Study of the Dynamics of a Condensing Bubble Using Lattice Boltzmann Method

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1 Study of the Dynamcs of a Condensng Bubble Usng Lattce Boltzmann Method Shahnawaz Ahmed 1, Sandeep Sreshth 1, Suman Ghosh 1 and Arup Kumar Das * 1 Department of Mechancal Engneerng, NI Rourkela, Rourkela, Inda Department of Mechancal Engneerng, II Kharagpur, Kharagpur, Inda 117 Receved: 6 October 013; Accepted: 15 January 015 Abstract Mesoscopc lattce Boltzmann method (LBM) s used to dscretze the governng equatons for a steam bubble nsde a tube flled wth water. he bubbles are kept at hgher temperature compared to ts bolng pont whle the lqud s kept subcooled. Heat transfer s allowed to take place between the two phases by vrtue of whch the bubble wll condense. hree separate probablty dstrbuton functons are used n LBM to handle contnuty, momentum and energy equatons separately. he nterface s consdered to be dffused wthn a narrow zone and t has been modeled usng convectve Cahn-Hllard equaton. Combned dffused nterface-lbm framework s adapted accordngly to handle complex nterface separatng two phases havng hgh densty rato. Developed model s valdated wth respect to establshed correlatons for nstantaneous equvalent radus of a sphercal condensng bubble. Numercal snapshots of the smulaton depct that the bubble volume decreases faster for hgher degree of superheat. he degrees of superheat are vared over a wde range to note ts effect on bubble shape and sze. Effect of ntal volume of the bubble on the condensaton rate s also studed. It has been observed that for a fxed degree of superheat, the condensaton rate s not exactly proportonal to ts volume. Due to the varaton n nterfacal confguraton for dfferent szed bubbles, condensaton rate changes drastcally. Influence of gravty on the rate of condensaton s also studed usng the developed methodology. 1. INRODUCION Multphase flows have been gven due attenton as they are often encountered n nature as well as n dfferent ndustral processes. Applcatons of multphase flows are plenty lke combuston, chemcal reactons, bolng, petroleum refnng and n other heat and mass transfer processes. Wth the mplcaton of engneerng knowledge n daly lfe problems, multphase flow s becomng more relevant for dfferent attractve felds lke fllng the fuel tank of a Formula 1 racng car wthn few seconds, ol jet splashng wthn the cylnders of racng engnes, etc. he condensaton of gaseous bubble plays a key role n many areas of techncal mportance. hs s commonly found n steam condenser n power plant, and other bochemcal and metallurgcal processes. Drect contact heat exchangers between gaseous bubble and surroundng mmscble lqud also have attractve ndustral applcatons due to hgh rate of heat transfer. Numerous examples are also found n several heat and mass transfer processes where heat transfer from gaseous bubble takes place wth the surroundng flud. Heat transfer from a gaseous bubble nsde a crcular tube s even more nterestng due to formaton and destructon of space and tme varant nterfaces. Examples are plenty where bubbles durng translaton through a tube causes heat transfer or phase change. Due to ts wde applcaton, bubble dynamcs n lqud column has been studed by number of researchers both expermentally by ung and Parlange [1], Kataoka et al. [], Polonsky et al. [3] and theoretcally by Bretherton [4], udose and Kawaj [5]. A lot of numercal works have also been reported to study the nterfacal dynamcs of the bubble n lqud ambent. In conventonal CFD models of two phase flow, a set of Naver- Stokes equatons are solved and the nterface s captured by usng volume of flud (VOF) or *Correspondng author: E-mal: arupdas80@gmal.com

