Theoretical Analysis of Bubble Formation in a Co-Flowing Liquid

Transcription

1 Journal of Chemal Engneerng of Japan, Vol. 35, No. 0, pp , 00 Researh Paper Theoretal Analyss of Bubble Formaton n a Co-Flowng Lqud W. B. CHEN AND REGINALD B. H. TAN Department of Chemal and Envronmental Engneerng, Natonal Unversty of Sngapore, Sngapore 960 Keywords: Non-Spheral Model, Bubble Formaton, Lqud Flow, Co-Flowng, Bubble Axs Translaton A realst non-spheral model for bubble formaton n a o-flowng lqud s presented. In the model, an nterfaal element approah s appled to desrbe the dynams of bubble formaton. The effet of flowng lqud veloty s modeled by a ombnaton of the bubble axs translaton and lqud pressure analyss of eah nterfaal element. The bubble shapes durng formaton are predted reasonably well by the present model. The effets of lqud veloty, gas flow rate, nozzle radus and gas hamber volume on the bubble growth rates are studed. The model predtons are ompared wth the expermental data n lterature and show good agreement. Introduton In many pratal hemal proesses, ontnuous operaton s more frequently used than bath and sembath operatons, n whh not only gases but also lquds are ontnuously fed nto a olumn. Therefore, bubble formaton n flowng lquds s of wde nterest n engneerng applatons. Bubble formaton n o-flowng or ounter-flowng lquds under onstant gas flow onons has been nvestgated both expermentally and theoretally (Chuang and Goldshm, 970; Sada et al., 978; Takahash et al., 980; Räbger and Vogelpohl, 98; Fawkner et al., 990; Oguz and Prosperett, 993). Several spheral, pseudo-spheral and non-spheral models have been reported for bubble formaton n flowng lquds. All the nvestgatons reported that the bubble volume dereased wth an nrease of superfal lqud veloty. One-stage spheral models Chuang and Goldshm (970) were the frst to propose a onestage model for bubble formaton at a nozzle submerged n o-flowng lqud. Ther model was based on the fore balane at bubble detahment. The bubble volumes were obtaned from the fore balane equaton or ts dmensonless form. Sada et al. (978) observed bubble formaton at a sngle nozzle n water flowng parallel to the nozzle. The gas flow rate and superfal lqud veloty ranged from m 3 /s to m 3 /s and from 0 m/s to.549 m/s, respetvely. It was assumed that the total fore atng on the bubble at the nozzle s the sum of the buoyany Reeved on November 5, 00. Correspondene onernng ths artle should be addressed to R. B. H. Tan (E-mal address: hetanbh@nus.edu.sg). fore and the drag fore. By usng a modfed Froude number, they obtaned the dmensonless orrelaton of bubble sze n the sngle bubblng regmes. Fawkner et al. (990) developed a theory to explan the varaton n bubble szes obtaned when a flowng lqud was pulsed nto a olumn nto whh a onstant flow of ar was sparged through a vertal nozzle at ts base. Ths one-stage model was based on that developed by Davdson and Shüler (960) for bubble formaton n a quesent lqud. The bubble was assumed to be spheral at all tmes durng ts formaton. The upward moton of the bubble was aounted for by supermposng the lqud moton on that obtaned from a balane between the upward fore due to buoyany and the drag fore due to nerta. Bubble detahment was assumed to take plae when the dstane between the bubble base and the nozzle enter was equal to the nozzle radus. The results predted by the model agreed well wth ther expermental data. Two-stage spheral models Takahash et al. (980) observed the bubble formed n o-urrent, ounter-urrent, and rossurrent flowng lquds under a onstant flow onons. To estmate the bubble volume, a two-stage spheral model based on the model of Takahash and Myahara (976) n quesent lqud was proposed, nvolvng an emprally formulated orrelaton fator. Newton s seond law of moton appled both at the end of the frst stage and the seond stage were gven to determne the bubble volume. Räbger and Vogelpohl (98) onduted expermental and theoretal studes on bubble formaton n flowng Newtonan lquds. A two-stage model based on Ramakrshnan et al. (969) n a quesent lqud was developed to alulate the bubble volume. The orrelatons obtaned from the model showed a good agreement for the maxmum bubble pressure and the detahment of bubbles. 95 Copyrght 00 The Soety of Chemal Engneers, Japan

