Theoretical Analysis of Two-Phase Bubble Formation in an Immiscible Liquid

Transcription

1 Theoretial Analysis of Two-Phase Bubble Formation in an Immisible Liqui W. B. Chen an Reginal B. H. Tan Dept. of Chemial an Environmental Engineering, National University of Singapore, Singapore 11960, Singapore Introution Two-phase bubble formation ouple with phase hange at a submerge nozzle is stuie theoretially. The whole bubble onsists of a apor an a liqui phase. The inner apor phase is assume to be surroune by a thin onensate layer, in whih the apor onenses partially as the two-phase bubble grows. The interfae element approah is applie to esribe the ynamis of bubble formation. The effet of heat transfer from bulk apor within the bubble to the surrouning bulk liqui is relate to pressure analysis of apor within the two-phase bubble ia the mass flux of onensation. The results ompute from this moel agree well with the experimental ata. The ispersion of gases through submerge orifies or nozzles plays an important role in enhaning the transport rates between phases in many hemial an physial proesses. In many inustrial heat exhange operations, a onensable gas is often injete into a liqui through a submerge orifie or nozzle to inrease the rate of heat transfer between the gas an liqui, as iret ontat heat transfer has the avantage of high heat transfer oeffiient. Denekamp et al. 197., Cho an Lee 1990., an Chen an Tan 001. have stuie vapor bubble formation at submerge nozzle in its own liqui, that is, the proesses stuie involve only one omponent. When the isperse an ontinuous phases are forme by two ifferent immisible omponents instea of one omponent, a two-phase partile an be observe uring bubble formation. It ontains a liqui an a vapor phase, the vapor aumulating in the inner part of the partile. Being neither a pure rop nor a pure vapor phase, this two-phase bubble has been referre to as a robble Suhoff et al., The latter system iffers from the former, one-omponent, system in that the onensate remains within the onfines of the bubble envelope. Owing to their wie appliation in inustrial heat-transfer proesses, two-phase bubbles have been extensively stuie both experimentally an theoretially. Sieman an Hirsh photographe onensing isopentane bubbles in water uring their free rise perio, or after their release from the nozzle. By analyzing onseutive pitures of eah run, the instantaneous surfae area, volume an vapor ontent of a ris- Corresponene onerning this artile shoul be aresse to R. B. H. Tan. ing robble oul be etermine an the heat flux an transfer oeffiients obtaine. Jaobs et al propose a moel for the ollapse of a bubble rising through a ol ontinuous immisible liqui. Wanhoo investigate the onensation of a rising two-phase bubble in immisible liqui an erive an analyti expression for the steay rate of heat transfer from a two-phase bubble onensing in an immisible liqui meium. Wanhoo et al measure experimentally the veloity of rise an the rag of a single vapor bubble ollapsing in another immisible liqui. Suhoff et al reviewe the publishe stuies sine 1963 on the onensation or evaporation of a robble an summarize the overall heat-transfer oeffiients measure for various systems. More reently, Kalman an Ullmann onute a series of experiments of iret-ontat onensation to the initial volume of a onensing bubble release from an orifie an the instantaneous shapes of robble uring the ollapse proess. Their experimental results showe that the initial volume of the onensing bubble oul be reasonably preite by employing Ruff s two-stage moel Ruff, 197. that was originally evelope for nononensing bubbles. Most of the above stuies eal with the ollapse of onensing robble after its release from the nozzle. However, it is known that the perio of bubble formation an ontribute signifiantly to the overall heat an mass transfer of the proess. Few reporte investigations have been publishe on the formation of two-phase bubbles. Terasaka et al propose a metho to measure the heat-transfer oeffiient for iret-ontat onensation uring two-phase bubble formation an theoretially estimate the iret-ontat heat-transfer oeffiient. Terasaka et al propose a robble for August 003 Vol. 49, No. 8 AIChE Journal

