MODELING AND MEASUREMENT OF INTERFACIAL AREA CONCENTRATION IN TWO-PHASE FLOW. Mamoru Ishii and Takashi Hibiki

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1 MODELING AND MEASUREMEN OF INERFACIAL AREA CONCENRAION IN WO-PASE FLOW Mamoru Ishii and akashi ibiki School o Nuclear Enineerin, Purdue University 4 Central Drive, West Laayette, IN , USA ishii@purdue.edu and hibiki@purdue.edu ABSRAC his paper presents experimental and modelin approaches in characterizin interacial structures in as-liquid two-phase low. For the modelin o the interacial structure characterization, the interacial area transport equation proposed earlier has been studied to provide a dynamic and mechanistic prediction tool or two-phase low analysis. A state-o-the-art our-sensor conductivity probe technique has been developed to obtain detailed local interacial structure inormation in a wide rane o low reimes spannin rom bubbly to churn-turbulent lows. Newly obtained interacial area data in 8 8 rod bundle test section are also presented. his paper also reviews available models o the interacial area sink and source terms and existin databases. he interacial area transport equation has been benchmarked usin condensation bubbly low data. 1. INRODUCION In the present state-o-the-art, the two-luid model is the most detailed and accurate macroscopic ormulation o the thermo-luid dynamics o two-phase systems (Ishii, 1975; Ishii and ibiki, 5). he existence o the interacial transer terms is one o the most important characteristics o the twoluid model ormulation. hese terms determine the rate o phase chanes and the deree o mechanical and thermal non-equilibrium between phases, thus they are the essential closure relations which should be modeled accurately. he present thermal-hydraulic system analysis codes such as RELAP5, RAC, and CAARE use one-dimensional orm o the two-luid model equations, where the interacial area concentration is speciied by empirical correlations based on steady-state, ully developed conditions. his approach suers ollowin shortcomins: 1) Durin relatively ast transients, where the relaxation time or the development o the low structure is similar or reater than the characteristic time o the transient itsel, the dynamic nature o the transient cannot be accurately predicted. Analysis o developin lows also suers the same drawback. ) he correlations or interacial area concentration are based on low reimes which are in turn speciied by constitutive equations as discrete unctions o local supericial velocities o two phases. he compound errors in this approach are lare. his also leads to artiicial discontinuities and numerical instabilities in the numerical solutions o the ield equations. 3) he low reime transition criteria and correspondin interacial area concentration correlations have a limited rane o applicability in terms o boundary conditions, luid properties, low eometry and scale. o overcome these problems, the interacial area concentration should be speciied by a transport equation which models the evolution o the eometric structure considerin the physical mechanisms o local interactions between the two phases (Ishii, 1975; Ishii and ibiki, 5). he transport equation can capture the dynamic chanes in the low structure, especially the transitions amon low reimes. ence the transport equation can be used or systems with wide rane o scales and boundary conditions. Followin this introduction, this paper presents the basic idea behind the derivation o the interacial area transport equation and discusses the two-roup approach includin modiied two- 1

