Re-Aeration on Stepped Spillways with Special Consideration of Entrained and Entrapped Air

15 Pages·2017·3.02 MB·English

geosciences
Article
Re-Aeration on Stepped Spillways with Special
Consideration of Entrained and Entrapped Air
DanielB.Bung1,* ID andDanielValero1,2 ID
1 HydraulicEngineeringSection(HES),FHAachenUniversityofAppliedSciences,Bayernallee9,
52066Aachen,Germany;[email protected]
2 DepartmentofArGEnCo,ResearchGroupofHydraulicsinEnvironmentalandCivilEngineering(HECE),
UniversityofLiege(ULg),4000Liège,Belgium
* Correspondence:[email protected];Tel.:[phone]
(cid:1)(cid:2)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:1)
(cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)
Received:4August2018;Accepted:3September2018;Published:5September2018
Abstract: Aswithmosthigh-velocityfree-surfaceflows,steppedspillwayflowsbecomeself-aerated
whenthedropheightexceedsacriticalvalue. Duetothestep-inducedmacro-roughness,theflow
field becomes more turbulent than on a similar smooth-invert chute. For this reason, cascades
are oftentimes used as re-aeration structures in wastewater treatment. However, for stepped
spillwaysasfloodreleasestructuresdownstreamofdeoxygenatedreservoirs,gastransferisalsoof
crucialsignificancetomeetecologicalrequirements. Predictionofmasstransfervelocitiesbecomes
challenging,astheflowregimediffersfromtypicalpreviouslystudiedflowconditions. Inthispaper,
detailed air-water flow measurements are conducted on stepped spillway models with different
geometry, withtheaimtoestimatethespecificair-waterinterface. Re-aerationperformancesare
determinedbyapplyingtheabsorptionmethod. Incontrasttoearlierstudies,theaeratedwaterbody
is considered a continuous mixture up to a level where 75% air concentration is reached. Above
this level, a homogenous surface wave field is considered, which is found to significantly affect
thetotalair-waterinterfaceavailableformasstransfer. Geometricalcharacteristicsofthesesurface
wavesareobtainedfromhigh-speedcamerainvestigations. Theresultsshowthatboththemeanair
concentrationandthemeanflowvelocityhaveinfluenceonthemasstransfer. Finally,anempirical
relationshipforthemasstransferonsteppedspillwaymodelsisproposed.
Keywords: gastransfer;steppedspillways;skimmingflows;self-aeration;air-waterinterface
1. Introduction
Stepped spillways and cascades are known to be effective energy dissipaters downstream of
reservoirsduetotheincreasedflowresistancecreatedbythesteps[1]. Inaddition,thestep-induced
macro-roughness leads to higher turbulence compared to smooth-invert chutes, which in turn
significantlyaffectstheflowstructure. Similartosmooth-invertspillways,steppedspillwayflows
become self-aerated depending on the discharge, slope and spillway extension [1–3]. Once the
self-aerationhasbeentriggered,thewatersurfacebecomesmoreandmoredisturbed,andbubbles,
entrainedinwater,aswellasdroplets,ejectedtotheair,aretransportedalongthechute. Thisinvolves
asignificantspecificair-waterinterfaceacrosswhichgastransfercantakeplace. Inparticular,oxygen
transferisofhighinterestaswaterinreservoirsoftensufferfromdeoxygenation,therebeingalevel
ofdissolvedoxygen(DO)contentrequiredtokeepgoodecologicalconditionsindownstreamrivers
andstreams.
Chanson[4]statedthatthere-aerationpotentialofsteppedspillwaysishigherthanforsmooth-invert
chuteswithanidenticalslope.Moreover,there-aerationinaskimmingflowregime(i.e.aflowregime
wherewaterflowsdownthechuteasacoherentstreamoverthepseudo-bottomformedbythestepedges)
Geosciences2018,8,333;doi:10.3390/geosciences8090333 www.mdpi.com/journal/geosciences
Geosciences2018,8,333 2of15
becomeshigherforlowerdischarges. Severalstudieshavebeenconductedinthepasttodescribethis
re-aerationpotentialE,whichrelatestheoxygendeficitupstreamofthestructuretotheremainingdeficit
atthedownstreamend,asfunctionsofthestructure’sgeometry[5–12].
However,allofthesestudiesonlyfocusontheglobalaerationperformanceanddonotinvestigate
thegastransfermechanismitself. Inordertophysicallydescribethegastransferprocess,knowledge
ofthespecificair-waterinterfaceandafeasiblegastransfermodelareessential. Thelattertypically
dependsonflowconditions,forexample,meanairconcentration,velocityandturbulence,andneeds
tobeestimatedbymeansofexperiments.
Inthispaper,thegastransfervelocityisdeterminedforsteppedspillwayflowsbyconducting
detailedmeasurementsoftheair-waterflowpropertiesandtimeseriesofdissolvedoxygencontent.
In contrary to earlier studies, the air-water mixture is assumed to consist of a bubbly flow region
(entrainedair)belowawavysurface(entrappedair),followinginsightsof[13–15].
2. Background
2.1. GasDiffusion
Gastransferbetweenwaterandairisdrivenbyanexistingconcentrationgradientgrad(C )
gas
andmaybedescribedbyFick’s1stlaw,accordingtowhichthediffusivefluxI (inmg/(m2s))is
gas
proportionaltothisgradientanddirectedfromhighconcentrationstolowconcentrations:
