1Flow in drainage systemsThis chapter outlines the basic design requirements of surface drainage systems,withspecialreferencetothetraditionallyappliedopenchannels,andthemorerecently introduced linear drainage channels. Since the adequacy of both systemsrequires an efficient transportation of rainwater from the area to be drained, thechapter includes the two prevailing methods for calculating the rainfall runoff, theLloyd-Davies formula, and the Wallingford procedure, with examples to illustratetheir application.1.1 Highway drainage systemsTheneedforremovalofstormwaterfromroadsurfaceswasrealizedduring early days of road construction. Romans, for instance, introduceddrainagetotheirroadsystems.Later,industrialadvancementrequiredraisingthestandardofhighways,whichcouldnotbeachievedwithoutincorporating an efficient method of disposal of surface water, in additionto cambering the road. Thus a surface drainage system became a necessity,and its design is now an integral part of modern highway design.The traditional, and still vastly popular, method for the disposal of surpluswater from road surfaces is by means of open channels built adjacent to thekerb.Thesechannelsarepreferablytriangular,withthekerbfaceasthevertical side, and a limited width of the carriageway, with a certain amount ofcross fall, as the sloping side (Fig. 1.1). Other shapes, such as rectangular ortrapezoidal, are less common. The allowable width of flow that can be allowedin roadside channels is normally governed by the safety requirements, anddependsonthetypeofroadconstruction.Forinstance,intrunkroadsitshould not exceed 15m if there is a hard shoulder, and 10m if there is a hardstrip. For shopping precincts the value is limited to 05m. The limitation onthe width of flow restricts the cross-sectional area of flow. Hence, in order tokeep the area of cross section of flow within the prescribed limits, the flow inthe channel has to be intercepted, at appropriate points, by means of gullies10131/7/0311:06Page 1with grated tops, or by openings in the kerb, called kerb inlets. In this way, thewater is removed from the road surface, and is directed, sometimes throughFrenchdrains,toareceivingchannelthatmaybe,forinstance,anaturalwatercourse in the countryside, or a specially designed storm sewer in theurban area (see Chapter 5 of reference [1.1]).An alternative system of drainage that is becoming popular is the lineardrainage system. Linear drainage channels are closed conduits, constructedby assembling precast units, and admit water through an inlet mechanismincorporated into the system. A linear drainage channel may be one of thefollowing two types:(a) Channels with grated top, flush with the road surface (Fig. 1.2). Suchachannelismadeofprecastunits,eachoflengthoftheorderof 2Design of linear drainage systemsPlanCross sectionRoad (pavement)Hard shoulderwith cross fallKerbFlow from gullypot or kerb inletChannelKerb Figure 1.1 A conventional drainage system.0131/7/0311:06Page 205 to 10m, and of the cross section, preferably, either circular with avertical neck, or of a U shape (Fig. 1.3).(b) Channelscombinedwiththekerb,theso-calledcombinedkerb-drainage channels (Fig. 1.4). Such a channel is also made of units of thesame length as in (a) above, and manufactured in variously shapedcross sections. Eachunitofacombinedsystemconsistsoftwoparts,thetopandthebase blocks (Fig. 1.5). An orifice, to allow free access of water into the channel,is located in the top block. At a pedestrian crossing, the kerb is dropped, andthe top block dispensed with. In this situation, only the base block providesthe necessary drainage capacity (see Chapter 11).The depth of flow in a linear drainage