2 Pseudo-spheral model Terasaka et al. (999) expermentally nvestgated the effets of lqud veloty, nozzle dameter, gas hamber volume and gas flow rate on volumes, shapes and growth rates of bubbles formed at a nozzle submerged n a o-urrent upward flowng lqud. A modfed pseudo-spheral model, based on that of Terasaka and Tsuge (990) for bubble formaton n a quesent lqud, was proposed. Although the bubble volumes, bubble growth rates and shapes were well predted by the model, ths model stll ontaned the same nadequaes present n the orgnal model of Terasaka and Tsuge (990), that s, the equatons of moton for expanson and translaton of an equvalent spheral bubble were utlzed for a model of bubble formaton. Non-spheral models Oguz and Prosperett (993) employed a boundary ntegral method to predt the bubble shapes and volumes n a flowng nvsd lqud under onstant pressure n a gas hamber onneted to a nozzle. The pressure n the bubble was taken to be spatally unform, whle the lqud pressure at the bubble nterfae was not requred to be unform and the bubble was not assumed to grow spherally. The effets of nozzle length, gas flow rate and lqud flowng veloty on the behavor of growng bubble were well predted by the model. Ths model s lmted to onstant pressure bubblng onons and low gas flow rates. In ths study, a realst non-spheral model for bubble formaton, whh s based on nterfaal element approah, s developed to predt the bubble shapes, growth rates and bubble volume and tme at detahment. Model predtons are ompared wth the expermental results of Terasaka et al. (999).. Model Development Non-spheral models of bubble formaton have been suessful n predtng bubble formaton n a quesent lqud. In partular, the nterfaal element approah employed by Marmur and Rubn (976) and Tan and Harrs (986) has yelded good agreement wth expermental results for bubble growth rate, detahment tme, bubble volume at detahment and hamber pressure flutuatons. The nterfaal element method was also suessful n modelng of bubble formaton wth lqud ross-flow (Tan et al., 000). Our present theoretal model for bubble formaton n a o-flowng lqud wll be based on ths modfed nterfaal element approah. Thermodynam expressons relatng the hamber and bubble pressures and the nstantaneous bubble volume are solved to provde the orfe gas flow rate. The effet of a parallel flowng lqud s taken nto aount by ombnaton of bubble axs translaton and pressure analyss of the surroundng lqud.. Physal system and bas assumptons The bubblng system under onsderaton onssts a gas that s fed nto a hamber wth volume V, at a onstant flow rate Q and pressure P o. Gas flows through the sngle nozzle R o nto the bubble at a flow rate q. Ths flow s assumed to be ontrolled by the pressure n the hamber and that n the bubble, P and P b, whh are both assumed to be unform wthn ther volumes. A lqud flows upward at a veloty U l parallel to the nozzle axs. The followng bas assumptons are made: (a) The bubble remans symmetral about ts vertal axs durng the growth and s a volume of revoluton around ts vertal axs. (b) The nfluene of gas and lqud vsostes at the nterfae s neglgble. () The growth of the bubble s unaffeted by the presene of other bubbles. (d) The gas s deal and the flow s adabat. (e) The upward flowng lqud s sothermal, unform, nvsd and rrotatonal. (f) There s no energy exhange or mass transfer aross the gas-lqud nterfae.. Equatons of moton.. Interfaal element representaton of the bubble nterfae The nterfaal element approah dvdes the bubble nterfae