hydraulic design of urban drainage systems

7/27/2019 Hydraulic Design of Urban Drainage Systems

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CHAPTER 4HYDR ULCDESGNOFURB NDR N GESYSTEMSBenChieYenDepartment of Civil &Environm ental EngineeringUniversity of Illinoisat Urbana-ChampaignUrbana, Illinois

A Osman AkanDepartment of Civil and Envirom ental EngineeringO ld Domin ion UniversityNorfolk, Virginia

U INTRODUCTIONGenerally speaking urban drainage systems consist of three parts: the overland surfacef low sys tem, th e sewer ne twork , and the underground porous media dra inage sy s tem.Some elements of these components are shown schematically inFig.14.1. Traditionallynodesign is considered for the urban porous media drainage part. Recently porous mediadrainage facilities such as infiltration trenches have been designed fo rflood reduction orpollution control in cities with high land costs. For example, preliminary work on thisaspectofurban porous m edia drainage designcan be found inFujita (1987), Moritaetal.(1996), Takaaki and Fujita (1984) and Yen and Akan (1983). Much has yet to be devel-opedtorefine andstandardizeonsuch designs;nofurther discussiononthis undergrou ndsubject willb egivenin this chapter.From a hydraulic engineering viewpoint, urban drainage problems can be classifiedinto tw o types: (1) design and (2) prediction fo r forecasting or operation. The requiredhydrauliclevelof the latter isoften higher thanth e former.Indesign, adrainage facilityis to be built to serve a llfuture events not exceeding a specified design hydrologic level.Implicitly the size of the apparatus is so determined that all rainstorms equal to andsmaller than the design storm are presumably considered and accounted for. Sewers,ditches, and channels in a drainage network each has its own time of concentration andhence its own design storm. In the design of a network all these different rainstormsshould be considered. O n the other hand , inrunoff prediction the drainage app aratus hasalready b een bu iltorpredetermined, itsdimensions know n,andsimu lationof flow froma particular single rainstorm event is mad e for the purpose of real-time forecasting to beused fo r operation an d runoff control, or sometimes for thed etermination of the flow ofapast eventforlegal purposes.Thehydrologic requirements forthesetw otypesofprob-lems aredifferent. In thecase ofprediction, agiven rainstorm withi ts specific temporaland spatial distributions is considered. For design purposes, hypothetical rainstorms withassigned design return period or acceptable risk level and assum ed tempo ral and spatial


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F I G U R E 14.1 S c h e m a t i c o f c o m p o n e n t s o f u r b a n c a t c h -m e n t . ( F r o mM e t c a l f E d d y , I n c .e t a l . , 1 9 7 1 ) .

distributions of the rainfallare used. Table 14.1 lists some of these tw o types of designand prediction problems.In the case of sanitary sewers, fo r design purposes th e problem becomes th e estima-tion of the critical runoffs in both quantitya nd quality,from domestic, commercial, and

T A B L E 14.1 T y p e s o f U r b a n D r a i n a g e P r o b l e m s ( a ) D e s ig n P r o b l e m sT y p e

S e w e r sD r a i n a g e c h a n n e l sDetention/retentionstorage p o n d sM a n h o l e s a n dj u n c t i o n sR o a d s i d e g u t t e r sI n l e t c a t c h b a s i n sP u m p sC o n t r o l g a t e sor v a l v e s

Design P u r p o s ePipe s i z e (ands l o p e )d e t e r m i n a t io nC h a n n e l d i m e n s i o n sGeometric d i m e n s i o n s( a n d o u t le t d e s i g n )G e o m e t r ic d i m e n s i o n sG e o m e t r ic d i m e n s i o n sG e o m e t r ic d i m e n s i o n sC a p a c i t yC a p a c i t y

H y d r o I n f o r m a t i o nS o u g h tP e a k d i s c h a r g e , Q p f o rd e s i g n r e t u r n periodP e a k d i s c h a r g e , Q p f o rd e s i g n r e t u r nperiodD e s i g n h y d r o g r a p h , Q ( t )D e s i g n h y d r o g r a p h , Q ( t )D e s i g n p e a k d i s c h a r g e ,G,D e s i g n p e a k d i s c h a r g e ,G ,,D e s i g n h y d r o g r a p hD e s i g n h y d r o g r a p h

R e q u i r e d H y d r a u l i cL e v e lL owL ow tomoderateL ow to m o d e r a t eL owto m o d e r a t eLow to m o d e r a t eL owModerateto h i g hM o d e r a t e to h i g h


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TABLE 14.1 (continued) TypesofUrbanD rainage P roblems (b) PredictionP roblemsT y p e

Real-timeo p e r a t i o nP e r f o r m a n c ee v a l u a t i o nS t o r m e v e n ts i m u l a t i o n

F l o o d l e v e ld e t e r m i n a t i o nS t o r m r u n o f fq u a l i t yc o n t r o l

S t o r m r u n o f fm a s t e rp l a n n i n g

P u r p o s e

Real-timer e g u l a t i o no f f l o wS i m u l a t i o n f o re v a l u a t i o n o fa s y s t e mD e t e r m i n a t i o no f r u n o f f a ts p e c i f i c l o c a t i o n sf o r p a r t i c u l a r p a s to r s p e c i f i e d e v e n t sD e t e r m i n a t i o no f t h e e x t e n to f f l o o d i n gR e d u c e a n dc o n t r o l o f w a t e rp o l l u t i o n d u e t or u n o f f f r o mr a i n s t o r m sL o n g - te r m , u s u a l l yl a r g e s p a t i a l s c a l ep l a n n i n g f o rs t o r m w a t e rm a n a g e m e n t

H y d r o I n p u t

Predicted a n d / o r j u s tm e a s u r e d r a i n f a l l ,n e t w o r k d a t aS p e c i f i c s t o r me v e n t , n e t w o r k d a t aG i v e n p a s t s t o r me v e n t o r s p e c i f ie di n p u t h y e t o g r a p h s ,n e t w o r k d a t aS p e c i f i c s t o r mh y e t o g r a p h s ,n e t w a r k d a t aE v e n t o r c o n t in u o u sr a i n a n d p o l lu t a n td a t a , n e t w o r k d a t a

Long-term d a t a

H y d r o I n f o r m a t i o nS o u g h tH y d r o g r a p h s , Q ( t , j c ( )

H y d r o g r a p h s , Q ( t , x)

H y d r o g r a p h s , Q ( t , j c . )

H y d r o g r a p h s a n ds t a g e sH y d r o g r a p h s Q ( t , J t1)P o l l u t o g r a p h s , c ( t , X 1 )

R u n o f f v o l u m eP o l l u t a n t v o l u m e

R e q u i r e d H y d r a u l i cL e v e lH i g h

H i g h

M o d e r a t e -h i g h

H i g h

Moderate t o h i g h

L o w

industrial sources over the service period in the future. For real- time control problems itinvolves simulation andprediction of the sanitary runoff inconjunction with th e controlmeasures.The basic hydraulic principles useful for urban drainage have been presented inChapter 3 for free surface flows, Chapters 2 and 12 for pipe flows, and Chapter 10 forpump systems. In the following, more specific applications of the hydraulics to urbandrainage components will be described. However, the hydraulic design for drainage ofhighwayand street surfaces,roadsidegutters,andinletshasbeen described inChapter13 ,design of stable erodible open channels in Chapter 16, and certain flow measurementstructures adaptable to urban drainage in Chapter 21; therefore they are not included inthis chapter.

74.2 HYDR ULICS OFDR IN GE CH NNELSFlows in urba n drainage channels u sually are open-channel flows with a freewater surface.However, sewer pipes, culverts, and similar conduits under high flow conditions couldbecome surcharged, and pressurized conduit flows do occur. Strictly speaking, the flow is


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alwaysunsteady, thatis ,chan ging w ith tim e. Nevertheless, in anumberofsituations, suchas inm ost casesofflow insa nitary sewersand forsome rainstormrunoffs, changeof flowwithtimeisslow enough thattheflowcan be regarded asapproximately steady.

14.2.1 Op en-Chan nel FlowOpen-channel flow occurs on overland, ditches, channels, and sewers in urban areas.Unsteady flow inopen channels can bedescribedby a momentum equation given belowin both discharge (conservat ive)and velocity (nonconservative) forms together with itsvarioussimplified approximate models:

_LM+ i a f P f i ] + i f Uj1*,+- .+s,=o . (i4.i)gA dt gAdx(A ) gA Ja ^1 dx fdynamicwavequasi-steady dynamic wave noninertiakinematic wavelf+(2p-l) +(p-l) +M+f-5.+a (14.2)

wherex = flow lo ngitud inal direction m easured h orizontally (Fig. 14.2);A = flow cross-sectionalarea normaltox \y = vertical direction; Y = depthofflowof the cross section,measuredvertically; Q =d ischarge throug h A;V = QIA, cross-sectional average veloci-ty alongxdirection; S0 = channel slope, equal to tan 6,9 = angle between channel bedandh orizontal plane; S f = friction slope; a = perimeter boundingth ecross sectionA ;ql= lateral flow rate (e.g., rainorinfiltration)peru nit lengthofchannel andunit lengthofperimeter a, beingpo sitiveforinflow ; Ux =^-com ponentvelocity oflateralflow when

F I G U R E 14.2 S c h e m a t i c o f o p e n , c h a n n e l f lo w .


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joining the main flow; g = gravitational acceleration; t = time; M = CgA)-1 I (U xV )q 1 da and (3 = Boussinesq momentumflux correction coefficientfor velocity distri-bution:

P =lu 2dA (14.3)u = ^-component of local (point) velocity averaged over turbulence.The continuity equation is +f=i* (14-4)

If the channel is prismatic or very wide, such as the case of overland flow, Eq. (14.4) canbe w ritten as f+s(vy)=U-* (14-5)wherebis thewater surface widthof the cross section.In practice, it is more convenient to set the x andy coordinates along the horizontallongitudinal direction and gravitational vertical direction, respectively, when applied toflow on overland surface and natural channels for which S0 = tan 0. For human-madestraight prismatic channels, sewers, pipes, andculverts,it ismore convenientto set thex -y directions along and perpendicular to the longitudinal channel bottom. In this case, theflow depth h is measured along the y direction normal to the bed and it is related to YbyY= hcos 6,whereas th echannel slope S0 =sin 0.Thefr ic t ion slope S f is usually estimated by using a semiempirical formula such asManning's formula

riiy\-y\ n2Q\Q\S f=PR- =R (14.6)or theDarcy-Weisbach formula

-ife-M-jwheren = Manning's roughness factor,Kn = 1.486 fo rEnglish unitsand 1.0 for SIunits;/ = the W eisbach resistance coefficient; andR = thehydrau lic radius, which isequaltoA divided by the we tted perimeter. T he absolute sign is used to accou nt for the occurrenceof flow reversal.Theoretically, thevalues ofnand / forunsteady nonuniform open-channel andpres-surized conduitflowshavenotbeen e stablished.They dependon thepipe surface rough-ness and bed form if sediment is transported, Reynolds number, Froude number, andunsteadiness and nonuniformity of the flow (Yen,1991). One should be careful that fo runsteady n onu niform flow, the friction slope isdifferent fromeither the pipe slope, the dis-sipated energy gradient, the total-head gradient, or the hydraulic gradient. Only for steadyuniform flow w ithou t lateral flow are these different gradients equal to one another.Atpresent, w e can only use the steady un iform flow values ofna n d / g i v e nin the lit-erature as approximations. T he advantageof/is itstheoreticalbasis from fluid mechan-ics and its being nondimensional. Its values fo r steady uniform flow can be found fromthe Moody diagram or theC olebrook-W hiteform ula given in Chap. 2, as well as in stan-


