ASSESSMENT OF AN INTERFACIAL SHEAR TERM FOR ADIABATIC DISPERSED AIR-WATER TWO-PHASE FLOW WITH THE TWO-FLUID MODEL

Transcription

1 ASSESSMEN OF AN INERFACIAL SHEAR ERM FOR ADIABAIC DISPERSED AIR-WAER WO-PHASE FLOW WIH HE WO-FLUID MODEL S. L. Sharma,. Hibii, M. Ishii School o Nuclear Engineering, Purdue University 400 Central Dr., West Laayette, IN , USA sharma55@purdue.edu, hibii@purdue.edu, ishii@purdue.edu J.P. Schlegel Nuclear Engineering Program, Missouri University o Science and echnology 301 W 14 th St, Rolla, MO 65409, USA schlegelj@mst.edu J.R. Buchanan, Jr., K. J. Hogan Bechtel Marine Propulsion Corporation, Bettis Laboratory 814 Pittsburgh-McKeesport Blvd, West Milin, PA 15122, USA jac.buchanan@unnpp.gov, evin.hogan@unnpp.gov P.W. Guilbert ANSYS UK Ltd Milton Par, Abingdon, Oxordshire, OX14 4SA,UK paul.guilbert@ansys.com ABSRAC In commercially available Computational Fluid Dynamics (CFD) codes such as ANSYS CFX and Fluent, the interacial shear term is missing in the ield momentum equations. he derivation o the two-luid model (Ishii and Hibii, 2011) indicates the presence o this term as a momentum source in the right hand side o the ield momentum equation. he inclusion o this term is considered important or proper modeling o the interacial momentum coupling between phases. For separated lows, such as annular low, the importance o the shear term is understood in the one-dimensional (1-D) orm as the major mechanism by which the wall shear is transerred to the gas phase (Mishima and Ishii, 1984). For dispersed two-phase low CFD simulations it is important to assess the signiicance o this term in the prediction o phase distributions. In the irst part o this wor, the closure o this term in threedimensional (3-D) orm in a CFD code is investigated. For dispersed gas-liquid low, such as bubbly or churn-turbulent low, bubbles are dispersed in the shear layer o the continuous phase. he continuous phase shear stress is mainly due to the presence o the wall and the modeling o turbulence through the Boussinesq hypothesis. In a 3-D simulation, the continuous phase shear stress can be calculated rom the continuous luid velocity gradient, so that the interacial shear term can be closed using the local values o the volume raction and the total stress o liquid phase. his orm also assures that the term acts as an action-reaction orce or multiple phases. In the second part o this wor, the eect o this term on the volume raction distribution is investigated. For testing the model two-phase low data measured at Purdue University is assessed. he interacial shear term is assembled in ANSYS CFX. Simulation results are presented to assess the eect o the interacial shear term on the phase distribution. 3204

2 KEYWORDS wo-luid model, Interacial shear, CFD, Dispersed gas-liquid low 1. INRODUCION In recent years, CFD has emerged as a very useul tool or engineers and scientists, providing valuable inormation on the temporal and spatial distribution o ey variables in a low ield. Most commercial CFD codes can be applied with conidence to solve a variety o single-phase low problems; however, considerable research eort is still necessary beore CFD can be applied to the study o gas-liquid twophase lows with the same level o conidence. While Direct Numerical Simulation (DNS) is impractical or industrial low problems involving signiicant void raction, an Eulerian-Eulerian approach (two-luid model) is widely used and considered well suited or industrial scale two-phase low problems [1]. Present commercial two-phase CFD tools already have Eulerian-Eulerian modeling capability using the two-luid model, which was developed by Ishii [1-3] by time averaging the instantaneous transport equations (continuity, momentum, and energy) or each phase. his time averaging results in interacial transer terms in the two-luid model which require closure relations. he predictive capability o the twoluid model relies heavily on the accuracy o these closure relations. In the present wor, the ocus is on adiabatic bubbly gas-liquid low. In case o adiabatic two-phase low, the three dimensional ormulations o two-luid model rigorously derived by Ishii [1,2] can be simpliied as Continuity equation: Momentum equation: α ρ v t α ρ + t + ( α ρ v ) = Γ t ( α ρ v v ) = α p + α ( τ + τ ) ; (1) + αρg+ v Γ + M α τ ; i i i (2) where the subscript i stands or the values at the interace; Γ is the mass generation term; M i is the interacial orce term; and τ i is the interacial shear stress. he interacial orce terms in the momentum equation are generally considered to include a drag, lit, wall, and dispersion orce or bubbles by liquid eddies. A thorough literature survey has been done to determine the best available hydrodynamic models or each o the orces. he best available models chosen and used or the current simulations are listed in able I. It is important to mention here that or some cases, especially those with high liquid velocity, some o the model coeicients were modiied. O speciic concern was the lit orce coeicient, which was changed to capture the void raction trend seen in the measurements. his was done to isolate the eects o the interacial shear term rom other modeling uncertainties and will be discussed in a later section o the paper. he interacial shear term, ( α τi), appears in the two-luid model [1], but is missing in presently available commercial CFD codes. he term arises in the macroscopic interacial momentum transer development o the two luid model. Whereas most o the drag and non-drag terms that comprise M i 3205

