protect the capillary sampler from plugging by hydrate particles. The grid (Figure 3b) was made of stainless steel (316) and it was fixed by a PTFE support. Such arrangement as shown in Figure 4 panels a and b allows sampling and analyzing the composition of the gas 5784

dx.doi.org/10.1021/ie101959t |Ind. Eng. Chem. Res. 2011, 50, 5783–5794


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Figure 1. Schematic flow diagram of the new apparatus. DAU, data acquisition unit; DW, degassed water; EC, equilibrium cell; G, gas cylinder; GC, gas chromatograph; HPT, high pressure transducer; LB, liquid bath; LPT, low pressure transducer; PP, platinum probe; RS, ROLSI sampler; SD, stirring device; SW, sapphire windows, TR, temperature regulator; V1, V2, V4, V5, feeding valves; V3, V6, purge valves; VP, vacuum pump; VS, isolation valve for LPT; WP, high pressure pump.

Figure 2. Cross section of the equilibrium cell: B, tightening bolts; MB, main body; PG, protection grid; PP, platinum probe; RS, ROLSI sampler; SD, stirring device; SW, sapphire windows; V3, purge valve.

phase in equilibrium with the gas hydrate and liquid phases in a way that hydrate particles do not enter or plug the capillary. A motor-driven turbine agitation system (Top Industrie) enables stirring of the cell contents at a speed up to 1500 rpm to increase the fluids contact and enhance water conversion into hydrate. The sample supplying system consists of gas cylinders and

a high-pressure syringe pump (Teledyne Isco, model 260D) which also allows measuring the amount of liquid supplied to the system. The composition analyzing system consists of an electromagnetic online microsampler (ROLSI) developed in our research laboratory. The details of this sampler device are given elsewhere.16 The ROLSI sampler is connected to a gas chromatograph (GC, Varian, CP3800). The GC uses a thermal conductivity detector (TCD) and a PORAPAK-Q packed column (length, 2 m; o.d., 1/8 in., 80/100 mesh). The TCD was calibrated for CO2 and CH4. The obtained calibration curves were fitted to second-order polynomial equations. The composition of the gas phase at a given equilibrium condition was determined from the peak area ratio of the unknown sample and the coefficients of the corresponding polynomial equation for each compound. In this way, the gas phase composition is analyzed by gas chromatography for CO2 and CH4, while the composition of water in the gas phase is considered negligible in the experimental conditions used in this study. However, it is also possible to measure the water content of the gas phase using a “dilutor technique”.19 The operating conditions of the gas chromatograph were as follows: carrier gas used was helium; gas rate, 25 mL/min; TCD oven temperature, 393 K; TCD wire temperature, 448 K; and column oven temperature, 323 K. Temperature was controlled using a thermostatic ethanol bath (Tamson Instruments, TV400LT). The bath is equipped with a glass window, which allows the visual observation of the cell content throughout the experiments. One platinum temperature probe (Pt100) inserted in the cell interior was used to measure the temperature inside the cell within measurement uncertainty, which is estimated to be less than 0.02 K with a second order 5785


Industrial & Engineering Chemistry Research polynomial calibration equation. The temperature probe was calibrated against a 25-Ω reference platinum resistance thermometer (TINSLEY Precision Instruments). The 25 Ω reference platinum resistance thermometer was calibrated by the Laboratoire National d’Essais (LNE, Paris) on the basis of the 1990 International Temperature Scale (ITS 90). The equilibrium pressure was measured using two calibrated pressure transducers (Druck, type PTX611). Pressure ratings for low (LPT) and high (HPT) pressure transducers are 8 and 40 MPa, respectively. Both pressure transducers are maintained at constant temperature (temperature higher than the highest temperature of the study) using an air-thermostat

Figure 3. (a) Lateral section of the equilibrium cell showing the position of the protection grid: B, tightening bolts; MB, main body; PG, protection grid; SD, stirring device; SW, sapphire windows; V1 and V2, feeding valves. (b) Detail of the protection grid for the capillary sampler: SD, stirring device.

