International Conference on Gas Hydrates (ICGH) (6th : 2008)

MODELING OF NATURAL GAS HYDRATE FORMATION ON A SUSPENDED WATER DROPLET Zhong, Dong-Liang; Liu, Dao-Ping; Wu, Zhi-Min Jul 31, 2008

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Proceedings of the 6th International Conference on Gas Hydrates (ICGH 2008), Vancouver, British Columbia, CANADA, July 6-10, 2008.  MODELING OF NATURAL GAS HYDRATE FORMATION ON A SUSPENDED WATER DROPLET  Dong-Liang Zhong*, Dao-Ping Liu, Zhi-Min Wu College of Power Engineering University of Shanghai for Science and Technology 516 Jungong Road, Yangpu District, Shanghai, 200093 CHINA  ABSTRACT After reviewing the documents about the studies of hydrate formation kinetics in the world, this paper analyzed the process of hydrate formation on a suspended water droplet, which was based on the hydrate formation with water spay method, proposed a corresponding mathematical model, and solved it. Afterwards, the discussion about this model was presented. The results indicated that equilibrium time diminished with the decrease of the water droplet radius, and prolonged with the increase of sub-cooling degree, 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. Therefore, shortening the duration time of the second period of hydration was obviously able to accelerate the hydrate formation on the water droplet. Keywords: water droplet; gas hydrate; formation kinetics; mathematical model NOMENCLATURE A Surface area [m2] -1 c Specific heat capacity [kJ/kg•K ] hi Convection heat transfer coefficient [W/m2•K] ΔH Hydration heat per unit mass [kJ/kg] M Molar mass [kg/mol] n Molar number [mol] Q The heat transferred to gas phase [kW] Q1 Hydration heat generated inside the hydrate film [kW] Q2 Hydration heat generated outside the hydrate film [kW] r Radius of the water droplet covered by the  hydrate film [m] t Growth time of the hydrate [s] Teq The phase equilibrium temperature [K] T The temperature of the water droplet [K] ΔT Subcooling [K] Vw The volume of the consumed water inside the hydrate film per unit time [m3/s] ZV The volume expansion coefficient ρ Hydrate density [kg/m3] λ Thermal conductivity [W/m•K] Subscripts w Water Gas hydrate H  *Corresponding author, Phone: +86-21-55270305 Fax: +86-21-55270305 E-mail: azhongdl@hotmail.com  i  0  Inside of the gas hydrate Initial time  INTRODUCTION Gas hydrates, also called clathrate hydrate is a sort of nonstoichiometric crystalline composed of water and gases with small-sized molecules [1], like CH4, C2H6, CO2, H2S, etc. Currently, gas hydrate technology is being widely used in the fields of storage and transportation of natural gas, seawater desalination, carbon dioxide sequestration, and cold storage air-conditioning [2], etc., therefore, it is substantially important to perform studies on gas hydrate. Kinetics research of gas hydrate started much later than the thermaldynamic research. The process of gas hydrates formation is quite complex, because it refers to heat and mass transfer among gas phase, liquid phase, and solid phase, meantime, a large amount of hydration heat is generated, and the whole process is influenced by many factors, so the kinetics research has progressed slowly. Recently, advanced observation and measuring methods have been adopted in the researches of hydrate formation kinetics worldwide. Sugaya and Mori [3] investigated the formation of refrigerant hydrate (CF3CH2 and CClFCH3) with a high-speed photomicrography camera. Sun Chang-yu [4] et al. studied Freon-12 hydrate formation in a circulating flow loop system using light scattering technique, and Ma Chang-feng [5] et al. utilized a microscope camera to investigate the hydrate formation process of bubbles suspended in water. Ju Dong Lee [6] et al. conducted studies of gas hydrate formation and decomposition on water droplets using an 89.4% methane-10.6% ethane mixture, and a 90.1% methane-9.9% propane mixture. Ryo Ohmura [7] reported the visual observation of the formation and growth of structure-H hydrate crystals on a water drop partially exposed to methane gas and partially immersed in a pool of a liquid large molecule guest substance (LMGS). However, they did not present a mathematical  model for the formation of gas hydrate on water drops, and quantitative descriptions of the process were absent. This paper analyzed hydrate formation process on a suspended water droplet on the basis of hydrate formation with water spay method, proposed a corresponding mathematical model, and presented the solution and discussion about the model, hoping to provide some valuable information for researchers in the circle of gas hydrates. DESCRIPTION