AERATION PERFORMANCE OF TRIANGULAR PLANFORM

LABYRINTH WEIRS

By Peter R. Wormleaton1 and Ebrahim Souftane

ABSTRACT: A high level of dissolved oxygen is vital for the maintenance of healthy streams and rivers.Structures in rivers can increase dissolved oxygen levels by creating turbulent conditions where small air bubblesare carried into the bulk of the flow. Plunging overfall jets from weirs are a particular instance of this, and theaeration properties of such structures have been studied widely in the laboratory and field over a number ofyears. On the other hand, labyrinth weirs, where the weir sill is cranked in planform thus increasing their length,have received little or no attention in this context. They have a proven hydraulic advantage over straight weirsof increased discharge at the same head for design conditions. However, they also serve to modify the combinedoverfall jet as individual jets from adjacent sections of the weir collide. This paper describes an experimentalinvestigation into the nature of these jets and how they affect the aeration performance of a triangular planlabyrinth weir. It is demonstrated that the aeration efficiency of these labyrinth weirs generally is better thantheir equivalent-length linear weir and that this advantage becomes more pronounced as the weir included anglebecomes smaller and also at lower overfall drop heights and higher discharges. These results point to the possibleadvantage of these type of weir in situations where both hydraulic and aeration performance needs to be opti­mized.

d

Flow

>w

Triangular labyrinth weir

dl-._--.

\ I I Iill'r\ III/\ I!

\1 I11I1

\: \ 11//

\\iI Ii I\II III!1\' ii,'\1\ ,I!

\Il'/IiPlan Elevation

1. Sketch of Colliding Jets over Labyrinth Weirs

Flow direction.,j"

FIG.

Flow)

" ·1/

'!'~~''"A /'r Flow·lines

Colliding overfall jets

ticular, the effect of varying the included angle e and the lon­gitudinal angle of slope of the weir sill <1>. The shape of theweir sill dictates the behavior of the jet. This in turn is believedto alter the air entrainment and contact time in both the jetitself and the downstream receiving pool and hence the aera­tion performance of the weir as a whole.

FIG. 2. Notation Diagram for li'langular labyrinth Weir

JOURNAL OF ENVIRONMENTAL ENGINEERING / AUGUST 1998 / 709

INTRODUCTION

Labyrinth weirs are those for which the weir crest is notstraight in planform (Fig. 1). The increased crest length oflabyrinth weirs gives them the clear advantage of reducingupstream levels for a particular discharge (or conversely in­creasing discharge for a given head) over low head ranges.Taylor (1968) and Hay and Taylor (1970) carried out experi­ments and demonstrated the advantage of these weirs in situ­ations where channel widths are limited. Cassidy (1983)showed how these weirs can be used to effectively increasethe length of a dam spillway crest.

As the upstream head increases the advantage in hydraulicperformance of these weirs relative to normal weirs becomesless marked because of the collision and interference of thejets from adjacent sections (Fig. 1). Eventually, the spaces be­tween the weir sections on the downstream side become filledwith water and the weir effectively becomes broad crested.The writers have investigated whether these colliding jetswould increase the turbulence and air entrainment in the fall­ing water and thus make the weirs more effective aerators.

Gameson (1957) was the first to report on the aeration po­tential of weirs in rivers. Since then a number of laboratoryinvestigations into weir aeration have been carried out, notablyVan der Kroon (1969a,b), Apted and Novak (1973), Avery andNovak (1978), and Nakasone (1987). Investigations also havebeen reported on the aeration performance of existing hydrau­lic structures and these are reviewed by Wilhelms et al. (1993).Gulliver and Rindels (1993), in particular, discuss problemsassociated with field measurements of oxygen transfer and thedegree of uncertainty involved. Much of this work has dealtwith straight weirs and free overfalls, among other structures,and none has concentrated specifically on the aeration perfor­mance of labyrinth weirs.

This paper describes an experimental investigation into theperformance of triangular labyrinth weirs (Fig. 2), and, in par-

'Sr. Lect.• Queen Mary and Westfield Coli., Univ. of London, MileEnd Rd., London, El 4NS, U.K. E-mail: p.r.wormleaton@qmw.ac.uk.

'Managing Dir., Pars Engrg., 52 Mersea House, Harts Ln., Barking,Essex IGl1 8LW, U.K.

Note. Associate Editor: Hilary 1. Inyang. Discussion open until January1, 1999. To extend the closing date one month, a written request mustbe filed with the ASCE Manager of Journals. The manuscript for thispaper was submitted for review and possible publication on November24, 1997. This paper is part of the JourlUll of Environmental Engineer­ing, Vol. 124, No.8, August, 1998. ©ASCE, ISSN 0733-9372/98/0008­0709-0719/$8.00 + $.50 per page. Paper No. 17070.

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(1)

(3)

(2)

(4)E = 1 _ .!. = (Cd - C.)r (Cg - C.)

Aeration Mechanism at Weirs

Clearly, the transfer efficiency represents the change in DOconcentration in the water flowing over the weir as a propor­tion of the upstream DO deficit. The values of rand E areseen to depend on the effective transfer coefficient KL = kLAIV and the contact time te , which can be considered as meanrepresentative values for the weir in (3).

Oxygen transfer due to a free falling overflow jet from aweir is a complex process, which depends on the geometryand hydraulics of the jet itself and the receiving pool. Thegeometry of the jet is in turn dependent on that of the weir.Realistic values for KL , and hence r and E, can be found onlyby measurement, although they may be extrapolated to similarsituations if the mechanisms of oxygen transfer are properlyappreciated.

terms of oxygen transfer efficiency E, as defined in (4), whichis used in this paper when it is more appropriate

The mechanisms by which air is entrained and transferredinto water because of a free falling jet are several and complex.Nevertheless, if we are to understand fully the aeration per­formance of weirs, it is necessary to identify these mechanismsand relate them to characteristics of that performance.

Gameson (1957) demonstrated three phases of oxygen trans­fer for a free falling water jet plunging into a receiving pool.These are (1) in the jet itself during fall-this is regarded assmall; (2) via the free surface of the receiving water-thisdepends on the intensity of surface agitation in the receivingpool; (3) in the air-water biphasic flow that exists within thereceiving pool because of air entrainment-this phase is gov­erned by the air entrainment rate and the bubble contact timein the water. Gameson observed that the last phase was themost significant contributor to the oxygenation process.