2 118 Study of the Dynamcs of a Condensng Bubble Usng Lattce Boltzmann Method level set methods. But nterface reconstructon usng VOF becomes complcated for three dmensonal cases whereas level set method volates mass conservaton for ts applcaton n large topologcal changes. Recently, lattce Boltzmann method (LBM) has emerged as an alternate and promsng tool for smulaton of complex two-phase flow problems as compared to the conventonal CFD solver for Naver Stokes equaton. It takes care about the features of the mcro-scale or meso-scale as well as conserves the macroscopc varables. LBM can handle multphase systems and complex nterfaces wth ease and effcency. Frsch et al. [6] frst proposed mcroscopc lattce gas cellular automaton and could reproduce the flud flow stuatons. McNamara and Zanett [7] used Bhatnagar-Gross-Krook [8] collson operator to make the methodology more robust and devod of statstcal nose. Chen and Doolean [9] and Succ [10] have reported a thorough summary of the LBM approaches and ts subsequent development over the years. Several approaches from Shan and Chen [11], Swft et al. [1], Gunstensen and Rothman [13] have been proposed to model nterfacal dynamcs n two-phase flow usng LBM. One of the earlest mult component model s developed by Rothman and Keller [14] usng color method. hey have used two dfferent partcle dstrbuton functons to represent two phases. Later Shan and Chen [11] have used potental method to smulate two-phase flow. However, these are phenomenologcal models. Swft et al. [1] made an mprovement n the model by suggestng a free energy approach. In ths model, the equlbrum dstrbuton s defned based on thermodynamcs and conservaton of energy s satsfed. Lee and Ln [15] developed a stablzed scheme of dscrete Boltzmann equaton for multphase flows wth large densty rato. However, t does not completely recover the lattce Boltzmann equaton for nterface to Cahn Hllard equaton. On the other hand, large densty rato between the fluds can cause numercal nstablty. o overcome ths dffculty, Inamuro et al. [16] proposed a model, based on the free energy method for multphase flows wth large densty rato. In a recent study by Zheng et al. [17], another model for large densty rato of two fluds has been proposed. In ther model the lattce Boltzmann equaton for nterface recovers Cahn-Hllard equaton. hey have modeled bubble dynamcs for dfferent range of assocated non-dmensonal parameters. But stll a proper descrpton of nterfacal behavor of the bubbles for dfferent magntudes of superheats and subcoolng s not clearly establshed. hs paper thus ams at numercally understandng the physcal behavor of the bubbles durng condensaton and bolng. Dffused nterface based lattce Boltzmann model as proposed by Zheng et al. [17] has been taken as the tool for the smulaton. Recently, Dong et al. [18] combned the model of Zheng et al. [17] wth necessary energy transport equaton to model nucleate bolng phenomena. Followng the approaches gven by Zheng et al. [17] and Dong et al. [18], n ths paper, we have studed the phenomenon of condensaton and bolng.. MODEL DESCRIPION A basc model to track the two-phase flow s developed based on lattce Boltzmann methodology. Dffused nterface concept s consdered to track the nterfacal behavor. As the phenomena s assocated wth flow of two fluds or phases wth dfferent denstes, ther respectve denstes are consdered as r L and r H respectvely. o decouple the nterfacal dynamcs from the bulk moton modfed momentum equatons are consdered. In these equatons, two derved propertes n and f are used whch can be defned as [19]: = ρ + ρ = ρ n ρ L H H L, (1) Usng the above derved quanttes, the flow can be descrbed by mass and momentum conservaton equatons and an nterface evoluton equaton as [19] n ( ) +. nu t = 0 () nu.( nuu) μ po μ Fb μ u t + = (3) Journal of Computatonal Multphase Flows

3 Shahnawaz Ahmed, Sandeep Sreshth, Suman Ghosh and Arup Kumar Das 119 ( ) θ μ (4) + = u M t where, P s the pressure tensor, q m s the moblty, m f s chemcal potental of the flud and F b s the body force. he nterface s consdered to be dffused wthn a narrow zone and t has been modeled usng convectve Cahn-Hllard equaton (Eq. (4)). In order to explan Eq. (4), n lattce structure, a modfed lattce Boltzmann equaton s adopted usng probablty dstrbuton functon g as: g + δ +Δ = + δ + Ω ( x c t, t t) g(,) x t (1 q) g( x c t,) t g(,) x t g where, last term of Eq. (5) s the collson operator and can be wrtten as: eq g ( x, t) g ( x, t). Ω g = + w τ t f s dmensonless relaxaton tme, c s lattce velocty and q s a constant coeffcent to determne the mplct parameter. w s the correspondng weght and f s the mass transferred due to phase change. he value of t f depends on the physcal propertes of the flud pars and accordngly the value of mplct parameter s chosen. he value of q s related wth t f as: (5) (6) 1 q = τ (7) After calculatng the drectonal g for the lattce, the macroscopc order parameter f s evaluated as: = g (8) Eq. (5) recovers Cahn-Hllard equaton wth second order accuracy as shown n the followng equaton: + ( u) θ μ + O( δ ) = 0 (9) t In ths equaton, moblty q M s defned usng lattce propertes as: M θ M = q τ Γ q 1 (10) Here, G s used to control the moblty of the flud par. In the present model DQ5 lattce structure has been consdered durng the propagaton of the nterfacal nformaton. Equlbrum dstrbuton functon of g can be calculated n DQ5 structure as: he coeffcents n Eq. (11) are taken as: eq g = A + B+ Cc u B 1 = 1, B = 0 ( 1) (11) C = 1 q (1) A = Γμ Volume 7 Number 015