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thermally controlled by a proportional-integral-derivative controller (WEST instrument, model 6100). After calibration against a dead weight balance (Desgranges & Huot 5202S CP, Aubervilliers, France), pressure accuracy was estimated to be within (0.002 MPa using a second order polynomial calibration equation. The apparatus is also equipped with a safety pressure relief valve. The data acquisition unit (AOIP, PC10) was coupled to a personal computer to record automatically pressure, temperature, and time data. The data acquisition software (AOIP, Ltc10) also allows adjusting the rate of data acquisition. Continuous recording of pressure, temperature, and time allows detecting any subtle changes in the system and true equilibrium conditions. 2.2. Experimental Procedure. Research grade carbon dioxide of purity 99.995 mol % supplied by Air Liquide and CH4 of purity 99.995 mol % supplied by Messer Griesheim were used without further purification. Water used was distilled and deionized. Gas mixtures composed of CO2 and CH4 were prepared in the equilibrium cell. After the equilibrium cell was well cleaned and evacuated with a vacuum pump, a specified amount of each gas was introduced into the volume-calibrated cell from the corresponding gas cylinders through a pressure-regulating valve. Once temperature and pressure were stabilized, the valve in the line connecting the vessel and the gas cylinder was closed. The partial and total pressures were registered from the pressure transducer reading. After gas mixing, the composition of the gas mixture was determined from the temperature, pressure, and cell volume data. The composition of the feed gas was also analyzed and checked by gas chromatography (GC). Several samples were taken through ROLSI sampler and analyzed by gas chromatography for evaluating measurement. A good agreement between both PVT and chromatographic methods was obtained. Molar concentrations of CH4 and CO2 were measured within an experimental uncertainty of less than 1%. An amount of about 10% (by volume) of the equilibrium cell was subsequently filled with water using the high-pressure syringe pump. All amounts of substances supplied to the cell were quantified. An isochoric pressure search method13-15 was followed to measure the hydrate dissociation conditions. The cell was immersed into the temperature-controlled bath and the temperature was decreased to form hydrate, while agitating at a constant speed of 1500 rpm. The temperature of the system was kept constant for at least 24 h to overcome the metastable period and allow complete hydrate formation. Hydrate formation in the cell was detected by a noticeable pressure drop. Once hydrate formation was completed and equilibrium conditions were reached, at least seven samples of the gas phase were taken through the ROLSI sampler for measurement repeatability and then composition of the gas phase in equilibrium at a given temperature, and pressure was determined through GC. Temperature was then increased in steps at a sufficiently slow rate.14 At every

Figure 4. Lateral (a) and cross sections (b) of the grid support: GS, PTFE grid support; PG, protection grid. 5786



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temperature step, temperature was kept constant and the composition of the gas phase was analyzed successively about every hour until an average of at least five measurements with a concentration difference of less than 0.001 mol fraction was achieved. At this point (typically reached in 4 h), it was assumed that equilibrium had been reached, and the average concentrations were registered as the compositions of the gas phase at the corresponding temperature and pressure condition. Temperature and pressure data were collected twice per minute throughout the experiment. In this way, a pressure-temperature diagram was obtained for each experimental run, from which the hydrate dissociation condition was determined.20 The point at which the slope of pressure-temperature data plots changes sharply was considered to be the point at which all hydrate crystals have dissociated and, hence, it is reported as the hydrate dissociation condition (HDC).20 There is a fundamental difference between hydrate formation and dissociation conditions. Since the gas and aqueous phases are initially disordered on a molecular level, the initial hydrate formation is affected by a metastability period, while hydrate crystals are ordered structures in nature and they are quickly dissociated when taken out of their pressure–temperature (p-T) stability region.1 For measuring an equilibrium condition at a higher pressure, the pressure of the system was increased by successively supplying water to the equilibrium cell until achieving the desired pressure. In this way, several p, T, and gas phase composition equilibrium data were obtained in parallel to the hydrate dissociation conditions (HDC) from each experimental run. The estimated uncertainty in the dissociation pressure and temperature measurements is expected to be within (0.05 MPa and (0.3 K, respectively. The maximum uncertainty in all measurements is attributed to the uncertainty in measuring compositions of the gas phase, which are measured by gas chromatograph. This uncertainty is estimated to be less than 1% as explained above. Other uncertainties for the liquid and hydrate phases are expected lower than 1%. Therefore the overall uncertainty is expected to be less than 1%. The following quantities were measured for every experimental run: total volume (V), each total quantity of CO2, methane, and water (n1t, n2t, n3t, respectively), mole fractions of gas phase (y1 and y2) and the equilibrium temperature (T) and pressure (p). The compositions of the hydrate and aqueous phases are then determined using a material balance approach reported by Ohgaki et al.3 in combination with the experimental data and the volumetric properties evaluated from the equations of state for gas mixtures. Considering that all types of molecules have the G same gaseous molar volume, vG i may be replaced by vm and the volume-balance in the equilibrium cell is given as V ¼