3 arise rom the normal components o the macroscopic interacial momentum transer, the interacial shear comes rom the tangential components. For this reason, most one-dimensional treatments developed or two-phase low neglect the interacial shear term unless considering either annular low or horizontal lows [3]. As such, the interacial shear is included in satisying the macroscopic jump conditions when summed across all phases and ields. For three-dimensional treatments, it is important to implement the interacial shear term and assess its signiicance. able I. Summary o Interacial Forces and Constitutive Relations or Momentum Equation Applied in Benchmar Cases Interacial Forces Phases Nature Coeicient Drag Force Gas and Liquid Interacial orce Ishii and Zuber (1979) [5] Wall Lubrication Force Gas and Liquid Interacial orce Antal et al. (1991) [6] C W1 = 1, C W 2 = 5 Lit orce Gas and Liquid Interacial orce Hibii and Ishii (2007) [7] urbulent dispersion orce Gas and Liquid Interacial orce Bubble-induced turbulence Liquid Reynolds stress Bertodano (1992) [8] C D,1 = Sato et al. (1981) [9] C = Sato, G INERFACIAL SHEAR ERM CLOSURE he ield momentum equations in CFX do not include the interacial shear term ( α τi). he inclusion o this term is important or proper modeling o the interacial momentum coupling between phases [1,3]. his can be explained by considering the momentum equations or each phase and simpliying them in an area-averaged orm. Gas phase: For the gas phase, several simpliications can be used. For example, due to the relatively small viscosity and density o the gas phase compared with the liquid phase, the total shear and gravitational terms can be neglected. 3-D orm 1-D orm = α p+ α τ + α ρ g+ M α τ ( ) g g g g g ig g = + αg p M ig αg τ dp D = αg + M ig αg τ dz z dp D = αg + M ig αg τ dz z gi (3) (4) 3206

4 Liquid phase: 3-D orm 1-D orm = α p+ α τ + α ρ g+ M α τ ( ) i dp = α + α τ + α ρ α τ D Mi g dz z dp D 4τ w = 1 αg + M i + α ρ g α τ dz D i z z (5) (6) It is then important to understand how the interacial shear term can be closed. Since the bubbles are dispersed in the continuous phase shear layer, the interacial shear stress can be approximated by the continuous phase shear stress, μ t τ τ = τ + τ (7) i ( μ t ) ( μ t ) ( μ t ) α τ α τ + τ (8) g1 g1i g1 α τ α τ + τ (9) g2 g1i g2 α τ i α τ + τ (10) he above orm also assures that this term acts as an action-reaction orce or the phases such that: α τ α τ α τ = (11) g1 g1i g2 g2i i 0 his term should be included in the ield momentum equation o CFD codes because this term becomes very important in separated lows such as annular lows or horizontal stratiied lows. his is due to the very high void gradients across the interace between the separated phases. o illustrate the importance o the interacial shear term, it is useul to show how the one-dimensional orm o the term evolves through various low regimes. In the case o annular two phase low in a tube, the one-dimensional (area averaged) axial contribution becomes 1 ( ) R α 1 α Pi αg. τi = i(2 rdr) lim (2 ) 2 i rdr i z A τ π = τ π = τ r π R δ 0, (12) r A 0 δ where Pi is the interacial wetted perimeter o the gas core. Here, τ i is generally closed through constitutive relations in terms o an interacial riction actor such as the Wallis correlation [10] or a rough, wavy ilm. In the case o dispersed gas-liquid low, the continuous phase shear stress is mainly due to the presence o the wall. In a 3-D simulation in CFX, the continuous phase shear stress can be calculated rom the velocity gradient, so that the interacial shear term can be closed using the local value o the void gradient and liquid phase total stress term given in equations (7) through (11). Again, it will be useul to understand how these terms can be closed in an area-averaged two-luid model ormulation, especially or dispersed low regimes lie bubbly or churn-turbulent low. In its one-dimensional orm, the constitutive relation in the dispersed phase requires the void and shear stress distribution. A power law distribution or shear stress and void raction is assumed such that 3207