nto a number of small elements. Beause of assumpton (a), the three-dmensonal bubble an be smply analyzed n a two-dmensonal way: wth vrtual axal dreton z* and vrtual radal dreton r* n the r* z* plane, whh s a vrtual spae and does not take the o-flowng lqud veloty nto aount, and only one sde of the bubble needs to be onsdered. Fgure (a) shows the axsymmetr bubble surfae represented by a sngle urve n the r* z* plane, whh onssts of a number of small elements. The enter of eah element s represented by a blak rle on the urve. For a spef element, the end ponts of the element are the mdponts between pont and eah of ts two nearest neghbors on both sdes. Equatons of moton are developed from fore balanes arred on eah element. The equatons are wrtten n fnte-dfferene form and solved numerally to obtan the nstantaneous oordnates of all elements. The poston of the bubble surfae at eah nstant durng ts growth an be determned by the nstantaneous poston of eah element... Bas equatons Fgure (b) shows a threedmensonal dfferental element of ths nterfae and the fores atng on t. The surfae fore s due to pressure dfferene between the gas n the bubble and the lqud, and the lne fores are due to surfae tenson. For a stat nterfae, these fores are n equlbrum. However, n dynam bubble formaton, the resultant of these fores s equal to the rate of hange n the lqud momentum. Fgure (b) shows also an ntal dfferental nterfae element da 0, whh grows to da af- 953

3 Fg. (a) Two-dmensonal nterfaal element representaton of bubble surfae: (b) Fores atng on an nterfaal element, n ylndral oordnate ter tme t, and the volume of lqud dsplaed by the dfferental element durng ths tme perod. To avod the extremely omplated soluton of the Naver- Stokes equatons for the moton of lqud, lqud momentum an be alulated usng the nertal mass and the veloty of the nterfae element: where r*, z* are vrtual radal oordnate from the axs of the bubble and axal oordnate from nozzle horzontal level respetvely, as shown n Fg. (a), P s the pressure dfferene between bubble pressure P b and the lqud pressure P l at eah nterfae element. β s an angle defned by: F d Um = ( ) () β = tan z * r * ( 5) where m s the added mass of lqud aelerated by the moton of the nterfae element at a veloty Ũ, and m = ( αρ + ρ ) V ( ) l b where α s the theoretal added mass oeffent [= ], ρ b, ρ l are the densty of bubble and surroundng lqud, respetvely. V s the volume of lqud dsplaed by the element sne the begnnng of ts movement. Resolvng Eq. () to gve oordnates of spefed dfferental nterfae element n spae as a funton of tme, onsderng unt angle of revoluton, κ about the bubble axs, we obtan a set of dfferental equatons of moton n ylndral oordnates: d r * Pdr * σd r * sn β Uz* m 3 ( )= ( ) () dz * r * Pdz * σ d( r * os β) sn β d Ur* m = ( ) ( 4) To aount for a parallel flowng lqud wth upward veloty U l, the apparent vertal translaton of eah nterfae element, dz*/, s defned relatve to the unform lqud veloty U l. The relatonshp between the vrtual oordnates r*, z* and the fxed ylndral oordnates r, z yelds (Terasaka et al., 999): and dz * dz = Ul ( 6 ) dr * = ( 7) dr.3 Thermodynams of the system The equatons of moton are based on a dfferental fore balane at the gas-lqud nterfae. Ths fore balane nvolves the pressure dfferene between the gas wthn the bubble and the lqud. Hene, alulaton of the nstantaneous pressure wthn the bubble and the pressure dstrbuton around the surroundng lqud are requred. The pressure wthn the bubble s determned by ts thermodynam relaton to the pressure n the gas hamber and the flow rate through the nozzle. Aordng to the bas assumptons made on the physal sys- 954 JOURNAL OF CHEMICAL ENGINEERING OF JAPAN