High-speed photography was used, during the early stagesof the current work, to help identify the influence of jet char­acteristics on the air entrainment processes. Interpretation ofthese photographs indicates that various air entrainment mech­anisms, which depend on the jet characteristics, are involvedas a water jet plunges into the receiving pool. Four majormechanisms were identified by Tsang (1987) during the pres­ent study and these are shown in Fig. 3. These mechanismsdirectly determine the air-water interfacial area and the resi­dence time of the entrained bubbles in the water body. Theoxygen mass transfer therefore would be governed by the pre­dominant air entrainment mechanism under the prevailing hy­draulic conditions. Ervine et al. (1980) previously reportedsimilar observations in their work on air entrainment by freefalling circular water jets.

Initially, water jets with relatively smooth surfaces issuedfrom the weir and entrained air mainly at the surface of thereceiving pool according to the type A mechanism. As the dropheight increased, the surface of the jets first became roughenedand then the jet oscillated during fall, entraining air as de­scribed, respectively, by type B and C mechanisms. This re­sulted in greater air flow into the receiving pool. With increas­ing drop height, the jet eventually would break up into discretedroplets and the type D air entrainment mechanism prevailed.The breakup of the jet reduced its penetration depth into thepool and hence also the depth of the biphasic zone. This ef­fectively reduced contact time te between the bubbles and thesurrounding water, and so aeration by mechanism D was ob­served to be less effective than by mechanisms B and C. Thisconfirmed the above observations of Gameson (1957). Itshould be noted that the "breakup length" of the jet (i.e., the

AERATION AT WEIRS AND OVERFALLS

Oxygen Transfer Process

Oxygen transfer and dispersion into water essentially is gov­erned by the processes of molecular diffusion, turbulent mix­ing, or both. At the water surface it requires a difference inactive partial pressures between the air and the water. Lewisand Whitman (1924) suggested that two laminar films or layerslie on either side of the air-water interface. Both films offerresistance to the passage of the oxygen molecules into thewater. However, for a slightly soluble gas, such as oxygen inwater, the resistance of the waterside is very much the greaterof the two, and so this effectively controls the transfer process.The mass transfer rate (dmldt) of gas molecules across aninterface is known to be proportional to the concentration gra­dient across the interface, and this can be expressed as

dm dCdi = Vdi = kLA(Cg - C)

where kL = bulk liquid film coefficient. The values of Cg andC are the saturation concentration of oxygen in water at pre­vailing ambient conditions and the actual concentration of ox­ygen in the water at time t-the difference being proportionalto the concentration gradient. The term A is the air-water con­tact area and V is the volume of water associated with this. Inquiescent conditions A would represent the water surface area.The dispersion of oxygen into the water then would be con­trolled by the rate of molecular diffusion, which is relativelyslow, and a concentration gradient would develop in the waterdownward from the surface. In terms of (1), this can bethought of as a restriction of the effective volume V to a layernear to the surface. This limitation in V would result in a rapidrise in the oxygen concentration C near the surface and a con­sequent rapid reduction in the mass transfer rate. However, ifthe water is agitated then turbulent mixing will disperse theoxygen through the body of the water and this is a far morerapid process than molecular diffusion. Depending on the de­gree of agitation and mixing, V can be taken as a much largerproportion of the water volume. Indeed in many cases oxygenconcentration can be assumed as evenly distributed throughoutthe whole water volume. In agitated conditions, the value ofthe air-water interface area A also becomes ill defined, al­though it would be greater than the surface area in quiescentconditions. Uncertainties in the values of A and V often makeit convenient to think in terms of a combined transfer coeffi­cient, KL = kLAIV, which can be taken as applicable to a par­ticular set of hydraulic conditions. Assuming that KL is con­stant over the contract time teo integration of (1) gives

(Cg - C,) (kLAte )(C

g- C

2) = exp V = exp(KLtJ

where C j and C2 = concentrations of oxygen in the water atthe start and end of the contract time te • Although other the­ories such as Higbie (1935), Danckwerst (1951), and Dobbins(1956) offer alternative models to the two-film theory, leadingto different theoretical values for KL , there is nevertheless gen­eral agreement on a formulation similar to (2) to describe theoverall process.

If (2) is considered to be integrated across a weir then theleft side becomes the ratio of the upstream and downstreamdissolved oxygen (DO) deficits and we have

The ratio r is called the deficit ratio and it can be used toquantify the oxygen uptake at weirs. However, the aerationperformance of weirs often is expressed more conveniently in

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~\111/ smooth Jet~, ~_\'~ ~~::~ of

I \ I Smooth receIVing _ /---==t ,,"';.... -pooisurface ". l> J'(I"., "\()'o. • •• II ',. .

. \~(-, '.\~:~ L~rg~ b~bble spatial Well d¢ined '.1,1 ~.~,.: • ·t, • :.I,' dlstnbutlon biphaslc zones ~ !,.~)t,." •• I., ''':'.,;r.,'.';i • '•.. ~ ---Deep bubble I ~ ••~',

: '.' • • penetration depth .: •

Type A

Type B

100

95

l908

E

I 85

c 80875

70

0 20 40 60 80

Upstream 00(%)

TypeC

Type D

FIG. 3. Labyrinth Weir Aeration Mechanisms [after C. C. Tsang(1987)]

Discrete Air pockets

O-waterdroplets---- enfgulled by

_____ ' ---. surtace rollerSplashing of 0 " /'

dO. 0 poolsurfilce .; <>.

_---.A ~•..: '. Intense SUrface_~~V'-./\-"':-~ -- V - '--""\.."" - agitation 61 • •

: €V ,', Entrained air bubbles --"'~ .•• ------with shallow penetration -

depth

Cd(%) =0.289Cu(%) + 69.53

By comparison with (5), this gives values for oxygen trans­fer efficiency E of 0.711 and for C. of 97.8%, confirming thatthe oxygen transfer efficiency is sensibly independent of theupstream DO deficit. It also reinforces the use of oxygen trans­fer efficiency as a useful indicator of the aeration behavior ofstructures.

However, it should be noted that if the upstream DO valueapproaches saturation then the oxygen transfer efficiency willbecome indeterminate and meaningless as a measure of theaeration performance. Thus upstream DO deficit was main­tained at more than 20% for all laboratory tests described inthis paper.

Temperature

Comparative evaluations of oxygen uptake at weirs requirethat oxygen transfer efficiency is corrected to a reference tem­perature, most commonly 20oe. To obtain a correction for­mula, a series of preparatory tests, described by Tsang (1987),were designed to study temperature dependence by deliveringhot water into the hydraulic system, which was blended withcold tap water to achieve test temperatures ranging from 14to 23°C. Three drop heights for the overfall jet were used inthe experiments, namely, 850, 1,050, and 1,150 mm. The datawere then tested against available correction formulas as de­scribed in the following.