2 roup as momentum equation. Furthermore, the measurement technique and the available experimental data are summarized, and newly obtained data in 8 8 rod bundle test section are also presented. Finally, the evaluation o the interacial area transport equation with respect to the database is discussed.. FORMULAION OF INERFACIAL AREA RANSPOR EQUAION.1 wo-roup interacial area transport equation he Boltzmann transport equation describes particle transport by an intero-dierential equation o the particle-distribution unction. Since the interacial area o the luid particle is closely related to the particle number, the interacial area transport equation can be ormulated based on the Boltzmann transport equation (Ishii and ibiki, 5). In a as-liquid two-phase low system, a wide rane o bubble shapes and sizes exist dependin on the iven low reime. hereore, to develop the interacial area transport equation describin the bubble transport in a wide rane o two-phase low reimes, the model must take account o the dierences in the transport characteristics o dierent types o bubbles. hese variations in shape and size o bubbles cause substantial dierences in their transport mechanisms due to the dra orces as well as the bubble interaction mechanisms. In the two-roup ormulation, the bubbles are treated in two roups, namely roup-1 bubbles consistin o spherical and distorted bubbles and roup- bubbles consistin o cap, aylor and churnturbulent bubbles. he roup-1 bubbles exist in the rane o minimum bubble size to maximum distorted bubble size limit, D d,max, whereas the roup- bubbles exist in the rane o D d,max to maximum stable bubble size limit, D max. he boundary bubble sizes, D d,max and D max are iven by σ σ (1) D d,max = 4 and D max = 4, Δρ Δρ where σ : surace tension, : ravitational acceleration, Δρ : density dierence. he two-roup interacial area transport equation can be obtained by averain the interacial area transport equation over the volume rane o each bubble roup as ai1 a i1 α 1 () + ( ai1vi1) = + ( α1v1) η ph 3 α 1 D Vc sc a α i1 1 χ ( α1 1) ηph S S ph AdV i D + v + + V Sm1 α 1, min ai ai α + ( aivi) = + ( αv) 3 α D V sc a α i1 1 max + χ ( α1 1) η ph SAdV i D + v + Vc Sm1 α 1, where a i : interacial area concentration, v i : interacial velocity, α: void raction, t : time, v : as-phase center o mass velocity, η ph : rate o volume enerated by nucleation source per unit mixture volume, χ: coeicient accountin or the contribution rom the inter-roup transer, D sc : surace equivalent diameter o a luid particle with surace area, A i, D Sm : Sauter mean diameter, S and S ph : particle source and sink rates per unit mixture volume due to -th particle interactions and that due to phase chane, respectively, V min, V c and V max : volumes o minimum bubble, critical bubble and maximum bubble, respectively. he two-roup interacial area transport equation is simpliied to a one-roup interacial area transport equation in the bubbly low reime.

3 . wo-roup momentum equation he two-luid model should be modiied or the two-roup interacial area transport equation (Ishii and ibiki, 5). In eneral, the pressure and temperature or roup-1 and roup- bubbles can be assumed to be approximately the same. owever, the velocities o two roups o bubbles are not the same, thereore it is necessary to introduce two continuity and two momentum equations in principle. Based on the above assumption, the density o the as phase is the same or roup-1 and roup- bubbles. he continuity equation or roup-1 and - bubbles are thereore iven by ( α ρ) (3) 1 + ( α ρv ) = Γ Δm , ( αρ) + ( α ρv ) = Γ + Δm 1, where ρ : as density, Γ : mass eneration or as phase, Δm 1 : inter-roup mass transer due to hydrodynamic mechanisms. he momentum equation or roup-1 and - bubbles are iven by ( α ρv ) (4) 1 1 μ + ( α ρ ) α p α ( ) 1 v 1v 1 = α ρ + Γ Δm v α + M + p p α ( ) ( ) i1 1 i1 i1 i1 1 1, ( α ρ v ) μ ( α ρ ) α p α ( ) v v ( m ) ( p p ) + = α ρ + Γ +Δ v α + M + α 1 i i i i, where p : pressure, μ : viscous stress, : turbulent stress, M i : eneralized interacial dra..3 One-dimensional interacial area transport equation he simplest orm o the interacial area transport equation is the one-dimensional ormulation obtained by applyin cross-sectional area averain over Eq.