OF THE HYDRATE FORMATION ON A WATER DROPLET Gas hydrates might form on the surface of the water droplets which were in the direct contact with hydrate-forming gases at a certain temperature and pressure, and the formation process was able to be divided into three periods: (1) gases contacted water droplets directly, and then the hydrate film formed quickly on the surface of water droplets after partial dissolution of the gas into water phase, covering the entire surface of water droplets. (2) Gases permeated through the porous hydrate film to the interface between the hydrate film and water droplet, and continued to form hydrates. Because the volume of the formed hydrates was larger than that of consumed water, partial water molecules diffused to the outer surface of the hydrate film. (3) Hydration on the inner and outer surfaces of the hydrate film occurred simultaneously, and a large number of hydrates formed. The schematic of hydration process of a water droplet was presented in Figure 1.  r2  gas film  r1  hydrate film unreacted w ater droplet  Figure 1 Schematic of hydration process of a water droplet  MODELING If hydration heat released from the hydrate growth surface was removed very fast, the hydrate formation process would be controlled by the mass transfer rate of hydrate-forming gases onto the growth surface. Contrarily, the formation process would be controlled by heat transfer rate [8]. Since in the water spray reactor the sprayed water droplets were super fine, the hydrate film formed on a water droplet at the beginning was very thin and of a kind of porous medium, so the resistance of mass transfer was considerably low. Consequently, the mathematical model to describe hydrate formation on a suspended water droplet which was established under such condition was controlled by heat transfer rate. First period of hydrate formation Hydration heat released from the inner side of the hydrate film was totally transferred to the unreacted cooler water, so an equation based on the energy conversation law was formulated as 4  r2  2  dr2 dt   H  H  4 r2 hi (Teq  T w )  r2  r0 , T w  T w 0 , r2  r2 eq , T w  T eq ,  2  The temperature of the unreacted water droplet at any time was determined by the lumped parameter method, expressed as:  T w  Teq  (T w 0  Teq ) exp(   hi A w 0   w c wV w 0  t)  (4)  Combining Eqs (1) and (4), equation (5) is formulated as  dr2 dt   H  H  hi (Teq  T w 0 ) exp(   hi A w 0   w c wV w 0  t ) (5)  Integrating Eq.(5) on the basis of Eqs.(2) and (3), so equation (6) is obtained.    c V T hi A r2  1  exp(  t) w w w0  r0 (6)  w c wV w 0   H  H A w 0   (1)  t  0  (2)  at t  t eq  (3)  at  respectively. Subscripts w , H , i and 0 represent water, gas hydrate, inside of gas hydrate, and initial time, respectively.  Therefore, the time for water droplets to reach the phase equilibrium temperature is deduced as:  t eq    Where, r2 represents the radius of the water droplet covered by the hydrate film, t is the growth time of the hydrate,  H is the density of the hydrate,  H is the hydration heat per unit mass, hi is the  convection heat transfer coefficient, Teq is the phase equilibrium temperature of the hydrate under the experiment pressure, Tw is the temperature of the water droplet, and teq and r2eq represent the elapsed time and the radius of the unreacted water droplet when the water droplet reached phase equilibrium temperature,   w c wV w 0 hi A     H Aw 0  ln 1  ( r2 eq  r0 ) H   w c wV w 0  T    (7) When the temperature of the water droplet reached the phase equilibrium temperature, hydration heat generated on inner reaction front should be equivalent to the amount of heat which was absorbed by the water droplet to reach the phase equilibrium temperature, hence, 4 4 4 3 3 3 (   r0    r2 eq )  H  H   r2 eq  w c p (Teq  T w 0 ) 3 3 3  (8)  So r23eq   H  H  was  the   r0 , the outer radius 3  H  H  T  wc p  of  generated  r1 eq  [   H  H (1  Z V )  3  And the heat transferred to gas phase from outer reaction front was the total hydration heat, so  H  H  T  wc p  where  T  Teq  T w 0 ,  ZV  Q  ng M H H  (12)  hydrates  Z V ]  r0  3  ,  was the volume  Equation (13) was obtained from the combination of equations (9) to (12).  ng   4  h0 Z V  H  T  M H H  1 h0 (  expansion coefficient, i.e. Z V  V H / V w .  1    r2  1 r1  )  ZV H  1 r1  2  (13) Second period of hydration formation During this period, heat Q transferred to gas phase from the water droplet was the total hydration heat from inner and outer reaction fronts, i.e. Q  Q1  Q 2 . According to the analysis with mass  transfer theory, it was known that if the volume of the consumed water inside the hydrate film per unit time was V w , the volume of the water diffused outside hydrate film would be V w ( Z V  1) . The  Because  of  h0 (  4 h0 r1  T   ng   r2    1  Mw  w  (10) Therefore,  r1  1 r1  ,  2  acquired,  where  water droplets and gas occurred by stoichiometric coefficient n, equation (14) was formulated.  