5 m r α αw r τi = τw, 1 R = α α R o g i i g z w n (13) R 1 α n + 2 4τ w ( α ). τ = τ (2 πrdr) α A =, (14) r n m D n + 2 where Cτ = is the distribution parameter. he value o m is theoretically equal to unity in the n + m + 1 case o single phase low. I it is assumed it to be close to unity or two-phase low then C τ can be expected to be close to unity. his gives the closure in one dimensional orm or the shear term as 4τ w ( αg). τi αg (15) z D In a horizontal channel this term will contribute to the slip between the dispersed and continuous phases even under a steady state conditions. he inclusion o this term does not alter the overall momentum balance o a two phase mixture because o the macroscopic momentum jump condition. However it indicates that the momentum interaction between phases is aected by the wall shear stress through the interacial shear and void gradient distributions [2,3]. So in one-dimensional orm, the right hand side (R.H.S.) o ield momentum equations (M.E.) are: R.H.S.o Gas M.E. dp 4 αg τw D = α g + M ig (16) dz D R.H.S.o Liquid M.E. dp 41 ( αg ) τw D = ( 1 αg ) + α ρ g+ M i (17) dz D It can be seen that or the gas phase, the presence o wall is elt through this interacial shear term. In the present wor, an attempt is made to assess the importance o this term in the 3-D orm o the phase distribution in a cross-section or dispersed two-phase low. he missing interacial shear term in CFX is calculated in a 3-D orm or each phase rom the product o the continuous phase total stress term and the volume raction gradient o the respective phase. his term is then added implicitly to the respective phase momentum equation to account or the importance o this term and its eect on the phase distribution or dispersed low. he continuous phase total stress is calculated rom a Boussinesq hypothesis using the velocity gradient. For the assessment o the interacial shear term, adiabatic upward bubbly air-water two-phase low data measured in two dierent geometries at Purdue University are utilized. he irst test section is o narrow rectangular geometry with cross-sectional dimension o 200mm (x-direction) by 10mm (y-direction) and was approximately 3m in height [4]. he second geometry considered is a round tube with an inner diameter o 50.8 mm and 3.061m length [11]. Conductivity probe ports were installed at three locations along the low duct length to measure the local two phase parameters such as void raction, interacial area concentration, and gas velocity. In the case o the rectangular channel, probes were located at z/d H =35, 88 and 142 away rom inlet. In the case o the pipe geometry the probes were installed at z/d H =6, 3 and Uniorm injection method or gas and liquid was used at the inlet. Further details o the test acilities are presented in more detail by Kim [4] and Hibii et al. [11]. he low conditions or the database used in this study are summarized in able II and Figure

6 Supericial Liquid Velocity, <j > [m/s] Flow Regime ransition (Neural Networ) Bubbly-Cap ransition (Mishima and Ishii, 1984) est Conditions Bubbly Run 2 Run 1 Cap urbulent Churn urbulent Supericial Gas Velocity, <j g0 > [m/s] (a) (b) Figure 1. est conditions in a j g j map or (a) Narrow Rectangular channel [4] and (b) Pipe [11]. able II. Summary o est Conditions Run 1 Run 2 Run 3 Run 4 <j g,0 >=8 m/s <j >=0.63 m/s <j g,0 >=9 m/s <j >=2.84 m/s <j g,0 >=26 m/s <j >=2.01 m/s <j g,0 >=0.518 m/s <j >=5 m/s Supericial Liquid Velocity, <j > [m/s] Flow Regime ransition (Mishima and Ishii, 1984) est Conditions Run 4 Bubbly Run 3 Slug Churn urbulent Supericial Gas Velocity, <j g > [m/s] Rectagular channel 200mm x 10mm Rectagular channel 200mm x 10mm Pipe, 50.8mm ID Pipe, 50.8mm ID 3. PREPARAION OF HE BENCHMARK he preparation o the benchmar starts with the geometry and mesh generation. he irst geometry is a rectangular duct that has dimensions o 200mm 10mm 2340mm. he domain is ocused on onequarter o the duct cross-section as shown in Figure 2. his was selected to save computational time based on assumed symmetric conditions. Based on a mesh sensitivity study it was ound that hexahedral mesh coniguration o , with a total o 97,200 computational cells, was suicient to give details o the phase distribution in the present geometry. A mesh with relatively ine cells near the wall region is used to capture the high gradient o two-phase low parameters near the wall. While selecting the mesh size near the wall, it was ensured that the value o y + is greater than 30 or irst the irst node, as the law o the wall has been used or the present simulation. y x (a)ransverse Mesh (b) Axial Mesh Figure 2. Mesh used or Flow Conditions or Rectangular Channel Geometry 3209

