122 246 THS LIBRARY Michigan State University MSU LIBRARIES Fe RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES wil] be charged if book js returned after the date Stamped below. This thesis was contributed by Mr..W. H. Urquhart under the date indicated by the department stamp, to replace the original which was destroyed in the fire of March 5, 1916. Titi IB HYwuURAULGI Q ’ QORKULASA for ORITF¥I O08 8S und VUHANNA LSB \ ) t. we . er aon _& oy k, W. Powell ie He UPQunurt WiCuIgun Zericulturul Collese Departmeit of Givil sanwgineering,. dyii, THESIS PReaPAC Lo Thue Object Of tui tnegis isa to lavestigate tio laws Which govern tne flow of water torouan orifices und in open Onunnels. It was iutended to investigute also tue flow or water over Weirs sud taroughn pipes of Various Kinds, put tie tine at our disposal proved tu be insufficient. No original exporinents were i1.ude but liusgteud we used tic data of sucn stundard authorities us, Darcy and Bazin, Kuttor, Floley aud Ytearns, Cunuingnus, Hittingor, koil, Legler, LaNICcca, LeVveille, M.Poirsve, Huapnareys und Abbott, aid tue Alasourl xiver Comission for chunnels, und Hamilteun bnitn Jdf., sllia, Jwid uu alig, ior orifices, In oru@er ty Fuclultate tuo work it wug aiviuea 1uto tue investigaticu of counnelg and investigutica of orifices, Tne forner part of tne work wae carried on by ar. Urqunaurt aid tue dutter by ar. Powell, ne problem Of circulur orifices was s0 daurgo tout time was mt found for tue Consideraticu of square anu Pectungulvr orifices uid ticrerore tuey do uel upyear in tulsa work, In tue Cuse of beti Oritices uaud Channels tue ideu was to determine, first tuo Luctor’ wulCh Couteroiled tiie discuarge, und $1.60) tO derive aii eupirioul Formula wuicn Would give tie discnuurge .8 @ definite functicu or Lunctlivis of tuosa Luctory. in tne case of Ciuiitne@ls, innumerable Lormulus uve Deeu propcsed at Various 6168 uad ti0 probie@u Wus te Ontuin a better oue if pOssiblso. Tne cused Was SOLO ILL AllLereut in tid uastteve of ore wih J > on ificos Bd Yl dbubsizoutls OL tio Lluw Oo: LLOW Wisaci ure ub ai ZOV 4 tenable secon tO be in tus Lorm of tubios,. Tne Compubatiouas an tias worn were Fuciliituted py tis use OF Burkinurdt’s Aribtinciuster,. BIsLiouiamrny T4tl0 Hydraulics Flow of Wuter aii shVers aud Otucr Unannels Hydraulice of Creat saivers Piow of water Sxperine:nts at Koorkee Apj-ii0d loecnanics New Hydraullios Hydrauiic Tables, cvoerricients und Formulae Least squares Treatise on Hyaruulics Hydraulics Discussion of Autter’s roriula Flow of @ater in nivers Hydraulic sxperiments wito Large Aportures Kxperinent ca Feictacudesyy Orifice CoOeLficients of Wisonurge Olrculus’ Oririces Contracti.., of Jets AUL“OP Huwildton oilbs, Jef. Gungulliet and Lutter Je de Révy oohmoer Cunni nga Goodman oulii Van NOV1140 es rliman Merriman povay tiptieUeise, VOde Y, De 3526 Aoi eGete, Vole &S, Pe 173 AetiaVebeys VOd, Oe Pe 4y infiese NOWS, Vol. 56, Pe 526 tLe WOWS, VOle 60, be 49 BUBINBOPLiign, wil, LL, 1,04, The works just s@utbicned nave been CousgulbO0d und muca Of toe duta inoiuued in thicuw o4a8 oeen used in tuo work wnichn foilows. LuiVawslualivis Qe Unle'Qidua LOW OF Basan In OP init Oi Ansnld. tne Luvesbigution of tio ..0oVOln0..5 OF Water ia Open cnunnelis nas peeuw vefore nyuruuiicians Lor muy yeursa, Tne first invostiguters sought te expressy tue lawa of Flow by Meuns Of mAtoemuticul principles but were wisuccessrul in ob- taining uccurate results. Galileo is said to nuve pee tus first iavestigutor upru tue Piow Of wurer in Pivots, out some Of his stato.euts nave beeu proved erronvous, such as tae irregularities of rivers caused no retardation of riow. Torricelli discovered ti:ui, oxoept Lor roslistuncss, tno vee aocity of jetta of wuter issuing frum simil orifices was equal to thut of podies LTaliling 61.0 Suue distuice in 8,:uce, a Zn order to muke Bu lVestigaticn Of tne various fore pulae for too flow of water in ope. cna.nwels, 1t will ve nec- @seary tO Culsiaer briefly tac Conaltious Wuer 41410n water flows, tuc elements wiscn a.0uld enter i400 tuo Cumpututicous of INOuls VOlOCIL1ES, uind tne Clircwastances Wiicu alrect tue cute of tiow, Tno inclinaticn or slope is geuerusly cvasidered to bo one OF tue cuiel elem@iuts wuilcu afrect tue Velocity und rute o7 flow. Ina longs, stPulgnt, unilora channel, tus accoleruting force aue to tne slope serves coly tO huintuln @ uniform volocity:; tue neud 1a u giveu Jeougtu belnug expended i: Cverconing tho ro- Glatuuces. If tue au. dy of water ts iucreusod und 3i0pe uimlaisned tae veiocity 18 increased, but the area Of tie cross-section Wili .o Gg@Greased untid tis resistuices @yual bie Force due to the nead in unit dengta. From this we s8e0 thut tue VYesooity unu dlsciuarye depends won tne urea O1 tie Ciuunued and Lio regidctunces toxwzetucr Witn tue inciliution, Tere id wa EPsut uncertalnty us to tno manner in wilch toe urea ui resgistiss Corves outer nto tue formulae for Coiputiig mean veiociaty. Tne more Cuigu0n Way beinue to uge sows function of tne hydruulic fuulus, or ureu of cross section divided oy tio wetted perineter. 1NO Lirst abblezpt te uiscover tue duw oY Wid tue velocity of Flow depends upcoa t..0 Fusl OF BLOpe uli Cross-section ,@8 mude by Branhins, #10 Observed tnat tuo water in streans uo- quires & GunBtant Velocity. He points tu tne friction of the Water 2Zainst tue wetted porrileter a8 tus Force .i201i Opposes tne acoeleruticn uid ascsunes biut tne resistance ig proporticnsl tO tue hydraulic rudlus. sPallms und Onezy ure to be regarded &s the aut.ors of the fourzuila, Ve 01i8 wnere ¥ @ moan volocity Ce & Ccistant Kom nydraulic Prudius BS Ancliiuubtliciu or giope, It bpeCcume vory eviaent to tuv eurlier lAnuvestigatora tuut the laws must be derived fron @ex,crimentul data, und alter work With tnut principle involved, we huve @ foruula by Prony; RB ® ave b Ve in wich "a® ad "p* are coerriciouts of friction deduced from experineats. Kytelwein used tne saue form but Lounu different Vuiues of ®*a® and *"b*,. Many otaers derived furmula but in every Gase tne coefficients were CCistant Values, Tne mauy Curiiue up to tuls tine did not recognize tue infiuenace of rcuanness Of wetted perimeter, or desxree aL slope upon tne coerficiunts., A now understauding cl tie subject was opened up by tio lavestisgutaous of Wurcy uw susin, TAeY Cue structed a: CaAperimental Gaim ovor 1800' scot long, Varying ia Oross-section, 840pe3 um lini, iutorliaa, All iBasurenents were Dade With tuo ,Proutesot Cure aud HHVve Dee bie bDB81sS O1 Leauy Coie putaticns, Tio principal fucts verived frou sazin's results are ie woe ceerficicns *c® varies Wits tiv dvgree of roughness O01 wetted porinotor. ii. “2c coerriciont fo" Vurlog Wibli aA. Bauzin's fCcriuula has tuo Lollowing form vs Vt in Woich ®"A® aud "B® ure Constunts Varying Wits tue surfaces. bazin diviued tne experiments iste Tive categories aud ti0 FOllowii, 1d a tubde or Fosults. . —BYsh R35 COOtLI0 ,2008. : — Oategory Channel A B _ ) . Cenent z ! Ourefully Pluied sourd | 0.000046 0.900001 57 | Bmooth Auhlar II Brick 0.000056 Q.00000405 Unplauned sourd | Iii ! Kubble | 9.000073 0.00018 3 Iv hartn 0.000085 0.000107 Y j Carrying gruvel _ | Pe 0u0aee 0.000214 This formula 18 not unlversuily &plicuble ultucugn it Gould be ind® se if the numper of cutesories wero increused and Variutions Considered for ual posaible Cuses, The Loriwula next in inpertunce for peucticaud use 18 tnut derived by Gunguillot aiid Kulter, bettcr xuown ua Aubtee'’s Fore mula. In tie developi.e..t cf tueir Lornula they used Bazins a3 vagisg uid Gideuvored tu omoody 1::16 t..0 Bifects of tue siOope us Weal wy u fFedubica betwoe.,, tue Coeiricients Of FLeuuacas, Tne coerficiont "c"® in tno form v @ CaS was put equal to y ; Ye where ®y*® und "xX" ure Variunios depending Uson tic sL0P0 Qua FOuLInNeSS. ‘i..6 Vurlution or *y* ig expressed cy *y® a (a+ &+ aM) und Variuticn of "x" by "a @ (a + yn. n 8B a+ ksi © By (8 + B) The deteriinutica of tae Various Ccunstuats in this formula Fornulae Va Will be outiined. The fora c ®@ Cun be eapressed as i > | a Asay dn Wuilch 13 a stralgat line by pistting Vuliues of*)* CG y y i C and i. A number of experiments were go ploLted and straigat dines draw: theougl peints, wvifferent iines boing druwn ror Gifferent Sivupes, nese exporimentsa so plotted were soi of Humphreys aud ADDOL uu scone cr busin, Accorul.g to Kutter, toe Warlous lines druwii turcuss tue plotted puluts iatereect in & COuw.0N polnt. This point is woere s/h w/1 neter » 1.611 Leet. This is tho constant "L* in tue general formula, We purpose ty Bu0W LY Glagru: Of plotted Vuiued of *®2* ana tnat tuesoe 41268 have 010 CQumon intersectio: uu cur atutouent is vOorne out by L108 stutlenont OF Herschui 1 118 DLOK Of FLOW OF Hater. AsdUWILE that tus polnat of 1intersecticia 1a7R # 1,611 feet, Kutter makes tie Tollowing statemeats, Z. Vuiues of "6G" iancreuse with decreuse of incsineatiou Wnen Kis greuter than 3.28 feet. II, Values of *c*® tnoreuse wlta increase of slope wien kK is loas than 3.28 Leet. From tno plotted results of *3* and *)* S Rm the values of *y* are ceusured on tue Verticus axis. By plotting tie Walues of "y® as Ordinutes uad 84" ag abscissa & number OF points were Obtalned wiuicu were usauned to lie won & stralent i356. From wuere tits sinc 80 drawa later=- BSected t1.0 uxlsa Of Orudiiutes, tuo Aistarce to tue Orlgonu Was measured as (a + 3) and tio Coustant "2" = 9.00251 was the tunguit n made by tne line and @ horizontal tiecugsn point of intersection, Tne Wasues of (a + 4) wid (fb) Dedng Know, tue Vuiue of ®a® is Getermined * 41,6, "the vusucs of "“n® were Obtéluca by & wruphical metnod, 48 foliows, plot Vasues of ©)" 49 ubsoissae und tuose OF *)" aa ordinutes, Assule a serios oth Values of *n* und plot trou: 6 upon the ordinate for *}* «2 1,531 Yeot. Pilot derived Vuiues of (a #1) upon tue axis ot ordamtes aud connect tuo points wita tuose of n° vy straigot lines, For euch Kumeli.e foud the pre- liidnasy Value of *n* inaicuted by tae pouition of tue point with relation to tiove dijnos, und tuking Known vu.ue of “u" solve for *y*- a+ pZ+p, Tase reciprovad of *y*® und plot on axis of ordinates und arow & 21ne8 tnrougn 16 aud g@uging. c.n6 polat Wisere dine so drawn intersscts tie series of "n* plotted on ordi- nute er z 10611 Elves Yulue of *n* to ve used. Tae Concsusions Of Kutter ure us follows concafnliug tue coefricieat *o*, Tho cverricioent iucreuses 1.9ita tne increuse of tie iyursulic fudius. 2eWitu decrease Of rousiness Of Wetted perineter, Se With uccreuse of lucsination wien KH 1s .reater tiuu 3.