Research Article Studies of Two-Phase Flow at a Chute...

Research ArticleStudies of Two-Phase Flow at a Chute Aeratorwith Experiments and CFD Modelling

Penghua Teng1 James Yang12 and Michael Pfister34

1Hydraulic Engineering Royal Institute of Technology (KTH) 100 44 Stockholm Sweden2Fluid Mechanics Vattenfall RampD 814 70 Alvkarleby Sweden3Civil Engineering Haute Ecole drsquoIngenierie et drsquoArchitecture de Fribourg (HEIA-FR) 1705 Fribourg Switzerland4Laboratory of Hydraulic Constructions Ecole Polytechnique Federale de Lausanne (EPFL) 1015 Lausanne Switzerland

Correspondence should be addressed to Penghua Teng tengpenghuabyvkthse

Received 17 May 2016 Accepted 25 July 2016

Academic Editor ShengKai Yu

Copyright copy 2016 Penghua Teng et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The chute aerator of a spillway is a structure in such a sense that air is in the intense emulsification entrained into the high-velocity water flow Correctly predicting the air entrainment and two-phase flow pattern at the aerator would contribute to reliablespillway operation Based on experimental data 2D numerical simulations are preformed to predict streamwise air concentrationsin the aerated flow in which a two-fluid model is used Depending on the air bubble size relatively good agreement is seen withthe experiments in the air cavity zone The simulations give rise to higher air concentration downstream of the cavity which ispresumably due to underestimation of the interfacial forces in the two-fluid model

1 Introduction

Spillways are important hydraulic structures for dam safetyIf the water flow velocity exceeds for example 20ms andthe cavitation index drops below a certain limit damage mayoccur due to cavitation in the chute bottom which affectsthe safety of the spillway [1] Hence protecting spillwaysfrom cavitation damage is a primary goal of engineeringdesign The use of aerators is probably the only economiccountermeasure for the purpose An aerator entrains air intohigh-speed flow alleviates the negative pressure near thechute bottom and thus avoids the risk of cavitation

Driven by engineering practice researchers have investi-gated aerators both in the laboratory and through prototypeobservations [2ndash8] Kramer andHager [9] examined throughexperiments flow velocity air concentration and air bubblesize distributions they concluded that the bubble rise velocityin chute flows depends on the Froude number Pfister andHager [7 8] analyzed the effects of geometrical parameters onstreamwise distributions of air concentration downstream ofaerators

Formany years physical models have been themajor toolto study the characteristics of the aerated flow Computa-tional Fluid Dynamics (CFD) has emerged as an importantalternative in multiphase flow modelling Both methods areundoubtedly complementary to each other With CFD it ispossible to obtain in detail air-water flowfields of the aeratedflow so as to understand the effects of governing parametersnecessary for a project in question

The Volume of Fluid (VOF) method is an interfacetracking scheme addressing the topological changes of theair-water interface in free-surface flows [10] To describeits hydraulic performance the VOF model is often used tosimulate the aerated flow of a spillway [11ndash14]

In an air-water flow exchange behaviors between thedispersed air phase and continuous water phase affect forcesbetween the phases Hence the correct modelling of forcesand turbulence is of prime importance for capturing thephysics The two-fluid model differs from the VOF modelin such a way that the momentum and continuity equationsare solved for each phase Furthermore the interaction forcebetween phases the drag force the virtual mass force and

Hindawi Publishing CorporationModelling and Simulation in EngineeringVolume 2016 Article ID 4729128 11 pageshttpdxdoiorg10115520164729128

2 Modelling and Simulation in Engineering

12 3

4

56

7

89

10

Figure 1 Chute model configuration [7] (with permission from ASCE) (1) flow meter (2) slide valve (3) jet-box (4) approach flow zone(5) aerator (6) air duct (7)measurement range (8) automatic position system (9) fiber-optical probe and (10) cavity piezometer

the turbulent dispersion force are modelled in the momen-tum equations

The two-fluid model was used to simulate the complexhydrodynamics of air-water flow in industrial applications[15ndash17] Zhang [18] carried out three-dimensional (3D) mod-elling with the model He focused on evaluations of suchparameters as the diameter of air bubble wall function andinterphase exchangemodels Zhang et al [19] performed two-dimensional (2D) simulations using the model in which theturbulence dispersion force was included in the momentumequations They concluded that the inclusion of the turbu-lence dispersion force gave better results of air concentrationin the flowwhich agreedwell with the experimental data [20]

Physical model tests of an aerator were performed atthe Laboratory of Hydraulics Hydrology and Glaciology(VAW) ETH Zurich [7 8] Based on its configurationsCFD modelling is performed using the two-fluid modelThe simulations time dependent and in 2D examine thetransport of air in the flow Included in the study areevaluations of air cavity length air-entrainment rate and airconcentration distributions The effect of bubble diameter adominating factor in the two-fluid model is also consideredon the momentum exchange between air and water Withregard to the experimental results the purpose of the study isto evaluate the suitability of the two-fluid model in solutionof the two-phase flow at the aerator and to learn about theair-water features of the flow

2 Physical Model

The experimental set of data was obtained from a hydraulicmodel test of an aerator conducted in a flume 03m wide and60m long at VAW ETH Zurich (Figure 1) [7] One aeratorconfiguration without offset but with a deflector is selectedfor the numerical modelling (Figure 2) The chute bottomangle is set herein to 120572 = 30∘ with the horizontal plane thedefector angle is 120593 = 813∘ with the chute bottom The chutelength upstream of the aerator is 20mThe height of defector(119904) is 00133m The water depth of the approach flow is ℎ

0

= 0084m The approach flow Froude number is defined as

119865 = 1198810(119892ℎ0)05 = 752 where 119881

0is the mean approach flow

velocity and 119892 is the acceleration of gravityThe air supply to the air cavity below the jet occurs

through a lateral duct on each side of the chute its massflux is measured with a thermoelectric air flow anemometerTo measure the spatial distribution of air concentration(denoted as 119862) a dual-tip fiber-optical probe is adopted witha sampling frequency of 1MHz Its measurement is based ondifferent refraction indices between the sapphire tip and thesurrounding phase If the tips are in the air phase the lighton the tips is reflected and detected Otherwise it is ldquolostrdquo inthe water phase The 119862 value at each point is obtained froma period of typically 20 s The measurement points have astreamwise spacing of 02m the interval perpendicular to thechute bottom is 2 to 5mm

For the approach flow the Weber and Reynolds numberis defined as We = 119881

0(120590119892ℎ

0) and Re = (119881

0ℎ0)120583119908 where 120590

is the surface tension and 120583119908is the kinematic water viscosity

If they exceed the minimum values that is We = 110 and Re= 17 times 105 the effects of surface tension and viscous force arenegligible [21ndash25] In the tests We = 236 and Re = 13 times 106both exceeding the limits

3 Numerical Model

A numerical model was set up to simulate the water-air flowat the aerator The computations are based on the two-fluidmodel in ANASYS 15 [26ndash28]

31 Two-Fluid Model The two-fluid model is based on theEuler-Euler approach Both separate and interacting phasesare allowed to be modelled In the model the continuity andmomentum equations are formulated for each phase