(cid:0) (cid:1)
I = −D ×grad C , (1)
gas gas gas
withD thediffusioncoefficient(inm2/s). D dependsonthetemperatureandisameasureforgas
gas gas
transfervelocity. Consideringmassconservationandassuminganisotropicdiffusivity,adiffusion
equationisfoundas
∂C
gas = D ×∇2C . (2)
gas gas
∂t
Fortheone-dimensionaldiffusionperpendiculartotheair-waterinterface,Equation(2)becomes
∂C ∂2C
gas gas
= D × . (3)
∂t gas ∂n2
TosolveEquation(3),thediffusionlengthnneedstobeknown,which,however,dependsonthe
viscosityandthediffusivityitself[16]. Thefollowingempiricalgastransfermodelbecomesthusbetter
suitedforpracticalapplications:
dC A
L = k× ×(C −C ), (4)
S L
dt V
with C the gas concentration in the liquid phase, C the saturation concentration, A the interface
L S
available for gas transfer in the water volume V and k the mass transfer coefficient (in m/s).
Thereciprocalvalueofkcanbeconsideredastheresistanceinvolvedfrombothphases(index“L”for
liquidand“G”forgas)[17]:
1 1 1
= + , (5)
k k H ×k
L G G
withH =C /C thedimensionlessHenrycoefficient,dependingonsalinity,temperatureandpressure.
G G L
It is worth noting that k is about 40 to 1000 larger than k , according to [18], for water and air
G L
(dependingonturbulence),andH isapproximately32foroxygen. Themasstransfercoefficientkin
G
Equation(4)canthusbereplacedbyk ofthewaterphaseonly:
L
dC
L = k ×a×(C −C ), (6)
L S L
dt
witha=A/Vthespecificair-waterinterface.
Geosciences2018,8,333 3of15
2.2. MassTransferCoefficient
Besides some classical, conceptional approaches, such as the two-film model of Lewis &
Whitman[17],thepenetrationmodelofHigbie[19]andthesurface-renewalmodelofDanckwerts[20],
morecomplex,hydrodynamicmodelsweredevelopedlater,suchastheLarge-EddymodelofFortescue
&Pearson[21],theSmall-EddymodelbyLamont&Scott[22]andtheModified-Small-Eddymodelby
Moog&Jirka[23],ortheturbulence-basedmodelbyGualtieri&Gualtieri[24].
The experimental studies of other researchers show a significant influence on the mean flow
velocity,Reynoldsnumberandmeanairconcentration[25],whileotherspinpointedtheinfluenceof
bubblesizes[26].
Itmustbenotedthatallofthesemasstransfermodelshavebeendevelopedforspecificflow
conditionsthathardlycomparetosteppedspillwayflows. Thechaoticflowregimeinskimmingflows
onasteppedspillwayinvolvessignificantlyhigherturbulenceandamoreintensegasexchangethan,
forinstance,risingbubblesinverticalpipes(caseforsomeofthesemodels). AspointedoutbyDemars
&Manson[27],estimationoftheair-waterinterfaceinself-aeratedflowsbecomesnon-trivialdueto
bubblesandspray. Amoredetailedexperimentalinvestigationofthere-aerationonsteppedspillways
andderivationofasuitablemasstransferequationisnecessary.
2.3. SpecificAir-WaterInterface
With consideration of a simple continuity concept and assumption of spherical bubbles,
Chanson[28]proposedtoestimatethespecificair-waterinterfaceby:
F (z)
a(z) =4× B . (7)
u(z)
DerivationofEquation(7)isbasedonastatisticalrelationbetweentheSauterdiameter,typically
measuredbyanairconcentrationprobe,andthediameterofasphericalbubble. F isthenumber
B
ofdetectedbubbles(ordroplets)perunitoftime(inHz),whichcanbedirectlyextractedfromthe
phasedetectionprobesignals,andu(z)isthestreamwiseflowvelocity. Toombes[29]andToombes&
Chanson[30]suggestedapplyingthisapproachforlargerbubblesaswell,despitethefactthatthe
shapecandifferfromanidealsphere.
However,itisknownthattheair-watermixtureinasteppedspillwayflowmayberegardedin
differentways. Whileclassicalapproachesassumeahomogenouscontinuumbetweenthebottom
andh ,thatis,theflowdepthwherethetime-averagedairconcentrationis90%[31,32],somerecent
90
studies distinguish a lower region with entrained air bubbles from an upper region where the air
contentmainlycomesfromairpocketsbeingentrappedbetweensurfacewaves[13,33–35]. Similar
flowstructureswerereportedforsmooth-invertchutes[14,15]. Indifferentstudies, withdifferent
geometricalconfigurations,somecharacteristicelevationsofthemaximumwavetroughextensions
onsteppedspillwaysareproposed. Forthesetupanalyzedinthispaper,Bung[13]suggestedh ,
75
thatis,theflowdepthwith75%meanairconcentration,asthelowerextensionofthesurfacewaves.
Figure1ashowsanexemplaryframetakenfromahigh-speedvideointhequasi-uniformflowregion
ofamodeltestwith1:2slope(ϕ=26.6◦),s=6cmstepheightandaspecificdischargeofq=0.07m2/s
(compare[36]). Thelevelofh ,detectedwithanairconcentrationprobe,isalsoindicated. Itcanbe
75
seenthatforhigherelevations,theflowstructurechangesandtheapplicationofEquation(7)becomes
questionableforhigherlevels. Figure1bshowsthedistributionofthedimensionlessspecificair-water
interfacea/max(a)assumingEquation(7)alongthefullz-coordinateuptothelevelcorrespondingto
99%airconcentration. Thechangeoftheair-waterflowstructureforthesurfacewaveregionbecomes
obviousagain.
Geosciences2018,8,333 4of15
Geosciences 2018, 8, x FOR PEER REVIEW 4 of 15

