Kinetic of hydrate formation of propane and its mixture with methane in a circulating flow reactor

Fluid Phase Equilibria 298 (2010) 38–47

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Kinetic of hydrate formation of propane and its mixture with methane in a circulating flow reactor M. Sarshar a,d , J. Fathikalajahi b,c,∗ , F. Esmaeilzadeh b a

Fars Engineering Research Center, Institute of Engineering, Shiraz, Iran Chemical and Petroleum Engineering Department, School of Engineering, Shiraz University, Shiraz, Iran Petroleum & Chemical Engineering Department, Sultan Qaboos University, Oman d School of Chemical Engineering, Shiraz University of Technology, Shiraz, Iran b c

a r t i c l e

i n f o

Article history: Received 20 December 2009 Received in revised form 14 June 2010 Accepted 27 June 2010 Available online 7 July 2010 Keywords: Kinetics Gas hydrate Circulating flow reactor Modeling Simulation

a b s t r a c t Kinetics of hydrate formation for propane and its mixture with 73% methane have been studied experimentally and theoretically at pressure up to 2 MPa and temperature up to 277.65 K in a 10 m circulating flow reactor. A mathematical model has been developed for the process of hydrate formation based on crystallization, mass transfer and thermodynamics concepts. The amounts of gas consumptions due to hydrate formation are measured experimentally and predicted by the model. The agreement between the experimental measured gas consumptions and predicted values by the mathematical model are very good and the average deviation errors in the prediction of gas consumption are less than 10%. © 2010 Elsevier B.V. All rights reserved.

1. Introduction Hydrate formation is the focus in the production of hydrates for storage and transport of natural gas, gas separation, water desalination, exploitation of gas hydrate deposits and deposition of CO2 hydrates on the sea floor. Also, hydrate prevention is an important issue in gas transportation lines where hydrate formation is considered a major problem. Gas hydrates are crystalline solids that are more properly called clathrate hydrates to distinguish them from stoichiometric hydrates found in inorganic chemistry. Natural gas hydrates are composed of water and gas. The gas molecules (guests) are trapped in water cavities (host) composed of hydrogen-bonded water molecules. Typical natural gas molecules include methane, ethane, propane, and carbon dioxide [1]. Sum et al. [2] highlighted the recent hydrate literature focusing on the thermodynamics, kinetics, structural properties, particle properties, rheological properties, and molecular mechanisms of formation. Falcone et al. [3] performed a critical review of multiphaseflow loops around the world, highlighted the pros and cons of

∗ Corresponding author at: Petroleum and Chemical Engineering Department, Sultan Qaboos University, Oman. E-mail addresses: [email protected] (M. Sarshar), [email protected] (J. Fathikalajahi), [email protected] (F. Esmaeilzadeh). 0378-3812/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2010.06.016

each facility with regard to reproducing and monitoring different multiphase-flow situations. The authors suggested a way forward for new developments in this area. Sarshar et al. [4] developed a set of mathematical models to predict the hydrate formation rate and the aqueous phase composition in the flow loop. They validated the model with experimental data in a 10 m loop using a gas mixture of 73% methane and 27% propane at a temperature of 277.15–278.15 K and pressures of 2–3 MPa. Davies et al. [5] developed and applied a transient multiphaseflow simulator which incorporated hydrate formation kinetics and thermodynamics to predict plugging in multiphase oil production lines. The model (CSMHyK v. 2.0) was shown to predict the formation of hydrate plugs in two industrial scale flow loops by combining well known engineering correlations with state-of-the-art measurements. They further developed the simulator to allow the study of hydrate formation in systems with varying concentrations of salt or monoethylene glycol. Balakin et al. [6] developed a computational model for the size evolution of hydrate particles in a pipeline pump system with turbulent flow. Their model was based on the population balance principle, and the simulation results were validated with experimental data. They found that the particle size was significantly influenced by aggregation and breakage, related to shear in the flow, and that these effects were comparable to those of growth and nucleation, related to diffusion processes.