The continuity equation for each phase is

120597

120597119905(120572119894120588119894) + nabla sdot (120572

119894120588119894

997888V119894) = 0 (1)

where subscript 119894 = 119886 and 119908 referring to the air and waterphase 120572

119894is the phase volume fraction 120588

119894is the phase density

997888V119894is the phase velocity and 119905 is the time

Modelling and Simulation in Engineering 3

Approach flowzone

Jet zone

Far-fieldzone

o

R

x

S

120572

zu

zh0

V0

120593

Figure 2 Aerator configuration

The momentum balance equation for each phase is

120597

120597119905(120572119894120588119894

997888V119894) + nabla sdot (120572

119894120588119894

997888V119894

997888V119894)

= minus120572119894nabla119901 + nabla sdot

997888120591119894+ 120572119894120588119894119892 + 119866 +119872

(2)

where 119901 is the water pressure 120591119894is the shear stress 119866 is the

interphase force and 119872 is the interfacial force between airand water phase119866 depends on water-air friction pressure cohesion and

other effects and is written as a simple interaction term of thefollowing form

119866119886119908= 119870119886119908(997888V119886minus997888V119908) (3)

where 119870119886119908

= 119870119908119886

= interphase momentum exchange coeffi-cient between the water and the air phase119872 consists of several independent forces

119872 = 119872119889119908+119872119897119908+119872V119898119908 +119872119905119889119908 (4)

where119872119889119908

is the drag force119872119897119908

is the lift force119872V119898119908 isthe virtual mass force and119872

119905119889119908is the turbulence dispersion

force In the two-fluid model the air phase is assumed toform bubbles The viscous stress creates bubble skin dragthe pressure distribution around the bubble creates formdrag The latter becomes significant when the relative bubbleReynolds number (Re

119903) increases119872

119889119908is written as

119872119889119908=119870119863Re119903

24 (5)

where119870119863is the drag coefficient Re

119903is obtained from

Re119903=120588119908

10038161003816100381610038161003816997888V119886minus997888V119908

10038161003816100381610038161003816119863

120583119908

(6)

where D is the bubble diameter 119870119863is based on the Schiller

and Naumann model [29]

119870119863=

24 (1 + 015Re119903

0687)

Re119903

Re119903le 1000

044 Re119903gt 1000

(7)

If bubbles accelerate relative to the water virtual masseffects occur 119872V119898119908 is due to the inertia of the water massencountered by the accelerating bubbles [30] defined as

119872V119898119908 = 05120572119886120588119908 (119889997888V119908

119889119905minus119889997888V119886

119889119905) (8)

The virtual mass effect is significant when the secondary-phase density is much smaller than the primary-phase den-sity119872119905119889119908

acts as a turbulent diffusion in dispersed flows andis based on de Bertodano [31]

119872119905119889119908

= minus119872119905119889119886= 119870119905119889120588119908119896119908nabla120572119886 (9)

where 119896119908is the water turbulent kinetic energy per unit of

mass and 119870119905119889is the turbulence dispersion coefficient

119872119897119908

acts on air bubbles mainly due to velocity gradientsin thewater flowfield FromDrew andLahey [30] it is writtenas

119872119897119908= minus119870119897120588119908120572119886(997888V119908minus997888V119886) times (nabla times

997888V119908) (10)

where119870119897is the lift coefficient

32 Grid Boundary Conditions and Grid IndependenceThe chute included in the simulations comprised 20 and506m upstream and downstream of the aerator The upperboundary of the domain is parallel to the chute bottom and ata distance of 12m from itTheheight of the transversal grooveis119867 = 003m the duct width is 119861 = 006mThe bottom of theduct is open to atmosphere allowing air to be sucked into thejet cavity

A quadrilateral mesh is generated in the Gambit softwareThe geometrical size and numerical grids are shown inFigure 3 The boundary conditions are defined in Figure 4The upstream boundary consists of both water and air inheight A velocity boundary is set to the water phase belowa pressure inlet (the atmospheric pressure) is given to the airphase above the water Both the top and the duct bottom arespecified as pressure inlet the downstream boundary is set aspressure outlet The other boundaries are treated as wall

Simulations are performed in 2D and time dependentlyusing the software Fluent in ANSYS 15 Zhang [18] comparedthe Standard 119896-120576 and Realizable 119896-120576 models for aerated


12

m

2m

506m

(a) Entire domain

B H

(b) Local refinement at the aerator

Figure 3 Numerical grid

Pressure inlet

Velocity inlet

Pressureoutlet

Pressure inlet

S1S2

S3S4

S5S6

S7

Figure 4 Numerical boundary conditions

flow The results have shown that there are small differencesbetween them However the Realizable k-120576 model is moresuitable to represent features of turbulent free-surface flow[32] Hence it is chosen in the simulations The parameterschosen for use in the two-fluid model are summarized inTable 1

To guarantee the numerical quality a check of gridindependence is necessary Three quadrilateral grids areexamined with the number of cells being about 9900(coarse) 33000 (medium) and 66000 (fine) respectivelyTheminimumcell size is 1mmat the aeratorThevariable 119911

119906refers

to the position corresponding to 119862 = 09 in the upper edgeof the nappe (Figure 2) For the grid independence checkthe 119862 values are compared as a function of 119885 = 119911119911

119906 The

comparison for location 119909 = 0211m is shown in Figure 5The results indicate that the medium size grid is sufficient tomodel the aerator flow

4 Results and Discussions

The flow field is usually divided into three zones thatis approach flow zone air cavity zone and far-field zone(Figure 2) The latter two zones are divided by the reattach-ment point (R) defined as the intersection of 119862 = 09 with

Medium gridFine gridCoarse grid

0

05

10

15

0 05

Z

C

10

Figure 5 Check of grid independence (at location 119909 = 0211m)

the chute bottom The main issues of concern for modellingeither physical or numerical include the air demand cavitylength and air concentration in the cavity zone at the end ofthe cavity and in the far field of the jet

A coordinate system (119909 119911) is defined The 119909-direction ison the chute bottom while the 119911-direction perpendicular tothe chute bottom is on the downstream face of the deflector(Figure 2) Several cross sections denoted as S1ndashS7 and allperpendicular to the chute bottom are used to describethe flow (Table 2) The distance between two neighboringlocations is 02m

41 Selection of Bubble Diameter The air phase exists pre-sumptively in the form of bubbles The air-water exchangebehavior is a dominating issue in modelling aerator flow


Table 1 Summary of parameters in the two-fluid model

Drag force model Turbulence model Virtual mass force model Turbulence dispersionforce model Lift force model

Schiller and Naumann Realizable 119896-120576 Drew and Lahey de Bertodano Drew and Lahey

Table 2 Locations of cross sections

S1 S2 S3 S4 S5 S6 S7119909 (m) 0211 0411 0611 0811 1011 1211 1411

and is influenced by the bubble dimensions In the two-fluidmodel this behavior is described by 119870