- Jt +, WAG Aucrease of Lucdanabioa wiOu 18 4055 buds 9ec6 VIOb. Tuo FosOWA ae 1s a tubio Of Vusues OF Fn* as assigucd to Gifferent surraces; = 9,009 for weil piuned tinver # 0,010 for newt couent @ 0.011 Lor COmemt Qnieetuatd Buiud #= 0.012 for ilipduned tisper 0,043 ror usnlur und brick #2 9.015 Ler sewers wii Crldulta 0.017 for cunals in firm gravel 0.025 Lor Culiude uni rivers free frou st.068 0.030 for Caimlad uid Sivers Wits sale stones - - ~-” -e e % 0.035 for Gunuis and rivers 1.1: bad order, | Tue Values Of *n"* wust be ussuneda by "guess anu ailow® Betsca, Walch may be All rigit Lor experienced wan but is poor for a novico, Take for exuuple tue flow of tne Mississippi river using "1" & 6.05, wa obtain 4 veloulity or 5.60 levt por secund sor @ Bpecific Case wid tiids udlig “a*S Q,.010 we cdtuln u& Velooity of 42.55 Teet per socend. Zils gives more tuun double tie dlscaurge for difforent Surlacus Wui0s 13 Vory nurd to believe. Very rew prominent engineers believe tiut 1f We Savuld line tne slusgissip,1 With CoWout bud discHurge WoUkl bo double Lor bus sui urea und slope Wid0u 4008 tO Si0W tiu.t uw soparate formula la necessary Lor durge rivers, fin all tue formulas so fur discussed, we Tiad taut tue velocity 13 ussumed toe vury us tue ayaraullc Pfudius or as tae square root of tie aydraulic ruulus anu tne suse nolds teu roe 6urding tie variation uue to slope. Kutter's own lnveatigaticn GROWS tiut a great uud uncortulin error is involved iu assuming taat the velocity varies us tao VR. fae Todlowiug Giasgrun und couputution 1s now liutroauced to substuntiate tne statement previcusly mude Conceriing the intersection of zines arawn turouga plottea values of "L"and i" and to aisprove tno staleneut of Autter tnat it is at a Sounk woen do = 4-641, Tho cata used belny tuken from that used in derivation of Kutter's Lormula. fae LTollowlng atu Wasa usod ln tue prepurubion of Plate I Adlustrating tuo elvect of cuange of siope upon tue Value of coe efficient "Gc". haperimonat Slope Neat Cement (9.0015 O72}. « O65 Bazin #24 Noat Cement |0.0049 Bott Le 99 [1.76 | 1.63] 1.52 +5 3 5411.29 Bazin #2 Smooth Bazin ¢30 2 |.0e45|.0a7 Smooth Smooth LO. B%pOriIBENS LVLOpe . — pours 0.006 wr 2205 |deby 14s 70 l4204 |e 09 |e 26 a boar ° ‘ Basin 20 a __ S| .oios | 010s .010e siete Dats Uupluned ; | vourds POON) | Mita he bel dati ihete the tl het d test Bazin #19 a | —_———} 020 31,010, 2095|,801 21.00 v41.00%5 Loors sdisaisaip- pi HiVerS D.00U0L 12 A... via . - . $~ail2 . _ Baons niver| 0004] AL. 07 pia jad ach Bl Woh a _ tOeup.'s sernude Lur tio #eiow OF water, aude Poriula Has boOm GOrAVed LPuu & W1IdG® Pu.we Of OrApEric mental duta and 16 ids suid 40 be Guo Lest cf bne KnOWL Lormulue vy bany nyadruulic eugiueerse., It is a wodiTICation Of Hagen's rermude und Was pubiisiied 1n & paper reud perore tie vociet, of w.nslneers 111 sugiana in 16676 WOLCULL dle V~= velocliy or riow in Loet por secunda, R® hyuraulic radius. Lo lengthy of Ciuiuneld, H® fail dn loigta FL" 8 ® cosecunt Of uugao or Buiove (}) £4 44. X,U,ii, ure CCeLLAGICiubs UGPviirciai, MpeCad Wis seta § sO wl v.10 vurLucn. Ten V. kh. vis FOr siuull vaiues of "H® wore accurute results ure optuined by Substituting Lor tuc andoax *ax* tuc Vusue (A + to / asti) e & Tne foiloWiig tuble is used 1 CGennection with tars Ceriula, Thrwip *s beriubia, Suriace iN u a YX 4 Wroveit Leon »,lpe 1.60 UeVO4757| 0.65 0.UsL6 v.07 iveted bneet * Lede] UUOD674| 0.077 Pi New Cast rive 1.65 1(0.0UD547 eee 2.0 0.UUG7 92/1 0.635 eee eae Pure Coenoat del O,.UU400 rel 2.95 [(G.00c429| WO | Bricxwork( su00tn) 2eUU O.0U/74o| 0.61 Ueldect Uend Brickwork( rougn) 2000 | G.000049] 0.625] O.Ulez4 U.50 Unplained pourds 2.00 | 0.000451| Ocbdn |] GeO Z54Y|! 0.50 Gimil gravel in Ceue nt 2-00 | O.O0lisl | 0.66 0.05956) 0.60 Lurge graveld in ceneiut 2.00 | 0.01415 | 0.705} 0.07590} 1.00 Hacuaerea masonry 2.00 0.01347 Oe 60 0.07520} 4.00 Barta (uo Ve vetation)) 2.00 0.41536 V2 Barta (steusy, ouch) | 2.00 | O.0ci44 | 0.76 ; Thrupp recogaizes tno uct that tne velocity Vuries as dif- fereut powers oF "a" ua "8" fur ailrore..t suPaces, Tie reiuticnu GXistinug vetwoen Tnarup,'s "8" um taut gouerpalay wed is Thrupp's "S"= 4 aS Generaiiy wed. Working with tue sane idea tuut "V" varies adiiferoutay for vurious surfaces, &@ number of coerficivats fer “S" aud "H*® nuve been de- fived aid @ COonparisol 15 Ssub.u.itted, Goerri@ients ror Radius | Goerricients forbic Surface Theupp | Urqunart THFUPP Urquhart loves | ti3 7? : Pure Cement 0.61 U.62 Le ¥5 4.72 Brickwork (snmoota) 0.62 0,612 €.00 200 Beickwork ( rough) 0.625 0.76 €.00 2.00 UVaplaned sourdas 0.615 | we 2.00 4.6e Smail gravel iA cepoent 0.66 O73 2-00 2.60 22. Goel icicuba 4 ’ RIVET ITT aoe Surlace farwupp Urguurt sRPrupp Urqunurt & yowosld & +OwSid Large gravei in couent 0.705 0.75 2.0 2.50 Hauwered masonry 0.66 Veld a&e0U 2656 sar trl Lio Begetutioiu) Ue72 0.65 «00 2.0 bartha (rounn, stony) Uels 0.73 2,00 2.00 DeniVaTionN OF FUKUI The aerivation or & new gonerui formula Was how unuertaken after @ study of wie Various formula Aoroin outiined and discussed, Tae Licst work to be undertusen Was Lo Ting 1f possivic avw tuo velocity Varies Wibi tue Aydruulic Pudius, ine Vveiocities and radii us duia taken with CUlstant siope were usea iu bic Compu- taticn, ne form of Gxprossion G8 aecided upon Was ¥* BR aad frou tuis we aave Vo KE that, 2s OR Vi ~ 208 Vo « 406 Hy, 10K HK Vaiu.s of "Z" must veo aerived fur Varicus surruces aud snupes of conduit as wus found in working tuis cut. On tue Lol- owing puge 18 BiVoOu @ Buiy.deo OF tuo Worn neCesuury. This lie Ciudes, radius, LOgaritiua of Padius, VGioclly, Ouerlbiuus oF voelocit,, wud Computed Value of "Z" ror neat conent conawit or Channel. Compubation fer 84" ln term 2 se Lon Vi Lon B Lor 408 a> 106 Ri neat Ce).81i6 6 Radius {ioe radius | Velocity L0G. VeLlocit a 0.505 (9.701568 5072 04570545 0.655 0.605 {9.781755 4.16 0.619094 0.605 0.662 (92655784 4.60 0.662756 0.739 0.750 9e87506L 4.67 0.68752Y 0.600 0.809 Fe VO7T44Y 5ed2 0.709270 0.672 0«667 ¥e¥ZGOL1Y Seay 0.723456 0.465 dim annette 43, —_—_ Radius 404 Kuuius Velocity 106. Velocity a 0. v4y ¥e¥/1266 5eld GO. 129066 0.466 Yee 309 305% 0.525146 0.251 Ye SAYOT4 4e SY 0.642465 0.6560 0. 522 90901646 5004 0.702452 0.45% 003575 9e5/4052 5266 0.754548 0.75% e450 965.5466 6006 0.763904 0.448 Oo%74 9e675176 Oo5d 0.525582 0.702 0.526 Gefa+o 50 6.63 0.654422 0.554 0.555 Ye 7466 5% 7.22 0.552450 0.560 0.595 774547 Fetdh 0.664518 0,621 0.632 Y,60U7 47 7065 02652549 0.578 0.665 Ye6cc5ce . 186 Ved ¥D4RZ 00545 RnG9O. eh OOP. omen DL erereeralamree Dt DOGS, . AVOruge 92649 Tue ,Peceagliug wLOtacd aS CubslNOd Gud Sunu0WR Wus used Lor LLiding Values OF tiw eXponouh "2" Coe Varauvuus Guses, Tne fFeuults OF tiuis work is shown in tav Vollowiug taviuationu: vo — —_ _ - —_— ——— ~_ vusfuce Of cuannels Vaiue or coefficient "3" Nout corant (senicirculas ) 0.610 Neat cee.ut (rectungudar ) 6.610 Pluned bourdgs (svuicircudae ) 0.759 Planned suuras ( rectungudus ) 0.7590 Unpduced sourag 0.675 Brick skucury (sncoth) 0.012 Belcx suguoury ( reugn) 0./60 Small gravel in cenont ( semicircular ) 0.750 Smui2 " ° ( rectungulue ) 0.600 Lurge gravel in CuLienut 0-750 Rubble uscury (clean) 0.750 Kubble susgcury (uirty) 0.750 BUrtii, AMSLUPyY Sldewullis 0.70 Smuil xivers (reguiur) 0.650 _ Irregudur niverse (rou ) 06750 HOre tiun 350 Waiues Of tia Coelricivat °2° were cunputea tid $16 Aae6un Values Cuund wus 84" @ 0.7512. FEOW G CuLpuPiaon of %, W14tO ~nucrease of 1Asecdanuabiogd Wie KH 1d Osa bids Decd VOL. Tao Foslowlsue 18s a tubsoe of Vusues OF 8n® ag aggslgucd to Gifferent surraces;: N®@ 0,009 for weil pluned Cinver N# 0,010 for nout coieut N@ 0,014 FOP COmeus QNieebusld Busi # 90,012 for iipdu.ned tinbor # 9,015 ror agnlur und vrick ® 0,015 Lur seWePs Wid CUndults 0.017 for cunais in Lirh gruvel # 0.025 tor caiula aud rivers free frou sbues wm Se Se Se Se mw ee 8 * 0.0450 Tor Gamulds uid Pivers With sue stones # 9.035 for canuis and rivers iu baa order, | Tue Values Of *na* wast be ussuned by *“guees ana uilow® BOt¢aca, Wich may be G21 rigit Lor experienced man but is poor for w novico. fake for exauple tne flow of tne Mississippi civer using 81" & 0.05, we cbtalna u velocity or 5.690 feet per socund ror @ specific Cause und tad udlig ®a*S 9.010 we cbtulno & Veloolty of 41.55 Loot per socond. dud’ BiV8s More uuu double ths dlscnurge for difforout surlaces Wich 13 Vory nurd to bedieve. Very rew prominent @hglneera believe tiut if We shauuid line the sisgiasip,1 With Cemout tue disCcu.rge Wouki bo double Lor tuo suis urea und slope Wudon £008 to Si0W7 tu.t uw soparate formula ia necessary Lor durge rivers, ia aii tue Lormulas so Lur discussed, we Liad tuut tue velocity 13 asaumed to Vury us tue ayuruullc rudlus or ag tue square root of Lio AYydYaulic PuulUs anu tne suse nolds teue re- Gurding tne variation aque to slope. Kutter's own investiguticn gnoWs tiut a great uud uncertuin error ia involved in wsswaing taal the velocity Varics ay tae /R. fae Fodlowiug Gdiagru, und computution is now litroauced to Substuntiate tne statement previcusly muae Cconceruing tae dnterc.ection Of sines aruWi turocugh plottea values of "L"and i" and to aisprove tic stalenenut of Autter tat it is at a Sonik woen do = 4-631, Tho Gata used belny tuken frou that used in derivation of Kuttier's Loruula, fae Loidowing Gaia Was usod li tue prépurpution of Plate I ALlustrating tue elrvect of cuange of siope upon tue Value of cc= efficient "csc", kxperimeat Slope Neat Coment 0015 Bazin #24 O72}. 2W066 « WEE Heat Coment OO4y Sobt Levy 76 |1.63/ 4.52 +5 [16 dP (2054) 1.29 Bazin #2 bmoota Bazin #50 2h |.ce%|.0a7 Ssmootn Snootn S ae SL be LO feeiiness pete SOB seaee ae 3 pe soe nae eee es ye ae Ore ae eee RA ZAR EE IA 20. BXperinent pope Unplaned a to . wounds p.0U6 |i Za05 | dab dash dealt Lda TO dat bad aS Bazin 20 a 7 | Uapluned a. | pourde =P. 0043) _AlizaO batt i deGO |i 26 leetZ leat has hati daz T Bazin ¢19 a . s1SSaisBip- pi kiver DeowoL le a $—aRule Bis AO | 1) Ue £ , ~ Baone KiVer G00004) Bl. 207 fe Bi jeePt lectd lol Zod. & Toeupy's secrwuda Lur tiw eiow of water. Suds Pormla fuss DOOu Gerlived LPruin & W1dG@ Puiu;e OL GAPE“ mentad data und 16 is suid tO bO bu0 vest cf tue KnOWa Fornulue vy ibany nyadPuulic engivreerg., IZt is wa modizigation of Hawgen's reormuda und Waa pubiisiied in & paper reud verore t1.60 bocl1et; of nnelimeers 11. Bigiana in 1667. ijOGULA UL, V@= vesocity or riow in reet por secend, R® hydraulic radlus. Le®™ Jengtn Of Gime, H® fxui1 ia leugtsa 8" 8S * cosecunt Of ulsio OF Sudve (}) f4 aa Xp U_ii, UFO COOLLACICUIS UOpeinsii, Lois ino Luburo CL Wi0 surluce. x tea V . 