7 he second case is a pipe geometry. A two degree azimuthal wedge with a radius o 25.4mm is considered with symmetry assumed in the azimuthal direction. Ater a grid sensitivity analysis, 36 nodes in the radial direction and 130 uniorm nodes in the axial direction were ound to be suicient to capture the physics and satisy the requirement or the wall unction approach (Figure 3). Figure 3. Cross section o Mesh used or Pipe Geometry he inlet boundary conditions are based on inormation at the irst measurement location. his is port 2, z/d H =35, in the rectangular channel and port 1, z/d H =6, in the case o the pipe. he local measurements o void raction and gas velocity are used to speciy the inlet boundary conditions and domain initialization. For the rectangular channel, the liquid velocity proile is derived rom the gas velocity proile by subtracting the slip velocity. In the short gap direction, or y-direction, a 1/7 th power law variation is assumed or the liquid velocity proile. In the case o the pipe the liquid velocity proile was also measured at port 1 and was used as the inlet condition. A constant relative pressure o 0 Pa is used or the outlet boundary condition or both cases. A ree slip wall boundary condition is speciied or the gas phase, with a no slip wall boundary condition or the liquid phase. However, the scalable wall unction is applied or the treatment o liquid low near the wall [12]. he ε turbulence model is used or the turbulence model [13-23]. In the current benchmar study ANSYS CFX-15.0 with a new customized executable solver has been used or three-ield two-luid model simulations. he three ields include the liquid phase and two gas ields or a two-bubble-group approach. High Resolution is selected as the advection scheme and 10-5 is taen as the Root Mean Square (RMS) residual target or convergence o the simulation. he coupled void raction option under Multiphase Control options has been used in the simulations. he time step or run was selected on a case-by-case basis to obtain aster convergence. 4. RESULS AND DISCUSSION 4.1. Run 1 Run 1 belongs to a bubbly low condition with relatively low liquid and gas velocities, as shown in Fig. 1(a). he liquid and gas phase were injected uniormly at the inlet. he void raction proile at port 2 which was used as the boundary condition is mostly uniorm, but does contain eatures rom the injection spargers, as seen in Figure 4. he local phase distribution prediction without the interacial shear term is compared with the measurements at port 4 and port 6. he igure shows the local prediction o void raction along the line y = 1mm, which is near the wall, and at y = 5mm, along the centerline. he prediction shows typical bubbly low wall-peaing phenomenon. he variation in the y-direction along the lines x = 3mm, which is near the wall, and x = 100 mm, which is the centerline, shows that the overall trend o the void raction distribution in y-direction is airly well predicted by CFX. Figure 5 shows the eect o addition o the interacial shear term (IAS) in the ield momentum equation. As can been seen rom phase distribution plot, the eect is mostly limited to the near-wall region. his is expected, as both interacial shear and void gradient is expected to be high near the wall region. Ater addition o the IAS, the void raction is redistributed away rom wall. his is most notable in the y=1mm line and conirmed in the x=3mm and y=90mm lines in Figure