119886119908

119870119886119908=120588119886119872119889119908

6120591119886

119863119860 (11)

where 119860 is the interfacial area 120591119886= (1205881198861198632)(18 120583

119886) and 120583

119886is

the air kinematic viscosityChanson and Toombes [33] carried out an experiment in

an open channel to study air-water flowpropertiesThey indi-cated that the probability of air bubble chord sizes between0 and 4mm in the flow region is more than 65 Takahashiet al [34] conducted a study of interactions between free-surface and cavity recirculation in a stepped channel Theresults showed that the majority of bubble diameters is below5mm Chen et al [35] studied bubble diameter distributionsin an aerated flow by means of physical model tests Theyshowed that the bubble sizes are commonly in the range of05ndash3mm in the vicinity of the chute bottom and 3ndash5mmclose to the free surface

In the VAW laboratory tests no systematical measure-ments were made of the air bubble sizes However fragmen-tary tests demonstrated that they were usually below 4mmWith reference to the abovementioned observations119863 = 1 23 and 4mm are selected to examine their effects on the flowcharacteristics

42 Cavity Length and Air Demand When the water flowsover the aerator then the deflector separates the flow fromthe bottom and a cavity region beneath the nappe of the jet isgenerated in which air is sucked inThe cavity length and theair supply are two characteristic parameters of interest Dueto the high velocity and turbulence a well-defined interfacebetween the air and water does not exist for the lower nappesurface The cavity length denoted as L (m) refers to thedistance from the aerator to the reattachment point 119877 onthe chute bottom (Figure 2) To describe the air-entrainmentcapacity of an aerator an air-entrainment coefficient (120573) isdefined as the ratio of air discharge (119876

119886 m3s) to water flow

discharge (119876119908 m3s)

Other conditions being given the bubble diameter 119863 isa primary parameter in the two-fluid model In literaturereview limited information has been found of its effects onthe cavity length and the air demand Table 3 compares theaveraged results of 119871 from the experiments and the numericalsimulations The simulated 119871 values are in relatively good

Table 3 Comparison of 119871rsquos between experiments and simulations

Experiments Simulations119863 = 1mm 119863 = 2mm 119863 = 3mm 119863 = 4mm

119871 (m) 1120 1175 1161 1152 1155

agreement with the experimental one themaximal differenceis 5 for119863 = 1mm

As the black-water core stretches to the end of the cavitythe air supply to the aerator is equal to the air entrained intothe flow from the lower nappe surface Table 4 shows the120573 results corresponding to the different 119863 values With theincrease of 119863 the 120573 value becomes smaller 119863 = 1mm givesthe largest difference119863 = 4mm is closer to the experimentalresult with an error of 9 All the simulations overestimatethe air demand 119863 is a parameter that does affect the airentrainment

43 Cavity-Zone Air Concentration Air is entrained into thewater by strong emulsification in the cavity zone particularlyat its end near the reattachment zone If the aerator geometryis given the state of air entrainment governs the air capacityof the aerator which forms an initial condition for thestreamwise transport of the air in the water Figure 6 showsthe 119862 distributions as a function of 119885 for the four locationsS1 S2 S3 and S4 (Figure 4) Each diagram compares thedistributions between the experiments and the simulationsThe119863 values vary from 1 to 4mm

For the upper edge of the nappe the four 119863 values leadto almost identical results at S1 and differ however from theexperimental result the range of the so-called black-waterzone is underestimated As the streamwise distance increasesfrom the aerator the simulated 119862 values corresponding tothe larger diameters decrease and approach gradually theexperimental ones At S4 the result of 119863 = 4mm is in goodagreement with the experimental data

For the lower edge of the nappe the numerical and testresults are more close to each other along the cavity Somesmall differences are seen for119863= 1mm in the beginning of thecavity (eg at S1) and 119863 = 4mm towards downstream (egat S3 and S4)

44 Lower EdgeConcentration Similarity To further examinethe air concentration of the lower nappe edge a characteristicthickness called the air-entrainment thickness 120575 is usuallyused to describe its development [20 36] For a given crosssection it refers to the difference between 119862 = 30 and 60that is120575 = 119911

30minus11991160 A normalized thickness 120578 is then defined

as 120578 = (119911 minus 11991160)120575 For the four locations Figure 7 compares

the changes of the 119862 values as a function of 120578




119876119886

(m3s) 00393 00629 00550 00460 00430120573 0228 0365 0319 0267 0249

0

05

10

15

0 05

Z

C

10

ExperimentsD = 1 mmD = 2 mm

D = 3 mmD = 4 mm

S1 (x = 0211 m)

(a) Cross section S1

0

05

10

15

0 05

Z

C

10


D = 3 mmD = 4 mm

S2 (x = 0411 m)

(b) Cross section S2

0

05

10

15

0 05

Z

C

10


D = 3 mmD = 4 mm

S3 (x = 0611 m)

(c) Cross section S3

0

05

10

15

0 05

Z

C

10


D = 3 mmD = 4 mm

S4 (x = 0811 m)

(d) Cross section S4

Figure 6 119862 distributions


minus4

minus2

0

2

4

Experiments

0 05C

10

D = 1 mmD = 2 mm

D = 3 mmD = 4 mm

120578

S1 (x = 0211 m)


0 05C

10


D = 3 mmD = 4 mm

minus4

minus2

0

2

4

120578

S2 (x = 0411m)



D = 3 mmD = 4 mm

minus4

minus2

2

0

4

0 05C

10

120578

S3 (x = 0611m)



D = 3 mmD = 4 mm

0 05C

10minus4

minus2

2

0

4

120578

S4 (x = 0811 m)


Figure 7 Similarity of 119862 distributions

Figure 8 is a plot of all the experimental 119862 values forthe lower edge The 119862 distributions are similar for differentlocations and can be assembled into the expression suggestedby Hager [37]

119862 (120578) = 1198981exp (minus119898

2(120578 + 119898

3)2

) (12)

where11989811198982 and119898

3are constants119898

1= 104119898

2= 016 and

1198983= 187

45 Cavity End The air content at end of the cavity evolvesfrom the air entrainment along the lower edge of the jetat the same time it reflects also the air contribution fromthe backflow upstream of the reattachment point 119877 and itsinteraction with the jetrsquos lower edge

Section S5 is located at the end of the cavity Figure 9compares along the whole cross section the 119862 distributionsbetween the experiments and the modelling Figure 10 shows


S1S2S3

S4Equation (12)

minus4

minus2

2

0

4

120578

0 05C

10

Figure 8 Similarity of 119862 distributions for experiments

0

05

10

15

0 05

Z

C

10

S5 (x = 1011 m)


D = 3 mmD = 4 mm

Figure 9 119862 distribution at S5

the similarity profile of119862 that is the change of119862 as a functionof 120578

From the experiments one can see that the black-water zone becomes small at S5 and is about to disappeardownstream of itThe thickness of the black water is relativelywell produced its predicted position is however lower Again

S5 (x = 1011 m)

minus4

minus2

2

0

4

120578

0 05C

10


D = 3 mmD = 4 mm

Figure 10 Similarity of 119862 distribution at S5

the high air concentration region close to the chute bottomedge is overestimated in the simulations