(). he exact mathematical expressions or the area-averaed source and sink terms involve many covariances that may urther complicate the one-dimensional problem. In the current one-dimensional ormulation or adiabatic two-phase low, the covariance terms are nelected because o relatively uniorm low parameter distribution over the low channel. For example, the one-dimensional one-roup interacial area transport equation is expressed as ai a i α (5) + ( ai vi ) = ( ΦB ΦC ) + a { + ( α v ) 3 }, z α z where : area-averaed quantity, a : interacial area concentration-weihted mean quantity, : void raction-weihted mean quantity, Φ B and Φ C : rates o chane o interacial area concentration due to bubble breakup and coalescence, respectively. owever, or subcooled boilin low, the phase distribution pattern may not be assumed uniorm, resultin in many covariances in the one-dimensional interacial area transport equation. o avoid the covariances, a simple model can be introduced to ormulate the transport equation or subcooled boilin low (ibiki et al., 3). In this case, the bubbles mainly exist near a heated wall, whereas almost no bubbles exist ar rom the heated wall. hereore, the low path may be divided into two reions, namely (i) boilin two-phase (bubble layer) reion where the void raction proile can be assumed to be uniorm, and (ii) liquid sinle-phase reion where the void raction can be assumed to be zero. hus, the one-dimensional interacial area transport equation can be obtained by averain the interacial area transport equation over the bubble-layer reion and applyin a actor o AB A C 3

4 to the interacial area concentration in the bubble-layer reion. ere A B and A C are the area o the bubble-layer reion and the cross-sectional area o the boilin channel, respectively. 3. MEASUREMEN ECNIQUE he local time-averaed interacial area concentration can be measured usin our-sensor conductivity probe method (Kataoka et al., 1986) based on the dierence in conductivity between the two phases. he probe is composed o a sensor and an electrically conductive casin. Usin the casin as the common round, the characteristic rise and all o the impedance sinals can be obtained when an interace passes throuh the probe sensor. With the acquired sinals rom the sensor, the local time-averaed void raction can be easily obtained by dividin the sum o the time raction occupied by the as phase by the total measurement time. One o the most important eatures o the conductivity probe is that it can measure the local interacial velocity with multiple sensors. his is o reat importance because the local time-averaed interacial area concentration can be obtained rom the interacial velocity throuh the mathematical relation. he our-sensor conductivity probe is made o our sensors. With one common upstream sensor and three independent downstream sensors, three directional velocities at a local point can be obtained by measurin the time delay between the sinals rom three pairs o double-sensors. When the directions o the three independent probes are chosen, the equation or the time-averaed interacial area concentration can be simpliied as (Kataoka et al., 1986), (6) a i =, Ω + + v s1 v s v s3 where Ω : time interval, v sk : passin velocity o the -th interace over probe k. In Eq.(6), no hypothesis or bubble shape is needed or calculatin the local interacial area concentration rom the interacial velocity. hereore, the our-sensor probe can be utilized in a wide rane o two-phase low reimes where bubbles are no loner spherical in shape. I we consider spherical bubbly low with no correlation between bubble velocity and movin direction, Eq.(6) can be siniicantly simpliied to enable double sensor probe method to measure the interacial area concentration (Kataoka et al., 1986; ibiki et al., 1998). o minimize the inite probe size eect on the measurement, a miniaturized our-sensor conductivity probe has been developed (Kim et al., ). A schematic diaram o the probe desin and its eometrical coniuration is shown in Fi.1. One o the important eatures o the probe desin is that the our sensor probe accommodates a built-in double-sensor probe or measurin small bubbles. hereore, the probe is applicable to a wide rane o low reimes, spannin rom bubbly ~5 mm ~5-7 mm Epoxy and conductive ink Stainless-steel tubin Epoxy Stainless-steel tubin, OD: ~3 mm hermocouple wires Coated sensor body < 5-micron ~.5 mm Sensor tip (uncoated) 1 Not scaled 3 ~.5-.8 mm Fi.1: Coniuration o the our-sensor conductivity probe (Kim et al., ). 4