r2  r2 eq at t  t eq  1  r1  )  Z V  H   T   Teq  T g . Because the reaction between  Q 2  Q1 ( Z V  1) 4  H (T eq  T1 )  1  was  M H H  n  ng   Q1   r2    2  relationship between the hydration heat generated inside and outside hydrate film was expressed as:  (9)  1  dr2   Z V  4  r2  2  dr2  (14)  dt  (15)  nh0 M w  T   r1 eq 2     w M H  H Z V r2  2  dt  was deduced,  total hydration time for a water droplet was Concerning a suspended static water droplet, the heat transferred to ambient gas phase was formulated as Q  4  r1 h 0 (T1  T g ) 2  (11)  t   w M H HZV 3 nh 0 M w  T r  3 1 eq  ( r2 eq  r2 )  t eq 3  3  (16)  Therein, reaction time for second period of hydrate formation was  3 nh 0 M w  T r1 eq  ( 0  r2  r2 eq )  3  ( r2 eq  r2 ) 3  3  (17)  MODEL SOLVING AND DISCUSSION Relation between the end time teq of first period and water droplet radius r0 Figure 2(a) and Figure 2(b) indicated that teq diminished with the decrease of water droplet radius r0 at the same sub-cooling degree. It was showed in Figure 2(a) that when r0 was at the value of 2.0mm, 1.5mm and 1.0mm, teq was at 90.0s, 70.0s and 45.0s, respectively; in Figure 2(b) when r0 was at the value of 0.2mm, 0.15mm, 0.1mm and 0.06mm, teq was at 9.2s, 7s, 4.6s and 2.8s, respectively. Therefore, it should be noted that the smaller the water droplet, the shorter the first stage lasts. In addition, it was shown in Figure 2 that despite of the size of the water droplet, hydrate film formed on the water droplet stopped at the same relative position by the end of the first period, that is, the thickness of the hydrate film was about 0.261r0 at the end of the first stage. Accordingly, it was indicated that in water spray devices hydrate film would quickly formed on water droplets as soon as water droplets with the diameter of micron grade left the nozzle, which was one of the advantages of producing hydrate with water spray method. Relation between the end time teq of the first stage and the subcooling degree ΔT As illustrated in Figure 3, teq has increased with the rise of subcooling degree ΔT when the initial water droplet radius was set at a fixed value. When the initial radius r0 of the water droplet was at the value of 2.0mm, teq was 73.0s, 82.0s, 90.0s and 121.0s corresponding to the subcooling degree Δ T of 1.0℃, 2.0℃, 4.0℃ and 6.0℃, respectively. Besides, by the end of the first period the thickness of the formed hydrate film has varied from the  subcooling degreeΔT, for example, the thickness of the hydrate film formed on the water droplet was 0.13mm, 0.26mm, 0.52mm and 0.78mm corresponding to the subcooling degree ΔT of 1.0 ℃ , 2.0 ℃ , 4.0 ℃ and 6.0 ℃ , respectively. Therefore, the reaction time went up with the increase of subcooling degree ΔT, i.e. teq has increased. 100 2mm 1.5mm 1mm  90 80 70 60  Time (s)   w M H HZV  50 40 30 20 10 0 0.6  0.8  1  1.2 1.4 Radius (m)  1.6  1.8  2 -3  x 10  (a) 10 0.2mm 0.15mm 0.1mm 0.06mm  9 8 7 6  Time (s)  t2   5 4 3 2 1 0 0.4  0.6  0.8  1  1.2 1.4 Radius (m)  1.6  1.8  2  2.2 -4  x 10  (b) Figure 2 Diagram of the relation between teq and water droplet radius r0 (ΔT= 4℃) Relation between the end time t2 of the second period and the subcooling degree ΔT When the initial radius r0 of the water droplet was at the value of 0.2mm, 0.15mm and 0.1mm, respectively, by the end of the first period the radius of the unreacted water droplet had been calculated with Eq.(7), i.e. r2=0.833 r0, as shown  in Fig. 4. Seen from the diagram, t2 did not change with the variation of the radius of the water droplet under the condition of the same subcooling degree, it was evidently indicated that t2 was at a fixed value. t2 turned out to be 6800s when ΔT was at the value of 4.0 ℃ . It concluded that blindly decreasing the radius of water droplets could not be able to obtain a desirable effect on hydration formation in water spray reactors.  140 ΔT=1℃ ΔT=2℃ ΔT=4℃ ΔT=6℃  120  Time (s)  100  80  60  40  17500s, 6800s and 4000s corresponding to the subcooling degree ΔT of 1.0℃, 2.0℃, 4.0℃ and 6.0℃, respectively. It was indicated that increasing the subcooling degree could be able to shorten the process of complete hydration of the water droplet. Actually, subcooling degree could not be lifted infinitely, because the requirements for the insulation of the experimental equipment and the performance of refrigeration system would be much higher, and tiny water droplets might have freeze into ice particles before hydration occurred owing to the huge subcooling degree, which has exceeded the research scope of this paper. Consequently, it should be noted that under a specific situation a proper subcooling degree has to be set in the research of hydration on water droplets.  