channel is also restricted by theavailable depth of the channel, requiring the flow to leave the channel atsuitablylocatedoutfalls,andconveyedtoareceivingchannel.Thus,theoutfalls of a linear drainage system serve the same purpose as the gullies ofthe traditional system. The growing popularity of linear drainage systemsismainlyduetotheirrequirementoflessamountofexcavationand,possibly, minimum obstruction with the services. A linear drainage systemis also more flexible in the sense that it is applicable to large paved areas3Flow in drainage systemsFigure 1.2 A linear drainage channel with grated top.0131/7/0311:06Page 34Design of linear drainage systemsCircular pipeGratingU-shaped channel GratingInvertFigure 1.3 Preferred cross sections for linear drainage channels of type (a) with grated tops.OrificeOrificeBaseTopBaseFigure 1.5 Units of combined kerb-drainage channel (type (b)).Figure 1.4 A combined kerb-drainage system of type (b).0131/7/0311:06Page 4such as motorways and airport runways, as well as to restricted areas suchas car parks and shopping precincts.The method of computing flow in linear drainage systems also applies toroofgutters.Aroofgutter(Fig.1.6)admitsrainwaterfromtheroofsurfaceatauniformrateoveritsentirelengthand,hence,therateofdischarge increases linearly along the length of the gutter.1.2 Design criteria of drainage systemsThebasicdesignrequirementofadrainagesystemistomaintainbalancebetween the volume of water to be removed, and the flow-carrying capacityof the system. The volume of water required to be removed by a drainagechannel is routinely determined from the runoff formula of the type(1.1)where Crcoefficient of runoff, I average intensity of rainfall (mm/hr),Assurface area to be drained (m2), and Qmaxrate of discharge (m3/s).Thecoefficientofrunoffistheratioofthevolumeofwateractuallytransported by the drainage channel to the total amount of rainfall on thesurface,anditsvaluedependsonthepermeabilityofthesurfacetobedrained. In the case of an impervious surface, the coefficient of runoff mayrepresent the loss of water due, for instance, to cracks in the surface, andponding.Atypicalurbandrainageschemeconsistsofamainchannel,whichisusually a pipe, with several tributary channels. The total area to be drainedQC IAmaxr s 36 1055Flow in drainage systemsFigure 1.6 A roof gutter with two vertical outlet-pipes.0131/7/0311:06Page 5is divided into sub-areas, each sub-area acting as a catchment area for a pipelength that is defined by a manhole at each end. Hence, the only parameterin Eq. (1.1) to be determined by external means is the intensity of rainfall.This is discussed in the following section. In a highway drainage scheme,theareatobedraineddependsonthelengthofthedrainagechannel,which is, generally, not known at the start of the computation. Hence, thereis an element of trial and error in computing the area to be drained, and thevolume of flow to be carried by the channel. This is dealt with in Chapter 9.Ithasbeencommonpracticetodesignahighwaydrainagesystem on the assumption that the drainage channel flows full, that is, it occupiestheavailableareaofcrosssectionentirely.Thisallowstheconditionofuniformflowtoprevail(seeChapter3),andthedischarge-carryingcapacity of the channel is obtained by multiplying the mean velocity of flowwiththeareaofcrosssectionofthechannel.Oneimplicationofthismethod is that the capacity of the channel is independent of the length ofthe channel, which is not the case of a linear drainage channel.In the event of rainfall, there is a regular inflow of water into the channel,causingtherateofflowinthechanneltorise,usually,fromzeroattheupstream end, to a maximum value at the downstream end. The