A number of empirical equations have been proposed todescribe the dependence of oxygen transfer efficiency on tem­perature. Gameson et al. (1958) suggested an equation of theform

In(1 - ET) = [1 + a(T - 20)] (6)

In(1 - E 20)

where ET and E20 = oxygen transfer efficiency values at TOCand 20oe, respectively. Gameson et al. (1958) obtained a valueof 0.0165°e- 1 for the constant a from a series of readings ona weir, with drop heights ranging from 0.6 (2 ft) to 3.3 m(0.75 ft) falling into a receiving pool 88 mm (3.5 in.) deep.

Tebbutt et al. (1977) used an equation of the following formto correct their readings for aeration on stepped spillways. Thevalue of a they used was 0.0335°C- 1

•

(r - 1) E (1 - E )T = T 20 = [1 + a(T - 20)] (7)

(r20 - 1) E20(1 - ET)

Holler (1971) proposed an exponential equation of the form

In(1 - ET ) _ (T-20) (8)In(1 - E 2O) - 'Y

FIG. 4. Relationship between Upstream and Downstream DOConcentrations

(5)

difference in level between the weir sill and the point ofbreakup) is not at all well defined and the jet breaks up overa considerable length. Thus the change from mechanism B orC to mechanism D is not sudden and takes place over a rangeof drop heights. It does not entail a reduction in aeration ef­ficiency, but a significant decline in the rate of increase inaeration efficiency with drop height.

FACTORS AFFECTING DEFICIT RATIO ANDAERATION EFFICIENCY

Upstream DO

Eq. (3) indicates that DO deficit ratio r and hence oxygentransfer efficiency E should be independent of the upstreamDO value Cu' However, the analysis is a very simplified andidealized description of the complex flow and aeration mech­anisms that occur when a jet plunges over a weir into a bodyof water. The writers therefore thought it prudent to confirmthe independence of oxygen transfer efficiency and upstreamDO level, prior to the main body of tests.

A set of readings was taken of deficit ratio for a model linearweir, with 320 mm sill length, under constant drop height,discharge, pool depth, and temperature conditions. The up­stream DO concentration Cu was varied over a range from 0to 80% of its saturated value and variation in the downstreamDO value Cd was noted. The results are illustrated in Fig. 4,which shows a linear relationship between Cu and Cd' A re­lationship between Cu and Cd may be derived from (4) as

Cd =(1 - E)Cu + EC.

The best-fit line from Fig. 4 is

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where 'Y = a constant, which Holler determined as 1.0241,using aeration data from experiments on a stirred containerprovided by Elmore and West (1961).

Gulliver et aI. (1990), using a semiempirical approach in­corporating fluid density, viscosity, and surface tension, pro­vided an equation of the form

In(1 - E)In(l _ E:) = 1.0 + ex(T - 20) + I3(T - 20)2 (9)

The values of ex and 13 given by Gulliver et aI. (1990) were0.02103°C- 1 and 8.621 X 1O-~oC-2, respectively. It should benoted that Gulliver's values were obtained from standard tem­perature variations in water properties and not from fitting toobserved aeration data.

The writers considered all of the foregoing forms of thecorrection formula and found that (7), with a best-fit ex valueof 0.0355°C- 1

, gave the least standard error of estimate (SEE)for the experimental data, equal to 1.249 X 10-2. Eq. (6) gavean SEE of 1.334 X 10-2 with a best-fit ex value of 0.0222°C- 1

•

The formulas of Holler et aI. (1971) [see (8)] and Gulliver etaI. (1990) [see (9)] both fitted the data very well, without anyadjustment of the parameters, giving SEE values of 1.493 X10-2 and 1.348 X 10-2, respectively. The writers chose to cor­rect their data according to (7) although, over most of thetemperature range used in the experiments, the corrected £20

values derived from each of the equations differed by less than1%.

Receiving Pool Depth

The residence time of entrained air bubbles in a water bodydirectly affects the oxygen mass transfer. This residence timeis related to the bubble flow path and hence the bubble pen­etration depth into the receiving water. Work by Jarvis (1970),Apted (1975), and Avery (1976) has demonstrated that "op­timurn" aeration can be achieved when flow paths of entrainedair bubbles in the receiving water are uninterrupted, Le., whenthe pool depth is greater than the bubble penetration depth (seeFig. 3). For consistency, all tests reported in this paper werecarried out under these conditions.

Water Quality

The presence of surface active agents, organic substances,and suspended solids in water have all been observed to affectthe aeration process. Surface active agents in particular appearto modify the process by reducing surface tension, formingdiffusion inhibiting films at the air-water interface, and affect­ing the hydrodynamic characteristics of the flow. The affect ofwater quality often is generalized by the use of a "water qual­ity factor" in equations for the deficit ratio, for instance Ga­meson (1957) and Markofsky and Kobus (1978). Avery andNovak (1978) used a similar constant to allow for the effectsof different concentrations of sodium nitrate in their water.

Tap water was used for all of the experiments reported inthis paper. Salt content was consequently low and was moni­tored constantly during the experiments to prevent any buildupof residues caused by the deoxydant chemicals added to thewater. Therefore the presence of chemicals or pollutants doesnot effect the results.

EXPERIMENTAL SETUP

Fig. 5 shows the laboratory arrangement for the currentstudy. The main recirculating system allowed water to bepumped from a large sump into a header tank where the leveland hence the flow over the test weir could be set by adjustingan overflow weir. The design of the header tank enabled quickand easy replacement of weirs for testing. The water jet fromthe test weir plunged into a receiving tank, whose height couldbe adjusted using a pulley arrangement. The water depth inthe receiving tank was controlled by an adjustable weir. Thewater then discharged through a pipe into a flow-measuringtank before returning to the sump.

The recirculatory system allowed the DO level of the waterarriving at the model weirs to be varied easily by controlleddosing of sodium sulphite in the presence of cobalt chlorideas the reaction catalyst. The dosing rate was regulated usinga variable-speed peristaltic pump to maintain an upstream DOdeficit of more than 20% without causing residues of excesschemicals to build up in the water. A circular plastic distributorwas installed in the end of the dosing pipe, beneath the foot­valve of the main recirculatory pump, to ensure good mixing

Honey-comb baffle r-------------------~

Deoxgenationchemical input

Release 1vale

Weir overfall

To DIS DO meterAdjustable heightreceiving tank

Flow measuring tank

Underfloor sump

Flexible hose

.........,.....~~Test weir} To UIS DO meter

Weir tank and header channel

, """-----,'----J..--...;.,

\

Flowcontrol

FIG. 5. Laboratory Weir Aeration Apparatus

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of the chemicals with the water. A 12.5-m-Iong Kenaflex hoseconnected the footvalve to the main pump, thus allowing suf­ficient time for the deoxygenation reactions to take place andto prevent the buildup of sodium sulphite residues in the water.The presence of any excess sodium sulphite was monitoredwith a potassium iodide indicator.