∑nGi vGm þ ∑nLi vLi þ nH vH

ð1Þ

where the subscript m denotes the gas mixture and the superscripts H, L, and G represent the hydrate, liquid, and gas phases, respectively. The volumetric properties for the gas mixture (vG m), liquid (vLi ) and the molar volume of ideal hydrate (vH) were calculated using the CSMGem thermodynamic model.1 In addition, eqs 2-4 are derived according to the material balances for the three present components in the system as follows: nt1

¼

nG1

þ nL1

þ zn

H

where z is water-free mole fraction in hydrate phase and q stands for hydration number. The total amount of each component once measured, can be partitioned into gas, liquid (aqueous phase), and hydrate (z) phases for CO2 and CH4 (n1t, n2t, respectively), and liquid and hydrate phases for water (n3t). The expressions for mole fractions of CO2 in the gas phase (y1) and CO2 (x1) and CH4 (x2) in the liquid phase are given by eqs 5-7:

ð3Þ

nt3 ¼ nL3 þ qnH

ð4Þ

ð5Þ

∑

ð6Þ

∑

ð7Þ

x1 ¼ nL1 = nLi x2 ¼ nL2 = nLi

The values of x1 and x2 can be estimated from a suitable thermodynamic model. In this work, the CSMGem thermodynamic model,1 which is based on the Gibbs energy minimization, was applied. This thermodynamic model has been developed by taking into account that for a system of known phases, the following three criteria should be satisfied to ensure that the Gibbs energy is at a minimum: temperature equilibrium of all phases, pressure equilibrium of all phases, and equality of chemical potential of a component in each phase.1 The Gibbs energy minimization method allows for calculations of the formation conditions for any phase (including the hydrate). It also allows for the calculation of phases present at any T and p (whether hydrates are present or not). Therefore, one can perform all thermodynamic calculations with every phase and not just the hydrate. In the cases where the model did not converge, these values were evaluated applying the apparent Henry’s constants of each pure gas system: xi ¼ fi ðT, pÞ=Hi ðT, pÞ

ð8Þ

where f (T,p) is the fugacity coefficient, H (T,p) is the apparent Henry’s constant, and subscript i refers to the ith component in the mixture. Equations 1-7 were solved using the numerical approach described in the next section.

3. MATHEMATICAL APPROACH FOR SOLVING THE MASS BALANCE EQUATIONS 3.1. Newton’s Method. For obtaining the values of the seven G L L L H unknown variables including nG 1 , n2 , n1 , n2 , n3 , n , and z, the Newton’s method17 has been used. Among the numerical methods for solving nonlinear equations (such as interval halving and fixed point iteration, etc.) the Newton’s numerical method17 can be reliably extended to solve systems of nonlinear equations, which are generally required in scientific and engineering problems.21 Consider the system of N nonlinear equations with N unknowns as follows:

ð2Þ

nt2 ¼ nG2 þ nL2 þ ð1 - zÞnH

∑

y1 ¼ nG1 = nGi

f1 ðxi - 1 , xi , :::Þ ¼ 0 f2 ðxi - 1 , xi , :::Þ ¼ 0

i ¼ 1, :::, N i ¼ 1, :::, N

ð9Þ ð10Þ

333 333 333 fN ðxi - 1 , xi , :::Þ ¼ 0

i ¼ 1, :::, N

ð11Þ

where f stands for the function and x stands for the unknowns. Solving for values of the unknowns is followed up by presenting 5787



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Table 3. Experimental (pexpt) and Predicted (ppred) Dissociation Pressures for Carbon Dioxide þ Methane Clathrate Hydrates in the Presence of Pure Water ppred (MPa) using

mole fraction of H2O

mole fraction of

expt

a,39-42

ARD %

CSMGem model1

ARD %

4.18

3.7

3.98

1.2

5.86

6.9

5.67

3.5

8.27

8.84

6.9

8.71

5.3

277.9

2.72

2.97

9.2

2.78

2.2

0.959 0.973

279.8 285.0

3.61 6.09

3.94 7.62

9.1 25.1

3.68 7.34

1.9 20.5

0.744

0.877

279.0

2.72

2.85

4.8

2.73

0.4

0.744

0.956

280.1

3.21

3.42

6.5

3.26

1.6

0.744

0.972

283.2

4.70

5.39

14.7

5.12

supplied to the system

T (K)

0.206

0.871

279.3

4.03

0.206

0.961

282.0

5.48

0.206

0.975

285.5

0.476

0.877

0.476 0.476

CO2 in the gas feed

p

(MPa)

HWHYD model

AAD % a

ppred (MPa) using b

9.7

8.9 5.1

Assuming sI. b ARD = 100 (|pexp - ppred|)/pexp.