2 mation moel to stuy the effet of operating onitions on two-phase bubble formation behavior at single nozzle submerge in water. The equations of motion for expansion an translation of an equivalent spherial bubble were use. The objetive of this artile is to stuy the hyroynamis of two-phase bubble formation ombine with the mehanism of heat transfer, an to gain theoretial unerstaning of two-phase bubble formation with onensation at a single submerge nozzle. A realisti nonspherial moel for bubble formation ouple with phase hange is esribe. The moel attempts to alulate the instantaneous shape of the robble uring its formation an to etermine the instantaneous volumes of both whole bubble an onensate, frequeny of bubble formation, as well as overall heat-transfer oeffiient. The effets of gas flow rate an temperature ifferene between injete gas an the suboole liqui will be isusse. A bubble formation moel, whih an preit bubble shapes, bubble growth, an etahment uner ifferent thermal an hyroynami onitions, woul signifiantly a to our unerstaning of some ispersion proesses, whih involve onensable gas an liqui iret ontat. The present moel shoul provie useful preitions of the harateristis of a phase-hange bubble in the single-bubbling region an provie the basis for esign an optimization of suh proesses. Moel Development Physial system an basi equations The system uner onsieration onsists of a onensable gas that is injete vertially upwars through a single nozzle R submerge in a liqui bulk temperature T. o l of epth h at a onstant flow rate. The following basi assumptions are mae:. a The bubble remains symmetrial about its axis uring the growth an is a volume of revolution aroun its entral axis. b. The influene of gas an liqui visosities at the interfae is negligible.. The growth of the bubble is unaffete by the presene of other bubbles.. The gas is ieal, isothermal, an inompressible an its flow is aiabati.. e The isperse phase is a pure substane. There are no nononensable gases in the isperse phase.. f Vapor onensate an ontinuous liqui are immisible or the issolution of vapor onensate into ontinuous liqui is negligible.. g Energy exhange aross vapor-onensate interfae is ominate by mass transfer ue to onensation an the energy exhange by heat transfer is neglete. h. The gravitational effet on the thikness of the onensate layer is negligible an the thikness of the onensate layer is assume to be uniform, as it is muh smaller than the raius of the robble. Figure 1 shows the nonspherial moel for two-phase bubble formation at a submerge nozzle. The nonspherial bubble formation moel propose by Tan an Harris is moifie to alulate the movement of the interfae elements. This approah assumes the instantaneous bubble shape is axially symmetri an ivies the interfae between Figure 1. Nonspherial bubble formation with phase hange. the two-phase bubble an the bulk liqui into a finite number of small elements. Assuming an invisi liqui, eah element at the bubble interfae moves as a result of fores ue to pressure ifferene an surfae tension. In ynami formation, the resultant of these fores is equal to the rate of hange in the liqui momentum, assuming that the gas momentum is negligible. The momentum of the liqui may be alulate using the ae mass onept an the veloity of the interfae. Thus, a fore balane at eah element generates a set of ifferential equations of motion in ylinrial oorinates r Pry r sin. s Um z. i 1. t z r Pzq ros. y s Um r. i. sin t where r an z are the raial oorinate from the axis of the bubble an the axial oorinate from the orifie horizontal level, respetively, P is the pressure ifferene between bubble pressure P an the liqui pressure Pl at eah interfae element. is an angle efine by z y1 stan. 3 r The seon term in the lefthan sie of Eqs. 1 an is the fore ue to surfae tension. There are two kins of surfae tensions surfae tension between vapor an onensate, an interfaial tension between onensate an liqui. ating on a two-phase bubble, as a two-phase bubble onsists of ouble interfaes. The onensate thikness is very small when ompare with the bubble equivalent raius, an the surfae ten- AIChE Journal August 003 Vol. 49, No