46 Far-Field Evolution It is the aerated air bubbles closeto the bottom that protect the chute bottom from cavitationdamage It is therefore of practical significance to examinethe streamwise development downstreamof the reattachmentpoint 119877 It provides information with respect to the effectivelength of the chute provided by the aerator in question

Figure 11 shows the119862 profiles at two typical cross sections(S6 and S7) in the far filed that is the change of119862with119885The119862 distributions behave in a similar manner for the examined119863 range For both the upper edge and lower boundary of theflow 119862 is overestimated all the simulations lead to a muchlarger 119862 value close to the chute bottomThe average 119862 valueat each location is well reproduced

47 Discussions As seen from (11) 119863 is a parameter thataffects the diffusion between the air and water the momen-tum exchange becomes more intense with augment in 119863In the simulations 119863 is a constant value in the entirecomputational domain In actual situations it may vary inthe different flow regions In the approach flow with theincrease of the surface turbulence air starts to be entrainedfrom the free surface at a location upstream of the aeratorCompared with the flow in the cavity zone the turbulenceintensity in the flow is low [6] As a result the C distributionof the approach flow might still be overestimated even withthe smallest diameter simulated (119863 = 1mm)

Previous model tests showed that the roughness of thesurface water increases along the cavity zone [9]This impliesthat the exchange of momentum between the water and airbecomes stronger a larger bubble diameter should be used to


0

05

10

15

0 05

Z

C

10


D = 3 mmD = 4 mm

S6 (x = 1211 m)


0

05

10

15

0 05

Z

C

10


D = 3 mmD = 4 mm

S7 (x = 1411m)


Figure 11 119862 distribution at S6 and S7

describe the process This is the reason why good agreementis obtained for the upper edge with119863 = 4mm For the lowertrajectory the simulations show only small deviations fromthe experiments The turbulence intensity is high but notas high as that for the upper edge The result is somewhatinsensitive to 119863 but smaller 119863 values than 4mm give betterresults for the majority part of the cavity

In the far-field zone the 119862 values near the chute bottomare overestimated in the simulations Kramer and Hager [9]studied the air transport process in chute flowsThey pointedout that air detrained after the reattachment point 119877 andthe interfacial force between bubbles and water dominatedthe process of detrainment in the zone This implies that theeffects of the interfacial forces are underestimated in the two-fluid model

5 Conclusions

To model the air-water flow at an aerator is a challengingissue especially if the water flow velocity is high which is thecase inmost spillway installations Based on the experimentaldata of ETH Zurich numerical simulations are performedto reproduce the characteristics of the aerated flowThe CFDmodel is the two-fluid model in ANSYS 15 the parametersof interest include air concentration cavity length and air-entrainment rateThe bubble diameter affects themomentumexchange between thewater and air and a range between 1 and4mm is investigated

In terms of the cavity length the experiments and CFDgive similar results irrespective of the bubble size The

bubble diameter affects the amount of air entrained into theflow the air flow rate from the 4mm simulations is closeto the measured one Along the cavity zone the surface-water roughness increases a somewhat larger bubble size issuitable to describe the exchange between the two phasesFor the lower jet trajectory the simulated air concentrationis relatively nonsensitive to the bubble size However sizessmaller than 4mm lead to results closer to the experimentsThe air concentration in the far field is overestimated theexperiments showed lower air contents The contributingreason is presumably the underestimation of the interfacialforces in the two-fluid model Besides the use of a singlebubble diameter is not sufficient to represent the complexwater-air exchange in the aerator flow but seems to beadequate as a first step

Notations

119860 Interfacial area (m2)119886119894 Constants (minus)119861 Groove width (m)119862 Local air concentration (minus)119863 Bubble diameter (mm)119865 Froude number (minus)119866 Interphase force (N)119892 Acceleration of gravity (ms2)119867 Groove depth (m)ℎ0 Depth of approach flow (m)

119870119886119908 Interphase momentum exchangecoefficient (minus)

119870119863 Drag coefficient (minus)


119870119905119889 Turbulence dispersion coefficient

(minus)119870119897 Lift coefficient (minus)

119896119908 Water turbulent kinetic energy (minus)

119871 Cavity length (m)119872 Interfacial force (N)119872119889119908

Drag force (N)119872119897119908 Lift force (N)

119872119905119889119908

Turbulence dispersion force (N)119872V119898119908 Virtual mass force (N)11989811198982 and119898

3 Constants (minus)

119901 Water pressure (Nm2)119876119886 Air discharge (m3s)

119876119908 Water discharge (m3s)

Re Reynolds number (minus)Re119903 Relative Reynolds number (minus)

119904 Deflector height (m)119905 Time (s)1198810 Approach flow velocity (ms)

997888V119894 Velocity of phase 119894 (ms)

997888V119886 Velocity of air bubble (ms)

997888V119908 Velocity of water (ms)

We Weber number (minus)119909 119909-coordinate (m)119885 Normalized 119911-coordinate (minus)119911 119911-coordinate (m)11991130 119911 (m) at 119862 = 30

11991160 119911 (m) at 119862 = 60

120572 Chute bottom angle (∘)120573 Air-entrainment coefficient (minus)120593 Deflector angle (∘)120590 Surface tension (Nm)120572119894 Volume fraction of phase 119894 (minus)

120572119886 Air volume fraction (minus)

120588119894 Density of phase 119894 (kgm3)120588119908 Water density (kgm3)

120588119886 Air density (kgm3)

120583119908 Kinematic water viscosity (m2s)

120583119886 Kinematic air viscosity (m2s)

120591119894 Shear stress of phase 119894 (Pa)120575 Air-entrainment thickness (m)120578 Normalized air-entrainment

thickness (minus)

Competing Interests

No potential conflict of interests was reported by the authors

Acknowledgments

The numerical modelling presented here was carried outas part of a PhD Project ldquoHydraulic Design of ChuteSpillway Aeratorsrdquo funded by Swedish Hydropower Centre(SVC) SVC has been established by the Swedish EnergyAgency Energiforsk and Swedish National Grid togetherwith Lulea University of Technology (LTU) Royal Instituteof Technology (KTH) Chalmers University of Technology

(CTH) and Uppsala University (UU) (httpwwwsvcnu)The authors are indebted to Ms Sara Sandberg of SVC forproject coordinations

References

[1] H Falvey Cavitation in Chutes and Spillways EngineeringMonograph 42 United States Department of the InteriorBureau of Reclamation Denver Colo USA 1990

[2] P Volkart and P Rutschmann ldquoRapid flow in spillway chuteswith and without deflectors a model-prototype comparisonrdquoin Proceedings of the Symposium on Scale Effects in ModellingHydraulic Research Esslingen am Neckar Germany 1984

[3] H Chanson ldquoFlow downstream of an aeratormdashaerator spac-ingrdquo Journal of Hydraulic Research vol 27 no 4 pp 519ndash5361989

[4] P Rutschmann and W H Hager ldquoAir entrainment by spillwayaeratorsrdquo Journal of Hydraulic Engineering vol 116 no 6 pp765ndash782 1990