(a) (b)
Figure 1. Flow structure in stepped spillway flows: (a) High-speed video frame taken for slope 1:2, s
Figure1. Flowstructureinsteppedspillwayflows: (a)High-speedvideoframetakenforslope1:2,
s = 6=c m6 ,cmq,= q0 .=0 70.0m7 2m/s2/,s,a nadndh h75 ddeetteerrmmiinneedd wwitihth ththe ecocnodnudcuticvtiitvyi tpyropbreo; b(eb;) (dbi)stdriibsutrtiiobnu toiof nthoef the
75
dimensionless air-water interface a/max(a) according to Equation (7). Shading of the markers from
dimensionlessair-waterinterfacea/max(a)accordingtoEquation(7). Shadingofthemarkersfrom
black (at the inception point of self-aeration) to white (at the onset of quasi-uniform flow).
black(attheinceptionpointofself-aeration)towhite(attheonsetofquasi-uniformflow).
3. Methodology
3. Methodology
3.1. General Comments (Data Base)
3.1. GeneralComments(DataBase)
The current paper presents a re-analysis of air-water flow data and oxygenation data published
Tinh e[3c7u–4r0re],n rtesppaepcetirveplrye. sWenhtislea threes-ea nstauldyiseiss coofnasiidr-ewreadt ear cfloontwinudoautas aainr-dwoaxteyrg menixatutiroen udpa ttoa hp90u, tbhleis hed
in [37c–u4r0re],ntr esstupdeyc taicvceoluyn.tsW fohri lhe75 tahse tshee sutpupdeire esxcteonnt soifd tehries dbuabbcloyn ltaiyneur.o Iuns oardire-rw toa taecrcomunixt tfuorr ethue pairt-o h ,
90
water surface roughness due to the surface waves [13], an idealized, simple sinusoidal surface is
thecurrentstudyaccountsforh astheupperextentofthisbubblylayer. Inordertoaccountforthe
75
considered. Air bubbles being entrained within these surface waves are not regarded. An analytical
air-watersurfaceroughnessduetothesurfacewaves[13],anidealized,simplesinusoidalsurfaceis
derivation of relevant equations and necessary approximations to describe the surface geometry and
considered. Airbubblesbeingentrainedwithinthesesurfacewavesarenotregarded. Ananalytical
its volume beneath the waves is given in Appendix A.
derivationofrelevantequationsandnecessaryapproximationstodescribethesurfacegeometryand
All data has been gathered on a physical model with slopes 1:3 and 1:2, step heights s of 3 cm
itsvoalunmd e6 bcmen aenadth dtihscehwaragvese sofi sq g=i v0e.0n7,i n0.0A9p apnedn 0d.i1x1 Am.2/s in the Hydraulics laboratory of Wuppertal
AUlnlidveartsaithy a[3s7b].e Tehneg cahtuhtee rwedidtohn waasp 3h0y csmic aalnmd iotsd teoltawl ditrhops lhoepigehst1 w:3asa 2n.d341 m:2., Ssktiempmhienigg fhlotsws woafs3 cm
and6focumnda nfdor dailslc hmaordgeel stoesftsq =(co0m.0p7a,r0e. 0[941a–n43d])0. .1W1amter2 /dsisicnhathrgeeH wyads rraeugluiclasteladb owriathto ar yfloafp Wvaulpvep, ertal
Univecrosnittryol[l3ed7] w. Tithh ea cfhlouwt emwetiedrt hanwd apsum30pecdm inatnod anit soptoenta hledardo ptanhke ibgehfotrwe baesin2g.3 4comnv.eySekdim inmtoi nthge flow
spillway. The chute was part of a closed water circuit.
wasfoundforallmodeltests(compare[41–43]). Waterdischargewasregulatedwithaflapvalve,
controlledwithaflowmeterandpumpedintoanopenheadtankbeforebeingconveyedintothe
3.2. Air-Water Flow Measurements
spillway. Thechutewaspartofaclosedwatercircuit.
Air-water flow measurements are conducted using a double-tip conductivity probe in the flume
3.2. Acier-nWtrealtienreF. Tlohwis Mpreoabseu croemnseinsttss of two needle tips with a diameter of 0.13 mm. Both tips are aligned in
flow direction with a lateral spacing of 1 mm which aim to pierce up entrained bubbles and ejected
Air-waterflowmeasurementsareconductedusingadouble-tipconductivityprobeintheflume
droplets. By a change of resistivity depending on the surrounding medium, phase changes can be
centredlienteec.tTedh ibsyp aropbpelycionng sai stthsroesfhtowldoinngee tdeclehntiipqsuew [i4t4h].a Fduiratmheert ienrfoorfm0.a1t3iomn mon. tBhoisth tytpipes oaf rperaolbige nceadn bine flow
directfioounnwd iitnh [a45la].t eWraitlhs pthaec iansgsuomf1ptmionm thwaht i(c1h) abiomth ttoippsi epriceercue ptheen straamine ebdubbbulbebs laensda n(2d) ebjuecbtbeldesd arroep lets.
Byactrhaannsgpeorotefdr eins ias tsilvipit-yfredee wpeany,d cirnogsso-cnortrheelastiuornr oouf bnodthin sgigmnaelds iyuiemld,sp ahna essetimchaaten gofe sthcea fnlowbe vdeleotceictyte. dby
applyRineglevaatnhtr epsahraomldeitnegrst, escuhcnhi qaus eth[e4 4l]o.cFalu, rttihmeer-ainveforargmeadt iaoinr coonncthenistrtaytpioen,o fthper oSbauetcear ndibaemfeotuern danidn [45].
Withbthuebbalses cuomunptt iroante tchaant b(e1 )dibroecthtlyt idpesteprmierinceedt hfreomsa mthee sbigunbablsle. sAadndditio(2n)alb puabrbalmesetaerrse, tfroarn esxpaomrpteled, in a
the specific air-water interface according to Equation (7), can be indirectly estimated from this data.
slip-freeway,cross-correlationofbothsignalsyieldsanestimateoftheflowvelocity.Relevantparameters,
For all configurations, tests are conducted from the inception point of self-aeration to farther
suchasthelocal, time-averagedairconcentration, theSauterdiameterandbubblecountratecanbe
downstream where quasi-uniform flow conditions set in. For the larger steps, data is gathered at step
directlydeterminedfromthesignals.Additionalparameters,forexample,thespecificair-waterinterface
edges and above the centre of the step cavity. For the smaller steps, all data is obtained at step edges
accordingtoEquation(7),canbeindirectlyestimatedfromthisdata. Forallconfigurations,testsare
only.
conductedAfsr otmhe theextirnaccteiponti oonf psouirnftacoef swelafv-aese ra(Ftiiognurteo 1faar) thfreormd ocwonndsturcetaivmityw hpreorbeeq udaastai- uannidfo rtmhe flow
conditdieotnesrmseitnianti.oFno orft htheel aarvgaeilrasbtleep asi,r-dwaatateirs ignatethrfearceed aabtosvtee ph7e5 disg neosta dnidreacbtloyv peotshseibclee,n atr ceoonftitnhueusmte pupca vity.
Forthteo shm75 aisl lceornsstiedpesr,edal ilnd tahtias isstuodbyt aainnde da athtrsetee-pdiemdegnessioonnally s.inusoidal wave field (with waveheights H
Astheextractionofsurfacewaves(Figure1a)fromconductivityprobedataandthedetermination