3 sion efine in Eqs. 1 an is expresse as s q l. 4 where an are surfae tensions between vapor an l onensate, an surfae tension between onensate an liqui, respetively. The term m in Eqs. 1 an is the ifferential ae mass i mis lq b. Vi 5. where is the ae mass oeffiient, taken as 0.5, the value for a bubble translating in an infinite meium Walters an Davison, This is regare to be an average value uring the whole formation proess. Vi is the volume of liqui isplae by the element sine the beginning of its movement. l an b are the ensity of the bulk liqui an the two-phase bubble, respetively. The mean ensity of the two-phase bubble is estimate by V q V s. 6 b where,, V, V, an Vb are vapor ensity, onensate ensity, vapor volume, onensate volume an whole bubble volume, respetively. The vapor volume is obtaine from the appliation for the mass balane V b tyv V s. 7 where is the onstant flow rate of gas injete into the bubble, t is the bubble growth time. Taking the vapor within the two-phase bubble as a nonsteay state open system, from the appliation for the mass balane an the first law of thermoynamis for a nonsteay-state open system, the erivation of pressure hange of the vapor within the two-phase bubble an be expresse as ž / 3 P RT g P ja P V s y y q y1. t V R T M V t V a g o. 8 where, R g, T, j, A, M, an ao are the aiabati gas onstant, gas onstant, vapor temperature, mass flux of onensation, surfae area of onensate, moleular weight of isperse phase, an ross-setional area of the nozzle, respetively. The liqui pressure Pl at eah interfaial element is om- pute by Pls Psq lg hy z. 9. where P is the system pressure above the bulk liqui. s Effet of heat transfer an onensation During the robble formation, the isperse vapor partially onenses to form a onensate layer when the boiling point of vapor is higher than the temperature of the bulk Figure. Physial moel for heat transfer from bulk vapor within robble into surrouning bulk liqui through a thin onensate layer. liqui. Figure illustrates the mehanism of heat transfer with onensation uring the two-phase bubble growth. The vapor within the robble is rapily mixe so that the temperature of the bulk vapor is assume to be onstant at its boiling point T. The overall heat-transfer oeffiient U for heat transfer from the bulk vapor within the robble into the surrouning bulk liqui through a thin onensate layer is alulate generally as s q q q 10. U H H H H g l where H g, H, H, an Hl are the heat-transfer oeffiients for onvetion in vapor phase, iret-ontat onensation, onution in the onensate layer, an onvetion in bulk liqui, respetively. Compare with H an H, heat-transfer oeffiients for onvetion in vapor an liqui phases are muh higher an their ontribution to U an be neglete. Thus, the overall heat-transfer oeffiient is estimate as s q 11. U H H The alulation of iret ontat heat-transfer oeffiient H was propose by Terasaka et al to be H s L( 1. T T yt l. 1r y1r. where L is the latent heat of vaporization. Also, the heat-transfer oeffiient H is given by k Hs August 003 Vol. 49, No. 8 AIChE Journal

4 where k an are the thermal onutivity an the thik- ness of the onensate, respetively. The thikness of the onensate layer is muh smaller than the raius of the robble. Assuming uniform thikness, the thikness of the onensate layer is approximate as V s 14. Both the thikness of the onensate layer an the mass flux are taken as funtions of time only. They are relate by the heat flux ontinuity onition A T ytl jls T Us q H k The instantaneous onensation mass rate m is efine by m s ja 16. The total aumulate onensate volume V growth time t an be obtaine 1 t H at bubble V s mt Rewriting Eq. 17 in ifferential form Initial onitions an numerial solution V ja s 18. t At the beginning of the omputation, the bubble shape is assume to be hemispherial, with its raius equal to that of the nozzle. Initially, both thikness an volume of the onensate film are zero. The ynami equations for robble formation an be solve with the following initial onitions r s R, V s0, s0, P s P q ghq o s l R o TH js 19. L An expliit finite ifferene metho propose by Tan an Harris is extene to solve the equations of bubble formation. The omputational proeure is summarize in the following steps:. 1 Initialize all parameters.. Inrease bubble growth time by t.. 3 Solve the equations of motion for eah element to yiel new oorinates of eah element.. 4 Compute the bubble volume an heat-transfer area by numerial integration of oorinates of all elements.. 5 Apply straightforwar Runge-Kutta metho to solve the ifferential Eqs. 8 an 18 simultaneously to obtain the instantaneous vapor pressure P an onensate volume V.. 6 Compute the instantaneous vapor volume V, onensate thikness, an onensation mass flux j by solving Eqs. 7, 14 an 15, respetively.. 7 Compute the pressure ifferene for eah element at the bubble interfae.. 8 Return to step an repeat steps 6 until the nek loses an etahment is attaine. Results an Disussion The two-phase bubble formation moel is applie to hexanerwater system an vinyl aetaterwater system. The surfae tension of hexanerwater s q s0.0504q l Figure 3. Two-phase bubble growth sequenes for experimental onitions: system=hexane/water, = m 3 /s, R o= m, T=7.0 K.. a Experimental photograph from Terasaka et al ; b ompute bubble shapes by present moel. AIChE Journal August 003 Vol. 49, No