8 4.2. Run 2 he second case to be discussed here belongs to a inely dispersed bubbly low. his case is at relatively high liquid velocity (j =2.84 m/s). For this case, measurement data are only available or ports 2 and 6. First, the simulation was carried out with the interacial orce models included in the able I, without the IAS. It was ound that most o the bubbles were swept to the near-wall region by the lit orce, which created a physically unrealistic void proile. he physics-based lit coeicient model o Hibii and Ishii was originally benchmared with experimental data or mainly low liquid velocity conditions [7]. At high liquid low rates, the eect o turbulence on the lit coeicient in the presence o wall region has not been understood clearly in literature. Investigations o this eect are not in the scope o the present wor. hereore, the lit coeicient was lowered to 5 to reproduce the measured void raction proile at port 6 or this highly turbulent bubbly low case. Since it is diicult to predict the trend at all locations o the domain, the center lines (y = 5mm, x = 90mm) were targeted. he local phase distribution results with a lit coeicient o 5 show good prediction in both wide (W = 200mm) and narrow (G = 10mm) directions as shown in Figure 6. he eect o addition o the IAS on the local void raction distribution is shown in Figure 7. he eect o the addition o the interacial shear term is limited to the near wall region. he two cases discussed previously belong to the rectangular channel geometry, where three dimensional eects were present. However, to see strictly two dimensional eects, two-phase low data measured in a 50.8mm ID pipe geometry was used or a second assessment. 0.5 Port 2 Exp (y=1mm) Exp (y=2mm) Exp (y=3mm) Exp (y=4mm) Exp (y=5mm) 0.5 Port 4 Exp (y=1mm) Exp (y=5mm) CFX (y=1mm) CFX (y=5mm) 0.5 Port 4 Exp (x=3mm) Exp (x=90mm) CFX (x=3mm) CFX (x=90mm) 0.5 ransverse Position, x/w [-] 0.5 ransverse Position, x/w [-] 0.5 ransverse Position, y/g [-] 0.5 Port 6 Exp (y=1mm) CFX (y=1mm) 0.5 Exp (y=5mm) CFX (y=5mm) 0.5 ransverse Position, x/w [-] 0.5 Figure 4. Local phase distribution at Port 2, 4 and 6 (Run 1) Port 6 Exp (x=3mm) Exp (x=90mm) CFX (x=3mm) CFX (x=90mm) ransverse Position, y/g [-] 3211

9 ` (a) (b) Figure 5. Local phase distribution at (a) Port 4 and (b) Port 6 with and without addition o interacial shear term IAS (Run 1) 3212

10 Port 2 Exp: (y=1mm) Exp: (y=2mm) Exp: (y=3mm) Exp: (y=4mm) Exp: (y=5mm) Port 6 Exp (y=1mm) Exp (y=2mm) Exp (y=3mm) Exp (y=4mm) Exp (y=5mm) CFX (y=1mm) CFX (y=2mm) CFX (y=3mm) CFX (y=4mm) CFX (y=5mm) Port 6 Exp (x=3mm) Exp (x=90mm) CFX (x=3mm) CFX (x=90mm) 0.5 ransverse Position, x/w [-] 0.5 ransverse Position, x/w [-] Figure 6. Local phase distribution at Port 6 (Run 2) 0.5 ransverse Position, y/g [-] Figure 7. Local phase distribution at Port 6 with and without addition o interacial shear term (Run 2) 4.3. Run 3 For Run 3, the initial simulation with the set o interacial orces included in able I showed signiicantly high wall peaing proile as shown Figure 8. he physics based lit coeicient model o Hibii and Ishii [7] (using a lit coeicient o 9 or this case) caused all bubbles to move towards the wall. his trend was not seen in the measurements at Port 2 and Port 3. his is a relatively high liquid velocity case (j =2m/s). urbulence may have eect on the lit coeicient. he lit coeicient was simply reduced to match the void proile seen in the measurements, as the main objective o the tas was to see the eect o addition o the interacial shear term. A lit coeicient o 4 gave a void proile similar to that seen in the measurements. his case was then run with the IAS implemented. Figure 9 shows, at both port 2 and 3, that the addition o the term has small eect on the void proile and is limited to near wall region. 3213

11 Exp: Port-1 (Inlet) Exp: Port-2 CFX: Port-2 EXP: Port-3 CFX: Port-3 Exp: Port-1 (Inlet) Exp: Port-2 CFX: Port-2 EXP: Port-3 CFX: Port ransverse Position, r/r [-] ransverse Position, r/r [-] (a) (b) Figure 8. Local Phase Distribution at Port 1, 2, and 3 with (a) Hibii-Ishii lit coeicient model, lit Coeicient=9 (b) with Lit Coeicient=4 (Run 3) (a) (b) Figure 9. Local Phase Distribution at (a)port 2 and (b) Port 3, with and without addition o interacial shear term (Run 3) 4.4. Run 4 he last case which will be discussed here is Run 4, which is a inely dispersed bubbly low at relatively high liquid low rate. he void distribution results with the Hibii and Ishii [7] lit orce coeicient model (lit coeicient ~ 7) predicted very high void raction near the wall region (Figure 10(a)), i.e. all bubbles were moved to the near-wall region by the lit orce, which was not seen in the experimental data. hereore the lit coeicient was reduced to 1 to match the measured void raction proile trend. As can be seen rom Figure 10(b) the void raction then matched the trend. Ater addition o the IAS, the volume raction is redistributed away rom wall. However, overall only slight dierences were seen in the void distribution at port 2 and 3 near the wall region as shown in Figure