oftheavailableair-waterinterfaceaboveh isnotdirectlypossible,acontinuumuptoh isconsidered
75 75
inthisstudyandathree-dimensionalsinusoidalwavefield(withwaveheightsHandwavelengthsλ),
Geosciences2018,8,333 5of15
Geosciences 2018, 8, x FOR PEER REVIEW 5 of 15
leadingtoaninterfacialsurfaceS oftheair-watermixture. ThelatterisaddedtotheinterfaceA
ent
and wavelengths λ), leading to an interfacial surface of the air-water mixture. The latter is added
formedbytheentrainedairbubbles. Aschematicillustrationoftheemployedconceptisshownin
to the interface Aent formed by the entrained air bubbles. A schematic illustration of the employed
Figure2. Inordertodeterminethespecificair-waterint𝒮𝒮erface,A =A +S needstoberelatedtothe
concept is shown in Figure 2. In order to determine the speciftioct air-wenatter interface, Atot = Aent +
totalvolume(uptotheupperwaveextent).
needs to be related to the total volume (up to the upper wave extent).
𝒮𝒮

Figure 2. Schematic illustration of the considered flow structure over the step cavities consisting of a
Figure2.Schematicillustrationoftheconsideredflowstructureoverthestepcavitiesconsistingofa
continuous air-water mixture up to h75 and non-aerated, sinusoidal surface waves over h75.
continuousair-watermixtureuptoh andnon-aerated,sinusoidalsurfacewavesoverh .
75 75
3.3. Surface Wave Characterization
3.3. SurfaceWaveCharacterization
To estimate the interface being available for gas transfer due to surface waves, a simple three-
Tdoimeesntsimionaatel sitnhuesoiindtaelr wfaacvee fbieelidn igs aasvsuamilaebdl. eThfiosr wgaavse ftierladn issf cehrardaucteertizoeds ubryf aeqcueawl waavveesl,enagtshism ple
three-λd iimn ebnostiho, nsatlresaimnuwsisoei daanldw taravnesvfieerlsde idsiraesctsiuonm, eadn.d Tah wisawveahveeighfite lHd. iSsimchilaarra csuterrfaizcee dwbayvees qual
waveclehnargatchtesrλistiincs bwoetrhe, asstsruemamedw tois aenaanlydtictarlalny sdveesrcsriebed tihreec steiolfn-a,earantdiona pwroacveeshs ebiyg [h3t], Hsh.eSdidminilga rligshutr face
on the fluid mechanics of self-aeration. Hence, it is herein proposed that surface waves are connected
wavescharacteristicswereassumedtoanalyticallydescribetheself-aerationprocessby[3],shedding
to the upstream non-aerated region, holding similar structure. As no closed solution for the surface
light on the fluid mechanics of self-aeration. Hence, it is herein proposed that surface waves are
area of the supposed wave field exists, a numerical integration was performed to find an empirical
connectedtotheupstreamnon-aeratedregion,holdingsimilarstructure. Asnoclosedsolutionfor
fitting function. The description of the underlying assumptions and solutions can be found in
thesurfaceareaofthesupposedwavefieldexists,anumericalintegrationwasperformedtofindan
Appendix A.
empiricalfittingfunction. Thedescriptionoftheunderlyingassumptionsandsolutionscanbefound
Characteristic wavelengths and waveheights are obtained from high-speed videos analyzed in
inAp[p46e]n, dwixhAich. extracted the free surface of the air-water mixture applying an image processing
Ctehcahrnaiqcuteer. iFstriocmw thaivse dleantag, tthhse amneddiwana wveahveeliegnhgttshsa raendo bwtaavineehdeigfhrotsm, gihviegnh i-ns pTeaebdle v1i,d heaovse abneeanly zed
in [46o]b,tawinheidch (Teaxbtlrea 1c)t eadndt hcoensfrideeeresdu rrfeapcreesoenftathtievea oirf -twhea twerhomlei xsttruurcetuarep ptol ydientgermanineim thaeg aeirp-wroacteers sing
technmiqiuxteu.reF rsoumrfacthe iisn dSeacttaio,nth 4e.1m. edianwavelengthsandwaveheights, giveninTable1, havebeen
obtained(Table1)andconsideredrepresentativeofthewholestructuretodeterminetheair-water
Table 1. Median wavelengths and waveheights extracted from high-speed videos with use of an
mixturesurfaceinSection4.1.
image processing technique; corresponding median absolute deviations (MAD) given in brackets.
Table1.M edianwavelengthsandwaSvloephee i1g:2h tsextractedfromhigh-speedvideSolospwe i1th:3 useofanimage
prSotceeps sHineigghtetc [hcmni]q ue;corresp3o ndingmedianabso6lu tedeviations(MAD3 )giveninbrackets.6
Discharge [m2/s] 0.07 0.09 0.11 0.07 0.09 0.11 0.07 0.09 0.11 0.07 0.09 0.11
6.1 7.5 Sl8o.p9e 1:27.1 7.3 6.6 6.7 5.9 7.S1l o pe15:.39 5.7 6.6
λ [cm]
(1.6) (1.4) (2.3) (1.7) (1.4) (1.8) (1.3) (1.2) (1.9) (1.3) (1.1) (1.6)
StepHeight[cm] 3 6 3 6
1.4 1.3 1.3 2.6 2.0 2.1 1.3 1.3 1.2 3.9 3.1 2.9
DischargHe [[mcm2/]s ] 0.07 0.09 0.11 0.07 0.09 0.11 0.07 0.09 0.11 0.07 0.09 0.11
(0.6) (0.8) (0.6) (1.0) (1.3) (1.1) (0.8) (0.5) (0.7) (1.4) (1.2) (1.2)
6.1 7.5 8.9 7.1 7.3 6.6 6.7 5.9 7.1 5.9 5.7 6.6
λ[cm]
(1.6) (1.4) (2.3) (1.7) (1.4) (1.8) (1.3) (1.2) (1.9) (1.3) (1.1) (1.6)
3.4. Dissolved Oxygen Measurements
1.4 1.3 1.3 2.6 2.0 2.1 1.3 1.3 1.2 3.9 3.1 2.9
H[cm]
The re-aerat(i0o.n6) pot(e0n.8t)ial o(f0 .t6h)e di(f1f.e0)rent (s1t.3e)pped(1 .s1p)illw(0a.y8) mo(d0e.5l) setu(0p.7s) has (1b.e4e)n d(e1t.e2)rmin(1e.d2)
by direct oxygen measurements. The absorption method, according to [47], has been applied by
3.4. Driesdsoulcviendg Othxey gdeisnsoMlveeadsu orxeymgeennt scontent through sodium solfite addition. In order to speed up the
deoxygenation process, cobalt sulfate was added as a catalyst. Before starting the tests, accurate
Tmhiexirneg- aoef rtahtei ochnepmoicteanlst wiaalso efntshuereddif bfeyr ceinrctusltaetpinpge tdhes pwialltwera iny tmheo tdaenlksse wtuitphs smhaasllb peuemnpdse. tBeersmidiense, dby
directmooxryeg seondmiumea ssuurlfeimte ewntass. aTdhdeeadb tshoarnp ttihoenormeteictahlolyd ,naececdoerdd itno gentosu[r4e7 ]f,uhlla sdeboexeyngeanpaptiloiend. Obyxyrgeednu cing
thedicsosnoclvenetdraotixoyng ewnacso nmteenastutrherdo uwgihths otdwiuo midseonltfiictael adopdtiitciaoln .pIrnoboersd e(rHtaochsp eHeQd1u0)p tuhpestdreeaomxy gaennda tion
process, cobalt sulfate was added as a catalyst. Before starting the tests, accurate mixing of the
chemicalswasensuredbycirculatingthewaterinthetankswithsmallpumps. Besides,moresodium