4 tem, the thermodynam equatons desrbng the bubblng system are derved as bellow. A mass balane on the hamber yelds V d ρ = ρ Q ρ q () 8 a where ρ a and ρ are the gas denstes at supply and hamber onons, respetvely. Smlarly, applyng mass balane to the gas wthn the bubble, dρbv b = ρ q ( 9) or the molar hange of gas wthn the bubble an be expressed as dn dn dn, and dn 0 0 = = ( ) n out out Applyng open system energy balane (or the frst law of thermodynams for an open system) to the gas wth the bubble, the nternal energy hange of gas wthn the bubble, de b, an be expressed n terms of the heat nteraton aross the gas-lqud nterfae, dθ, work nteraton aross gas-lqud nterfae, dw, and energy balane due to a non steady flow (the last four terms n the rght-hand sde of Eq. ()), deb = dθ + dw + ( hndnn + dek, n ) ( houtdnout + den,out ) Here [ ] ( ) deb = ndev + evdn = nndev + evdnn ( ) and e v s molar nternal energy of gas wthn a bubble. As we assume that there s no heat transfer aross the gas-lqud nterfae, dθ = 0 ( 3) dw = PbdVb ( 4) The ontrbuton of the knet energy of gas through the orfe s taken nto aount n the equaton. Ths hange n knet energy s assumed to arse from the trappng of the upward flowng gas by the bubble envelope, and may affet the bubble pressure sgnfantly due to the small bubble volume. Ths hange of the knet energy may be expressed as: de k,n ρb q = q a o ( 5) de k,out = 0 ( 6) where a o s the ross-setonal area of the orfe. Substtutng Eqs. () (6) nto Eq. (), t follows that ρv q nncvdt + Pb dv = RgTdnn + q a o ( 7) where C v s onstant-volume heat apates. Rewrtng Eq. (7) together wth an deal gas law equaton P V = nr T ( 8) b b g The pressure hange of the gas wthn the bubble s obtaned dp b 3 ρq q P dv b = γρ + ( γ ) b 9 Vb γ ao ( ) where γ s adabat gas exponent. A smlar dervaton of pressure hange wthn the hamber was obtaned dp γ = PQ a Pq V ( ) ( 0) In ths ase, the knet energy hange between nflow and outflow s taken to be neglgble. Ths an be justfed by the fat that the effet of gas knet energy on hamber pressure s small when the hamber s large. If the hamber volume s small, the gas flow rates nto the hamber and out through the orfe are approxmately the same. Hene, the net hange of gas knet energy s lose to zero. The pressure drop through a nozzle an be desrbed as follows (Terasaka, et al., 999) P P k q k q = + ( ) b where k and k are expermentally obtaned. Equatons (8), (9), (9), (0) and () make up a set of smultaneous dfferental equatons and an be solved to obtan P b, P, ρ b, ρ, and q va an explt fnte tme-dfferene method. VOL. 35 NO

5 Fg. Analyss of lqud pressure dstrbuton: (a) geometry for estmaton of lqud pressure dstrbuton, (b) dstrbuton of dmensonless modfed pressure: (---) potental flow for spheres and ( ) approxmaton for flud spheres.4 Calulaton of lqud pressure The effet of the parallel lqud flow (veloty U l ) s estmated va the lqud pressure dstrbuton at the bubble nterfae as shown n Fg.. The entrod, C, of the bubble envelope s loated by the expresson: z n zds S = ( ) For eah tme step, the angle θ for eah nterfae element s defned as shown n Fg. (a). Fgure (b) shows the values of the pressure oeffent C p for a steady flow around a flud sphere (Hamele et al., 967). The authors provded the steady state solutons of the Naver-Stokes equatons for flows around rulatng flud spheres by usng fnte-dfferene methods. Ther results showed that the potental flow soluton provded a good approxmaton at the front part of a bubble (Clft et al., 978). The dashed lne s the analytal soluton for an deal potental flow, whle the sold lne was used n the present model as a lose approxmaton to Hamele et al., (967) soluton. In the present model, C p s estmated as follows: 9 = sn, ( 3a) 4 Cp θ θ C p = 0, θ > 38. ( 3b) Thus, lqud pressure dstrbuton on the surfae of a bubble an be alulated: ϕl = ϕo + ρlul Cp ( 4) where ϕ l and ϕ o are non-gravtatonal pressures wth referene to the orfe, ϕ = P + ρ gz ( 5) ϕ o l l l = P o ( 6) Therefore, lqud pressure dstrbuton an be omputed by: Pl Po ρlul Cp ρlgz = + ( 7) where P o s absolute lqud pressure at the orfe, P P ρ gh ρu o s l l l = + ( 8) and P s s system pressure above the bulk lqud.. Numeral Soluton. Intal and boundary onons The bubble s ntally assumed to be a hemsphere of radus equal to the nozzle radus, and ts nterfae s represented by N equally spaed ponts. Eah pont represents a md-pont of an nterfaal element. The ntal pressures n the bubble and hamber are both assumed to be the sum of hydrostat pressure 956 JOURNAL OF CHEMICAL ENGINEERING OF JAPAN