4 10 Study of the Dynamcs of a Condensng Bubble Usng Lattce Boltzmann Method As the nterface s related to surface tenson across t, the surface tenson force can be calculated as: F = P = μ p s where, p o s the statc pressure and s gven by, p 0 = nc s c s s the speed of sound. he chemcal potental m f descrbed n Eq. (9) s computed by usng followng relatonshp: o (13) ( ) 3 * μ = A 4 4 κ (14) where A s the ampltude parameter used to control the nteracton of energy between the two phases and k s the curvature. Due to the ncorporaton of dffused nterface, varaton of f near the nterface s calculated as: ( ) = * tanh ζ W z s the coordnate whch s perpendcular to the nterface and W s the thckness of the nterface layer. hs can be expressed as: (15) W = κ / A * (16) Here, the expected order parameter s: * ρ = H ρl (17) he pressure tensor s calculated as: ( ) = ( ) κ * * n p A (18) Surface tenson of a flud par s related wth the derved propertes as: σ = κ ζ ζ d (19) hus the potental form of surface tenson related term s ndependent of the average densty and densty dfference. It s obvous that m f z f s related wth the surface tenson coeffcent and the wdth of nterface layer due to the ncorporaton of dffused nterface concept. For dscretzaton of mass and momentum conservaton equaton DQ9 lattce structure s used as proposed by Zheng et al. [17]. Lattce Boltzmann mplementaton of Eq. ( and 3) can be gven as: wth collson parameter as: f ( x+ c δt, t+ δt) = f ( x, t) +Ω f (0) Ω = f eq f ( x, t) f ( x, t) + τ n 1 w cu. 1 ( ) + μ + δ τ c u c c c ( F ) t B n s s (1) Journal of Computatonal Multphase Flows

5 Shahnawaz Ahmed, Sandeep Sreshth, Suman Ghosh and Arup Kumar Das 11 he equlbrum dstrbuton functon n Eq. (1) can be wrtten as: eq 3 9 f = wa + wn + 3cα uα u uαuβ cα cβ () where, the coeffcents are as follows: 9 15( μ + n / 3) A1 = n 4 4 A = 3( μ + n/ 3) (3) 4 w = w = = w = = 9, 1 9, 1 1,3,4,5 6,7,8,9 36 he energy equaton employed s gven as δ ρ τ + = u 1 G t x 3 x ρ ( ρ ρ ) Ja L L G. (4) he energy equaton n terms of partcle dstrbuton functon s recovered usng Inamuro et al. [16] model. hey proposed a model for the dffuson system n whch there s the smplest dstrbuton functon h among other thermal models. he LBM equaton s: ( x+ c δt, t+ δt) = h( x, t) +Ω h h (5) where, and Ω = heq, ( x, t) h( x, t) ρg + ( τ ρ ρ ρ ) h w Ja l g. ( 1 3 ) eq h = w + + c uα l (6) (7) A vapor bubble of volume V b s ntroduced n a lqud. In tme nterval Dt the mass transferred nto the bubble s expressed as: V' m = ρ t' dv' G 1 = h fg 1 = h fg dv b' dv' dt ' λl S' x' V' b ' λl x' ds' V ' dv' (8) where, r G stands for vapor densty, for temperature, h fg for the latent heat and l l for thermal conductvty of lqud. Based on the phase order parameter, the phase-change s consdered as: ρ ρ ρ ρ = Δ Δ = ( L Δm) ( G + Δm) L G t Δ t Δ t = Δ m Δ t (9) Volume 7 Number 015