sulfitewasaddedthantheoreticallyneededtoensurefulldeoxygenation. Oxygenconcentrationwas
measuredwithtwoidenticalopticalprobes(HachHQ10)upstreamanddownstreamofthespillway
Geosciences2018,8,333 6of15
andalltestswererepeatedatleastonce. Timeseriesofinstantaneousdissolvedoxygenconcentrations
CDO(tG)eoasrcieenecexs p20e1c8t, e8,d x tFoORfo PlElEoRw REaVnIEeWx p onentialfunctiongivenby 6 of 15
downstream of the spillwCay a(ntd) a=ll Ctests w−er(cid:0)eC repeat−edC at(cid:1) le×aset− oknLc×ea. TT×imt,e series of instantaneous (8)
DO S,p,T S,p,T 0
dissolved oxygen concentrations CDO(t) are expected to follow an exponential function given by
where CS,p,T is the saturation concentration at local conditions (with atmospheric pressure(8p) and
temperature T) and kLaT is the aeration coefficient at temper−a𝑘𝑘t𝐿𝐿u×r𝑔𝑔e𝑇𝑇×T𝑡𝑡 . For comparison of aeration
where CS,p,T is the saturation c𝐶𝐶o𝐷𝐷n𝐷𝐷c(e𝜕𝜕n)t=ra t𝐶𝐶io𝑆𝑆,n𝑝𝑝, 𝑇𝑇a−t l�o𝐶𝐶c𝑆𝑆a,𝑝𝑝l ,𝑇𝑇co−n𝐶𝐶d0i�ti×on𝑒𝑒s (with a,tmospheric pressure p and
performancetestswithdifferentlocalconditions,C andk a (yieldedbyanonlinearregression
analytseims)pcearnatubree nTo)r manadl izkLeadT itso tahes taaenrdaatirodn tceomefpfiecrieantut Sra,ept, Totefm20pe◦rCaLtuaTnred Ta. pFroers csoumrepoafri1so0n1 3ohf Paaeraastion
performance tests with different local conditions, CS,p,T and kLaT (yielded by a nonlinear regression
analysis) can be normalized to a standard temperature of 20 °C and a pressure of 1013 hPa as
k a = k a ×1.02420−T (9)
L 20 L T
(9)
20−𝑇𝑇
and and 𝑘𝑘𝐿𝐿𝑎𝑎20=𝑘𝑘𝐿𝐿𝑎𝑎𝑇𝑇×1.024
C 1013
C =C × S,St,20 × , (10)
S,20 S,p,T
CS,St,T p (10)
respectively. In Equation (10), CS,St𝐶𝐶,T𝑆𝑆,20re=fe𝐶𝐶r𝑆𝑆s,𝑝𝑝,𝑇𝑇to×𝐶𝐶th𝑆𝑆,e𝑆𝑆𝑡𝑡,2s0t×an1d0a1r3d,ized saturation concentration for
p=10r1es3phecPtaivaenlyd. Itne mEqpueartaitounr e(1T0),, wChS,Sitl,Te rCefers to =th9e. 0s9tamn𝐶𝐶𝑆𝑆dg,𝑆𝑆a𝑡𝑡/r,𝑇𝑇dLizfoedr 𝑝𝑝fsualtlusrtaatinodna crodnccoenntdraittiioonn sf.oAr pn =e x1e0m13p lary
hPa and temperature T, while CS,St,20 = S9,S.0t,92 0mg/L for full standard conditions. An exemplary result
resultfromasingleoxygenationtestisshowninFigure3a. Figure3bshowstheresidualsbetweenthe
from a single oxygenation test is shown in Figure 3a. Figure 3b shows the residuals between the fitting
fittingandthelaboratorydata,provingagoodfitting.
and the laboratory data, proving a good fitting.