6 Fg. 3 Effet of o-flowng lqud veloty on the bubble shapes durng formaton for expermental onons: N /Water, Q = m 3 /s, R 0 = m, V = m 3, U l = , 3.0 0, m/s, respetvely (from left to rght). Expermental photographs are from Terasaka et al. (999) at the nozzle and the exess pressure due to surfae tenson P ( t 0) P ( t 0) P b o σ R = = = = + ( 9) Hene, the orfe flow rate s assumed to be zero ntally, qt ( 0) 0 30 = = ( ). Computatonal proedure After ntalzaton and tme nrement, the new oordnates of eah element are omputed by solvng dfferental equatons of moton va an explt fnte tme-dfferene method (Refer to Appendx). The new bubble volume at tme t s then obtaned by a numeral ntegraton of the new oordnates of eah element, and used to evaluate the new hamber and bubble pressures. The pressure of surroundng lqud s evaluated by Eq. (7). When the radal dstane between an element on the nek and the bubble vertal axs s redued to zero, bubble detahes. The bubble nterfae s dvded ntally by 50 ponts, whh s suffent to adequately represent the bubble shape. Sne growth s non-spheral, some 0 ponts on the bubble surfae may tend to bunh together whle others tend to move further apart durng the ourse of the omputaton. Non-unform pont spang s undesrable sne exessvely large spang between elements results n an naurate representaton of the bubble surfae whle bunhng of element ponts unneessary nreases omputaton tme. Ths problem s overome by the ntroduton of a smple subroutne whh nserts adonal ponts when the spang between elements beome too large, and deletes ponts f pont spang falls wthn a spefed range. 3. Results and Dsusson In ths seton, the model predtons of sngle bubble formaton n o-flowng lqud are ompared wth the expermental observatons by Terasaka et al. (999). The followng equpment varables are nput to the model: orfe radus, gas hamber volume. The system varables are: gas-lqud nterfaal surfae tenson, densty of lqud, densty of gas, and the adabat onstant for the gas. The operatng varables are: system pressure, gas flow rate, o-flowng lqud veloty, and lqud heght above the nozzle. The expermental measured parameters k and k for dfferent nozzles are obtaned from Terasaka et al. (999). VOL. 35 NO

7 Fg. 4 Bubble growth rates n o-flowng lqud for bubble formaton at onons: N /Water, Q = m 3 /s, R 0 = m, V = m 3, U l = Expermental data from Terasaka et al. (999) 3. Bubble shapes durng formaton Fgure 3 shows the effet of o-flowng lqud veloty on the bubble shapes durng formaton and detahment tme. The onons for these runs are based on the followng onons: System = N /Water, System pressure = atmospher, Q = m 3 /s, R 0 = m, V = m 3, and U l = m/s, m/s and m/s, respetvely, orrespondng to an expermental study by Terasaka et al. (999). The bubble shapes at any lqud flow veloty wthn the range of our study are not onsderably dfferent, and the detahment tme for bubble growth dereases wth an nrease of o-flowng lqud veloty, whh agrees very well wth the expermental observatons. In Fg. 3, the tme nterval between two onseutve ontours s 4 ms, and the fnal shape shows nek losure and hene detahment. These sequenes learly show that the bubbles are approxmately spheral only n the early stages of formaton, eventually beomng noteably non-spheral and detahng naturally when the nek loses. The omputed bubble shapes by the present model agree approxmately wth the shapes photographed by a hgh-speed vdeo amera by Terasaka et al. (999). For