M. Sarshar et al. / Fluid Phase Equilibria 298 (2010) 38–47

Nicholas et al. [7] modeled gas hydrate/ice deposition from a dissolved-water phase in a liquid condensate system using mass and energy balance. The same modeling parameters were used to model three flow loop experiments (1.89 and 2.83 L/min flow rate for deposition tests and a 1.89 L/min for dissociation test) with acceptable accuracy. Relative changes in both temperature and pressure drop were modeled using an ice deposit with a 67% void fraction. Nicholas et al. [8] conducted a set of experiment to investigate hydrate/ice plugging and deposition mechanisms from water dissolved in a liquid condensate system, using a single-pass flow loop. They observed two different hydrate/ice plugging mechanisms. Hydrate/ice deposition from a dissolved-water phase resulted in a lengthwise uniform/dispersed deposit and a gradual pressure drop increase. Further, they observed that the dispersed deposit acted as insulation at the flow loop wall, and the deposit began to propagate downstream. However, cooling below the liquid water saturation curve resulted in free water coalescence, and a localized hydrate/ice restriction in the flow loop. This localized restriction resulted in a rapid pressure drop increase. Aspenesa et al. [9] investigated the wettability alteration of pipeline surfaces from contact with oil, and the adhesion energy between water and solid in the presence of oil. They determined contact angles for model systems as a function of solid material and oil composition. The results show that both the presence of petroleum acids in the oil, and low surface free energy of the pipeline material, lead to more oil-wet systems and consequently reduced adhesion energy between water and solid. Sarshar et al. [10] established a method for CO2 conversion to hydrate using a tubular circulating flow reactor. This method was useful for bioethanol plants which produced CO2 as a byproduct of ethanol fermentation. In this regard, hydrate formation experiments were carried out at 279 K and 3.5–5 MPa to determine the rate of CO2 hydrate formation. Further, a model was developed for prediction of the rate of hydrate formation. The predicted hydrate formation rate was validated with the experimental data at different operating conditions. Talaghat et al. [11] investigated, experimentally, simple gas hydrate formation in the presence of kinetic inhibitors in a flow mini-loop apparatus. A rate equation based on the Kashchiev and Firoozabadi model [12] was developed to predict the gas consumption rate during hydrate formation. Good agreements were found between the predicted and measured induction times in the presence of polyvinyl pyrrolidone (PVP) and l-tyrosine. During the pipeline shut down however, hydrate particles may settle in bends and build hydrate plugs [13]. Shabani et al. [13] designed and constructed an experimental setup to study the flow of such plugs at start up operations. They performed the experiments using model fluid and model hydrate particles. The propagations of initial plugs in a bend were recorded with impedance probes along the pipe. They simulated the evolutions of the plugs by numerical integration of the incompressible mass balance equations, with an imposed mixture velocity. They calculated the slip between particles and carrier fluid using a drag relation together with a particle–fluid force balance. Serrano et al. [14] developed a three-phase injector/reactor at Oak Ridge National Laboratory for the continuous synthesis of gas hydrates. The reactor received water and a hydrate-forming species and rapidly formed hydrate with a residence time of a few seconds. The reactor was designed to maximize interfacial area between reactants, thus minimizing mass transfer barriers and thermal effects that negatively affect conversion of reactants into hydrate. The reactor was tested for ocean carbon sequestration and in the laboratory for coal-bed methane produced-water treatment using liquid carbon dioxide.

39

Zhong et al. [15] analyzed the process of hydrate formation on a suspended water droplet and proposed a corresponding mathematical model. The results indicated that equilibrium time diminished with the decrease of the water droplet radius, and prolonged with the increase of subcooling degree. Further, the reaction time for the second period reduced with the increase of subcooling degree, but was free from the effect of the variation of the water droplet size. The first period of the hydration on the water droplet was quite short, while the second period was considerably longer. Hemmingsen et al. [16] summarized the experimental efforts for the last decade within StatoilHydro on the study of hydrate plugging risk in under inhibited systems. They used the flow simulator as the main experimental equipment. The overall results for under inhibited systems with ethylene glycol or methanol showed that the plugging potential increased up to a maximum at concentrations around 10–15 wt.%. At higher concentrations the plugging potential reduced compared to the uninhibited system. Azarinezhad et al. [17] introduced the laboratory tests at subzero conditions as well as an economical evaluation and a pipeline transportation simulation on one of the West Siberian oil fields. These simulations demonstrated that the concept was viable, and suggested that HYDRAFLOW technology could offer significant benefits over existing flow assurance strategies. Boxall et al. [18] provided a means to predict when and where hydrate plugs would form in oil dominated flow lines. The predictions showed good agreement to data for hydrate formation in three flow loops with five oils. Boxall et al. [19] performed a series of experiments to better understand the blockage potential for an oil dominated system as an important step in moving from hydrate prevention to hydrate management. The experiment was performed by varying the water cut, fluid velocity, and gas–liquid volume fraction using the ExxonMobil (XoM) flow loop in Houston, TX, USA. The XoM large loop was a three pass, four inch internal diameter flow loop with a sliding vane pump capable of generating liquid velocities of up to 4 m/s. The systems that were studied included a range of water cuts from 5 to 50% in a light crude oil (Conroe crude) and a gas phase of either pure methane or 75% methane and 25% ethane. They compared the results with the hydrate plug prediction tool, CSMHyK, integrated into the multiphase-flow simulator OLGA5® . They used the comparison between the model predictions and the flow loop results as the basis for improving hydrate formation and plug prediction [19]. Duchateau et al. [20] proposed a new procedure that consisted in characterizing the efficiency of KHI at second hydrate formation. This procedure limited the stochastic character of hydrate formation using the persistence of precursory hydrate structures in water that previously experienced hydrate formation and decomposition. It was shown that the presence of these precursors strongly increased the reproducibility of measurements as compared to systems containing ‘fresh water’ and allowed unambiguous discrimination between blank (uninhibited system) and KHI tests. Turner and Talley [21] demonstrated that the non-adhesive hydrate slurries exhibited low viscosities in a field-scale flow loop when formed under appropriate conditions. The factors that favored formation of low-viscosity hydrate slurries included high Reynolds number and capillary number and high mass transfer and heat transfer rates. High liquid loading and high superficial fluid velocities were found to be conducive to the formation of low-viscosity hydrate slurries. Amin and Pack [22] developed a flow loop to address the low seawater temperature conditions that led to problems such as hydrate formation/plugging, wax precipitation and sand accumulation.