3 vis YOr siGil values of "x"® wore accurute rosults ure ootuinued by substituting Lor tus anudox *a* tic Vuiue (4 + To / Medi) e ix Tne folloWiny tuble is wed ih Coanection with btais Ccrisulda, Thrwip *s seri uba., Suriace Ni U0 pS xX 4 Wrovg4t iron »1lpe 4.50 UeVO47E7| U.09 U.UL6 V.07 uiveted bacet * ete] UQUVOD674| 0.077 ee New Gast ripe 1.65 bores ie 2200 ((O.bUG752/40.65 | OO | Leud ripe 1675 | Ge0Uz24] v.02 ars Pure Conout 4074 |( 0.00400 furel 4.99 6. VOO429| W.GL Bricawork( su.0cothn) 2eUU U.0U/74o| 0.6L UeUdecr+ Ulead Brickwork( rough) 2000 | G.000645] 0.625] O.Ul224) 250 Unplained vourds 2,00 0.006454!) Oco4dD] GeUZs4Y¥! UL 50 GSimll gravei iu Ceue us 2-00 QO.01isd Veb6 0.05956 0.60 Lurge gravei in ceneut 2.00 | 0.01425 | 0.705} 0.07590) 4.00 Hucaerea susoury 6.00 6.01347 0.60 O.07o20} 4,00 Barta ( uo Ve,.e0tatica) 2.00 0.01536 Vele Barta (stuuey, Pouca) | 6.00 O.02L44 | Oefc Tharupp reCoguizes tie ruct that tne velocity varies as dif- ferent powers of "x* uid 88" Lur aifrere..t surFaces, Tie re@lubici CXisting between Tarupi.-'s "8" um tiut goueralay used is Thrupp's "S"2 2 4s EBeNerailiy wed. 8 Working with tue suse idea tuut "V" Varies diifercutsy Lor vurlious surfaces, & nuiber of coerficivats for "6" ana "Hn" nuve peen de- Pived aid @ COonpuriso: is Ssub..itted, Surface Pure Cement Uaplaned sourdas Goerri@ients for Radius | Voerricients forbig Theupp | Urqunar Thrupp |vrquiurt tosh like 0.62 U.62 de ¥D i-72 Brickwork (sneota) O«64 0.612 &.00 2-00 Beickwork ( rougii ) 0.625 0.76 2.00 aen8 e615 e 2.00 1.6 0.6144 ced . . 0.2606 Q.73 2-00 2.60 Smail gravel in cement 22. pOoAd AGA SIL. mamaeson COSAAAGAGUED Avi DdOp Surface farcupp Urquiurt sar upp Urgunurt & yowoild & yOwveil Large gPravei in couenut G.709 0.75 2.u0 2.50 Hamwered masonry 0.66 U.fd 2eUU 2058 bartn( no Begetublon) Vef2 0.65 “=.00 2.0 burtia (rougn, stony) Veils Odd 2,00 2,00 DBKLVATLGN OF PORKMULA Tne aerivation o: uw uew gonerui formula Was new unuertaken after & study of tio Various forwula horcin OULiined und discussed, Tae First work to be undertusen Was to Ting If possivic Aw tue velocity Vuries With tue Aydraulic rudius. nue Vveiocities anda radii us dutu takea wits Culstant slope were used i tic Compu- tation, tne form of Cxpressicyi G48 aecided upon wag ye & aud from tiis we nave VV. KE that, Zs ace Vi = Jog V~ ° 40g H,- LOU Values of "2" ust be aerived fur Varicus surruces uuu Saupes of conduit 43 wus Pound in Worsing tuis ous’, On tue Lol- dOWlng puge 18 Elven @ Bu.yio OF buco Worn nOCesuury. This la- Gludes, radius, LOgaritium CY Pudius, VGiccily, Logard bius or VelOCit,, uud Computed value of "Z" ror neat couent conduit or Channel, Compubaticoan fer 82" la term 2 = LOK Vi Lon F Lor 408 at. ~ 10K Ki neat Ce6i..61i6.- Radius ([i0e% radius | Velocity 106. Velocit 4 0.505 9.701566 5ef2 06570545 0.655 0.605 92784755 4.16 0.619093 0.605 0.662 92635784 +200 0.662756 0.730 0.750 9e87506L 4.67 0.68752y 0.600 0.609 Fe 9O7 449 Dede 0.709270 0.672 0-667 ¥e 956019 Dee 0.723456 0.465 O.945 9. ¥6l421 bebe 0.741152 06754 43, dn sty aman in tn _~ in > ee —_ " _ ne —_—_eS ee ” — Radius 40% Huuius Velocity L0¢%. Veioocaty 2 On vay ¥0¥77266 501d G.729066 0.1466 90825509 jeb* 0652 51% 0.251 Yo sx¥OT4 be SY 0 .64%2465 0.650 0.522 905071656 450 Q4 0.702452 0.45% 0.375 9ed1 4058 5066 0.74558 0.754% U. 450 Y 0 53466 6406 0.763904 0.498 0.474% 9.675778 e542 0.815562 0.702 0.526 Gefi45 50 6.834 0.6 54%421 0.554 0.55% Yo 7466 5% 7.22 0.652450 0.560 0.595 Ha 7T745h7 Zo%d 0.869518 0.621 0.652 ¥,60U717 74635 0.652545 0.578 0.065 PeEecbce ; 1086 QoS yD4RZ 02545 BaGPO ab eae eel Tce ann tee OGG TM swosnme AVOruKe 9.649 Rne ,peceaiig wtucd us Cubsi nod aud su0WaA Wug used Lor findsug Yaiues of tre exponout "8" Loe Vurivus Guses, ne reuults cr tiuis Work 18 shown in the vollowiisgs taniuatiou: tae 9 —~ —_— —_ — vurface of cuannels Values o1 coelficieat #3" Nout votant (senioirculas) 0.610 Neat cese.ut (rectungulur ) 0.640 Pluned pourds (svuicirculae ) 0.759 Planned suuras ( rectunxulus ) 0.750 Unpduced sv0urag 02675 Beiok abevury (sKn00th) 0.612 Belck sdagoury ( rcugn) 0./60 Small gravel in ceuent (senicircular) 0.750 Smail " e §6(eectangulusr) 0.600 lurge gravel in Coueut 06750 Rubble wascury (cioan) 0.790 ktubble Agsciwy (ulrty) 0.750 BUrti, MASLLLy sidewalls 0.740 Gmail Kivers (rogulur) 0.650 | Irregudur bivers (rougis ) 0.750 Moree tiun 350 Waiues of tuo Cocerricivoat "2° were computed tid $16 B6un Values Cuund wus 84" @ QO,751l2. FPO & CULwuPliaon OF LF. tho Vaaues ol tudo Coelracicat 16 Lina taut For Buusad C.c1nes’ it Varies aa tne dexree Of Fuugiuiess and aiso lAncreases Wita increase in rougiseaas OL Wetted periueter. Vide Clerc LOdoul 80" dik wud V® 07 ib Tae loss of neud in @ Gauinel due LO Peoslstuiles say be expresseda by tue formula, Hs maf yaece "1" Jf d03b nead | K2G( FL" u Leictics: Lactor. Feu bas Cun be derived t= iii ids «oor ee iW = 0 7H. Tania 18 aBsuiug Git tue fFogistunces ure properbtiouus to tne VR but 1t has been tound tut tuoy ace more neariy proportioned to Ré and furuula, V @ p ROVS «will pe wed us a bugis for furtner investigation. Thie Cui uso be Written, vV @ p Bt |Re wWoere tne Yasue of, "c® m= pe , Tae izverty 1s tucen of lutroaucin, into tho foruula ror "0" a constant "B* waion will vary With ulllerent degrees of rougnness Of wetted perimeter, and tie feriula ag it wow stands 18, °c® » p7/R( lem). A GPiul Vaiue of *p* ia now ussuned us 50 wid tne FollOWilis data in metric measure used as a pasis for Curticr cuiculation, R21.0 meters, © = 112°0; K 8 0°0205 meters, 0 ® 60°45 now, 9. 8 22% #«, #$£%nence (1+ m) @ i 324 a9 7 | A1s0, ° @®4#e2l , wie ln) Fl + se2) 50 Ve If we diviae ache * 0-466 Wilda equals 40208 and tnorefore "SC" Cau pra be @xpressecda by tne Lfoiiowing *cC*® = 50/8( 2 ~ rd for metric ieusure RH ii @8 ©C® w 66/R(2 + dig) COP buglisi isusure, a LIS Preuss Guard 78 Cai Sains U VULuO aoe LF tii8 » 2°¢ i he Gai a3 a —_ — _ bumWut fue. Vurus cl Yaad pn sO phOLbdde G 66 7a THO 350 Gullngs Uo 2a bi.0 ClLpUubaticn OF tuo Virautacau Of "K9 wor UWJOE Lu Lindiia, Vus.ues cl “a for t.0 Vurlous surbucey oF £40 Ciiliie NOiBs A BALd4O PULO OY G5.43 CLiguUGublouu ds 2usOPboud nwOPO BucWasig $BLEOG Values cL "un", GGipubod Vuiuos us "G8 Leon vow OS, tho “i, * Gi POCspPLCas OF “JR, Wu" OL 8" wat: ti0 asad uvortgo, GiViued by (py) uiia £00 Values POLLO Gd Ov pUububa. isk 23 @ GUubIO BAVA Bd Vusues cr "g8 us derived for tie Vuricug surruces, G ° Guupututacn of “i » eal 7 2°6 aor (eut Cet) oo rete nmenrnmenreeonmre Pp 0 R YR SR 66 VR m 66 15600 | 0.605 | Oetdu hed sa 60.366 decks a4%6e2 42565 404/70 O.vle bec yO de +60 diieyY | Ge 266 O.f/6 bect5 e904 dedfid bu 5ed de G5% 42006 Oevyd &0330 de dd bine wy Ve yey Ve YUE ere ae 3'tO de je 5 12005 Uarl¥ Ofc 9 onl 3 20380 44055 43.6.5 0.466 0.040 wel? 2eb15 ea) +00) 0.463 Oey? 1,042 ae dad 40245 Lode | 00767 | Oey42 | 1.000 | Zeddy | 2.052 129 O06 59 0. yo7 de O45 2eh40 a. Uy d 442, 4.022 4.005 | Ooyyds | 22255 be dy 42 50h O.e208 0. {Uy do 40 226/0 de dod 126.9 | Ovjce | 06754 | dezcd | zen45 | 10465 b5c ot 0.450 0.610 46053 eet aed ¥% 157.2 | 0.652 | O.bye | 2620 | 26550 | sede? 447 2Y Bur )e Vaplusied Bourds, nie BR RB” 1973 B O78 ee y 3 ©.86 JR y Q 081% | 07655 | 2.6295 | 0045 | U4Y94 | 24626 | 2.85 | 75-72 2e3Y | 8095 | 1.6755 ” 8 0 5408 Ze! 75.6% e412 | 6565 | 1.92235 e ® e6415 054 | 52.2% e499 05554 401544 6 ° of0604% | 5-42 6< 286 06488 | .ylyv2 | 1.7652 " « of/S64 | 5-42 64.47 o7G0 | «3595 | 166055 " e 05506 | 6e46 | 90.07 0 52% | 626% | deov4 | ODN | 605505 | oe 5SSY | Ho4S | 85027 etyS | ot6452 | dbe75i2 8 " 0/056 | 5.54% 65257 6612 | .9176 | 1.7556 | * * 07623 | 6626 | 85062 0705 | «¥%02 | 465002 | * . 05504 | 6076 | 55022 06UG | 05655 1 2.5293 | ® s "6466 | 7.42 | 64,22 0500 | 28855 | 2.7i4c | O0S5y¥ | .O7242 |.70712 |! 7657 | 54635 0624 | .¥2uU0 247660 « a 0166 | b.74 64045 0255 | oo (8735 | 20659355 | 0059 eO59h2 | 20% | 5698 | 560024 ebe5 | e 9205 | 1.7665 " " 01593 | Tee 66.85 06866 | - 9405 | 12.8065 , * « e8d02 | 7-72 | 86675 o 542 | eds | 1.694% |.0082% | .072492 [65659 | 52835 b5¢ 0.953 0.4506 0.009 0.i4%2 0.059 0.402 0.019 0.0235 0.4216 0, doy 0,312 0.509 0.do4 1.06 2e%e 0.40 0.65 7ed2 Yea 2026 3026 5019 %.06 2.26 O45 F057 405% 4.05 5eS% 0257 Z2eyl H#el5 0055 1287 O06] c2ee4 5.02 4.45 0,14 ed 02550 5e61 2045 5.50 > . ° edd £053 £203 _ Bazan 49 O.41 «0081 5 e128 66026 52610 52850 06298 0.122 5.20 2.06 2.45 ool 1-522 (e542 00255 72550 0.020 02 56% Gely2 Oo27 4eb% 2.54 168 60155 66452 5.097 8.300 Us246 Dood 0.115 5.01 1,07 2041 off 6eol4O 90077 BeF9Z ¥eIU0 04552 0.046 06554 Beeld 0255 %-04% =p — o % ’ atirnnee _ Bauin 22 Komen puseury Brick e524 210076 12.295 We2 99 LL¥d 1Z.ZEO VYY4 06556 GeOL5 12621 xe75 Ou 0467 160177 Woz27 LyHkI9 1600 0.650 06556 06577 525 2.04 ce, «580 180660 dood62 1654956 16.550 06516 06124 062350 2.78 066 Unt 0662 ZLeOVZ WOW AHS ZUMO C6994 6547 00692 %.70 2.59 Bu prea 6625 2000 det RO. vagiil 33 - - oe VEs9GA LY ; erCs R 8 ki Gam B nk Uc P B Gui UaPr BB xen Deer gured al er e424 09006656 ¥.045 8-562 6eS562d BeYOH Oe tiD OetO4% 0.440 9659 ae62 14.55 e745 35052 dieZ4d 1504%O dy do 7 0.3424 0-072 Oed09 2650 Ve5dS Vell 0552 252075 Wrel¥o We Soc dtylS Ood7¥ Ve2dy Oodn7 ido BSd 2ots 1.04 wens aves - . . - Ftleiey « bLearns —_- QE BRICK, _ 1.016 ,00UUGLS 2.445 2.266 2510 0465 Oedo? UA0G7 0.065 sde4d d50d5 3.04% 00656 .000U246 2550 orth 06522 20545 06169 O,c2¥Y U.007 50.70 41.60 4. 97792 Q22°Y HOTTOCT OPH ST OPH°ET HETOT a Qh OS N9°2 OfF°R BL°T BG9°H 929°Q OB1T°N OFT*K O9TTK OTA°K 296° " CO*2 29°AT 92°00 O2° 292°H9 260°Q 27G2°9 O7T°SG OG4°SG OGL°S 2n0°9 GE9600° Geer T = yap 2°& Geer 60°0 loleo Loot Lnorva OF2°9 096°9 a9G°) L9G°2L a €S°T 646°tT 22° 0 29°0T OCT°O OFO® OF46°0 O6H°G ONG*G OFT°A@ OGG n O2°T Phot Ol°92 nese FKk9O°O FFO°T FKhS°T 096% OOLTH O9T°S LOR*E GLETT)® 64°O ° HEmMa Ion wy Y Try b bara) q IN vv q — Orr q os ¥5 —— T st 435 np eds 1 - —— sO — ania AN eae Sipe tet Peasant Pye eer oe PR rene seen ene ee ee ae bene eS eee as : 1 yt heathennens pa J re t ee are / . i ————————' : ——— na ee 47 Rt al —a | - nm i peel ; “= em | | ‘Shy Pe AP ae a | ~~ ~ behest ee Ree: onl SE 6 a ee BBs a2 | + “TH a ) ad / a | . oA : ; — aw ; a THF / Lage ; ; ad / el wee - Pr et re | | Dar” ©) aaa See 5 Sanur aS e80te ee } be: ; bee hs Pre se Bs. SEE . et ae | 2 So.) ppfteth ct er. eesti , | Sian", | Ses Spaeareces =Ser ren © pastepese reacowsces es rat a cess Ts . seae ewe \ i | Hs tiesearcis ZRRORS ed VexocsT y A perets as ComPured Ay FoR MULE Benue fstab iat Rt pews et age | Ban a eee es yy weeeseeeey 2 Pea a / } TS | seat a bern POSEY) FoscbeLs a sacha La. Sseeted 15 Sel MEO, hc co } ra mee) URQUHART. 