large gas flow rates, gas flow veloty nto a bubble nreases, and the frequeny of bubble generaton also nreases. A sgnfant pressure wake s aused by the prevous rsng bubble, so that the bubble shapes may beome slghtly elongated along the axs. In our present model, the effet of the presene Fg. 5 Effet of o-flowng lqud veloty on the bubble growth rates. Expermental data from Terasaka et al. (999) of others bubbles s not taken nto aount. Hene the omputed bubble shapes appear to be more rounded and less sharpened at the top. However, the general agreement on both shape and sze s reasonable. 3. Bubble growth rates Fgure 4 shows the bubble volume omputed at varous tmes durng the formaton perod. The onons for ths run are orrespondng to Fg. 3 wth U l = m/s, and a omparson s made wth bubble volume expermentally obtaned by Terasaka et al. (999). The fgure ndates that the results omputed by the present model orrelate well wth the expermental data. 3.. Effet of lqud veloty Fgure 5 shows the bubble growth rates for several values of o-flowng lqud veloty. The onons for these runs are orrespondng to Fg. 3. The symbols represent the bubble volume and tme at detahment for dfferent lqud flow velotes. Clearly, the nstantaneous bubble growth rate s vrtually unaffeted by the o-flowng lqud veloty. The man effet of lqud flow veloty s to affet the detahment tme, and thereby the bubble volume at detahment. Wth nreasng lqud flow veloty, the bubble s predted to detah earler and the bubble volume at detahment s onsequently smaller. These results agree wth the expermental observatons and trends reported by Terasaka et al. (999). 3.. Effet of gas flow rate Fgure 6 shows the effet of gas flow rate on bubble growth rates for the onons: System = N /Water, System pressure = atmospher, R 0 = m, V = m 3, U l = m/s, orrespondng to the expermental run n Terasaka et al. (999). The bubble has a longer 958 JOURNAL OF CHEMICAL ENGINEERING OF JAPAN

8 Fg. 6 Effet of gas flow rate on bubble growth rate n oflowng lqud. Exoermental data from Terasaka et al. (999) Fg. 8 Effet of gas hamber volume on bubble growth rate n o-flowng lqud. Expermental data from Terasaka et al. (999) Fg. 7 Effet of nozzle radus on bubble growth rate n o-flowng lqud. Expermental data from Terasaka et al. (999) growth tme and the fnal bubble volume s smaller when the njeted gas flow rate s lower as expeted emprally. Ths trend s also observed expermentally for sngle bubble formaton n a quesent lqud. It an be noted agan that the model predtons math the expermental data rather well Effet of nozzle radus Fgure 7 shows the effet of nozzle radus on bubble growth rates for the onons: System = N /Water, System pressure = atmospher, Q = m 3 /s, V = m 3, U l = m/s, orrespondng to the expermental run n Terasaka et al. (999). When the nozzle radus s larger, the rate of the nrease of bubble volume s slower. An nrease n nozzle radus auses the reduton of a pressure drop through the nozzle, whh means the drvng fore for bubble formaton s redued. Therefore, bubble growth rate s slower n the ase of larger nozzle radus Effet of gas hamber volume The effet of gas hamber volume on bubble growth rate s shown n Fg. 8 for expermental onons: System = N / Water, System pressure = atmospher, Q = m 3 /s, R 0 = m, U l = m/s, orrespondng to expermental run n Terasaka et al. (999). The bubble begns to expand earler n the ase of smaller gas hamber volume. Whle, the fnal bubble volumes at detahment are almost same n ths onons. 3.3 Varaton of bubble volume wth lqud flow veloty and gas flow rate Fgure 9 shows the varaton of bubble departure radus wth o-flowng veloty for the onons: System = N /Water, System pressure = atmospher, R 0 = m, V = m 3, Q = ,.8 0 6, m 3 /s. Smulated results are ompared wth the expermental data avalable (Terasaka et al., 999) for the nfluene of o-flowng lqud veloty on bubble volumes at the three values of gas flow rate. For all gas flow rates studed, bubble volume at detahment dereases wth the nrease of lqud flow veloty. It an be seen that the model predtons follow the expermental trends rather well. VOL. 35 NO