[5] M A Kokpinar Air-entrainment in high speed free surface flows[PhD dissertation] Department of Civil Engineering MiddleEast Technical University Ankara Turkey 1996

[6] K Kramer Development of aerated chute flow [PhD disserta-tion] VAW ETH Zurich Zurich Switzerland 2004

[7] M Pfister and W H Hager ldquoChute aerators I air transportcharacteristicsrdquo Journal of Hydraulic Engineering vol 136 no6 pp 352ndash359 2010

[8] M Pfister and W H Hager ldquoChute Aerators II hydraulicdesignrdquo Journal of Hydraulic Engineering vol 136 no 6 pp360ndash367 2010

[9] K Kramer and W H Hager ldquoAir transport in chute flowsrdquoInternational Journal of Multiphase Flow vol 31 no 10-11 pp1181ndash1197 2005

[10] C W Hirt and B D Nichols ldquoVolume of fluid (VOF) methodfor the dynamics of free boundariesrdquo Journal of ComputationalPhysics vol 39 no 1 pp 201ndash225 1981

[11] D K H Ho K M Boyes and S M Donohoo ldquoInvestiga-tion of spillway behavior under increased maximum flood bycomputational fluid dynamics techniquerdquo in Proceedings of theConference on the 14th Australasian Fluid Mechanics pp 577ndash580 Adelaide University Adelaide Australia December 2001

[12] M C Aydin and M Ozturk ldquoVerification and validation of acomputational fluid dynamics (CFD)model for air entrainmentat spillway aeratorsrdquo Canadian Journal of Civil Engineering vol36 no 5 pp 826ndash836 2009

[13] J-M Zhang J-G Chen W-L Xu Y-R Wang and G-JLi ldquoThree-dimensional numerical simulation of aerated flowsdownstream sudden fall aerator expansion-in a tunnelrdquo Journalof Hydrodynamics vol 23 no 1 pp 71ndash80 2011

[14] V Jothiprakash V V Bhosekar and P B Deolalikar ldquoFlowcharacteristics of orifice spillway aerator numerical modelstudiesrdquo ISH Journal of Hydraulic Engineering vol 21 no 2 pp216ndash230 2015

[15] R F Mudde and O Simonin ldquoTwo- and three-dimensionalsimulations of a bubble plume using a two-fluid modelrdquoChemical Engineering Science vol 54 no 21 pp 5061ndash50691999

[16] S MMonahan V S Vitankar and R O Fox ldquoCFD predictionsfor flow-regime transitions in bubble columnsrdquo AIChE Journalvol 51 no 7 pp 1897ndash1923 2005


[17] D Zhang N G Deen and J A M Kuipers ldquoNumericalsimulation of the dynamic flow behavior in a bubble column astudy of closures for turbulence and interface forcesrdquo ChemicalEngineering Science vol 61 no 23 pp 7593ndash7608 2006

[18] X D Zhang Three dimensional numerical simulation of highvelocity flow in spillway tunnels [PhD thesis] China Instituteof Water Resources and Hydropower Beijing China 2004

[19] H-W Zhang Z-P Liu D Zhang and Y-H Wu ldquoNumericalsimulation of aerated high velocity flow downstream of anaeratorrdquo Journal of Hydraulic Engineering vol 39 no 12 pp1302ndash1308 2008

[20] Q S Shi High-Velocity Aerated Flow China Water and PowerPress Beijing China 2007

[21] W H C Maxwell and J R Weggel ldquoSurface tension in Froudemodels-correctionrdquo Journal of the Hydraulics Division ASCEvol 96 p 845 1970

[22] P Novak and J Cabelka Models in Hydraulic EngineeringPitman Boston Mass USA 1981

[23] J Peakall and J Warburton ldquoSurface tension in small hydraulicrivermodelsmdashthe significance of theWeber numberrdquo Journal ofHydrology New Zealand vol 35 no 2 pp 199ndash212 1996

[24] P Novak A I B Moffat C Nalluri and R NarayananHydraulic Structures E amp FN Spon London UK 1997

[25] V Heller ldquoScale effects in physical hydraulic engineering mod-elsrdquo Journal of Hydraulic Research vol 49 no 3 pp 293ndash3062011

[26] J B Joshi ldquoComputational flowmodelling and design of bubblecolumn reactorsrdquo Chemical Engineering Science vol 56 no 21-22 pp 5893ndash5933 2001

[27] A Sokolichin G Eigenberger and A Lapin ldquoSimulation ofbuoyancy driven bubbly flow established simplifications andopen questionsrdquo AIChE Journal vol 50 no 1 pp 24ndash45 2004

[28] K Ekambara M T Dhotre and J B Joshi ldquoCFD simulationsof bubble column reactors 1D 2D and 3D approachrdquo ChemicalEngineering Science vol 60 no 23 pp 6733ndash6746 2005

[29] L Schiller and Z Z Naumann ldquoA drag coefficient correlationrdquoVdi Zeitung vol 77 pp 318ndash320 1935

[30] D A Drew and R T Lahey Particulate Two-Phase FlowButterworth-Heinemann Boston Mass USA 1993

[31] MA L deBertodanoTurbulent bubbly flow in a triangular duct[PhD dissertation] Rensselaer Polytechnic Institute Troy NYUSA 1991

[32] Fluent Inc Fluent Version 62 Userrsquos Guide Fluent Lebanon PaUSA 2006

[33] H Chanson and L Toombes ldquoStrong interactions betweenfree-surface aeration and turbulence in an open channel flowrdquoExperimental Thermal and Fluid Science vol 27 no 5 pp 525ndash535 2003

[34] M Takahashi C A Gonzalez and H Chanson ldquoSelf-aerationand turbulence in a stepped channel influence of cavity surfaceroughnessrdquo International Journal ofMultiphase Flow vol 32 no12 pp 1370ndash1385 2006

[35] X P Chen R Z Xi D C Shao and B Liang ldquoNew concept ofair entrainment effect onmitigating cavitation damagerdquo Journalof Hydraulic Engineering vol 34 no 8 pp 70ndash74 2003

[36] M S Yuan ldquoCalculation of 2-D air concentration distributiondownstreamof an aeratorrdquo Journal ofHydraulic Engineering vol22 no 12 pp 9ndash16 1991

[37] W H Hager ldquoUniform aerated chute flowrdquo Journal of HydraulicEngineering vol 117 no 4 pp 528ndash533 1991

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



12 3

4

56

7

89

10

Figure 1 Chute model configuration [7] (with permission from ASCE) (1) flow meter (2) slide valve (3) jet-box (4) approach flow zone(5) aerator (6) air duct (7)measurement range (8) automatic position system (9) fiber-optical probe and (10) cavity piezometer

the turbulent dispersion force are modelled in the momen-tum equations

The two-fluid model was used to simulate the complexhydrodynamics of air-water flow in industrial applications[15ndash17] Zhang [18] carried out three-dimensional (3D) mod-elling with the model He focused on evaluations of suchparameters as the diameter of air bubble wall function andinterphase exchangemodels Zhang et al [19] performed two-dimensional (2D) simulations using the model in which theturbulence dispersion force was included in the momentumequations They concluded that the inclusion of the turbu-lence dispersion force gave better results of air concentrationin the flowwhich agreedwell with the experimental data [20]