(a) (b)
Figure 3. Exponential regression of oxygenation curves for one exemplary model run with 1:2 slope,
Figure3.Exponentialregressionofoxygenationcurvesforoneexemplarymodelrunwith1:2slope,
s=6sc m= 6a cnmd aqn=d 0q .1=1 0.m112 /ms2:/s(:a ()a)F iFtittitningg ttoo tthhee eexxppeerriimmeenntatla lddataat afofro lrocloalc acolncdointidonitsi oannsd arnesdulrteinsug lting
standardized oxygen transfer parameters (note: only every 2nd data point is shown for better
standardizedoxygentransferparameters(note:onlyevery2nddatapointisshownforbetterlegibility);
legibility); (b) Residuals resulting from the curve fitting.
(b)Residualsresultingfromthecurvefitting.
4. Results
4. Results
4.1. Specific Air-Water Interface
4.1. SpecificAir-WaterInterface
The specific air-water interface according to Equation (7) has been determined between the
Tinhceepstpioenc ipfiocinat iorf- wsealft-eareriantitoenr faancde tahcec qouradsiin-ugnitfoorEmq fuloawtio rneg(io7n) fhoars 0 b≤ eze ≤n hd75 einte arlml cionnefdigubreattwioenes.n the
incepTtihoen repsouilntst ionf dsiemlfe-anesiroantiaol nnuamndbetrhse aqreu iallsuis-utrnatiefodr imn Ffligouwrer e4g foior n1:f2o srlo0p≤e aznd≤ Fhigurien 5a flolrc o1:n3fi sglouprea.t ions.
75
It can be observed that the total amount strongly depends on the flow regime. While for smaller step
TheresultsindimensionalnumbersareillustratedinFigure4for1:2slopeandFigure5for1:3slope.
heights and larger discharges a more stable skimming flow regime is found, a more tumbling flow is
Itcanbeobservedthatthetotalamountstronglydependsontheflowregime. Whileforsmallerstep
found for larger step heights and lower discharges. This leads to a more chaotic flow regime and
heightsandlargerdischargesamorestableskimmingflowregimeisfound,amoretumblingflow
stronger aeration. For all configurations, the specific air-water interface follows a distribution as
isfoundforlargerstepheightsandlowerdischarges. Thisleadstoamorechaoticflowregimeand
shown in Figure 1b with a maximum slightly below h75. The highest air-water interfaces being
strongobersearevreadt iionn t.hFe ocrurarlelncto tnefistgs uarrae tiino nthse, tohredesrp oefc i3fi5c0 amir2/-mw3a. terinterfacefollowsadistributionasshown
inFigureI1nb woridtehr atmo adxeitmerummines litghhet lygabse ltorawnshf7e5r. Tvehleochitiyg hfersotma irt-hwe aateerraitniotenr feafcfiecsiebneciinesg o(sbhsoewrvne din
thecusurrbesneqtuteesnttslya)r, ethine tthoetaol radier-rwoafte3r5 0inmte2rf/amce3 .from entrained air bubbles is required and can be
Icnalocurdlaetredto bdye terminethegastransfervelocityfromtheaerationefficiencies(shownsubsequently),
thetotalair-waterinterfacefromentrainedairbubblesisrequiredandcanbecalculatedby
Geosciences2018,8,333 7of15
x=(cid:90)Ltoty(cid:90)=bz=(cid:90)h75
A = adxdydz, (11)
ent
x=Li y=0 z=0
whereL isthedistancefromthespillwaycresttotheinceptionpointofself-aeration,L isthetotallength
i tot
ofthespillwayandbisthewidthoftheflume.TheresultingsurfaceareaS ofthesupposedsinusoidal
Geosciences 2018, 8, x FOR PEER REVIEW 7 of 15
air-watermixturesurfaceaccordingtoEquation(A7)isaddedtoA fromEquation(11)todeterminethe
ent
totalavailableair-waterA interface.Descriptionoftheassumedsurfacewavegeometryandderivation
tot
ofrelevantfunctionsisgivenintheAppendixA.ResultsforbothinterfacesA andS,aswellasthetotal