9 Fg. 9 Varaton of bubble volume at detahment wth oflowng lqud veloty. Expermental data from Terasaka et al. (999) Conlusons In ths paper, a non-spheral model for bubble formaton n o-flowng lqud has been developed. The bas onepts and equatons of the nterfaal element method, whh s appled to desrbe the dynams of bubble formaton, are dsussed n detals. Thermodynams expressons of the bubblng system are solved together wth the equaton of a pressure drop through a nozzle to obtan the gas flow rate through the nozzle and the pressure of gas wthn the bubble. The effet of a parallel flowng lqud s taken nto aount by pressure analyss of surroundng lqud. Model predtons are ompared wth the expermental results avalable n lterature for dfferent onons of lqud flow veloty, gas flow rate, nozzle radus and gas hamber volume. Bubble shapes, bubble growth rates and the varaton of bubble volume wth lqud flow veloty and gas flow rate are presented. The smulated results agree well wth the expermental data of Terasaka et al. (999). Appendx: Fnte dfferene forms of equatons of moton Equatons (3) and (4) nvolve dfferentaton wth respet to tme t and vrtual spae. In order to solve these smultaneous dfferental equatons of moton for eah element, these equatons are rewrtten to be wth respet to tme t and fxed spae, then expressed as fnte dfferene forms. The relatonshp between the varables wth respet to vrtual and fxed spaes s as follows: U r * dr * dr = = ( A.) U z* dz * dz = = U ( ) l A. dz * dz Ul tan θ = = dr * dr ( A.3) sn θ = dz U ( dz U ) + ( dr) l l ( A.4) Fg. 0 Varaton of bubble volume at detahment wth gas flow rate. Expermental data from Terasaka et al. (999) osθ = dr ( dz U ) + ( dr) l ( A.5) Substtutng the above expressons nto Eqs. (3) and (4), The varaton of bubble volume at detahment wth gas flow rate s shown n Fg. 0; the onons are: System = N /Water, System pressure = atmospher, R 0 = m, V = m 3, U l = 0, , 3.0 0, m/s. At a onstant lqud flow veloty, bubble volume at detahment nreases when the gas flow rate nreases. It an be noted agan that our model predtons math the expermental trends losely. d r Pdr σ d r sn β = Uz* m A.6 ( ) ( ) ( ) ( ) ( ) ( ) ( ) l l r P dz U + σ d r os β dz U + dr d = Ur * m [ ] ( ) ( A.7) 960 JOURNAL OF CHEMICAL ENGINEERING OF JAPAN

10 U z Applyng the dfferenng formula as follows: dz z z = =, et. ( A.8) t z + z z = ( A.9) where the supersrpt refers to the poston at tme t t, and denotes the poston at tme t t. Subsrpt denotes element on the bubble nterfae, startng from the top of the bubble. Fnally, onvertng Eqs. (A.6) and (A.7) to fnte dfferene form wth respet to fxed spae: [( + ) ( ) ] P r + r r + r 8 r r r r σ sn θ sn θ ( ) ( ) m z z m z z mu mu l l = + t t t { rp z z + U t l r + r+ r + r + σ osθ + osθ ( ) + + r + r z z + ( ) U t l ( A.0) m = r r m r r ( A.) t Equatons (A.0) and (A.) are arranged to yeld explt expressons for fxes poston of element z m z m z z t mu mu = + + L + m t t ( ) ( A.) m r r m r r t m L = + + A. 3 ( ) ( ) where L and L are the left-hand-sdes of Eqs. (A.0) and (A.), respetvely. Equatons (A.) and (A.3) together wth Eq. () an be solved to obtaned three unknowns: r, z and m. Sne m s tself omputed from r and z, an teratve proedure s requred. Nomenlature A = bubble surfae shown n Fg. (b) [m ] A 0 = ntal bubble surfae shown n Fg. (b) [m ] a 0 = ross-setonal area of the orfe [m ] C = entrod pont of bubble envelope C p = pressure oeffent defned n Eq. (3) and Fg. (b) [ ] C v = onstant-volume heat apates [J/(mol K)] E b = nternal energy of gas wthn bubble [J] e v = molar nternal energy of gas wthn bubble [J/mol] F = total fore atng on nterfaal element [N] g = aeleraton due to gravty [m/s ] h = heght of lqud