5 to churn-turbulent low reimes. Alon with the probe desin, the sinal processin is constructed, such that it can identiy and separate the local parameters into two roups, namely: roup-1 or spherical and distorted bubbles and roup- or slu and churn-turbulent bubbles. hrouhout the benchmark experiments, very ew missin interaces are observed or the roup- bubbles unless the probe is traversed very close to the wall o the low duct. he measurement area o the our-sensor probe used in the experiments is less than. mm and the tip distance o the double-sensor probe or small bubbles is.4 mm. his allows the measurable bubble diameter to be as smal1 as 1 mm. he diameter o the sensor electrode is.13 mm, however, the sensor itsel has a diameter o less than.5 mm. he details o the double sensor and our-sensor probe techniques can be ound in ibiki et al. (1998) and Kim et al. (), respectively. 4. DAABASE OF INERFACIAL AREA CONCENRAION 4.1 Existin database Extensive data o axial development o low parameters in adiabatic and boilin two-phase low have been obtained over a wide rane o low conditions includin downward low. he measured low parameters include local void raction, interacial area concentration, interacial velocity, bubble Sauter mean diameter, liquid velocity and liquid turbulence. he covered experimental conditions are est section eometry: Round pipe, conined channel, annulus and rod bundle, est section size: 1 mm to 1 mm, Flow reime: Bubbly, cap-bubbly, slu and churn-turbulent lows, Flow condition: Supericial as velocity,, up to 1 m/s and supericial liquid velocity,, rom -3.1 m/s (i.e. downward low) to 5. m/s (i.e. upward low), Flow direction: Vertical upward and downward lows, hermal condition: Adiabatic and diabatic lows, Gravity condition: Normal and micro ravity conditions, Pressure condition: Atmospheric pressure he extensive review o the existin database can be ound in ibiki and Ishii (1, ) and ibiki et al. (6). In what ollows, newly obtained data taken in 8 8 rod bundle test section are shown. 4. Database or 8 8 rod bundle eometry Recently, axial developments o local low parameters in air-water two-phase low were measured in 8 8 rod bundle test acility. he test section consisted o a 3m lon vertical low channel with square cross section havin sides o lenth 14 mm. It housed an 8 8 array o 64 rods each havin a diameter o 1.7mm arraned at a pitch distance o 16.7 mm. he rods were held in place by six spacer rids located at zd = 31, 61, 91, 11, 151 and 18 where D : hydraulic diameter o a typical subchannel (=14.8 mm), z : axial distance rom the inlet o the test section. he narrowest and widest aps in the center subchannel were, respectively, 4. mm and 1.9 mm. he measurement ports were located at the axial distances o zd =7, 86, 94, 116, 14, 137 and. he detailed inormation on the experimental acility and instrumentation can be ound in Paranape et al. (8). Fiure shows the measurement rid or local low data. At each axial location in the low channel, the local low data were taken at the center o the sub-channels and the narrowest ap between the rods. he local data were taken in 5 axial locations in the low channel under supericial as and liquid velocity conditions which covered bubbly, cap bubbly, cap-turbulent and churn-turbulent low reimes. Fiure 3 shows low visualization in various low reimes. 5

6 d W S x Subchannel Center Rod Gap Center Fi.: Measurement rid in a rod bundle eometry. (a) bubbly low =. m/s, =. m/s. (b) cap bubbly low =.14 m/s, =. m/s. (c) Cap turbulent low (d) Churn turbulent low =.8 m/s, =.1 m/s. = 8.8 m/s, =. m/s. Fi.3: Flow visualization in various low reimes (Paranape et al., 8). he representative data at zd = in two bubbly and one cap-turbulent low reime conditions are shown in Fis.4-7. In Fis.4-6, the data are shown alon the center line o the rod bundle rom the bundle center, x. he distance in the iures is normalized by the hal width o the rod bundle casin desinated by W. In Fi.7, the data is also plotted alon the diaonal line o the rod bundle rom the bundle center, d. he distance in the iure is normalized by the hal width o 6