5 Figure 5. Two-phase bubble growth rates for bubble formation at onitions: system = hexane/ water, = m 3 /s, R o= m, T =7.0 K. Figure 4. Two-phase bubble growth sequenes for experimental onitions: system=vinyl aetate/ water, = m 3 /s, R o = m, T=18.9 K. s0.064 Nrm. For vinyl aetaterwater system, s q l. s0.0187q s Nrm Prakoso et al., 001. Two-phase bubble shapes uring formation Figure 3 shows the two-phase bubble growth sequenes for the ase of a hexanerwater system, s y6 m 3 rs, R o s y3 m, T s7.0 K, orresponing to the experimental onitions in Terasaka et al The bubble growth time interval between two onseutive ontours is 5 10 y3 s, an the final shape shows nek losure an, hene, etahment. Both the experimental an ompute bubble shapes iniate that bubble shapes uring formation are non-spherial. A bubble starts off as a hemisphere with its raius the same as that of the nozzle, then beomes approximately spherial as it grows, an finally attains the shape of an irregular ellipsoi with a nek. The ompute instantaneous bubble shapes are in very goo agreement with the experimental vieo images taken by Terasaka et al Figure 4 shows another example of two-phase bubble formation; the onitions are: vinyl aetaterwater s y6 m 3 rs, R os y3 m, T s18.9 K, whih are the experi- mental onitions in Prakoso et al This figure shows a similar tren of bubble shapes uring formation as that in Figure 3. The system stuie in Figure 4 iffers from that in Figure 3 in that the onensate of vinyl aetate is slightly soluble in water. The effet of this issolution rate on bubble formation has been neglete in our moeling. The preite shapes orrelate well with the experimental vieo images taken by Prakoso et al exept that the experimental etahment time is a little shorter than preite by our moel. This is probably ue to the effet of onensate issolution, whih woul hasten etahment by aelerating the thinning of the bubble nek in the final stages. Two-phase bubble growth rates The omparison of the bubble growth rates between the results ompute by the present moel with the experimental ata at onitions orresponing to Figure 3 are plotte in Figure 5. The volumes of the growing bubbles in Figure 5 were measure by Terasaka et al using high-spee vieo amera 1,000 frames per seon., with ata apture an image analysis. The figure iniates that the preite rate of growth by the present moel follows losely the experimental ata Terasaka et al., Figure 6 shows the variation of bubble growth rates with temperature ifferent T for the onitions hexanerwater, s y6 m 3 rs, R os y3 m, orresponing to the experimental run in Prakoso et al Simulate results are ompare with the experimental ata available Prakoso et al., 001. for the effets of temperature ifferene T on the two-phase bubble growth rates. It iniates the two-phase bubble grows faster an etahes earlier for the ase of lower temperature ifferene. It an be seen that our moel preits the experimental trens reasonably well August 003 Vol. 49, No. 8 AIChE Journal

6 layer is also smaller by several orers of magnitue than the average raius of the whole bubble at the same bubble growth time. Therefore, the assumption mae in Eq. 14 is reasonable. It is worth remarking that vieo images suh as those in Figures 3 an 4 annot provie an aurate iniation of the onensate layer thikness, sine the bubble urvature an light refration woul istort the true liqui film thikness. Figure 6. Effet of temperature ifferene T on twophase bubble growth rates. Figure 7 shows the effet of gas-flow rate at onstant temperature ifferene T on bubble growth rates for the onitions hexanerwater, R os y3 m, T s15.7 K, orresponing to the experimental run in Prakoso et al The two-phase bubble volume is smaller when the injete gas-flow rate is lower as expete empirially. This tren was also observe experimentally for one omponent bubble formation without onensation. From the experimental urve of s3. m 3 rs, it appears that elaye release ours, uring whih a bubble remains attahe to the nozzle without growing signifiantly MCann an Prine, It an be note again that the moel preitions math the experimental ata rather well. Volume an thikness of onensate layer Figure 8 shows the ompute volume an thikness of onensate layer uring bubble