increasein discharge causes both the depth and velocity of flow to also vary, notably,if the channel is of constant cross section. The varied-flow theory takes thisinto consideration. The theory assumes, however, that the flow entering thechannellaterallyisconstant,thatis,therateofthelongitudinalflowincreases linearly.In a normal situation, there is no lateral inflow at the upstream end of a drainage channel, so that the discharge rate Q at a distance x from theupstream end is given by the equation(1.2)whereL isthelengthofthechannel,andQ0isthedischargeatthedownstream end of the channel (Fig. 1.7).If U0and A0are, respectively, the velocity and the cross-sectional area offlowattheoutlet,thenthecapacityofthechannelQ0isdefinedbytheequation(1.3)Hence,theabove-mentionedbalancebetweenthevolumeofwaterto be drained and the capacity of the transporting channel is translated intotheequalityofQmaxdeterminedfromEq.(1.1),andQ0calculatedfrom Eq. (1.3).Q U A0 0 0Q QxL06Design of linear drainage systems0131/7/0311:06Page 6The focus in the design of a traditional drainage system is on the spacingbetweentheoutlets,whichisdeterminedbytheefficiencyofthegullygrating, or of the kerb inlet. The gully efficiency is defined as the ratio of thevolume of flow entering the gully to the volume of flow approaching it. Thesame applies to the efficiency of the kerb inlet. Gully gratings are classifiedinto various grades. For instance, a gully may either be of a heavy duty (Dtype)withefficiencyvaryingfrom84%to99%,orofamediumduty(Etype)withefficiencyvaryingfrom70%to98%[1.2].Itisfoundthatthemain parameters on which the efficiency of a gully grating depends are thewidth of flow, and the longitudinal and side slopes of the channel estimatedimmediately upstream of the gully. A linear drainage channel is providedwith one or more outfalls which are units specially built for the purpose ofexpelling water. The efficiency at which the water enters and leaves a lineardrainage system is usually taken as 10, and the focus in design is on thecapacity of the channel.From the varied flow theory, it is found that the location of the maximumdepth of flow in a drainage channel varies with the gradient of the channel.For instance, if the channel is laid level then the depth of flow is maximumat its upstream end. If a small gradient is introduced, then the location ofthe maximum depth moves downstream. A small increase in the gradientcauses the deepest cross section of flow to move considerably downstream.Having reached the downstream end at a certain gradient, the maximumdepth stays there at all steeper gradients.Forachannelofconstantcrosssection,andlaidatauniformslope, themagnitudeofthemaximumdepthincreaseswiththelengthofthechannel, a longer channel admitting more water. Since the depth of flowcannotexceedtheavailabledepthofthesystem,thecapacityofthe7Flow in drainage systemsDirection of channel gradientxQmax QLChannel Figure 1.7 Inflow to a linear drainage channel.0131/7/0311:06Page 7drainage system must necessarily be related to the length of the channel.Hence, before deciding upon the capacity of a linear drainage system, it isimportanttofindboththelocationandthemagnitudeofthemaximumdepth for, although, the capacity is defined by the rate of discharge at theoutlet, the limiting depth of flow may, or may not be at the outlet.Asapreliminarytothecomputationofvariedflowinadrainage channel(tobediscussedinlaterchapters),Iconsiderthesimplecaseof a rectangular area to be drained by means of a rectangular channel, andcomparetheresultsoftheanalysisusingboththeuniformflowandthevaried flow theories. I assume that the channel is laid at a uniform