7 the bundle diaonal lenth desinated by S. he low conditions in Fis.4 and 5 are =. m/s and =. m/s, and =. m/s and =1. m/s, respectively, correspondin to bubbly low reime where only roup-1 bubbles are observed. he low condition in Fis.6 and 7 is =.5 m/s and =. m/s, correspondin to cap-turbulent low reime where two roups o bubbles are observed. Comparin Fis.4 and 5, the eect o increased supericial liquid velocity on the spatial distribution and manitude o low parameters can be clariied, whereas the comparison between Fis.4 and 6 provides some insihts on the eect o increased as velocity and low reime transition on the spatial distribution and manitude o low parameters. Open and solid symbols in Fis.4 and 5 indicate local low parameter measured at subchannel center and rod narrowest ap, respectively. As the supericial liquid velocity increases, the bubble size becomes smaller due to bubble breakup and the bubbles tend to mirate toward the rod narrowest ap, which is similar to wall peakin observed in a round pipe test section (ibiki and Ishii, 1999). his can be explained as ollows. he lit orce actin on a bubble is enerally enhanced by increased velocity radient. he increased liquid velocity enhances the as velocity radient between the subchannel center and rod Void Fraction, <α> [-] Location rom Center, x/w [-] Interacial Area Conc., <a i > [m -1 ] Location rom Center, x/w [-] Gas Velocity, <v > [m/s] Location rom Center, x/w [-] Sauter Mean Diameter, <D Sm > [mm] Location rom Center, x/w [-] Fi.4: Local parameter data alon center line in bubbly low at Flow condition: =. m/s and =. m/s. zd =. 7

8 Void Fraction, <α> [-] Location rom Center, x/w [-] Interacial Area Conc., <a i > [m -1 ] Location rom Center, x/w [-] Gas Velocity, <v > [m/s] Location rom Center, x/w [-] Sauter Mean Diameter, <D Sm > [mm] Location rom Center, x/w [-] Fi.5: Local parameter data alon center-line in bubbly low at zd =. Flow condition: =. m/s and =1. m/s. narrowest ap, which may also indicate the aumented liquid velocity radient. In addition to this, the lit orce direction is lipped over around the bubbles size o 5 mm. I the bubble size is smaller than 5 mm, the lit orce tends to push the bubbles toward a wall. For the bubble size smaller than 5 mm, the absolute value o the lit coeicient decreases with increased bubble size (ibiki and Ishii, 7). hus, the increased supericial liquid velocity decreases the bubble size resultin in the increased absolute value o the lit coeicient. Such combined eects enhance the void raction at the rod narrowest ap. As the bubbles mirate toward the rod narrowest ap, the interacial area concentration near the rod narrowest ap is increased. he as velocity at the subchannel center is hiher than that at the rod narrowest ap, since the liquid velocity is bound to be zero at the rod surace. For the lower liquid velocity condition, the bubble Sauter mean diameter is comparable to the rod narrowest ap (4 mm). Open and solid symbols in Fi.6 indicate local low parameter measured at subchannel center and rod narrowest ap, respectively. Circle and trianle symbols indicate local low parameter or roup-1 and - bubbles. As can be seen rom Fi.6, as the supericial as velocity increases, roup- bubbles, namely cap bubbles are ormed. Unlike the low supericial as velocity condition shown in Fi.5, the dierence in low parameter distribution between the subchannel center and rod narrowest 8

9 Void Fraction, <α> [-] Group at Center Group at Gap Location rom Center, x/w [-] Interacial Area Conc., <a i > [m -1 ] Group at Gap Group at Center Group at Gap Location rom Center, x/w [-] Gas Velocity, <v > [m/s] Group at Center Group at Gap Location rom Center, x/w [-] Sauter Mean Diameter, <D Sm > [mm] Group at Center Group at Gap Location rom Center, x/w [-] Fi.6: Local parameter data alon center-line in cap-turbulent low at Flow condition: =.5 m/s and =. m/s. zd =. ap is insiniicant, which means that the low parameter distribution is inluenced by the presence o the bundle outer casin. Since the size o the roup-1 bubble exceeds 5 mm, the bubbles around the rod narrowest ap may be squeezed by rods. Since lare bubbles tend to rise in the center o the channel, the void raction at the subchannel center becomes hiher. he as velocity shows the power-low distribution over the whole bundle test section in a lobal sense. he as velocity o roup-1 bubble is slihtly