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Delahaye et al. [23] proposed an original rheological characterization of CO2 hydrate slurries in the flow. They used an experimental loop to produce CO2 hydrate slurries by gaseous CO2 injection in pre-cooled water. They measured pressure drops as a function of volumetric flow rates for various fractions in solid particles. In order to improve the description of the rheological behavior of the studied slurry, an empirical model based on the method of the capillary (Ostwald) viscometer was applied from their experimental data provided a Herschel–Bulkley (HB) type equation integrating the solid fraction of the slurry. Mokhatab et al. [24] proposed crucial questions as well as provided significant information on the best method to prevent and remediate hydrates in deepwater production operations. Tang et al. [25] observed three types of flow patterns in the reactor with variation of the gas entrainment rate, i.e., single bubble regime, intermediate regime, and jet regime. The flow pattern under the free suction state of the ejector was in the jet regime. They observed the micro bubble generated from ELR with a static mixer could shorten the induction time for hydrate formation significantly. Robøle et al. [26] developed a test facility to implement total control of both flow conditions and system chemistry. This made it possible to undertake investigations on the phenomena that occurred in multiphase flow. The test facility was 200 m long, 75 mm diameter, high pressure flow loop which can be tilted upwards or downwards. The fluids were several combinations of real hydrocarbon oil, gas and water (with salts included). The partial pressures of CO2 and H2 S in the experiments were below 2 and 0.005 MPa, respectively. The maximum operating temperature and pressure were 413.15 K and 11 MPa absolute, respectively. Matthews et al. [27] indicated both the plugging tendency and the nature of plugging mechanism in the flow loop observed in the field. The flow loop results, in conjunction with transient flow simulation, indicated the most probable locations for the plug formation. Palermo and Goodwin [28], constructed a pilot flow loop which had operating characteristics designed to make the loop represent a production system as closely as possible in a controlled, re-circulating system. Its main outlines were a diameter of 50 mm,

a length of 140 m and a maximum pressure of 10 MPa. It was shown that THI (threshold hydrate inhibitor) had a marked effect on the formation conditions at concentrations of 500 ppm and above. A relationship between the subcooling temperature corresponding to hydrate appearance and THI concentration was determined. Paytavy et al. [29] studied methane hydrate formation in a laboratory loop where the liquid blend saturated with methane was circulated up to a pressure of 7.5 MPa. They investigated the effects of pressure, liquid velocity, liquid cooling temperature ramp and liquid hydrocarbon amount on hydrate formation kinetics. Then, both thermodynamic conditions (pressure and temperature) at the maximum values of methane hydrate growth rate were predicted. Finally, the efficiency of some kinetic additives and some surfactants developed to avoid either nucleation or crystal growth and agglomeration of methane hydrates were evaluated. Kini et al. [30] measured CH4 + C3 H8 hydrate composition as a function of pressure, temperature, and vapor composition. Giavarini et al. [31] investigated the formation kinetics of propane hydrate, both from melting ice (at 274.15 K and 0.4 MPa) and from water (at 275.15 K and 0.36–0.48 MPa) in a stirred vessel. Lee et al. [32] examined methane–propane clathrate hydrate crystal growth within an enclosure partially filled with liquid water under different subcooling conditions both with and without the presence of n-heptane. Luoa et al. [33] carried out methane hydrate formation experiments in the presence of tetrahydrofuran (THF) in aqueous solution in a transparent bubble column in which a single pipe or a sintered plate was used to produce bubbles. They observed and investigated morphologically kinetics behaviors of hydrate formation on the surface of the rising bubble, the mechanical stability of the hydrate shell formed on the surface of the bubble and the interactions among the bubbles with hydrate shell. Further, they developed a kinetic model to correlate the experimentally measured gas consumption rate data. In this work, the kinetics of the hydrate formation of propane and its mixture with methane in a circulating flow reactor has been studied theoretically and experimentally. A mathematical model of the hydrate formation in the reactor has been developed and tested for propane and a mixture of 73% methane and 27% propane