Physical model tests of an aerator were performed atthe Laboratory of Hydraulics Hydrology and Glaciology(VAW) ETH Zurich [7 8] Based on its configurationsCFD modelling is performed using the two-fluid modelThe simulations time dependent and in 2D examine thetransport of air in the flow Included in the study areevaluations of air cavity length air-entrainment rate and airconcentration distributions The effect of bubble diameter adominating factor in the two-fluid model is also consideredon the momentum exchange between air and water Withregard to the experimental results the purpose of the study isto evaluate the suitability of the two-fluid model in solutionof the two-phase flow at the aerator and to learn about theair-water features of the flow

2 Physical Model

The experimental set of data was obtained from a hydraulicmodel test of an aerator conducted in a flume 03m wide and60m long at VAW ETH Zurich (Figure 1) [7] One aeratorconfiguration without offset but with a deflector is selectedfor the numerical modelling (Figure 2) The chute bottomangle is set herein to 120572 = 30∘ with the horizontal plane thedefector angle is 120593 = 813∘ with the chute bottom The chutelength upstream of the aerator is 20mThe height of defector(119904) is 00133m The water depth of the approach flow is ℎ

0

= 0084m The approach flow Froude number is defined as

119865 = 1198810(119892ℎ0)05 = 752 where 119881

0is the mean approach flow

velocity and 119892 is the acceleration of gravityThe air supply to the air cavity below the jet occurs

through a lateral duct on each side of the chute its massflux is measured with a thermoelectric air flow anemometerTo measure the spatial distribution of air concentration(denoted as 119862) a dual-tip fiber-optical probe is adopted witha sampling frequency of 1MHz Its measurement is based ondifferent refraction indices between the sapphire tip and thesurrounding phase If the tips are in the air phase the lighton the tips is reflected and detected Otherwise it is ldquolostrdquo inthe water phase The 119862 value at each point is obtained froma period of typically 20 s The measurement points have astreamwise spacing of 02m the interval perpendicular to thechute bottom is 2 to 5mm

For the approach flow the Weber and Reynolds numberis defined as We = 119881

0(120590119892ℎ

0) and Re = (119881

0ℎ0)120583119908 where 120590

is the surface tension and 120583119908is the kinematic water viscosity

If they exceed the minimum values that is We = 110 and Re= 17 times 105 the effects of surface tension and viscous force arenegligible [21ndash25] In the tests We = 236 and Re = 13 times 106both exceeding the limits

3 Numerical Model

A numerical model was set up to simulate the water-air flowat the aerator The computations are based on the two-fluidmodel in ANASYS 15 [26ndash28]

31 Two-Fluid Model The two-fluid model is based on theEuler-Euler approach Both separate and interacting phasesare allowed to be modelled In the model the continuity andmomentum equations are formulated for each phase

The continuity equation for each phase is

120597

120597119905(120572119894120588119894) + nabla sdot (120572

119894120588119894

997888V119894) = 0 (1)

where subscript 119894 = 119886 and 119908 referring to the air and waterphase 120572

119894is the phase volume fraction 120588

119894is the phase density

997888V119894is the phase velocity and 119905 is the time


Approach flowzone

Jet zone

Far-fieldzone

o

R

x

S

120572

zu

zh0

V0

120593



120597

120597119905(120572119894120588119894

997888V119894) + nabla sdot (120572

119894120588119894

997888V119894

997888V119894)


997888120591119894+ 120572119894120588119894119892 + 119866 +119872

(2)




119866119886119908= 119870119886119908(997888V119886minus997888V119908) (3)

where 119870119886119908

= 119870119908119886


119872 = 119872119889119908+119872119897119908+119872V119898119908 +119872119905119889119908 (4)

where119872119889119908





119903) increases119872


119872119889119908=119870119863Re119903

24 (5)



Re119903=120588119908

10038161003816100381610038161003816997888V119886minus997888V119908

10038161003816100381610038161003816119863

120583119908

(6)



119870119863=

24 (1 + 015Re119903

0687)

Re119903

Re119903le 1000

044 Re119903gt 1000

(7)


119872V119898119908 = 05120572119886120588119908 (119889997888V119908

119889119905minus119889997888V119886

119889119905) (8)



119872119905119889119908

= minus119872119905119889119886= 119870119905119889120588119908119896119908nabla120572119886 (9)



119872119897119908



997888V119908) (10)






12

m

2m

506m

(a) Entire domain

B H



Pressure inlet

Velocity inlet

Pressureoutlet

Pressure inlet

S1S2

S3S4

S5S6

S7




119906refers


119906 The





0

05

10

15

0 05

Z

C

10










S1 S2 S3 S4 S5 S6 S7119909 (m) 0211 0411 0611 0811 1011 1211 1411


119886119908

119870119886119908=120588119886119872119889119908

6120591119886

119863119860 (11)


119886) and 120583

119886is






discharge (119876119908 m3s)




119871 (m) 1120 1175 1161 1152 1155













119876119886

(m3s) 00393 00629 00550 00460 00430120573 0228 0365 0319 0267 0249

0

05

10

15

0 05

Z

C

10


D = 3 mmD = 4 mm

S1 (x = 0211 m)


0

05

10

15

0 05

Z

C

10


D = 3 mmD = 4 mm

S2 (x = 0411 m)


0

05

10

15

0 05

Z

C

10


D = 3 mmD = 4 mm

S3 (x = 0611 m)


0

05

10

15

0 05

Z

C

10


D = 3 mmD = 4 mm

S4 (x = 0811 m)




minus4

minus2

0

2

4

Experiments

0 05C

10

D = 1 mmD = 2 mm

D = 3 mmD = 4 mm

120578

S1 (x = 0211 m)


0 05C

10


D = 3 mmD = 4 mm

minus4

minus2

0

2

4

120578

S2 (x = 0411m)



D = 3 mmD = 4 mm

minus4

minus2

2

0

4

0 05C

10

120578

S3 (x = 0611m)



D = 3 mmD = 4 mm

0 05C

10minus4

minus2

2

0

4

120578

S4 (x = 0811 m)




119862 (120578) = 1198981exp (minus119898

2(120578 + 119898

3)2

) (12)

where11989811198982 and119898


1= 104119898

2= 016 and

1198983= 187




S1S2S3

S4Equation (12)

minus4

minus2

2

0

4

120578

0 05C

10


0

05

10

15

0 05

Z

C

10

S5 (x = 1011 m)


D = 3 mmD = 4 mm




S5 (x = 1011 m)

minus4

minus2

2

0

4

120578

0 05C

10


D = 3 mmD = 4 mm








0

05

10

15

0 05

Z

C

10


D = 3 mmD = 4 mm

S6 (x = 1211 m)


0

05

10

15

0 05

Z

C

10


D = 3 mmD = 4 mm

S7 (x = 1411m)