hiher than that o roup- bubble. his can be explained by the dierence in dritvelocity in roup-1 and - bubbles. In the tested condition, the Sauter mean diameter o roup- bubble exceeds the hydraulic diameter o the subchannel. Due to the presence o the rods, the drit velocity may be reduced considerably. his can be understood by the act that the drit velocity o a slu bubble is in proportional to the square root o channel diameter. For air-water pipe low at room temperature and atmospheric pressure, the drit velocities or small and slu bubbles becomes equal at D =5.8 mm. In the channel smaller than 5.8 mm, slu bubble risin velocity is smaller than small bubble risin velocity. Fiure 7 shows the local low parameter measured at subchannel center alon the diaonal line o the bundle test section. Interestinly the void raction o roup-1 bubbles has a sliht peak near the bundle outer casin. he distributions o low parameters alon the diaonal line are consistent with those alon the subchannel center line. 9

10 Void Fraction, <α> [-] Group 1 Group Location rom Center, d/s [-] Interacial Area Conc., <a i > [m -1 ] Group 1 Group Location rom Center, d/s [-] Gas Velocity, <v > [m/s] Group 1 Group Location rom Center, d/s [-] Sauter Mean Diameter, <D Sm > [mm] Group 1 Group Location rom Center, d/s [-] Fi.7: Local parameter data alon diaonal line in cap-bubbly low at Flow condition: =.5 m/s and =. m/s. zd =. 5. SINK AND SOURCE ERM MODELING AND EIR EVALUAION o close the interacial area transport equation, extensive eorts to model various sink and source terms o the interacial area transport equation have been made. Maor modeled sink and source terms are summarized in able 1. o demonstrate the capability o the interacial area transport equation, the one-dimensional one-roup interacial area transport equation is evaluated in condensation systems usin ai (7) a v = Φ Φ Φ + α v. i i a I RC CD z 3 α z In Eq.(7), the bubble breakup and coalescence terms are calculated by ibiki and Ishii s model (a), whereas the bulk condensation term is computed by Park et al. model (7). Fiure 8 compares the interacial area transport calculation with condensation data taken in a vertical concentric annulus (Park et al., 7). he low condition is p (inlet pressure) =.13 MPa, (inlet liquid temperature) = 98. C and G (total mass lux) = 139 k/m s (Zeitoun, 1994). As can be seen rom Fi.8, the interacial area concentration chanes due to bubble coalescence, bubble breakup and void raction chane are neliibly small in the tested condition. he condensation in heat transer-controlled reion without bubble number density chane, Φ, is the 1 C

11 maor contribution to the interacial area concentration chane, whereas the contribution o the inertia-controlled reion with bubble number density chane, Φ IC, is about 16.8 %. he interacial area transport equation can predict the interacial area chane alon the low direction successully. able 1: Maor modeled sink and source terms o the interacial area concentration. Cateory Investiator Details One-roup Wu et al. (1998) One-roup bubble coalescence and breakup Bubble ibiki and Ishii (a) One-roup bubble coalescence and breakup Interaction ibiki et al. (1) One-roup bubble coalescence and breakup Yao and Morel (4) One-roup bubble coalescence and breakup wo-roup ibiki and Ishii (b) wo-roup bubble coalescence and breakup Bubble Fu and Ishii (3) wo-roup bubble coalescence and breakup Interaction Sun et al. (4) wo-roup bubble coalescence and breakup Phase Chane Park et al. (7) Bulk condensation Riznic and Ishii (1989) Flushin Wall ibiki and Ishii (3) Active nucleation site density Nucleation Situ et al. (8) Bubble departure size Source Situ et al. (8) Bubbly departure requency IAC Chane, <a i > - <a i > z/d=1 [m -1 ] p =.13 MPa, =98. o C, G=139 k/m s Experimental Data (Zeitoun, 1994) Predicted IAC Chane eat ranser Inertia Void ransport Random Collision urbulent Impact Axial Position, z/d [ - ] Fi.8: Contribution o bubble coalescence, breakup, void transport, and condensation to interacial area transport (Park et al., 7). 