6 1 Study of the Dynamcs of a Condensng Bubble Usng Lattce Boltzmann Method Eq. 8 s normalzed by means of V V = b b V bo tu, t = d = = x x, x de b e (30) V bo s the bubble volume at an ntal stage, d e s the equvalent dameter of the bubble, U s termnal rsng velocty and s the temperature of surroundng lqud. he dmensonless form of the Eq. 8 s wrtten as: ρ G dv dt λ ( = l b ) h U fg d e x (31) Mass transfer due to phase change can be defned as: ρ = ρ ρ ρ l Ja ( l g ) g Pe x (3) where, r l and r g are the denstes of lqud and gas (bubble), respectvely, and Ja and Pe are Jacoban and Peclet number gven by: 1 Ja = h c pl jg ' ' ( b ) (33) ρ UdC Pe = λ L e pl ρ G When f < 0, the latent heat term can be added nto Eq. 8 to obtan soluton ρl( ρl ρg) Ja for bolng and same term must be added when f > 0, to obtan soluton for condensaton. hus we obtan: l (34) h ( x+ cδ t, t+ Δ t) = h ( xt, ) h eq, ( xt, ) h ( xt, ) ρ + + w G ( ρ ρ ) τ ρ Ja. L G L (35) 3. RESULS AND DISCUSSION After the valdaton of the developed numercal code effect of dfferent subcoolng of the lqudgas par on bubble shape s nvestgated. he phenomenon of condensaton s studed n detal wth sngle bubble surrounded by lqud. he numercal results are valdated wth the correlaton gven by Mkc et al. [0]. Mkc et al. [0] proposed correlaton for nstantaneous radus of a sphercal evaporatng bubble. In order to valdate, water and water vapor are consdered as contnuous and dscrete phases respectvely. Intally the vapor s consdered to be at 10 C whle water s at saturaton temperature (100 C). Smulatons have been done for dfferent ntal bubble radus and fnal rad have been obtaned. Fnal radus of the bubble has also been found out from Mkc et al [0] correlaton and tabulated n able 1. From the percentage error one can get the predctablty of the model. Journal of Computatonal Multphase Flows

7 Shahnawaz Ahmed, Sandeep Sreshth, Suman Ghosh and Arup Kumar Das 13 Usng the developed methodology, the phenomenon of condensng bubble s studed n detal. he vapor bubble s kept at saturaton temperature of 100 C whle surroundng water s at lower temperature. Frst, evoluton of an unbounded vapor bubble n subcooled water wthout gravty s demonstrated. Effect of wall wthout gravty stuaton s studed next. Parametrc studes such as volume and degrees of subcoolng on the bubble shrnkage are establshed. Fnally the condensaton and dynamcs of a vapor bubble n water s smulated under the acton of gravtatonal pull. For the smulatons n ths secton t has been consdered that the temperature of the surroundng lqud s not changng durng condensaton. hs replcates Stefan s boundary condton n lqud state. At frst, to establsh the extncton of nterface usng the developed mode a very smplfed case of unbounded condensng bubble s smulated. An 8mm dameter saturated (100 C) vapor bubble s placed n the atmosphere of subcooled lqud water. Water temperature s consdered to be kept at 60 C. A doman sze of lattces are consdered wth perodc boundary condton to replcate unbounded doman. Propertes requred for smulaton has been taken for standard water. Numercal smulaton s performed and the contours of the bubble after, 4, 6, 8, 10 s are shown n Fgures 1(a-f) respectvely. As the saturated bubble s at hgh temperature, heat s transferred from bubble to subcooled water. As a result vapor bubble loses ts heat and starts condensng. Due to condensaton vapor transforms nto lqud. It s observed that wth tme the sze of vapor bubble decreases and vapor gets converted nto water. hs happens due to the subcoolng between water and saturated vapor. Water vapor gets converted nto lqud water by losng necessary latent heat from tself and rsng the temperature on mmedate lqud near the nterface. Next, the effect of condut wall durng the shrnkage of bubble s smulated usng the proposed methodology. A 7mm dameter saturated vapor bubble s placed n subcooled (60 C) water nsde a tube havng 1mm dameter. Effect of gravtatonal acceleraton s neglected. A mesh sze s employed for the smulaton. In ths case wall boundary condtons are employed at the left and able 1. Valdaton wth Mkc et al. [0] Numercal Fnal Mkc et al. Intal Radus Radus (1970) Percent Error 3 mm 3.48 mm 3.76 mm 7.44 % 3.8 mm 4.6 mm 4.6 mm 7.3 % 4.8 mm 5.1 mm 5.34 mm 4.11 % (a) (b) (c) (d) (e) Fgure 1: Condensaton of an unbounded vapor bubble wthout gravty. (a. ntal, b. t = 1 s, c. t = s, d. t = 3 s). Volume 7 Number 015