20  4  x 10 4 1.3  1.4  1.5  1.6 Radius (m)  1.7  1.8  1.9  Figure 3 Diagram of the relation between teq and subcooling degree ΔT (r0=2mm) 7000 0.2mm 0.15mm 0.1mm  3 2.5 2 1.5 1 0.5  5000  0  4000 Time (s)  3.5  x 10  6000  ΔT=1℃ ΔT=2℃ ΔT=4℃ ΔT=6℃  2 -3  Time (s)  0 1.2  -0.5 3000 2000  0  0.2  0.4  0.6  0.8 1 1.2 Radius (m)  1.4  1.6  1.8  2 -3  x 10  Figure 5 Diagram of the relation between t2 and subcooling degreeΔT (r0=2mm)  1000 0 -1000  0  0.2  0.4  0.6  0.8 1 Radius (m)  1.2  1.4  1.6  1.8 -4  x 10  Figure 4 Diagram of the relation between t2 and radius of the water droplet (ΔT=4℃) Relation between the end time t2 of the second stage and the subcooling degree ΔT It was observed that t2 had reduced with the increase of the subcooling degree ΔT, as shown in Figure 5. When the initial radius r0 of the water droplet was at the value of 2.0mm, t2 was 40000s,  Complete hydration time t of the water droplet Eq.(16) indicated that the total hydration time t of a suspended water droplet consisted of the end time teq of the first period and the end time t2 of the second period. Through solving the above mathematical model we have found that the duration time of the first period was considerably short, varying from 1.0s to more than 100.0s, but the duration time of the second period was much longer, changing from 1.0 hour to more than 10.0 hours. As a result, in order to reduce the complete  hydration time of a water droplet, it is proposed to focus on lowering the duration time of the second period, like increasing the subcooling degree, etc., so as to accelerate the process of gas hydrate formation. CONCLUSION The process of hydrate formation on a suspended water droplet was analyzed based on the hydrate formation with water spay method, and a corresponding mathematical model to describe this process was proposed in this paper. The process of the hydrate formation was divided into two periods, in the first period hydration heat generated on the inner side of the hydrate film was totally transferred to the unreacted cooler water; and in the second stage, hydration heat released from the inner and outer reaction fronts on the water droplet was completely transferred to the ambient gas phase. Through the model solving and the discussion of the calculated results, it was found that the end time teq of the first period had diminished with the decrease of water droplet radius r0, and had increased with the rise of the subcooling degree ΔT; the end time t2 of the second period had reduced with the increase of the subcooling degree Δ T, but was free of the influence of the variation of the water droplet. In addition, the duration time of the first period was pretty short, but the duration time of the second period was much longer. Thereby, in order to accelerate the hydration of the water droplet, shortening the duration time of the second period was substantially necessary and important. REFERENCES [1] Sloan E D. Clathrate of natural gas. New York: Marcel Dekker, 1998.  [2] Fan SS. Technology in the storage and transport of natural gas hydrate. Beijing: Chemical Industry Press, 2005. [3] Sugaya M, Mori YH. Behavior of clathrate hydrate formation at the boundary of liquid water and fluorocarbon in liquid or vapor state. Chemical Engineering Science1996;51: 3505-3517. [4] Sun CY, Chen GJ, Guo TM. R12 hydrate formation kinetics based on laser light scattering technique. Science in China (Series B) 2003;46 (5): 487-494. [5] Ma CF, Chen GJ, Guo TM. Kinetics of hydrate formation using gas bubble suspended in water. Science in China (Series B) 2002;45 (2):208-215. [6] Lee JD, Susito R, Englezos, P. Methane-ethane and methane-propane hydrate formation and decomposition on water droplets. Chemical Engineering Science 2005; (60):4203-4212. [7] Ohmura R, Sadatoshi M, Sinya I, Ebinuma T, Narita H. Formation and Growth of Structure-H Hydrate Crystals on a water drop in contact with methane gas and large-molecule guest-substance liquid. Proceedings of the 5th International Conference on Gas Hydrate, Norway, 2005. [8] Gnanendran N, Amin R. Modelling Hydrate Formation Kinetics of a Hydrate Promoter-Water-Natural Gas System in a Semi-batch Spray Reactor. Chemical Engineering Science 2004;59:3849-3863.  

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