where J refers to the Jacobian Matrix and is calculated by the expression below:21 Dfm Ja ðm, nÞ ¼ ð14Þ Dxn xNa

In the above equation m and n are the numbers of particular function and related variables, respectively. Equations 13 and 14 are repetitively used until the following convergence criteria are satisfied: 3 2 2 3 ε1 x0ða þ 1Þ - x0ðaÞ 7 6 6 7 6 x1ða þ 1Þ - x1ðaÞ 7 6 ε2 7 7 6 6 7 7 6: 6: 7 7 e6 7 6 ð15Þ 7 6: 6: 7 7 6 6 7 7 6: 6: 7 5 4 4 5 xNða þ 1Þ - xNðaÞ εN

Figure 5. Dissociation conditions measured in this work for various methane þ carbon dioxide clathrate hydrates at various CO2 load mole fractions: 0.206 ((), 0.476 (2), and 0.744 (b).

where ε is the defined tolerance of the Newton’s numerical method. 3.2. Constraint Handling. According to the mass conservation law, some constraints should be taken into account for solving the system of equations as follows:

the preceding functions as Taylor series and appropriate mathematical derivations and rearrangements as 2

3

2

3

x 0ða þ 1Þ x 0ðaÞ 6 6 6 x 1ða þ 1Þ 7 7 6 x 1ðaÞ 7 7 6 7¼6 7 6l 7 6l 7 4 5 4 5 x Nða þ 1Þ x NðaÞ 2 Df1 Df1 3 2 33 f1 ðxði - 1Þa , xðiÞa , :::Þ 6 6 Dxi - 1 Dxi 6 Df2 7 6 Df 2 6 f2 ðxði - 1Þa , xðiÞa , :::Þ 76 76 -6 76 Dxi - 1 Dxi 3 3 6l 56 l 4 l fN ðxði - 1Þa , xðiÞa , :::Þ 6 4 DfN DfN Dx Dx 3 3 i-1

i

3 37 7 7 7 37 7 7 7 5 3

-1

i ¼ 1, :::, N

ðxði - 1Þa , xðiÞa , :::Þ

xNða þ 1Þ ¼

xNa - Ja-1

f ðxNa Þ

ð13Þ

nG1 e nt1

ð16Þ

nG2 e nt2

ð17Þ

nL1 e nt1

ð18Þ

nL2 e nt2

ð19Þ

nL3 e nt3

ð20Þ

Because the mentioned Newton’s method is not able to handle the following constraints, we have considered an objective function that is supposed to be minimized by an appropriate optimization method as follows: N

ð12Þ where a denotes the estimated value of an unknown variable. Finally, the following expression is used applying the Jacobian matrix:21

FFðxi - 1 , xi , :::Þ ¼

∑i fi ðxi - 1 , xi , :::Þ

ð21Þ

where FF stands for the primary objective function before introducing the constraints. It should be noted that the other constraint (0 e z e 1) is satisfied intrinsically through solving the equations. 5788



Figure 6. (a) Comparison of the dissociation conditions measured in this work with available literature data for various methane þ carbon dioxide clathrate hydrates at selected CO2 load mole fractions: (a) 0.206 this work ((); 0.210-0.2502 ()); 0.2645 (0); 0.2746 (); 0.2008 (Δ); pure CO2 hydrates1 (dashed line); pure CH4 hydrates12 (solid line). (b) 0.476 this work ((); 0.390-0.5002 ()); 0.490-0.5045 (0); 0.5390.5456 (); 0.6008 (Δ); pure CO2 hydrates1 (dashed line); pure CH4 hydrates1 (solid line). (c) 0.744 this work ((); 0.670-0.8502 ()); 0.7305 (0); 0.7776 (); pure CO2 hydrates1 (dashed line); pure CH4 hydrates1 (solid line).

3.3. The Differential Evolution Strategy. The optimum values of a number of mathematical model parameters have to be obtained by appropriate optimization methods. Almost all of the traditional optimization algorithms have the possibility of

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Figure 7. (a) P-y phase diagram of the methane þ carbon dioxide þ water system under hydrate formation conditions at different equilibrium temperatures. This work: 273.6 K (9); 275.2 K (b); 276.1 K ((); 278.1 K (2), and 280.2 K (þ). Literature data: 275.2 K9 (O); 277.2 K9 ()); 279.2 K9 (Δ); 281.2 K9 (0); 283.2 K9 (); 285.2 K9 (-); 274.2 K10 (solid line); 276.2 K10 (thin dashed line); 278.2 K10 (medium dashed line); 280.3 K3 (thick dashed line). (b) P-z phase diagram of the methane þ carbon dioxide þ water system under hydrate formation conditions at ∼280.2 K. This work, 280.2 K (b); literature data, 280.3 K3 (O).