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dard hydraulics and fluid mechanics references. Its ma jor disadvantage is that for a givenpipe andsurface roughness,th evalueof /var ies notmerely withtheReynolds num berbutalso withthe flowdepth. Inother words,as the flowdepthin thesewer changes duringastorm runof f , /mu s t be recomputed repeatedly.Manning 's nwas originally derived empirically. Itsmajor disadvantageis its trouble-some dimensionoflengthtoone-sixth pow er thatisoften misunderstood. Itsm ain advan-tage is that for flows with sufficiently high Reynolds number over a rigid boundary witha given surface roughness in a prismatic channel, the value of n is nearly constant over awide range of depth (Yen,1991).Values ofncan be found in Chow (1959)or Chap. 3.Other resistancecoefficients and form ulas, such as C hezy's orH azen-W illiams's,havealso been used. They possess neither th edirectfluidm echanics justificationas/norinde-pendence of depth as n. Therefore, they are not recommended here. In fact, Hazen-Williams' may be considered as a special situation of Darcy-Weisbach's formula. A dis-cussion of the preference of the resistance coefficients can be found in Yen(1991).Equations (14.6) and (14.7) are applicable to both surcharged and open-channelflows. For the open-channel case, th epipe is flowing partially filled and the geometricparameters of the flow cross section are computed from the geometry equations givenin Fig. 14.3.The pair of mom entum and continuity equations [Eqs. (14.1) and (14.4) or E qs. (14.2)and (14.5)] with ( 3 = 1and nolateral flow isoften referred to as theSa int-Ven ant equa-t ionsorfu l l dynamic wave equat ions .Actually, theyare not anexact representation of theunsteady flow because they involve at least the following assumptions: hydrostaticpressure distribution overA ,uniform velocity distribution overA(hence (3=1), andneg-ligible spatial gradient of the force due to internal stresses.

Those interested in the more exact form of the unsteady flow equ ations should refer toYen (1973b, 1975,1996).Conversely, simplified form s of the mom entum equation, name-ly ,th e noninert ia (misnomer diffusion wave) andk inemat ic wave approximat ions of thefull momentum equation [Eq. (14.1)] are often used for the analysis of urban drainageflow problems.Am ong the approximations shown in E qs. (14.1) or (14.2), the quas i-s teady dynamicwave equat ion is usually less accurate and m ore costly in com putation than the noniner-tia equation, an d hence, is not recommended fo r sewer flows. Akan and Yen (1981),among others, compared theapplication of thedynamic wave, noninertia, andkinem aticwav e equationsfor flowroutinginnetworksandfound th enoninertia approxim ation gen-erally agrees well withthedynamic wave solutions, whereas thesolutionof the kinemat-ic wave approximate is clearly different from the dynamic w ave solution, especially w henthe downstream backwatereffect is important. Table 25.2 of Yen (1996) gives the properform of theequationsto beused fo r different flow conditions.

F l o w A r e a A = - ^ - ( - s i n )H y d r a u l i c R a d i u s R *-^ -(I--~^-)D e p t h h = Y 11" 008" " 5W a t e r S u r f a c e W i d t h 8 = D s i n - |-4 , i n R a d i a n sF I G U R E 14.3 S e w e r p i p e fl o w g e o m e t ry . ( F r o m Y e n , 1 9 8 6 a )


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T A B L E 14.2 Theoretical Comparison of Approximations to D ynamic Wave EquationKinematic Noninertia Quasi-steady Dynamicwave dynamic wave wave

Boundaryconditions required 1 2 2 2Account f o rdownstream N o Y e s Y e s Y e sbackwater e f f e c t and flow reversalDampingof floodpeak No Yes Yes YesAccount for flow acceleration No No Only Yesconvectiveacceleration

Analytical solutions do not exist fo r Eqs. (14.1) and (14.2) or their simplified formsexcept for very simple cases of the kinem atic wa ve and noninertia approxim ations.Solutions are usually sought numerically as described in Chap. 12. In solving the differ-ential equations, in addition to the initial condition, boundary conditions should also beproperly specified. Table 14.2 shows the boundary conditions required for the differentlevels of approximations of the momentum equation. It also shows the abilities of theapproximations in accounting for downstream backwater effects, flood peak attenuation,and flow acceleration.For flows that can be considered as invariant with time the steady flow momentumequations which are sim plifiedfrom Eq. (14.2) fordifferent conditions are given in Table14.3.Thelateralflowcontribution, mq,can b efrom rainfall (positive) orinfiltration (neg-ative) or both. Instead of these equations, the following Bernou l li to tal head equ at ion isoften usedfor flowprofile computations:

T A B L E 14.3 Cross-Section-Averaged One-D imensional Momentum EquationsforSteady FlowofIncompressible Homogeneous FluidPrismatic channel

Constantpiezometricpressure distributionK = K =1

P = constantK=K = IPrismaticorwidechannelD e f i n i t i o n s :

D h=A/ w a t e r su r f a c e w i d th ; K a n d K * = p i e z o m e t r ic p r e s s u r e d i s t r i b u t io n c o rr e c t i o n f a c t o r sf o r m a ina n l a t e r a l f l o w s ;


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o c V2 a V2- f +Y2 + yb 2=-^+Yl+ybl +he+ hq, (14.8)where the subscripts 1 and 2 = the cross sections at the two ends of the comp utational reach,Ax,of thechannel, Y +yb thestageof thewater surface wherethechannelbed elevationat section 1 isybl andthata tsection 2 is yb2 =ybl + S0Ax;hqis theenergy head from thelateralflow, ifany;theenergy head loss he = SeAxwhereSeis theslopeof theenergy line;an d a = theC oriolisconvective kinetic energyfluxco rrectioncoefficient due tononuni-formvelocity distribution over the cross section (Cho w, 1959; Yen, 1973). If there are otherenergylosses, they shouldbeaddedto the right-handsideof theequation. Methodsofback-water surfacep rofile com putation using these equationsarediscussedinChap.3 .If the flow is steady and uniform, Eqs. (14.1) and (14.4) or Eqs. (14.2) and (14.5)reducetoS0= S f andQ = A V .Hence,forsteadyuniform flowusing M anning's formula,

Q = 0.0496S 0 2 D ~(t)S2^(|))5/3 (14.9)where < |) is inradian s (Fig. 14.3). Correspondingly, theD arcy-W eisbachform ula yields

Q=lD5fl( n (141Q)Figure 14.4is aplotofthese tw o equations thatcan beused to find$.

14.2.2 Surch arge FlowSewers, culverts,andother drainag e pipes som etim esflow full with water underpressure,often knownas surcharge flow (Fig. 14.5). Such pressurized conduit flow occurs underextreme heavy rainstorms or under designed pipes. There are two ways to simulateunsteadysurcharge flow in u rban drainage: (1) The standard transient pipe flow approachan d(2) the hypotheticalpiezom etric open slot approach.14.2.2.1 Standard transient pipe flow approach In this approach, the flow is con-sidered as it is physically, that is , pressurized transient pipe flow. For a uniform sizepipe, the flow cross-sectional area is constant, being equal to the full pipe area A ^hence 3A/3.X = O. The con tinuity and m om entum eq uations [E qs. (14.4) and (14.2)with ql = O] can be rewrit ten as

Q=AfV (14.11)

+ {+-}=-Sf 8r dx ( g YJ /whereP0 =the piezometric pressure of the flow and y = the specfic weight of the fluid.If th e pipe has a constant cross section and is flowing full with anincompressible fluidthroughout its length, then 3V/3jt = O. By further neglecting the spatial variation of P,integration of E q. (14.12)over the entire length, L, of the sewer pipe yields

P exit V 2 yi ( i av"i-f =-*i--*IH+ (1413)e n t r a n c e v


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F I G U R E 14.5 S u r c h a r g e f l o w i n a s e w e r . ( A f t e rP a n s i c , 1 9 8 0 ) .or

Jr?-*--*- +^ (1414)where H 1 1 = thetotal headat the entranceof thepipe, Hd = th ew ater surface outside th epipe exit, andKu andKd = the entrance and exit loss coefficients, respectively (Fig. 14.5).Equations(14.11) and(14.12)canalsobederived as aspecialcaseo f the commonly usedgeneral, basic, closed conduit transient flow continuity equation fo r waterhammer andpressure surge analysis, see, e.g., Chaudhry, 1979; Stephenson, 1984; Wood, 1980; Wylieand Streeter, 1983.

< ( R a d i a n s )F I G U R E 14.4 C e n t r a l a n g l e c j > o f w a t e r s u r f a c ei n c i r c u l a r p i p e ( f r o m Y e n , 1 9 8 6 a ) .

( D e g r e e s )


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T+-?=


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b = ngD2 /4c 2 (14.19)The surge speed in apipe usually ranged from a fewhundredfeet per second to a fewthousandfeetper second.For anelastic pipe w ithawall thicknesseand Young's modu luso felasticity E p J assuming no pressure forcefrom the soil acting on the pipe, the surge speed cisH) +)]

where E f is the bulk m o d u l u s of elast ic i ty andp/ is the bu lk density , respectively, of theflowing water (Wylie and Streeter, 1983). Special conditions of pipe anchoring againstlongitudinal expansionorcontraction andelasticity relevantto thesurge speed c aregivenin Table 14.4 where c o = Poisson' s rat iofor the pipe w all m aterial, that is, -co is the ratioof th elateral u nit strain toaxial unit strain, and a is aconstanttoaccountfor therigiditywith respect to axial expansion of the pipe. For small pipes, E q. (14.19) m ay give toosmall a slot width, which would cause numerical problems. Cunge et al. (1980) recom-menda w idth of 1 cm or larger.The transition between part-full pipe flow and slot flow is by no means computation-ally smooth and easy, and assumptions are necessary (Cunge and Mazadou, 1984). Oneapproach is toassumeagradual w idth transition from thepipeto theslot. Sjoberg (1982)suggested two alternatives for the slot width based on two different values of the wavespeed cin E q. (14.19). For thealternative ap plicable toh/D>0.9999 ,his suggested slotwidthbcan beexpressed as

L =10-6+ 0.05423exp [-(/i />)24] (14.21)H efurtherproposed to compute the flow areaAand hydraulic radiusRwhen the depth his greater than the pipe diameter DasT A B L E 14.4 Special C ond itions of Surge Speed in Full Pipe Eq. 1 4 . 2 0 )FactorP i p e Anchor

E l a s t i c i t y E

T | = (l + ( D ) + aFreedomofpipelongitudinalexpansionV a l u e ofaxialexpansionfactor a

Rigid pipe,= 00c > = E f / P f

ConditionDD +eEntirely f r e e( e x p a n s i o n j o i n t s a tb o t h e n d s )1

A ir entrainmentP 7 = pw V w + pf l aE- E 1 + V J i (E J E J - 1

Onlyoneendanchored1- 0.5 co

Entirelengthanchored1 - c o2

N o airP / = P wEf= EW

S u b s c r i p t w d e n o t e s w a t e r ( l i q u i d ) ; S u b s c r ip ta = a i r ; s u b s c r i p t / = f l u i d m i x t u r e ;Y = v o lu m e .