Fig. 1. Flow reactor setup for the hydrate formation [10]. T1: reactor temperature, T2: gas temperature, T3: chiller temperature, P1 and P2: reactor pressure sensors, F: magnetic flow meter, PR: pressure regulator, DP1: differential pressure sensor, Pump 1: circulating pump, Pump 2: water injection pump, Pump 3: ice water pump

M. Sarshar et al. / Fluid Phase Equilibria 298 (2010) 38–47

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at pressure up to 2 MPa and temperature up to 277.65 K. The kinetics parameters of hydrate formation are investigated at different operating conditions. 2. Materials and methods A circulating flow reactor [4,10] is used to study the kinetics of hydrate formation. The reactor was made of 316 SS schedule 80 pipe with an ID of 10.6 mm and a pressure rating of 10 MPa. The reactor layout, shown in Fig. 1, is 1.5 m wide by 3 m long with a total length of 10 m [4]. Methane and propane are provided by Arish Gas Gostar Company (Ltd.) and their purities are 99.9%. Tap water is used as the aqueous phase. The procedure for the preparation of the methane and propane mixture, the instrumentations and the procedure of the gas flow measurement were described in [4]. The experimental method includes temperature, pressure and flow adjustments to provide a suitable condition for hydrate formation and growth. The experimental procedure is as follows: Reactor pressure is increased to the experimental values of 1 and 2 MPa for propane and 73C1 + 27C3 , respectively. Initial temperature of the reactor content is 289.15 K, which is outside the hydrate formation zone. To move to the formation zone, the temperature is decreased to the experimental values of 276.65 and 277.65 K for propane and 73C1 + 27C3 , respectively. A few minutes later, hydrate growth is started and continued through the rest of the operation. Within a few minutes of entering the zone, the induction period is started for the formation of a stable nucleus which must be passed to start the growth. The required time to pass the induction period is the induction time. Afterwards, hydrate formation is carried on, reducing the reactor pressure due to gas consumption. To overcome the pressure loss, gas is injected to fix the reactor pressure at the experimental value. At the mentioned operating conditions, the process of hydrate formation is monitored by measuring the gas consumption.

Fig. 2. Schematic representation of the pipe element [4].

2- Water is saturated by the gas 3- Primary nucleation is the dominating mechanism of hydrate nucleation 4- Gas components are dissolved in the aqueous phase, then reacted to form hydrate 5- Hydrate crystals are formed and suspended in the stream 6- Gas consumption rate is increased by the formation of hydrate crystals Based on the model representation of Fig. 2 and taking into account the above assumptions, the following partial differential equations are developed for the concentration of the gas components, water and the hydrate through the reactor [4]. g

g

∂Cj

=−

∂t

1 ∂(Q Cj ) − xrj S ∂Z

j = 1, . . . , n

∂C w 1 ∂(QC w ) =− − nw · r S ∂Z ∂t

(2)

1 ∂(QC h ) ∂C h =− +r S ∂Z ∂t

(3)

where the initial and boundary conditions are as follows: At t = 0 as the initial condition C g = C g (0) C w = C w (0)