above the orfe (nozzle) [m] k = fator defned by Eq. () [Pa s/m 3 ] k = fator defned by Eq. () [Pa s /m 6 ] m = added mass of nterfaal element [kg] n = molar number of gas wthn bubble [mol] n n = molar number of gas njeted nto bubble [mol] P a = gas pressure at nlet to hamber [Pa] P b = pressure of gas wthn bubble [Pa] P = pressure of gas n hamber [Pa] P l = pressure of surroundng lqud [Pa] P o = lqud pressure at nfnty [Pa] P s = system pressure above the bulk lqud [Pa] Q = gas flow rate nto hamber [m 3 /s] q = gas flow rate through orfe (or nozzle) [m 3 /s] R o = orfe (or nozzle) radus [m] R g = gas onstant [J/(mol K)] r = true radal oordnate wth respet to the true vertal [m] r* = vrtual ylndral radal oordnate from axs of bubble [m] t = bubble growth tme [s] Ũ = veloty of nterfaal element [m/s] U l = lqud o-flow or ross-flow veloty [m/s] U r* = horzontal veloty of element based on vrtual oordnate [m/s] U z* = vertal veloty of element based on vrtual oordnate [m/s] V b = bubble volume at any nstant [m 3 ] V = gas hamber volume [m 3 ] V = volume of lqud dsplaed by the element sne the begnnng of ts movement [m 3 ] W = work nteraton aross gas-lqud nterfae [J] z = true axal oordnate wth respet to orfe (nozzle) horzontal level [m] z* = vrtual axal oordnate from orfe (nozzle) horzontal level [m] α = added mass oeffent [ ] β = angle defned n Eq. (5) γ = adabat gas onstant [ ] P = pressure dfferene between the bubble pressure P b and the lqud pressure P l [Pa] ρ a = gas densty at supply [kg/m 3 ] ρ b = densty of vapor nsde bubble [kg/m 3 ] ρ = densty of vapor nsde hamber [kg/m 3 ] ρ l = densty of surroundng lqud [kg/m 3 ] σ = surfae tenson [N/m] ϕ l = non-gravtatonal lqud pressure on bubble surfae [Pa] ϕ o = non-gravtatonal lqud pressure at nfnty [Pa] Θ = heat nteraton aross gas-lqud nterfae [J] Lterature Cted Chuang, S. C. and V. W. Goldshm; Bubble Formaton Due to a Submerged Capllary Tube n Quesent and Coflowng Streams, Trans. ASME, J. Bas Eng., 9, (970) Clft, R., J. R. Grae and M. E. Weber; Bubbles, Drops, and Partles, Aadem Press, New York, USA (978) Davdson, J. F. and B. O. G. Shüler; Bubble Formaton at an Orfe n an Invsd Lqud, Trans. Instn. Chem. Engrs., 38, (960) Fawkner, R. D., P. P. Kluth and J. S. Denns; Bubble Formaton at Orfes n Pulsed, Flowng Lquds, Trans. Instn. Chem. Engrs., 68, (990) VOL. 35 NO

11 Hamele, A. E., A. I. Johnson and W. T. Houghton; Numeral Soluton of the Naver-Stokes Equaton for Flow Past Spheres: Part II. Vsous Flow around Crulatng Spheres of Low Vsosty, AIChE J., 3, 0 4 (967) Marmur, A. and E. Rubn; A Theoretal Model for Bubble Formaton at an Orfe Submerged n an Invsd Lqud, Chem. Eng. S., 3, (976) Oguz, H. N. and A. Prosperett; Dynams of Bubble Growth and Detahment from a Needle, J. Flud Meh., 57, 45 (993) Räbger, N. and A. Vogelpohl; Bubble Formaton n Stagnant and Flowng Newtonan Lquds, Ger. Chem. Eng., 5, (98) Ramakrshnan, S., R. Kumar and N. R. Kuloor; Studes n Bubble Formaton-I Bubble Formaton under Constant Flow Conons, Chem. Eng. S., 4, (969) Sada, E., A. Yasunsh, S. Katoh and M. Nshoka; Bubble Formaton n Flowng Lqud, Can. J. Chem. Eng., 56, (978) Takahash, T. and T. Myahara; Bubble Volume Formed at Submerged Nozzles: Constant Flow Conon, Kagaku Kogaku Ronbunshu,, (976) Takahash, T., T. Myahara, S. Senza and H. Terakado; Bubble Formaton at Submerged Nozzle n Courrent, Counterurrent and Crossurrent Flow, Kagaku Kogaku Ronbunshu, 6, (980) Tan, R. B. H. and I. J. Harrs; A Model for Non-Spheral Bubble Growth at a Sngle Orfe, Chem. Eng. S., 4, (986) Tan, R. B. H., W. B. Chen and K. H. Tan; A Non-Spheral Model for Bubble Formaton wth Lqud Cross-Flow, Chem. Eng. S., 55, (000) Terasaka, K. and H. Tsuge; Bubble Formaton at a Sngle Orfe n Hghly Vsous Lquds, J. Chem. Eng. Japan, 3, (990) Terasaka, K., H. Tsuge and H. Matsue; Bubble Formaton n Courrently upward Flowng Lqud, Can. J. Chem. Eng., 77, (999) 96