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A =(n D2/4) +(h- D )b (14.22)R = D / 4 (14.23)

A slight improvement to Sjoberg's suggestion to provide a smoother computationaltransition is to use A =A9999+b(h - 0.9999Z)) (14.24)for h iD >0.9999andassume thatthetransition startsa th iD=0.91.B etweenh iD =0.91and0.9999, real pipe areaA and surface widthBare used. How ever, forh /D > 0.91,Riscomputed from Manning's formula using pipe slope and a discharge equal to the steadyuniformflow ath =0 .9 ID ,Q91;thus,forh / D > 0.91

R = (A91 IA)R91 (14.25)Because of the lack of reliable data, neither th e standard surcharge sewer solutionmethod nor the Preissmann hypothetical open-slot approachhasbeen verifiedfor a singlepipe or anetworkof pipes. Past experiences withwaterhamm er andpressure surge prob-lemsin closed conduitsm ay provide some indirect verificationof the applicability of thebasic flow equationsto unsteady sewer flows. Nevertheless, direct verificationishighlydesirable.Junand Yen(1985) perform edanumerical testing and found there is no clear superi-ority of one approach over the other. Nevertheless, specific comparison between them isgivenin Table 14.5.They suggested that if the sewers in a netw ork are each divided into

manycom putational reachesand asignificantpartof the flowdurationisund er surcharge,th estandard approach saves com puter time. C onversely,iftransition between open-chan-nel andpressurized con duitflowsoccurs frequently and thetransitional stability problemis important,theslot model wouldb epreferred.

T A B L E 14.5 Comparison Between Standard Surcharge Approach and Slot ApproachItemConceptFlow equations

Discretization for solutionWatervolume within pipe

Discharge i npipe a tg i v e n timeT r a n s i t i o n betweeno p e n channelf l o w a n ds u r c h a r g e f lo w

S t a n d a r d Surcharge ApproachDirect physicalTwod ifferent sets, one equationf o r surchargeflow, twoequationsf o r open-channel f l o wWhole pipe length forsurcharge flowConstant

Same

S p e c i f i c criteria

H y p o t h e t i c a l Slot Approac hConceptualSameset of twoequations(continuity andmomentum)f o r surcharge andopen-channelflowsD ivide into A J C sVaries slightly with slotsize, inaccurate if slot is toowide, stability problems ifslotis toonarrowV a r i e s slightly with A J C , thusa l l o w s transitiontoprogressw i t h i n pipeSlotwidth transitiont oavoidnumericalinstability


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TABLE14.5 ( C o n t i n u e d )I t e mP a r t f u l l o v e r p i p e l e n g t hT i m e a c c o u n t i n g f o r t r a n s i t i o nP r o g r a m m i n g e f f o r t s

C o m p u t a t i o n a l e f f o r t

S t a n d a r d S u r c h a r g e A p p r o a c hA s s u m e e n t i r e pipe l e n g t h f u l lo r f r e eY e s , s p e c i f ic i n v e n t o r y o fs u r c h a r g e d p i p e s a t d i f f e r e n t t i m e sM o r e c o m p l i c a te d b e c a u s e of twos e t s ofequationsand t i m e a c c o u n t ira n d c o m p u t e r s to r a g e f o r t r a n s i t i o n

D e p e n d i n g m a i n l y o n a c c o u n t i n gf o r t r a n s i t i o n t i m e s

H y p o t h e t i c a l S l o t A p p r o a c hA s s u m e f u l l o r f r e e A x b y A rNo, i m p l i c i tR e l a t i v e ly s i m p l e b e c a u s e o fig o n e e q u a t io n s e t a n d n os p e c i f i c a c c o u n t in g a n ds t o r a g e f o r t r a n s i t io nb e t w e e n o p e n - c h a n n e l a n df u l l - p i p e f l o w sD e p e n d i n g m a i n l y on s p a c ed i s c r e t i z a t i o n A x

74.3 FLOWINASEWER

14.3.1 Flow in a Single SewerOpen-channel flow in sewers and other drainage conduits are usually unsteady, nonuni-form, andturb ulen t. Sub criticalflowsoccur moreoften than supercritical. Forslowlytimevarying flow suchas the case of the flow traveling time through the entire length of thesewer much smaller than th e rising time of the flow hydrograph, the flow can often betreated approximatelyasstepwise steady without significant error.The flow in a sewer can be divided into three regions: the entrance, the pipe flow, andthe exit. Figure14.7showsa classification of 10different cases ofnonuniform pipe flow

s u b c r i t i c a l s u p e r c r i t ic a l t o s u r c h a r g e

s u p e r c r i t i c a l s u b c r i t i c a l t o s u r c h a r g e

s u p e r c r i t i c a l t o s u b c r i t i c a l s u r c n a r g e t o s u p e r c r i t i c a l

s u b c r i t i c a l t o s u p e r c r i t i c a l s u r c h a r g e t o s u b c r i t i c a l

s u p e r c r i t i c a l j u m p t o s u r c h a r g e s u r c h a r g eF I G U R E 14.7 C l a s s i f i c a t i o no f f l o w i n a s e w e rp i p( A f t e r Y e n , 1 9 8 6 a ) .


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F I G U R E 14.8 T y p e s o fsewer entrancef lo w .( A f t e r Y e n , 1986a).

based on whether the flow at a given instant is subcritical, supercritical, or surcharge.There arefour casesofpipe entrance co ndition, asshownin Fig. 14.8 an dbelow:C a s e P i p e entrance hydrau lic cond itionI Nonsubm erged entrance, subcriticalflowII Nonsubm erged entrance, supercritical flowIII Submerged entrance, air pocketIV Submerged entrance, water pocket

Case I is associated with downstream control of the pipe flow. Case II is associatedwithupstream control. In C ase III, the pipe flow und er the air pocket m ay be subcritical,supercritical, or transitional. In Case IV, the sewer flow is often controlled by both theupstream anddow nstream conditions.Pipe exit conditions also can be grouped into four cases as shown in Fig. 14.9 andbelow:C a s e P ipe exithydraulic conditionA Nonsubm erged, free fallB Nonsubmerged, continuousC Nonsubmerged, hydraulic jum pD Submerged

e a s e l

( b ) c a s e I I

( c ) caseH I

( d ) c a s e I V


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F I G U R E 14.9 T y p e s o fsewere x i t f l o w . ( A f te rY e n , 1986a).In CaseA, thepipe flow is under exit control. InCase B, the flow isunder upstreamcontrolif it is supercritical an ddow nstream controlifsubcritical.InCaseC , thepipeflowisunder upstream control while th ejunction water surface is under downstream control.In CaseD , thepipe flow isoften under downstream control,but it canalsobeunder bothupstream anddownstream control.Thepossible combinationsof the 10cases ofpipeflowwiththeentrancean dexit con-ditions are shownin Table 14.6 fo runsteadynonuniform flow. Some of these27possiblecombinations arerather rare fo runsteadyflow and nonexistentfo r steadyflow, for exam-ple, Case 6. For steady flow in a single sewer, by considering the different mild-slopeMandsteep-slope 5backwater curves (Chow, 1959) asdifferent cases, thereare 27po ssiblecasesinadditionto theuniformflow, ofwhichsixtypes w ere reportedbyB odhaine(1968).

T A B L E 14.6 PipeFlow Conditions Pos sible Poss ible ExitC a s e P i p e F low Entrance Cond itions Cond itions1 Subcritical I, III A, B2 Supercritical II, III B , C3 Subcritical > hydraulic drop-supercritical I, III B , C4 Supercritical > hydraulic jump > subcritical I I , I I I A , B5 Supercritical > hydraulic jum p > surcharge I I , I I I D6 Supercritical s u r c h a r g e II, III D7 Subcritical > surcharge I , I I I D8 Surcharge-> supercritical IV B , C9 Surcharge > subcritical I V A , B10 Surcharge IV DS o u r c e : F r o m Y e n( 1 9 8 6 a ) .

c a s e

c a s e B

c a s e C

c a s e D


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Thenonun iform pipe flows shown in Fig. 14.7 are classified without considering thedifferent modes of air entrainment.The typesof thewater surface profile, equivalent tothe M, S ,and A (adverse slope) types of backwater curves fo r steady flow, are also nottaken into account. Additional subcases of the 10 pipe flow cases can also be classifiedaccording to rising, falling, or stationary water surface profiles. For the cases with ahydraulicju m p or drop, subcases can be grouped according to the m ovem ent of the ju m por drop, be it moving upstream or downstream or stationary. Furthermore, flow withadverse sewer slope also exists because of flow reversal.Duringrunoff, thechangeinmagnitudeof the flow in asewercanrangefromonlya fewtimes dry weather low flow in asanitary sewer to asmuchasmanyfold for aheavy rain-storm runoff in astorm sewer.Th etime variationofstorm sewerflow isusually m uch m orerapidthan thatofsanitary sewers. Therefore, theapproximationofassum ing steadyflow ismore acceptablefo rsan itary sewers thanforstorman d combined sewers.In thecaseof aheavy stormrunoff enteringaninitiallydrysewer,as the flowenters thesewer, boththedepthan ddischarge starttoincrease asillustratedinFig. 14.10attimes J1 ,t2 ,andt3for theop en-channel phase.As the flowcontinuesto rise, thesewer pipebecomescompletely filled andsurchargesasshowna t t4and 5inFig. 14.10.Surchargeflowoccurswhenthesewerisunderdesigned, w henthe flood exceeds thatof thedesign returnperiod,whenth eseweris notproperly maintained,orwhen storagean dpum ping occur.U n d e r surcharge conditions, the flow-cross-sectional area and depth can no longerincrease because of the sewer pipe boundary. However, as the flood inflow continuesto increase, the discharge in the sewer also increases due to the increasing differencein head between the upstream and downstream ends of the sewer, as sketched in thedischarge hydrograph in Fig. 14.10. Even under surcharge conditions while the sewer

F I G U R E 14.10 T i m e v a r i a t io n o f f l o w i n a s e w e r . ( A f t e rY e n , 1 9 8 6 a ) .