3. Mathematical modeling of gas hydrate formation Modeling gas hydrate formation on the kinetics base can be a very interesting subject in the case of using inhibitors, because in comparing kinetics inhibitors to thermodynamic inhibitors a small dose will be required. This can be very important when one considers gas hydrates for storage and transport of natural gas and desalination of water. Due to the importance of hydrate formation in the gas pipelines and evaluation of low dosage inhibitors for hydrate formation, kinetics models of gas hydrate formation have become an issue of concern. The following models are based on the previous work of Sarshar et al. [4,10] which were applied to hydrate formation from carbon dioxide and a mixture of methane and propane. Mixed gases at system temperature and a pressure higher than the equilibrium hydrate formation pressure in the presence of free water, form hydrates which are suspended in the flow stream [4]. In this work, Sarshar et al. [4] methodology is followed to develop and implement the model for the new system. 3.1. Development of the mathematical models An element of the pipe segment that is used for the development of the mathematical models of gas hydrate formation in a reactor is shown in Fig. 2 [4]. Based on the operational conditions of the reactor, the following assumptions are taken into account for the model development. 1- Temperature is constant

(1)

Ch

(4)

=0 For all (t) as the boundary condition

g CZ=0 w CZ=0

= C g (0) w = CZ=L

(5)

h h CZ=0 = CZ=L

where L is the loop length. The solution of the above equations requires the driving force and reaction rate are to be expressed accurately. 3.2. The reaction rate The hydrate formation rate is directly proportional to the easily measured gas consumption rate. Kashchiev and Firoozabadi [12] developed the following equation for the rate of gas consumption in the hydrate formation reaction in a batch system. This expression is used as the rate of reaction for each element of the flow reactor (Fig. 2). r=

b h G3m J 3m t Mwh

(6)

3.3. Nucleation rate expression Nucleation rate is expressed by Eq. (7), which has been developed by Kashchiev and Firoozabadi [34]. J = A exp

 g  kT

exp

 −W  kT

(7)

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M. Sarshar et al. / Fluid Phase Equilibria 298 (2010) 38–47

Fig. 3. Nuclei formation on a solid substrate.

A is defined based on the mechanism of attachment of the hydrate building units to the nucleus. For heterogeneous nucleation on a solid substrate or at the solution/gas interface without nucleation active center, the parameter of A is calculated by Eq. (8) [34]. 1/3

A=

g

z ε(4 c)1/2 h DCe n˙ 1/3

(8)

aW

In the above equations, Z is the Zeldovich factor and is calculated by Eq. (9) [34].



Z=

W 3kT n˙ 2

(9)

n˙ is the number of building units constituting a nucleus and W is the work of cluster formation. They are calculated by Eqs. (10) and (11) [34]. n˙ =

3 8c 3 h2 ef

(10)

27g 3

W=

3 4c 3 h2 ef

(11)

27g 2

In the above equation  is the surface energy of the hydrate–solution interface and is a function of wetting angle of the nuclei on the solid substrate which is calculated by Eq. (13). The shape of the formed nuclei on a solid substrate is the same as a cap and is characterized by the wetting angle [34]. The schematic representation of nuclei formation on a solid substrate is shown in Fig. 3. ef =  =

(12)

1 4

(2 + cos ) (1 − cos )

2

1/3 (13)

The shape factor (b) of the nuclei on a solid substrate is calculated by b = 1/3 , where is determined by Eq. (13).

Fig. 4. Schematic representation of gas hydrate formation from aqueous phase.

unit in the supersaturated aqueous phase and the hydrate phase as shown in Fig. 4 and defined by Eq. (16). g =



aq

(16)

j=1

Anklam and Firoozabadi [36] derived a general expression (Eq. (17)) for the supersaturation of the crystallization of multicomponents gas hydrates in the aqueous solutions based on the above definition. g =





ni (T, Pe , xe )kT ln

i=1

g fˆi (T, P, y)



g fˆi (T, Pe , y)

+nw(T, Pe )vw(T, Pe ) − vh(T, Pe )

(17)

3.6. Determining the gas consumption The gas consumption for the hydrate formation reaction is calculated at each time step by integrating the hydrate concentration over the length of the reactor. Hence, the following equation is used to calculate the consumption.



Z=LR

GC(t) = 3.4. Growth rate expression

aq

Xj g + nw w − h

CH (t, z)

dR2 4

dZ

(18)

0

If the hydrate crystal is assumed as a sphere, its radius (rc(t)) is defined by Eq. (14) based on the power law growth rate.

3.7. Method of the solution

rc(t) = (Gt)m

The required parameters for the solution of the mathematical models consist of the reactor dimensions, physical properties, equilibrium values and the kinetics parameters. The physical properties are taken from the literature and the reactor dimensions are given to the program. The equilibrium values are calculated based on the Parrish and Prausnitz method [37] and the kinetics parameters are calculated by the expressions given in Section 3. The only parameter that is optimized to match the experimental data is the angle of , in which the nucleus makes with the solid substrate (Fig. 3). The major parameters are calculated and given in Table 1. The developed equations in the modeling section are solved using backward difference time domain method. The following steps are used in the simulation algorithm which is shown in Fig. 5.