getting trapped at local optimum(s) depending upon the degree of nonlinearity and initial guess. There is no guarantee for these optimization methods to find the global optimum solution but the population-based search algorithms are supposed to do so.22,23 Such optimization techniques which are designed based on natural phenomena have generated lots of interest recently like the simulated annealing (SA),24 the evolution strategies (ES),25 the genetic algorithms (GA)24-27 and the differential evolution (DE)18 that have been developed to overcome the problem of traditional methods. The prominent characteristics of the DE optimization strategy can be found elsewhere.18,22-24 5789



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Optimization of the batch fermentation process,22 nonlinear chemical processes optimization,23 optimization of process synthesis and design problems,29 and optimal design of shell-and-tube heat exchangers30 are among the various applications of this technique. These researches have shown that the DE is an efficient, effective, and robust evolutionary optimization method. Recently, this strategy has been proven to have high capabilities in phase behavior predictions and literally tend to converge to the global optimal value of the objective function.31-34 The computational steps, different strategies, and recommendations to select the operational parameters of the DE algorithm are well-established in previous works.28-35 3.3.a. Problem Formulation. The concept of the penalty functions are considered to deal with the optimization problems subjected to several constraints which penalize infeasible solutions (i.e., eliminate the unexpected results). On the basis of this method, the penalty function takes a finite value when a constraint is violated and a value of zero when constraint is satisfied. The penalized function is generally written as36 L

M

∑ Ægl æ þ m∑¼ 1 jhmj l¼1

OF ¼ FFðxi - 1 , xi , :::Þ þ PN½

ð22Þ

where OF is the objective function, gl is the inequality constraint, h is the equality constraint, and L and M refer to the number of inequality and equality constraints, respectively. In eq 22, PN denotes the penalty parameter which is defined by the user. The value of this parameter depends on the order of magnitude of the model parameters and usually lies between 1 and 106.31,34 The higher selected values for PN, the more effects of constraints on computations would be expected. The bracket-operator Ææ denotes the absolute value of the operand, if operand is negative. The constraints of the problem are considered to be inequality constraints and written as follows: nt1 - nG1 g 0

ð23Þ

nt2 - nG2 g 0

ð24Þ

nt1 - nL1 g 0

ð25Þ

nt2 - nL2 g 0

ð26Þ

nt3 - nL3 g 0

ð27Þ

The final formulation of the objective function is given by 5

OF ¼ FFðxi - 1 , xi , :::Þ þ ð1Þ

∑ Ægl2æ l¼1

ð28Þ

where, g1 ¼ maxf0, nt1 - nG1 g

ð29Þ

g2 ¼ maxf0, nt2 - nG2 g

ð30Þ

g3 ¼ maxf0, nt1 - nL1 g

ð31Þ


ð32Þ


ð33Þ

17

Therefore, Newton’s numerical method has been coupled with the differential evolution (DE) optimization strategy18 to

obtain the values of the unknown variables. The calculation steps are continued until the convergence criterion of the Newton’s method and all of the constraints of the objective function are satisfied.