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diameter remains constant,theflow isusual ly n onu niform . Thisis due to the effects ofthe entrance and exit on the flow inside the sewer, and hence, the streamlines arenot parallel.As th e flood startstorecede, th eaforemen tioned flow process isreversed. The sewerwill return from surcharged pipe flow to open-channel flow, shownat t6 and I 1 in Figure14.1.Sinceth erecession isusually butnotalways more gradual thantherisingof theflood, th ew atersurfaceprofilein thesewerisusu ally more gradu al duringflow recessionthanduring rising.The d ifferences in thegradientof the water su rface profiles during therisingandreces-sion of the flood bear importance in the self-cleaning an d pollutant-transport abilitiesofthe sewer. D uring the rising period, w ith relatively steep grad ient, the flow can c arry notonly the sediment it brings into the sewer but also erodes the deposit at the sewer bottomfrom previous storms.For agiven dischargeandgradient, th e amountof erosion increas-eswiththeantecedent du rationo fwettingand softening of the deposit. D uringthe reces-sion,witha flatterwatersurface gradientanddeceleration of the flow, thesediment beingcarried intothe sewerby the flow tendsto settle ontoth e sewer bottom.If th estormis notheavyand the flood is not severe, th erisingfloww illnotreachsur-charge state.The flood may rise, fo r example, to the stage at r3 showninFig.14.10andthen starts to recede. The sewer remains under open-channel flow throughout the stormrunoff. For such frequent small storms, the flow in the sewer is so small thatit isunableto transport out the sedimen t it carries into the sew er, resulting in depo sition to be cleanedupby later heavy stormsorthroug h artificial means.For a single-peak flood entering a long circular sewer having a diameter D and pipesurface roughness k ,Y en (1973a) reported that fo r open-channel flow, the attenuationofthe floodpeak,Qpx ,at adistancex downstreamfrom thepipe entrance (x = O) and thecor-respondingoccurrence timeofthis peak, tpx ,can bedescribed dimensionlessly as

n ( ( V/H17 ( K V o . 4 2 ( Qn v-16|-=exp -0.0771*- A ^MH=Q p0 pl (D][DJ (Dj (D ^ V )r j?132~Mxp(f*- iM\

66( -W^={6.03.oglo[{|)-0,8]-52o}(A) []

f O4-4 l0-^ a 0 . 8 2 ~ | 0 . 5x[slMF- iM (1427)where Qp0 and t p 0 = thepeak dischargeand its timeof occurrence atx = O ,respectively;Q bis the steady base flow rate an dRb = hydraulic radius of the base flow; tg = the timeto the centroid of the inflow hydrographatx = Oabove th ebase flow; g = th egravita-tional acceleration; and Vw = (QJA1) + (gAJB1)112 = the wav e celerity of the base flow,whereAb = th ebaseflowcross = sectional area andBb = th ecorresponding water-sur-facew idth.Inboth equations,thesecond nondim ensional parameterin theright-hand sidek lD is apipe property param eter;th e third parameter R 1 J D is abase flow parameter; th efourth nondim ensional parameter represents th e influenceof the flood discharge; where-as the f i f t h andlast nondim ensional parameter reflectstheshapeof theinflow hydrograph.Thesingle- peak hydrograph showninFig. 14.10is anideal casefor thepurposeofillustration. Inreality, because th ephase shift of the peak flows inupstream sewersandth etime-varyingnatureofrainfall andinflow,usuallyth ereal hydrograph sarem ultipeak.


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Because the flow is nonuniform and unsteady, the depth-discharge relationship, alsoknown as the rating c ur v einhydrology,isn onunique. Even if we arewilling to considerthe flow to besteadyuniformas anapproximation,thedepth-discharge relation isnonlin-ear,andw ithinacertain range, no nun ique,asshow n nondimensionallyandideally inFig.14.11for acircular pipe.T henon un ique depth-discharge relationship fo rnonun iformflow,aidedby thepoor qualityof thewaterand restricted accessto thesewer, makesit difficultto measu re reliably the tim e-vary ing flow in sewers. A m ong the m any simple and sophis-ticated mechanical or electronic measurement devices that have been attempted on sewersand reported in the literature, the simple, mechanical Venturi-type meter, which has sideconstriction instead of bottom constriction to minimize the effect of sediment clogging,still appears to be the mo st practical m easurem ent m eans, that is, if it is properly designed,constructed,andca librated and if it islocatedat asufficient distance from theentrancean dexit of the sewer. On the other hand,thehydraulic performance graph described in Sec.14.6.1can beusedtoestablish th erating curvefor asteady nonuniformflow.

DhDameeRoPemercGrae

A r e a

Discha rge Hidrau l i c Rad ius

V e l o c i t y

Hydraul ic E lemen t s R e l a t iv et o Ful l C r o s sSec t i on ,F I G U R E 14.11 R a t i n g c u r v e f o r s t e a d y u n i f o r m f l o w i nc i r c u l a r p i p e .


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Flow insewersisperhapsone of themost complicated hydraulic phenom ena. E venfo ra single sewer, there are anumberof transitional flow instability problems. One of themis thesu rge instabilityof the flow inpipes of anetwork.Theother four types of instabil-ities that could occur in a single sewer pipe are the following: The instability at thetransition between open-channel flow and full conduit flow, the transitional instabilitybetween supercritical flow and subcritical flow in the open-channel phase, the water-sur-face roll-wave instability of supercritical open-channel flow, and a near dry-bed flowinstability.Further discussiononthese instab ilities can befound in Yen(1978b , 1986a). Itis important to realize the existence of these instabilities in flow modeling.14.3.2. Discretization of Space-Time Domain ofa Sewer forSimulationNo analytical solutions are known for the SaintVenant equations or the surchargedsewer flow equation. Therefore, these equations fo r sewer flows are solved numericallywithapp ropriate initial andboun dary conditions.Thedifferential termsin thepartial dif-ferential equations are approximated by finite differences of selected grid points on aspaceand time domain,aprocess often knowna s discretization.Sub stitutionof the finitedifferences into a partial differential equation transforms it into an algebraic equation.Thus,th eoriginal set ofdifferential equationscan betransformed intoa set offinite dif-ference algebraic equationsfor numerical solution.Theoretically, the computational grid of space and time need not be rectangular.Neither needthe spacean d timedifferences A J Cand A fbekept constant. Nonetheless, it isusuallyeasier fo rcomputer coding tokeep A J CandA tconstant throughoutacom putation.For su rcharge flow, Eq. (14.14) dictates the application of the equation to the entire leng thof th e sewer, and the discretization applies only to the time domain. In an open-channelflow,it is normally advisable to subdivide the length of a sewer into two or three compu-tational reaches of Ax, unless the sewer is unusually long or short. One computationalreach tends to carry significant inaccuracydue to the entrance andexitof the sewer andisu sually incapable of sufficiently reflecting the flow inside th e sewer. Conversely, toomany computational reaches would increase the computational complexity and costswithoutsignificant improvemen t inaccuracy.The selectionof the timedifference Ar isoften anun happy compromise of three crite-ria. The first criterion is the physically significant time required for the flow to passthrough the computational reach. Consider a typical range of sewer length between 100and 1000ft anddivideitintotwo orthree A J C , and ahighflowvelocityof5-10ft/s, asuit-able compu tational time interval w ould be approximately 0.2-2 min. For a slowly vary-ingunsteadyflow,this criterion is not importantand larger computationalArw ill suffice.For a rapidly va ryin g unsteady flow , this criterion should be taken into ac count to en sureth ecomputationisphysically meaningful.Thesecond criterionis asufficiently smallAr toensure num erical stability.Anoften-usedguideis the Courantcriterion

Ac/Ar> V+ V g A I B (14.28)In sewers, which usually have small A J C compared to rivers and estuaries, this criterionoften requiresa Arless th an halfam inuteand sometimes 1 or 2 s.Th e third criterion is the tim e interval of the av ailable in pu t data. It is rare to have rain-fall orco rresponding inflow hydrograph data with a time resolution as short as 2, 5, oreven 10m in. Valuesfor A rsmaller th anthedata time resolutioncanonlyb e interpolated.This criterion becom es imp ortant if the in-between v alues cann ot be reliably interpolated.


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In a realistic application, all three c riteria sho uld be con sidered. U nfortun ately, in m anycomputations only the second numerical stability is considered.There are many, m any num erical schemes that can be adopted for the solution of theSaint-Venantequ ationsortheir approx imateforms [Eqs.(14.1)-(14.5)].Theycan beclas-sified as explicit sch emes , implicit sch emes ,and the m e t ho dof charac teri s ti c s .Manyofthese m ethods are described in Chap. 12, as well as in Abbott and B asco (1990),Cungeetal. (1980), Lai (1986), and Yen (1986a).

14.3.3 Initialand Boundary ConditionsAs discussed previouslyandindicated inTable 14.1,boundary conditions, in addition toinitial conditions, must be specified to obtain a unique solution of the Saint-Venantequa-tionsortheir approximate simplified equations.The initialco ndi t ionis, of course, the flow condition in the sewer pipe w hen com pu-tations start,t=O,thatis, either thedischarge Q ( X , O), or the velocity V(x, O),paired w iththedepthh(x,O). For acombined sewer, thisisusuallyth edry-weatherflow orbaseflow.For a storm sewer, theoretically, this initial condition is a dry bed with zero depth, zerovelocity, and zero discharge. H owev er, this zero initial condition imposes a singularity inthe numerical computation.To avoid this singularity problem, either a small depth or asmall discharge is assum ed so that the co m putation can start. This assum ption isjustifi-able because physically there is dry-bed film flow instability, and the flow, in fact, doesnot start gradually and smoothly from dry bed. Ba sed on dry-bed stability consideration,an initial depth on the order of 0.25 in., or less than 5 mm , appears reasonable.

However,insewers, this small initial depth usuallyisun satisfactory because negativedepth is obtained at the end of the initial time step of the computation. The reason is thatthe con tinuity equation of the reach often requires a water volume much bigger than theamountof water in the sewer reach w ith a sm all depth. Hen ce, an initial discharge, or baseflow,thatpermits the com putation to start is assumed. F or a storm sewer, the m agnitudeof thebaseflowdependson the characteristics of the inflowh ydrograph, the sewer pipe,the numerical scheme, and the sizeof Ar and A J C used. For small A x and Ar, a relativelylarge base flow is required, but m ay cau se a sign ificant error in the solution. In either case,it is notuncommon thatin the first fe wtim e stepsof thecomputation,th ecalculated d epthanddischarge decrease as the flood propagates, a result that contradicts the actual physi-cal process of rising depth and discharge. Nonetheless, if the base flow is reasonablyselected and the numerical scheme is stable, this anomaly would soon disappear as thecomputationprogresses. An alternative to this assumed base flow approach to avoid thenumerical problem is to use an invertedP riessmannhypothetical slot throughout the pipebottom and assigning a small initial depth, discharge or velocity to start the computation.Currey (1998) reported satisfactory use of slot width between 0.001 and 0.01 ft.Asto boun dary conditions, whe n theSaint-Venantequations are applied to an interiorreach of a sewer not connected to its entrance or exit, the upstream cond ition is simp ly thedepthand discharge (or velocity) at the d ow nstream end of the preceding reach, wh ich areidentical w ith the depth and discharge a t the upstream of the present reach. L ikewise, thedownstream condition of the reach is the shared values of depth and discharge (or veloc-ity) with th e following reach. Therefore, th e boundary conditions for an interior reachneednot beexplicitly specified be cause theyare implicitly accountedfor in the flowequa-tionsof the adjacent reaches.For theexterior reaches con taining either thesewer entranceor theexit, theupstreamboundarycond itions required depend on w hether the flow issubcriticalor supercritical asindicated in Table 14.7.