12 Exp: Port-1 (Inlet) Exp: Port-2 CFX: Port-2 EXP: Port-3 CFX: Port-3 Exp: Port-1 (Inlet) Exp: Port-2 CFX: Port-2 EXP: Port-3 CFX: Port ransverse Position, r/r [-] ransverse Position, r/r [-] (a) (b) Figure 10. Local phase distribution at Port 1,2 and 3 with (a) Hibii-Ishii Lit Coeicient Model, Lit coeicient=7 (b) Lit coeicient=1 (Run 4) (a) (b) Figure 11. Local phase distribution at (a) Port 2 and (b) Port 3 with and without Addition o interacial shear term (Run 4) 5. CONCLUSION A CFD model was prepared or assessing the interacial shear term ( α τi) within the ramewor o ANSYS CFX or rectangular channel and pipe geometries. Benchmar simulations against adiabatic airwater upward bubbly low data were carried out with applicable interacial orces including drag, lit, and bubble dispersion orce. Sato s model [9] was used or modeling bubble-induced turbulence or the liquid phase. he main conclusions rom this eort are 3215

13 1. For the higher liquid low rate cases analyzed here the lit coeicient had to be reduced signiicantly in order to reproduce the measured void proile. he possible eect o turbulence on lit coeicient at high liquid low rate has not been signiicantly investigated in the literature and this eect should be addressed in uture research. 2. he addition o the interacial shear term ( α τi) resulted in small eects limited to the near-wall region, as the continuous phase stress is high in this region. 3. Additional cases with higher void ractions need to be assessed in near uture, as the interacial shear term is expected to have more inluence under those conditions. Notice his report was prepared as an account o wor sponsored by an agency o the United States Government. Neither the United States Government nor any agency thereo, nor any o their employees, nor any o their contractors, subcontractors or their employees, maes any warranty, express or implied, or assumes any legal liability or responsibility or the accuracy, completeness, or any third party s use or the results o such use o any inormation, apparatus, product, or process disclosed, or represents that its use would not inringe privately owned rights. Reerence herein to any speciic commercial product, process, or service by trade name, trademar, manuacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or avoring by the United States Government or any agency thereo or its contractors or subcontractors. he views and opinions o authors expressed herein do not necessarily state or relect those o the United States Government or any agency thereo. REFERENCES 1. M. Ishii and. Hibii,. hermo-luid dynamics o two-phase low.2 nd Ed., Springer, New Yor (2011). 2. M. Ishii., hermo-luid dynamic theory o two-phase low. Eyrolles, Paris (1975). 3. M. Ishii and K. Mishima, wo-luid model and hydrodynamic constitutive relations. Nucl. Eng. Des. 82, pp (1984). 4. S. Kim, Interacial area transport equation and measurement o local interacial characteristics, Ph.D. hesis, Purdue University, West Laayette, IN (1999). 5. M. Ishii and N. Zuber, Drag coeicient and relative velocity in bubbly, droplet or particulate lows. AIChE J. 25, pp.843 (1979). 6. S. P. Antal, R.. Lahey Jr., and J. E. Flaherty, Analysis o phase distribution in ully developed laminar bubbly two-phase low. Int. J. Multiphase Flow 17, pp (1991). 7.. Hibii and M. Ishii, Lit orce in bubbly low systems, Chem. Eng. Sci. 62, pp (2007). 8. M. Lopez de Bertodano, urbulent Bubbly Flow in a riangular Duct, Ph.D. hesis, Rensselaer Polytechnic Institute, roy New Yor (1992). 9. Y. Sato, M. Sadatomi and K. Seoguchi, Momentum and heat transer in two-phase bubble low I, Int. J. Multiphase Flow 7, pp (1981). 10. G. B., Wallis, One-dimensional wo-phase Flow, McGraw-Hill Boo Co.,USA (1969) 11.. Hibii, M. Ishii, Z. Xiao, Axial interacial area transport o vertical bubbly lows. Int. J. Heat Mass ranser 44, (2001). 12. ANSYS CFX Release 15.0, ANSYS CFX-Solver heory Guide, ANSYS Inc.(2013) 13. M. A. Lopez de Bertodano, R.. Lahey Jr, O. C. Jones, Development o a ε model or bubbly two-phase low, ASME J. Fluids Eng. 116, pp (1994). 3216

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