4. RESULTS AND DISCUSSION Hydrate dissociation conditions (temperature and pressure) for the CO2 þ CH4 þ H2O systems were measured at different compositions of CO2 in the gas feed. The compositions of the methane þ carbon dioxide gas mixtures along with the experimental hydrate dissociation conditions in the presence of pure water are presented in Table 3 and Figure 5. In addition, the hydrate dissociation pressures were predicted at the corresponding equilibrium temperature, CO2 mole fraction in the gas feed, and water mole fraction introduced to the system using two hydrate thermodynamic models: CSMGem1 (which is based on the Gibbs energy minimization) and HWHYD39-42 (which is based on fugacity equality of each components throughout all phases present). The predicted hydrate dissociation pressures are also listed in Table 3 along with the absolute relative deviation (ARD) from the obtained experimental values. As can be seen, the AAD of the predicted hydrate dissociation conditions by HWHYD39-42 and CSMGem1 models are on average 9.7% and 5.1%, respectively, which are considered to be in acceptable agreement with those values measured in this work. For comparison purposes, Figure 6 panels a-c show the hydrate dissociation conditions measured in this work along with some selected experimental data from the literature.2,4-6,8 Our results generally indicate that the hydrate dissociation pressures of the CO2 containing gas mixtures are greater than those of pure CO2 hydrates. It is also observed that as the relative amount of CH4 to CO2 increases in the feed gas at a given temperature, the equilibrium pressure conditions shift to higher pressures. As can be seen, the agreement between our data and the experimental data reported in the literature is generally acceptable. The satisfactory agreement demonstrates the reliability of the new apparatus and method used in this work. To the best of our knowledge, no hydrate structural transition has been reported in the literature for CO2 þ CH4 hydrates. Therefore, considering that the CO2 þ CH4 þ H2O system most likely forms structure I hydrates1,5,37,38 similar to pure CH4 or CO2 hydrates, and applying Gibbs phase rule, there are two degrees of freedom at liquid water-hydrate-gas (Lw-H-G) equilibrium conditions for this system.1 In this work, temperature and pressure were controlled while the compositions of CH4 and CO2 in the gas phase under Lw-H-G equilibrium were measured and compared with available literature data.3,9,10 As shown in Figure 7 panels a and b, the mole fraction of CO2 in the gas and hydrate phases, respectively, measured in this work are found to agree with the experimental data reported in the literature within experimental uncertainties. Furthermore, it is observed that the mole fractions of CO2 in the gas and hydrate phases generally decrease as pressure increases. It is also observed that the composition of CO2 in the gas phase increases as temperature increases, which translates in less amount of CO2 trapped in the hydrate phase at high temperatures. Compositional analysis of the hydrate and liquid phases for the ternary CO2 þ CH4 þ H2O system was determined using the material balance approach described in the experimental section. The nonlinear mass balance equations were solved using the numerical Newton’s method17 combined with the DE optimization strategy18 to handle the required constraints, as 5790


a

5791

0.165 0.165 0.165 0.115 0.115 0.115 0.057 0.057 0.165 0.165 0.165 0.115 0.115 0.115 0.057 0.057 0.165 0.165 0.165 0.115 0.115 0.115 0.057 0.057 0.165 0.165 0.165 0.115 0.115 0.057 0.057 0.165 0.165 0.165 0.115 0.115 0.057 0.057 0.165 0.165

1.43 5.236 8.185 1.65 5.355 8.48 1.696 5.182 1.43 5.236 8.185 1.65 5.355 8.48 1.696 5.182 1.43 5.236 8.185 1.65 5.355 8.48 1.696 5.182 1.43 5.236 8.185 5.355 8.48 1.696 5.182 1.43 5.236 8.185 5.355 8.48 5.182 8.395 8.185 8.185

273.6 273.6 273.6 273.6 273.6 273.6 273.6 273.6 275.2 275.2 275.2 275.2 275.2 275.2 275.2 275.2 276.1 276.1 276.1 276.1 276.1 276.1 276.1 276.1 278.1 278.1 278.1 278.1 278.1 278.1 278.1 279.2 280.2 280.2 280.2 280.2 280.2 280.2 282.2 284.2

2.234 2.416 2.44 1.844 1.941 2.048 1.51 1.607 2.583 2.712 2.766 2.123 2.22 2.4 1.792 1.865 2.813 3.025 3.027 2.318 2.503 2.69 1.985 2.174 3.416 3.631 3.802 3.037 3.319 2.45 2.58 3.565 4.486 4.655 3.541 4.109 3.139 3.481 5.767 7.19

-b -b 0.096 0.549 0.392 0.294 0.884 0.801 0.338 -b 0.179 0.65 0.586 0.366 0.831 0.752 0.264 0.239 0.238 0.644 0.4 0.312 0.877 0.784 0.233 0.225 0.148 0.457 0.273 -b 0.786 0.266 0.307 0.245 0.727 0.42 0.86 0.788 0.276 0.107

zCO2 (mole fraction) (water free base)

yCO2 (mole fraction)

0.141 0.125 0.081 0.345 0.288 0.22 0.63 0.545 0.166 0.129 0.086 0.384 0.302 0.228 0.657 0.565 0.179 0.134 0.096 0.405 0.315 0.232 0.669 0.579 0.202 0.139 0.103 0.323 0.233 0.694 0.609 0.202 0.147 0.108 0.344 0.235 0.62 0.49 0.114 0.115

hydrate phase

gas phase

-b -b 0.0006 0.0003 0.0006 0.0003 0.0001 0.0001 0.0008 -b 0.0009 0.0003 0.0006 0.0003 0.0002 0.0003 0.0007 0.0009 0.0007 0.0004 0.0004 0.0004 0.0002 0.0004 0.0021 0.0009 0.0008 0.0004 0.0004 -b 0.0004 0.0009 0.0079 0.007 0.0005 0.0005 0.0004 0.0002 0.0016 0.0012