gradientS0,whichmayhaveavaluerangingfrom0to1/10.Letotherdatabeasfollows:Width of channel B 02mDepth of channel h 024mThe Mannings formula of the uniform flow theory (Chapter 3, Eq. (3.5)),gives the mean velocity of flow(3.5)in which R is the hydraulic radius, and n the coefficient of roughness. Forthe channel flowing at full capacity,A area of cross section of flowarea of cross section of the channel 02 024 0048m2P wetted perimeter 02 048 068mR hydraulic radius A/P 00706mFor n 0011, say, we obtain from Eq. (3.5)The capacity of the channel Q0(in litres/second) is then given byThe values of Q0at various slopes calculated by this equation, which is theoutcome of the uniform flow theory, and the corresponding values obtainedbytheapplicationofthevariedflowtheoryareshowninTable1.1.Q AU S S0 0 048 0 15 527 745 3 U S S0 07060 011 15 5272 /3 0 0UR Sn2 30/8Design of linear drainage systems0131/7/0311:06Page 8Comparing these two sets of results, it is found that, for flat gradients, that is, up to about 1/100 in the example considered, the capacities calculated bythe uniform flow theory are on the conservative side. At steeper slopes, thedifferenceinthecapacitiesisnotverysignificant.However,thetablealsoshows that, for a given slope, there is a maximum allowable length of thechannel, that is, if the length of the channel exceeds this value then the depthof water will rise above the specified limit of 024m. Therefore, these lengthsmust represent the theoretical maximum spacing between outlets.1.3 Intensity of rainfallRainfall records suggest that, given the total amount of rainfall in an area,storms of shorter duration are more frequent than those of longer duration.This led to the evolution of the Bilham formula [1.3](1.4)whereN10numberofstormsin10years,D durationofstormsinminutes, and r total rainfall in inches.By a simple transformation, Eq. (1.4) can be written as(1.5)where Dstorm duration in hours, Nstorm return period in years, andI intensity of rainfall in mm/hr.IDND1 14 14 2 541 3 55 ( )/ N D r103 551 25 0 1 ( )9Flow in drainage systemsTable 1.1 Comparison between channel capacities obtained from uniform and variedflow theories (X distance of deepest cross section of flow from the upstream end)S0Uniform flow Varied flowU (m/s) Q0(l/s) L (m) X (m) U0(m/s) Q0(l/s)0 0 0 149 0 100 4801/1000 0491 2357 216 148 114 5741/500 0694 3333 265 217 122 5861/200 110 5270 378 357 137 6581/100 155 7453 520 520 149 7151/50 220 10540 745 745 214 10271/20 347 16665 1188 1188 342 16421/10 491 23568 1685 1685 487 23380131/7/0311:06Page 9TheBilhamformulaiswidelyusedintheUnitedKingdom,althoughpreferenceissometimesgiventoaformuladepictingthelocalrainfallrecords, for instance, the Birmingham formulaHowever, in order to incorporate the local conditions in a general way, it has been necessary to modify the Bilham formula, the modified versionis known as the Lloyd-Davies formula, or the rational formula.1.3.1 The Lloyd-Davies formulaThefeaturewhichdistinguishestheLloyd-Daviesformulafromtheoriginal Bilham formula is the replacement of the duration of rainfall by thetimeofconcentration,andtheapplicationoftheBilhamformulaforeach pipe length of the scheme, individually. This is to incorporate the evidenceavailable from the records of intense storms that indicate the storm givingthehighestdischargefromagivencatchmentareaoccurswhentheduration of the rainfall is equal to the time of concentration. The time ofconcentrationisdefinedasthesumoftheentrytime (timetakenbythewater to flow across the surface) that is, from the farthest point of the areaconcerned to the point of entry to the channel, and the accumulated time offlow ofwaterwithintheconfinesofthechannel,thatis,betweentheupstream and downstream