8 14 Study of the Dynamcs of a Condensng Bubble Usng Lattce Boltzmann Method rght walls. he shapes of bubble at dfferent tme nstants are shown n Fgures (a-d). Due to wall effects the condensng bubble no longer keeps ts sphercal shape. he shape oscllates ntally and gets stable after lttle ntal teraton. hough the fnal shape of the condensng bubble s not sphercal stll a promnent decrease n bubble volume s promnently observed. Due to 40 C of subcoolng saturated vapor gets converted nto lqud and the volume of bubble keeps on decreasng. o establsh the effect of subcoolng on bubble shrnkage, smulatons have been made for dfferent degrees of subcoolng. It s consdered that the bubble s at 7 mm dameter (Fgure 3 a) ntally and placed nsde an unbounded doman of 1 1 mm area. Perodc boundary condtons on all sdes of the doman are consdered to replcate unbound stuaton. Gravtatonal acceleraton s consdered to be neglgble. hree dfferent degrees of subcoolng.e. 10 C, 17 C and 5 C are consdered for the surroundng lqud whereas vapor bubble s kept at 100 C. After performng the numercal smulaton on bubble contours obtaned after 10s for three dfferent subcoolng are defned n Fgures 3 (b-d). It can be observed from the fgure that the bubble dmnshes faster for hgher values of subcoolng. hs s obvous as hgher subcoolng allows more possblty of condensaton and transformaton of vapor nto lqud. In Fgure 4 bubble volume after 10s are depcted for dfferent subcoolng. From the fgure also t s clear that for hgher subcoolng bubble volume decreases faster. he ntal volume of saturated vapor bubble s also vared by keepng other parameters constant. Doman sze s kept as 1 1 mm wth perodc boundary condtons. Gravtatonal pull s consdered neglgble for the smulaton. Intal bubble dameter s vared as 5mm, 6mm and 7mm respectvely. Water havng 40 C subcoolng s used as contnuous phase. Results are noted after tme nterval of 10s and depcted n Fgures 5 (a-f). It can be observed from the fgure that bubbles havng hgher dameter gets more chance to condense. As a result the condensaton rate s faster. In bgger bubbles (7 mm) as compared to the smaller ones (5 mm and 6 mm). After the parametrc varatons, smulatons have been made to understand the dynamcs of condensng bubble under the acton of gravtatonal pull. A saturated (100 C) vapor bubble of 5 mm dameter Fgure (6 a) s placed nsde a 1 mm dameter tube flled wth subcooled lqud water (60 C). A lattce resoluton s generated for ths smulaton. he shapes of vapor bubble after 0.09 s, 0.18 s, and 0.36 s are shown n Fgures 6 (b-d) respectvely. As the bubble moves up under the acton of buoyancy, the shape no longer remans sphercal. A bullet shaped bubble moves up n the lqud column. At the same tme t can be observed that due to subcoolng the vapor bubble gets condensed and decrease ts volume. (a) (b) (c) (d) Fgure : Condensaton of a vapor bubble wthout gravty wth wall. (a. ntal, b. t = 0.5 s, c. t = 1 s, d. t = 1.5 s.). (a) (b) (c) (d) Fgure 3: Effect of varous degrees of subcoolng on condensaton. (a. Intal, b. Superheat = 10 C, c. Superheat = 0 C, d. Superheat = 30 C.). Journal of Computatonal Multphase Flows

9 Shahnawaz Ahmed, Sandeep Sreshth, Suman Ghosh and Arup Kumar Das Bubble volume mm Subcool Δ Fgure 4: Bubble volume account for dfferent subcoolng after equal tme nterval. (a) (b) (c) (d) (e) (f) Fgure 5: Effect of ntal volume on condensaton wthout gravty. (a. 5mm, 0s, b. 6mm, 0s, c. 7mm, 0s, d. 5mm, 10s, e. 6mm, 10s, f. 7mm, 10s.). (a) (b) (c) (d) Fgure 6: Dynamcs of a condensng bubble wth subcoolng under gravtatonal acceleraton. (a. Intal, b. t = 0.16 s, c. t = 0.4 s, d. t = 0.4 s.). Volume 7 Number 015