5 Conclusions




Notations










119872119905119889119908


3 Constants (minus)









11991160 119911 (m) at 119862 = 60








thickness (minus)

Competing Interests


Acknowledgments



References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Approach flowzone

Jet zone

Far-fieldzone

o

R

x

S

120572

zu

zh0

V0

120593



120597

120597119905(120572119894120588119894

997888V119894) + nabla sdot (120572

119894120588119894

997888V119894

997888V119894)


997888120591119894+ 120572119894120588119894119892 + 119866 +119872

(2)




119866119886119908= 119870119886119908(997888V119886minus997888V119908) (3)

where 119870119886119908

= 119870119908119886


119872 = 119872119889119908+119872119897119908+119872V119898119908 +119872119905119889119908 (4)

where119872119889119908





119903) increases119872


119872119889119908=119870119863Re119903

24 (5)



Re119903=120588119908

10038161003816100381610038161003816997888V119886minus997888V119908

10038161003816100381610038161003816119863

120583119908

(6)



119870119863=

24 (1 + 015Re119903

0687)

Re119903

Re119903le 1000

044 Re119903gt 1000

(7)


119872V119898119908 = 05120572119886120588119908 (119889997888V119908

119889119905minus119889997888V119886

119889119905) (8)



119872119905119889119908

= minus119872119905119889119886= 119870119905119889120588119908119896119908nabla120572119886 (9)



119872119897119908



997888V119908) (10)






12

m

2m

506m

(a) Entire domain

B H



Pressure inlet

Velocity inlet

Pressureoutlet

Pressure inlet

S1S2

S3S4

S5S6

S7




119906refers


119906 The





0

05

10

15

0 05

Z

C

10










S1 S2 S3 S4 S5 S6 S7119909 (m) 0211 0411 0611 0811 1011 1211 1411


119886119908

119870119886119908=120588119886119872119889119908

6120591119886

119863119860 (11)


119886) and 120583

119886is






discharge (119876119908 m3s)




119871 (m) 1120 1175 1161 1152 1155













119876119886

(m3s) 00393 00629 00550 00460 00430120573 0228 0365 0319 0267 0249

0

05

10

15

0 05

Z

C

10


D = 3 mmD = 4 mm

S1 (x = 0211 m)


0

05

10

15

0 05

Z

C

10


D = 3 mmD = 4 mm

S2 (x = 0411 m)


0

05

10

15

0 05

Z

C

10


D = 3 mmD = 4 mm

S3 (x = 0611 m)


0

05

10

15

0 05

Z

C

10


D = 3 mmD = 4 mm

S4 (x = 0811 m)




minus4

minus2

0

2

4

Experiments

0 05C

10

D = 1 mmD = 2 mm

D = 3 mmD = 4 mm

120578

S1 (x = 0211 m)


0 05C

10


D = 3 mmD = 4 mm

minus4

minus2

0

2

4

120578

S2 (x = 0411m)



D = 3 mmD = 4 mm

minus4

minus2

2

0

4

0 05C

10

120578

S3 (x = 0611m)



D = 3 mmD = 4 mm

0 05C

10minus4

minus2

2

0

4

120578

S4 (x = 0811 m)




119862 (120578) = 1198981exp (minus119898

2(120578 + 119898

3)2

) (12)

where11989811198982 and119898


1= 104119898

2= 016 and

1198983= 187




S1S2S3

S4Equation (12)

minus4

minus2

2

0

4

120578

0 05C

10


0

05

10

15

0 05

Z

C

10

S5 (x = 1011 m)


D = 3 mmD = 4 mm




S5 (x = 1011 m)

minus4

minus2

2

0

4

120578

0 05C

10


D = 3 mmD = 4 mm








0

05

10

15

0 05

Z

C

10


D = 3 mmD = 4 mm

S6 (x = 1211 m)


0

05

10

15

0 05

Z

C

10


D = 3 mmD = 4 mm

S7 (x = 1411m)





5 Conclusions




Notations










119872119905119889119908


3 Constants (minus)









11991160 119911 (m) at 119862 = 60








thickness (minus)

Competing Interests


Acknowledgments



References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










12

m

2m

506m

(a) Entire domain

B H



Pressure inlet

Velocity inlet

Pressureoutlet

Pressure inlet

S1S2

S3S4

S5S6

S7




119906refers


119906 The





0

05

10

15

0 05

Z

C

10










S1 S2 S3 S4 S5 S6 S7119909 (m) 0211 0411 0611 0811 1011 1211 1411


119886119908

119870119886119908=120588119886119872119889119908

6120591119886

119863119860 (11)


119886) and 120583

119886is






discharge (119876119908 m3s)




119871 (m) 1120 1175 1161 1152 1155













119876119886

(m3s) 00393 00629 00550 00460 00430120573 0228 0365 0319 0267 0249

0

05

10

15

0 05

Z

C

10


D = 3 mmD = 4 mm

S1 (x = 0211 m)


0

05

10

15

0 05

Z

C

10


D = 3 mmD = 4 mm

S2 (x = 0411 m)


0

05

10

15

0 05

Z

C

10


D = 3 mmD = 4 mm

S3 (x = 0611 m)


0

05

10

15

0 05

Z

C

10


D = 3 mmD = 4 mm

S4 (x = 0811 m)




minus4

minus2

0

2

4

Experiments

0 05C

10

D = 1 mmD = 2 mm

D = 3 mmD = 4 mm

120578

S1 (x = 0211 m)


0 05C

10


D = 3 mmD = 4 mm

minus4

minus2

0

2

4

120578

S2 (x = 0411m)



D = 3 mmD = 4 mm

minus4

minus2

2

0

4

0 05C

10

120578

S3 (x = 0611m)



D = 3 mmD = 4 mm

0 05C

10minus4

minus2

2

0

4

120578

S4 (x = 0811 m)




119862 (120578) = 1198981exp (minus119898

2(120578 + 119898

3)2

) (12)

where11989811198982 and119898


1= 104119898

2= 016 and

1198983= 187




S1S2S3

S4Equation (12)

minus4

minus2

2

0

4

120578

0 05C

10


0

05

10

15

0 05

Z

C

10

S5 (x = 1011 m)


D = 3 mmD = 4 mm




S5 (x = 1011 m)

minus4

minus2

2

0

4

120578

0 05C

10


D = 3 mmD = 4 mm








0

05

10

15

0 05

Z

C

10


D = 3 mmD = 4 mm

S6 (x = 1211 m)


0

05

10

15

0 05

Z

C

10


D = 3 mmD = 4 mm

S7 (x = 1411m)





5 Conclusions




Notations










119872119905119889119908


3 Constants (minus)









11991160 119911 (m) at 119862 = 60








thickness (minus)

Competing Interests


Acknowledgments



References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














S1 S2 S3 S4 S5 S6 S7119909 (m) 0211 0411 0611 0811 1011 1211 1411


119886119908

119870119886119908=120588119886119872119889119908

6120591119886

119863119860 (11)


119886) and 120583

119886is






discharge (119876119908 m3s)