𝑥𝑥=𝐿𝐿𝑡𝑡𝑡𝑡𝑡𝑡 𝑦𝑦=𝑏𝑏𝑧𝑧= ℎ75 ent (11)
air-waterinterfaceA fromallthetestedconfigurations,arelistedinTable2.
tot
𝐴𝐴𝑒𝑒𝑛𝑛𝑡𝑡= � � � 𝑎𝑎d𝑥𝑥d𝑦𝑦d𝑧𝑧,
where Li is the distancTe afbrolem2 t.hTeo staplilalwir-awy ac𝑥𝑥tr=ee𝐿𝐿rs𝑖𝑖tin t𝑦𝑦toe= rt0hfae𝑧𝑧c =ien0fcoerpatilolnco pnofiingtu oraf tsieolnf-sa.eration, Ltot is the total
length of the spillway and b is the width of the flume. The resulting surface area of the supposed
sinusoidal air-water mixture surface according to Equation (A7) is added to Aent from Equation (11)
Slope1:2 Slop𝒮𝒮e1:3
to determine the total available air-water Atot interface. Description of the assumed surface wave
StepHeight[cm] 3 6 3 6
geometry and derivation of relevant functions is given in the Appendix A. Results for both interfaces
DiAscehnta argned[ m2,/ sa]s we0l.l0 7as the0 .0to9tal a0i.r1-1wate0r .0in7terfa0c.0e9 Atot f0r.o11m all0 t.0h7e tes0te.0d9 conf0i.g11uratio0.n07s, are0 .l0is9ted i0n.1 1
ETnatbralien 2ed. air
bubblesAent𝒮𝒮[m2] 7.37 7.31 6.93 12.67 12.60 13.21 5.65 7.05 6.89 17.58 18.32 16.06
(Equation(11))
Table 2. Total air-water interface for all configurations.
Air-watermixture
surfaceS[m2] 1.95 1.58 1.34 3.24 2.29 2.58 2.55 2.62 2.06 10.67 7.88 5.71
Slope 1:2 Slope 1:3
(Equation(A7))
Step Height [cm] 3 6 3 6
Atot[mD2i]scharge [m9.23/s2] 8.89 0.078 .270.09 150..9111 01.40.78 9 0.1059. 79 0.118 .200.07 9.607.09 80..1915 02.087. 25 0.0296 .200.112 1.77
Entrained air bubbles Aent [m2] 7.37 7.31 6.93 12.67 12.60 13.21 5.65 7.05 6.89 17.58 18.32 16.06
(Equation (11))
Itisinterestingtonotethatforlessstableskimmingflowregimes(i.e.,forlargerstepsandlower
Air-water mixture surface
1.95 1.58 1.34 3.24 2.29 2.58 2.55 2.62 2.06 10.67 7.88 5.71
discharges)[m, 2t]h (Eequtoattioanl (aAi7r))- water interface increases significantly with a relevant contribution of the
𝒮𝒮
surface waves.AWtot [imth2] considera9t.3io2 n8o.8f9 th8is.27t ot1a5l.9s1 ur1f4a.8c9e A15.79 in8T.2a0 bl9e.627 an8.9d5 th2e8.2t5o ta2l6.w20 ate21r.7v7 olume,
tot
consistingoftheair-watermixturevolume
It is interesting to note that for less stable skimming flow regimes (i.e., for larger steps and lower
discharges), the total air-water interface increases significantly with a relevant contribution of the
surface waves. With consideration of this txo=(cid:90)taLlt ostuy(cid:90)r=fabce Atot in Table 2 and the total water volume,
consisting of the air-water mixtureV vmoliuxm=e h75dxdydz (12)
x=0 y=0
𝑥𝑥=𝐿𝐿𝑡𝑡𝑡𝑡𝑡𝑡 𝑦𝑦=𝑏𝑏 (12)
plusthefluidvolumeV beneaththesurfacewavesforz>h accordingtoEquation(A8)(V =V +V),
75 tot mix
thetotalspecificair-waterinterfacea is𝒱𝒱fo𝑚𝑚u𝑚𝑚𝑥𝑥n=da�sfoll�owℎs7i5ndT𝑥𝑥adb𝑦𝑦lde𝑧𝑧3:
tot
plus the fluid volume beneath the surface 𝑥𝑥w=a0ve𝑦𝑦s= f0or z > h75 according to Equation (A8) ( tot = mix
+ ), the total specTiafibc la𝒱𝒱eir3-.wTaottearl isnpteecrfifiaccea aitro-t wisa ftoeurnindt earsf faoclelofwors ailnl Tcoanbfileg 3u:r ations. 𝒱𝒱 𝒱𝒱
𝒱𝒱
Table 3. Total specific air-water interface for all configurations.
Slope1:2 Slope1:3
StepHeight[ cm] 3 Slope 1:2 6 3 Slope 1:3 6
Step Height [cm] 3 6 3 6
Discharge[m2/s] 0.07 0.09 0.11 0.07 0.09 0.11 0.07 0.09 0.11 0.07 0.09 0.11
Discharge [m2/s] 0.07 0.09 0.11 0.07 0.09 0.11 0.07 0.09 0.11 0.07 0.09 0.11
atot[ma2to/t m[m32]/m3] 150 1501 32 132 106106 197197 190190 171777 9911 9988 8811 22232 3 1961 96 1531 53