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TABLE 14.7 Some Types of Specified Boundary Conditions for Simulation of ExteriorReachesofSewersLocation

Subcritical flow

Supercriticalflow

U p s t r e a m E n d o f SewerEntranceReach ( x = O )One ofG ( O , O/ z ( 0 , f }V ( O , Of o r all t to be simulatedTwo of the above

Downstream E n d o f SewerE x i tReach (x = L)One of/ z ( L , t ) , e.g.ocean tides, lakesQ ( L , t ) , release hydrograph< 2 ( / i ) ;rating curveV ( h ) ; storage-velocity relationf o r all t to be simulatedNone

For a sewer that is divided intoMcom putational reaches andM + 1 stations, there isacontinuity equationand am om entum equation writteninfinite d ifference algebraic formfor each reach. There are 2(M + 1) unk now ns, namely, the depth and discharge (orveloc-ity) at each station. The 2(M+1) equations required to solve for the unknowns comefrom Mco ntinuity equ ationsand Mmom entum equationsfor the Mreaches,plusthe twoboundary conditions. If the flow is subcritical, one boundary condition is at the sewerentrance(x =O) and theotheris at thesewer exit(x=L).If the flow is supercritical, bothboundary con ditionsare at theupstream end,th e entrance, one ofthem often is acriticaldepth criterion. If at one instant ahydraulic jum p occurs in an interior reach inside th esewer, tw o upstream boundary conditions at the sewer entrance and one downstreamboundary conditionat thesewer exit shouldb especified. If ahydraulic drop occurs insideth e sewer, one boundary condition each at the entrance and exit of the sewer is needed;the drop is described with a critical depth relation as an interior boundary condition.Handling the moving surface discontinuity, shown schematically in Fig. 14.12, is not asimple m atter.Themovingfront may travel from reach to reach slowlyin different Ar,oritm aymove throughtheentire sewerin one A r .If, for anyreason,it isdesired tocomputeth e velocityof the moving front Vwbetween tw ocomputational stations / and / + 1 in asewer, thefo llowing equationcan beusedas an approximation;

u p e r c r i t i c o l t o u b c r i t i c o l u b c r i t i c a l t o u p e r c r i t i c a l

S u p e r c r i t i c a l t o S u b c r i t ic a l S u b c r i t i c a l t o S u p e r c r i t i c a l

F I G U R E 14.12 M o v in g w a t e r s u r f a c e d i s c o n t i n u i t y i n as e w e r . ( A f te r Y e n , 1986a).


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AV A VV n= ' . ^V + (14-29)A - + 1 A

14.3.4 Storm Sewer Designwith Rational MethodThe most important components of an urban storm drainage system are storm sewers.A numberofmethods existfo rdesigningthesizeofsuch sewers. Somearehighly sophis-ticated, u sing theSaint-V enantequ ation s, wh ereas others are relatively simple. In contrastto stormrunoff prediction/simulation models, sophisticated storm sewer design methodsdonot necessarily prov ide a better design than the simp ler methods, m ainly because o f thediscrete sizes of commercially available sewer pipes.If th epeak design discharge Qp for asewer isknown,the required sewer dimensionscan becomputedbyusing Manning'sformula such that

A RW = - e- (14.30)K n VS0which can be obtained from E q. (14.6) by assuming the friction slope S f is equal to thesewerslopeS0 .All other sym bols in the equation hav e been defined previously. For a cir-cularsewer pipe, th eminimum required diameter dr isf n Q T/8d,= 3 . 2 0 8 fM (14.3Ia)L Kn ^o J

wherekn = 1 for SIun itsand 1.486forEn glish units.If theD arcy-Wesibachformula (Eq.14.7)isused,r / 11/5dr= 0.811- -G,2 (14.3Ib)L O O J

These tw oequationsareplotted inFig. 14.13fo r design applications. The assumptionS0= S f essentially implies that around th e time of peak discharge, th e flow can well beregarded approximately as steady uniform flow for the design, despite th e fact that th eactual spatial and temporal variations of the flow are far more complicated as describedin Sec. 14.3.1.In sewer designs, thereare anumberofconstraintsa ndassum ptions thatarecomm onlyused in engineering practice. Those pertinent to sewer hydraulic design are as follows:1. Free surface flow exists for the design discharge, that is, the sewer is under "gravityflow" or open-channel flow. The design discharge used is the peak discharge of thetotal inflow hydrographof the sewer.2. The sew ers are com m ercially available circular sizes no smaller than, say, 8 in. or 200mm indiameter.In theU nited States,the commercial sizesin inchesareusually8, 10,12, and from 15 to 30 inches with a3-in. increment, an d from 36 to 120 in. with anincrement of 6 in. In SI units, commercial sizes, depending on location, include mostif not all of the fo llow ing : 150, 175, 200, 250, 300, 400, 500, 600, 750, 1000, 1250,1500, 1750, 2000,2500,and3000mm.3. The design diameter is the smallest comm ercially av ailable pipe that has a flow capac-ity equal to or greater thanthe design discharge and satisfiesall the appropriate con-straints.4. To prevent or reduce perm anent deposition in the sewers, a nominal m inimum per-missible flow velocity atdesign discharge or atnearly full-pipe gravity flow is speci-


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F I G U R E 14.13 R e q u i r e d s e w e r d i a m e t e r , ( m o r f t )fied.A m inimum full-pipe flow velocity of 2ft/s or 0.5m /sat the design discharge isusually recommended or required.

5. To preve nt the occurrence of scour and other undesirable effects of high velocity =flow, a maximum permissible flow velocity is also specified. The most commonlyusedvalue is 10ft/s or 3m/s.How ever, recent studies indicate that due to the improvedqualityofm odern concrete andother sewer pipe m aterials, theacceptable velocitycanbe considerably higher.6. Storm sewers mu st be placed at a depth that will allow sufficient cushioning to preventbreakage due to ground surface loading and will not be susceptible to frost. Therefore,minimum cover depths must be specified.7. The sewer system is a tree-type network, converging toward dow nstream.8. Thesewersarejoined atjunctionsormanholes w ith specified alignment,fo r example,the crow ns aligned, th e inverts aligned, or the centerlines aligned.9. At any jun ction or manhole, the dow nstream sewer cannot be sm aller than any of theupstreamsewers at that jun ction , unless the jun ction has significantly large detention


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storage capacity or pump ing.Therealso is evidence th at this con straint is unnec essaryfor very large sewers.Various hydrologic and hydraulic methods exist for the determination of the designdischarge Qp.Am ong them th e rational method isperhaps th e most widely and simplestused method fo rstorm sewer design. W ith this m ethod, each sewer isdesigned individu-ally an dindependently, except thattheu pstream sewerflow timem ay beusedto estimateth etim eo fconcentration.T hedesign peak dischargefor aseweriscomputedbyusingtherational form ula

Q 1 1 = I ^ C f J (14.32)where i = the intensity of the design rainfall;C = therunoff coefficient (see C hap. 5 foritsvalues);andaissurfa ce area.Thesubscriptj represents they'thsubarea upstreamto bedrained. Note that^a. includes all the subareas upstream of the sewer being designed.Each sewerhas its owndesign ibecause each sewer has its own flow timeof concentra-tion anddesign storm.Theon ly information needed from upstream sewersfor thedesignof acurrent seweris theupstreamflow time for thedetermination of the timeof concen-tration.Therational form ulaisdimensionally hom ogenousand isap plicableto anyconsistentmeasurement units.Therunoff coefficient Cisdimen sionless. It is apeak discharge coef-ficientbut not arunoff volumefraction coefficient. How ever, inE nglish units usuallytheformula isused w iththe area aj inacres and rain intensity iin inches per hour.The con-version factor1.0083isapproximated asunity.Theprocedureof therational methodisillustratedin thefollowinginEn glish unitsforthe design of the sewers of the simple example drainage basinA shown schematically inFig. 14.14. The catchment properties are given in Table 14.8. For each catchment, thelength L0 and slope S0 of the longest flow pa thor better, the largestLJvS0 shouldfirst beiden tified.As discussed Sec. 14.7, anumberofformulasa re available to estimatethe inlet time or time of concentration of the catchment to the inlet. In this example, Eq.(14.86) is used with K = 0.7 for English units and heavy rain, that is, t0 =0.7(nL 0A/S^)-6. The catchment overland surface texture factor T V is determined fromTable 14.16The design rainfall intensityis computedfrom th e intensity-duration-frequency rela-tionfo rthis location, i Q Q T0 .2i(in./h) = - (14.33) T - Z JT A B L E 14.8 CharacteristicsofCatchmentsof Example D rainage Basin ACatchment Area Longest Overland Path Inlet T im e R u n o f f( a c r e s ) Length L 0 S l o p e S u r fa c e T e x t u r e t 0 C o e f f i c i e n t( f t ) N ( m i n ) C

I 2 250 0.010 0.015 6.2 0.8 3 420 0 0081 0 016 9 3 0 7 3 400 0 012 0 030 11 7 0 4IV 5 640 0 010 0 020 12 9 0 6V 5 660 0 010 0 021 13 1 0 6

T o t a l a r e a = 1 8 a c r e s


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F I G U R E 14.14 S e w e r d e s i g n e x a m p l e d r a i n a g e b a s i n A . ( a ) L a y o u t ( b ) P r o f i l e s .where td = the rain duration(min)w hich is assumed equal to the time of concentration,tc ,of thearea described, and Tr = thedesign return period inyears. For this example, Tr= 10years. D eterm inationof / for the sewers isshowninTable 14.9a.T heentries inthistable areexplained asfollow s:

Column L Sewer numb er identified by the inlet num bers at its two ends.C o l u m n2. The sewer num ber immediately upstream, or the num ber of the catchmentthatd rains directly through man hole or junc tion into the sewer being con-sidered.Column3 . The sizeof the directly drained catchmen t.Column 4 . Valueof the runoff coefficient fo r each catchment.Column5. Productof Cand thecorresponding catchment area.Column6 . Summationo f C}a.for all theareas drainedby the sewer;it isequalto thesu m ofco ntributing valuesinColumn 5.