The DO levels in the flow were measured by membraneelectrodes (pHOX systems model IC660) based on a temper­ature-compensated Mackereth-type galvanic cell design. Cali­brated flow cells were used to house the electrodes. These cellswere designed to maintain sufficient flow around the elec­trodes, thereby avoiding local oxygen depletion and preventin.gthe formation of bubbles in the flow. Water samples were SI­

phoned from two points in the header tank upstream of theweir and from five points at different depths in the receivingtank. The individual upstream and downstream samples weremixed in two separate manifolds before being fed through totheir respective DO sampling flow cells.

EXPERIMENTAL PROGRAM

The dimensions of the weirs tested are given in Table 1.Each weir had a total crest length of 320 mm. In the first seriesof tests, weirs H1-H5 had horizontal sills with varying in­cluded angles. Weir L6 is, of course, a linear rectangular weirand was used as a control and for comparison with other pub­lished results. In the second series of tests, weir sill slope wasvaried with the included angle kept constant at 60°. WeirsS7-S9 had sloping sills with their levels increasing in thedownstream direction. Weirs SlO-SI2 had decreasing sill lev­els in the downstream direction.

The model weirs were built out of I-cm marine plywood inthe form of detachable units, which could be substituted easilyin the test rig. One-millimeter-thick aluminum plates were fit­ted along the whole crest length to ensure sharp-edged con­ditions. These were beveled at 45° on the downstream side toensure that overflow jets sprung clear at all flows.

Each weir configuration was tested under flow rates Q vary­ing from approximately 1.0 to 4.0 Lis. The drop height Hd ,

defined as the difference between the water levels upstreamand downstream of the weir, was varied between 700 and1,500 mm. The depth in the receiving tank was maintainedthroughout at greater than the bubble penetration depth to en­sure optimum aeration conditions.

RESULTS AND ANALYSIS

Linear Weir {L6)-Comparison with Published Data

A number of empirical equations have been published giv­ing deficit ratio relationships for linear sharp-crested weirsbased on either field or laboratory data. The writers have con­verted these to give oxygen transfer efficiency according to

TABLE 1. Experimental Program and Details

Half·crest Breadth Width SillWeir length b Included d w slopeno. (mm) angle a (mm) (mm) <l> Remarks(1 ) (2) (3) (4) (5) (6) (7)

HI 160 30 154.5 83.0 0 Series 1-horizontalsills

H2 160 45 147.8 122.5 0H3 160 60 138.6 160.0 0H4 160 90 113.1 226.3 0H5 160 120 80 277.1 0L6 160 180 0 320.0 0 Linear weirS7 160 60 138.6 160.0 5.7 Series 2- sloping si1lsS8 160 60 138.6 160.0 11.3S9 160 60 138.6 160.0 21.8SIO 160 60 138.6 160.0 -5.7SII 160 60 138.6 160.0 -11.3S12 160 60 138.6 160.0 -21.8

(4). Gameson (1957) and Gameson et al. (1958) first reportedon the aeration characteristics of river structures and proposeda formula for oxygen transfer efficiency Er at water tempera­ture TOC, which can be expressed in metric units as

Er = 1 - [1 + 0.361a,.bw(1 + 0.046T)H,J-l (10)

The U.K. Water Research Laboratory (WRL) (1973) lateramended Gameson's equation to

Er =I - [I + 0.038awbw(1 + 0.046T)Hd(1 - O.OllHdWI (11)

where aw = a water quality factor, which has a suggested valueof 1.8 for clean water in the WRL equation; and bw = anaeration coefficient. Butts and Evans (1983) applied this re­lationship to a number of river structures and found bw valuesbetween 0.79 and 1.3 for sharp-crested weirs. The writers haveadopted a bw value of 1.0 suggested by the WPRL for labo­ratory sharp-crested weirs. Although no allowances were madein either of these relationships for the flow rate or receivingpool depth, these factors were later recognized to have someinfluence on the aeration process.

Nakasone (1979, 1987) proposed the following empiricalequation, based on both field and laboratory data, to describethe aeration performance of weirs and cascades:

E20 =1 - exp[-aH~q~d~] (12)

This equation accounts for the flow per unit weir length qwand the depth of the receiving pool d•. The coefficients a, p,'Y, and 8 were defined for different ranges of flow and dropheight, and their values reflect the greater dependence of E20on drop height rather than either flow or pool depth. It shouldbe noted that the data generally were collected for cases wherethe receiving pool depth was less than the optimum bubblepenetration depth, and so its inclusion in the equation is nec­essary.

Avery and Novak (1978) studied the aeration behavior inthe laboratory of four rectangular weirs with crest lengthsranging from 100 to 300 mm, under a range of flows between1.0 to 5.0 Lis and drop heights from 800 to 2,450 mm. Theyderived the following equation in terms of a weir Froude num­ber Fw , and a weir Reynolds number, Rw :

(13a)

or

(13b)

where

Fw= (gH:)O'2~ and R

w=&

2qp v

where qp = discharge per unit perimeter of the jet at impactwith the receiving pool. This equation applies to solid jetsonly.

Comparisons are shown in Figs. 6(a and b) between (11),(12), and (13) and the observed values of r20 and E2O , respec­tively, for the linear weir (L6 in Table 1). Both of these com­parisons have been shown in order to highlight the inherentdanger of using equations derived on the basis of deficit ratioto calculate aeration efficiency. It can be shown from (4) that,ignoring terms of '<?O(Arlr)2 in the binomial expansion of {I+ (Arlr)}-', the percentage error in aeration efficiency is re­lated to that in deficit ratio by

[t:.E] % =_1_. [Ar] % (14)E (r - 1) r

This shows that a percentage error in r leads to an increasingpercentage error in E as r becomes smaller and nearer to unity.

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4 -r------------------.,

errors in E for smaller values of both r and E. This is clearlythe case here, where errors in r20 [Fig. 6(a») are magnified atthe lower end of the range and reduced at the upper end whentransposed into E 20 values [Fig. 6(b»). The writers have usedaeration efficiency throughout this paper, so the foregoingcomments need to be borne in mind when comparing the per­formance of weir aeration formulas.