(14)

In which m > 0 is a number and G (m1/m /s) is the growth constants and is calculated by model kinetics considerations.



G

m2 s



g





= 2ε h DCe eg/kT − 1 ,

m=

1 2

(15)

Eq. (15) is used for the growth by undisturbed volume diffusion of dissolved gas toward a spherical crystallite [34]. Diffusivity is calculated based on the Wilk Chang expression [35]. 3.5. Driving force of the gas hydrate formation As discussed previously, various definitions are presented for the driving force of hydrate formation but, the supersaturation (g) is one of the best definitions among them. The supersaturation is defined as the difference between Gibbs free energy of a hydrate

Step 1: The partial differential Eqs. (1–3) and their initial and boundary conditions (4–5) are converted to discrete equations

M. Sarshar et al. / Fluid Phase Equilibria 298 (2010) 38–47

43

Table 1 Operating conditions and major parameters of the Eqs. (1–21). Parameter

Definition

Propane

using backward finite difference time domain. Reaction rate is calculated by Eq. (6). The discretized concentrations of gas, water and hydrate are defined by Eqs. (19–21), respectively.



Ci+1,k+1

−1 1 = Ci,k + 1 + A¯ A¯



B¯ Ci+1,k + ri,k A¯

(19)

Reference

−5

8.8 × 10 0.461 10.6 10 1.0 ± 0.01 0.34 0.74 276.65 0.216 0.294 0.33 20 × 10−3 24.55

Reactor cross-section area (m ) Water molecule surface area (nm2 ) Reactor diameter (mm) Reactor length (m) Operating pressure (MPa) Calculated equilibrium pressure (MPa) Flow rate (m3 /hr) Operating temperature (K) Hydrate volume (nm3 ) Water molecular volume (nm3 ) Driving force Surface energy (J/m2 ) Angle of nucleus formation

S aw d LR P Pe Q T vh vw g/kT 

73C1 + 27C3

−5

2

8.8 × 10 0.461 10.6 10 2.0 ± 0.01 0.52 0.74 277.65 0.216 0.294 1.16 20 × 10−3 89.5



W Ci+1,k+1 =

−1 W 1 Ci,k + 1 + A¯ A¯

H Ci+1,k+1 =

−1 H 1 Ci,k + 1 + A¯ A¯



– (36  vw2 )0.5 – – – Hydrate equilibrium calculations – – [1] [38] Eq. (17) [11] Optimization

 

W Ci+1,k +

B¯ W n A¯

H Ci+1,k −

B¯ W n A¯



ri,k

(20)

ri,k

(21)

where −Q t A¯ = S Z B¯ = xw t homogeneous nucleation; B¯ = (aw/vw)t gas–liquid interface nucleation; B¯ = (4/d) nucleation on solid substrate; i and k stand for time and length steps, respectively. Step 2: At each time step, the rate of gas consumption is calculated by Eq. (6). Step 3: Nucleation rate and the growth rate are calculated by Eqs. (7) and (15), respectively. Step 4: Supersaturation or driving force is calculated by Eq. (17). 4. Results and discussion In this section, the results of the calculations are discussed separately for propane and 73C1 + 27C3 mixture. 4.1. Prediction of propane hydrate formation Kinetics parameters of propane hydrate formation reaction are analyzed at 276.65 K and different operating pressures of 0.4–1.0 MPa. Nucleus size, work for nucleus formation and the nucleation rate are calculated and shown in Figs. 6–8 as a function of pressure and wetting angle. Fig. 6 indicates that the required number of clusters forming a stable nucleus is directly proportional to the wetting angle and is indirectly proportional to the pressure.

Fig. 5. Algorithm of the solutions.

Fig. 6. Nucleus size as a function of pressure and wetting angle at 276.65 K.

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M. Sarshar et al. / Fluid Phase Equilibria 298 (2010) 38–47

Fig. 7. Work for nucleus formation as a function of pressure and wetting angle at 276.65 K.