M a n h o l eG r o u n d

N o t e : A l l e l e v a t i o n s a r e i n f e e t( 1 0 - Y r F l o o d L e v e l )

( C h a n n e l B e d )


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C o l u m n 7. Valueso f inlet time to the sewer for the catchments drained, that is, theoverland flow inlet time for directly drained catchments, or the time ofconcentration for the imm ediate u pstream connecting sewers.C o l u m n8. The sewer flow time of the immediate upstream connecting sewer asgivenin Column 9 in Table 14.9&.C o l u m n9 . The timeofconc entration tc foreachof the possible critical flow paths,tc = inlet t ime (Column 7) + sewer flow t ime (Column 10) for eachflow path.C o l u m n 1 0 .Thedesign rainfall duration tdis assumed equal to the longestof thedif-ferent times of concentration of different flow paths to arrive at theentrance of the sewer being considered, for example, for Sewer 31, td isequal to 13.9m in from Sewer21,w hich is longer than thatfrom directlycontributing C atchme nt V (13.1m in).C o l u m n IL The rainfall intensity i for the duration given in Column 10 is obtainedfrom theinten sity-dura tion relation for thegiven location, inthis case,E q.(14.33) for the 10-year design return period.Table 14.9/?shows the design of the sewers for which the Manningn = 0.015, mini-mum soil coveris 4.0 ft, andm inimum nom inal design velocity is 2.5ft/s. Theexit sewerofthesystem (Sewer 31)flowsintoacreek fo rwhichth ebottom elevation is 11.90ft, thegroundelevationof itsbankis21.00 ft, and its 10-year floodwater level is20.00ft.C o l u m n L Sewer num ber identifiedby itsupstream inlet (manhole) num ber.C o l u m n2 . G round elevationat theupstream manholeof the sewer.C o l u m n3 . Lengthof thesewer.C o l u m n4 . Slope of the sewer, usua lly follows approximately the average groundslope alongthe sewer.C o l u m n5. D esign discharge Qpcom puted according to E q. (14.32); thus, the productof Columns6 and 11inTable 14.90.C o l u m n6 . Requ ired sewer diameter, as com puted by using E q.(14.31)or Fig. 14.13;

for Manning's formula withn = 0.015 anddrin ft, Eq. (14.3Ia) yields o Ydr=[0.0324-j

in whichQp, in ftVs, isgiveninColumn5 and S0is inColumn4.C o l u m n 7. The nearest commercial nominal pipe size that is not smaller than therequired size is adopted.Column8. Flow velocity com puted asV = Q/A that is, it is calculated as Colum n 5multipliedby4 /n anddividedby thesquareo fColumn7. Asdiscussed inYen (1978b),there are several w ay s to estimate the average v elocity of theflow through the length of the sewer. Since the flow is actually unsteadyand nonuniform, usu ally the one used here, u singfull pipe cross section,is agood approximation.Column9. Sewer flow time is com puted as equal to LIV , that is, Column 3 dividedby Column 8 and converted into minutes.


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TABLE 14.9 Rational Method D esignofSewersofExample D rainage BasinA(a)D esign Rain Intensity

UpstSewerTi(m(8

Inlet Time(min)

(7 )- L C H

(6 )CA< 5 )

R u n o f fC o e f f i c i e n tC j(4)

Areaai

(acres)(3 )

Directly DrainedCatchment o rContr ibu t ingUpstream Sewer( 2 )

Sewer

(1 )6.29.311.712.99.311.713.112.9

3.71.2

7.9

10.9

1.62.11.23.03.71.23.07.9

0.80.70.40.6

0.6

2335

5

IIImIVii1 2

V21

11-21

12-2121-31

31-41


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TABLE 14.9 (Continued)( b ) Sewer Design

S L

( f t )( 1 0 )

SewerFlowT i m e( m i n )

( 9 )

FlowV e l o c i t yV( f t / * )

( 8 )

DiameterU s e dJ n

( f t )( 7 )

RequiredDiam.d r( f t )( 6 )

DesignDischargeQ P

( f t 3 / * )( 5 )

S l o p eS

( 4 )

LengthL

( f t )( 3 )

U p s t r e a mM a n h o l eG r o u n dE l e v .( f t )( 2 )

Sewer

( 1 )3.651 0 . 44.007.207.80

1.40.91.0

5.46.66.7

2.001.002.502.752.50

1.980.992.432.532.50

17.15.233.044.444.4

0.00810.02900.01000.01440.0156

450360400500500

35.0041.5031.9028.7028.70

1 11 22131(31


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C o l u m n 1 0 .P roduct ofColumns3 and 4; thisis the elevation difference between th etw o endsof the sewer.C o l u m n 1 1 .The upstream pipe crown elevation of Sewer 11 is computed from thegroundelevation minus the minimum soil cover, 4.0 ft, to savesoilexca-

vationcost. In this example, sewers are assumed invert aligned except thelast one (Sewer 31), which is crown alignedat itsupstream (23.85 ft forupstream ofSewer31 anddow nstreamofSewer 21) toreduce backwaterinfluence from th ewater level atsewer exit.C o l u m n 1 2 . Pipe invert elevation at the upstream end of the sewer, equal toColumn11m inus Column7.C o l u m n 1 3 .P ipe crown elevation at the downstream end of the sewer, equal toColumn 11m inus Column 10.C o l u m n 1 4.P ipe invert elevationat thedow nstreamend of thesewer, equaltoColumn13m inus Colum n 7. For the last sewer, the dow nstream invert elevationshould be higher thanthe creek bottom elevation, 11.90ft .The above examp le dem onstrates that, in the rational m ethod, each sewer is designedindividually and independently, except th e computation of sewer flow time for the pur-pose of rainfall duration determination for the next sewer, that is, the values of t f inColum n 8 of Table 14.9a are taken from those in C olumn 9 of Table 14.96.The pro file of the exam ple designed sewers are sho w n as the solid lines in Fig. 14.14.If thewater levelof the creek downstreamofSewer 31is ignored, theoretically acheap-erdesign couldb eachievedbyputtingtheexit Sewer 31o n aslightly steeper slope, from0.0144to0.0156 to reduce the pipe diameter from 2.75 to 2.50 ft. The new slope can beestimated from HH* 1434)

This alternativeisshow n withthe parentheses inTable 1 4 . 9 Z ? and asdashed lines in Fig.14.14b.How ever, one shou ld be aw are that the w ater level of a 10-year flood in the creekis 20.00ft andhence,th e last seweris actually surchargedand itsexit issubmerged. Thesewer will not achieve the design discharge unless its upstream manhole is surcharged byalmost4 ft(20.00-16.05).Therefore, th eoriginal designof2.75 ftdiameter is asafer andpreferred option inview of thebackw atereffect from th e tailwater level in the creek. Infact, Sewer 21-31m ayalsobe surchargeddu e to thed ownstream backwater effect.Sometimes, a backwater profile analysis is performed on the sewer netwo rk to assessthedegreeofsurchargein the sewersandm anholes.In suchananalysis, energy losses inth epipesandmanholes shouldberealistically accou nted for. H ow ever,theintensity-dura-tion-frequency-based design rainfall used in the rational method design is an idealistic,conceptual, simplistic rain and the probability of its future occurrence is nil. The actualperformance of the sewer system varies withdifferent actual rainstorms, each hav ing dif-ferent temporal and spatial rain distributions. But it is impossible to know th e distribu-tionso fthese futurerainstorms, whereasth e ideal rainstorms adoptedin thedesignof therational method are used as a consistent measure of protection level.Although designingsewers using the rational method is a relatively simple and straightforward matter, check-ing th eperformance of the sewer system is a far more complex task requiring thoroughunderstandingof thehydrologyand hydraulicsofw atershed runoff. For instance, check-ing thenetwork performancebyusinganunsteadyflowsimulation model w ould requiresimulation of theunsteadyflow invarious locations in thenetw ork accountingfor losses


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in sewer pipes as well as in manholes and junc tions(thelatter will be discussed in the nextsection).Moreover, for a given sewer netw ork layout, by using different sewer slopes, alterna-tive designsof thenetwork sewers can be obtained.Acost analysis shouldbe conductedto select the most economic feasible design. This can bedone witha system optimizationmodel sucha s Illinois Least-Cost Sewer System Design Model (ILSD) (Yenet al., 1984).74.4 HYDRAULICSOFSEWERJUNCTIONSThere are various auxiliary hydraulic structures such as junctions, manholes, weirs,siphons, pumps, valves, gates, transition structures, outlet controls, and drop shafts in asewer network. Information relevant to design of most of these apparatuses are welldescribed in standard fluid mechanics textbooks and references, particularly in theGerman text by Hager(1994)a ndFederalHighw ay Adm inistration (FHW A,1996).In thissection, themost important auxiliary component inm odeling thesewer junction s aredis-cussed. For sewers of com mo n size and length, the headloss for the flow through a seweris usually two to five times the velocity head. Thus, the head loss through a junction iscomparable to thesewer pipe loss, and is not aminor loss.

14.4.1 Junction ClassificationsA sewer junction usually has three or four sewer pipes joined to it. U nder normal flowconditions, onedownstream pipereceives the outflow from thejunction and other pipesflow into the jun ction. How ever, junc tions w ith only two or more thanfour joining pipesare not uncom m on. The m ost upstream junction s of a sewer network are usually one-wayjunctions having onlyone sewer connected to ajunction.The horizontal cross sectionofthe junctioncan becircular or squareor may be another shape. Thediameter or horizon-tal dimension of a jun ction norm ally is not smaller than the largest diameter of the join-ing sewers. Toallow theworkers room tooperate, usually junction s are not smaller than3 ft (1 m ) indiameter. Forlarge sewers, theaccessto thejunctioncan be smaller thanthediameter of the largest joining sewer.Sewers may join a junction withdifferent vertical and horizontal alignments, and theym ay have different sizes and slopes. Vertically, the pipes mayjoin at the junction withtheir centerlines or inverts or crowns aligned, or with any line of alignment in between.There is no clearly preferred alignment that could simultaneously satisfy the requirementsof good h ydraulics at low and high flows w ithou t com plicating either construction cost ordesign. The bottom of the jun ction is usually at or slightly lower than the low est invert ofthe joining sewers.In the horizontal alignment, often th e outflow sewer is aligned with one (usually themajor) inflow sewer in a straightlinewith other sewers joining at an angle. For cities w ithsquare blocks, right-angle junction s aremost common. Typical sewer benching and flowguidesin junct ions areshowninFig.14.15.Withthe alignment of the joining pipes and the shape and dim ensions of jun ctions notstandardized, theprecise, quantitative hydraulic characteristics of thejunctions varycon-siderably. As a result, there are many individual studies of specified junctions, but a gen-eral comprehensive quantitative description ofjunctions is yet to be produced.For the purpose of hydraulic analysis, junc tions can be classified according to the fol-lowing scheme (Yen, 1986a):


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D i r e c t l y O p p o s e d L o l e r o l W i t h D e f l e c t o r( H e o d L o s s e s A r e S l i l l E x c e s s i v e W i t hl h i s M e t h o d . B u t A r e S i g n i f i c o n t l yL e s s T h o n W h e n N o D e f l e c t o r s E x i s t )

B e n d W i t h S t r o i g h t D e f l e c t o r

B e n d W i t h C u r v e d D e f l e c t o r

I n l i n e U o s t r e o m M o i n 9 (J L o t e r o lW i t h D e f l e c t o r

F I G U R E 14.15 J u n c t io n b e n c h i n g o f s e w e r s a n d f l o w g u i d e s .