The Avery and Novak (1978) equation provides the best fitfor the linear weir data over the experimental range of dis­charges and drop heights. The standard error for E20 is 0.011,thus the Avery and Novak equation will give an E 20 valuewithin the range of :::t:0.022 of the observed value with 95%probability. The WRL equation in (11) does not take accountof variation in discharge. This clearly affects the calculatedE20 values, which it increasingly underestimates with higherdischarge and drop height. Nakasone's (1987) equation in (12)grossly underestimates the E20 value and this again increaseswith depth and discharge. One explanation for this is that theexperimental weir width Bw is 0.32 m whereas the upstreamchannel width B is 0.65 m giving a BwlB ratio of 0.49. As theBwlB ratio decreases, the relative narrowing of the weir causesthe overflow streamlines to distort with a consequent narrow­ing and concentration of the overfall jet. This leads to a greaterpenetration depth of the jet into the receiving water, thus in­creasing aeration in the downstream pool. The Nakasone(1987) equation was developed from data for relatively sup­pressed weirs and so will underestimate aeration from narrowweirs. This underestimation would be expected to increase asthe discharge increases leading to greater bending of thestreamlines. Such an effect is seen clearly in Fig. 6. This in­crease in aeration with narrowing of the weir was observed byboth Nakasone (1987) and Van der Kroon (1969a,b). Byadopting the value of discharge per unit length qp at the pointof impact, the Avery and Novak (1978) formula automaticallyaccounts for the narrowing of the overfall jet. However, it ismore difficult in use than the discharge per unit weir lengthqw used by Nakasone.

Horizontal Sill Labyrinth Weirs (H1-H5, L6)

This series of tests was carried out with triangular labyrinthweirs all having the same total sill length of 320 mm but withthe inclined angle 6 varying from 30 to 120°. The linear weirL6, with included angle of 180°, completes the series.

The aeration performance of these weirs is expressed interms of the aeration efficiency E, which is shown to vary withdrop height H d and discharge Q in Figs. 7(a-f). All of thesegraphs show an increase in aeration efficiency with dropheight, irrespective of the included angle 6. Generally, a biggerdrop height leads to greater bubble penetration depths into thedownstream pool and longer contact times te• This in turn in­creases DO deficit ratio and hence aeration efficiency as in­dicated by (3) .

On the other hand, the way in which aeration efficiencyvaries with discharge changes significantly as the includedweir angle 6 becomes smaller. In Figs. 7(e and f), for weir H5(6 = 120°) and the linear weir L6, respectively, the aerationefficiency is reduced as the discharge increases over the wholerange of drop heights tested. Generally, this is observed inlinear weirs and is predicted by Avery and Novak's (1978)formula in (13). However, Figs. 7(a-d) show that as the valueof 6 becomes smaller, this relationship is increasingly reversedfor higher drop heights, and aeration efficiency becomesgreater as the discharge is increased. The reason for this re­versal can be found by consideration of the jet characteristicsin terms of the four aeration mechanisms identified earlier andshown in Fig. 3. With the larger included angle 6, the jet re­mained generally solid and coherent over the whole range ofdrop heights tested. Aeration was therefore mainly by mech-

4

0.8

A AA

••

•

3

0.6

2

0.4

••

•• f AA A A

A A AA AA

A A A

A~. AAAA

AI AA.4A :A

• • ••• ••• •• ••• • • • • • •.. --..... . .... : ...

Observed Deficit Ratio r20

• Avery & Novak

• Nakasone

A WPRL

--- Line of Equality- ----_.__._-

o

•

0.8.,-----------------"

3

: • Avery & Novak

• Nakasone

0.6A WPRL

--Line of Equality0

~>-UI: AGIU

ffiI: • •0 0.4 •! • •GI • •<C •'tl • •GI •'Ii • •:i • ••.!:! • • •III • •0 • • •• •

0.2 • • •• •• •• ••

Thus 10% errors in r values of 1.5 and 4.0 (which are nearthe extremes of the observed range) lead to 20 and 3.3% errorsin E, respectively. Because (11)-(13) were all derived to op­timize r values, it follows that they would tend to give larger

Observed Aeration Efficiency E20FIG. 6. Comparison of Aeration Formulas for RectangularWeir: (a) DO Deficit r2Q; (b) Aeration Efficiency E••

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0.8WelrHl

Included engle = 30075

0.7

~0.65c

Iw 0.6c0

~ 0.55~ ..... 1.01 lis

""'2.12 lis0.5

....... 3.04 lis 1

0.45__ 4.01 lis

04

600 800 1000 1200 1400 1600

Drop Height (mm)

0.8 _._----_ ..__.--

WelrH2Included Angle =45

0.75

0.7

'"g 0.65.!!.2iii 0.6c

i 0.55C

1 ..... 1.01 lis

0.5 ..... 2.04 lis' ....... 3.00 lis

0.45 !~4.00Vs

0.4600 800 1000 1200 1400 1600

Drop Height (mm)

1600

..... 1.021/..,

..... 2.00 lis I

"""'3.11 lis__ 3.96 lis

140012001000800

0.4 1-------+----->--------+-----------1

600

0.75

0.8 -.------------.-------WelrH4

Included Angle =90

0.5

0.45

0.7

~.~ 0.65

isw 0.6

:5i 0.55C

0.8WelrH3

0.75 Included Angle = 60

0.72.10

'".i 0.651.01

"E .. Break-up Lengthsw 0.6c0,..e 0.55~ ..... 1.01 lis

0.5 t 4.01 ..... 2.10 lis~ -b- 3.00 Us I

0.45 3.00 ,--4.0~

0.4600 800 1000 1200 1400 1600

Drop Height (mm) Drop Height (mm)

1 ~1.20l/s1..... 2.01 lis

"""'3.12 lis 1

__ 4.28 lis

0.75 ------..-~-----.--.--Linear Weir L6

0.7 Included Angle = 180

0.35

0.65

0.4

i)' 0.6

i.2 0.55iii.§ 0.5

! 0.45c;.: 1.03 lis I: 2.01 lis,1 3.01 lIs 1

--4.00 lis0.45

0.75 . ----------.-.-----.~.-~WelrH5

Included Angle = 1200.7

0.65

'""c"~

0.6

wc0 0.55~"C 0.5

0.4 I---~------>----------->-------'

600 700 800 900 1000 1100 1200 1300 1400 1500 1600

Drop Height (mm)

0.3 !-----------+------.---------+-----+-------;600 700 800 900 1000 1100 1200 1300 1400 1500 1600

Drop Height (mm)