7 ieee eae % pre ee Brisioin re Rachst | ; 4 inte | ‘ : \ We Pas i | | . rf jaro TENG aS pace Tasos } ; Sty ee + ; 4 ai-* erate / | | | cP — See | a * tt - Pai ticiisios aks BP capacesntd potreeli | | | a (Sam oe . ivy PERCENT: Pd F eee | hme tt es _: | +— Ses best in SLEtty iizeoks <4 oy me : | z Base ro on $0 ee ee , : - =, ailedes mtpPhanuUbsci OL kdute Ii. The COSuLbts OF bid Corp, uPlise. ch bt Ouby-Tive series OF BuUuglUES “re Biwct.. OY LOUIS OL u uluUgPuL. The ieui percentage Of error ios Loeund Leeuw euv.. gerisy is plotteu us apsClssu, Tne results aS dorivod Trew tie theroe Loriuula uco ,lotted ihn tio BunG HOPIZLiutud Lin@® Lor euch serics uid tus SsorPles ure ut Oyu Spaces Vecticully. ae Vuricus pelts so plotted vy Counectea by dines, D@rived Loriuulu, mouvy Lull iine hutter's « , udtted " Huzin's ® » Got anuddusn. His wetsod cL dO.cusbPutle,a Was uged as 1t Wus tuougnt CO uukO tu@ results Cleuror tilda aay OFUSP WetacU,. DATA. 206 Q4itu Used li tne agerivublican und Computation of forumulu wus sulected frua a greut wuss of experinents. Oiudty SuCi: Guts a8 Naud & pureiutiy Doe... buntsi witas O10 Kreatest cure beung used. all iiiforimtion Cencorning t16 Churucter OL tie ded, tne number OF experineits uad 10b.0u5 Ol LO&sureueit belnyg lire vostiguted whenever possiodio. wGLOpe LOUsureiusents wore 4130 tuken Wits greut care, Tne Varicus .otneods of meusure,.ent were by LOaSurlii, .»reVirusdy tue quantity cs wutor, Fitot tube, Liouts, und Current weters., Tne Gdubu dmCirdes, uubiority, .ean Aydruulic Pudiugs, 8sLOLO Ulld Loan VOlOoLlLy. 32. Data used in tue derivation of formula, Location Mean Mean Ooerricient AusSnority Hydrau-e slope Veloc- and 11ic na- ity. Ge _yV_ Description — dius. fh _faat, Re Test Channel "Darcy and Bazin 0.366 0.00150 3.02 128.9 Neat vement Fe 0.503 " 4.72 135.6 semicirculur *"Recnercnes Hy- | drauliques® 0.605 # 4.16 1458.0 0.652 ° 4.60 143.7 0.750 " 4.87 145.2 0.809 " 5.i2 147.1 0.867 « 5.29 146.7 0.915 . 5-51 148.6 O.s4Y . 5-75 152.5 0.992 . 5-912 1535.3 1.029 . 6.06 154.2 1.034% ® 6.11 155.1 3S. MOuUiL OU Li LOcution Hyé@rau- & LOpe Veliocc- GVoertricient aud AuLiority 11G wudlus of ity Desoriptioa in reet vurlace® Je@t peor Ow Vo a - / BC 6 Vee Test Giuiwod WULCY UVelto 0. 004Yy 505% 416.5 Heat Ueieiis ia C.e5d . we, 39 ddbel HOC LULZ Udae baci Ue S22 " yet 126.Y ye 02375 ° 5606 Loe e4 Oetau * C.0o 45504 0.474 * 0054 asded U.se5l6 ® 605) d50eD 00556 " 7.42 15602 0.595 ® 7Te4a 157.2 0.632 . 1.65 Lo7e2 0.665 . 7206 15/08 0.696 . 3.07 5c 2 fest viunnel burcy 0.030 Oeti252 4.67 875 GuPGsuldy pluipda lid 0.043 . 2050 ¥0.0 Bourds buzin 0.053 ° 2266 ye hectungular #29 0,062 . 5000 Yoe5 0.074% * 5059 106.4 3. MmO4n Mean Locuticn Hydruw Slope Veldoc- voerriciet aid Aubsority 210 nadius or ity Description Li Leet KBurluce Leet ,er 0 = Vv BOCs YEH Catto 425 0.02 0.0047 0.40 76.5 0.052 “ 4.40 6329 0.066 . 405d O94 0.075 " io7% ¥207 0.004 a 1.94 716 0.091 e Zeil 102.2 0.093 “ 2.16 105.2 sudbury F¥teley 1.863 0O.0001606 2.529 146.2 2 Og 0.0001596 2.6072 147.9 Conduit isi 2.111 0.000150 2.605 155-6 Plaster of vtearngs Pure Qenent Test Chunnel *Durcy 0.490 0.0015 2.64 107.6 Planued sourde and 0.537 « bee 115.6 yemicirculargs Bazin® 0.632 ° Jel 120.6 #e2e 0.727 " 4.04 125.0 0.746 ad %.25 iz 4.2 0.856 « %.5L 425.8 O.9eL " %.64% 124.7 0.964 & 4.57 1252 4.2625 " 5.00 125.2 1.096 ® 5029 150. L.iey s 5045 152.3 1.146 ° 504 43505 Tost Unwell “parcy 0.029 0. 00%7 0.7¢ 16.5 Pluned Busaras uid 0.052 0.00487 4.50 6500 ” eng O28 066 . L. dt BY% HOVuicwe dtuyin® “ 2g 0.075 s 1.74% Y2e7 ms 0.064 " 1.94 ¥7-6 0.092 ° e2eid 202.1 0.095 " 2.16 103.2 33, Locubtich MBUiL MBUL1 Coerlicisat UNG Autnucelty uyuraudiG woldpe Velocity Gs Vi pesori;, tion MULLS VES T8s% Cnuinel “DuPcy 0.2345 O.u04y 5057 YVed UN, dun@u ovutdd UG O. ae a 4, 43 106.3 neCbLUENGULU? busi" Ve 40 s 5205 11002 O.446 « 5. b> dlc .3 ec U0550 s Pe 115.7 0,612 e 6.26 414.3 Ue 604 s Oe9VU Lit+ed 0.403 s 676 i505 Oefts e 7200 L160d Onfli/ 9 (<0 116.7 0.dVUG% ° fet 1215.0 UedsSY . 7Tedy dic. Test Giaasmned *Durcy 0.147 0.00524 252 401.4 Ubpduned buarud and Oc 5d " 4.42 101.4% OCCU UIEE B4Z).1" UVecby ® 5055 duU9 8 U0 393 e Ge2t 109.7 Be We 5d s Gel dd jek 0.460 « (47 415.26 Ue U6 « Le 245.2 0.542 s 7-73 115.6 U.572 8 Oe U4 446-Y U.004 6 5e<6 417-4 0.640 e 6.57 LLY.0 Test viuuuned *Durcy Veloso 0. OU4S eefa Se¥5 Unpiuied beuras aud V.eld ¥ 53f9 101.2 ROCGuisg ual bazii." 0. 542 ® i. 55 406.2 0.402 a u.85 LOY.4 od Ue 52 8 5.29 d1i2.2 Veb504 e 56d 1415.0 Oe 547 8 543 424.5 Webs] " 6.23 116.4 0.626 ” 6-45 116.4 U.e662 e 60/4 417.8 e696 e 6.y¥0 417-Y O.727 8 (edb 119.6 36, LOC. Lien Os: BUN Cverricisit wii Aubnority Ayuraulic w LOO Veiod lty Gos y besceeii.$10n 1uUii Us Leet. iw TOSt Wisuitect @ yufGy 0.29 0.002 03 240s ¥5ee Ulpluswted ounrag “wa UO. 3603 ® ae 0Y yf od nectung lus buslne® 0.455 e 3eib 402.06 U.5a5 o 3eDS 106.5 36 O.OUL 8 belo 100.Y Ue bbe « %.15 112.5 U. {04 6 +. 54 415.5 Uef>o¥ a #51 dined O.0ud e + Ja 44906 UVed46 * +.00 116.3 V.a60 # 5. Os 119.0 Ue Yca e bead 41509 Onazidly Guuual SpDurc, U.4l O.UUd. 5el 5 L0uU.0 ABLGOP WUSLLLY sid O27 s jo3 421.0 oWO0tLIY aresseag KBuZen® YU. . Oeil 410,U Oe/f " 6efD 1224.0 e359 Spillwuy Grospols “burcy 0.52% Q.101 12.29 67.9 ASNLUP ssuUSciPry Bid Vetus " Lo, id 74%.5 aliny deposit Basin® 0.550 “ LG 0 OG (7-2 0.662 s 2k, Oy 6l.& #32 Test Onunacl 0.192 0,004 eel GYef Bricawors Burcy Vedat " 3ebO Y503 UMoOCLA bad Ye 599 ® edo YOe6 kectungular Bazin Ue 424 e 4.72 20507 Vetod ” 20 h0 105.2 75 (1. 40 . 5633 LU507 eyed " Jee 206, 4 22 OLU " OeOL 109.0 ; Uebud " 6.45 40724 2eOY7] « Oe +f 110.6 Ye floy * oe LOY9.7 O.77Y¥ " Geld 108 2 f wOCu £10 MOT m@an VOeLLicloit uisi aAubsority yuruulle wLOQpPe Velocity VOsSCR2 pp b104 buud us 0 = re sudbury Ccniuit *rbeiey 1,016 0.Gu90L4 0. +45 127.25 Ginnocin Bbrici: un tac Yeo 0.0NU0E46 U.900 119.6 WEll wmude jiints Lteauaras"® 1,QUe¢ UVeUU0U705 OO. f6Y 120.9 1450 O.¥5f 0.050746 1.064 42569 0.775 Oe0USUYSS LeULOS 125.6 0.650 QO.0U0J125 1,241 427-5 Oeb7T7 O.CO00L5yY6 1.L4Y 119.7 0.675 O,UuLl655 Led 125.9 Vety5 QeQUGLO4HO 1.07 120. Veo wd JeULULFOL 1.ydY dz7.4 Vest O.FUuL7 a5 Levit 126.0 0. (oe UV. VUUAE OS de sy 42 526 Gerospeis neser= “arcy 0.424 0.057 ¥eu4 Teed Vil? Agudur bas4h Q.beU e abe 4o (oe Musou yY, MGaviy pudii" Oe f%5 * L504 0426 rectangular Veb5d . 150 VS Obey WII bolunis Fleut Quanlnge 2.2 0.QU0151 weal 4i2eo aqueduct bitin. 2(2 Q. GUGLY> Zep 12fley SELoOth DrICK isVUPKS®@ 2e¥4 Ved Q a ad 104525 46650 ce V4 OO. UUUEUS eely 412.6 2e¥y O.U0GZ54 50 BU 416.4% 5e65 OeUUU47 5 —- #5 116.2 +20 QO.UULUG2Z5 sou 121.0 Test Ghunnel "pDurcy Ue +54 0.0025 2ei7 76.0 Buudi gouvel eid 0.546 . “Hey 6.0 senicircular Bazin" 0.6149 s 2069 Geel Oe7 5d s 3+05 64.0 Ve ic " yen o322V 0.026 bd Je 032 © “te 0 UV. YOO “ yest Oe OU. Ybo 8 Seld Ove V 1.0d2 “ Ded 06-0 38 andi, laa “pete LOCabica wae Ahi NOUD Voerricient Usha Aubuuraty tiyaruulic ULOpO VeLocity C5 VO DOSCeaybtica uA us Y kw T@st Unmuisives. ®Durcy 0.259 0.U0%Y <.id 61.7 Datel gruvedr wm LW. 2357 " eeF2 70.5 KCC Gungudur bagi" Ve +00 5 Jbetu 72.5 ht Ve 5eV " 5e04 760k Vedas # +. 14 T7.2 Ve b+ ® 4.44 18.26 Ue JU0 ” +. 0% 193 Ue lO eeu OVef/ e705 ° bed S206 0.6 52 * 5-<0 é2e4 Veofad " bes O5ek Q.9¥L0 # 5edD7 &3.4 Test Cau mel "Darcy O.294 Y.004y ae7¥ %/ Burge eruvel uid Oe4L7 ” cot 2D0d nectanugulsar Badia" Ue5dU " mien aoe on Location Mean HO0an Coefricient ana Autiucrity Hydraulio wulope Velocity 0 - _ VO DSacriytion Hada Us 7 RB Aa?r, at bem Kutter 4.22 0.000462 2,822 63.7 Lrre-ulur ved 72907 0, 0U0d6 5.2150 66.6 70f8 0.000993 72524 $503 Veser Se 96 0. 00016 5% ae 410 M2. q Barth Sohnward &.79 0,000149237 350470 84.7 6054 0.0005¥56 4.087 61.3 6ef5 0.0004107 4%,950 ¥5e6 6049 Q0,0004110 5.162 102.0 D , dH Q0,000200 %. 06% 93.2 9-95 0.000200 4%. 389 97-9 10.52 8.0002167 %.756 99.2 41.06 0.0002167 5.166 205.5 12.64 0.0005326 7.92% 96.5 15255 9.000550% 7.-3Ga 92.9 yaone at tivei8 3068 0.00004 0.56% 4564 Raconnuay 76 8 0.815 58.9 * 0.966 58 e7 yee e 4.6021 64.8 40.87 ® 10355 S569 22.61 8 4.910 &8.6 11.54 e 2.942 89 «3 eo: # 2.254 972s i # 263509 98.0 15.83 . 20519 94.5 Seine at Villevert 5266 0.000127 2.09 7862 Furis ne Poirde 7,08 0.000153 2.26 7507 Bo 43 0.000145 2e%ls 7i.7 9.48 0.000140 32310 92.5 10.92 0.000140 Sel” 95-6 12,.a9 0.000140 g08ae 92.4% 14.50 0.000150 02 35a 94.0 45-02 0.000140 4,512 9803 15.93 0.000172 4.662 89.5 46.865 0.000142 %.400 102.2 48.39 0.000103 4%.689 407.6 cote i etl ian amie _ - -_ aa pe we _ _ —_ — 0.000167 3 Locaticn MIO Hydra Ue Mean Coerricisnt und Aubaority 120 Nudius Slope Velocity 0 = 5 on besorij; tion RB Seine at Linke ry 7et9 0.00009 203540 9303 Poissy 7263 0.000087 2313 S¥.5 11.24 0.000057 2.562 ayaa 12.45 0.000060 2. 559 80. L327 0.000050 2.5f2 91.2, 4.20 0.000054 2.5395 936 15-86 - 0.000062 2.919 92.7 46.85 0.Q0U067 32402 92.4 47-87 0.000075 32550 Yhed Bayou LaFrourcne Humpa~- 12.560 0.00003655 2,807 12967 Line gravel Preys and 15.04% 0.00003732 2.8%3 425 48 Aboott 12.47 0.0000438% 2.789 4196 25073 0.00004468 5.076 136, Missouri River Missouri 5.65 0.0001137 43,02 115.8 kiver Og 6,60 0.0001209 <.97 496 22 6.15 0.0001150 3.39 11.0.7 6.07 0.000116 4.25 106.06 $.05 Q.000117Q0 4.40 101.9 42.590 0.0U001L70 3.78 203.3 10479 0.0001183 4.63 202,90 S035 0.0001196 3.02 95.5 & 205 Q0,0001210 2.96 95.0 id «SO 0.0002372 % 486 107.3 42.30 0.0001516 4.97 ° 11.60 0.00015 5-92 7245 12.60 0.00UL5 389 $2.4 7072 0,000.1556 pot 69, 7 24.130 | 0.0001615 72 ¥3 oP 25.40 0.0001627 5.14% 402.7 45.0 0.0001672 4.22 90.5 44-7 4.3% 97-6 Tables For Practicul vee. Tae following Tabics will fucilitate tiv use Of tne Poriula for ta. uniform flow of water ina rivers aiid gaaliler Gnannelg as derived io tulsa work viz. Vs (@ ¢ DR) (RY + wm) 9%7/R, in wnich Vw m@un Velocity Re nean uydraulio radius 3 a slope Ra & Coefficient of rcugnnesy cf porimeter, a, und DB, ure nunberical vcustaots depending upon co.i- diticn und waterial of wetter perimeter ¥Y = Variable power of R X= Vuriasie §& * 3 fuble I, Cuntuing & sist of meteriale and cunstants to be used, AnoOWins tae wateriai und shape of ocnhunnei all valusea of “at, *b*, Sy", fm", and *x* Gun be selected from tuls table and velocity canputed vy Sint e pudtipiacution, Table II consists of values of iu . Tho 10ft hand colunn contains Yuluss of KR and all otner columns values of RY , toe vale ue of *y*® appeuring at tue top of the colum, Values of RY are given for every 90.05° and Otner Values may ve round by interpola- tion. able III consists of vaiues of 8* , Tne ef, hand colunn Cuntaing values of & from 0.00001 uw to 0.02. The otner coluans contain values of 8* , tue vulue of "x" appeuring ut tne top ar tne colum, Values not apnoesaring in tne table can be found by interpolation as in case of values of kK’, TABio J £6. Material o b Y m xX Neat Cerent (semoiroular) 103.0 20.0 0.430 1.155 0.58 Heat Cenent (recta:igulaur ) $20.7 13-0 0.2230 12.350 0.56 Planed Boards (senicirculur) 99.8 ¥eO 04250 C.925 O57 Planed voards (rectangular) 90.0 15.0 0.250 0.920 0.57 Unpluned Boards 4506 14.9 0.175 0.866 0.55 Brick Masonry (sooth) 5205 1504 0.212 0.420 0.50 Brick Hasoury (fcugh) 1.3 7-5 0.260 O0.800 90.50 Guell Gravel(sesiciroular) 51.3 269 06230 0.375 02485 Small Gravel (rectangular) 2&.2 Se7 0.200 0.290 0.385 Large Geavel in cecent 29e5 Heb 04250 04250 0.40 Rubble Mascarm (ciean) 15502 0«=—6- 406 5 S250 =206200 = 20268 Kkubble Masonry ( dirty) 21200 14.6 0.250 -0.100 0.42 Barth, Masciry walla 790% 2:9 e200 -0.220 0.55 Siuwll Rivers, (rectungulur) 71.5 1.4 0.150 -0.240 0.50 Irvreguiar Rivers Fl.5 1.3 0250 -0.420 0.50 Rs Xe | Hydraulic Kudius & Variable power Y= 0.100 0.120 BTS 0.798 0.776 0.45 0.20 0.25 3 0:40 0.45 0.50 0.55 0,60 0.65 0.70 0.75 0.60 0.85 0.90 0.95 1.00 1.05 1.20 4.25 1.20 1.25 1°35 i°ho 2.45 1.50 4055 4.60 1,65 1.70 1.75 4.80 0.827 0.851 0.872 0.867 0.900 0.912 0.923 0.955 0.942 0.950 0.958 0.965 0.972 0.978 0.954% 0.990 0.995 4.000 4.2 005 4.009 1.914 1.016 1.0cz 4,026 4.052 1.03% 1.20356 2.042 1.045 1. OMS 1.051 12055 1.058 1.060 1.064 1.066 1.069 1.072 1.977 2.080 0 1,067 4.992 1.096 0.812 0.858 0.859 0.876 0.691 0.904 0.916 0.927 0.946 0.954% 0.954 0.962 0.969 O.