1. According to the geometry: (a) one-way junction , (b) two -wa y junction , (c) three-wayjunction merging(two pipes flow intoonepipe)ordividing (one pipeflows intotw o pipes), and (d) four- or more-way junction m erging, dividing, ormerging anddividing.2. According to the flow in the joining pipes: (a ) open-channel junction (with open-channel flow in all joining pipes), (b) surcharge junction (with all joining pipes sur-charged), and (c) partially surcharged junction (with some, but not all, joining pipessurcharged).3. According to the significance of the junc tion storage on the flow: storage junction orpoint junction .

Hydraulically, the most important feature of a junc tion is that it imposes backwatereffects to the sewers connected to it. A jun ctio n provides, in addition to avolume how-ever small of temp oral storage, redistribution and dissipation of energy, and mixing andtransfer ofmom entumof the flowand of thesediments andpollutantsit carries. The pre-cise,detailed hydraulic description of the flow in asewer junction is rather complicatedbecauseo f thehighdegreeofmixing, separation, turbulence, andenergylosses. However,correct representation of the junction hydraulics is important in realistic simulation andreliable comp utation of the flow in a sewer system (Sevuk and Yen,1973).

D e p r e s s e d F l a t

H a l f F u l l


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14.4.2 Junction Hydraulic EquationsThe c ontinuity equation of the water in a jun ction is

ds1,Q1+Qj = (14.35)where Q1 = th e flow into or out from th ejunctionby the /-thjoining sewer, being posi-tiveforinflow andnegativeforoutflow; Q. = th edirect, temporally variable water inflowinto (positive) or the pum page or overflow or leakage out from (negative) the junc tion, ifany;s = th estorage in thejunction;andt =time.For atwo-way junction,th eindex i =1,2; for athree-way junction , / = 1, 2, 3, and so on.The energy equation in a one-dimensional analysis form is

T / 2 p N fly T / 2_i+_i+Zjj+fi =,_+lG_ (14.36)where Z1 ., P 1 , V 1 = thepipe invert elevation above the reference d atum ,piezom etric pres-sure above the pipe invert, and velocity of the flow at the end of the section of the /thsewer w here it m eets the jun ction , respectively; H j =the net energy input per unit volum eofth edirect inflow expressed inwater head; K 1 = th eentrance orexit loss coefficientfo rthe /th sewer; Y th e depth ofwater in thejunction;and g = th e gravitational accelera-tion. Thefirst summ ation term in E q. (14.36) is the sum of the energy input and output bythe joining pipes. The second term at the left-hand side of the equation is the net energybrought in by the direct inflow. The first term to the right of the equal sign is the energystored in thejunctionas itsw ater depth rises.T he last term is the energy loss.Themom entum equationsfor the two horizontal orthogonal directions x andzare

1(2 ,)=Jg^JA (14.37)and

2(avfe)=JgydA (14.38)wherepx andpz = th ex and zcom ponents of the pressure acting on thejunction bound-ary,respectively, and A = the solid and w ater boundary su rface of the junction . The directflow Q J is assumed entering the junction without horizontal velocity component. Theright-hand side term of Eqs. (14.37 and 14.38) is thex or z component force, where theintegration is over the entire junction bound ary surface A. The left-hand side term is thesu m ofmom entumof the inflow and outflowof the joining pipes. Note that for a three-way merging junction, two of the Q. s arepositive and one Q.is negative, whereas for athree-way dividing junction, two of the Q. s are negative.

Joliffe (1982), Kanda and Kitada (1977), Taylor (1944), and others suggested the useofm om entum approach to deal with high velocity situations. To illustrate this ap proach,consider the three-way junction show n in Fig. 14.16.The control volume of water at thejunctionenclosed by the dashed line is regarded as a point, and there is no volum e ch angeassociated with a change of depth within it. One of the two merging sewers is along thedirection of the downstream sewer, whereas the branch sewer makes an angle (pwith it.Whenoneassumes thatthe pressure distribution ishydrostatic and the flow issteady, theforce-momentum relation can be written as


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F I G U R E 14.16 C o n t r o l v o l u m e o fj u n c t io nf o r m o m e n t u m a n a ly s i s .

Jh 2A 2+Jh3A 3cos 9 -Jh sin(p- Y 1 A 1 +F (14.39)= P Q1V 1- P Q2V 2 - P Q3V 3cos c p

whereA = the flow cross-sectional area in a sewer; h = depth o f the centroid of A; y= thespecific w eight ; p = the density of water; Q = th e discharge; V = Q I A = thecross-sectional mean velocity; and F = the sum of other forces that are normallyneglected.Some oftheseneglected forces are the co m ponent of the w ater w eight in thecontrol volume along the small bottom slope, the shear stresseson the walls and bo t-tom, and the forcedue to geometry of thejunct ionif the sewersare notinv ert alignedor the longitud inal sewers are of different dimensions. The subscripts 1,2, and 3 iden-tify th e sewers shown in Fig. 14.16, and brepresents th e exposed wall surface of thebranch in the control volume shownasab in the figure.For the special case ofinvertalignedsewers withth ebranch (pipe3)joiningatright angle, (p = 90,E q. (14.39) canbe simplified as

A 2(gh 2+\q)= A1^h1 + V f) (14.40)or

&JW(SW)+ 1T (1441)Q 1 LUJ(sW)+ i J ( 'Based onexp erimental resultsof invert-aligned equal-size pipes merging with(p = 90,Joliffe (1982) observed thattheup stream d epth / I1 = h2andproposed that

7r=7T= -b (14-42)hcl hclwhere hc l= the critical depth in the do wn stream sewer,F3 = the Froude number of theflow in thebranch sewer,and

^=0.999- 0.482(- 1- 0.381(^-T (14.43)(Qi) (Qi)

b=0.514- 0.067[ ]+0.197[ [- 0.122^T (14.44)(Qi) (Qi) (Qi)Theequation d escribingtheloadofsedimentorpollutants, exp ressed interm sofcon-centration c, can bederived from theprinciple ofco nservationasJcds+ I Qf1 + Q f j =G (14.45)


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wh ere G = a source posit ive) or sink nega tive) of the sedim ent or po llutan ts in thej u n c t io n .Equations 14.3514.44)are the theoretical basic e qua tions for sewer ju nc tions. Theyare applicable tojunct ion s under surcharge as wel l as openchannel f low s in thejo iningpipes. However, more specific equations can be written for the pointtype and storagetype junc t ions.14.4.3 Experiments on Three Way Sewer Junctions andLoss CoefficientsProper handl ing of f lo w in se we r networks required informat ion on the loss coeff icients at the jun ct io ns . U nfortunate ly , there e x is t s prac t ica l ly n o u s e f u l q u a n t i t a t i vein formation o nenergy an dm o m e n tu m l o s se s o f u n s t e a d y flow p a ss i n g t hr o u g haj u n c t ion. Therefore, s teady f low inform at ion on s e w e rj u n c t i o nlosse s a r e co m m o n ly u se das ana p p r o x i m a t i o n .Table 14.10 sum m arizes the experim ental conditions of threeway m erging, surcharging,topopen sewer jun ctio ns condu cted by Johnston and Volker 1990),Lindyal l 1984),and Sangster et al. 1958, 1961).Also l isted in thetable are theexperiments by Blaisdellan d Mason 1967), Serre et al. 1994), andR amamu rt hy and Zhu 1997); these experiments were notconductedo n opentop sewer junctionsbut on threeway merging closedpipes. They are listed in the table as an example because these tests wereconducted withdifferent brancha ndma in diam eter ratios andw i t hdifferent pipe alignm ents. Hence, theym ay provide he lpful informat ion fo r sewer jun ct ion s. There ex is t s considerablymore information on merging or dividing branched closed conduits than on sewer junctions. T he reader m ay look elsew here e.g., Fried and Ide lch ik, 1989; M iller, 1990) fo rinformation about centerlinealigned threeway joining pipes as an approximation tosewer junct ion s.The loss coeff icientsK21 an d 31 for them erging f low aredefined as

fv fv2_\\tz h>z\\tzh \\t L L fS 1446)

jFigure 14.17 shows th e experimental results of 1987) and Sangster et al. 1958) andLindv all for the case of iden tical pipe size of the m ain and 90 m erging lateral. The corresponding curves suggested by Miller 1990) and Fried and Idelchik 1989) for threeway identical closed pipe junctions are also shown as a reference. The values of the losscoefficients in a sewer junction that is open to air on its top are expected to be slightlyhigher than the enclosed pipe jun ction cases given by M iller because of the water volum eatth ejunct ion above th e pipes.Theeffect of therelative sizeof thejoining branch pipeis showninFig. 14.18.T heexperimentaldata of Sa ngster et al. 1961)have identical upstream pipe sizes,D2 =D3 fo r fourdifferent valuesoflateral branchtodo wn stream m ain pipe area ratio, A3M 1.T hedataofJohnstonan dVolker 1990)onsurcharged circular opentop sewerjunctionare notplottedin Fig. 14.18becau se the m ainline pipe area ratio A 2M 1 = 0.41 instead of unity in the figure. Conversely,asacom parison,thesmoothed curveof ^ T2 forA3M 1 = 0.5 of thethreeway pipe junctionofSerreet al. 1994)with A1= A 2 is plottedinFig. 14.18a,andtheir experimental curves of ^ T3forA3M 1 = 0.21 and0.118areplottedinFig. 14.18b.Also shownin thefigure,a s reference,are thethreeway pipe junction curves fo r different values of A3M 1 suggested by Fried andIdelchik 1989)and M iller 1990)for identical size of m ain pipes,A2 =A1. The experimentsof Sangsteret al. 1961)indicatedthat for agiven A3M 1,th eeffect O fA2M 1 on theloss coefficients ism inor. Therefore, their curves shouldbecom parable with thoseofFriedandIdelchik,


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TABLE 14.10 Experimental StudiesonThree-way JunctionofMerging Surcharged ChannelsP i p e AlignmentatJLone r t i c a lC h a n n e lS l o p eS h a p e o f Channelsy p e o f Junctione f e r e n c e

Straighone 90

Straigh90me

Centerlslight dlateralStraighmerginby 15Straigh90 me

Straigh90 em

Flushed bottom

Center aligned

Flushed bottom

Centeror topalignedCenter aligned

Same height

Horizontal

Horizontal

Horizontal

(Horizontal)

(Horizontal)

Circular,D = 3.0 in.3.75 in.4.75 in. or5.72 in.Circular,D mam = 144 mm,A > / A n a i n =1A0.686, or0.389Circular,A n a i n d = 7 0^Hl,/Uu p / A n a i n , = 0.64, > b / D m a i n , =0.91Circular,DJD^ = 0.25 ~ 1.0Circular,^ m a i n =4441 11'/V/U, = 0.14,0.23,0.34, or 0.46Rectangular,4.1 4mm high,main width91.5m m,branch width20.4 mm,70.5 mm,or91.5 mm