6. CONCLUSIONS In relation to the modelin o the interacial transer terms in the two-luid model, the concept o the interacial area transport equation has been proposed to develop a constitutive relation or the interacial area concentration. he interacial area transport equation can replace the traditional low reime maps and reime transition criteria. he chanes in the two-phase low structure can be predicted mechanistically by introducin the interacial area transport equation. he eects o the boundary conditions and low development are eiciently modeled by this transport equation. he successul development o the interacial area transport equation can make a siniicant improvement in the two-luid model ormulation and the prediction accuracy o three-dimensional two-phase low simulations. his paper reviewed the current status o the interacial area transport equation development and presented newly obtained interacial area concentration data in 8 8 rod bundle test acility. Althouh much eorts have been done to develop it, urther modelin and experimental work are required to establish the interacial area transport equation. 11

12 ACKNOWLEDGEMENS his work was perormed under the auspices o the US Nuclear Reulatory Commission and the US Department o Enery. he authors would like to express their sincere appreciation to NRC and DOE stas or their support on this research. he ratitude is also extended to Messrs. S. Paranape and Y. Lian at Purdue University or their help in the bundle experiment. REFERENCES X. Y. Fu, M. Ishii, wo-roup interacial area transport in vertical air-water low I. Mechanistic model, Nucl. En. Des., 19, (3).. ibiki, M. Ishii, Interacial area concentration in steady ully developed bubbly low, Int. J. eat Mass ranser, 44, (1). ibiki, M. Ishii, Experimental study on interacial area transport in bubbly two-phase lows, Int. J. eat Mass ranser, 4, (1999).. ibiki, M. Ishii, One-roup interacial area transport o bubbly lows in vertical round tubes, Int. J. eat Mass ranser, 43, (a).. ibiki, M. Ishii, wo-roup interacial area transport equations at bubbly-to-slu low transition, Nucl. En. Des.,, (b).. ibiki, M., Ishii, Interacial area concentration o bubbly low systems, Chem. En. Sci., 57, ().. ibiki, M. Ishii, Active nucleation site density in boilin systems, Int. J. eat Mass ranser, 46, (3).. ibiki, M. Ishii, Lit orce in bubbly low systems, Chem. En. Sci., 6, (7).. ibiki et al., Local measurement o interacial area, interacial velocity and liquid turbulence in two-phase low, Nucl. En. Des., 184, (1998).. ibiki et al., Interacial area transport o bubbly low in a small diameter pipe, J. Nucl. Sci. echnol., 38, (1).. ibiki et al., Modelin o bubble-layer thickness or ormulation o one-dimensional interacial area transport equation in subcooled boilin two-phase low, Int. J. eat Mass ranser, 46, (3).. ibiki et al., Interacial area concentration in boilin bubbly low systems, Chem. En. Sci., 61, (6). M. Ishii, hermo-luid Dynamic heory o wo-phase Flow, Eyrolles (1975). M. Ishii,. ibiki, hermo-luid Dynamics o wo-phase Flow, Spriner (5). I. Kataoka et al. Local ormulation and measurements o interacial area concentration in two-phase low, Int. J. Multiphase Flow, 1, (1986). S. Kim et al., Development o the miniaturized our-sensor conductivity probe and the sinal processin scheme, Int. J. eat Mass ranser, 43, (). S. Paranape et al., Global low reime identiication in a rod bundle eometry, Proc. ICONE-16, Paper No (8).. S. Park et al., Modelin o the condensation sink term in an interacial area transport equation, Int. J. eat Mass ranser, 5, (7). J. R. Riznic, M. Ishii, Bubble number density and vapor eneration in lashin low, Int. J. eat Mass ranser, 3, (1989). R. Situ et al., Bubble departure requency in orced convective subcooled boilin low, Int. J. eat Mass ranser (8, in print). X. D. Sun et al., Modelin o bubble coalescence and disinteration in conined upward two-phase low, Nucl. En. Des., 3, 3-6 (4). Q. Wu et al., One-roup interacial area transport in vertical bubbly low, Int. J. eat Mass ranser, 41, (1998). W. Yao, C. Morel, Volumetric interacial area prediction in upward bubbly two-phase low, Int. J. eat Mass ranser, 47, (4). O. M. Zeitoun, Subcooled low boilin and condensation, Ph.D. hesis, McMaster University, Canada (1994). 1