At any fixed wetting angles, nucleus size is reduced sharply with increasing pressure up to dew point pressure of 0.5 MPa. Afterwards, it remains nearly constant with respect to pressure. A similar trend is observed in Fig. 7 where formation energy of nucleus is shown as a function of pressure and wetting angle ( ). It is shown that the required energy of nucleus formation is directly proportional to the wetting angle and indirectly proportional to the pressure. Formation energy of nucleus at any fixed wetting angles is reduced sharply with respect to pressure up to a dew point pressure of 0.5 MPa. After this, it remains nearly constant with respect to pressure. Pressure effects for larger wetting angles are more significant than the lower angles which can be explained by the wetting tendency of the nucleus on the substrate surface. The pressure has little effect on the nuclei with high wetting tendency or lower wetting angle. In Fig. 8 it is shown that, nucleation rate is directly proportional to pressure and indirectly proportional to wetting angle. Large nucleation rate is observed in the regions where nucleus size is small and formation energy is low. The above kinetics parameters are most affected by driving force and effective surface energy. Driving force is a function of pressure and temperature as expressed by Eq. (17) and effective surface energy is dependent on the wetting angle and hydrate–solution interface energy as shown by Eq. (12). Dependency of the driving force on the pressure at constant temperature of 276.65 K is shown in Fig. 9 and dependency of the factor on wetting angle is shown in Fig. 10. Driving force is sharply increased as the pressure rises up to dew point pressure of 0.5 MPa as shown in Fig. 9. At pressure above

Fig. 8. Nucleation rate as a function of pressure and wetting angle at 276.65 K.

Fig. 9. Driving force of hydrate reaction as a function of pressure at 276.65 K.

Fig. 10.

factor as a function of wetting angle.

0.5 MPa, the driving force is almost unchanged, indicating the negligible effect of pressure on the liquid fugacity. In Fig. 10 it is shown that the factor is directly proportional to the wetting angle, hence it can be concluded that the effective interfacial energy is highly reduced for the nucleus with higher wetting tendency or lower wetting angle. Simulation of propane hydrate reaction is carried out at an average value of 276.65 and 1.0 MPa. The predicted consumptions based on different angles of nucleus formation are given and are compared with the experimental data in Fig. 11. Best fit to the experimental data is achieved with = 24.55. Prediction error for the best fit is equal to 9.6%, which is calculated by Eq. (22), as the

Fig. 11. Measured and calculated propane consumptions at 1.0 MPa and 276.65 K in a hydrate reactor (AADE =24.55 = 9.60%).

M. Sarshar et al. / Fluid Phase Equilibria 298 (2010) 38–47

Fig. 15. Driving force of hydrate reaction as a function of pressure at 277.65 K.

Fig. 12. Nucleus size as a function of pressure and wetting angle at 277.65 K.

Fig. 13. Work for nucleus formation as a function of pressure and wetting angle at 277.65 K.

average absolute deviation error (AADE).



100 Cal.(i) − Exp.(i) n Cal.(i) n

AADE =

45

(22)

i=1

4.2. Prediction of 73% methane + 27% propane hydrate formation Kinetics parameters of 73% methane + 27% propane hydrate formation reaction are analyzed at 277.65 K and different operating pressures of 1.0–7.0 MPa. Nucleus size, work for nucleus formation and the nucleation rate are calculated and represented in Figs. 12–14 as a function of pressure and wetting angle. Fig. 12 indicates the required number of clusters forming a stable

Fig. 14. Nucleation rate as a function of pressure and wetting angle at 277.65 K.

nucleus is directly proportional to the wetting angle and is indirectly proportional to the pressure. At any fixed wetting angles, nucleus size is reduced sharply with increasing pressure up to 4 MPa. Afterwards it is gradually changed with respect to pressure. A similar trend is observed in Fig. 13 where formation energy of nucleus is shown as a function of pressure and wetting angle ( ). It is shown that the required energy of nucleus formation is directly proportional to the wetting angle and indirectly proportional to the pressure. Formation energy of nucleus at any fixed wetting angles is reduced sharply with respect to pressure up to 2 MPa. It then remains nearly constant with respect to pressure. Pressure effects for larger wetting angles are more significant than the lower angles which can be explained by the wetting tendency of the nucleus on the substrate surface. The pressure has little effect on the nucleus with high wetting tendency or lower wetting angle. In Fig. 14 it is shown that, nucleation rate is directly proportional to pressure and indirectly proportional to wetting angle. A large nucleation rate is observed in the regions where nucleus size is small and formation energy is low. The above kinetics parameters are most affected by driving force and effective surface energy. Driving force is a function of pressure and temperature as expressed by Eq. (17) and effective surface energy is dependent on the wetting angle and hydrate–solution interface energy as shown by Eq. (12). Dependency of the driving force on the pressure at constant temperature of 277.65 K is shown in Fig. 15, and dependency of the factor on wetting angle is shown in Fig. 16. Driving force is sharply increased as the pressure increases up to 4.0 MPa as shown in Fig. 15. At pressure above 4.0 MPa, the driving

Fig. 16.

factor as a function of wetting angle.