ends of the channel. The normal range of theentrytimeistakenfrom2to4minutes,thelargervalueisappliedtoexceptionally large paved surfaces, and an accuracy of within 1/2 minute inthe estimated entry time is considered adequate.As mentioned above, a drainage scheme consists of a main channel anditsseveraltributaries,andatributaryofthemainchannelmayhaveits own tributaries. The main channel, which is usually the longest, and thetributaries are made up of various lengths of pipes separated by manholes(see Fig. 1.8). Since the volume of flow increases with the distance from thehighest point of the area to be drained, heavier pipe sections are requiredastheflowprogresses.Itis,therefore,usualtofindthatdifferentpipelengthsofaparticularschemehavedifferenttimesofflowand,hence,differenttimesofconcentration.Normally,pipesareselectedfromexperience, assuming that they run just full, that is, without surcharge. Thisallowsthetimeofflowtobecalculatedfromauniformflowformula.Alternatively, the velocity of flow can be obtained from the design tables[1.4] based on the Colebrook-White formula (see Chapter 4).Each pipe length is identified by two numbers, say, m and n, separatedby a dot. The first number m identifies the channel (main or tributary) toIDD40 minutes 2020 12010Design of linear drainage systems0131/7/0311:06Page 10which the length belongs, and the second number n identifies its locationwith respect to the most upstream manhole in the channel. Thus, assumingthat both m and n start from 1, a length labelled as 2.3 means that it belongsto channel 2 in the scheme, and is third from the most upstream manholein the channel (Fig. 1.8).Denoting the time of concentration and the time of flow for a length m.nbyTm.nandtm.nrespectively,andassumingthattheentrytimeforalllengths is the same, we havethat is,(1.6)and(1.6a)where tm.iis the time of flow in the length m.i. TheprocedurebasedontheLloyd-Daviesformula,andadoptedin thedesignofastormsewersystemisdescribedinreference[1.5].In thefollowingexample,theprocedureisillustratedingreaterdetail.Thevelocities of flow are calculated from the Mannings formula (Eq. (3.5)).T T tm n m n m n . . . 1T tm m . . 1 1 entry time T tm n m iin. . entry time 111Flow in drainage systems2.12.23.12.34.15.14.26.1 6.21.21.11.31.41.5To naturalwatercourseManholeFigure 1.8 Plan of a sewer system.0131/7/0311:06Page 11Example 1.1: The plan of a sewer system consisting of several lengths ofcircular pipes is shown in Fig. 1.8. If the estimated entry time is 2 minutes,andthecoefficientofroughnessn 0011,determinethetimesofconcentration for all lengths involved.Solution: Table 1.2, which represents the solution, consists of the followingcolumnsColumn 1 pipe reference number m.n.Column 2 Lm.nlength of the pipe m.n.Column 3 dm.nassumed diameter of the pipe m.n.Column 4 S0,m.nlongitudinal gradient of the pipe m.n.Column 5 Rm.nhydraulic radius of the pipe m.n.Assuming that the pipe runs full, Rm.ndm.n/4Column 6 Um.nmean velocity of flow in the pipe m.n.U1 10 0170 0111 33. (0 0375) m/s,and so forth.2/ 3 UR Snm nm n ,m n../.) (2 3012Design of linear drainage systemsTable 1.2 Determination of time of concentration from the Lloyd-Davies formula(Example 1.1)1 2 3 4 5 6 7 8Ref. length Lm.ndm.nS0,m.nRm.nUm.ntm.nTm.nm.n (m) (mm) (mm) (m/s) (min) (min)1.1 60 150 0017 375 133 075 2751.2 60 200 0017 500 161 062 3372.1 50 150 002 375 144 058 2582.2 50 200 0017 500 161 052 3103.1 50 150 0025 375 161 052 2522.3 60 250 0025 500 226 044 3541.3 80 300 003 750 280 048 3854.1 60 150 0025 375 161 062 2625.1 60 150 0017 375 133 075 2754.2 60 250 002 625 202 049 3111.4 50 350 003 875 310 027 4126.1 50 150 0027 375 167 050 2506.2 50 200 002 500 174 048 