10 16 Study of the Dynamcs of a Condensng Bubble Usng Lattce Boltzmann Method Bubble volume mm Subcool Δ Fgure 7: Bubble volume account wth the presence of gravty for dfferent subcoolng at 0.3 s Effects of parameters lke degree of subcoolng are also nvestgated on the shrnkage and dynamcs of vapor bubble under gravtatonal acceleraton. Intal dameter of the saturated (100 C) vapor bubble s consdered as 5mm and the condut dameter s taken as 1 mm. A lattce resoluton s used for the smulatons. o observe the effect of subcoolng, contnuous phase temperature has been vared. Results obtaned from the numercal smulaton after 0.3s are depcted n Fgure 7. It can be observed that bubble volume decreases fast at hgh degree of subcoolng. 4. CONCLUSIONS In ths paper dynamcs of a condensng bubble s smulated usng dffused nterface based thermal lattce Boltzmann model. Followng salent ponts can be mentoned as major fndngs: ) Saturated vapour bubble dmnsh wth tme sgnfyng condensaton due to surroundng subcooled lqud. ) Condensaton depends on the surface area of the bubble. As a result t has been observed that bgger bubble condenses at a faster rate. ) Rate of condensaton depends on the degree of subcoolng. Hgher subcoolng results n faster condensaton. v) Presence of wall n the vcnty of condensng bubble forms t n bullet shape whle condensng. v) Condensng bubble under the nfluence of gravtatonal pool dmnshes faster. v) akng the queue from present exercse, numercal smulaton of multple condensng bubbles of equal/unequal sze n neghborhood, whch can be consdered as a close smulaton of a complex bubble column reactor, can be attempted. REFERENCES [1] K.W. ung, and J.Y. Parlange, Note on the moton of long bubbles n closed tube nfluence of surface tenson, Acta Mechanca, 1976, 4, [] Y. Kataoka, H. Suzuk, and M. Murase, Drft-flux parameters for upward gas flow n stagnant lqud, J. Nucl. Sc. echnol. 1987, 4, [3] S. Polonsky, L. Shemer, and D. Barnea, Relaton between the aylor bubble moton and the velocty feld ahead of t, Int. J. Multphase Flow 1999, 5, [4] F.P. Bretherton, he moton of long bubbles n tubes, J. Flud Mech., 1961, 10, [5] E.. udose, and M. Kawaj, Expermental nvestgaton of aylor bubble acceleraton mechansm n slug flow, Chem. Eng. Sc., 1999, 54, Journal of Computatonal Multphase Flows

11 Shahnawaz Ahmed, Sandeep Sreshth, Suman Ghosh and Arup Kumar Das 17 [6] U. Frsch, B. Hasslacher and Y. Pomeau, Lattce-Gas Automata for the Naver-Stokes Equaton, Phys. Rev. Lett., 1986, 56, [7] G.R. McNamara, and G. Zanett, Use of the Boltzmann equaton to smulate lattce-gas automata, Phys. Rev. Lett., 1988, 61, [8] P.L. Bhatnagar, E.P. Gross, and M. Krook, A model for collson processes n gases, Phys. Rev. 1954, 94, [9] S. Chen, and G. Doolen, Lattce Boltzmann method for flud flows, Annu. Rev. Flud Mech. 1998, 30, [10] S. Succ, he Lattce Boltzmann Equaton for Flud Dynamcs and beyond. Clarendon Press, Oxford, 001. [11] X. Shan, and H. Chen, Lattce Boltzmann model for smulatng flows wth multple phases and components, Phys. Rev., 1993, 47(3), [1] ] M.R. Swft, W.R. Osborn, and J.M. Yeomans, Lattce Boltzmann smulaton of nondeal fluds, Phys. Rev. Letters, 1995, 75, [13] A.K. Gunstensen, and D.H. Rothman, Mcroscopc modelng of mmscble flows n three dmensons by a lattce boltzmann method, Europhys. Lett., 1998, 18(), [14] D.H. Rothman, and J.M. Keller, Immscble cellular automaton flud, J. Stat. Phys., 1988, 5, [15]. Lee, and C.L. Ln, A stable dscretzaton of the lattce Boltzmann equaton for smulaton of ncompressble twophase flows at hgh densty rato, J. Comput. Phys. 005, 06, [16]. Inamuro, M. Yoshno, H. Inoue, R. Mzuno, and F. Ogno, A lattce Boltzmann method for a bnary mscble flud mxture and ts applcaton to a heat transfer problem, J. Comput. Phys., 00, 179, [17] H.W. Zheng, C. Shu, and Y.. Chew, A lattce Boltzmann for multphase flows wth large densty rato, J. Comput. Phys., 006, 18, [18] Z. Dong, W. L, and Y. Song, A numercal nvestgaton of bubble growth on and departure from a superheated wall by lattce Boltzamann method,. Int J. Heat Mass ransfer. 010, 15, [19] N. akada, M. Msawa, A. omyama, and S. Hosokawa, Smulaton of bubble moton under gravty by Lattce Boltzmann Method, J. Nucl. Sc. echnol., 001, 38 (5), [0] B.B. Mkc, W.M. Rohsenow, and P. Grffth, On bubble growth rate, Int. J. Heat Mass ransfer, 1970, 13, Volume 7 Number 015