¥76 0.982 0.989 0.994 4.2000 4.006 1.010 1.015 1.029 1.025 1.0¢y 953 i oke 4.046 1.049 1.053 1.057 1.060 1.064% 1.067 1.9070 1.073 1.079 4.085 1.091 4.0908 4.202 1.2106 0.442 0.773 0.807 0.635 0.556 0.874% 0.889 0.902 0.9214 0.925 0.935 0. 944 0.953 0.961 0.968 0.975 0.962 0.968 O95 2.000 1.006 1.012 42016 1.022 4.027 1.050 1.035 2.059 1.04% 1.047 1.050 1.054% 1.058 1.061 1.065 1.068 1,072 1.074 4.077 1.039 1,086 1.092 1.097 1.203 1.108 0.150 0.475 0.708 0.752 0.786 0.812 0.645 0.55% 0.672 0.687 0.902 0.914 0.926 0.937 0.448 0.95% 0.967 0.976 0.96% 0.9¥2 1.600 1.007 1.014 2.022 1,028 1.034 1.040 1.046 1.052 1.057 1.063 1.068 1.073 1.078 1.083 1.087 1.092 1.096 1.101 1.2105 1.409 1.118 1.126 1.234 1.240 1.147 0.66% 0.717 0.754% 0.765 0.810 0.832 0.852 0.3/0 0.866 0.90: 0.92 0.927 0.939 0.952 0.962 0.972 0.962 02790 1.000 1.009 1.017 4.025 1.052 2.040 1.047 1.054% 1.061 1.007 1.073 4.060 1,986 1.069 1.097 1.103 1 2 d08 42324 1,419 i124 1.329 1.239 Le hh 40457 4-466 e274 Q.20 0.632 0.684 0.725 0.756 0.786 0.612 0.843 0.652 0.872 0.687 0.903 0.917 0.954 0.944 0.¥56 0.9646 O97 0.959 4.000 4.010 ieOLY 1.028 4.037 1.045 4.054 1.062 1.070 1,077 1.054% 4.092 4.099 42205 4.1412 1.418 4.425 4.242 1.2357 de d43 4.249 4.160 42272 1.161 4.192 4.201 0.23 0.569 0.646 0.692 0.727 0.758 0.766 0.610 0.832 0.853 0.872 0.889 0.905 Oey22 0.946 0.950 0.963 0.977 0.966 4.000 1.022 1.0.22 1,033 1,043 4.052 i. 062 41.072 1.082 1.089 1.098 1.106 1.214 1.122 1.2140 1.137 Le 45 a. 152 1.159 1.21266 1.473 4.186 2.199 2.212 do223 1.235 0-25 0.562 0.622 0.669 0.797 0.740 0.769 0.745 0.819 O.S4.L 0.862 0.860 0.598 0.915 0.932 0.946 0.960 0.974% 0.967 1.000 1.012 1.024 1.035 1,046 1.057 1.068 1.07% 2.088 1.097 1.107 1.216 1.2325 2.354 1.242 1.150 1.258 12166 1,174 4.2652 1.369 1.204% 41.248 1.2 1.2 1.257 47, 0,26 0.5459 0.621 0.658 220897 0.734 0.761 0.788 0.613 0.835 0.856 0.876 0.694% 0.942 0.928 0.944 0.956 0.973 0.986 1.000 1.013 32025 4.037 2. O48 1.060 4.072 4.061 2.091 4.2202 22322 1.128 4.230 d.d5y 2.248 41.1357. 1.165 1.173 1,182 1.190 40197 1.213 1.228 1.242 1.256 1.268 0.50 0.316 0.387 Do 7 0.500 0.548 0.592 0.632 0.672 0.707 0.742 0.775 0.906 10.936 0.866 0.895 0.923 0.949 0.975 21.000 1.02% 1.049 1,072 4.062 Les 1.140 1.162 1.976 1.204 1.225 22245 1.265 12234 1.308 1,323 1. 342 1.357 1.376 1.596 1.4148 2. 449 1.483 4.547 4.549 4.582 5020 4,20 %.60 4.50 5,00 5450 64900 6,50 7290 7250 $90 &.50 9,00 9250 40.60 45.00 420.09 25.00 1.301 1,495 42205 2,412 1.416 ii23 +2130 4,137 1285 i 149 1.254% ) 1,160 1.165 1.170 1.175 1.186 1.196 1.206 1.225 1.223 16252 1.239 1.246 1.252 16859 1,322 1. 350 42580 0.110 2.211 4.425 2.1420 L,i2h 4.128 1.256 1.144 1.154% 2.258 1.465 1.272 1.177 1.183 l. 458 16194 12206 1.218 12229 1.239 1.248 1.257 10265 1.273 1.2621 LeZSP 1.347 1.2390 224285 0.112 1.423 d2327 ded2ze 4.127 1.4351 4.459 Leh47 42357 1.161 4.2166 1.1475 21,2362 1.167 46192 2.394 14.240 1.222 1.25% Leet 1.253 2.262 1.270 10279 1.0257 1.29% 1.354 de 599 0.250 1.254% 14,362 1.4607 1.173 1.2379 Ae bVL 1.202 1.212 Leei2 2e2 5k 1.240 1.249 1.257 40285 2.673 1.242 44508 1.327 Le 535Y 1 1.553 4.560 1.379 1.390 1.402 1.413 1.501 4.507 4.4354 1.620 — Lhoe 1.476 0.475 1.162 1,190 1.197 1.205 1.238 1.226 1.259 1.252 1.263 1.275 1.255 1.296 1.306 1.326 2 55 1.348 1.508 1.423 1.439 1.45% 1.469 1.483 40497 1.606 1.689 1.757 aad Ii, Joni nues., 0.20 1.210 1.229 L.22y 1.237 1.246 2.202 deoZ7 i £292 2-506 rece 1.532 2-345 4.557 1.3568 1.460 4 e406 4451 1.4558 1.495 3, 516 1.535% 46 55¢ 4.569 4,585 1.729 4.822 0.23 1.246 1.257 1.267 1.276 1.267 1.3507 2.325 Le 543 2.559 1.3575 1.592 1.406 1.5241 1.45% Le bes 2.4050 1.510 1.5356 i «564% £2590 44615 4636 42658 1.678 4699 1-864 4-992 0.25 1.270 b.282 1.29% 1, 305 1.316 26536 1.356 Le3t/ 1.595 Le bli L426 1. 448g bg HER 1.480 LAYS 25352 1.565 1.597 1.627 1.6655 1.662 4.707 40752 1.756 16179 1.968 48, ew 0626 9.50 12282 4.295 1.3507 4.319 4.332 244255 1.574 16395 42415 1.612 1,643 1.673 1.703 1.732 1.789 1.64% 1.897 i. 949 1.434% 2.000 12452 2.054% 4470 2098 1.457 2.245 1.504% 2,192 125235 2.237 1.558 2. bo dD4 2 hse 4.627 2.550 2.659 2.646 4.689 2.739 4.747 2 829 10745 2.716 207724 52000 1.796 3.082 4.820 2.022 Jelh3 3.673 2edld 2el77 4.472 +6703 20097 2 256 2-509 5,000 tat ated, 49, Table Iii 8B a B81O0pe oh Aw Variable power. a —_ a - —_ a — —_ ¥ 0.585 0.4.40 O42 £0.50 0655 0.57 0.58 0.68. meal. amb, ani 00002 »Q0003 «00004 » 80005 «00006 #80007 «00006. oCQOUY eQ001 e C002 29003 »Q004 0905 «0006 «C007 e 9008 «6009 »Q0L0 eO0L5 e002 BD eOC25 «9050 20G55 0040 e0045 00590 00055 «0060 2065 ~0970 ©0075 «0020 eVO0E5 0090 «0095 202100 001552 001514 Oe Cry 0222 00237 0251 00264 +0277 . 02655 00377 20440 00492 .0536 00575 20010 00642 20656 06999 e316 «00914 eOY9¥6 » OL68 03.21.50 eid¥2 o2R¥Y 043504 e334 04395 52439 oat 7Y 03520 01559 0 b595 04651 02665 oLbyy 0 00 5a 20474 e810 e 02 05 20216 e240 o 02 42 002512 09552 00589 00457 00514 ©0547 «9501 e591 © 06 32 09742 «0661 e OYLO «0979 etO¥9 eAb5L oick O1 ote 45 91292 0455% 04374 o 1412 0 4450 et 455 ed 519 04555 01585 e01260 eOl¥22 0VL56 0166 0179 20190 eOLy9 «U2 OFQ 09279 50552 20374 e O412 O43 o O47 5 00577 0Q515 005495 00656 00554 0307 20872 eO¥Y20 (eO09s4 01034 «1.669 oAlZ% 012165 2 £206 o i244 edZda e3416 «3350 01352 ot415 0 3446 a = S001. 01189 20200 .00794% .004526..00173 .0ul42 ~01520 01064 6 O04 00545 00654 ~ 00708 ~00775 2008 57 600885 «V0950 »QL00 eUL42 -0175 ~02.00 ~0223 20245 002.65 00233 ~ ¢ 03500 20516 00558 « 0445 9 G4200 00548 00242 29053 20672 oG/06 0742 .0775 0806 00557 0606 e0o5 e0Y2a eO¥49 00975 e 4000 «00260 000425 e 00 56... 06431 «00476 e0O5i6 e 00554 000596 2006 39 eVOxe 4 091415 0014559 00153 eO1L69 oGlE4+ eOL¥5 «0205 20224 002/99 204528 «9470 ©0410 o O44 20450 09512 00542 00572 0599 00627 00653 06/6 20705 00726 09750 00772 00794 «00209 eV02Z64 0G0 512 0005535 266592 eQO4HLS 00462 200493 000525 GOTT y¥ eCO962 eOLLS 001451 +0146 eOL59 »0172 20176 01285 0 O245 000126 0015S e BOd 48 eO0c61 200420 © 00356 e00569 e004e 1 e OOD aQUK7S «00716 eOO9O0L 202.05 eQied 00135 oO Lae «01.60 20166 oQ162 oe 50 0272 0 Q509 oO 544 «0374 204507 0455 00465 o U48 9 00515 005359 » 0562 00556 2 0606 «0630 06652 09672 » 0692 0900496 e 0006 34 0008 43 eQ01021 eO0OL19 200134 00347 « VUL6% 00177 » ODLYD «00206 o OO4r22 200459 200569 200645 «00699 «00784 690817 eV09l2 00999 09146 0170 e092 | 20212. 20255. 0254 0272 eQ292 29509 0425 00354 00559 00357 0059 0406 oG422 00% 46 IO. CIRCULAR _OPISIC: * NOTATION All linear distaness are in Pnglish feat. Allimearsurss of area are in English square fet. All measures of eapacity are in English euble feet. Ds rlscity ,esllied the coffieient of velority hoa bean plsec7d by warious authorit- ies between .97and unity or even shove. Haville in his t: bles publishei in 1876 cives the walus .974 . Merrie: nS Hy, Irculics Isao gives .98 as an averare of the rasultsa obtaine4s -t MeGili University. In “act as tebtrr ax arimentel wotis: hevwea core in ose cni the eritfic-:s erploy?! nave b come more nearly "frietion-= less", the valuss hove gromn near to unity. This sits weisht to the resulta obtained in the Ohio Stats experirents refered to sbove in thich Cy was found to have a range betveen .¢C996 an: 099999 or for all practical purposes, unitye It he: been assume) in shat follows that the cofficient of walocity, Cv, is unity for a staniard ( that is a perfectly frictienless orifice). As Rill bo seen hovever 1° ‘hia asaumetion satBorract it. Joes not affect the walitity of the formula sines it waul3 be trken cora of in tra cofticiznt Cy . THR COVPICIEN? OF CONTPACTION, The regio of the area oF the cortracted seetion to tre area am to orifice 13 e71192 the coe efficient of contraction, or, Co . It is tha one whose determine rfion has eaused the most difficulty, as the diract meraurement of ‘ks contracted seation with suffiele: t acasrsoy for saientife fe purposen ‘sa alwost irroasible. The earlier rriters gawe “ix- ei walues to “his constant, som: of them being as fallo78; — IF, Nevton « © == #© @ © @ #,707 Poleni = = © = ws ew = @,6F7 Borda « © e© ee ee & ofa! Vichellotti - = «= = .f4° Fosaut « «@ ~ © @= @ @ @ £65 DuBuat - © «© ww ew wm 0 667 Venturi « = © = = © = 637 Fytelwein «= = @© = = = e610 Rayer ae wee ew em om ohT7 It in interesting in this case to notios the dscrease in value of the cosfficient as the orifices became mora perfect. Phe last wolue is interesting because it is feunded on @ rational basis by assuming that"the velocities of the particles of water *xhich arproach the erifica from all sider,nre inversely as the squares of their distances from ita center?! From this he derived (al- thouch we were unable to find hia work) that the ratio betreen the diemeter of the orifice sni of the eentracted section vas 18.7854 and.7854 gives us .¢6I7. Ifm fixed coefficient of contract ion were to be used this is probably ae good as any,but it has been repeatedly shewn by experiment that Ci is a variable, rhich 2s we shall shox depends upon three governing conditions. RANRINR! 