2981.5 50 400 0025 1000 310 027 4390131/7/0311:06Page 12Column 7 tm.ntime of flow in the pipe m.n Lm.n/Um.nColumn 8 Tm.ntime of concentration in pipe m.n.From Eq. (1.6), T1.1t1.120 275 minutes.From Eq. (1.6a), T1.2T1.1t1.2275 062 337 minutes,and so forth.The next step in runoff calculations is to find the intensity of rainfall foreach length of the sewer system. This is done by substituting the respectivetimes of concentration for the duration of rainfall in the Bilham formula.The surface area As,m.ncontributing to the runoff drained by the length m.nis determined as follows:(1.7)(1.7a)where as,m.narea to be drained exclusively by the length m.nAs,m.n1cumulative surface area drained by all lengths of the pipe m,upstream of m.nA*s,m.ncumulative surface area drained by all other pipes discharginginto the manhole between the pipes m.n 1 and m.n.The total runoff in each individual length is then obtained from Eq. (1.1).This is compared with the capacity of the channel length obtained from Eq.(1.3). If the capacity of the channel is found to be less than the runoff thenthe pipe diameter has to be revised.Example1.2:TheareatobedrainedbyeachpipelengthofthesewersystemshowninFig.1.8isgivenincolumn4ofTable1.3.Forastormfrequencyof1year,determinetherunoff(litrespersecond)foreachchannel length, and check the value against the flow capacity of each pipe.Solution:ThesolutionispresentedinTable1.3whichcontainsthefollowing columns:Column 1 pipe reference number m.n.Column 2 Dm.nduration of rainfall for the pipe m.n Tm.nColumn 3 Im.nintensity of rainfall for the pipe m.n obtained from Eq. (1.5).A as m s m , . , . 1 1A A a As m n s m n s m n s m n , . , . , . , .* 1t1 1600 75 601 33 minutes,and so forth.13Flow in drainage systems0131/7/0311:06Page 13Hence, for N 1,Column 4 as,m.nsurface area drained, exclusively, by the pipe m.n.Column 5 As,m.ncumulative surface area drained by the pipe m.n.For instance, fromEqs (1.7), As,1.1as,1.11000m2As,1.2As,1.1as,1.21000 800 1800m2A*s,1.3As,2.33000m2Therefore, As,1.3As,1.2as,1.3A*s,1.31800 1600 30006400m2It can be verified that the last entry in column 5 is the sum of all entries incolumn 4.Column 6 Qmax,m.nrunoff carried by pipe m.n.Taking Cr10, we obtain from Eq. (1.1)Q, max 74 l/sand so forth.1 01000360020 56., IDDm nm nm n .../( )1 14 14 2 541 3 55 14Design of linear drainage systemsTable 1.3 Determination of intensity of rainfall and maximum discharge from theLloyd-Davies formula (Example 1.1)1 2 3 4 5 6 7 8Ref. length Dm.nIm.nas,m.nAs,m.nQmax,m.nAm.nQ0,m.nm.n (min) (mm/hr) (m2) (m2) (l/s) (m2) 1000 (l/s)1.1 275 74 1000 1000 2056 1767 23501.2 337 66 800 1800 3350 3142 50582.1 258 76 1200 1200 2533 1767 25442.2 310 69 600 1800 3500 3142 50583.1 252 77 1200 1200 2567 1767 28452.3 354 64 1200 4200 7583 4909 6126*1.3 385 62 1600 7600 13089 7069 197924.1 262 75 1200 1200 2533 1767 28455.1 275 73 1000 1000 2055 1767 23504.2 311 69 800 3000 5833 4909 5466*1.4 412 59 1200 11 800 19667 9621 298256.1 250 77 1200 1200 2600 1767 29516.2 298 70 1400 2600 5128 3142 54671.5 439 57 1500 15 900 25617 12566 38956* See text.0131/7/0311:06Page 14Column 7 Am.narea of cross section of the pipe m.n d2m.n/4Column 8 Q0,m.nflow capacity of the pipe m.n Um.n Am.nwhere Um.nis taken from column 6 of Table 1.2.It can be seen from Table 1.3 that pipes labelled 2.3 and 4.2 (denoted byasteriskincolumn8)haveinsufficientcapacity,thusrequiringtheirdiameters to be revised. All other pipe lengths are capable of carrying thevolume of runoff.1.3.2 The Wallingford procedureTheWallingfordprocedure[1.6]isamodifiedformoftheLloyd-Daviesformula. It calculates the peak discharge from the formula(1.8)in which Asis in hectares (1 hectare 104m2). This