formation at onitions orresponing to Figure 3. It an be seen that the volume of onensate is smaller by several orers of magnitue than that of the whole two-phase bubble. The thikness of onensate O erall heat-transfer oeffiient an onensation ratio Figure 9 shows the instantaneous overall heat-transfer oeffiient U for bubble formation at onitions orresponing to Figure 3. The instantaneous overall experimental heattransfer oeffiients were obtaine from experimental observations by a heat an mass balane Terasaka et al., as follows ž / L V Us y 0. A T t Overall heat-transfer oeffiient ereases rapily at the beginning of bubble growth. The iret ontat heat-transfer oeffiient H is equal to the overall heat-transfer oeffiient U at the very beginning of bubble growth, when the onensate film has not been generate. To etermine whether inertial an buoyany effets on the one han, or heat-transfer effet on the other han, is more ominant in the two-phase bubble formation proess, it is neessary to estimate the vapor onensation ratio. The vapor onensation ratio is efine as the ratio of onensate mass to the mass of the whole two-phase bubble at etahment. The former an be alulate from the onensate ensity an volume. The latter is equal to the mass of vapor injete through the nozzle uring eah bubble formation perio. A high vapor onensation ratio iniates a ominane of heat transfer over inertial an buoyany effets, an vie versa. Figure 7. Effet of gas-flow rate on two-phase bubble growth rates. Figure 8. Volume an thikness of onensate layer uring bubble formation at onitions: system = hexane/water, = m 3 /s, R o = m, T=7.0 K. AIChE Journal August 003 Vol. 49, No

7 ement approah suessfully moels the physial phenomena of both hyroynamis an heat transfer uring a two-phase bubble formation. The metho generates realisti bubble shapes uring formation an subsequent etahment. The simulate results in terms of instantaneous bubble shape, bubble volume, bubble growth rate as well as overall heat transfer are in goo agreement with the experimental ata from Terasaka et al an Prakoso et al Notation aosross-setional area of the nozzle, m Assurfae area of onensate, m C sonstant-pressure heat apaity, Jr mol K. p C sonstant-volume heat apaity, Jr mol K. e smolar internal energy of vapor within two-phase bubble, Jrmol E sinternal energy of vapor within two-phase bubble, J Ek,inskineti energy of gas injete into bubble, J gsaeleration ue to gravity, mrs Figure 9. Instantaneous overall heat-transfer oeffiient at onitions: system=hexane/water, = m 3 /s, R = m, T=7.0 K. o Figure 10 shows the effet of temperature ifferene between vapor an bulk liqui on the vapor onensate ratio at bubble etahment, for several values of gas-flow rates. It shows a signifiant proportion of vapor onensation 50 90%. ours uring the formation proess. It also shows an inrease in vapor onensation ratio with inrease of temperature ifferene an erease of gas-flow rate, iniating an inrease in ominane of heat-transfer effet over inertial an buoyany effets. Conlusion Two-phase bubble formation at a submerge nozzle in an immisible liqui is analyze theoretially. The interfaial el- hsheight of liqui above the nozzle trip, m hsmolar enthalpy of onensate, Jrmol hinsmolar enthalpy of gas injete into bubble, Jrmol Hsheat-transfer oeffiient for onution in onensate layer, Wr m K. H siret ontat heat-transfer oeffiient, Wr m K. Hgsheat-transfer oeffiient for onvetion in bulk vapor phase, Wr m K. Hlsheat-transfer oeffiient for onvetion in bulk liqui phase, Wr m K. jsonensation mass flux, kgr m s. k sthermal onutivity of onensate, Wr m K. Lslatent heat of vaporization, Jrkg msonensation mass rate, kgrs m sifferential ae mass of the element, kg i Msmoleular weight of isperse phase, kgrmol nsmolar number of vapor within two-phase bubble, mol n smolar number of onensate, mol n smolar number of gas injete into bubble, mol in P sliqui pressure on bubble surfae, Pa l P ssystem pressure above bulk liqui, Pa s P s vapor pressure of two-phase bubble, Pa Pspressure ifferene between bubble pressure an liqui pressure at interfae, Pa sgas onstant flow rate, m 3 rs rsraial oorinate from axis of bubble, m R sgas onstant, Jr mol K. g R osnozzle raius, m tsbubble growth time, s Tlsbulk liqui temperature, K T stemperature of vapor within the two-phase bubble, K Usoverall heat-transfer oeffiient, Wr m K. Urshorizontal veloity of element, mrs U s vertial veloity of element, mrs z V s whole volume of two-phase bubble, m 3 b V sonensate