-b -b 0.994 0.9898 0.9932 0.9902 0.9872 0.9871 0.9921 -b 0.9961 0.9889 0.9919 0.9889 0.9875 0.9884 0.9931 0.9957 0.9931 0.9883 0.9883 0.9883 0.984 0.9882 0.989 0.9917 0.9924 0.9871 0.987 -b 0.9859 0.992 0.9898 0.9917 0.9849 0.9881 0.983 0.9848 0.9919 0.9921 0.151 -c -c 0.376 0.291 -c 0.626 -c 0.171 -c 0.109 0.392 0.295 0.245 0.634 0.589 0.183 0.128 -c 0.664 0.408 -c -c -c -c -c 0.095 -c 0.232 -c 0.604 -c -c -c -c 0.255 -c -c -c -c


xCO2 xCH4 xH2O (mole (mole (mole fraction) fraction) fraction) -b -b 0.0055 0.0099 0.0062 0.0095 0.0127 0.0128 0.0071 -b 0.003 0.0108 0.0075 0.0108 0.0123 0.0113 0.0062 0.0034 0.0062 0.0113 0.0113 0.0113 0.0158 0.0114 0.0088 0.0074 0.0068 0.0124 0.0126 -b 0.0137 0.0071 0.0023 0.0013 0.0146 0.0114 0.0167 0.015 0.0065 0.0067

gas phase

aqueous phase

0.296 -c -c 0.538 0.478 -c 0.765 -c 0.32 -c 0.225 0.574 0.52 0.419 0.786 0.732 0.397 0.335 -c 0.397 0.584 -c -c -c -c -c 0.193 -c 0.392 -c 0.786 -c -c -c -c 0.409 -c -c -c -c

zCO2 (mole fraction)(water free base)

hydrate phase




xCO2 xCH4 xH2O (mole (mole (mole fraction) fraction) fraction)

aqueous phase

CSMGem model1 predictions (compositional analysis)

7.1 9.0 1.0 0.6 3.0 26.7 2.1 2.3 7.5 3.5 4.2 2.2 4.5 64.0 29.5 7.8 0.4 0.8 8.5 9.7


gas phase

2.0 21.9 13.5 5.3 25.7 11.7 11.3 14.5 5.4 2.7 50.4 40.2 38.4 46.0 30.4 43.6 0.0 2.6 20.3

zCO2 (mole fraction) (water free base)

hydrate phase

24.2 0.0 15.7 38.0 0.0 21.3 0.0 45.4 0.0 0.0 21.0 0.0 18.6 34.5 55.9 47.6 0.0 31.6 27.1

66.7 0.0 100.0 0.0 0.0 66.7 0.0 133.3 0.0 0.0 14.3 0.0 25.0 50.0 37.5 125.0 0.0 80.0 38.8

0.2 0.0 0.2 0.3 0.0 0.2 0.0 0.5 0.0 0.0 0.1 0.0 0.2 0.4 0.4 0.6 0.0 0.3 0.2

xCO2 xCH4 xH2O (mole (mole (mole fraction) fraction) fraction)

aqueous phase

absolute relative deviations (ARD %)

The HWHYD 39-42 model does not converge for the studied conditions. b Some/all of constraints are not satisfied. c No three phase flash convergence using CSMGem model 1.

0.048 0.048 0.048 0.116 0.116 0.116 0.181 0.181 0.048 0.048 0.048 0.116 0.116 0.116 0.181 0.181 0.048 0.048 0.048 0.116 0.116 0.116 0.181 0.181 0.048 0.048 0.048 0.116 0.116 0.181 0.181 0.048 0.048 0.048 0.116 0.116 0.181 0.181 0.048 0.048 AAD %

CO2 CH4 H2O p (mol) (mol) (mol) T (K) (MPa)

feed

experimental data (compositional analysis)

Table 4. Three-phase Equilibrium Data for Gas Mixtures of CO2 þ CH4 in the Presence of Water at Different Temperatures and Pressures

Industrial & Engineering Chemistry Research ARTICLE



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Figure 8. Pressure-composition diagram for the methane þ carbon dioxide þ water system under Lw-H-G equilibrium at (a) 273.6, (b) 275.2, (c) 276.1, (d) 278.1, (e) 280.2 K. This work: yCO2 (2); zCO2 (b). CSMGem model1 predictions: yCO2 (Δ); zCO2 (O).