formula is applicable tourban areas of up to 150ha, with uniformly distributed impervious surfaces,and to rainfall of duration in excess of 5 minutes. The runoff coefficient Crissplit into the volumetric runoff coefficient Crvand the routing coefficient Crr:(1.9)ThevolumetriccoefficientCrvdependsuponthepermeabilityofthesurface.Itsvaluerangesfromabout06forareasmadeupofrapidlydraining soils to about 09 for heavy soils, or paved surfaces. An averagevalueof075isthereforerecommended.TheroutingcoefficientCrrdepends upon the shape of the time-area diagram [1.7], and on the variationof rainfall with the time of concentration. The recommended value for Crris 13, which leads to the usual practice of taking Cr10.As was done in the application of the Lloyd-Davies formula, the intensityof rainfall is calculated on the basis of the time of concentration, but by aprocedurerecommendedbytheMeteorologicalOffice[1.8].Thecentralparameter involved in this procedure is MTD, the depth of flow associatedwith the specified values of the return period T and the duration of stormD.Thereturntimeisspecifiedbythedrainageauthorityusingtheprocedure,andthedurationofstormiscalculatedbytheprocedureillustrated in Example 1.2.For a given region, the basic data available are as follows:(a) M560, the depth of rainfall for T 5 years, and D 60 minutes. ThevalueisobtainedfromamapoftheUnitedKingdom(Fig.1.9),showing contours of various depths of rainfall.C C Cr rv rrQC IAC IAr sr s max 0 362 7815Flow in drainage systems0131/7/0311:06Page 1516Design of linear drainage systemsThe values on the contours arein mm of rainfall9876543216 5 4 3 2202020202020202020202020202020202020202020202020202020202020 20202020202020202020202020202020202020202020202020202020202020242020161616161616161616161616161616161616161616161616161616161616120 2040 60 80 100010 20 30 40 50 6012121212121212121212 121212121212121612161216161616161611312337892 3 4 5 6 1National gridIrish gridKilometresStatute miles2Figure 1.9 Map showing rainfall depths of 5-year return period and 60-minuteduration (M560) in the United Kingdom.0131/7/0311:06Page 1617Flow in drainage systems(b) The conversion factor r defined by(1.10)Its value is obtained also from a map of the United Kingdom (Fig. 1.10).(c) The ratio z1is defined by the equation(1.11)Corresponding to the value of r determined in (b), and the specified D,the value of z1is read from a graph shown in Figs 1.11(a) and 1.11(b).(d) The parameter z2, defined by the equation(1.12)The value of z2is obtained from Tables 1.4(a) and 1.4(b).(e) The area reduction factor (ARF), which depends on the extent of theareacontributingtoflow,andthedurationofrainfall.Itsvalueistaken from the graph shown in Fig. 1.12.Finally, the intensity of rainfall is then given by(1.13)The following example illustrates the procedure.Example1.3:ForacertainlocationinEngland,M560 min20 mm,andr 420%.Determinetheintensityofrainfalloveranareaof2km2for T 2 years, and D30 minutes.Solution: For r 042 and D30 minutes, we obtain from Fig. 1.11(b)Hence,fromEq.(1.11),thedepthofrainfallofthespecifieddurationofD30 minutes isFrom Table 1.4z2080M M5 30 5 600 8 0 8 20 16 0 min minmm ( )z10 8 IMDT D zMMT DD25 minzMMD155 60 minrMM day min5 25 600131/7/0311:06Page 1718Design of linear drainage systems1 2 3 4 5 6987654321130353535353535303030303030303030302510101015202025252535452530001020 40 60 8010020 30 40 50 60304020252530303535353540404031233278942 3 4 5 6National gridIrish gridr 100KilometresStatute miles1Figure 1.10 Ratio of 60-minute to 2-day rainfall of 5-year return period (r).0131/7/0311:06Page 1819Flow in drainage systems5 min 1 hr 2 hr 4 hr 6 hr 10 hr 24 hr 48 hr 10 min15 min 30 minDuration