119871 (m) 1120 1175 1161 1152 1155













119876119886

(m3s) 00393 00629 00550 00460 00430120573 0228 0365 0319 0267 0249

0

05

10

15

0 05

Z

C

10


D = 3 mmD = 4 mm

S1 (x = 0211 m)


0

05

10

15

0 05

Z

C

10


D = 3 mmD = 4 mm

S2 (x = 0411 m)


0

05

10

15

0 05

Z

C

10


D = 3 mmD = 4 mm

S3 (x = 0611 m)


0

05

10

15

0 05

Z

C

10


D = 3 mmD = 4 mm

S4 (x = 0811 m)




minus4

minus2

0

2

4

Experiments

0 05C

10

D = 1 mmD = 2 mm

D = 3 mmD = 4 mm

120578

S1 (x = 0211 m)


0 05C

10


D = 3 mmD = 4 mm

minus4

minus2

0

2

4

120578

S2 (x = 0411m)



D = 3 mmD = 4 mm

minus4

minus2

2

0

4

0 05C

10

120578

S3 (x = 0611m)



D = 3 mmD = 4 mm

0 05C

10minus4

minus2

2

0

4

120578

S4 (x = 0811 m)




119862 (120578) = 1198981exp (minus119898

2(120578 + 119898

3)2

) (12)

where11989811198982 and119898


1= 104119898

2= 016 and

1198983= 187




S1S2S3

S4Equation (12)

minus4

minus2

2

0

4

120578

0 05C

10


0

05

10

15

0 05

Z

C

10

S5 (x = 1011 m)


D = 3 mmD = 4 mm




S5 (x = 1011 m)

minus4

minus2

2

0

4

120578

0 05C

10


D = 3 mmD = 4 mm








0

05

10

15

0 05

Z

C

10


D = 3 mmD = 4 mm

S6 (x = 1211 m)


0

05

10

15

0 05

Z

C

10


D = 3 mmD = 4 mm

S7 (x = 1411m)





5 Conclusions




Notations










119872119905119889119908


3 Constants (minus)









11991160 119911 (m) at 119862 = 60








thickness (minus)

Competing Interests


Acknowledgments



References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119876119886

(m3s) 00393 00629 00550 00460 00430120573 0228 0365 0319 0267 0249

0

05

10

15

0 05

Z

C

10


D = 3 mmD = 4 mm

S1 (x = 0211 m)


0

05

10

15

0 05

Z

C

10


D = 3 mmD = 4 mm

S2 (x = 0411 m)


0

05

10

15

0 05

Z

C

10


D = 3 mmD = 4 mm

S3 (x = 0611 m)


0

05

10

15

0 05

Z

C

10


D = 3 mmD = 4 mm

S4 (x = 0811 m)




minus4

minus2

0

2

4

Experiments

0 05C

10

D = 1 mmD = 2 mm

D = 3 mmD = 4 mm

120578

S1 (x = 0211 m)


0 05C

10


D = 3 mmD = 4 mm

minus4

minus2

0

2

4

120578

S2 (x = 0411m)



D = 3 mmD = 4 mm

minus4

minus2

2

0

4

0 05C

10

120578

S3 (x = 0611m)



D = 3 mmD = 4 mm

0 05C

10minus4

minus2

2

0

4

120578

S4 (x = 0811 m)




119862 (120578) = 1198981exp (minus119898

2(120578 + 119898

3)2

) (12)

where11989811198982 and119898


1= 104119898

2= 016 and

1198983= 187




S1S2S3

S4Equation (12)

minus4

minus2

2

0

4

120578

0 05C

10


0

05

10

15

0 05

Z

C

10

S5 (x = 1011 m)


D = 3 mmD = 4 mm




S5 (x = 1011 m)

minus4

minus2

2

0

4

120578

0 05C

10


D = 3 mmD = 4 mm








0

05

10

15

0 05

Z

C

10


D = 3 mmD = 4 mm

S6 (x = 1211 m)


0

05

10

15

0 05

Z

C

10


D = 3 mmD = 4 mm

S7 (x = 1411m)





5 Conclusions




Notations










119872119905119889119908


3 Constants (minus)









11991160 119911 (m) at 119862 = 60








thickness (minus)

Competing Interests


Acknowledgments



References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










minus4

minus2

0

2

4

Experiments

0 05C

10

D = 1 mmD = 2 mm

D = 3 mmD = 4 mm

120578

S1 (x = 0211 m)


0 05C

10


D = 3 mmD = 4 mm

minus4

minus2

0

2

4

120578

S2 (x = 0411m)



D = 3 mmD = 4 mm

minus4

minus2

2

0

4

0 05C

10

120578

S3 (x = 0611m)



D = 3 mmD = 4 mm

0 05C

10minus4

minus2

2

0

4

120578

S4 (x = 0811 m)




119862 (120578) = 1198981exp (minus119898

2(120578 + 119898

3)2

) (12)

where11989811198982 and119898


1= 104119898

2= 016 and

1198983= 187




S1S2S3

S4Equation (12)

minus4

minus2

2

0

4

120578

0 05C

10


0

05

10

15

0 05

Z

C

10

S5 (x = 1011 m)


D = 3 mmD = 4 mm




S5 (x = 1011 m)

minus4

minus2

2

0

4

120578

0 05C

10


D = 3 mmD = 4 mm








0

05

10

15

0 05

Z

C

10


D = 3 mmD = 4 mm

S6 (x = 1211 m)


0

05

10

15

0 05

Z

C

10


D = 3 mmD = 4 mm

S7 (x = 1411m)





5 Conclusions




Notations










119872119905119889119908


3 Constants (minus)









11991160 119911 (m) at 119862 = 60








thickness (minus)

Competing Interests


Acknowledgments



References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










S1S2S3

S4Equation (12)

minus4

minus2

2

0

4

120578

0 05C

10


0

05

10

15

0 05

Z

C

10

S5 (x = 1011 m)


D = 3 mmD = 4 mm




S5 (x = 1011 m)

minus4

minus2

2

0

4

120578

0 05C

10


D = 3 mmD = 4 mm








0

05

10

15

0 05

Z

C

10


D = 3 mmD = 4 mm

S6 (x = 1211 m)


0

05

10

15

0 05

Z

C

10


D = 3 mmD = 4 mm

S7 (x = 1411m)





5 Conclusions




Notations










119872119905119889119908


3 Constants (minus)









11991160 119911 (m) at 119862 = 60








thickness (minus)

Competing Interests


Acknowledgments



References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

05

10

15

0 05

Z

C

10


D = 3 mmD = 4 mm

S6 (x = 1211 m)


0

05

10

15

0 05

Z

C

10


D = 3 mmD = 4 mm

S7 (x = 1411m)





5 Conclusions




Notations










119872119905119889119908


3 Constants (minus)









11991160 119911 (m) at 119862 = 60








thickness (minus)

Competing Interests


Acknowledgments



References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















119872119905119889119908


3 Constants (minus)









11991160 119911 (m) at 119862 = 60








thickness (minus)

Competing Interests


Acknowledgments



References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Research Article Studies of Two-Phase Flow at a Chute...

Documents

Transcript of Research Article Studies of Two-Phase Flow at a Chute...