FIG. 7. Variation In Aeration Efficiency with Drop Height and Discharge for Horizontal Triangular labyrinth Weirs over a Range of In­cluded Angles: (8) Weir H1, Included Angle =30; (b) Weir H2, Included Angle =45; (c) Weir H3, Included Angle =60; (d) Weir H4, In­cluded Angle =90; (e) Weir H5,Inciuded Angle =120; (f) Weir L6,Inciuded Angle =180

anisms A, B, or C in Fig. 3, and the amount of air entrainedper unit volume of flow decreased as the discharge rate in­creased. However, as the included angle became smaller therewas an increasing tendency, caused by the coIlision of adjacentoverfalls, for the combined jet to break up and fragment intodiscrete droplets. This led to aeration mechanism D as shownin Fig. 3. The breakup length of the jet was observed to be­come shorter for lower discharges, in line with the observa­tions of Horeni (1957) and also for smaller included angles O.A similar effect can be observed readily when pouring waterfrom a jug spout. As discussed earlier, the onset of jet breakupand aeration mechanism D results in a reduction in the rate ofincrease of aeration efficiency with drop height. This is illus­trated in Fig. 7(c) for weir H3 (0 = 600

) where approximatebreakup lengths are indicated. Because the observed breakuplengths are shorter for lower discharges, mechanism D be-

comes effective at lower drop heights than for higher dis­charges, leading directly to the reversal of aeration efficiencyvariation with discharge as shown in Fig. 7(c).

Fig. 8 shows the variation in aeration efficiency with in­cluded weir angle for three drop heights, 0.7, 1.1, and 1.5 m,and two discharges, 1.0 and 4.0 Lis, which have been inter­polated from the observed data. This shows an increase inaeration efficiency as the included angle becomes smaller forall drop heights at the higher discharge. Little or no breakupof the jet was observed at this higher discharge for any in­cluded angle 0, thus aeration mechanism D did not occur.Moreover, it was noted that as the included angle becamesmaller, the overfall jets from the two adjacent halves of theweir increasingly collided, and this possibly led to greater mix­ing in the combined jet and increased air entrainment. Thecombined jet also was observed to become more coherent or

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Weir Angle

FIG. 8. Variation in Aeration Efficiency of Horizontal Sill Weirswith Included Weir Angle for Different Flow and Upstream HeadValues

0.9

o

O.B

150

0.70.6

-+-. Hd=O.7m:Q=1.0I/s- Hd=O.7m:Q=4.01/s- ... Hd=1.1 m:Q=1.0I/s- Hd=1.1 m:Q=4.0I/s- 0-. Hd=1.5m:Q=1.0I/s____ Hd=1.5m:Q=4.0I/s

0.50.4

._," .... -. -.-...-

40

·10.1----------------.....

0.9·r---------------------~

50~--------------...,

ability that the equation will predict E20 within ::!:O.0476 of itstrue value. This equation has a similar dependence on the dropheight Hd to that of Avery and Novak (1978) in (13). However,it shows less dependence on flow than Avery and Novak,which partly is caused by the fact that their relationship isexpressed in terms of discharge per unit length at impact qpand also that it was developed only for solid jets (i.e., nobreakup). The formula shows that the oxygen transfer effi­ciency should increase as the included angle e decreases,which is in line with the observations. Comparisons betweencalculated and observed values of the aeration efficiency E20

shown in Fig. 10 indicate that this equation tends to overes­timate E 20 values below approximately 0.55.

Weir Angle

FIG. 9. Percentage Difference In Aeration Efficiency betweenHorizontal Sill Labyrinth Weirs and Equivalent-Length LinearWeir for Different Flow and Upstream Head Values

ObMrYed Aemlon Eflk:lency E20

FIG. 10. Comparison of Observed Aeration Efficiency E20 Val­ues with Those Calculated from (14)

180

-'-

150120

;~1iJ ~ - -_ .. _ .. _ .. _

90

............... _._.-. .....

60

.... ~.\

\ .. ,

0.4 I_+-' Hd=O.7m:Q;;;1-:01tSl_ Hd=O.7m:Q=4.0I/s I

l-..·Hd=1.1 m:Q=1.0I/sl

0.35 _ Hd=1.1 m:Q=4.01/s-0-' Hd=1.5m:Q=1.0I/s---- Hd=1.5m:Q=4.0I/s

0.3 ~===+====+===+-----1f---~30

0.8

0.75

0.7

0.65

~ 0.6cGlUlEW 0.55c0

~~ 0.5

0.45

"compact" as the included angle became smaller, resulting insomewhat greater penetration and bubble contact time in thereceiving pool. At the lower discharge, on the other hand,breakup of the jet was observed as the drop height increasedand this occurred at smaller drop heights as the included anglereduced. This explains the relative lack of improvement inaeration efficiency for smaller included angles at the lowerdischarges with higher drop heights. However, at the lowerdrop height (Hd = 0.7 m), breakup did not occur to the sameextent and both the lower and the higher discharges exhibiteda similar improvement in aeration efficiency with reduction inincluded angle. Fig. 9 shows the percentage difference be­tween the aeration efficiency for the labyrinth weirs and thestraight weir. It is clear that as the included angle becomessmaller there is a significant increase in aeration efficiencyexcept for the higher drop heights and lower flows. This figureindicates that the aeration advantages of labyrinth weirs gen­erally increase with discharge and decrease with drop height.

Two empirical approaches were adopted in order to derivea relationship for calculating the aeration efficiency in termsof drop height Hd , discharge Q, and weir included angle e. Inboth cases all of the 216 readings for weirs H1-H5 and L6were used. The first of these involved a simple multiple re­gression equation based on all of the available data. This gavethe relationship

£20 = 1 - [1 + 1.48H~·3'Q-O.l31 (sin ~r0342rl (15)

where the drop height H d is in meters, the discharge Q is inliters per second, and the included angle e is in degrees. TheSEE for this formula was 0.0238, which indicates a 95% prob-

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Eq. (15) was based on Avery and Novak's (1978) fonnulaand was obtained in order to detennine the effect of the in­cluded angle on this well used relationship. However, an al­ternative empirical equation that more closely fits the observeddata is given in the following equation:

from the fact that (15) and (16) were both originally derivedin tenns of the oxygen deficit ratio r20' Optimizing on r20 willlead to an increasing spread of £20 values in the lower rangesas described earlier. Nevertheless the comparisons do indicatea degree of overestimation of both r20 and £20 at lower values.