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M. Sarshar et al. / Fluid Phase Equilibria 298 (2010) 38–47

Fig. 17. Measured and calculated CH4 + C3 H8 consumptions at 2.0 MPa and 277.65 K in a hydrate reactor (AADE =89.5 = 10.05%).

force is gradually changed, indicating the slight effect of pressure on the liquid fugacity. In Fig. 16, it is shown that the factor is directly proportional to the wetting angle, and is concluded that the effective interfacial energy is highly reduced for the nucleus with higher wetting tendency or lower wetting angle. Simulation of 73% methane + 27% propane hydrate reaction is carried out at the average value of 277.65 and 2 MPa. The predicted consumption based on different angles of nucleus formation is given and compared with the experimental data in Fig. 17. Best fit to the experimental data is achieved with = 89.5. The prediction error for the best fit is equal to 10.05%, calculated by Eq. (22) as the average absolute deviation error (AADE). 5. Conclusion In this study, hydrate formation in a flow reactor has been studied theoretically and experimentally. All operating parameters such as temperature, pressure and flow are measured and monitored. The hydrate formation rate has been monitored in the experiments by measuring the gas consumption rate. In all experiments, different values of the induction time that usually happen in hydrate studies are observed. The experimental gas consumptions for hydrate formation reaction are calculated by numerical integration of the consumption rates with respect to time. The theoretical studies are carried out by developing a set of partial differential equations for the hydrate formation process in the reactor. The PDEs are developed based on the concepts of mass transfer, nucleation and equilibrium thermodynamics. All parameters such as kinetics, transport and equilibrium are calculated by the mentioned equations of Section 3. The only parameter which is optimized to match the experimental gas consumption is the wetting angle of the nucleus formation on the substrate. The predicted gas consumptions for propane and 0.73C1 + 0.27C3 at operating temperatures of 276.65–277.65 K and 1–2 MPa are compared with the experimental data to validate the model prediction. Gas consumptions are predicted based on the different values of the angle of nucleus formation to show the effect of the wetting angle on the model prediction. The best prediction belonged to the optimized value of the wetting angle. By the comparison of the predicted and experimental consumptions, it is shown that, the ability of the model to predict the process of hydrate formation is excellent.

A b B c C D d f g G GC J k m Mw Na n n˙ nw P Q rc R r ri S T t vw W X y Z z

kinetics parameter shape factor parameter of Eqs. (19–21) parameter of Eqs. (7–9) (36)1/3 concentration (mol/m3 ) diffusivity (m2 /s) pipe diameter (m) fugacity (MPa) Gibbs free energy (J/mol K) growth rate constant gas consumption (mol) nucleation rate (m−3 s or m−2 s) Boltzman constant parameter of Eqs. (14–15) molecular weight Avogadro number mole number (mol) number of hydrate building units hydration number pressure (MPa) volume flow rate (m3 /s) crystal radius (m) gas constant hydrate formation rate (mol/m3 s) hydrate radius (m) pipe cross-section area (m2 ) temperature (K) time (s) molecular volume of water (m3 ) work for cluster formation (J) hydrate phase composition gas phase composition longitudinal coordinate (m) Zeldovich factor equations (8) and (9)

Greek letters ε kinetics parameter  chemical potential (J/mol) wetting Angle  density (kg/m3 )  surface energy (J/m2 ) h volume of a hydrate building unit (m3 ) gas component parameter of Eqs. (12) and (13) Subscripts e equilibrium h hydrate I counter p nucleation active center w water ef effective Superscripts aq. aqueous phase g gas phase h hydrate phase W water Acknowledgments

Nomenclature a mass transfer area (m2 ) aw surface area of a water molecule (m2 ) A¯ parameter of Eqs. (19–21)

The financial support of Research and Development of the Iranian Oil Company is greatly appreciated. Also, the cooperation of Fars Engineering Research Center is acknowledged.

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Jamshid Fathikalajahi, PhD, is a Professor of Chemical Engineering, School of Oil, Gas and Chemical Engineering, Shiraz University. He is presently at Chemical and Petroleum Engineering Department, Sultan Qaboos University. His main research interests are kinetics and thermodynamics, mass transfer.

Feridon Esmaeilzadeh, PhD, is an Associate professor of Chemical Engineering, School of Oil, Gas and Chemical Engineering, Shiraz University. His main research interest is thermodynamics.