volume, m 3 V s volume of liqui isplae by the element sine the begini ning of its movement, m 3 V s volume of vapor within the two-phase bubble, m 3 Ws work interation aross vapor-onensate interfae, J zsaxial oorinate from orifie horizontal level, m Figure 10. Effet of temperature ifferene on vapor onensation ratio. Greek letters sae mass oeffiient, imensionless sangle efine by Eq. 3, imensionless saiabati gas onstant, imensionless sonensate thikness, m sheat interation aross vapor-onensate interfae, J smean ensity of two-phase bubble, kgrm 3 b sonensate ensity, kgrm 3 sbulk liqui ensity, kgrm 3 l sensity of vapor within the two-phase bubble, kgrm August 003 Vol. 49, No. 8 AIChE Journal

8 ssurfae tension, Nrm ssurfae tension between vapor an onensate, Nrm lssurfae tension between onensate an liqui, Nrm Literature Cite Chen, W. B., an R. B. H. Tan, A Moel for Steam Bubble Formation at a Submerge Nozzle in Flowing Suboole Water, Int. J. Heat an Flui Flow,, Cho, S. C., an W. K. Lee, A Moel for Steam Bubble Formation at a Submerge Orifie in a Flowing Liqui, J. Chem. Eng. Jpn., 3, Denekamp, J., A. Kogan, an A. Solan, On the Conensation of an Injete Vapor Bubble in a Suboole Liqui Stream, Prog. Heat Mass Transfer, 6, Jaobs, H. R., H. Fannar, an G. C. Beggs, Collapse of a Bubble of Vapor in an Immisrible Liqui, Pro. Sixth Int. Heat Transfer Conf., Toronto, Kalman, H., an A. Ullmann, Experimental Analysis of Bubble Shapes During Conensation in Misible an Immisible Liquis, J. Flui Eng., 11, MCann, D. J., an R. G. H. Prine, Regimes of Bubbling at a Submerge Orifie, Chem. Eng. Si., 6, Prakoso, T., K. Terasaka, an H. Tsuge, Effet of Operating Conitions on Two-Phase Bubble Formation Behavior at Single Nozzle Submerge in Water, J. Chem. Eng. Jpn., 34, Ruff, K., Formation of Gas Bubbles at Nozzles with Constant Throughput, Chem. Ing. Tehn., 44, Sieman, S., an G. Hirsh, Diret Contat Heat Transfer with Change of Phase, AIChE J., 11, Suhoff, B., M. Plishke, an P.-M. Weinspah, Diret Contat Heat Transfer with Change of Phase-Conensation or Evaporation of a Drobble, Ger. Chem. Eng., 5, Tan, R. B. H., an I. J. Harris, A Moel for Non Spherial Bubble Growth at a Single Orifie, Chem. Eng. Si., 41, Terasaka, K., W.-Y. Sun, T. Prakoso, an H. Tsuge, Measurement of Heat Transfer Coeffiient for Diret-Contat Conensation uring Bubble Growth in Liqui, J. Chem. Eng. Jpn., 3, Terasaka, K., T. Prakoso, W.-Y. Sun, an H. Tsuge, Two-Phase Bubble Formation with Conensation at Nozzle Submerge in Immisible Liqui, J. Chem. Eng. Jpn., 33, Walters, J. K., an J. F. Davison, Initial Motion of Gas Bubble Forme in Invisi Liqui, J. Flui Meh., 17, Wanhoo, R. K., Conensation of Single Two-Phase Bubbles in an Immisible Liqui: Heat Transfer Charateristis, Chem. Eng. Comm., 105, Wanhoo, R. K., S. K. Sharma, an G. K. Raina, Drag Coeffiient an Veloity of Rise of a Single Collapsing Two-Phase Bubble, AIChE J., 43, Appenix: Derivation of Vapor Pressure Change The total mass balane for the vapor within the two-phase bubble is ns n y n A1. where nin an n are the molar number of gas injete into bubble an onensate, respetively, an ž / in nin P s t R g T A. n V V ja ja s s s s A3. t t M M t M M where P an T are the pressure an temperature of the vapor, respetively. V, A, an are volume, surfae area, an ensity of the onensate. Applying open system energy balane or first law of thermoynamis for open system. to the vapor within the twophase bubble, the internal energy hange of vapor within the two-phase bubble E an be expresse in terms of heat interation aross the vapor-onensate interfae, work interation aross the vapor-onensate interfae W, an energy balane ue to nonsteay flow the last three terms in the righthan sie of Eq. A4. Here, E s q W q hinninyhnqek,in A4. E s ne q e n A5. is the sensible enthalpy inrease, whih, in general, is negligible ompare with latent heat effets from mass transfer an onensation, so s0 W sy PV / A6. A7. Ek,inq t A8 ž. a o Substituting the above expressions into Eq. A4, it follows that / nc T q PVs RTnq g t A9 ž. a o Rewriting Eq. A9 together with ieal gas law equation PVs nr T g A10. After substituting Eqs. A an A3, the pressure hange of vapor within the two-phase bubble is obtaine ž / 3 P RT g P ja P V s y y q y1. t V R T M V t V a g o where is the aiabati gas exponent s C p C Eq. 8. A11. an C an C are onstant-pressure an onstant-volume p heat apaities, respetively. Manusript reeie July 3, 001, an reision reeie Feb. 4, 003. AIChE Journal August 003 Vol. 49, No