0.35

0.3 '.:::... -_- --_- --l

~ ~ u ~ M ~ M ~ U ~ M

ObHfved A8n1t1on Etnclency DO

FIG. 11. Comparison of Observed Aeration Efficiency Eao Val­ues with Those Calculated from (15)

where units are as in (15). The SEE for this equation, from all216 readings, is 0.0183, indicating a 95% probability of theequation giving an £20 value within ::!::0.0366 of that observed.It is interesting to note here that the nonlinear dependency onthe drop height H d is all contained in the index of the flow Q.The comparison between the observed and calculated £20 val­ues using (16) is shown in Fig. 11. This shows a better fit than(15) at higher aeration efficiency values, but the tendency tooverestimate £20 at values less than approximately 0.55 re­

mains. The relatively large error in the lower £20 values results

Sloping Sill Labyrinth Weirs (S7-S12, H3)

This series of tests was carried out with triangular labyrinthweirs all having the same total sill length of 320 mrn andincluded angle e of 60°. However, the sill sloped at a seriesof angles <P, as shown in Table 1, where <P is defined in Fig.2. Positive values of <I> indicate increasing sill level in thedownstream direction and vice versa. Weir H3 completes theseries as the appropriate horizontal sill weir.

The variation in aeration efficiency with sill slope <I> isshown in Fig. 12. It is apparent from this graph that weirs witha decreasing sill level in the downstream direction (i.e., neg­ative <I> values) have a lower aeration efficiency than level sillweirs. This is caused by the fact that a major proportion ofthe flow falls in two colliding jets over the downstream ex­tremities of the weir sill, as illustrated in Fig. 13(a). It hasbeen observed earlier that when the colliding jets flow overthe upstream of the weir (i.e., near to the included angle), theytend to coalesce into a more compact and solid combined jet,which actually improves aeration efficiency. However, colli­sion of the more remote jets from the downstream extremitiesof the weir appears to cause their premature breakup. Thisleads to a lowering of the aeration perfonnance of the weir.This perfonnance only becomes more comparable with thehorizontal weir at higher drop heights simply because the hor­izontal weir itself becomes more prone to breakup of the jet.At low drop heights the reduction in aeration efficiency of thenegatively sloping weirs is more noticeable at higher flows.This is because, as pointed out earlier, the jet breakup length

(16)

0.8

0.75

!I 0.70.86

0.6

I 0.55

0.5

I0.45 ..0.4

0.65,/

~III

I

I i

\ II \\ \

~ Flow direction

Flow predominantly 'concentrated'

~']~'''~ ..,",..-- - .....~:~-..

Plans

~FIOW direction ~

/Flow predominantly over downstreamextremities of weir leading to more remotejets and premature break·up of combined jet.

I

.--Hd=O.7m:Q=1 Va I

I

..... Hd=O.7m:Q=4Vs

-+- Hd=1.5m:Q=1 Vs II-Hd=1.5m:Q=4Vs

0.35

§'•~c

i

Elevations-60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60

DIS - UIS Sill Levels (mm)

FIG. 12. Variation in Aeration Efficiency with Sill Slope forWeirs with 600 Included Angle

a) Negative 4Jvalues b) Positive 4J values

FIG. 13. Sketch of Flow Patterns over Sloping 5111 TriangUlarLabyrinth Weirs

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increases with flow. Therefore there is less tendency for jetbreakup with the horizontal sill weir at higher flows and sothe relative reduction in aeration efficiency of the negativelysloping weirs due to premature breakup of the opposing jetsis increased.

Where the sill level increases in the downstream direction(i.e., positive <I> values) there appears to be little effect on theaeration efficiency except for a noticeable improvement forhigher flows at low drop height. Observations of the jet indi­cated that it was "concentrated" near to the upstream of theweir as shown in Fig. 13(b). In the case of higher flows at lowdrop height the jet did not break up and the additional con­centration of the jet led to an increase in bubble penetrationdepth in the downstream pool thus raising aeration efficiency.Where the jet reached breakup length, as in the other cases inFig. 12, the additional concentration did not translate into in­creased penetration depth and hence aeration efficiency.

It is apparent from the results in Fig. 12 that sloping theweir sill does not lead to significantly increased aeration effi­ciency, except in the case of high flows at low drop height.Sloping sill weirs generally would not therefore be recom­mended and consequently empirical equations describing theirperformance were not sought.

CONCLUSIONS

o A series of laboratory experiments have been carried outto measure the aeration performance of triangular laby­rinth weirs over a range of flows between 1 and 4 Liswith drop heights from 700 to 1,500 mm. The total weirlength was kept constant at 320 mm, while the includedangle 8 was varied between 30 and 180° (linear weir). Ina second series of experiments the longitudinal slope ofthe weir sill <I> was varied between ±21.8° (1 in 2.5) withconstant included angle of 60°.

o The performance of the linear weir (8 = 180°) was com­pared with three published formulas, Avery and Novak(1978) [see (11)], WRL (1973) [see (12)], and Nakasone(1987) [see (13)]. Avery and Novak (1978) gave the bestfit with an SEE in £20 of 0.011. Nakasone did not fit thedata well because it does not allow for adjustment of theoverfall jet geometry because of weir side contractions.

o The aeration efficiency increased with drop height in allcases.

o Decreasing discharge led to higher aeration efficiencyover the whole of the range of drop heights, provided weirincluded angles 8 were 120° and greater. However, forsmaller included angles, this variation of aeration effi­ciency with discharge was reversed at higher drop heightsand aeration efficiency increased with discharge. The rea­son for this is that the overflow jet had an increasing ten­dency to break up at lower discharges as weir includedangle 8 decreased. This led to reduced bubble penetrationinto the receiving pool and so reduced aeration efficiencycompared with the higher flows, where the overflow jetremained solid.

o The aeration efficiency for triangular labyrinth weirs hasbeen compared with that of a linear weir with the samesill length. The aeration efficiency increases as the in­cluded angle 8 becomes smaller down to 30°. The aerationadvantage of the labyrinth weir becomes greater at smallerdrop heights and larger discharges. For a drop height of700 mm and discharge of 4 Lis, a triangular labyrinth weirwith included angle of 30° and sill length of 320 mm hasan aeration efficiency 47.8% higher than the equivalent­length linear weir.

o Two empirical equations were derived in order to expressthe aeration efficiency of triangular labyrinth weirs interms of drop height, discharge, and weir included angle.

718/ JOURNAL OF ENVIRONMENTAL ENGINEERING / AUGUST 1998

Both of these equations indicated an increase of aerationefficiency as the weir included angle decreased.

o Triangular labyrinth weirs with a longitudinal sill slopedecreasing in the downstream direction were found tohave worse aeration performance than their equivalenthorizontal sill weirs. Where the longitudinal sill slope in­creased in the downstream direction, an improvement inperformance was only seen for higher flows at lower dropheights. It was concluded that triangular labyrinth weirswith longitudinally sloping sills in general offer no sig­nificant advantages over equivalent horizontal sill weirs.

o The writers have provided explanations for the aerationperformance of triangular labyrinth weirs in terms of ob­served overfall jet characteristics. To facilitate these ex­planations the complex aeration process was broken downinto four aeration mechanisms. More work is needed toidentify precisely and quantify these mechanisms, partic­ularly if the results are to be extrapolated to the design oflabyrinth weirs with different geometries or scale.