(a)
Figure4.Cont.
Geosciences2018,8,333 8of15
Geosciences 2018, 8, x FOR PEER REVIEW 8 of 15

(b)

(c)

(d)

(e)

(f)
FigureFi4g.uSrpe e4c. iSfipceaciirfi-cw aairt-ewraintetre rinfatecrefaac(ei na (min2 m/m2/m3)3)f oforra allllc coonnffiigguurraattiioonnss wwiitthh ssloloppe e1:12:;2 d;odwownsntrsetarmea mof ofthe
the inception point of self-aeration and from z = 0 (pseudo-bottom) to h75; data in quasi-uniform flow
inceptionpointofself-aerationandfromz=0(pseudo-bottom)toh ;datainquasi-uniformflowregion
region assumed similar to the last two profiles: (a) s = 3 cm, q = 0.07 7m52/s; (b) s = 3 cm, q = 0.09 m2/s; (c)
assumedsimilartothelasttwoprofiles:(a)s=3cm,q=0.07m2/s;(b)s=3cm,q=0.09m2/s;(c)s=3cm,
s = 3 cm, q = 0.11 m2/s; (d) s = 6 cm, q = 0.07 m2/s; (e) s = 6 cm, q = 0.09 m2/s; (f) s = 6 cm, q = 0.11 m2/s.
q=0.11m2/s;(d)s=6cm,q=0.07m2/s;(e)s=6cm,q=0.09m2/s;(f)s=6cm,q=0.11m2/s.
Geosciences2018,8,333 9of15
Geosciences 2018, 8, x FOR PEER REVIEW 9 of 15

(a)

(b)

(c)

(d)

(e)

(f)
FigureFi5g.uSrpe e5c. iSfipceaciifri-cw aairt-ewraintetre rinfatecrefaac(ei na (min2 m/m2/m3)3)f oforra allllc coonnffiigguurraattiioonnss wwitithh ssloloppe e1:13:;3 d;odwonwsntrsetarmea mof ofthe
the inception point of self-aeration and from z = 0 (pseudo-bottom) to h75; data in quasi-uniform flow
inceptionpointofself-aerationandfromz=0(pseudo-bottom)toh ;datainquasi-uniformflowregion
region assumed similar to the last two profiles: (a) s = 3 cm, q = 0.07 7m52/s; (b) s = 3 cm, q = 0.09 m2/s; (c)
assumedsimilartothelasttwoprofiles:(a)s=3cm,q=0.07m2/s;(b)s=3cm,q=0.09m2/s;(c)s=3cm,
s = 3 cm, q = 0.11 m2/s; (d) s = 6 cm, q = 0.07 m2/s; (e) s = 6 cm, q = 0.09 m2/s; (f) s = 6 cm, q = 0.11 m2/s.
q=0.11m2/s;(d)s=6cm,q=0.07m2/s;(e)s=6cm,q=0.09m2/s;(f)s=6cm,q=0.11m2/s.
Geosciences2018,8,333 10of15
4.2. Re-AerationPerfomance
Relevant re-aeration performance parameters are obtained by nonlinear regression of the
oxygenation data, as shown in Figure 3. As the experiments have been conducted on different
dayswithdifferentlocalconditions,thestandardizedre-aerationparametersarealwayspresentedin
thefollowing. Foreachconfiguration,fourindependentmodelrunswereperformedandtheaveraged
resultsfork a andC arelistedinTable4.
L 20 S,20
Table4.Standardizedre-aerationperformanceparametersforallconfigurations(eachaveragedfrom
fourindependentmodelruns,correspondingminimumandmaximumvaluesgiveninbrackets).
Slope1:2 Slope1:3
StepHeight[cm] 3 6 3 6
Discharge[m2/s] 0.07 0.09 0.11 0.07 0.09 0.11 0.07 0.09 0.11 0.07 0.09 0.11
(6.71) (9.27) (9.39) (8.51) (11.61) (10.53) (6.56) (8.14) (9.74) (7.74) (9.40) (10.92)
kLa20[1/h] 6.85 9.32 9.76 8.73 11.68 11.11 7.54 8.87 10.17 8.60 10.08 11.47
(7.04) (9.41) (10.13) (8.87) (11.81) (11.49) (11.33) (11.33) (11.33) (11.33) (11.33) (11.89)
(8.44) (8.57) (8.60) (8.46) (8.54) (8.61) (8.29) (8.39) (8.41) (8.26) (8.31) (8.34)
CS,20[mg/L] 8.61 8.68 8.70 8.63 8.67 8.75 8.43 8.48 8.48 8.42 8.43 8.53
(8.77) (8.78) (8.70) (8.80) (8.79) (8.95) (8.61) (8.61) (8.61) (8.61) (8.61) (8.86)
4.3. MassTransferVelocity
ThemasstransfercoefficientcanbedirectlydeducedfromTables3and4. Theresultsareshown
inTable5.
Table5.Deducedmasstransfervelocitiesforallconfigurations.
Slope1:2 Slope1:3
StepHeight[cm] 3 6 3 6
Discharge[m2/s] 0.07 0.09 0.11 0.07 0.09 0.11 0.07 0.09 0.11 0.07 0.09 0.11
kL[cm/h] 4.6 7.1 9.2 4.4 6.1 6.3 8.3 9.1 12.6 3.9 5.1 7.5
Obviously, k increases for higher discharges and is thus a function of the Reynolds number.
L
Theflowregime(i.e.,thestabilityoftheskimmingflow),whichwasaffectingthespecificair-water
interface, is found to have no significant influence on k . In addition, it is noticed that the air
L
concentration strongly affects the mass transfer as well (as it was also described in [38] under
consideration of an air-water mixture up to h ). With the average flow velocity u being found
90
intheaeratedflowregionandtheaverageairconcentrationonthestructure(uptoh ),thefollowing
75
relationshipisfound:
k
L =2.342∗e−5.579×C×10−5. (13)
u
The curve fitting reaches R2 = 0.96 when neglecting a single obvious outlier, as indicated in
Figure6.