mentioned earlier. The experimentally measured data, results of solving the material balance equations, and comparison with the predictions of CSMGem thermodynamic model1 are summarized

in Table 4 and illustrated in Figure 8 panels a-e at selected isothermal conditions. It should be mentioned that sketching such figures is essential in designing the absorption/desorption units 5792


Industrial & Engineering Chemistry Research (multistage operations) in CO2 capture processes. As can be seen in these figures, the predictions of the CSMGem model1 for the mole fraction of CO2 in the gas phase generally agree with our experimental data. However, the thermodynamic model1 shows considerable deviations when predicting the molar compositions of CO2 in the hydrate phase. This suggests that the parameters of the thermodynamic model may require readjustment using fully compositional analysis þ hydrate dissociation conditions experimental data, as such models have generally been developed for hydrates of hydrocarbons using mainly experimental hydrate dissociation data. As for the unspecified points shown in Table 4 related to the predictions of the CSMGem model,1 the model is not able to predict the three-phase equilibrium (Lw-H-G) behavior at the studied conditions. The HWHYD model39-42 does not converge at all for the studied conditions.

5. CONCLUSIONS We reported the details of an apparatus based on the staticanalytical method with gas phase capillary sampling, especially designed for measuring hydrate disociation conditions and the compositions of the existing phases in equilibrium with gas hydrate. An isochoric pressure-search method13-15 was used for measuring hydrate disociation conditions for the CO2 þ CH4 þ water system. The compositions of the gas phase in equilibrium with gas hydrate and aqueous phase were measured by gas chromatography technique. A material balance based approach3 was applied for determining compositions of the hydrate and aqueous phases. A mathematical algorithm based on the Newton’s numerical method17 coupled with the differential evolution optimization strategy18 was used for solving the material balance equations. We succeeded in comparing the experimental data generated in this work with the corresponding literature data. In addition, a comparison was made between the experimental data generated in this work with the predictions of two thermodynamic models, namely CSMGem1 and HWHYD.39-42 It was shown that the aforementioned thermodynamic models can acceptably predict hydrate dissociation conditions for the CO2 þ CH4 þ water systems studied in this work. While the CSMGem1 acceptably predicts the composition of the gas phase in equilibrium with gas hydrate and aqueous phases for the CO2 þ CH4 þ water systems studied in this work, it unreliably predicts the compositions of the hydrate phase. The HWHYD model39-42 did not converge for the studied conditions. Finally, it was argued that the latter thermodynamic models require reconsiderations in adjusting their parameters by using both hydrate dissociation and compositional data of the existing phases in equilibrium with gas hydrate, as these models have traditionally been developed for hydrocarbon systems. ’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected]. Tel.: þ (33) 1 64 69 49 70. Fax: þ (33) 1 64 69 49 68. Notes §

This manuscript was presented at the AIChE Annual Meeting 2010 in Salt Lake City, UT.

’ ACKNOWLEDGMENT Veronica Belandria thanks Fundayacucho of Venezuela for providing her a Ph.D. scholarship. Ali Eslamimanesh is grateful to Mines ParisTech for providing him a Ph.D. scholarship. The

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financial supports of Agence Nationale de Recherche (ANR) as part of the SECOHYA project and Orientation Strategique des Ecoles des Mines (OSEM) are gratefully acknowledged. The authors thank Takeshi Sugahara from Osaka University for fruitful discussions. The authors also thank Kevin Reigner and Tarik Jaakou for their technical assistance.

’ NOMENCLATURE AAD = average absolute deviation ARD = average relative deviation DE = differential revolution FF = primary objective function f = fugacity or mass balance equation g = inequality constraint G = gas H = hydrate H = Henry’s constant, (MPa) h = equality constraint HDC = hydrate dissociation condition J = Jacobian matrix L = number of inequality constraint or liquid m = number of function n = number of moles or number of variables M = number of equality constraints N = number of unknown variables OF = final objective function p = MPa PN = penalty parameter q = hydration number T = temperature, K V = total volume, (m3) v = molar volume, (m3/mol) x = mole fraction in liquid phase or unknown variables y = mole fraction in gas phase z = water free mole fraction in hydrate phase Greek letters

ε = convergence criterion of Newton’s method ∂ = derivative operator Subscripts a = the estimated values of unknowns i = ith unknown variables l = number of inequality constraints m = refers to gas mixture or the number of equality constraints N = Nth unknown variables W = water 1 = refers to carbon dioxide or number of equation 2 = refers to methane or number of equation 3 = refers to water Superscripts exp = experimental G = gas phase H = hydrate phase L = liquid phase pred = predicted t = total

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