• Triangular labyrinth weirs have been shown to have asubstantially better aeration performance than their equiv­alent-length linear weirs. Moreover, this advantage be­comes more apparent at low drop heights. This wouldmake either single- or multiple-cycle triangular labyrinthweirs particularly useful in situations where DO levels inrivers need to be raised without large loss of head. Theycould be used to increase the oxygen levels of dischargesat outfalls into a river or as control and aeration structuresin the river itself. Particularly, in the latter case their ex­tended sill length, which enables them to transmit a largerange of discharge with limited upstream level variation,would be an additional hydraulic advantage.

APPEND~I. REFERENCES

Apted, R. W. (1975). "The mechanics of aeration at weirs with specialreference to river quality," MSc thesis, University of Newcastle-upon­Tyne, U.K.

Apted, R. w., and Novak, P. (1973). "Some studies of oxygen uptake atweirs." Proc.• XV Congr., Paper B 23, International Association forHydraulic Research, Istanbul, Turkey, 177-186.

Avery, S. (1976). "The transfer of oxygen from air entrained by jetsentering a free water recipient," PhD thesis, University of Newcastle­upon-Tyne, U.K.

Avery, S., and Novak, P. (1978). "Oxygen transfer at hydraulic struc­tures." J. Hydr. Engrg., ASCE, 104(11), 1521-1540.

Butts, T. A., and Evans, R. L. (1983). "Small stream channel dam aerationcharacteristics." J. Envir. Engrg., ASCE, 109(3), 555-573.

Cassidy, J. (1983). "Labyrinth crest spillway planning, design and con­struction." Int. Cont on Hydr. Aspects of Floods and Flood Control,City University, London, U.K.

Danckwerst, P. V. (1951). "Significance of liquid film coefficient in gasabsorption." Ind. and Engrg. Chern., June, 1460.

Dobbins, W. E. (1956). "The nature of the oxygen transfer coefficient inaeration systems." Biological treatment of sewage and industrialwastes, Rheinhold Publishing Corp., New York, N.Y. 141-183.

Elmore, H. C., and West, W. F. (1961). "Effect of water temperature onstream reaeration." J. Sanit. Engrg. Div., ASCE, 87(6), 59-71.

Ervine, D. A., McKeogh, E. J., and Elsawy, E. M. (1980). "Effect ofturbulence intensity on the rate of air entrainment by plunging waterjets." Proc., ICE, 69(2), 425-445.

Gameson, A. L. H. (1957). "Weirs and the aeration ofrivers." J. Inst. ofWater Engrs., II (6), 477-490.

Gameson, A. L. H., Vandyke, K. G., and Ogden, C. G. (1958). "Theeffects of temperature on aeration at weirs." Water and Water Engrg.,63, 489-492.

Gulliver, J. S., and Rindels, A. J. (1993). "Measurement of air-wateroxygen transfer at hydraulic structures." J. Hydr. Engrg., ASCE,I19(3),327-349.

Gulliver, J. S., Thene, J. R., and Rindels, A. J. (1990). "Indexing gastransfer in self-aerated flows." J. Envir. Engrg., ASCE, 106(3), 503­523.

Hay, N., and Taylor, G. (1970). "Performance and design of labyrinthweirs." J. Hydr. Engrg., ASCE, 96(I1), 2335-2357.

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Higbie, R. (1935). "The rate of absorption of a pure gas into a still liquidduring short periods of exposure." Trans, AlChE, 31, 365-389.

Holler, A. G. (1971). "The mechanism describing oxygen transfer fromthe atmosphere to discharge through hydraulic structures." Proc., XIVCongr., International Association for Hydraulic Research, Paris, France,373-382.

Horeni, P. (1956). "Disintegration of a free jet of water in air." Vy­zkumny Ustav, Vodohospodarsky Prace a Studie, Sesit 93, Praha-Pod­baba (in Czech).

Jarvis, P. J. (1970). "A study of the mechanics of aeration at weirs,"MSc thesis, University of Newcastle-upon-Tyne, U.K.

Lewis, W. K., and Whitman, W. G. (1924). "Principles of gas absorp­tion." Ind. and Engrg. Chern., 16(12), 1215-1220.

Markofsky, M., and Kobus, H. (1978). "Unified presentation of weiraeration data." J. Hydr. Engrg., ASCE. 104(4),562-568.

Nakasone, H. (1987). "Study of aeration at weirs and cascades." J. Hydr.Engrg., ASCE, 113(1), 64-81.

Taylor, G. (1968). "The performance of labyrinth weirs," PhD thesis,University of Nottingham, U.K.

Tebbutt, T. H. Y., Rajaratnam, S. K., and Essery, I. T. S. (1977). "Reaer­ation performance of stepped cascades." J. Inst. of Water Engrs. andSci., 31, 285-297.

Tsang, C. C. (1987). "Hydraulic and aeration performance of labyrinthweirs," PhD thesis, University of London, U.K.

Van der Kroon, G. T. N., and Schram, A. H. (l969a). "Weir aeration­part I: Single free fall." H 20, 22, 528-537.

Van der Kroon, G. T. N., and Schram, A. H. (l969b). "Weir aeration­part II: Steps, weirs or cascades." H 20, 22, 538-545.

Water Research Laboratory (WRL). (1973). "Notes on water pollution-aeration of weirs." Report. No. 61, Department of the Environment,Stevenage, U.K.

Wilhelms, S. C., Gulliver, J. S., and Parkhill, K. (1993). "Reaeration atlow head hydraulic structures." Tech. Rep. No. W-93-2, U.S. ArmyCorps of Engineers, Waterways Experiment Station, Vicksburg, Miss.

APPENDIX II. NOTATION

The following symbols are used in this paper:

A = area of air-water interface;C = concentration of oxygen in water;

C, = saturated concentration of oxygen in water;d, = receiving pool depth;E = oxygen transfer efficiency = (Cd - C.)/(C. - C.);

Hd = drop height of weir overfall jet;kL =bulk liquid film coefficient;Q weir discharge;qp = weir discharge per unit jet perimeter at impact;qw = weir discharge per unit sill length;

r = DO deficit ratio = (C, - C.)/(C. - Cd);tc = air-water contact time;V = aerated volume of water;

a, ~, 'Y = constants;9 = triangular weir included angle; and

I\> = weir sill slope.

Subscripts

d = downstream; andu = upstream.

JOURNAL OF ENVIRONMENTAL ENGINEERING / AUGUST 1998/719

J. Environ. Eng. 1998.124:709-719.

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