8 YORNULA. The coefficient co depends not only on these three factors, but uren a fourth, namely the ratio of the arena of the ehannel of approach to the ares oeftthe orifice.It is apparent that the water in arproaching the orifice through the tank or other vease)] must have seme velocity ,and that this welocity enters into ths actual velecity through the orifice ani contracted seation.Rankin* r«cornised this when he gay as an empirical formulate PLATE ZZ. mo - =e 2 5/2 .616=1 61844, Whfoh in our notation would a. bate a, = ® 2 618-1610 Thie formila lesy-: Cy to a by dsteminsd indepentantly, and dees recornise ths ether three factora which influence the value of o, namely h, D, ania. THR VWALUR OF C, . Prof. John Geednan of the University of Laads has shown that the theoretical coffielent of contraction depends on the factor n alone and has (in Enginesring War. II, 1904 ) accorrlished what was lone consi:jieared impossible by giving us a rational expression for Co in terms of n for staniard orifioss. His derivation is substantially as fellors. Assure a cylindrical tank ( fir. 2 ) of radius Rand a circular orifices of radius of R,and area A in the center of ths bottome Then let a= ths theor- etical arsa of jat at contractsd section snd we the ~eirht of a column of water I foet high ani I scusare inch in section. Yhen ti:e orifices is o,24ne)} ° the weight of the tank and oontents is reduced by the veight of the column of water ever the orifices (plus)the total prossure en the bottom of the tank evuivalent to the kinetie energy of the approaching water ae it flows over the orifices plate, (i.e. the bottom of the tanke)" If h is larg? i+ may be sasumead thot the water arprosches the orifices in nyramiclal passoges, thes pyramids having their center at the cantor of the erifices. Then particals having equrl walecities will lia in hemispheres having the center of the orifice as a canter. The natation ehich folloxs will be understood from th: figure, FP, p, and P_being the pressures at the points hnving valocities ¥, v, ani V,. The nrea of the innermost shell is off, ani that of the orifices ic TR, therefore ; ve ¥-----9) 535. Alao since tha aren of the henicrheréc shells wary as the squares of their radii, the vwelacify of Sloe rerors the shells 2. Va a * , (7) v, 3 Also by Bernioulli's theorem the prescure head plus the vel- will wary inversely as ths squcre of the rodii, or ocity head is constant at esch section, therefora ; ashe eal and hence from (2) Repe ret v= ¥,") « r¥s. ( B.-1) oon anne l(5). The ksy to the whole Jerivation is the fact that shem the of ifice is opened ani the water flows inward along the bottom of the tank,the pressure at any point on the bottom ia reduced by an omount equ:l te the chang- in weleaity head between the ede ani the point in question. If we consijier a ring on the bottom with ite inner radius ea rand it? outer redius w r+der, its srea vill be Pfr ar and the unit pressure is reduce’ by (Rep) which is gained while the water is flowing from the outer edre of the tank to the eirele af rsdiius r, therefore the whole loss is Ya Ry fren ar dr which from (3) rives r= eR er onw¥ = (3 Le1)y dr, Tv (pty *op)dr = FER & )reR a eV (Bae! ~ BS + uN ani sine Renk, this ecuals a wR, T “p* | _- en* 2n* rast + )a ead (Mezn’ er) Thien, Srom(T) equals wV,R(ntenel) --+-...------«--- ---(4). Sgn* There i: alse the toe ty. e° veicn? of the column of «ater ovar ty? - W wacneneennne en (5 e the erifics 2 with ~ 26 36, The loca o° Sight mast equal tha pressure which the escape ing jet would exert on a plata across its path, that is, (wa V) Via wa 2o----- enn (6), 4 & Then (4) + (5) mint equal. (6) or wVfR (n*- “n+ I) - eViR, = wav ti Son & € If for WR, we put A, this becomes xV_A (n’- 2n +1) = waV- 3=soor € nt gE Therefore, Cy = ‘ = 2 Ispoe 2p +I) which in the ex- pression soucht. TH VALUE OF C . Thr tor’ abowa has disrepoard:d the velocity of approach far, V’ the actual velocity is equal to V + Va. The result so far in correct however, since the difference hetwean V, the aotual velocity, and VY, the velocity considerel, is equal te Wa, tha welocity in the upper part of the tank, both of which rere neglected, Po find Co we proceed as follo-s:- Va Age a Viz Ca av’ Thereforos=- Vy = CoA v’ Ten VeVv-vV, 2 V’- CC AW avi r-coa) a Therefore:- C, £ Y's ate yer Since as will be shorn later Cy, = Cy within a fer %, an’ é oe oe is uztually wery small, we may for all practically good ri use write this 6, "TI - Dd, THR COVYPICIFNT OF LOSS. If then we assume that Cy = I it *ould seem that ceGQ. G Rut axrerinientn have shern that this is not correct but that 2 must insteai writs cSChC.or if ClSCeCx wa may fut CSC. GQ ehere Cxis a wariabls thich re my coll the coefficient of loss,and vhich depends on both the heal ani tha dismeter ef the orifias. Its presence is undoubtedly due to the fant that rater is not a perfect fluid and doses not move toxrard the orifices vith the mathematical precision which is assured in the jterivation of the formuls for Co Prof.Hele-Shaw (in "nrin- eoringe Janef 189°.) hasshorn by actusl photographs that in reality tha moter does not move as rapidly from the sides of the the tank toy-r! the orifice as it doss from the region directly berind tho orifice. thus the formula gives too freat a loss of weirht due to velocity and therefor: C.«i11 bs larger than Ci,80 that in general Cxwill be leas than unity.Also since Cyis net exact- ly equal to unity ,C,*ill be still smaller. On the ether hand the theory is based on a relatively large value of h so that re are not sure of its truth in the case of small heads. Exserirsnts ehios that for these C,is sraller than Ch,and therefore for small herds C, will be larger. While the xhole derivation has been for a horizontal orifice experiments have shown that the coefficients are the same for a vertical as for a horizontal orifice where tiee head is relatively large. ¥here h is small we rust remember that ec in not equal to C so that for small he ds ell observed values of C haw heen reduced in the work that followesby the tabl* giv- enon pores "fo of Smith's Hydraulics. TIP VALTE OF n. Another iAiffieulty arises from the faot that most tanks are not circular and therefore R,and hence n,are not easily datermined. Prof.Gooiman says to take the distance from 38. the orifien to tho nesrest sige of the forifiecs plots" but in the experirents considered tha bottom of the tank was elese to the orifice and it was thoucht wise to avercge the distancts to the bettom ani tor fer a wnlue of Rao In avalusting Cs the curve on pilates IV may be ured with sufficiont exactness for values of nercester than & or I0 . The table below fis copied fror Gooij-e man's articre and will be ureful when = man, i.¢., fora cire cular xppreach. The values from Fenkine"s fernmila are as follovs - The netation ia ehanged to our avn. When C, 8 .97 the aeconi cok umn cives values ef ¢ . a e665 0645 of 72 e652 o> o6i1T e622 e640 e621 4 0654 a€15 e637 e612 5 e631 e612 e6<6 0607 6 e629 e610 6c 2£08 8 067:8 »609 26.2 e603 ZO 0627 0608 0620 e601 py basse Uadnetassasd catesomastnnaet O2/27- 9-770). ZW A Xtal 0) A : eee oa | | Val a ah a a oe ead a 2 nt-2n*y) a econ ashes gey B55" 3h Ellice mw te | a= zesbe tel : ly at UES OF c: K 2 aN ‘ i Seated isetaasestse a. 45 39. DERIVATION OF TH: VALUE OF Cc. ° M.TYOD. The gtneral method in the d*rivation of a formula fer tha vi.luea of Cy, wis first te find records of reliable experiments shich gave sufficient data for finding @, C,, and Cg . Then a eee Then values of h and C. were pletted and a search maida fer the equation ef a curve which vould pass through these points. After the form of the equetion was determined uron the values of the conztantea in the equation which rould make it pase through the largest number of points, i. 6. the one which would give the most probari« values,wee found by the method of least sauares, DATA. The records ef only three sats of experireants rere found in complete enough form to be available for this work. The first wers the Holyoke experirents earried on by Hamilton Seith Jre in T@A4-5 and recorde! in his hydraulice. There rere three orifioss used, 02, «05 ,and 010 feet in diameter respectively. These experiments were wery accurate but the heads dii net go above 5 feet and the arprorech was rectangular waking a determine ation of the wolue of na little uncertain. The second series sare performed by Horace Juid ani Roy 8. King at the Ohio Btate University an I9CA and recerded in Rng. Mews Vol. Sf, pare 327. Viva orifices were ysed with diameters of RF, I, Ii , 2, ani 2h inches rerpactively. The heade waried from 4 te 93 feet ani the anpreach waa circular. This was the most satisfactory group ef gata as no wis absolutely istermined. The third series was carriad on hy Theodore G. Fllia €.". at Holyoke, Mass, in 3574 ani ree corded in the Transactions of the A.S.C.2. Vol.&, pare 19. Three 6O orifices were used with diameters of.5, 1.007, ani 2.00 feet respectively. The heads varied from 1.15 fest to 17.7 feet. This was the least natiafactory series as the heads «ere most of them very lew compared with the diameter of the erifice:, and the orf- ice vas placed grite close to the bottom ef the tank making °n® quite uneertain. More than this the diseharre was measured ever a weir, thus introducing another chance for error. COMPUTATION OF Os. The tadle on page shoers in columns hanio the values as given by Gmith. Thene ore a:l means @f from tvo to six ebservations. n wan found by dividing thea distanes from the hottem of the tank to the surface of the water by the diareter of the orifice. C.was then taken frem the curve on PLATE IV. A, ie the oross-sectional area of the tank @p to the level of the water by thé’Atadted £9 sudstitution in the formula:- C.8 ‘ C. is determined. Then diviiing ec by the preduot of-----.--- C. and C. gives Cx . VARIATION OF Cx YVITH 1 . The values ef Cx and h were than plotted on PLATR V. It is raadily seen that the simplest curve of this form is the equilateral hyperbola. The vertical as - yuptoete seems te ve the Cx, aagis but the horizontal one is raised above the h axis. Wa may therefera write as a trial equation h( Oxe dB) Ba er Cx® be £ e To annly the method of least squares let hCx ~ hb @® a bea tyrioal observation equation. Tren for the unknewn a the normal equation will be formed by aiding the various equations of the form h B.- hb ea or if there are n equations wa «ill havese =O x - Ls Xh a nh w--.