D010025012051Z12346810r 030r 030012015018021024027Figure 1.11(a) Relation between z1and D for different values of r (012 r 03).5 min 1 hr 2 hr 4 hr 6 hr 10 hr 24 hr 48 hr 10 min 15 min 30 minDuration DZ1r 0306432105025045010r 030033036039042045Figure 1.11(b) Relation between z1and D for 03r045.0131/7/0311:06Page 1920Design of linear drainage systemsTable 1.4(a) Relationship between rainfall of arbitrary return period T and rainfallof return period of 5 years (England and Wales)M5Ratio (z2) of MTto M5rainfalls(mm)M1M2M3M4M5M10M20M50M1005 062 079 089 097 102 119 136 156 17910 061 079 090 097 103 122 141 165 19115 062 080 090 097 103 124 144 170 19920 064 080 090 097 103 124 145 173 20325 066 082 091 097 103 124 144 172 20130 068 083 091 097 103 122 142 170 19740 070 084 092 097 102 119 138 164 18950 072 085 093 098 102 117 134 158 18175 076 087 093 098 102 114 144 147 164100 078 088 094 098 102 113 142 140 154150 078 088 094 098 101 112 121 133 145200 078 088 094 098 101 111 119 130 140Table 1.4(b) Relationship between rainfall of arbitrary return period T and rainfallof return period of 5 years (Scotland and Northern Ireland)M5Ratio (z2) of MTto M5rainfalls(mm)M1M2M3M4M5M10M20M50M1005 067 082 091 098 102 117 135 162 18610 068 082 091 098 103 119 139 169 19715 069 083 091 097 103 120 139 170 19820 070 084 092 097 102 119 139 166 19325 071 084 092 098 102 118 137 164 18930 072 085 092 098 102 118 136 161 18540 074 086 093 098 102 117 134 156 17750 075 087 093 098 102 116 130 152 17275 077 088 094 098 102 114 127 145 162100 078 088 094 098 102 113 124 140 154150 079 089 094 098 102 111 120 133 145200 080 089 095 099 101 110 118 130 140WhenthisvalueissubstitutedinEq.(1.12),theresultisthedepthofrainfall of the specified duration of D 30 minutes and return period of 2 years. That is,M2 300 80 16 0 12 8 minmm 0131/7/0311:06Page 20ForD 30minutes,and2 km2astheareatobedrained,theareareduction factor is about 094, therefore The average intensity of rainfall is then obtained from Eq. (1.13):1.3.3 The formula for intensity of rainfall applied to gully spacingThe time of concentration used in the design of gully spacing is usually lessthan 5 minutes, which is out of the range of application of the Wallingfordprocedure. Hence, in such situations, an alternative formula recommendedby the Highway Agency is to be used (see Chapter 9).References[1.1] SALTER R.G.,HighwayDesignandConstruction,2ndedn,MacmillanEducation, London, 1988.I 12 03(30/60) 24 1 mm/hr M2 300 94 12 8 12 03 minmm 21Flow in drainage systems5 min 10 min15 min 30 min 1 hr 2 hr 4 hr 6 hr 10 hr 20 hr 48 hrDuration DArea AT (km2)0990980960940920900875085075070065060055100908070605040302010987654321080Figure 1.12 Area reduction factor ARFrelated to area and duration of rainfall.0131/7/0311:06Page 21[1.2] RUSSAM K.H., The Hydraulic Efficiency and Spacing of British Standard RoadGullies, Road Research Laboratory report LR 277, 1969.[1.3] BILHAME.G.J., The Classification of Heavy Falls of Rain in Short Periods, BritishRainfall 1935, HMSO, London, 1936, 262280.[1.4] WALLINGFORD H.R. and BARR D.I.H., Tables for the Hydraulic Design of Pipes,Sewers and Channels, 7th edn, Vol. II, Thomas Telford, London, 1998.[1.5] A Guide for Engineers to the Design of Storm Sewer Systems, Road Note 35,Road Research Laboratory, 1963.[1.6] Design and Analysis of Urban Storm Drainage, The Wallingford Procedure, Vol.4, The Modified Rational Method, Hydraulic Research, Wallingford, 1981.[1.7] WHITE J.B., Design of Sewers and Sewage Treatment Works, Edward Arnold,London, 1970, ch. 4.[1.8] Rainfall Memorandum No. 4, Meteorological Office, London, 1977.22Design of linear drainage systems0131/7/0311:06Page 22