Square,rectangular, orroundboxRoundbox

Square box

EnclosedpipejunctionEnclosed pipejunction

Enclosedrectangularconduit junction

Sangster et al.(1958, 1961)

Lindvall(1984, 1987)

Johnston andVolker(1990)

BlaisdellandMason (1967)Serreet al.(1994)

Ramamurthya n d Zhu(1997)


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S a n g s t e r e t a l . ( 1 9 5 8 )L i n d v a l l ( 1 9 8 7 )L i n d v a l l ( 1 9 8 4 )M i l l e r ( 1 9 9 0 )F r i e d l d e l c h i c k( 1 9 8 9 )

M a r s a l e k ( 1 9 8 5 ) M 1M a r s a l e k ( 1 9 8 5 ) M 3d e G r o o t B o y d ( 1 9 8 3 )S a n g s t e r e t a l . ( 1 9 5 8 )L i n d v a l l ( 1 9 8 4 ) T y p e 1M i l l e r ( 1 9 9 0 )F r i e d l d e l c h i k( 1 9 8 9 )

F I G U R E 14.17 E x p e r i m e n t a l headless c o e f f i c ie n t s f o r s u r c h a r g e d 3 - w a y s e w e r j u n c t i o n w i th i d e n t i c a lp i p e s i z e s a n d 9 0 m e r g in g l a t e r a l , ( a ) M a i n l i n e l o s s c o e f f ic i e n t K2 1 ; ( b ) B r a n c h l o s s c o e f f i c i e n t K3 1 .Miller, and Serre etal.How ever, Fig.14.18depicts considerable disagreement among the dif-ferent sources, indicatingtheneedformore reliable investigations.The joining angle of the lateral branch is a significantfactor affecting the loss coeffi-cients, particular on K ^ PThevaluesof theloss coefficients decrease if the joining anglemore or less aligns with the flow direction of the main, and increase if the lateral flow isdirected against themain.Thereferences ofFried andldelchik (1989) and Miller (1990)provide some idea on the adjustment needed for theKvalues due to the joining angle.

M i l l e r ( 1 9 9 0 ) S a n g s t e r e t a l ( 1 9 6 1 )

S e r r e e t a l (1 9 9 4 )M i l l e r

F r i e d l d e l c h i kS a n g s t e r e t a l ( 1 9 6 1 )

S e r r ee t a l

F I G U R E 14.18 Headless c o e f f ic i e n t s f o r s u r c h a r g e d 3 - w a y j u n c t i o n w i th 9 0 m e r g in g l a te r a l o f d i f f e r -e n t s i z e s , ( a ) M a i n l i n e l o ss c o e f f i c ie n t K2 1 . ( b ) B r a n c h l o s s c o e f f ic i e n t K3 1 .


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T o w n s e n d P r i n a 1 9 7 8M a r s a l e k1985

F I G U R E 1 4 .1 9 Headless c o e f f ic i e n t s f o r 3 - w a y o p e n - c h a n n e l s e w e r ju n c t i o n w i t h i d e n t ic a l p i p es iz e s a n d 9 0 m e r g i n gl a t e r a l , ( a )M a i n l i n e lo s sc o e f f ic i e n t K 2 1 ( B )B r a n c h l o s s c o e f f i c i e n t 31 ( A f t e r e n 1 9 8 7 ) .ListedinTable 14.11is asumm aryof experiments onsteadyflow inthree-way m erg-ingopen-channel jun ction s. M ostof thestudies were done with point-type junc tions.T heexperimental subcritical flow results of storage-type junctions by Marsalek (1985) andTownsendandP rins(1978) are plotted in Fig. 14.19for lateral join ing 90 to the same sizemainline pipes. Yevjevich and Barnes (1970) gave the combined main and lateral losscoefficient but not theseparate co efficients, makingtheresultdifficult to beusedinrout-

in g simulation.T hepointsin the figurescatter considerably,but theyaregen erally in thesamerangeof the loss coefficient values fo r surcharged three-way 90m erging junctionexceptK3 , forTownsenda ndP rins ' data.It isinteresting tonote thatthemost frequentlyencountered sewer junctions are three- and four-way box junctions with unsteady sub-critical flow in the joining circular sewers. None of theopen-channel experiments w asconducted under these conditions. All were tested with steady flow. It is obvious thatexisting experimental evidence and theory do not yield reliable quantitative informationonth elosscoefficients ofthree-way sewer junc tions. B efore more reliable informationisobtained,prov inciallyfo rdesignandsimu lationofthree joining identical size sew ers, fo rK 2 j acurve drawn between thatofLindvall an d thatof Sangsteret al. can be used as anapproximation. For 3},thecurve ofL indvall can beused.Forjoining pipes ofunequalsizes, the curv es of Sang ster et al. appear to be tentatively acceptable.

14.4.4 LossCoefficient for Tw o-W ay Sew er JunctionsTwo-way junctions are used fo r change of pipe slope, pipe alignment, or pipe size.E xperimental studies on tw o-w ay, surcharged, top-open sewer jun ction s are listed in Table


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TABLE 14.11 Experimental Studieson Three-W a yJunctionofMerging Open Channelsi p e AlignmentatJunction

ongitudinale r t i c a lC h a n n e lS l o p eS h a p e o fChannelsT y p e o fJunctionR e f e r e n c e s

Straight throughandone mergingchannel at 45or 135Straight throughandone mergingchannel at 51

Straight throughandone mergingchannel at 15,30, or 45Straight throughandone mergingchannel at 30,60, or90Straight throughandone 90merging pipe

Flushedbottom

Flushedbottom

Flushedbottom

Flushedbottom

Flushedbottomor crownaligned

Horizontal

0.0062, 0.012

Each channelslope variedindependentlyHorizontal

0.000080.000540.00107

Rectangular,identical width,B = 4in .

Trapezoidal,identical width ,B =7.2 in.

Rectangular ortrapezoidal(side slope 1:1)Rectangular,B = 5in.

Circular,A n a i n =6'25in-D bT = 1.87 in.

Point

Point

Point

Point

Squarebox

Taylor(1944)

Bowers (1950)

Behlke andPritchett(1966)Webber andGreated (1966)

Y e v j e v i c h a n dBarnes (1970)


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TABLE 14.11 ( C o n t i n u e d )

Straight throughandone mergingchannelat 3060,or 90

Straight throughandonemergingchannelStraight throughandonemergingchannel at 45or90Straight throughandone 90merging channelStraight throughandone 90merging channelStraight throughand one mergingchannel at15,45,70,or 90

Flushedbottom

FlushedbottomInvert dropacrossjunctionboxInvert drop(15 or18mm)FlushedbottomNot available

Flushedbottom

Horizontal

Horizontal

Lessthan0.01

Horizontal

Horizontal,0.0001,0.0075,0.005, or0.01Channel slopeadjustableto achieveequilibriumwater depthHorizontal

Rectangular,* m a i n= 1 0 0 , 2 0 0 ,400mmB br= 100mm;Circular, d i f f e r e n tsizesCircular,d i f f e r e n t sizesCircular,A n a i n =161 11'D b T = 102 mmRectangular,B = 457 mmCircular,equal diameter,D =6 9 m mRectangular,identical width,B = 0.5 ft

Circular,identical diameter

Point

Point

Rectangularbox

Point

Point

Point

Square boxorroundbox

KandaandKitada(1977)

RadojkovicandMaksimovic(1977)Townsend andPrins (1978)

Lin and Soong(1979)J o l i f f e (1982)

BestandReid (1984)

Marsalek(1985)


40/45

TABLE 14.12 Experimental Studies on Straight-Through T w o - W a yOpen-Top Junction of SurchargedP i p e AlignmentatJuh a n n e l S l o p eh a p e o f Channels Longitude r t i c a lT y p e o fJunctionR e f e r e n c e s

Straightthroughor 90 beStraight tor 45bejunctionbend dowo f junctiStraightor30 orbend injStraight

Straight

Straight

Straight t

Straight t

FlushedbottomFlushedbottom

FlushedbottomFlushedbottomCenteralignedFlushedbottomCenteralignedCenteraligned

Horizontal

0.0094-0.0192

0.002 and 0.010

Horizontal

Horizontal

Horizontal

Horizontal

Horizontal

Circular,D = 3.0, 3.75,4.75, or 5.72 in.Circular, identicaldiameter, D = 6in .

Circular,identicaldiameter,D = 102 mmCircular, identicaldiameter,D = 88 mmCircular, identicaldiameter,D = 144 mmCircular, identicaldiameter,D = 6inCircular, identicaldiameter,D = 88 mmCircular, identicaldiameter,D = 90 mm

Square,rectangular,orroundboxRectangularbox

Rectangularbox orroundboxRectangularbox orroundboxRound box

Square boxorroundboxSquare box

Roundbox

Sangster etal.(1958, 1961)Ackers (1959)

Archeret al.(1978)HowarthandSaul (1984)Lindvall(1984)Marsalek(1984)JohnstonandVolker (1990)Bo PedersenandMark(1990)


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14.12.All the experim ents were con ducted with the same size upstream and dow nstreampipes joinin g th e junction. Only Sangster et al. (1958, 1961) tested also th e effect ofdif-ferentjoining pipe sizes. These experimental results show thatfor astraight-through, two-wayjunct ion,thevalueof theloss coefficient isusuallynohigher than 0.2. Alignmentofthe joining pipes and benching in the junc tion are also im portant factors to determine thevalue of the loss coefficient.Figure 14.2Oa showsth e headloss coefficiet of a surcharged tw o-wa y open-top junc-tion connecting tw o pipes of identical diameters aligned centrally given by the experi-mentsofArcheret al. (1978), Howarthan d Saul (1984), JohnstonandV olker (1990) andLindvall(1984). No ticeable is the sw irl and instability ph enom ena w hen the jun ction sub-mergence (junction depth to pipe diameter ratio) is close to two and the correspondinghigh head loss coefficient. The ranges of loss coefficient given by Ackers (1959),M arsalek (1984), and S angster et al. (1958) are also indicated in Fig. 14.2Oa,but the dataonth evariation withthe pipe-to-junction-size ratiow as notgivenby these investigators.Sangster et al. (1958) also tested the effect of different sizes of joining pipes fo r sur-charged two-way junction. Some of their results areplotted inFig. 14.2Ob.They did notindicate a clear influence of the effect of the size of the junction box. However, B oPedersen and Mark (1990) demonstrated that the loss coefficient of a two-wayjunctioncan be estimated as a combination of the exit headloss due to a submerged d ischarging jetand the entrance loss of flow contraction. They suggested that th e loss coefficient Kdepends mainly on the size ratio between the junction and the joining pipes of identicalsize. For aninfinitely large storage jun ction,the theoretical limitofKis 1.5.For thejunc-tion-diameter to pipe-diameter ratio, D JD less than4, they proposed to estimate th eKvalues accordingtobenchingasshowninFig. 14.21.

14.4.5 Sto rage JunctionsFor a storage- (or reservoir-) type junction, th e storage capacity of thejunction is rela-tively large in comparison to the f

hydraulic design of urban drainage systems

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