- ~ eennnnannncananaaa(}) For the unknovn »b we will have the sum of the equations Gt ence nee ace acc Siren reece Bireneerercnne Rae djametep @ 0” faet 06739 of 475 74 of 49 4.47 1.0000 1.0394 20430 4°98 159 06°50 9.954 %.0000 1.0077 3.390 F264 297 A250 IX,B2 7.0092 F007? Nieameter a .05 feet 0.477 .A30I «4 06246 3.98 100003 1.0085 025365 2€°6%3 26 6-46 3.96 1.075% Y.0C0:.7 O.7-:0 26199 <9 e647 $042 [T.0CS7 6.9926 O.91G «6160 S93 e6=49 4.96 1.0092 0.9858 OeS9 06194 324 26548 SeC% T0002 C.99T2 T.740 61735 SO 06249 ToiF? YeOCOE Oe975F eetT3 06970 70 «=o f249 8.44 F.CCOI 0.9712 3.570 efO06O0 BR 26249 12.96 T,000F 0.9496 fe6S0 o605E FO 6250 6X4 F.000F 0.9680 Pinmmeter a .I0 feet Oe66T FITS 14) e8E38 4.23 {0012 0.9798 0.900 .6096 I? 46241 4.95 f.00%0 0.9758 Io.750 846042 28 4246 72.44 1.0007 06.96F7 30X80 6025 39 «66248 «I5.79 1.0004 0.9640 4.600 .60I3_ 54 .6249 16.05 %.0003 0.9619 e? the formte h’C @h*h gs ha or Th C,-dIH S akh « (2) - Then from (1) ShEnC,-b(EHS o nash and from (2) _pzh'C.enbZhi gsnath _ Z ATC, -nzb* C, ad [(ShF -nzh’y] b 8 CSANC, *Cc (tn)* - nxn” Therefare Thrrefor= Og 62. elzeo from (I) Lh*tn C, ~ osh*Sh 8 rth also from (2) In Tne, - oon*Sh = alsh YF thero“orea — Th*dh Cy, = Thih7¢, #8 a ( amh*e(xhy) therefore as ene Sant re Z nin” - Inthe folkering table the work fer finding the quantities ch rh’ ani Thc, is given fer the first erifice. ceceecemenndgemeter S 02 _ nh ex h” ney, spc, O.730 Y.0394 O.546F 0.768% 0.5676 2.030 1.0077 5.0049 2.4487 5.9504 3.190" 1.0022 I0.I76%_3,1970 30.1993 6 2559 _ e627 04138 16.7373 Therefore b a re ee Te ee 40 8025250 I9 -9.3494 mo x 0 245694396 eter @.9899 Also na Es SSP KG e IITA 16 062736 24158 = 108 0 5055-106 26429 | a @¥ 0444 ae 44 e on aei3376 = 99357 mw @ The fellewing pages give in a eendensed ferm the cemp- utation fer the values of b. The wealue ef a is not givzn,for as Will be shorn later, it is net used. -oF 63. A 6.457 £.0085 ei9I0 04407 eI926 0.536 140027 BIB SBA 2381 6720 06920 05184 07142 05145 0.979 . S9%2 .6630 .9208 06554 T0740 eO78I 360276 e70IS 2e96I3 £2730 g7re 704559 2 665T4 72563 3e570 «so 96GB «=: 1207449 «= 304615 =: 12 38:75 4.630 9699 BTABGD 4g SATO 2007509 13,702 4635220 34,9007 45.3876 _ a Diameter & ,I{0 ee ape anneal el Kernel Eee 0066I 9798 04369 = -o 6496 04282 6.900 9758 eBI00 3=—_- 8782. 7004 T0770 43s 9567 4=—-_s-209929 «2069242 2 B92 3.180 .9601 1®.2%°4 3.0685 9.7484 42600 096F9 RIS IS00 494247 2003838 1X.07% 35.5122 10.6884 34.2139 be 7z20698 = 335.253] o PS27382_ 15590 gudd ani King's Experirentn. The dicmater ef the arproach ros « feot therefore 3- Diameter @® 0626! ne onetime e 3769 » and therefora €.8 .f° 475 -« e03T3 C,s —~ x. a @ 20006 ae same a C.C.2 06: 5I2 Li s624FF DEE ani C,s eOe. et —j Ee 4.30 48085 «4 9734 16.6100 3.990 16.2629 8.99 .6I0t .974T 60.8202 6&.775Y 78.8884 Y6e%5 =o 6IOB. 976% «= 8005675 1643530 273 .9TSE 23060 e6108 e29768 556.5600 23.0478 $i3.927T1 35.33 06120 19770) I5%8.2060 34.568% %E2T.9965 47of5 e@BIXIX 69776 8 1906.8°25 4° 8678 879.7513 B4el5 eGIT@ 69789 2943.0655 $3eF053 =. £80.9639 66060 6304 69764 4467.0.100 6502235 456.8712 £4.70 -oGI80 ao P38 PI7420900 8303279 70579 E97 $38.37. _18605.5765 337.2795 18310.6041 & 527004 e 98Ta ae Diameter ® .0854 n a 5 : Sse “= 062458 , C.® T.00KL , Cxs ma So a ro 4.02 6322 9776 16.1864 3.9300 I5.7S84 B87 6078 69720 788769 «= 8. B2TE 7664732 07 43% 06092 oS 745 BOL .08 Se YA .90iT E649 DAO PhelS 06089 6S490 07H H = -BBATHGD 688.7062 35080 3 eRIT? «=o 876200 TZAI6ES50 ©=6035209TR = 8367022 61683 e6ITE o974 SABE SATES 8007300 F300 LG BPelO ofA 09K = 338707400 = 860 B4GR = 3330 eo SHG T1007) =e BOG =o 9730) 5500505625 960 B400 0 4B 724 IES 05048 06096 on 9779 85470008 — G00KKIS == B33T.4IT 308.94 fo. £19%323498 354.9105 _23B69.9385. b 8 1923.9.45 « 129166013 © 63165032 @ 69752 ys e@ 3, Diepeter gs of255 I « 15.9 C.2 «6240 Ci.2 720024 Cc £ TORTS yon ‘ns 1 8 srt ce en So mtn or re 4207 e608 : 09726 16.5649 309585 Z6.IIIO BOD KORA OTEK 74 3838 Be k838 72 oP6AS 16.83 .60°7 ,973T 27% 02409 16.0883 265 .£907 eeend ef EE 0270 49320625 “206193 452.6058 25066 af OE 9730 12716356 34.6972 1237-3074 43.61 TB 02% IQOT & 722 42.4325 TAS0.4876 AF. 0 of FB eT ED 382 e7400 SAE.OE $295 .27848 765.56 efTHO ef 20 4976 .7I36F Fie 6 549 Bade BES 92210 60N> 6 97: 6 8599-2700 8.6039 7 T.8TIS 30060 eee ORT 0E9FO — SAX.0910 —- 20935.63546 vsgoz2001 I~ITI9E7.56 eet Sree eke = = 29727 Diameter 8.2085 A= TOUTE B 9.6, 0.€.6227, C8 Y.0068, c,2 “etree 8 a 6.78 5960 49506 45.9664 6.4461 43.6976 II.54 8956 29500 I33.I7%6 10.9684 113.6976 T7089 98958 28h? Be aOhT TRePISR = 254e0305 ~HeS7 9 19690) 99504 BS °BeGt SP O54 520.3585 74077 TSO 069190 1708.9729 32.9967 1147.296% AGe27 o59CO 095CA@ 324009179 43.9843 298525718 §7eCL 05555 069.98 3341.9961 54.9079 317402279 G9e45 065955 09458 4°17 .135F4 65.9351 45772158 27%000 95955 09.9% BFA9000%” BP FBAG =8°10,4987 0035 ZIIUEB ATO Bize25h2 2105645200 b= 21056003 mi SA2 eC mw EF6LIRTA 209478, cela @ ee Swe iee E6, oL&64 lee z eg 12 Com 062335 9 Cie 2.06043 CX « ¢ - " 8 9 “as ? Diameatar = a ee 8.00 «6084 9728 2840000 408590 24.2950 ¢.0@ .6083 .971¢ 2.4414 & e8221 80.1049 17079 =o 6080 8722 316 4841 17.2777 507.3894 T7ePH =p BOP «976 $40.0976 22-800 624.7588 SES eAONA = 9715 13040654. 3500906 1267-24427 47.02 e6088 49724 221008804 4567222 2149.8601 57.70 = osFOFL «=o. PTS 3329 .2900 66.0140 3233.7394 69.998 26026 29717 4898 .6001 67.9743 4757.5204 2972. 401 222.0239 357.85 233.7322932 34727299 «2056742436 ds oe 244 ae - @ eee <0. = 9712 =, BELLIS EXPERIMSUTS . The value of n was determined in the same ranner as fer the @nith experiments. Fer values ef % leas than 10 the vale us of C wes corrected te « .- The valnes in brackets were then averaged since these rere determine’ hy fewer ebservations than the first three. Tables en per felloringj5- Diemetar »« 0.8 5 LF O—D—=nu”a—nO=E0EVW 201516 60049 06192 Bed 51.2 1.0022 .9¢78 A.1558 .fOPTO 1004 26228 83.2 1.0024 .9658 Be3476 ofO534 W4eR 26239 WAsd 1.0010 4.9693 eee 060076 1667 26542 133.6 1.0008 .9627 8.0100 60117 18.1 6243 145.0 1.0008 .94622 9.0600 .80192 R082 .6244 141.8 1.0007 29634 OeF15SO .60114 23.2 06246 185.0 2.0007 9518 WeV700 6 5999KE 7600 6246 214038 2.00086 69599 4209600 260102 28.0 .6247 224.5 200008 .9°16 1404700 260064 32.0 06 47 £483 1.0008 4.9410 U5e46CO 060077 33.0 c6B48 264.62 1.0008 .9411 1506500 .60535 33.8 6248 1.0005 .9¢84 29500 2,0004 9540 Diameter & 0.5 me Re ie 2e1516 9678 406294 BeOS 4 4803 Sel5S2 9 o9ASB 176277 = 460137 = -16 6800 6e3476 FOB «40029020 «527 39 OKD 76550 9620 56.8990 743042 5643927 1065150 69417 «-11068652 «09ND R «= 106.3308 1403030 612405752 «1367480 96.6326 205570 09622 27441342 45.9146 203,458 WEAoGESO TRO O663S 5907077 0538. a 2328 +0464 = 9594 65 ehca Diemster 2 1.0007 68 ee = 2.1475 256829 2.19 .5784 39.0 1.0132 .9868 Ze3607 65939 3.41 26043 84.6 1.0069 .9672 4.6092 259027 5.86 of186 = 9309 = 1.0052) 69.194 769705 .5E3518 9.02 26225 W4del 1.0074 29310 7eO172 459512 B07 E221 14765160034 69438 LOsfE19 9 5047 L129 96233 SDD 2-026 9510 | Diameter 2 1.0007 rr rege ee BO ge re 2.1473 69868 = 632630 -1..1322 142990 263607 BLT? 6.6779 2.2823 63902 4,8 91 o3474 ened 74 $eSE58 21 .°574 7.9705 6934006705209 = 704.45 593356 TDLTE 9438 62.6821 9.4723 59.1897 9510 0.3427 12.6135 B5eDo47 sg 27406433 3322467 259.7554 DB 75907554 = 3362407 @ 392.5154 » 09406 241406455 @ 35208 62/85. Diameter @ 2.0°6 167677 65907 1.41 25322 45.2 1.0328 21.0715 404725 06041 2.76 65559 68.4 1.0220 29920 ve595B 05963 2.82 05619 $8.2 1.0232 1.0789 508375 06107 3044 26018 119.0 1.5275 29923 6el233 66120 2699 66508 127646 1.0253 9 9205 BeII32 F125 4470 o6137 16001 1.01279 9 oor ww 063° 06155 Fe24 0f279 17003 23,0115 S862 6 Q. Piameter = £,.000 re ah nen arene Kone Ee een 1o7677 120725 301248 1.8941 503482 70°58 1.0289 6o 782 226701 £29310 43 P%5 8920 = - 2020122 404377 19.6521 $6555 oe 9OR3 2400597 6.7256 33.7678 6.9323 865 48.6706 628536 47.8182 Bo%4%2 09854 69.6090 Ge£214 5.5928 ee eed OEE 87 B90 9.5082 $3 26312 39.S€5) ne TTS S99 GSTI06 27 BELL Bs a 09675 VARIATION OF t WITH D. The valuesof Bb obtained abeve wore then plotted with the values of D, in the upper curves of PLATR WI. The points de net lie very elose toany regular curve but it will bs seen in general that for small diameters the val- ue of b appreaches unity, and its vwariaticn with D becomes less as D decreases. It was assumed that the curvs approaches a hor- izontal asymptote at a value of D which le greater than szere, anparently about 930 or .940 » I¢ is aleo assumed that the curve becomes tangent to ghe vertical axis atthe point P= 0 anit +b © ] . The simplest curve of this form which could be was of the farm x+fs o> and which with these variables and moved te the proper axis, viveat-aer & £, +) ° But nines » &8 an abstreaot mimber and dsia in feet. a cofficient must be applied to 3 tea make it in the prorer seale. Then we eo ae CURVE SHOW SVE Vee eee: ; | OF wT Dd Lr rm b LOL ; | alee at 2 FROM E.ot a ; ) y + = 0-06 yA D JaFrid | | | | } / | ) ; } ib egusas ssn cseegpecseCac08etee sasstzstes oeesetessctssdadeascazszle bsceeszus ties | i } t | } } } } : | } > 0] | : } ; | ; Hie | | | ieaeeaiit i ' ? plese eee aeezsaant Sates SGU CRSSAHSTEN SOARSSSSSs SES ea baeae: — ana tw Oe 1 | os: a | | | i t ) ! Ri SSuETHEEEE gS tatgd febsas fuel St sseaas cauaascas tonne O32” EO adlteetaat bee aens prstas 4 | SHOWING ache ee ce. | | bg 7 od ae | o | | eee AUER ae 8 | ; | | CURVE FROM. EQATION -- peestce eeneeecu wads Rdsacktwseh sUsenesees 2 SEuAn ert is ____+-Q= rat} | | 70. may writ- an ecuations e kdt+f = Fo mae or = * 2 fx e (led sat eerli-ersy2e (ren) se frat f= ft d+ theraforaje ( 1 +b) = 7/ $s » Anotioeds- i 2 D. hal ~f dD ? and also +P wifes Civ FP] tr Caeey therafore j- theretere j- Since f£ occurs in the second power in this ecuation , to solve directly hy least cquares would be a very tedious proe ceas ( it is deseribed on prre 183 of Merrimans Least Squsresb. The method employe: was tops:s the most probable hypertola thraugh the poinys for the Judd and King sxperiments (shich seem to be the most recular). Ryperbola go (Cbeg)del . Yn pt... > on 00626 09818 006146 000392 000389 20854 09782 oC BER 060729 000711 01255 oh 7" olf f£097 001575 001832 ol664 e712 016161 002769 002689 elCEL 6 9474 9754 204329 04201 eee 8? 262576 oo OO794 009422 206548 = 06954 02416 . TL, Assuming that the aszyrptotes of tre hyperbola and the cubic are the same, f Fill e.ual 1 - ¢ = .05384 and f° @ 0034106. Then our equaticn ia t= kD [ 20034166 - (1 - vf}= 00584(1 = bye Then the normal equotion i: te e = D’[.0034106 = (2 « v)/2 00584 T (Ll - wy PD [ .00742—(2=n)"/ Thereforatea fhe computation folloraste VALUE OF k . 2 002 e102 .003208¢ 00000000678 00000000438 008 =. 0366 = 0070708 §=40000001387% =.0000000107:: 010 90420) «0017296 200000006375 00000002992 00055 eC18% 40020796 .00000006375 .c0000002718 00554 e058 eC027956 00000081848 200000005699 01255 00273 20026656 00000012944 .00000012190 01664 0286 40025616 00808835613 .00000018455 08083 20526 60006136 6C00C00%7095 00000001793 08050 0406 20027676 200000145235 2006000077675 2.0009 2.0894 .0001174 .00000041445 .00000001402 (2.2700) £0725) (o0r3sig (o00ce4s:57) — bococe217¢) _ __ 20090643079 40000616979 Sinoe the experimon’s by Kllis were much lesa certain thenthese of the other experiments tne firet elicht values were given a weight of two, the next tra of one, anid the last ones rere omitted althgather heoause of the unoertsinty of a large orifica under so lew a hed. The *eifhting was accorplished by csirply by turning ths crank af tha adding machine onge or tric. Tk, es the ease might bo, shen the solumns were addcie By this rethod ke @ .O00004507 x 0584 2 14817 eo fO0002l ¢ then £s 10504 @ 039414 and Be Le gohA4 D K e Sines thia walue is somerhst approximate at any rate we may write it b2® 1 ~o6f D . e4t MT VALUES OF a - The values ef b computed bs thi- formula #112 in general be different from the moat probable values computed from the separate exp-riments, that is the hy- perbolas vill have diff-rent asymptotes. The most probable vale ues of a will then have to be computes for each size of erife ice on this basis. The equations were of the fom h( C-»d ) - and if b& be known, a ie the only variable and cen be sasily computed. The work is eas fellows, b being the constent come puted in each case frsm bel =~ .045 oe and then substitut- ° ed in the normal @quationies at Fn G.- dbz=h » velues of*a* will be n found en the next pare . 13. VALURS OF "a® . yo _ fh s...—. G.0200 «P26 604238 6350 + 00469 0.0800 .€800 14.9097 15.297 OelkO90 = 977-2 L084 12.073 3 & — eSOls 5 ~ 00172 O.Of26 09779 331.2795 338.370 9 + 60432 O.0P54 ef746 39409208 36456910 9 +007.28 0.1255 49707 «=. 34160740 2500400) 8 pele Col664 29885 347.7799 357.950 3 +1570 Ock0R3 69648 342e25E2 3600350 9 * 6008 0.8000 .9553 59.3877 62.685 7 +0659 1.0007 69493 s-33.2467 25.0867 @ ? 220009 094525903706 39. 585 —.010z the2795 Thess values ef °a® ware then plotted on tho lower part of PLATK VI. The curves xhich seemed to fit them most nearly ean @ parebola of” the formt~ 22 17D . Prom this the norgial is TayPS1IP , tharefore 12 Z oD. » ‘The values for 1D) 8 .2083 and fF 2 2.059 were not ured becsurne they vere soa far off the ourwe . The computatiin follozs om the next pare. 7& COMPUTATION OF °1%, OOD 0.02 00460 01414 200650 9.05 e2016 o22356 000022 0020 eO1LTE 03/62 000544 000626 20432 02503: 201078 0.0654 ,0538 oF VE 000695 001235 60852 03544 203019 001664 =o A570 4079 106404 _ OSC = 059 OTL MELD _120°07 90102-20003 401920 e208 92492} 22 .249232 C7045 @ 1 1 Z.Li06 Vt therefores= act i7D® ,.c7065 7D » VF = THR FINAL FORMULA. Then from the above jerivation Cc. 2 bt+a with b® 2 @ 406 ani cE \e a 8 V D i* < Therefore te Cc, & bra 2 de el b X + & e e h eS + + 4 73 The tnieaie had for its purpose the determination of the discharge in ths toarms of the determining factors. This has sew been accomplinhed as is shown by the folloring formulaeia--- q@2® @84 : -/f _— __ Cc a “~ ¥ z ‘ z zZ e Q J 2 ay Vt y-V/2e ln~y ) tw A/2en For erdinary heads thie is equal te Ayirh , and for en: case 4+ ean dba Aatarmined with the ald of the table whieh follows this parnacrapne -@8 6.6. C, Oe 2 ra + p= 2n tJ ) ana may be taltean from PLATZ Ve Ci= 4 and 42 usually nearly unitye C,® 1 oe 608 | et + p * 3 b ° fhe second term may be teken from the uprer curve on PLATE Vie and the expression O.%7/D from the lovor curvéee «x. Then s@ Ds (1. 00 Bo LE | so 2. v 70, Ratio ; in terms 6° the ratio 5 ® a ee ee oS aBAO4 1.25 .9938 2.2 99983 af 975% 1.5 09952 Len o 884 e625 02774 1.4 2960 ZeA 8986 o? e823 1.5 e965 265 e998 078 09645 Le6 8969 320 9991 o& e367 1.7 099735 505 3904 0675 e%h9E Lf ofS76 4.0 09895 of 9897 Lg o897E 4.6 8996 1.0 9928 2.0 09280 6.0 9997 deol 09933 cel o99B2 100 1.0005 eet On 8 9954 ~ Taken from pree 22 of Bnith's Ryéraulies. The values ef CG. obtaine! dy 1d. the formula are Gompared with those observed,also the valuas of o given by Hamilton Gnith in his table with the ebaerved waluas or Cc +s D 2 202 h aa re error £ (fy | Cbae tbe ls oy —es e e ee 6 ° Ue ° 2 2,639} 2.06077 0.9910 201467 1.66 2676 .630 0630 2000 200 .CO 20392] 220022 095900 20222 2672 2049/2626 2627 oCCl 220 004 avers = £8.26 19207 Probable error ® .67415/79.5°5 & 2.61 D2 G5 0437 | 1.0085 1.ORBB 60077 0076 e536 | 2.0077 1.0095 .00F2 2.68 oTEQ | 02970 2.0070 00100 0.99 02°91 9925 .9970 .0058 0.58 16740] 9782 09992 .0110 1.12 20740} 09712 66858 20146 1.569 4570] 09696 806844 40148 1.53 4.630] .9680 .9874 .0154 aver.: 3.06 210.729 - Propahie error R 2674571 .04) & sk ae _ Bvere=e0F 204 Provise G@rrse @ Geld avere.09 17 Probes rrs 832} 78 Ds S Obhacr.oomp. O@Tror > a ory. G oIT. * h Cc Cc tab. tab. ete) viv ast G f e900] -978F e610 e609 +00] «20 «04 1.755) 9667 2604 nA e200 00 «00 ' Bel&O} 29640 e605 2605 2090 .00 -O 4.600 2619 e601 2601 2060 206 200 AVers = 1.88 18.955 AVere=.04 20} Probable error » .67:57/ 4.734 * 1.47 Prob. errs = 0.06 D = 20626 : >» AelO | oO? 1 eSG20 20068 0090 Oebi0| 060) «604 0005 050 064 8.09 | 8761. 8799 2.0038 0639 06152] .610 .601 .CO9 1.50 2.25 16076 | «8763 09780 20027 0.28 04078] 661: 2598 .01Z 2.00 4.60 23262] 09767 0727 sCOZ1 0.21 0.044] 6612 0557 0014 2.3 5.29 35033 | 09764 2.9784 .0006 0.06 0.003] 612 .596 .Cl6 2.6 6.76 43.85 | 9753 9783 20007 0607 04008] 612 2595 2016 2.6 6476 $4.25 | O72 oA7ES cOCOF 0207 0.005] 6612 894 018 2.9 8.41 66.80 | 8762 oS7F2 oCC1B O61 02032] .610 0594 .016 2.6 6.76 64,70] -O7F1 e8781 20057 ayere = 0.20 1.156 AVOK., = 263° 83.8 Probable error * »67457.146 © 26 Probe erre * 1.75 ° p= 1.0007 “Leib [eOh6H LeOL00 eOe4L 2e44 BeO5a [ose o50n wld cehO Be 2ed8 | oP FITS e9793 60121 1.55 1.563 ]2589 596 2007 142 1.4 4of1| oP494 e8640 20146 1.454 2037210590 05°6 2004 0,7 0.5 7297) 9340 e583 2.0243 2.60 6e670/e583 0596 2:13 2.2 ae 7.92 | 8438 09583 20043 064% 00250 /e559 e896 2007 1.2 1.2 29288] 29510 29558 20048 0.5% 0.222 125°4 .595 .001 0.2 Oe) Qavere & 1.47 17.2070 Aavere = 2 RD ER or Prah. arvrar 2 caste ase = 2 ne ms 44 BeG7 720 oOT6T 00019 0256 17035 09743 29758 e015 15 £4065 08690 69754 .CCL4 056 S580 oG7EL oS752 0030 31 SoS 097B4 29789 00024 024 Bare oO76P 9750 COE elt TWe7Hh 69730 097.9 eO0RD 019 02.45% 99749 29745 0001 00250 0.023 Ce tlh CeSCR 0.05% 02024 0 e034 2608 ofCE eUli eS10 e017 e018 e017 0015 2e5 12 — 9.00 Vek 4 Gen 20lL 0.0007 9610 593 .018 3.0 9,00 AV OT eK 40567 eens VOT = 0925 OsE 57 Prom _» orrar ® 0674574107 © 0.22. Probe erre 1L.6F 2) 821255 $007 09706 oO TAD 0041S 04H O01E4 0608 6600 2008 262 1,69 B62 09926 09736 00C10 0610 0.010 06°38 2579 2069 1.58 2.25 16053 o9731 eB722 20009 00S Oc0Chmef59 2597 0012 2.0 4.00 22025 99730 09718 COIS 612 06014 2.669 .596 2.013 2.2 8.42 S3e66Ke9750 09714 60016 AMG 0.026 .609 595 2014 2.3 5.29 43061 09730 68713 20017 017 02025 2609 0594 2015 2.5 6.25 $2.20 9720 e711 20019 019 0.031 2609 0.994.015 2.8 E6628 70056 o9880 02712 00179 (2.62) Goi76) o61i 28°93 00735 (405) Al000) GRelO oF72% o8710 0016 216 00056 ofC8 2592 0016 2.6 6-7F . peacmenemcenenanmnnorn tN cent EO ISB aves 2010 56.90 a2FOd> Orrer 9 674572048 2.0018 Probe oxror = 1-85_ 80 9.08 9707 9716 .0069 0° ohS8 0FT8 0507 -QLL LeF&O Seed 172.79 94692 .9722 2.002) 2:2 0048 26°58 05°6 2012 2.0 4.06 2 e24 aS6B7 AS71146 o0O29 050 eDOCS e658 6BS6 2-01) 2.40 46C0 SRelZ oG6h3 9718 20035 233 0209 2658 2555 2013 2.) 4.41 47.02 e881 «9724 .0C45 044 0194 e608 0594 2015 ©£.5 6425 57072 09680 29715 oCC25 254 e116 e608 594 C14 2.5 5029 69.89 e9679 9712 oCC3S 2e54 0126 0608 0593 201: 2.5 €.25 e002 RTE o97LE 200354 “9 L2 08 25392 2016 2,6 cecensee-ensenemneens AVES, = 0028 0.8390 averse = 2014 42082 error 8 4°74 S 0.22 PFDs Brre @ 1,56 — De .<065 BeTl oP5CH eP6SE 20190 1.699 3.960 2596 2598 e002 23 C9 21054 09500 09676 o@S176 LeES 344235 2896 597 001 2 004 17059 08503 08666 00265 Le7TL 2eSE4 0896 0596 200% 20 OD 23eLF 99506 e9665 eOL8A 1.64 72690 0596 0596 of 00 oe GD 34077 09490 8057 0187 2.76 36058 2598 0895 20°09 2O 000 46027 99506 e955 of149n 1657 2.465 0596 0591 2002 03 oC@ S7of1 eBA9E eGEGS eOLSS LCi 2eBE° 0596 0894 oo C2 23 e09 G42 gO40° oBFSX 290185 1.63 2.660 2896 .593 .COS 25 025 95200 9493 pORSR sOLSS Jeoll 22756 9995 05°92 .COS 95 25 eee eeen acne one VOFe 8 1572 26.666 aver, Oo23 08) Preb . errer ® .6745/3.33 & 1.23 Probe erre £ Ocxl % Da .§ e S1. pon a ae ee Aver, Probe le error @ 06745717042 ® 2.82 3264 104.525 _ h ~Obser. comp. ™ ™ ie obser. @ os . 2015009778 09785 00107 eld 16232 0600 0598 2002 9.3 0.09 4016 09658 29673 20015 15 4023 e602 .597 2005 16 64 6o35 29693 69632 eCO61 663 6397 0605 0597 2007 1.2 1.44 7030 09617 09621 C004 204 002 2601 0597 0006 .7 48 B.O01 09622 .9615 20007 407 6005 .601 0596 0005 08 064 9.86 29654 9608 20026 427 073 0602 0896 006 1.0 1.00 16051 9618 o9601 0019 «20 040 e601 0896 6005 1.8 3 64 11.97 9599 .9595 00504 604 eC0R 0600 0596 0004 67 649 12.98 .9616 .9592 00724 625 4063 0602 0596 6005. 68° 664 14.47 29610 9588 of0%> 23 .053 0602 2596 .005 1.6 64 15046 09611 29585 2.0076 27 .073 .601 .896 .005 .& 64 25008 09684 09584 .0100 1.03 10082 .605 .606 .000 1.5 2.25 17026 29540 .9582 90042 944 94 0576 9596 207° 00 _... aver. eee 2 9, el? HVOF 8 78 99 80 Propable error 22674570268 8 0235 Probe @FFe © 0+6" _p 32,00 1.77 1.0715 1.0017 .06908 6.50 42.250 2260 162089 49836 00453 4.452 19.448 4.47 .9920 .9677 .0243 2.45 6.003 6.83 9923 .9627 .0286 2.99 8.940 6.93 .9885 .9597 40248 3.00 8.000 8.34 49854 49572 0282 2.94 86644 9264 .9862 9536 00306 3.20 10,240 SUMMARY OF EERORS IN 7, Rrroxr from formula. Krror from tables _p__ave.err, Pro,erre (PR) ave-epsPre. err. (PR.) 002 202k 261 eB1Z 0007? ~— 010 9.010 05 = 209 069 0476 209 ell 2012 1001088 AP 20161 04 06 2004 0062 30 026 0068 «= «2032075 3.063 085 625 025 1063 2.26 = 1.68 4522 01250 8 015 0023-2610 1655 2.403 0166 028 022 0048 = B14 1.56 26434 .208 72-2023 1.513 423 (2 044 +50 036 235 0123 -79 8 60 £360 20002472028 357324321022 2254 00.98 es aver. 12.850 1.14 ® aver. 12.406 a SEO SER rer 2 .674 428 & .8) - Pp = 79 CONCLUSTON. fhe table by Hamilton Saith Jr. ( page 59 of his Mydrau- lics) hap been widely quoted and is the chief basis, at the pre- sent time,ef estimates of the discharge ef oircular erifiascs. Ae will be seen from the comparisons above, our formula agrees mors closely with the results of experiment than the table, al- though derived partly fromthe same experiments (i.e. Smiths’ and Bliis'). We do not,of course, believe that eur formula is per- fect, but it is better than most emperi-al formulae in that it gives values at the extreme values ef the variables which are at least possible, if not absolutely eorrect. For an erifice eo” 63 zero diameter the cofficient of los: is unity. Por headea infinit- ly large , the cofficiant of loes varies between .940 and 1,700 with the diametar. The head can never be les: than one-half tne diameter or ,what is the samé thing , the diameter more than txise the head, and for this condition the cefficent of loss has @ maximum Walue ef 1.669 . But with the surface of the water at the top of the orifice the factor n would be unity and the coe efficient ef contraction .500 . If we assume the cofficient of aprroash as unity this makes the cofficient of discharge .84465 , whioh is probably te hich but isa very possible. For orifices of id inch diameter and upward GF , we believe that this formula given a little better reaults than any table yet published, As stated in the introtuction , it in , as fer as we have been able to find , the only formala so fnr published which gives the dinchares from a circular orifice in terms of all the factors which influence it and without any variable numerics] cofficient, to be determined from the tables.